. There is some disagreement about how exactly to characterize this equivalence, but we think it can be agreed that the equivalence
is true iff p is at least a necessary equivalence: the two sides of the biconditional have the same truth values in every possible world. Now consider the proposition that speeding is illegal. According to manifestation pluralism—and the second-order functionalist—this proposition is true-as- such, and it has the superwarrant property, where the superwarrant property manifests truth-as-such. is an unplain truth. According to manifestation pluralism, things are different with the truth attribution . This is a plain truth: it is true-as-such, but has no further truth property. In particular, it does not have the superwarrant property. The attribution is weakly grounded in the unplain truth of . One might be tempted to suppose that the truth attribution should inherit the truth properties of . But Lynch takes the inheritance view to be ‘hopeless’.5 It is hopeless, Lynch argues, in view of the fact that we use truth to generalize over propositions. Suppose I say: (G) Everything George says is true. This is a universal generalization over propositions: For every proposition p, if George says that p then p is true.
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Now suppose that George expresses propositions about all kinds of subject matter—about speeding, about prime numbers, about mountains. According to the inheritance view, truth attributions inherit their truth properties from their attributees. But the attributees of my truth attribution (G) have a variety of truth properties—superwarrant, supercoherence, correspondence. So (G) will have all those truth properties together—and that is not possible. That is not to say that there cannot be truths that possess all three properties; we are not ruling out the possibility that there are truths with the correspondence property that happen also to be superwarranted and supercoherent. But (G) ranges over propositions that aren’t correspondence-apt. So (G) itself is not correspondence- apt. So (G) cannot inherit all the alethic properties possessed by the propositions over which it ranges. According to the manifestation pluralist, then, the inheritance view must be rejected. But we think the inheritance view, or something very like it, is right. And it is hard to see how the manifestation pluralist—or the second- order functionalist—can resist it. Consider the proposition . Recall that a belief is superwarranted just when believing that p is warranted at some stage of inquiry and would remain warranted without defeat at every successive stage of inquiry. A proposition that is true in this way is one that is warranted to believe, and remains warranted to believe however our information is expanded or improved. Assuming with the manifestation pluralist that the proposition is superwarranted, it follows that the truth attribution is also superwarranted. No further information can undermine either or without undermining the other; if one remains warranted through every successive stage of inquiry, so does the other. This is because anyone who grasps the concept of truth will accept the conceptual equivalence of and .6 In general,
is superwarranted if and only if
is superwarranted. Similarly, if we work with the truth operator on propositions:
is superwarranted if and only if is superwarranted. It follows, then, that is true and are both true-as-such and superwarranted. They are not plainly true.
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This conclusion remains even if we accept that there is a sense in which the truth of
is dependent on the truth of
, and not vice versa. Both
and
are superwarranted, given the necessary equivalence between
and
. The inheritance view is sensitive to this dependence:
inherits its truth properties from
, and not vice versa. But, to state the obvious, if inherits the property of superwarrant, it has the property—it is not plainly true. The case of supercoherence runs parallel. Following Lynch, we can characterize coherence and supercoherence along the following lines. The coherence of a framework is, of course, a complicated matter, involving many ingredients (consistency, simplicity, completeness, mutual explanatory support, and so on). Assuming that we have a characterization of a coherent framework, suppose we go on to say that a proposition
coheres with a framework F if including
in F would make F more coherent.7 Now introduce the notion of supercoherence as follows:
supercoheres with F if and only if
coheres with F at some stage of inquiry and would continue to do so without defeat through all successive and additional improvements to F. That is,
supercoheres with F if adding
to F makes F more coherent, and will continue to do so through all subsequent improvements to F. Now consider the proposition
and the truth attribution
. Suppose that
supercoheres with F. So, when we add
to F, it makes F more coherent, and will continue to do so through all improvements to F. But suppose instead that we add
. Then
will supercohere with F. Adding
instead of
will also make F more coherent, and will continue to do so through all improvements. If either one supercoheres with F, then so will the other. But then
is not a plain truth. And, similarly, is not a plain truth. What about correspondence? Here, the discussion cannot be clear-cut, just because the relation of correspondence is (notoriously) unclear. Suppose that
corresponds to fact F. Given the correspondence between
and F, there will be a necessary connection between F and the truth of
. Any world that contains F is a world in which
is true; and any world that does not contain F is a world in which
is false. But given the necessary equivalence between
and
,
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43
there will also be a necessary connection between F and the truth of
. That is, any world which contains F is a world in which
is true, and any world which does not contain F is a world in which
is false. It seems natural to take this as saying that there’s a correspondence between
and F. But then it follows that
is not plainly true—like
, it is both true-as-such and has the correspondence property. At this point, the manifestation pluralist might appeal to the dependence asymmetry between
and
:
depends for its truth on the truth of
, but not vice versa. The truth attribution
is weakly grounded; but
itself is strongly grounded. Now this might support the thought that the correspondence between
and F is dependent on the correspondence between
and F.
corresponds to F only because (i)
and
are necessarily equivalent, and (ii)
corresponds to F. But even if this is so, the correspondence between
and F remains. Even if this correspondence is in some way derivative, still
has the correspondence property—it is not plainly true. Again, we can take the inheritance view as reflecting the derivative way in which
has the correspondence property: it inherits the property from its attributee
. And again we can state the obvious: If
inherits the correspondence property, it has the property. The manifestation pluralist might claim that there’s a relevant difference between
and
. has a proposition as its subject; has grass as its subject. Does this make a difference to the correspondence relation? This depends on one’s view of the equivalence between
and
. We might hold, with Frege and many others, that the necessary equivalence between
and
is an indication of the special, transparent way in which the truth predicate works when applied to individual propositions. The predicate ‘true’ is not an ordinary property-ascribing predicate. If we accept that the truth predicate is transparent in this way, it is very hard to see how
and
could differ in whatever truth properties they do have—in particular, if one corresponds to F, so does the other. If, on the other hand, one thought of ‘true’ as an ordinary property- ascribing predicate, then, as a correspondence theorist, one should think
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that
corresponds to a fact involving the proposition that
and the property truth (and not, say, a fact concerning grass and greenness). Now
corresponds to a different fact from
−- one fact involves a proposition and the property of truth, the other involves grass and greenness. But still
has the correspondence property. Again, according to the inheritance view,
will have the correspondence property in a derivative way. The proposition corresponds to the non-semantic facts, and that’s why corresponds to the semantic fact that is true. Still, has the correspondence property. It is not plainly true. We should observe again that truth attributions can take another form, employing a sentential operator, as with . Here there is no reference to propositions, and no use of a truth predicate (property-ascribing or otherwise). And again we will have a necessary connection, a correspondence, between the proposition and the fact that grass is green. To sum up, we’ve argued as follows: (1) A truth attribution shares its truth properties, whatever they may be, with its attributee. In addition, we’ve taken the inheritance view to handle the natural thought that the truth of
depends on the truth of
, but not vice versa. And we’ve observed what is obvious, that the inheritance view implies (1). Now, we also agree with Lynch that (2) (1) is incompatible with a pluralist view of truth properties, in virtue of truth’s role in generalizing over propositions, as exemplified by (G) above. (Of course, given (2), and given that the inheritance view implies (1), it follows that the inheritance view is also incompatible with pluralism about truth properties.) If we are right about (1) and (2), then it follows that we should reject pluralism about truth properties.
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One further remark. The manifestation pluralist might take a leaf out of the deflationist’s book, and regard a generalization such as (G) as equivalent to an infinite conjunction of conditionals: If George says that there are mountains then there are mountains, and if George says that grass is green then grass is green, and … . Similarly, a generalization such as ‘Something George said yesterday is true’ would be regarded as an infinite disjunction. Then the problem of dealing with generalized truth attributions reduces to the problem of dealing with truth-functional compounds. The manifestation pluralist will say, as we saw above, that truth-functional compounds are plainly true, while their atomic components are unplainly true. This is so whether the compounds are mixed (as with ) or unmixed. Lynch writes: “Compound propositions, mixed or not, are true because they are plainly true”.8 The plain truth of a conjunction is a matter quite independent of the discourses from which the conjuncts are drawn. But now problems emerge for truth-functional compounds, in parallel with the problems for truth attributions. Suppose that at some stage of inquiry, both
and are separately warranted, and would remain warranted without defeat at every successive stage of inquiry. Then it seems hard to deny that is warranted at the given stage and at every successive stage: that is, if
and are separately superwarranted, then is superwarranted. And if we accept a sense in which the truth of depends on the truth of
and separately, then this asymmetry is accommodated naturally by the inheritance view, which preserves the superwarrant of . Similarly with supercoherence. If adding each of
and separately to F makes F more coherent, and will continue to do so through all subsequent improvements, then so will adding to F. Suppose
corresponds to fact F, and to fact F*. Then the worlds which contain both facts F and F* will be exactly those worlds in which is true— we have a correspondence between and the facts. In each case, we have an unmixed conjunction which is superwarranted, or supercoherent or corresponds—they are not plainly true. The manifestation pluralist’s claim is that any true conjunction is plainly true. The plain truth of a conjunction is a matter supposedly independent of the mixed or unmixed character of the conjunction. But now
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we have seen that there are conjunctions that are not plainly true. So, the manifestation pluralist’s argument for the plain truth of conjunctions has broken down—we’ve now lost whatever motivation there was for supposing that, say, is plainly true. And, as we saw earlier, the conjunction cannot possess the correspondence property, given the first conjunct, and even if we allow that the conjunction is superwarranted, this fails to explain the truth of the second conjunct. The conjunction is true, but the manifestation pluralist has no adequate account of its truth. The point here connects to the case of truth attributions. Suppose George only ever says two things: “Speeding is illegal” and “There are mountains”. Now consider again: (G) Everything George says is true. Under the circumstances, (G) is equivalent to , which is in turn equivalent to . So, if manifestation pluralists don’t have an adequate account of the truth of this conjunction, they will not have an adequate account of the truth of (G).
4
Wherein Plurality?
Difficulties with moderate pluralism, some independently motivated commitments, and an attempt to accommodate the main motivations for pluralism, lead us to propose the following set of desiderata for a view of truth: 1. No wholesale deflationism (as per arguments we have provided elsewhere).9 2. No commitment to quietism about realist/anti-realist debates. 3. Acknowledgement of some plurality, so as (a) to make room for substantive realist/anti-realist debates, and (b) to accommodate the idea that, in some sense, no ‘one true size fits all’ when it comes to different discourses.
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4 . No plurality of truth properties (as per the above objections). 5. No commitment to a plurality of kinds of meaning. We’ll address these desiderata in turn. Regarding desideratum (1), we have argued in earlier work that the concept of truth is richer than deflationists can allow—it has links to other important ‘nodes’ in our conceptual scheme (such as meaning, belief, and assertion) that cannot be ‘disquoted away’.10 We agree with Frege, Davidson, Wright, and others, that the concept of truth is fundamental in our conceptual scheme, and though it may not be given an analytic definition, it can be illuminated by articulating its connections with other concepts (e.g., in the style proposed by Strawson11). It is not clear to us, however, whether accepting the conceptual robustness of truth commits one to there being a substantive (or ‘nonabundant’) metaphysical property denoted by the predicate ‘is true’ that has an underlying nature shared by all and only true things. So, despite rejecting wholesale conceptual (as well as linguistic deflationism), we have been prepared to leave the door open (at least provisionally) for metaphysical deflationism: the claim that nothing explanatory may be gained by invoking a property denoted by the predicate ‘is true’ (and its equivalents). Here, however, we are only concerned to deny that the conceptual robustness of truth requires recognizing different kinds of truth, each appropriate in different domains, and capturing diverse ways for statements to be true. Turning to desideratum (2), some deflationists—notably Horwich— maintain that “[t]he use of these labels [‘realism’ and ‘anti-realism’] within philosophy is an unholy mess – to the point that they surely lack determinate application”; indeed, Horwich says he would “like to avoid ‘isms’ altogether”.12 He thinks that in areas where anti-realism may seem tempting, “we can devise a coherent and attractive perspective combining the most plausible contentions of the self-styled ‘realists’ with the most plausible contentions of the self-styled ‘anti-realists’”, bypassing all concerns about metaphysically ‘spooky’ facts.13 According to this ‘quietist’ view, once we admit that it is legitimate to speak of truth, propositions, and even beliefs and facts, in a given domain, there remains no philosophically significant issue of metaphysical relevance to be settled regarding the discourse in question. We reject this quietism.
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This leads directly to our next desideratum (3). As a default position, we agree with Wright and others that there may well be a point to drawing metaphysical distinctions among various areas of discourse, and that we can be cautiously optimistic about the prospects of making philosophical sense of debates between realists and anti-realists in various domains. Moreover, we think that this requires acknowledging a certain plurality. However, we deny that satisfying desideratum (3) requires accepting a plurality of truth properties. In addition to the objections we, and others, have raised against moderate pluralism, there is a general worry about the invocation of a plurality of truth properties. Suppose we ask: Why is correspondence truth the right kind of truth for statements about middle- size physical objects? Why is it not appropriate for ethics? Why is superassertibility appropriate for comic discourse? And so on. Presumably, the pluralist would agree that the reason has something to do with the ontology of the relevant domain—with the relevant facts. Alethic pluralism is thus motivated by ontological pluralism, according to which domains of discourse may be objective to different degrees, or exhibit different degrees of mind-dependence. But if this is so, then it’s very unclear what the appeal to diverse truth properties adds, explanatorily speaking, to the already acknowledged ontological plurality. Hence desideratum (4). For related reasons, we propose desideratum (5). In agreement with Wright, we think one should deny (contra Dummett) that the plurality required for capturing realist/anti-realist debates is a plurality in kinds of meaning, or one that holds “at the level of the propositions” expressed by sentences in different discourses.14 Nothing in the semantic behavior of sentences in different discourses suggests that they possess different kinds of meaning. And, as with truth pluralism, if we acknowledge plurality at the level of ontology, it is not clear what an appeal to diverse meanings would add, explanatorily speaking. As a default position, then, we think that the debates between realism and anti-realism should be reconstructed neither in terms of kinds of truth properties nor in terms of kinds of meaning. Instead, we suggest, more conservatively, that the plurality be assigned to the relevant realms of facts—to the worldly conditions that could render statements in given domains true.
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In a critique of Wright’s Truth & Objectivity, Pettit proposes something along these lines: Under the envisaged scenario, there remains only one sort of truth: that which is defined by the platitudes-satisfying role. It is just that what truth involves in one area – what realizes the appropriate role – may be different from what it involves in another. The difference … will be explained by reference to the different subject-matters: the different truth-conditions, and the different truth-makers, in each discourse.15
Now, to gain traction against the truth pluralist, who insists on locating the relevant plurality in kinds of truth as opposed to kinds of meaning, it is important to note that the notion of truth-conditions (as well as that of subject-matter, and even content) is invoked in discussions of truth in two ways that can—and, we submit, should—be separated. A Davidsonian truth-conditional theory of meaning aims to yield as theorems meaning-specifying biconditionals, such as (W) “Wasser ist nass” is true if, and only if, water is wet. The right-hand side of the biconditional is a sentence that can be used by the theorist to specify the meaning of the sentence mentioned on the left-hand side. Here, the right-hand side picks out the worldly condition—water’s being wet—under which the mentioned sentence is true. However, as recognized by Davidson,16 that is not sufficient for the biconditional to be meaning giving. Crucially, the right-hand side must pick out that worldly condition in a way that is fit to capture the semantic place occupied by the mentioned sentence. Contrast (W) with (W′) “Wasser is nass” is true if, and only if, H2O is wet. Although (W′) is a true biconditional, and its right-hand side picks out the same worldly condition as the right-hand side of (W), (W′) is, intuitively, not meaning-giving. We must here set aside the difficult question whether—and how—a truth theory for a language L can, as Davidson
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hoped, do all that we may expect of a theory of meaning for L. Still, we must be careful with the familiar slogan: “The meaning of a sentence is given by its truth-conditions”. In a Davidsonian theory of meaning, when the meaning of a sentence is given by specifying its truth-conditions (as does the theorem (W)), it matters how those conditions are picked out. So, the notion of truth-conditions relevant to the familiar slogan is a semantic one, to wit: (i) truth-conditions as they figure in meaning-giving biconditionals— worldly conditions picked out in a way fit for specifying the meaning of a given sentence.17 However, when Pettit invokes truth-conditions as obviating the need for a plurality in kinds of truth, he is presumably thinking of truth- conditions in a different way (as suggested by his reference to ‘truth- makers’18). The notion of truth-conditions relevant here is a metaphysical one, to wit: (ii) worldly conditions (objects, properties, states of affairs—if any) identified by a metaphysician as revealing the underlying nature, ontological constitution, and so on, of elements in the relevant domain. Our aim in what follows will be to take a stab at clarifying the distinction we have in mind, and to explain how this distinction bears on what we take to be the best way to accommodate the motivations behind pluralism.19 Begin with truth-conditions as they figure in meaning-giving biconditionals. Truth-conditions thus understood feature in Davidson’s seminal “Truth and Meaning”, where he proposes that a theory of meaning for a natural language should take the form of a theory of truth for that language.20 Meaning-giving biconditionals are designed to capture the logical place occupied by individual sentences in the whole (potentially infinite) network of sentences of a language. Meaning-giving biconditionals are relatively neutral, metaphysically speaking. This relative neutrality is well-captured by Davidson himself, when he says:
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If we suppose questions of logical grammar settled, sentences like ‘Bardot is good’ raise no special problems for a truth definition. The deep differences between descriptive and evaluative (emotive, expressive, etc.) terms do not show here. … we ought not to boggle at ‘“Bardot is good” is true if and only if Bardot is good’; in a theory of truth, this consequence should follow with the rest, keeping track, as must be done, of the semantic location of such sentences in the language as a whole—of their relation to generalizations, their role in such compound sentences as ‘Bardot is good and Bardot is foolish’, and so on. What is special to evaluative words is simply not touched: the mystery is transferred from the word ‘good’ in the object-language to its translation in the meta-language.21
On the Davidsonian picture presented in “Truth and Meaning”, the truth-conditions cited in meaning-giving biconditionals have the following important features: (1s22) They are arrived at through systematic logico-semantic analysis of the relevant object language. To do its job, such an analysis will exhibit the truth-conditions of sentences as a function of the semantic values of their parts, in a way that reveals how they systematically interact with other sentences and sentence parts, how they embed in various constructions (such as conditionals, modal and propositional attitude contexts), and so on. (2s) Calling the worldly conditions that feature in the biconditionals ‘truth-conditions’ seems apt, given the involvement of truth in recovering logical structure, entailment relations among sentences, and so on. But to play its role here ‘truth’ need not be understood as denoting any robust (or specific) metaphysical property. At the same time, the use of the Tarskian truth schema to specify sentences’ meanings in no way commits one to deflationism about truth, and on some views is incompatible with it. (As is well known, Davidson himself has argued against deflationism about truth, for reasons we cannot rehearse here.23) (3s) Davidsonian semantic analysis can yield surprising results. It can reveal covert ambiguities and context-sensitivity; it can assign sentences logical forms that diverge radically from surface forms (think of Russell’s analysis of sentences containing definite descriptions, or Davidson’s own proposed analysis of action sentences);24 it can exhibit unexpected relations among sentences; and so on. The assignment of truth-conditional
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meaning to mentioned sentences is not a trivial matter; it is subject to substantive constraints, and the semantic contents it yields are not ‘thin propositions’ that are merely the shadows of all syntactically well-formed sentences. (e.g., the theory may rule out as meaningless sentences such as “Twas brillig and the slithy toves did gyre and gimble”, or even “Colorless green ideas sleep furiously”.) Thus, a Davidsonian theory is not a disquotational theory of meaning. (4s) However, there is no presumption that a Davidsonian semantic theory will yield in every case an analytic paraphrase. Take, for example, the sentence “Both Jack and Jill are Americans”. It’s reasonable to expect our semantic theory to unpack the meaning of this sentence in terms of the conditions that Jack is American and Jill is American—in contrast with the different set of conditions for “Jack and Jill lifted the piano”. But there is no presumption that the theory could do any better for “Jack went up the hill” than telling us that it is true iff Jack went up the hill. Similarly, for a sentence such as “There are ten mountains in the Taconic range”, the theory may do no better than offer a disquotational truth- condition. In general, there is no presumption that a truth-conditional analysis will reveal anything interesting about the meaning of semantic ‘atoms’ such as the English terms ‘dog’, ‘mountain’, ‘water’, ‘walk’, ‘love’, ‘blue’, ‘tall’; let alone ‘happy’, ‘beautiful’, ‘funny’, ‘good’, ‘wrong’, and so on. So, there is a sense in which the meaning-giving biconditionals are somewhat modest, semantically speaking. (5s) More importantly, as Davidson himself remarks, meaning-giving biconditionals are also ontologically or metaphysically modest. For they may not in general reveal facts about the existence, composition, or underlying nature of the worldly conditions that the semantic theorist invokes in her specifications of meaning. In propounding, for example, Harrison Ford was good in Blade Runner as the truth-condition of the English sentence “Harrison Ford was good in Blade Runner” (as well as its translations into other languages), we, as theorists of the object language, can remain relatively neutral on what makes for good acting. This means that settling on the biconditionals that specify the meanings of sentences in a given domain can leave room for substantive metaphysical questions and debates. Having settled on the meanings of mathematical sentences, for example, it is open to the semantic theorist (who may or
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may not herself be a metaphysician) to ponder the nature of mathematical facts—whether there are numbers, what kinds of things they are, and so on. Using the vocabulary of an area of discourse, and putting on a metaphysician’s hat, so to speak, one can ask questions such as ‘What in the world (if anything) makes something beautiful, morally right, funny?’. These questions are not in general questions about language, but are instead questions raised in the ‘material’ mode, using language. (6s) In the case of areas of discourse thought to involve commitment to ontologically problematic facts, there seems to be a great advantage to recognizing the availability of the semantic notion of truth-conditions. Acknowledging that the meanings of sentences across a wide array of discourses can be specified by giving their truth-conditions (using the appropriate biconditionals) allows us to accommodate undeniable logico- semantic continuities between the allegedly problematic areas and more straightforwardly ‘descriptive’ ones. (Unless a sentence such as “Hunting for fun is morally wrong” can be assigned truth-conditional meaning, it is entirely unclear how it can embed in conditionals or participate in logical inferences involving purely descriptive elements.) (7s) At the same time, the association of truth-conditional meanings with sentences of, for example, ethics does not automatically remove all worries about the problematic character of putative facts in the relevant domain. One can still be an anti-realist about ethics, even if ethical sentences can be assigned truth-conditional meanings. We would argue that securing truth-conditional meanings for sentences in an area of discourse does not mute all significant disputes between realists and anti-realists regarding its status. This brings us to the second notion of ‘truth-conditions’—the one that seems to be at work in the ‘scenario’ Pettit puts forward. On our way of carving things up, these are worldly conditions invoked when attempting to answer metaphysical questions about ontology, nature, constitution, and so on—questions such as, for example, What is pain? What is color? What makes a person happy? How are mountains to be individuated? When does S know that p? These conditions have the following important features: (1m25) They are semantically innocent. They are provided—and are offered in response to questions that are posed—in the ‘material mode’,
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as opposed to responding to questions about the meanings of sentences. Metaphysical questions can follow on the heels of assigning semantic truth-conditions to sentences, as when we learn from the semanticist— perhaps disappointingly—that “Torturing animals is morally wrong” is true iff, well, torturing animals is morally wrong, and we press: but what makes a practice morally wrong? However, this question (about the nature of moral wrongness) can arise prior to, and independently of, recovering the truth-conditional meaning of any sentence involving the phrase ‘morally wrong’. (2m) Except when one’s metaphysical inquiry concerns language, the metaphysical search for worldly conditions has nothing to do with the analysis of meaning—though, of course, one will typically have to make competent use of language to raise the relevant questions. Familiarly, when the metaphysician of mind tells us that pain is a certain configuration of brain states, or essentially a functional state, this is not offered as a meaning analysis. Similarly for the utilitarian reduction of the goodness of actions to their maximization of utility, and various other reductive accounts. (It may be thought that semantic externalism—a view that connects meaning with conditions in speakers’ external environment—gives the lie to the metaphysical neutrality of semantic analysis just suggested. But this is a misunderstanding. Semantic externalism maintains that, in the case of at least some terms, notably natural kind terms, their meaning is (partly) individuated in terms of the worldly substances to which users of the terms are causally related. So, crudely, a speaker could not mean water by their word ‘water’, unless they were causally related to (the substance) water. First, notice that this only provides a necessary, but not a sufficient condition on meaning water by ‘water’. It thus falls short of giving a semantic analysis of ‘water’ (or the conditions on competent use of the term). But, second, as far as semantic theorizing is concerned, the necessary condition is not to be specified metaphysically. All the externalist semantic theory is in a position to claim is that the meaning of ‘water’ is dependent on the nature of water, whatever that is. If water is in fact identical to the chemical substance H2O, then being H2O is constitutive of its metaphysical nature. But there is no expectation that ‘H2O’ should figure in an externalist semantic account of the term ‘water’.)
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(3m) The worldly conditions that figure in the truths of a semantic theory can be seen as truth-conditions only in the sense that we can think of them as making true some sentences/propositions/beliefs/and so on, and not others. But, so understood, they are conditions that are individuated metaphysically, not semantically. Consider: the worldly condition of H2O being wet is, metaphysically speaking, one and the same condition as that of water’s being wet. So this worldly condition—described either way—makes true the sentence “Water is wet”. But, for all that, “‘Water is wet’ is true iff H2O is wet” is not a meaning-giving biconditional for “Water is wet”. It will not be (or at any rate should not be) a theorem derivable from a truth theory for English.26 Where does all this leave us with respect to truth pluralism? Consider a metaphysical inquiry into what makes something illegal. Such an inquiry may conclude that the legality of this or that act depends in some systematic way on our legal practices, on certain aspects of history, and so on. Perhaps it will conclude that nothing is legal that would not be judged legal by an ideally placed judge, so that it makes no sense to suppose that the legality of an act could forever elude human judgment. We can summarize the results of this inquiry by saying: In legal matters (in contrast with other sorts of matters), truth is judgment-dependent. But the question is whether putting things this way really commits us to a distinct substantive truth property possessed by all true legal sentences (as contrasted with, say, sentences of everyday discourse about mid-sized dry goods). The explanatory gain in invoking a distinct kind of truth seems to us illusory; better, we think, to appeal to differences in what makes true statements in the legal realm true—what objects, properties, or states of affairs (including practices, history, and so on) make for the legality or otherwise of this or that act or practice. Our complaint, in short, is that nothing is added by invoking a metaphysical plurality of truth properties over and above whatever plurality is recognized in the worldly conditions that our metaphysicians have identified or proposed, as they investigate different domains of discourse. Pedersen and Lynch address what may seem like this complaint under the heading “the double-counting objection”. Distinguishing differences “at the level of subject matter” (which they understand as metaphysical differences) from differences “at the level of truth” (which they consider
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to be semantic differences), they address the objection that “to accommodate …the appeal of realism and antirealism with respect to different domains one only needs to buy into differences in subject matter”.27 Following Wright, they respond that the metaphysical difference will inevitably bring a semantic difference in its train. In accounting for the circumstance that is true it seems right to say that there is a fit between the proposition and reality, and that this fit is in no way due to us shaping, or somehow contributing to, what the relevant tract of reality is like. Matters change when we turn to .28
Our question is: What are we here adding to the claim that we have something to do with things being illegal, but nothing to contribute with respect to whether something is a mountain? What difference is there in what ‘accounts for the circumstance’ that the two different propositions are true that is not simply a matter of the difference in the constitution of the relevant facts? Of course, we can ascend to the ‘formal mode’, and instead of talking about what makes for the existence of mountains ask what accounts for the truth of . If we want to generalize over the whole domain, we may need to use the truth predicate, viz. “For all p, if p is a mountain-statement, then p is true iff …”. And, depending on how the condition is filled in, we may be able to say, for example, that mountain statements are true in a mind-independent way. But our point is that the possibility of characterizing the differences in the formal mode in no way betrays commitment to a new, additional difference—one that requires postulating differences in ways of being true, or the possession of divergent truth properties by statements in different areas of discourse. Indeed, when Pedersen and Lynch expound the ‘semantic difference’ between the truth of and , they themselves immediately resort to talk in the material mode: “Mountains are mind-independent entities while laws are social – and so, mind-dependent – constructs”.29 Unlike others who have worried about double counting, our objection is not motivated by a pluralist view of propositional content or a deflationist view of truth.30 If we are right, there is a way of making sense of
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disputes between realists and anti-realists that neither goes via a distinction at the level of propositions nor depends on deflating all truth.31 But it does not depend on invoking different kinds of truth, either. We thus endorse alethic monism: there is only one way for true sentences, propositions, beliefs, and so on, to be true. However, there may be multiple kinds of worldly conditions that make them true. The relevant plurality can be captured in the material mode; it doesn’t require any semantic or alethic ascent. Of course, given the equivalence of
and
, one can advert to a formal mode and speak of the truth of ‘x is red’ being a different sort of thing from the truth of ‘x is divisible by 2’—indeed, sometimes putting things in terms of truth may be unavoidable. It is the additional move, to a plurality of truth properties, each appropriate to a different domain of discourse, that we here oppose. This move, we maintain, is not forced on us by taking seriously debates between realists and anti-realists. Alethic plurality contributes no explanatory power; all we need is a plurality of kinds of worldly conditions.32 * * * A final remark. Wright has proposed the following analogy by way of making pluralism about truth plausible.33 Winning a game is a unitary concept. But in different games different things count as winning. So, there is a plurality of ways of winning. Similarly, Wright has suggested, there is a plurality of ways of being true, and thus a plurality of truth properties that are ‘satisfiers’ of a uniform notion of truth. A difficulty Wright considers for the analogy is that, given a game, it is typically obvious what constitutes winning in the game. By contrast, knowing what constitutes being true in any given domain typically requires extensive philosophical investigation and can be highly contentious. We suggest that our way of portraying things can readily absorb the disanalogy. As regards games, the key idea is what constitutes winning will vary from game to game. Regarding different domains of discourse, the idea should not be that what constitutes being true varies from area to area. Instead, recast in metaphysical terms—‘at the levels of facts’ rather than at the level of propositions or truth—the idea should be that the kinds of worldly conditions
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that make a statement true will vary. But then it stands to reason that it will not in general be obvious what the ‘realizers’ of truth in various areas are, in contrast with what constitutes winning a game. Access to what in the world makes true statements true requires a metaphysical investigation. When it comes to games, since they are invented, and in that sense ‘of our own making’, knowing what game is being played guarantees knowing what constitutes winning it. (That is part of what is instituted when the game is designed.) Not so for what renders statements true in a given domain of discourse. In general, engaging in an area of discourse does not bring into existence the relevant worldly conditions. And, whatever disagreements we may have cannot be settled—as they can in the winning case—simply through reflection on the rules of the discourse. Where disputes arise, they are metaphysical ones.34
Notes 1. Wright (2013: 124) 2. Here and throughout, ‘
’ is a name of the proposition that p. 3. See, for example, Lynch (2013). 4. See, for example, op. cit. 5. Lynch 2013. 6. The equivalence here is stronger than necessary equivalence. As Jeremy Wyatt has pointed out to us, it’s arguable that A and B can be necessarily equivalent without both being superwarranted for a subject S. Consider the necessarily equivalent propositions and . Suppose S grasps all the relevant concepts. It is possible that is superwarranted for S, but is not, since S may have warrant to believe that there’s water in the glass, but not that there’s H2O in the glass. But if S grasps the concept of truth,
and
will be conceptually equivalent for S, and if one is superwarranted for S, so is the other. 7. Lynch offers this definition of propositional coherence (with respect to moral propositions) in Lynch 2009, p.171. 8. Lynch 2013. 9. Bar-On and Simmons (2007), Bar-On et al. (2004). 10. Bar-On and Simmons (2007).
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11. Strawson (1992). 12. Horwich (2006: 195). 13. Ibid. 14. See Wright (2003: 136). Wright registers disagreement with ‘meaning pluralists’ by saying: “the realist/anti-realist debate is not a semantic debate in the end” (2013: 126). This can be confusing, since meaning pluralists like Blackburn sometimes describe their disagreement with truth pluralism by complaining that it introduces an unnecessary detour via the semantic property of truth. 15. Pettit (1996: 886) (our emphasis), cited with approval in Wright (1996: 101f.). 16. Davidson (1984: 171–180). 17. On some views (though not Davidson’s), truth-conditions so understood are what competent speakers have mastered (or internalized) and know, at least implicitly. For relevant discussion and references, see Bar-On (1996). 18. And witness his subsequent reference to “what it is for something to hold in physics – what the truth-condition is …” (ibid., our emphases). 19. It will be important to bear in mind that we are not suggesting, along the lines of e.g. two-dimensional semantics, that sentences have associated with them two kinds of meaning, or two sets of truth-conditions. 20. Davidson (1984: 17–36) 21. Davidson (1984: 31). 22. ‘s’ for semantic. 23. Davidson (1990, 1996). 24. Davidson (1980: 105–122). 25. ‘m’ for metaphysical. 26. This is perhaps why deflationists about truth are perfectly happy to allow that we do – and can, consistently with deflationism – speak of worldly conditions that we loosely refer to as truth-conditions. For discussion and references, see Bar-On et al. (2004). 27. See Pedersen and Lynch (2018). 28. op.cit. 29. op.cit. 30. Dodd (2013) advances a version of the double counting objection that draws on deflationism. 31. For we have here only sought to question the utility of invoking a plurality of truth properties over and above the property of truth that, by the moderate pluralist’s lights, all true items possess (regardless of domain).
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32. Asay (2016) also argues that all the plurality we need is to be found in the world, not in a plurality of truth properties. But Asay’s plurality is a plurality of truthmakers rather than truth conditions. And Asay is a primitivist about the concept of truth and a deflationist about the property; we don’t make these commitments. 33. See Wright (2013: VII), who follows Edwards (2011, 2013). 34. Our thanks to Jeremy Wyatt for many helpful comments, and to the participants in the Conference on Pluralism about Logic and Truth, University of Connecticut at Storrs, April 2015.
References Asay, J. 2016. Putting Pluralism in Its Place. Philosophy and Phenomenological Research 96 (1): 175–191. Bar-On, D. 1996. Anti-Realism and Speaker Knowledge. Synthese 106: 139–166. Bar-On, D., and K. Simmons. 2007. The Use of Force Against Deflationism: Assertion and Truth. In Truth and Speech Acts, ed. D. Greimann and G. Siegwart, 61–90. New York/London: Routledge. Bar-On, D., W. Lycan, and C. Horisk 2004. Deflationism, Meaning and Truth- Conditions, reprinted with Postscript in Deflationary Truth, ed. B.P. Armour-Garb and J.C Beall, 321–352. Chicago: Open Court Readings in Philosophy. Davidson, D. 1980. Essays on Actions and Events. Oxford: Oxford University Press. ———. 1984. Inquiries into Truth & Interpretation. Oxford: Oxford University Press. ———. 1990. The Structure and Content of Truth. Journal of Philosophy 87: 279–328. ———. 1996. The Folly of Trying to Define Truth. Journal of Philosophy 93: 263–278. Dodd, J. 2013. Deflationism Trumps Pluralism. In Truth Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C. Wright, 298–322. Oxford: Oxford University Press. Edwards, D. 2011. Simplifying Alethic Pluralism. Southern Journal of Philosophy 49: 28–48. ———. 2013. Truth, Winning, and Simple Determination Pluralism. In Truth Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C. Wright, 113–122. Oxford: Oxford University Press.
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Horwich, P. 2006. A World Without Isms. In Truth and Realism, ed. P. Greenough and M.P. Lynch, 188–202. Oxford: Oxford University Press. Lynch, M. 2013. Three Questions for Truth Pluralism. In Truth Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C. Wright, 21–41. Oxford: Oxford University Press. Pedersen, Nikolaj J.L.L., and M. Lynch. 2018. Truth Pluralism. In The Oxford Handbook of Truth, ed. M. Glanzberg. Oxford: Oxford University Press. Pettit, P. 1996. Realism and Truth: A Comment on Crispin Wright’s Truth and Objectivity. Philosophy and Phenomenological Research 56: 883–890. Strawson, P. 1992. Analysis and Metaphysics. Oxford: Oxford University Press. Wright, C. 1996. Précis to Truth and Objectivity and Response to Commentators. Philosophy and Phenomenological Research 56: 863–868, 911–941. ———. 2003. Saving the Differences. Cambridge, MA: Harvard University Press. ———. 2013. A Plurality of Pluralisms? In Truth Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C. Wright, 123–153. Oxford: Oxford University Press.
Truth Pluralism, Quasi-Realism, and the Problem of Double-Counting Michael P. Lynch
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Unity and Diversity
Philosophical tradition has long decreed that truth comes in only one kind. To talk about different kinds of truth is like saying that the Sphinx has a different kind of eye—it is just another way of saying that it has none at all.1 But philosophical tradition also decrees that traditions are there to be opposed. Truth pluralism—or the idea, roughly, that there are different kinds of truth—stands with the opposition in this case. Since its introduction onto the contemporary scene by Crispin Wright over two decades ago, truth pluralism has been connected to, and partly motivated by, two major explanatory projects. One project involves accommodating the intuitions that drive both realism and anti-realism (Wright 1992). By adopting truth pluralism, the suggestion goes, we may say that in some domains, statements or beliefs are true in a realist way, while in other domains, truth is understood in ways traditionally championed by “anti-realists.” Thus, for example, some statements are true M. P. Lynch (*) Department of Philosophy, University of Connecticut, Storrs, CT, USA e-mail: [email protected] © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_3
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because they represent—in some suitably thick sense of “represent”— mind-independent natural objects (Lynch 2009), while others—such as those concerning normative matters—are true in virtue of being superassertible, or supercoherent in some fashion or other (Wright 1992, 1998b). A second major explanatory project connected to truth pluralism involves accommodating two distinct intuitions concerning the relationship between our thought, talk, and the world. On the one hand is the idea that our thought and talk itself come in radically diverse kinds. Thought is not a “seamless web” (Blackburn 1998a, p. 157). Rather, different kinds of attitudes and speech-acts have different functions in our cognitive lives, and that fact offers a crucial constraint on our account of how we understand, or explain their content. Call this: semantic diversity: there are real differences in kind between the contents of our beliefs and indicative statements. Privileging semantic diversity is an obvious starting point for certain forms of classical expressivism, according to which the moral judgments differ from non-moral ones in part because the latter have representational content while the former do not. However, a typical complaint with such views is that, in their haste to privilege diversity, they tend to deny: cognitive unity: all such statements and beliefs are subject to a single type of cognitive normative assessment or correctness. Roughly put, a typical complaint against classical expressivism is that statements about values are subject not only to practical reason, but also theoretical reason. They figure in the premises of (valid) arguments and as the antecedents of conditionals; they are truth-functional; when we believe the contents of these statements, we can provide epistemic justifications for our holding of them; and so on. As we might put it, value judgments can be correct or incorrect just as judgments about the physical world can be. Expressivists, of course, have offered a range of responses to this challenge—a challenge that is sometimes bundled under the heading of the “Frege/Geach” problem.2 The truth-pluralist, can, in effect, be seen as doing the same. That is, while Crispin Wright (1992) initially introduced
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pluralism as an alternative to expressivism, the position can be understood as offering a response to the complaint that the root insight of expressivism is inconsistent with cognitive unity.3 Thought and talk are unified (one) because truth is correctness, but thought and talk are diverse (many) because correctness comes in different kinds. There are, naturally, different ways to pursue these points. Early statements of truth pluralism were typically understood as implying that there was no single concept of truth, and that the word “true” is ambiguous across domains (Pettit 1996; Sainsbury 1996; Tappolet 1997).4 Most current pluralists reject this view, however, emphasizing that there is a single concept of truth. Said concept is a functional concept, one which names a property which can itself be realized by very different underlying properties—a correspondence property for some kinds of beliefs and statements, and a coherence property for others.5 Suitably understood, this is still consistent with the idea that truth comes in different kinds. Just as the functionalist in the philosophy of mind can say that there are different kinds of pain—pain as realized by neural property Φ and pain as realized by neural property Ψ—the truth pluralist can say there are different kinds of truth: truth as realized by property α, and truth as realized by property β. The differences enter in at the level of the properties that determine or realize truth. As noted, these differences are often cashed out by appealing to metaphysically heavyweight properties of representation or idealized coherence. Strictly speaking, that is not necessary. Some pluralists (e.g., Beall 2013) are attracted to a view according to which there are different kinds of truth, but every kind of truth is “deflated” or minimal. Even so broadly sketched, pluralism has at least two prima facie benefits to anyone attracted to the expressivist insight of semantic diversity. First, there is the significant theoretical benefit of allowing us to say that some beliefs and statements have representational content and some do not—without complicating the semantics (see Lynch 2009). That is, all indicative statements can be seen as alike in having their content determined (at least in part) by their truth conditions. It is just that those conditions themselves come in different kinds. Second, the differences in the metaphysics—the differences in the realizing properties—can be appealed to in order to explain the differences in content between statements and
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beliefs of different kinds. Thus, rather than trying to distinguish, for example, all normative from non-normative beliefs in advance, so to speak—and thus possibly running into the problem cases of legal discourse, or discourse involving so-called thick concepts—the pluralist can sort beliefs by appealing to the properties that realize truth for the particular content believed.6 There is a tremendous amount of work—much of it technical—to be done in order to fully understand the details of such a view. Among other things, pluralists must say something about so-called mixed inferences and compounds, as well as the relationship between truth and logical consequence. They must also offer substantive theories about the properties that determine or realize truth. Such work is still ongoing.7 In this essay, however, I’m concerned less with the details of truth pluralism than I am with an objection to the view that comes up before we even get to those details. It is a criticism most recently and forcefully articulated by Simon Blackburn, whose own quasi-realist view shares the truth pluralist’s desire to reconcile semantic diversity with cognitive unity. It goes like this: There aren’t different kinds of truth; there are simply different kinds of truths. As Blackburn recently put it, while the differences between, say, ethical judgments and judgments about the physical world are significant and important, …they strike at the level of the proposition: they mark distinctions of subject matter, and perhaps eventually distinctions of objectivity or the possibility of cognitively fault-free disagreement. But why add to a distinction of content, another, mirroring distinction, one only applying to kinds of truth or conceptions of truth?” (Blackburn 2013, p. 265)
In Blackburn’s words, the truth pluralist is engaged in a kind of double-counting. The objection is intuitive, and a similar point has been pressed by a number of authors (Dodd 2013; Sainsbury 1996).8 Hence it seems incumbent on the truth pluralist to have something to say about it. On closer inspection, however, there appears to be at least two subtly different claims that can be made under its auspices. In what follows, I will attempt
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to disentangle them. As we’ll see, only one version of the point is compatible with Blackburn’s own quasi-realism; but it faces its own problems— problems that encourage the reintroduction of the very distinctions it was aimed against. As such, I conclude that truth pluralism need not fear being audited for double-counting—or at least no more than the quasirealist. Indeed, the quasi-realist, I will argue, has excellent motivations for adopting—or readopting, as we shall see—truth pluralism.
2
Subject-Matter First
The first way of construing the double-counting objection can be found in the Blackburn quote given just above. It is the idea that semantic diversity can be reduced, or solely explained as, diversity in subject-matter. Quine first made this point: There are philosophers who stoutly maintain that “true” said of logical or mathematical laws and “true” said of weather predictions or suspect confessions are two usages of an ambiguous term “true”…Why not [instead] view “true” as unambiguous but very general, and recognize the difference between true logical laws and true confessions as a difference merely between logical laws and confessions? (Quine 1960, p. 131)9
In Quine’s formulation, the truth pluralist’s error lies in part in positing “true” as ambiguous: it needlessly gives different meanings to the predicate. As we’ve already noted, the contemporary pluralist can agree—“true” should be understood as expressing a single concept. Thus, the second aspect of Quine’s point is more salient for our purposes. Namely, we don’t need to appeal to truth at all in order to explain the differences between confessions and logical laws (or morals and mathematics). In Quine’s hands, this point is clearly meant to fall out of his disquotationalist view of truth. As Quine would put it, to say that the statement “snow is white” is true is just another way to talk about snow. Truth is a device for disquotation. If so, then why semantically ascend to the lofty heights of truthtalk to explain the differences when we can look to the world itself?
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Why indeed. The answer is that sometimes there is no world to look to. That is, in some cases, we want to explain the differences between different kinds of thoughts, even though—and perhaps precisely because— we think that some of them fail to represent any independent reality. But how do we capture this possibility if we are not allowed to appeal to the semantic differences between the statements or beliefs in question? Consider, by way of illustration, a different sort of philosopher from Quine: someone who thought that truth is always and everywhere corresponding with propertied objects. This philosopher might say that what marks the differences between our language games is that in physics we have one type of object—electrons, protons, and the rest—and in morality we have another—hedons, or, as Dworkin (1996) once memorably put it, morons. Or, perhaps they might say: the difference consists in the fact that moral discourse is just never true; there are no objects to which it refers, poignant as it may be.10 Absent further garnishing, these are just the options the Quinean version of the objection leaves us as well. If all we have to say about the differences in our thought and talk is “look to the objects,” then we either end up with blanket realism or an error theory. That may be perfectly correct—I have nothing to say against such options here. The point, rather, is that such a strategy leaves some options out, options that Blackburn for one would be keen to insist upon, according to which the semantic diversity of our thought is greater than we are so far acknowledging.11 Of course, both Quine and our imagined correspondence theorist might try to recapture those options by saying this: some sentences are in the game of truth and others are not. The former has truth conditions, or express propositions. The standard usage of the latter amounts to, for example, planning to do something, approving of so and so, or commanding someone to do such and such. This, of course, is just classical expressivism. But to endorse that view is to explain semantic diversity, not in terms of different kinds of propositions, but by appealing to a difference between discourses that do express propositions and those that don’t—between those that are truth-conditional and those that aren’t. It just gives up on cognitive unity.
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As a result, Quine’s “subject-matter first” point cannot be the double- counting objection that Blackburn initially had in mind. For such a view, while certainly coherent and no doubt attractive to many, is not aiming to at the same explanatory goal of explaining both diversity and unity. Let us therefore look elsewhere for the real point behind Blackburn’s objection.
3
Content, Not Truth
We needn’t, I think, look far. Here is Blackburn invoking some heroes from Philosophical Valhalla: The point is that Ramsey and Wittgenstein do not need to work with a sorted notion of truth—robust, upright, hard truth versus some soft and effeminate imitation. They need to work with a sorted notion of a proposition, or if we prefer it a sorted notion of truth-aptitude. (Blackburn 1998b, pp. 166–167)
Like Quine before him, Blackburn takes issue with the truth pluralist’s view that appealing to different kinds of truth will help explain the differences. And, like Quine, he is motivated to take this view at least in part because of his allegiance to a version of deflationism about truth. In particular, Blackburn is attracted to Horwich’s minimalism. According to Horwich (1998), the concept of truth functions as a logical device for generalization: it allows us to overcome our merely medical limitations and make certain generalizations, such as “every proposition is either true or false.” We grasp that concept by being a priori disposed to grasp instances of the equivalence schema: ES: The proposition that p is true if and only if p. When we understand these facts—how we grasp the concept, and its function—we know all the essential facts about truth. Any other fact about the truth-role can be deduced from (instances of ) ES together with some non-truth-theoretic fact. Consequently, even if we grant that truth
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is a property—if a metaphysically transparent one, like the concept of being a logical conjunction—it is a property that does no significant explanatory work. Blackburn, as is well known, enlists minimalism about truth in the service of quasi-realism. He takes it that ethical, aesthetic judgments, statements, and the like do, in fact, express propositions. They just express unique kinds of propositions, each kind of which is still apt for the same kind of truth. The question remains how to explain that. We’ve just seen one way: explain differences in propositional content by appealing to differences in the objects and properties to which those propositions (or their structural elements) refer. But Blackburn wants a view that allows that moral judgments, beliefs, and so on might be true without it being the case that there are, in any serious sense, moral properties. He wants to say that some kinds of propositions are true in very different circumstances, as it were, than others. Which is just to say—isn’t it?—that they have different kinds of truth conditions. But that seems precisely to be the view that the truth pluralist is urging, minus the simple explanation for its obtaining that the pluralist can supply—namely, there are different kinds of truth conditions because there are different kinds of truth. Although it may be obvious, it is worth emphasizing that the quasi- realist is not simply repeating her minimalism in taking on this view. She is not simply saying: each proposition specifies its own truth condition. One can endorse that point without being a quasi-realist. Rather, she is insisting, in addition, that such truth conditions are themselves sorted into discourses, domains, and subjects—in short, kinds. That is the point of saying that moral judgments, beliefs, and statements are semantically distinct from the non-moral versions of such things. But what would explain this fact? Why should the conditions under which some contents are true differ in kind from others? At this point, given that the quasi- realist has excluded the answers above, it again seems hard to avoid the simplest answer: because they are subject to different kinds of truth.12 Indeed, this is precisely the view that Blackburn had himself defended earlier in his career: The problem is not with a subjective source of value in itself, but with people’s inability to come to terms with it, and their consequent need for a picture in which values imprint themselves on a pure passive, receptive witness…
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To show that these fears have no intellectual justification means developing a concept of moral truth out of the materials to hand: seeing how, given attitudes, given constraints upon them, given a notion of improvement…we can construct a notion of truth. (Blackburn 1984)
This passage (and the detailed discussion of the idea by Blackburn that follows) is striking. The idea is precisely that of the truth pluralist and motivated in a similar way: namely, by pointing out that those who embrace a “subjective source” for value can respond to the obvious objections by demonstrating that moral judgments are apt for a distinctive, but nonetheless real, kind of truth. To sum up so far: The double-counting objection was initially framed as the simpler alternative to truth pluralism. The thought seemed to be: well, we can accomplish all of our theoretical goals—explaining both semantic diversity and cognitive unity—just by appealing to something we already have for “free”: different kinds of propositions. We don’t need (“in addition”) kinds of truth. And that all goes swimmingly, save for the fact that the most obvious way to press the objection—Quine’s way— self-consciously fails (or ignores) that very explanatory task. The Quinean route ends up either getting unity at the price of diversity (by enforcing a blanket realism), or diversity at the price of unity (by enforcing a distinction between the truth-conditional and the non-truth-conditional). Consequently, we arrive at the quasi-realist’s actual suggestion: truth explains unity, kinds of propositions explain diversity. But this slogan is already seeming less simple, and indeed, less distinct, than it did at first blush. It is starting to sound like truth pluralism in disguise.
4
Quasi-Realism at Any Price
In part for the reasons given above, some expressivist sympathizers have pursued other options in recent years. One option is to go hybrid (Ridge 2006, 2009) and argue that ethical commitments are comprised of a conjunction or complex of a belief state and a desire or sentiment. Another is to take there to be one basic, non-doxastic kind of commitment of which ethical and non-ethical commitments are themselves species (Schroeder
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2008, 2010). Both of these suggestions have considerable technical merits (and complications) all their own, and of course warrant separate discussion. In any event, they are not the options taken by the quasi-realist. Blackburn aims, he suggests, to hew closer to the expressivist’s original core. Thus, we find him saying: There are propositions properly theorized about in one way, and ones properly theorized about in another. The focus of theory is the nature of the commitment voiced by one adhering to the proposition, and the different functional roles in peoples’ lives (or forms of life, or language games) that these different commitments occupy. (Blackburn 1998b, p. 167)
What makes A-propositions different from B-propositions is the differences in the commitments we take toward them in our lives, commitments that have different motivational and emotional profiles, for example. Call this the commitment-first approach. If it is to succeed, it must distinguish kinds of commitment without relying on either a simple, unexplained difference of kind between propositions, or on kinds of truth. In the halcyon days of the expressivism of yore, prior to eating the fruit of the Frege-Geach tree, we would just have said: well, some attitudes or commitments—like belief—just are the kinds of commitments that are truth-apt, whereas other kinds of attitudes or commitments are not. But again, that obviously isn’t the picture the quasi-realist is offering. That’s because the quasi-realist, remember, takes it that all kinds of commitments to propositions are truth-apt. They are, after all, commitments to propositions. Put another way, the quasi-realist is saying that we can take what we used to innocently call the attitudes of asserting and belief toward every kind of proposition. We deflate belief and assertion as much as we deflate truth, as Blackburn (1998a, b) himself would be the first to admit. Over the last decade, some philosophers attracted to quasi-realism, and at points Blackburn himself, have pursued another framework, one made famous by Brandom (Brandom 1998) and recently championed by Price (Price 2011, 2013). According to this view, what makes one attitude/speech-act/commitment different from another is its special normative and social role. The common basic idea is that if you, for example,
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believe or assert that p (as opposed to merely wondering whether p), then you become liable to certain pushbacks from your interlocutors if it turns out that not-p. Moreover, you incur certain obligations to provide them with evidence that p should they demand it, to stand by certain easily recognized implications of p, and so on. In short, what distinguishes one sort of commitment/attitude/speech-act from another is the type of status and responsibility accorded to the person who engages in that commitment/attitude/speech-act. They are not distinguished in terms of truth, reference, or direction of fit (whether word-to-world or world-to-word). Price, in particular, has urged that a view of this sort is the most intuitive extension of the quasi-realist view (see, e.g., 2011, 2013). And he has been explicit that it is designed to capture what I’ve called semantic diversity and cognitive unity. Moreover, he has urged that the view be adopted globally—that we become global expressivists, as he has put it. Of course, in order to take the view to help us with our present problem of distinguishing kinds of propositions without appeal to different kinds of truth, we’d need to be able to show at least two things. First, as Brandom (1998) in particular has tried to do, we would have to show that we could build a real theory of content out of this theory of the types of attitudes/commitments/speech-acts. Thus, for example, such a theory must have the resources to distinguish between contents that concern our own informational states and attitudes, and those that are wholly independent of them. Whether it can do so remains an open question, so let’s put this aside. Second, the Pricean pluralist must be able to distinguish ethical and non-ethical propositions while still maintaining the idea that, for example, ethical propositions can still be believed and asserted, albeit in a different way than propositions about the physical world are believed and asserted. It is, after all, the appeal to this “different way” of believing and asserting that makes the commitment-first approach different from the truth-conditional approach. The point here, to clarify, is not that the global, Pricean quasi-realist must simply distinguish between different attitudes (like belief and desire) that are operative in different domains of discourse; this is something that all players can grant. Rather, they must distinguish between different kinds of one type of attitude—for example, belief (Dreier 2004; Ridge 2006).
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Let’s pause here for a moment. The double-counting objection, in the hands of the quasi-realist, maintains that the truth pluralist unjustifiably distinguishes between kinds of truth when a distinction between kinds of truths can do. But now we see that the quasi-realist distinguishes between ways of asserting and believing (different kinds of assertion and belief ) when a simple distinction between assertions and beliefs would do. Double-counting! If, contra quasi-realism, we can appeal to truth, we can say, for example, that the differences between moral beliefs and other sorts of beliefs doesn’t lie in the fact that there is a special “moral way” of believing, but in the fact that moral contents are true in different kinds of conditions— precisely the view we saw Blackburn maintain in an earlier version of quasi-realism. Tit for tat, however, is a game of only limited interest. The quasi-realist double-counts, but the commitment-first approach can, after all, appeal to real differences in our practices. But that fact, while real, can sometimes obscure a simple point. These commitments, recall, are now agreed to also constitute moral beliefs in a deflated sense. But with what does this deflated sense contrast? Surely there must be a contrast if we wish to count not only different beliefs but also different kinds of belief. And it is not enough to say that, for example, non-moral beliefs aren’t motivational. For that would paint mathematical beliefs, modal beliefs, aesthetic beliefs, and beliefs about the physical world with the same wide brush. Thus, in the physical case in particular, it is tempting to add that such beliefs represent the world while moral beliefs don’t. But, as a number of authors (Chrisman 2008; Dreier 2004) have asked: how do we say that without giving up on our global deflationary stance toward truth, belief, assertion, proposition, and the like? Huw Price has replied, on behalf of the quasi-realist, that they can make truth-free distinctions between different kinds of discourses by appealing to a difference in kinds of representation: what he calls i-representation and e-representation. According to Price, i-representations represent by having “a position in an inferential or functional network” (Price 2013) while e-representations represent by covarying with, or tracking, the environment (Price 2013, p. 43). Price maintains that both notions can have their uses. But, most importantly for our purposes, appealing to the distinction allows the quasi-realist to say that “scientific
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claims nevertheless have a world-tracking, e-representational character that moral claims lack” (2013, p. 39), and that “while all assertoric vocabularies are i-representational, some may be much more e-representational than others” (2013, p. 152). Price puts this distinction to use in various subtle and interesting ways. But it is unclear whether it can help the quasi-realist press her contrast with the truth pluralist. One reason for thinking this is that it seems to be a truism about truth and representation alike that: Represent: A belief with the content that x is F is true if and only if the object represented by the concept x has the property represented by the concept F. Thus, one might think, if there are different types of representation, then we get not only different kinds of belief but also different kinds of truth, namely: Represent-I: An i-belief with the content that x is F is true if and only if the object i-represented by the concept x has the property i-represented by the concept F. Represent-E: An e-belief with the content that x is F is true if and only if the object e-represented by the concept x has the property e-represented by the concept F. How might the Pricean quasi-realist resist this charge? One way would be to just accept that Represent-I and Represent-E result in two distinct concepts of truth—concepts which “true” is simply ambiguous between. Alternatively, they can insist on their minimalism about truth. Truth is one thing, they can say, and representation is another. And all there is to say about truth as such is still simple, and requires no appeal to the different kinds of representation. But this point also won’t distinguish the Pricean quasi-realist in any significant way from the truth pluralist. As noted earlier, truth pluralism, in its principal formulations, isn’t an ambiguity view. It can indeed be described as the view that there are different kinds of truth—in the same sense in which the functionalist in the philosophy of mind can say that there are
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different kinds of pain. In Wright’s view and in others’, there is more than one property that can determine, realize, or manifest the property of truth as such. And it is difficult to see how Represent-I and Represent-E aren’t just saying exactly that—namely, what determines the truth of i-beliefs is i-representation and what determines the truth of e-beliefs is e-representation. But even if this is somehow resisted, the fact remains: while we started with the thought that there was a strong contrast between truth pluralists and quasi-realists, we have ended with the thought we might as well call representational pluralism. The bump in the carpet has been moved. Unsurprisingly, perhaps, Price himself suggests that the real difference between truth pluralism and representation pluralism lies in how proponents of each view tend to conceive of their project. Truth pluralists like Wright typically see themselves as revealing the relational properties that determine whether a belief or statement’s content is true. They are doing a bit of metaphysics in service of their semantics. It is this that Price wishes most to resist. For Price, representation-talk of either kind is just that: talk. It serves a purpose, but it drops out at what he calls the “upper level” (2013, 155ff). In saying that, for example, some assertions or attitudes are more e-representational than others, we are ultimately not making a distinction between different kinds of relational properties. Instead, we are making claims about the “talk not the ontology” (2013, 158); we are merely describing the roles such assertions play in our greater theorizing. This is the point of his insisting that his view represents a global expressivism or quasi-realism. Ultimately, then, the Pricean quasi-realist explains the differences between kinds of thought and talk only internally to our thought and talk themselves. Interestingly, Blackburn remains skeptical; and his skepticism is worth quoting in full: Locating thinkers or speakers in a landscape in which we only answer to statuses accorded to us or denied to us by fellow thinkers and speakers risks distorting our positions. For we do not just answer to each other. We answer to each other because of what we get right or wrong about the things we are involved with—the things we are talking about…If I am deciding whether an object is red or square or weighs five pounds, I am not primarily concerned with what other people will say is the case nor predicting the penalties if I am out of step with them…It is not like tuning up an
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orchestra, where my prime concern is to listen to whether my note is the same as the notes of other players. We are not after democratic harmony but getting the judgment right. (2013, p. 268)
Indeed. There are two key points here, both directed—rightly in my view—at ways in which the Pricean view misrepresents our actual practices. The first is the worry that in globalizing expressivism we run afoul of the claim that pragmatic views like this get things backward. In “answering to each other,” we take ourselves to be correct in our judgments because they are true—not vice versa. And second, when in particular we are concerned with matters involving the middle-sized dry goods of everyday life, or science, we take our judgments’ correctness, and—given the first point—their truth, to be a matter of whether we are responding, in making those judgments, to those dry goods themselves. Blackburn’s point here could be usefully put in terms of Price’s two kinds of representation. When I describe an object as red or square, I ordinarily take my description to be correct precisely because it is a response to how things are independently of me. And “responsiveness” here surely means, at least in part, that I am tracking the environment—I am e-representing the middle-sized dry goods around me in just the way one suspects that evolution has programmed me to do. And that seems, moreover, to capture part of the difference between talk of square things and talk of what is morally right and wrong. The latter is not plausibly governed by a norm that is cashed out in terms of responsiveness to a world independent of us, and the former is.
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Quasi-Realism and Truth Pluralism
As we saw above, the quasi-realist wasn’t always opposed to truth pluralism. In fact, the earlier Blackburn embraced it. What I’ve been suggesting, in effect, is that he was right to do so. For truth pluralism gives you both what we called earlier Cognitive Unity (and the cognitive surface that goes with it) together with Semantic Diversity—the view that the quasi-realist is so keen to insist on. And we don’t need to go in for the view that there is more than one concept of truth to make the point. We
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can, as suggested above, simply stick with the idea that truth is a functional property. To define truth functionally in the sense I am interested in is to define it by way of its connections to other related concepts. These connections are embodied in certain common truisms that have played a central role in the historical discussions over truth: the equivalence schema; the slogan we called “Represent”; the idea that what makes a proposition correct to believe is that it is true; that what valid inferences preserve is truth; that truth is a worthy goal of inquiry; and the like. The idea is that these truisms, or ones very much like them, jointly pick out the truth-role. Beliefs are true just when they are correct, when they are the sort of beliefs we aim to have during inquiry, when the world is as they portray it as being. That is the job description of truth, as it were. It seems to me that this is just the sort of view that both Blackburn’s quasi-realist and Price’s expressivist need. It allows us to say, against traditional theorists, that truth as such may be a very thin, functional property. But seeing truth as a functional property in this sense is entirely consistent with there being other properties, distinct from truth, which satisfy this description and so realize that thin property—which, in short, play the truth-role. And that is a helpful thought, because it opens to door to appealing to these other properties to explain what grounds the norm in different domains or discourses. It allows us new tools for addressing what plays the truth-role for different kinds of content and, therefore, giving us an easy explanation for what makes those kinds different kinds in the first place. They are different kinds because they are subject to being correct in different ways. Of course, the ultimate suitability of this suggestion rides in part on whether we can make sense of a property—distinct from correspondence—that could play the truth-role for domains like the ethical. One possibility, as already noted, is to say that truth is sometimes realized by an extremely deflated property. Another is to follow Blackburn’s earlier lead: to look at the “materials at hand,” and to see whether we can construct such a property from properties already in play in the moral realm—properties, for example, like coherence. This, in fact, was precisely the property that Blackburn put to such effective use in his earlier (1984) work.
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The basic recipe might be sketched as follows. Step one: look to what warrants moral beliefs. According to a leading theory of moral epistemology inherited from Rawls, a moral belief is warranted to the degree to which the moral framework to which it belongs is in a state of wide reflective equilibrium. We can, if we like, see this as a form of coherence theory: S’s moral judgment that p is warranted to the degree that it coheres with the rest of S’s moral and non-moral judgments. The term “coherence” can be taken here to pick out a family of epistemic desiderata. It is generally thought that a framework is coherent insofar as, and to the degree to which, its members display relations of mutually explanatory support, it is complete, and it is consistent. Call these coherence- making features. Such features themselves come in degrees: members of a framework can be more or less consistent, more or less mutually explanatory, and so on. A framework of judgments increases in coherence to the degree to which it exemplifies these features, on balance, to a greater degree. “On balance” because the features are not themselves isolated in their coherence increasing power. A framework would not be more coherent on balance, for example, simply by increasing its size (completeness) by including consistent but explanatorily unconnected judgments. Intuitively, by increasing its explanatorily isolated judgments, the coherence of the framework would on balance remain static or decrease. Step two: use these definitions to make sense of what it would be for a moral framework to improve in coherence. Framework F is more coherent at t2 than at t1 when, on balance, it has at t2 either more of the coherence-making features or some of those features to a greater degree. So, if completeness, consistency, and explanatory connectedness are coherence-making features, adding a consistent and explanatorily connected judgment to the system will increase that system’s coherence along those dimensions. Consequently, we can say that P coheres with moral framework F if, and only if, including P in F would, on balance, make F more coherent. Step three is to use this notion of improvement to build a property that could be said to realize truth (as opposed to warrant). This is not a trivial task. But neither is it hopeless. One suggestion, building on work by Wright and others, would be to say that if warranted moral judgments are those that are coherent, then true moral judgments are those that are, as we might say, supercoherent:
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SUP: The moral judgment P supercoheres with F if and only if P coheres with F at some stage of inquiry and would continue to do so through all successive improvements of information to F, both moral and non-moral. Call an individual moral judgment supercoherent just when it coheres with some moral framework. The suggestion, then, comes to this: when a moral judgment is true, it is true in virtue of being supercoherent. Obviously, there is much more to say about each step of this recipe, and about the strategy overall. I’ve tried to say some of it elsewhere (Lynch 2009). But whatever the details, I think that despite Blackburn’s own changes of heart over the years, the case remains strong for thinking that the quasi-realist should embrace truth pluralism. On the one hand, it secures the “quasi” in quasi-realism: the view is consistent with there being no distinctively moral properties in the world. Second, it is consistent with the thought that our value judgments have a different function than our judgments about the natural world (Blackburn 1984). But, unlike a more deflationist quasi-realism, the theory still allows for a sharp contrast between the moral and the non-moral: the door is open to holding that value judgments are true in virtue of supercoherence, and that judgments about the physical world are true in virtue of representing entities in the natural world. Yet the view would also secure the realist side of the equation, as supercoherence theories of what realizes truth in the ethical domain seem to allow for the possibility of moral error. A judgment’s being supercoherent is a significant cognitive achievement. Many of our current moral judgments may not be supercoherent. Indeed, it is possible that none of them are. And, as already noted, the view allows for a notion of improvement of moral judgment. The problem of double-counting turns out to be no more or less of a problem for truth pluralism than it is for any other view that wishes to capture both semantic diversity and cognitive unity. When the philosophical tax-man marches through the door, pluralists can be proud of their books. Or, at least, they can be as proud as any other philosophical accountant who toils to record the credits and debits incurred in our mad little business—the business of reconciling the manifest and scientific images of ourselves.13
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Notes 1. The analogy is Nietzsche (1968, p. 540). 2. Recent examples include (Bar-On and Chrisman 2009; Blackburn 1998a; Ridge 2006, 2009; Schroeder 2008, 2010). 3. See (Lynch 2013a). 4. See for example, (Wright 1998b) and (Lynch 2009, 2013b); in contrast, see (Kölbel 2008) (Cotnoir 2009). 5. The functionalist way of putting the point was first expressed by Pettit (1996) and Lynch (1998); Wright (2013) now interprets his view in a similar way. 6. Expressions of the pluralist position (see Wright 1992; Lynch 2001) have sometimes obscured this point by presenting the position as relativizing or indexing truth realizing properties to domains. As David (2013) has correctly noted, these flourishes are unessential to the truth pluralist’s point. See also Lynch 2013b. 7. A recent sampling includes: Cotnoir 2009, 2013a, b; Edwards 2008, 2018; Horton and Poston 2012; Jarvis 2012; Pedersen 2006, 2012a, b; C. Wright 2013; and C. D. Wright 2005, 2010, 2012. 8. One might also see Jamin Asay as making a similar point (Asay 2018). 9. Quine, interestingly, does not say who these philosophers were. Clearly, though, truth pluralism was, as it were, in the air at this time. 10. Here I refer only to traditional correspondence theorists (see, e.g., Russell 1966 and Fumerton 2002). Such a view should be distinguished from a position—which is intended to be a form of truth pluralism—that says that correspondence itself comes in different kinds (see, e.g., Sher 2004, 2005). 11. Of course Quine himself didn’t think there are moral or modal properties, and Blackburn would agree. These are not the sorts of things we expect to quantify over in our final, most rigorous theory of the world. 12. Wright makes a similar point in his 1998 response to Blackburn (Wright 1998a). 13. Many people have contributed to my thoughts on these matters over the years, including notably, Nikolaj J. L. L. Pedersen, Jeremy Wyatt, and Nathan Kellen—the editors of this volume—and Crispin Wright, Simon Blackburn, Huw Price, Mark Timmons, Mike Ridge, Mark Chrisman, Dorit-Bar On, and Cory Wright. While working on this paper, I benefitted from participation in the Pluralisms Global Research Network (National Research Foundation of Korea grant no. 2013S1A2A2035514). This support is also gratefully acknowledged.
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References Asay, J. 2018. Putting Pluralism in Its Place. Philosophy and Phenomenological Research 96 (1): 175–191. https://doi.org/10.1111/phpr.12303. Bar-On, D., and M. Chrisman. 2009. Ethical Neo-expressivism. In Oxford Studies in Metaethics, ed. R. Shafer-Landau, vol. 4, 132–165. Oxford: Oxford University Press. Beall, Jc. 2013. Deflated Truth Pluralism. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 323–338. New York: Oxford University Press. Blackburn, S. 1984. Spreading the Word. Oxford: Oxford University Press. ———. 1998a. Ruling Passions. Oxford: Oxford University Press. ———. 1998b. Wittgenstein, Wright, Rorty and Minimalism. Mind 107 (425): 157–181. ———. 2013. Deflationism, Pluralism, Expressivism, Pragmatism. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 263–277. New York: Oxford University Press. Brandom, R. 1998. Making It Explicit: Reasoning, Representing, and Discursive Commitment. Cambridge, MA: Harvard University Press. Chrisman, M. 2008. Expressivism, Inferentialism, and Saving the Debate. Philosophy and Phenomenological Research 77 (2): 334–358. Cotnoir, A. 2009. Generic Truth and Mixed Conjunctions: Some Alternatives. Analysis 69 (3): 473–479. Cotnoir, A.J. 2013a. Validity for Strong Pluralists. Philosophy and Phenomenological Research 86 (3): 563–579. ———. 2013b. Pluralism & Paradox. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 339–350. New York: Oxford University Press. David, M. 2013. Lynch’s Functionalist Theory of Truth. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 42–68. New York: Oxford University Press. Dodd, J. 2013. Deflationism Trumps Pluralism. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 298–322. New York: Oxford University Press. Dreier, J. 2004. Meta-Ethics and The Problem of Creeping Minimalism. Philosophical Perspectives 18 (1): 23–44. Dworkin, R. 1996. Objectivity and Truth: You’d Better Believe It. Philosophy & Public Affairs 25 (2): 87–139.
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Edwards, D. 2008. How to Solve the Problem of Mixed Conjunctions. Analysis 68 (2): 143–149. Edwards, D. 2018. The Metaphysics of Truth. Oxford: Oxford University Press. Fumerton, R.A. 2002. Realism and the Correspondence Theory of Truth. Lanham: Rowman & Littlefield Publishers. Horton, M., and T. Poston. 2012. Functionalism about Truth and the Metaphysics of Reduction. Acta Analytica 27 (1): 13–27. Horwich, P. 1998. Truth. New York: Oxford University Press. Jarvis, B.W. 2012. The Dual Aspects Theory of Truth. Canadian Journal of Philosophy 42 (3–4): 209–233. Kölbel, M. 2008. “True” as Ambiguous. Philosophy and Phenomenological Research 77 (2): 359–384. Lynch, M.P. 1998. Truth in Context. Cambridge, MA: MIT Press. ———. 2009. Truth as One and Many. Oxford: Oxford University Press. ———. 2001. Functionalist Theory of Truth. In The Nature of Truth, ed. M.P. Lynch, 723–750. Cambridge: MIT Press. ———. 2013a. Expressivism and Plural Truth. Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition 163 (2): 385–401. ———. 2013b. Three Questions for Truth Pluralism. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 21–41. New York: Oxford University Press. Nietzsche, F. W. 1968. The Will to Power. Edited by W. Kaufmann. Translated by R. J. Hollingdale. Vintage books ed. New York: Vintage Books. Pedersen, Nikolaj J.L.L. 2006. What Can the Problem of Mixed Inferences Teach Us About Alethic Pluralism? Monist 89 (1): 102–117. ———. 2012a. Recent Work on Alethic Pluralism. Analysis 72: 588–607. ———. 2012b. True Alethic Functionalism? International Journal of Philosophical Studies 20: 125–133. (Symposium on M. P. Lynch’s Truth as One and Many). Pettit, P. 1996. Realism and Truth: A Comment on Crispin Wright’s Truth and Objectivity. Philosophy and Phenomenological Research 56: 883–890. Price, H. 2011. Naturalism Without Mirrors. New York: Oxford University Press. ———. 2013. Expressivism, Pragmatism and Representationalism. New York: Cambridge University Press. Quine, W. 1960. Word and Object. Cambridge: MIT Press. Ridge, M. 2006. Ecumenical Expressivism: Finessing Frege. Ethics 116 (2): 302–336.
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———. 2009. The Truth in Ecumenical Expressivism. In Reasons for Action, ed. D. Sobel and S. Wall, 219–242. New York: Cambridge University Press. Russell, B. 1966. On the Nature of Truth and Falsehood. In Philosophical Essays, ed. B. Russell, 147–159. London: George Allen & Unwin. Sainsbury, R.M. 1996. Review: Crispin Wright: Truth and Objectivity. Philosophy and Phenomenological Research 56 (4): 899–904. Schroeder, M. 2008. Being For. Oxford: Oxford University Press. ———. 2010. Noncognitivism in Ethics. London: Routledge. Sher, G. 2004. In Search of a Substantive Theory of Truth. Journal of Philosophy 101 (1): 5–36. ———. 2005. Functional Pluralism. Philosophical Books 46 (4): 311–330. Tappolet, C. 1997. Mixed Inferences: A Problem for Pluralism About Truth Predicates. Analysis 57 (3): 209–210. https://doi.org/10.1111/1467-8284. 00077. Wright, C. 1992. Truth and Objectivity. Cambridge, MA: Harvard University Press. ———. 1998a. Comrades Against Quietism: Reply to Simon Blackburn on Truth and Objectivity. Mind 107 (425): 183–203. ———. 1998b. Truth: A Traditional Debate Reviewed. Canadian Journal of Philosophy 28 (Supplement): 31–74. ———. 2013. Plurality of Pluralisms. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 123–156. New York: Oxford University Press. Wright, C.D. 2005. On the Functionalization of Pluralist Approaches to Truth. Synthese 145: 1–28. ———. 2009. Truth, Ramsification, and the Pluralist’s Revenge. Australasian Journal of Philosophy 88 (2): 265–283. ———. 2012. Is Pluralism about Truth Inherently Unstable? Philosophical Studies 159 (1): 89–105.
The Metaphysics of Domains Douglas Edwards
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Introduction
Pluralist theories of various sorts commit themselves to the notion of a domain. Most prominently, truth pluralists suggest that there are different stories to tell about how sentences1 get to be true depending on the domain to which the sentence belongs.2 Indeed, some commitment to the idea is ingrained in the very motivations for the view. The truth pluralist’s dissatisfaction with truth monism is that one theory of truth does not seem to fit all the different kinds of thought and talk that we ordinarily take to be truth-apt, so we need to open ourselves up to the idea that, rather than having one global theory of truth, we have a number of different theories of truth which apply locally, for different kinds of thought and talk. There is also a practical reason why truth pluralists commit themselves to domains. Suppose for a moment that we hold that there are many distinct truth properties, and we have no division of sentences into different domains. On this picture, all sentences are capable of possessing all D. Edwards (*) Department of Philosophy, Utica College, Utica, NY, USA © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_4
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the different truth properties. Now, the different truth properties are distinct properties, so it is possible for a sentence to possess one of the truth properties but lack another. If we take falsity to be lack of truth, then we have a sentence which both has a truth property and lacks a truth property, which gives us a sentence that is both true and false. This complicates matters and makes life difficult for pluralist proposals that do not try to sort sentences into separate domains. Given that, as noted above, truth pluralists have independent reason to hold that sentences are sorted into domains, it seems natural to adopt them in response to this sort of problem. However, some pluralists are sceptical of the notion of a domain, and Lynch (2013a) ends up rejecting domains, choosing instead to start with specific sentences (or propositions, for Lynch), as opposed to sentences of a particular kind. This, to my mind, is a mistake. As just noted, a key component of pluralist views is that different kinds of subject matters are apt for different sorts of treatment. If this idea is given up, then a central motivation for the view is lost. As a result, I think that pluralists should take the notion of a domain seriously as a central aspect of the view. In this chapter, I give an account of what domains are and respond to some problems of domain individuation. I begin by discussing the semantic and metaphysical aspects of a domain, and then show how the two aspects combine to yield an account of domains in general. Following this, I will discuss two problems of domain individuation: the problem of mixed atomics, and the problem of mixed compounds.
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The Semantic Aspect of a Domain
I characterize domains as having two main components: a semantic component and a metaphysical component.3 The semantic component concerns the components of atomic sentences—singular terms and predicates—and the metaphysical component concerns the things referred to by singular terms and predicates, namely objects and properties. The discussion of the semantic aspect of a domain will thus concern the ways in which singular terms and predicates come in different kinds, and how this contributes to the notion of a domain.
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We will begin with predicates—linguistic items, typically of the form ‘is F’, which, when combined with a singular term, form an atomic sentence. My aim in this section is to substantiate the claim that predicates come in different kinds, and that predicate kinds are distinguished by the kinds of functional roles that predicates have. This will give us the grounds for understanding the semantic aspect of a domain. The general idea can be illustrated by thinking about how to distinguish moral predicates from physical predicates. Generally speaking, predicates typically refer to properties. One idea is that there is something distinctive about the properties that moral predicates pick out that renders them distinct from the properties that physical predicates pick out. One feature might be that moral predicates pick out normative properties, where one’s recognition that an action or object has a normative property entails that one has, in virtue of that recognition, a reason to act or to refrain from acting. This would be distinct from a physical predicate, which ascribes a property that has causal, but not normative, powers. Moral predicates will ascribe properties whose recognition gives one a distinctly moral reason to act or not to act. We can elaborate on what is meant by ‘moral’ here by thinking about some general ‘platitudes’ concerning the word ‘moral’. For example, Smith (1994) offers a set of platitudes which he takes to serve as analyses of moral concepts: To say that we can analyse moral concepts, like the concept of being right, is to say that we can specify which property the property of being right is by reference to platitudes about rightness: that is, by reference to descriptions of the inferential and judgemental dispositions of those who have mastery of the term ‘rightness’. (Smith 1994: 39)
Smith goes on to describe platitudes that concern the ‘substance’ of morality: For example, there are platitudes concerning the substance of morality: ‘Right acts are often concerned to promote or sustain or contribute in some way to human flourishing’ (Foot. 1958); ‘Right acts are in some way expressive of equal concern and respect’ (Dworkin 1977: 179–83; Kymlicka
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1989: 13, 21–9; 1990: 4–5), and the like. What these platitudes about substance force us to admit, at the very least, is that there are limits on the kind of content a set of requirements can have if they are to be moral requirements at all, as opposed to requirements of some other kind (Dreier 1990). (Smith 1994: 40)
Smith offers a detailed account of the kinds of features moral properties have, which demarcates them from other kinds of properties. This process is explicitly carried out by reference to the relevant concepts of these properties, but, for our purposes, we can interpret Smith’s thoughts in relation to moral predicates. The suggestion is that we can individuate kinds of predicates in accordance with the general functional roles that those predicates are taken to have. Whilst we have just talked about moral and physical predicates so far, the idea is intended to be general, in that we could construct a list of general features for all kinds of predicates. These are intended to mark fairly intuitive distinctions between kinds of subject matter that are already implicit, in that we generally have a good sense of when someone is making a moral claim, or an aesthetic claim, or a physical claim, or a mathematical claim, to take just a few examples. Indeed, philosophers should have a good sense of how to individuate these things, otherwise it might be difficult to distinguish aesthetics from philosophy of mathematics! The thought here is that this is, at least in part, done by the functions of the words involved, particularly the predicates, in relation to the types of features they are describing. However, the fact that we can often individuate different kinds fairly easily does not mean that there won’t occasionally be difficulties demarcating them. For example, compare religious predicates with moral predicates. Both are concerned with describing how to live well, and how to behave towards others and ourselves. In this way, they have similar functional roles, which may lead one to think that they are essentially the same. It is not much of a stretch to imagine that someone coming to their ethics class with the impression that moral predicates just are religious predicates. Perhaps, for instance, they think that the predicates ‘is right’ and ‘is wrong’ are equivalent in meaning to the predicates ‘is pious’ and ‘is sinful’, as they have only considered moral issues in religious contexts.
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However, with some philosophical work we can separate the religious predicates from the moral predicates. The predicates ‘is pious’ and ‘is sinful’, for example, concern a person’s relationship with God, and to hold that ‘is right’ and ‘is wrong’ are equivalent to those terms is just to hold that actions are right or wrong insofar as they affect an individual’s relationship with God in a particular way. However, we can show that ‘is right’ and ‘is wrong’ have a broader role to play in that we can consider the rightness or wrongness of actions under different background views, which show that the meanings of the predicates ‘is right’ and ‘is wrong’ are not tied to a person’s relationship with God, unlike ‘is pious’ and ‘is sinful’.4 The idea is, then, that the first part of the semantic aspect of a domain is understood in terms of kinds of predicates, with kinds of predicates understood in terms of functional roles. The functional roles are understood in terms of the features of the properties that the predicates are purported to pick out. We will say more about the semantic aspect again in a moment when we look at singular terms, but let’s say a bit first about the metaphysical aspect.
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The Metaphysical Aspect of a Domain
The account given above of the individuation of predicate kinds is neutral on the substantial metaphysical matter of whether there are any properties that correspond to these predicates, and, if so, what those properties are like. For every predicate kind F that is individuated, there are three available options: (i) There are objective F properties. (ii) There are projected F properties. (iii) There are no F properties. Option (i) holds that there are properties ‘out there’, so to speak, and this can explain why we have predicates of a certain kind: they are responsive, in some way, to existent properties. Option (ii) holds that there are
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properties that the relevant predicates pick out, but these properties exist because they are projections of the predicates, as opposed to being already existing things that our predicates respond to. Option (iii) is the error- theoretical option, where, perhaps due to the specification of the predicate kind, or alternatively because the world is uncooperative, there are no properties that correspond to the predicates of a certain kind. An error theorist about morality, for example, holds that there are no properties that correspond to our moral predicates.5 Our focus here will be on (i) and (ii). We can illustrate the differences between them using the following principle about properties: (P) The object referred to by ‘a’ falls under the predicate ‘is F’ iff the object referred to by ‘a’ has the property referred to by ‘is F’. Option (i) holds that there is a right-to-left order of determination on (P): it is because A has the property of being F that A falls under the predicate ‘is F’. For example, it is because this knife has the property of being metallic that this knife falls under the predicate ‘is metallic’. We have a property-to-predicate direction of explanation. Option (ii), on the other hand, takes the reverse view: there is a left-to- right order of determination on (P) as it is because A falls under the predicate ‘is F’ that A has the property of being F. One example here is the property of being cool6: Zack Morris has the property of being cool because Zack Morris falls under the predicate ‘is cool’, rather than vice versa. We have a predicate-to-property direction of explanation. In order to make sense of this idea, we need to make distinctions between kinds of properties. It is commonplace to distinguish between abundant and sparse properties in theories of properties.7 First, let us take the view of properties standardly called ‘predicate nominalism’.8 On this view, there are properties insofar as there are extensions of predicates: to have the property of being yellow is to be in the extension of ‘is yellow’; to have the property of being mammalian is to be to be in the extension of ‘is mammalian’, and that’s all there is to it. On this view, properties are taken to be abundant: there are as many properties as there are extensions of predicates.9 This allows us to be very permissive when it comes to
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assigning properties to predicates, for any predicate with consistent rules for its application will have an extension: a class of (actual and/or possible) objects that satisfy it. Thus, the predicate ‘is yellow or mammalian’ will have an extension—the class of objects that are either yellow or mammalian—and thus will express a property. On this view of properties, it is not the case that an object is in the extension of a predicate ‘is F’ because it has the property of being F; rather the object has the property of being F because it falls under the predicate ‘is F’.10 An opposing view is the idea that that properties form a distinct ontological kind, such as universals.11 On this view, things that share a property instantiate the same universal, and in order for two or more objects to instantiate a universal, it must be shown that they share some significant similarities, and that the universal plays some important causal- explanatory role.12 If we take the only properties to exist to be those that are universals, then this limits the number of properties to the number of universals. As universals require some significant similarities at the metaphysical level, the only properties that exist will have to meet this requirement, which renders properties sparse. There is also room for a mixed approach, originally due to David Lewis (1983), that allows for both abundant and sparse properties. Lewis holds that all properties are classes, but, in some cases—abundant properties— there won’t be anything more to say about why particular objects are members of a class, but in others—sparse properties—there will.13 Or, to put things in terms of predicates as we have been doing, in some cases there will not be an account of why a particular class of objects falls under a particular predicate, whereas, in other cases, there will. For example, consider the predicate ‘is yellow or mammalian.’ Some things are in the extension of this predicate, and are thus members of the class of things that are yellow or mammalian. However, this doesn’t imply that all of the members of this class share anything significant in common over and above their membership in the class. Contrast this now with, for example, falling under the predicate ‘is metallic’, and being a member of the class of metallic objects: membership of this class is not arbitrary, and each member of the class must meet some significant constraints in order to be a member of the class. Indeed, it is plausible to
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think that all metallic objects share some significant feature in common. Despite the fact that both properties are classes, there are significant metaphysical differences between them, leading to the former’s status as an ‘abundant’ property, and the latter’s status as a ‘sparse’ property. In particular, sparse properties ground genuine similarities between their bearers, and have a causal-explanatory role, whereas abundant properties do not ground genuine similarities between their bearers, and do not have a causal-explanatory role. The two ways of looking at properties broadly correspond to the two ways we considered the predicate-property relation above, with sparse properties being those mentioned in option (i) and abundant properties being those mentioned in option (ii). To relate this to our discussion of predicates, I am going to use the terminology of ‘responsive’ predicates and ‘generative’ predicates. Broadly speaking, responsive predicates respond to sparse properties, and generative predicates generate abundant properties.14
4
Relating Objects and Singular Terms
We have spoken so far about predicates and properties, but, to complete the picture, we need to also discuss singular terms and objects. The general function of singular terms is to refer to objects,15 and we can divide singular terms into kinds by the sorts of objects they are used to refer to. For example, we can distinguish between ‘the largest prime number’, ‘the New York-Massachusetts border’, and ‘the guilty party’ as different types of singular terms in a sense, namely mathematical, institutional, and moral. This is partly because of how they are used: ‘the largest prime number’ is used to refer to a mathematical object; ‘the New York- Massachusetts border’ is used to refer to an institutional object; and ‘the guilty party’ is used to refer to a moral object. Moreover, the case can be made that there is a distinction between responsive and generative singular terms, which lines up with the corresponding distinction between predicates. For example, if we take causal accounts of reference seriously, at least for physical objects, then the use of a singular term is responsive to the existence of a physical object which
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causally constrains its use. Such objects would be considered sparse objects. Now, not all objects will meet this condition: abstract objects, for example, do not have causal powers, and do not look suited for a causal theory of reference. However, we can give a different understanding of abstract objects by using the ‘Neo-Fregean’ approach of Bob Hale and Crispin Wright. Their idea is that to be an object is to be the referent of a singular term appearing in a true sentence: The lynch-pin of Frege’s platonism, according to our interpretation, is the syntactic priority thesis: the category of objects … is to be explained as comprising everything which might be referred to by a singular term, where it is understood that possession of reference is imposed on a singular term by its occurrence in true statements of an appropriate type. (Wright 1983: 53) [O]bjects, as distinct from entities of other types (properties, relations, or, more generally, functions of different types and levels), just are what (actual and possible) singular terms refer to. (Hale and Wright 2005: 171)
The basic idea for our purposes here is that we can secure reference to a mathematical object—the number 5, say—by noticing that its associated singular term appears in at least one true sentence. Thus, the truth of the sentence ‘the number 5 is prime’ is sufficient to secure the existence of a referent for the singular term ‘the number 5’: the number 5. We can use this idea to construct the notion of an abundant object, which pairs with the notion of an abundant property. If we hold that an object a is abundant, then it cannot be that the singular term ‘a’ refers to a because a exists, as we are explicitly denying that a has existence prior to ‘a’ referring to a. As a consequence, we need some grounds to establish in virtue of what it is that ‘a’ refers to a. Hale and Wright offer an account here in terms of truth: ‘a’ refers to a if ‘a’ appears in a true sentence. Singular terms can thus be responsive or generative: responsive singular terms respond to sparse objects, and generative singular terms generate abundant objects.
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Summarizing Domains
I have suggested that a domain has both a semantic and a metaphysical component, which are closely related. The semantic component of a domain is understood in terms of singular terms and predicates. The domain to which a singular term or predicate belongs is determined by the kind of singular term or predicate that the relevant singular term or predicate is identified to be. This is done by identifying the functional role of a singular term or predicate kind to demarcate different kinds of singular terms or predicates. The semantic aspect of a domain will thus be understood as a singular term and predicate kind. For example, the semantic aspect of the moral domain will be the singular terms and predicates identified as being moral singular terms and predicates in virtue of their playing the functional role associated with moral singular terms and predicates. The metaphysical aspect of a domain will be composed of the objects and properties that the singular terms and predicates in the semantic aspect of the domain refer to. Thus, the metaphysical aspect of the moral domain, for example, will be the objects and properties referred to by moral singular terms and predicates. This is not to say that all there is to objects and properties of different kinds is that they are referred to by different kinds of singular terms and predicates. This is because, as we have seen, the relationships between singular terms and objects, and predicates and properties, varies: in some cases, the nature of the objects and properties is dependent on the singular terms and predicates (the abundant model), whereas in other cases, the singular terms and predicates are dependent on the objects and properties (the sparse model). This account of domains is intended to be available to theorists of various sorts, not just truth pluralists. In the remainder of the chapter, I will focus on a couple of problems of domain individuation that have been posed for truth pluralism, though I think the problems remain general problems for those who wish to individuate different kinds of subject matter.
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The Problem of Mixed Atomics
The problem of mixed atomics (Sher 2005; David 2013; Wyatt 2013) puts pressure on the idea that we can assign sentences to domains. The first class of examples of mixed atomics takes atomic sentences that are composed of singular terms and predicates of different kinds. Two examples are ‘this crystal is beautiful’ (David 2013) and ‘Charlie is delicious’ (Wyatt 2013), where ‘Charlie’ is the name of a beet. Let us take Wyatt’s example, ‘Charlie is delicious’. What is this sentence about? It is about Charlie, a beet, a material object. The problematic element is what is being said about Charlie, namely that he is delicious, which is not a material property (we suppose). Here, we have singular term and predicate belonging to different kinds, so what domain does the sentence belong to? Wyatt (2013) claims that these examples should push the truth pluralist to admit that sentences like this belong to more than one domain, even if they are apt for only one kind of truth. I will argue though that the examples do not push the truth pluralist in that direction, and that truth pluralists can still say that atomic sentences only belong to one domain. By wearing its commitment to domains on its sleeve, truth pluralism undoubtedly leaves itself open to these kinds of concerns about assigning sentences to domains. However, many other theories in the area will face the same sorts of problems. Truth pluralists take the demands for truth- aptness to be very minimal, and focus their attention on what kind of truth a sentence is apt for. For other views, such as classical correspondence theories of truth, there is only one kind of truth, but the key question becomes whether sentences in some domains are capable of being true at all. Crucial to this sort of project is the ability to separate the truth-apt sentences from the non-truth-apt ones, and this involves separating sentences into different domains, as the examples of non-truth-apt sentences on such views often include sentences employing moral or aesthetic terms.16 With this in mind, suppose we take it that physical sentences are truth- apt, and aesthetic sentences are not. Now, take Wyatt’s example of ‘Charlie is delicious’: is this a physical or an aesthetic sentence, or both? It
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s eemingly cannot be both, as a sentence cannot both be and not be truthapt, so it seems as though Wyatt’s preferred option of a sentence belonging to two domains is not even in play here. The answer though seems fairly obvious: the aesthetic predicate ‘is delicious’ immediately renders the sentence non-truth-apt, regardless of the object involved. This suggests that the key factor in determining the domain to which a sentence belongs is the predicate, and this is a conclusion that can be reached independently of any commitment to truth pluralism. Pedersen and Wright suggest a solution in this spirit for truth pluralism itself: Consider the following sentences: (41) The Mona Lisa is beautiful. (42) Speeding is illegal. Prima facie, what determines the domain-membership of (41) and (42) is the aesthetic and legal predicates ‘is beautiful’ and ‘is illegal’, respectively. It is an aesthetic matter whether the Mona Lisa is beautiful; this is because (41) is true in some way just in case the Mona Lisa falls in the extension of the aesthetic predicate ‘is beautiful’ (and mutatis mutandis for (42)). (Pedersen and Wright 2013)
With this in mind, remember that an atomic sentence is composed of two elements: a singular term and a predicate. We can distinguish between two things: what a sentence is about, and how the thing the sentence is about is represented to be. We can grant that a sentence is about the object referred to by the singular term, for example, ‘Charlie is delicious’ is about Charlie, but what makes a sentence a sentence in that it is a bearer of content is that there is a way the object is represented to be: Charlie is represented to be delicious, and this representation occurs due to the attribution of a property to the object. So, it is not what a sentence is about that we should be considering for domain membership, it is rather how the thing the sentence is about is represented, by the use of a predicate to attribute a property.17 We can thus say of our examples that they belong to a single domain: because
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deliciousness is an aesthetic predicate, ‘Charlie is delicious’ is just an aesthetic sentence, and because wrongness is a moral predicate, ‘torture is wrong’ is just a moral sentence. They belong to these domains because of the functions of the predicates involved. Consequently, simple mixes of singular terms and predicates in a sentence should not motivate any move from the view that each atomic sentence belongs to a single domain. Atomic sentences are thus assigned to domains by the predicate they contain. The singular term is not relevant to domain individuation. This avoids the problem of mixed atomics, but there are independent reasons to favour this account. Many different kinds of thing can be said about the same object: a single chair can be blue, solid, beautiful, sad, dangerous, presidential, or singular, to name just a few examples. According to the truth pluralist all of these things said about it will be of different domains. Given that the nature of the singular term is held fixed, and the only thing that varies is the predicate, this is independent reason to think that domain-membership is determined by the predicate. It is also worth pointing out that singular terms will nevertheless be parts of domains. Each singular term will have its ‘home’ domain, depending on what kind of object to which it refers. For instance, physical singular terms will have a certain character, just as physical predicates will have a certain character. The central difference between singular terms and predicates is that singular terms can be part of sentences of domains other than their own, whereas predicates cannot. Whilst each singular term has its ‘home’ domain, it does not just appear in sentences of that domain, as detailed above. However, one might worry that this solution to the problem of mixed atomics is too simplistic. If we avoid the problem by holding that it is the predicate that is relevant for determining domain-membership, then what about predicates that appear to be members of more than one kind? I will now attempt to allay that concern by considering an example of a supposedly mixed predicate. The mixed atomics with supposedly mixed predicates that I’ll examine are ‘thick’ predicates, such as ‘courageous’, ‘kind’, or ‘lewd’. The sentence ‘Serena Williams is courageous’ might be considered mixed because ‘courageous’ expresses a property (or properties) with both moral and non- moral aspects. We can note again that truth pluralism is not the only view
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to face this sort of problem. There is an abundance of work on whether expressivism can deal with thick predicates, with the worry being—in that case—that the expressive and descriptive elements are inseparable, thus causing problems for the view that moral language is purely expressive.18 One strategy in response is to take sentences involving thick predicates to be compound sentences, as opposed to atomic sentences. Williams (1985) puts this approach nicely as follows: A statement using [thick] terms can be analyzed into something like “this act has such-and-such a character, and acts of that character one ought not to do.” It is essential to this account that the specific or “thick” character of these terms is given in the descriptive element. The value part is expressed, under analysis, by the all-purpose prescriptive term ought. (Williams 1985: 144)
A truth pluralist could adopt this solution by holding that we are no longer dealing with a problem of mixed atomics, but rather a problem of mixed conjunctions. In this case there will be no problem accounting for the domains of the atomics, as—in line with what was said above—the truth pluralist can use the properties attributed to identify the domains the atomic sentences belong to. The only problem might be how to account for the truth of the conjunction, but then the truth pluralist is in familiar territory and can apply her favourite solution to the problem of mixed conjunctions, which we will discuss below. However, this option has some downsides. For one thing, whilst it promises to specify the way a sentence involving a thick predicate is true, it does not answer the question of what domain it belongs to: is it still a mixture of a moral and a physical sentence, for example? Accordingly, it is worth exploring an alternative account, which is inspired by Michael Smith’s (2013) account of thick predicates. Smith suggests that the difference between thick and thin predicates is not as a difference in kind, but in degree. Thick predicates seem to be thick because they impose a certain amount of descriptive content on the entity that bears the property. For example, saying ‘Serena Williams is courageous’ attributes not only some positive moral feature to Serena Williams, but also some descriptive features, namely that she displays
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some sort of resistance in the face of danger, for example. This may suggest to some that there is some difference in kind between thick predicates, like courageousness, and thin predicates, such as goodness, as one imposes descriptive content whereas the other does not. However, this is a mistake. Even thin properties like being morally good impose some descriptive content on their bearers. For example, if an action is deemed to be morally good, then this at the very least implies that the action was carried out by an intentional agent, as opposed to simply being the result of happenstance.19 The movement of the branch of a tree in the breeze, for example, would not be something that could be morally good, on the grounds that it does not meet the requirements for being a bearer of the property of being morally good. This suggests that thin predicates like moral goodness do impose some descriptive content on their bearers, even if it is more minimal than the descriptive content imposed by thicker predicates. If we take this line, then we can say that all moral predicates will have a degree of thickness, with the thinner predicates being predicates like moral goodness, and the thicker predicates being predicates like courageousness. There will also be predicates thicker than courageousness which impose very specific descriptive features on their bearers, such as the property of being a good father.20 However, we would say that, despite the variations in degrees of thickness, these are all moral predicates. The fact that some moral predicates imply some descriptive content does not mean that they are not moral predicates, as predicates of pretty much any property imply some constraints on what can bear it. The status of these predicates as moral predicates will be determined by the functional role that these predicates play in the cultivation and evaluation of character, and the way we decide what we ought to do. Moreover, this explanation generalizes to thick predicates of any kind, and is not just intended for moral predicates. The key idea is that we look to the function of a predicate to determine the domain to which the sentence in which it occurs belongs. For example, we might take the predicates ‘is sublime’ and ‘is beautiful’ to be thick predicates, as, even though they seem to be aesthetic predicates, they have different descriptive content. However, the fact that they have different descriptive content does not affect their status as aesthetic predicates due to the function they have as ways to evaluate of pieces of art, for example. Once again, the fact that
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some predicates also have some descriptive content does not affect their key function. I have suggested that the examples of mixed atomics divide into two main classes: (i) those where object and property are from different domains; (ii) those where the property seems mixed. I argued that neither of these classes poses a problem for the account of domains given above.
7
The Problem of Mixed Compounds
A separate problem is presented by the problem of mixed compounds (Tappolet 2000). The thought here is that we can easily use logical connectives to form compound sentences using atomic sentences from different domains, such as ‘this cat is wet and this cat is funny’, to take Tappolet’s (2000) example. The problem was originally posed as a challenge to explain how such compounds are true on a pluralist account, but we can also view is as a challenge to the notion of a domain: to which domain do such mixed compounds belong? Some have seen this problem as requiring additions to the standard domains that truth pluralists offer, including perhaps different domains for each kind of compound (Kim and Pedersen 2018; Pedersen and Lynch 2018; Gamester Forthcoming). We do not need to go in this direction, though. In previous work (Edwards 2008), I gave a response to this problem as a problem for understanding how such sentences are true by saying that compound sentences (whether mixed or not) are true in virtue of their components being true in the way(s) specified by the logical connective in question. The truth of a compound is derivative, in the sense that it is dependent on the truth of its components. This suggests a difference between the ways that compounds are true (in virtue of their components) and the way that atomic sentences are true (in virtue of having their relevant truth property). We can extend this idea to show that there is a difference between considering the question of which domain an atomic sentence belongs to and considering the question of which domain a compound belongs to. The key idea is that domain membership is only relevant when we consider atomic sentences. This is because domains are composed of
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objects and properties, and atomic sentences are composed of singular terms and predicates that refer to those objects and properties. Accordingly there will be a direct connection between the content of a domain and an atomic sentence. Compound sentences, on the other hand, form an additional layer above this framework. They are composed of atomic sentences, and the characteristic of a particular compound that makes it a particular compound is not the kinds of terms that are contained in its atomic components, but the particular connective involved, determined by the conditions under which the compound is true. It is the truth conditions of a conjunction, for example, which mark it out from a disjunction. This is what makes it a conjunction, as opposed to another form of compound, and this has nothing to do with the kinds of atomic sentences that compose it. With this idea in mind, we do not face the same sorts of questions about domain-membership with compounds as we do with atomics. The essence of a compound sentence which makes it a compound sentence is not the kinds of atomic sentences that compose it, but the truth conditions it has. We do face a different sort of question though, namely if compound sentences do not belong to the conventional domains demarcated, where do they belong? In previous work (Edwards 2009, in response to Cotnoir 2009), I suggested that they belong to the logical domain, in that the conditions under which each compound is true are determined by one’s preferred logical system. On this account, each compound, whether mixed or not, is a member of the logical domain. I think that there is something to this general idea, though it needs some development. One thing to consider is whether we see the logical domain in this context as a regular domain amongst the other domains, or as something distinct. I mentioned above that one way to think about compound sentences is as being at a level of an additional layer above atomic sentences. This is because compound sentences are essentially manipulations of atomic sentences into new structures: they take atomic sentences as ingredients, and with the additional of some specified rules, they turn them into new, compound sentences. As this process involves operations on atomic sentences, as opposed to a different kind of atomic sentences, it is more natural on this model to say that compound sentences do not reside in a domain of similar status to other domains. As a consequence, if, as
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suggested earlier, we understand the metaphysical aspect of a domain as being the entities from which atomic sentences are built, compound sentences do not have a domain in this sense. This makes them an interesting class of sentences for a pluralist, as they are sentences that are capable of being true, but they do not have a domain. This is where we can draw upon the idea suggested above that compound sentences are, in a sense, ‘logical’ in nature. The generation of compound sentences involves the taking of atomic sentences, and connecting them in some way in accordance with some specified rules to make new sentences. It is the ‘connecting them in some way in accordance with some specified rules’ that contains the logical component here, as the rules by which compounds are formed are specified by the basic axioms of one’s background logical theory. Now, it is an open question for truth pluralists whether all logical principles apply so universally as to be applicable to sentences from all domains, with principles like the principle of excluded middle being a particularly controversial case, along with the notion of logical consequence.21 However, less controversial is the construction of compounds, and their subsequent truth conditions.22 What we have here are cases of uncontroversial principles, which apply universally, such that they allow one to take atomic propositions of any kind and connect them in specified ways to make new sentences.23 We can extract from this the way in which compound sentences are both domain-less, yet also logical in some sense. They are domain-less because they are constructions out of atomic sentences, and not themselves atomic sentences. However, they are logical in character because they are the results of the implementation of logical principles on atomic sentences, and they are true insofar as they meet the standards laid down by these logical principles.24
8
Summary
As noted earlier, the notion of a domain has been both a key and controversial aspect of pluralist theories. Problems of domain individuation, such as the problem of mixed atomics and the problem of mixed compounds, along with the general difficulties of ‘dividing up’ language into different domains have presented a significant challenge to the role of the
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notion of a domain in pluralist theories. My suggestion here has been that there is an account of domains available to pluralists, and that these problems of domain individuation have solutions, so domains can continue to play a key role in pluralist theories.
Notes 1. I will be using sentences, as opposed to propositions, as the main examples of truth-bearers. See Edwards (2018: Chap. 1) for more on this choice. 2. For examples of domain-based ontological pluralism, see Cotnoir and Edwards (2015) and Edwards (2018). See Lynch (2009) for an example of domain-based logical pluralism. 3. Note that Wyatt (2013) recommends talking of ‘topics’ and ‘domains’ as separate things, with Wyatt’s ‘topics’ loosely corresponding to my ‘metaphysical aspects’, and Wyatt’s ‘domains’ loosely corresponding to my ‘semantic aspects’. I choose to use the word ‘domain’ for both, because I do not think that these aspects can be separated enough to warrant them being called different things, as opposed to parts of the same thing. I hope it will become clear why below. 4. See Edwards (2018) for further development of this idea in relation to social and institutional predicates. 5. Examples of error theory in morality are Mackie (1977) and Joyce (2001). 6. See Haslanger (2012: 89–98) for an extended discussion of coolness. 7. For more on this distinction in relation to truth, see Edwards (2013). 8. The name is due to Armstrong (1978). 9. Note that this is not the most abundant view of properties available. As Lewis (1983) notes, if we take the view that properties are classes (class nominalism), then properties will be more abundant than on predicate nominalism, as there will be classes to which there is no predicate attached. 10. See Edwards (2014: Chap. 5) for more on this view. 11. See, for example, Armstrong (1978). See also Edwards (2014: Chap. 2). 12. This perhaps requires that universals are taken to be the immanent universals favoured by Armstrong (1978), as opposed to abstract universals, and I will assume that here. 13. See Edwards (2014: Chap. 6) for more on this idea. 14. Note that this terminology still applies if we are thinking about properties as classes, and classes as mind-independent. This is because, even if
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there is a vast number of classes, we still need to make sense of a predicate selecting a particular class, and thus having the particular extension it does, which will be dependent on our practices. 15. See, for example, Hale (1994). 16. By using ‘non-truth-apt’ here I am working through a case where the correspondence theory is paired with some form of expressivism, as opposed to a form of error theory. Regardless of whether the correspondence theory is paired with expressivism, error theory, or indeed a form of fictionalism, a distinction will need to be made between the sentences that are able to be true and those that are not. 17. Lynch (2009, 2013a) just holds that what matters is what a sentence is about, with aboutness encompassing both object and property, which leads straight into the problem of mixed atomics. 18. Eklund (2011) provides an overview. Also see Lynch (2013b) for an argument suggesting that expressivists should embrace alethic pluralism. 19. Compare this thought to the idea that it is a fact about truth that—no matter what theory of truth you have—truth is not a property that can be borne by shirt buttons. 20. As discussed by Stewart-Wallace (2016). 21. See, for example, Lynch (2009: Chap. 5) for discussion of these issues in relation to truth pluralism. 22. For instance, in the discussion above, the rules for the formation of a conjunction are those specified by the axioms of classical logic, where a conjunction is specified to be a compound proposition that is true if and only if each of its conjuncts are true. However, this does not commit one to classical logic in a substantial way, as what it takes for an atomic proposition to be true here can be determined in a number of different ways, in accordance with the general approach of truth pluralism. 23. The account given here is for truth-functional compounds only. 24. Thanks to Nikolaj J. L. L. Pedersen, Jeremy Wyatt, and Nathan Kellen for very helpful feedback on this paper. I’d also like to thank audiences at the Pluralisms Week conference at Yonsei University and the University of Albany philosophy colloquium. Portions of this article draw on material originally published in Chap. 4 of my book The Metaphysics of Truth (Oxford University Press 2018), used with permission of Oxford University Press. While working on this paper, I benefitted from participation in the Pluralisms Global Research Network (National Research Foundation of Korea grant no. 2013S1A2A2035514). This support is also gratefully acknowledged.
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References Armstrong, D.M. 1978. Universals and Scientific Realism. Cambridge: Cambridge University Press. Cotnoir, A.J. 2009. Generic Truth and Mixed Conjunctions: Some Alternatives. Analysis 69 (2): 473–479. Cotnoir, A.J., and D. Edwards. 2015. From Truth Pluralism to Ontological Pluralism and Back. Journal of Philosophy 112 (3): 113–140. David, M. 2013. Lynch’s Functionalist Theory of Truth. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 42–68. New York: Oxford University Press. Edwards, D. 2008. How to Solve the Problem of Mixed Conjunctions. Analysis 68 (2): 143–149. ———. 2009. Truth-Conditions and the Nature of Truth: Re-Solving Mixed Conjunctions. Analysis 69 (4): 684–688. ———. 2013. Truth as a Substantive Property. Australasian Journal of Philosophy 91 (2): 279–294. ———. 2014. Properties. Cambridge: Polity Press. ———. 2018. The Metaphysics of Truth. Oxford: Oxford University Press. Eklund, M. 2011. What Are Thick Concepts? Canadian Journal of Philosophy 41 (1): 25–49. Gamester, W. Forthcoming. Logic, Logical Form, and the Disunity of Truth. Analysis. Hale, B. 1994. Singular Terms. In The Philosophy of Michael Dummett, ed. B. McGuinness and G. Oliveri, 17–44. Boston: Kluwer Academic Publishers. Hale, B., and C.J.G. Wright. 2005. Logicism in the 21st Century. In The Oxford Handbook of Philosophy of Mathematics and Logic, ed. Stewart Shapiro, 166–202. Oxford: Oxford University Press. Haslanger, S. 2012. Resisting Reality: Social Construction and Social Critique. New York: Oxford University Press. Joyce, R. 2001. The Myth of Morality. Cambridge: Cambridge University Press. Kim, S., and Nikolaj J.L.L. Pedersen. 2018. Strong Truth Pluralism. In Pluralisms in Truth and Logic, ed. Nikolaj J.L.L. Pedersen, Jeremy Wyatt, and Nathan Kellen. Cham: Palgrave Macmillan. Lewis, D. 1983. New Work for a Theory of Universals. Australasian Journal of Philosophy 61 (4): 343–377. Lynch, M.P. 2009. Truth as One and Many. Oxford: Oxford University Press.
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———. 2013a. Three Questions About Truth. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 21–41. New York: Oxford University Press. ———. 2013b. Expressivism and Plural Truth. Philosophical Studies 163 (2): 385–401. Mackie, J.L. 1977. Ethics: Inventing Right and Wrong. Harmondsworth: Penguin. Pedersen, Nikolaj J.L.L., and M.P. Lynch. 2018. Truth Pluralism. In The Oxford Handbook of Truth, ed. M. Glanzberg. Oxford: Oxford University Press. Pedersen, Nikolaj J.L.L., and C.D. Wright 2013. Pluralist Theories of Truth. In The Stanford Encyclopedia of Philosophy, ed. E.N. Zalta, http://plato.stanford. edu/archives/spr2013/entries/truth-pluralist/. Sher, G. 2005. Functional Pluralism. Philosophical Books 46: 311–330. Smith, M. 1994. The Moral Problem. Malden: Wiley-Blackwell. ———. 2013. On the Nature and Significance of the Distinction Between Thick and Thin Ethical Concepts. In Thick Concepts, ed. S. Kirchin, 97–120. Oxford: Oxford University Press. Stewart-Wallace, A. 2016. An Old Solution to the Problem of Mixed Atomics. Acta Analytica 31 (4): 363–372. Tappolet, C. 2000. Truth, Pluralism and Many-Valued Logics. The Philosophical Quarterly 50 (200): 382–383. Williams, B. 1985. Ethics and the Limits of Philosophy. London: Routledge. Wright, C.J.G. 1983. Frege’s Conception of Numbers as Objects. Vol. 2. Aberdeen: Aberdeen University Press. Wyatt, J. 2013. Domains, Plural Truth, and Mixed Atomic Propositions. Philosophical Studies 166 (1): 225–236.
Strong Truth Pluralism Seahwa Kim and Nikolaj J. L. L. Pedersen
Earlier versions—or parts of this chapter—have been presented by Nikolaj J. L. L. Pedersen at the University of St. Andrews (January 2009); University College Dublin (October 2010); University of Tokyo (October 2012); Sungkyunkwan University (December 2012); Northern Institute of Philosophy (July 2013); Truth & Pluralism (Pacific APA 2014); University of Barcelona (LOGOS, July 2014); Yonsei University (December 2014); University of Toronto (April 2015); University of Connecticut (April 2015); Nanyang Technical University (August 2015); and Lingnan University (February 2016). An earlier version of the chapter was presented by both authors at the 1st Pluralisms Global Research Network Workshop at Yonsei University (January 2014). Thanks to the following people for helpful discussion: Dorit Bar-On, Jc Beall, Mandel Cabrera, Ben Caplan, Colin Caret, Roy Cook, Aaron Cotnoir, Doug Edwards, Filippo Ferrari, Tim Fuller, Sungil Han, Joe Hwang, Lina Jansson, Jinho Kang, Junyeol Kim, Sungsu Kim, Max Kölbel, Michael Lynch, Adam Murray, Franklin Perkins, Graham Priest, Gurpreet Rattan, Sven Rosenkranz, Stewart Shapiro, Gila Sher, Keith Simmons, Cory Wright, Crispin Wright, Jeremy Wyatt, Byeong-Uk Yi, Andy Yu, and Elia Zardini. Research for this chapter was supported by grant no. 2013S1A2A2035514 (Pedersen and Kim) and grant no. 2016S1A2A2911800 (Pedersen) from the National Research Foundation of Korea. We gratefully acknowledge this support.
S. Kim Scranton College, Ewha Womans University, Seoul, South Korea N. J. L. L. Pedersen (*) Underwood International College, Yonsei University, Incheon, South Korea e-mail: [email protected] © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_5
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Truth Pluralism
A survey of traditional theories of truth reveals significant differences in the details of the various theories. Some account for truth in terms of correspondence to reality, others in terms of coherence. Yet others favor a pragmatist story put in terms of what is believed at the end of enquiry or what it is useful to believe. Interestingly, despite their differences traditional theories share a very fundamental assumption: monism about truth. Advocates of correspondence, coherence, and pragmatist theories have traditionally approached the truth debate as a one-player game. One theory—and one theory only—can tell us what the nature of truth is.1 Recently, the fundamental assumption of monism has been challenged. Truth pluralists hold that there are different ways of being true. Propositions about riverbanks might be true in virtue of corresponding to reality while propositions about the law might be true in virtue of cohering with the body of law. According to pluralists, monists are thus wrong in maintaining that exactly one of the traditional theories correctly identifies a single way of being true, applicable across all truth-apt domains of discourse. Instead two or more accounts are correct, assuming that their scope is restricted to specific domains. Truth pluralism has attracted a great deal of attention in the last few decades, with Crispin Wright and Michael P. Lynch being the view’s two most prominent advocates. Wright and Lynch do not stand alone, however. A steadily increasing number of truth theorists endorse, develop, or defend some form of pluralism.2 Different versions of pluralism have emerged. Strong pluralists give up on the idea of truth-as-such by denying that there is a single truth property applicable across all truth-apt domains of discourse. Truth is many, not one. Moderate pluralists, on the other hand, hold on to the idea of truth-as-such. There is a single generic truth property that applies across all truth-apt discourse. Truth is one. However, at the same time truth is many because propositions belonging to different domains may possess the generic truth property in virtue of having distinct properties such as correspondence or coherence. This chapter presents and develops a form of strong pluralism. We spell this idea out in a strongly reductionist fashion (to be given a slightly more refined formulation in section more refined formulation in Sect. 3):
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(TR) There is a plurality of properties T1, …, Tn such that (i) for every proposition Φ, Φ’s being true is identical to Φ’s being Ti (1 ≤ i ≤ n), and (ii) there is no property among T1, …, Tn that satisfies (i) for every Φ. (TR) is the token-token identity theory in an alethic incarnation. For example, the truth of the proposition that there are mountains simply is this proposition’s corresponding to reality, the truth of the proposition that speeding is illegal simply is this proposition’s cohering with the body of law, and neither correspondence nor coherence reduces truth for every proposition.3 Given our endorsement of a reductionist, identity-theoretic form of pluralism we refer to T1, …, Tn as “truth-reducing properties” or “truth properties”. This chapter has two aims. The first aim is to present our favored version of strong pluralism in greater detail. In pursuing this aim we go against the current that has so far set the course of the pluralism debate. Strong pluralism has been somewhat neglected in the literature. It has been widely regarded as a non-starter due to a battery of seemingly devastating objections leveled against it.4 Among these objections the problem of mixed compounds is regarded as being particularly pressing—and difficult—for the strong pluralist to deal with. The second aim of the chapter is to give a strongly pluralist response to the problem of mixed compounds. We proceed as follows: in Sect. 2 we present the problem of mixed compounds. In Sects. 3 and 4 we present two complementary metaphysical parts of our view—a specification of the range of truth-reducing properties we endorse and an account of truth grounding. We rely on this work in Sect. 5 where we respond to the problem of mixed compounds.
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The Problem of Mixed Compounds
Some discourse is pure in that it concerns only one domain. 〈Mt. Everest is extended in space and there are trees〉 is an empirical proposition, and 〈2 + 2 = 4 and 3 + 6 = 9〉 is a mathematical proposition. (Angle brackets are used to represent propositions.) However, not all discourse is pure.
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Some discourse concerns more than one domain. Consider, for example, the following compound proposition:
( mix - Ù ) Mt.Everest is extended in space and Bob’s drunk driving is illegal. This is a mixed conjunction. It pertains to more than one domain, the empirical domain and the legal domain. The problem of mixed compounds emerges when we consider the issue of what to say about the truth of mixed compounds.5 Clearly (mix∧) is true—its conjuncts, after all, are both true. However, it is less clear how exactly the strong pluralist is going to account for this. According to our brand of strong pluralism, each conjunct belongs to a specific domain and within that domain truth is (token-)identical to a certain property.6 Let us suppose that the truth property for the first conjunct is correspondence, and that it is coherence with the body of law for the second. However, what truth property does the conjunction itself have? Given our favored form of pluralism this amounts to the question of which property is identical to truth of the conjunction. Let us consider two arguably implausible options. One option is to say that the conjunction is correspondence-true. (We sometimes use the label “F-true” or “F-truth” to signify that F is the truth-reducing property for a given proposition.) This is implausible because it completely neglects the contribution made by the second conjunct—a conjunct that is coherence-true. Another option is to say that the conjunction is coherence-true. However, this is implausible for the same kind of reason: it would completely neglect the contribution made by one of the conjuncts. A third option is to say that the conjunction possesses some property distinct from both correspondence and coherence. We favor a response of this kind. However, we recognize that anyone who does so faces two tasks: 1. Something must be said about the nature of the third property. What property is it? Also, while it is distinct from the truth properties of both of its conjuncts, does its nature or instances somehow depend on them, and if so, how?
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2. Since the strong pluralist rejects the idea that there is a truth property that applies across all truth-apt domains, an assurance must be provided that the property possessed by the compound is not such a property. In Sect. 5 we offer a solution that addresses both of these tasks.
3
omains, Truth Properties, and Truth D Grounding
The core idea behind both moderate and strong pluralism is that propositions belonging to different domains may be true in different ways. Domains are central to both forms of pluralism, as the domain membership of a proposition is a key factor in determining the way in which it is true. On our view logical form is another. In this section we introduce an account of domains and then offer an account of how the domain membership and logical form of a proposition determine which property reduces truth for that proposition. Once this account is in place, a response to the problem of mixed compounds naturally offers itself.
Propositions and Domains Despite the central role that domains play within the standard pluralist framework not much systematic work has been done on their nature. We hope to do at least a little better by outlining a systematic treatment of domains.7 We take the range of propositions that pluralists are interested in (and typically deal with) to be the following: Prop: (a) atomic propositions, and (b) complex propositions whose only constituents are atomics and truth- functional logical operators (i.e. those represented by sentential connectives).
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Let us account for the domain membership of propositions in Prop. Domains are sets of propositions individuated by their subject matter. This idea seems to line up with how pluralists think about domains. 〈2 + 3 = 5〉, 〈Mt. Everest is extended in space〉, and 〈Bob’s drunk driving is illegal〉 belong to different domains. Why? Because they concern different subject matters or are about different kinds of states of affairs. Domains constitute a hierarchy of sorts. There is a range of base-level domains (D1, …, Dn). These domains are the domains that pluralists talk about when they introduce their view—that is, the mathematical domain, empirical domain, ethical domain, legal domain, and so on. Atomic propositions belong to exactly one base-level domain.8 Other domains are characterized in terms of base-level domains and set-theoretic union. The domain of a disjunction or a conjunction is the union of the domains of its constituents. The domain of a negated proposition is the same as the proposition negated. That is: (DA)
For atomic Φ: D(Φ) = D(~Φ) = Di (where Di is one of the base - level domains D1, …, Dn)
(D∨)
D(Φj ∨ … ∨ Φj + m) = D(~(Φj ∨ … ∨ Φj + m)) = D(Φj) ∪ … ∪ D(Φj + m)
(D∧)
D(Φj ∧ … ∧ Φj + m) = D(~(Φj ∧ … ∧ Φj + m)) = D(Φj) ∪ … ∪ D(Φj + m)
A mixed compound belongs to a domain that is distinct from the domain of its constituents. However, pure compounds and their negations belong to the same domain as their constituents. That is:
( D Ú Ù - pure )
If D (F j ) = ¼ = D (F j + m ) = Di , then
D (F j ∨…∨ F j+m ) = D (~ (F j ∨…∨ F j+m )) = D (F j ∧…∧ F j+m ) = D (~ (F j ∧…∧ F j+m )) = D (F j ) ∪…∪ D (F j+m ) = Di ∪…∪ Di
= Di
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As a result, some atomic propositions and compounds belong to the same domain. For example, 〈3 < 8〉 belongs to the mathematical domain as does 〈22 = 4 and √4 = 2〉, 〈22 = 4 or √4 = 2〉, and any other compound with only mathematical constituents. Domains proliferate, and they do so along two dimensions. First, letting n be the number of base-level domains, domains proliferate into ones involving k base-level domains for every k ≤ n. For example, if there are three base-level domains, domains proliferate into 1-domains, 2-domains, and 3-domains—where “k-domain” signifies a domain involving exactly k base-level domains. Second, k-domains proliferate. Any two k-domains that differ with respect to at least one base-level domain are distinct. For example, D1 ∪ D2 is distinct from D1 ∪ D3 as well as from D2 ∪ D3 (and D1 ∪ D3 from D2 ∪ D3). How many domains are there? Answer: 2n – 1 (where, again, n is the number of base-level domains). Thus, if there are three base-level domains, there is a total of 7 domains: three 1-domains (D1, D2, D3), three 2-domains (D1 ∪ D2, D1 ∪ D3, D2 ∪ D3), and one 3-domain (D1 ∪ D2 ∪ D3).
Propositions and Truth Properties We provided a general statement of our reductionist, identity-theoretic form of pluralism in the opening section—in the form of (TR). Earlier we specified Prop as the range of propositions that pluralists are concerned with. Let us restate our general thesis, factoring in the focus on Prop: (T) There is a plurality of truth or truth-reducing properties T1, …, Tn such that (i) for every proposition Φ, Φ’s being true is identical to Φ’s being Ti (1 ≤ i ≤ n), and (ii) there is no property among T1, …, Tn that satisfies (i) for every proposition Φ. What properties are in the plurality of truth properties T1, …, Tn? There is a number of base-level truth properties Tj, …, Tj + m. These include the “standard” pluralist properties such as correspondence, coherence,
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superassertibility, and so on. We refer to these properties as “base-level truth properties” because they reduce truth for atomics: (TA) For atomic Φ, Φ’s being true is identical to Φ’s being Tk, where Tk is the base-level truth property that reduces truth for atomic propositions in Φ’s domain. To illustrate consider the atomic propositions 〈Mt. Everest is extended in space〉 and 〈Bob’s drunk driving is illegal〉. They belong respectively to the empirical domain and the legal domain. We can suppose that their domain membership determines respectively correspondence and coherence with the body of law as the relevant base-level truth properties. In addition to base-level truth properties there are compound-specific truth properties. For disjunctions we endorse the following truth property: Φi ∨ … ∨ Φi + n’s being true is identical to Φi ∨ … ∨ Φi + n’s being (T∨) a disjunction with at least one disjunct that has its truth property (as determined by its domain and logical form). Let “T∨” label the property characterized by (T∨). (T∨) gives us the following individually necessary and jointly sufficient conditions for a proposition to be T∨: (T∨*) Proposition Γ is T∨ if and only if (i) Γ has the form Φi ∨ … ∨ Φi + n (i.e. a disjunction), and (ii) at least one of Φi, …, Φi + n has its truth property (as determined by its domain and logical form). For conjunctions we endorse the following truth property: Φi ∧ … ∧ Φi + n’s being true is identical to the property of being (T∧) a conjunction with conjuncts that all have their truth property (as determined by their respective domains and logical forms). Let “T∧” label the property characterized by (T∧).
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(T∧) gives us the following individually necessary and jointly sufficient conditions for a proposition to be T∧: (T∧*) Proposition Γ is T∧ if and only if (i) Γ has the form Φi ∧ … ∧ Φi + n (i.e. a conjunction), and (ii) Φi, …, Φi + n all have their truth property (as determined by their respective domains and logical forms). Why endorse the compound-specific properties T∨ and T∧? Our motivation for doing so is to take seriously the idea that disjunctions and conjunctions are truth-functional compounds. The truth of a disjunction is a function of its being a disjunction and the fact that at least one of its disjuncts has its truth property. The truth of a conjunction is a function of its being a conjunction and the fact that all of its conjuncts have their truth property. What matters in each case are logical form and the fact that some constituent or all constituents have the relevant truth property. It does not matter what specific property it is or what domain the constituents belong to.
Hyperintensionality Before we move on to talk about truth grounding it is worth highlighting an interesting feature of our view: a commitment to a hyperintensional treatment of propositions. Some propositions belong to the same domain but have different truth-reducing properties. This is the case for any atomic proposition belonging to domain Di and any compound whose constituents all belong to that same domain. For example, consider 〈Mt. Everest is extended in space〉, 〈Mt. Everest is extended in space or there are trees〉, and 〈Mt. Everest is extended in space and there are trees〉. These propositions all belong to the empirical domain. However, while correspondence to reality is the truth-reducing property for 〈Mt. Everest is extended in space〉, T∨ reduces truth for 〈Mt. Everest is extended in space or there are trees〉 and T∧ does so for 〈Mt. Everest is extended in space and there are trees〉. This difference is a reflection of the fact that, on our view, which
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property reduces truth for a given proposition is not determined by the proposition’s domain membership alone. Logical form plays a role, too. We are committed to a hyperintensional treatment of propositions. By this we mean that we are committed to taking some intensionally equivalent propositions to be distinct propositions. Here we take two propositions p and q to be intensionally equivalent if and only if: for any world w, p is true if and only if q is true. Within our strongly pluralist framework this amounts to the following: propositions p and q are intensionally equivalent if and only if: for any world w, p has its truth-reducing property if and only if q has its truth-reducing property. Consider an atomic proposition p and the compound proposition p ∧ p. Let p belong to domain Di and let Ti be the base-level truth property for atomic Di-propositions. Recall that T∧ is the truth property for any conjunction, including p ∧ p. The two propositions in question, that is, p and p ∧ p, are intensionally equivalent. In any world w, p has its truth- reducing property Ti if and only if p ∧ p has its truth-reducing property, T∧. For, whenever p is Ti, it is also the case that p ∧ p is a conjunction whose conjuncts have their truth-reducing property (namely, Ti). Conversely, whenever p ∧ p is T∧, it is also the case that p is Ti. Now consider the following list of statements concerning propositions p and p ∧ p and truth-reducing properties: (p1) (p2) (p ∧ p1) (p ∧ p2)
Ti is the truth-reducing property for p. T∧ is the truth-reducing property for p. Ti is the truth-reducing property for p ∧ p. T∧ is the truth-reducing property for p ∧ p.
(p1) is true while (p2) is false, and (p ∧ p1) is false while (p ∧ p2) is true. This shows that, although p and p ∧ p are intensionally equivalent, they cannot be the same proposition. For, what would happen if they were? This would clash with our observations concerning (p1), (p2), (p ∧ p1), and (p ∧ p2)—observations that are non-negotiable commitments of our view. In light of these considerations we cannot adopt any account that treats p and p ∧ p as having the same content or as being the same proposition. Thus, for example, we cannot adopt a standard possible-worlds
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account of propositions. According to this account a proposition is identified with the set of possible worlds at which it holds.9 This would make p and p ∧ p the same proposition. We are committed to a hyperintensional treatment of propositions given features that are specific to our favored version of strong truth pluralism. We believe that a more general argument can be given—one that establishes the same commitment for truth pluralism in general, whether strong or moderate. However, we also think that this more general argument requires more work to cash out properly. For this reason, we leave it for another day. The issue of hyperintensionality has gone unnoticed in the literature. However, it strikes us as interesting and significant. Our argument shows that distinctively pluralist commitments impose a significant constraint on the pluralist’s theory of propositions and any account they might offer concerning their nature. This is worth noting.10
4
Truth Grounding
On the picture presented above, correspondence, coherence, superwarrant, and so on are truth-reducing properties only for atomic propositions. T∨ and T∧ are truth-reducing properties only for compounds of a particular logical form, that is, disjunctions and conjunctions, respectively. We now turn to the issue of how compound truth is grounded. Our account of truth grounding is complementary to our account of truth reduction. Both accounts are needed for a comprehensive metaphysics of truth. The issue of truth reduction for a given compound is the issue of specifying a property to which the truth of that compound is identical. The issue of truth grounding for a given compound is the issue of mapping how, or in what ways, the truth of the compound depends on the semantic status of its constituents. Given the characterization of T∨, if p ∨ q is T∨, this is so because at least one disjunct has its truth-reducing property. But this is to say that the instantiation of the truth-reducing property of a disjunct suffices to ground the instantiation of T∨, that is, disjunction-truth. Given the characterization of T∧, if p ∧ q is T∧, this is
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so because every conjunct has its truth-reducing property. But this is to say that the conjuncts’ having their respective truth-reducing properties jointly grounds conjunction-truth. We also want to say that these because claims cannot be reversed, that is, that truth grounding is asymmetric. Grounding is standardly taken to be (strongly) asymmetric, (strongly) irreflexive, and (strongly) transitive, that is,
( SA ) If éë p ùû ¬ éëq ùû, G then not : éëq ùû ¬ éë p ùû, D ( Strong asymmetry ) ( SI ) Not : éë p ùû ¬ éë p ùû, G ( Strong irreflexivity ) ( ST ) If éë p ùû ¬ éëq ùû, G and éëq ùû ¬ D, then éë p ùû ¬ G , D ( Strong transitivity ) “[p]” is read as the fact that p while “[p] ← [q]” is read as [p] is grounded by [q] (or [q] grounds [p]). Upper-case Greek letters denote (possibly empty) sets of facts. Strong asymmetry says that it is not the case that [p] is among the grounds of [q], provided that [q] grounds [p]. Strong irreflexitivity says that no fact is among its own grounds, and transitivity says that if [q], Γ ground [p] and ∆ grounds [q], then Γ, ∆ ground [p].11 These general observations about truth grounding and grounding’s features can be used to shed light on the grounds of truth in relation to specific compounds. Consider the following five compounds (parentheses used to indicate scope): (∨1) Mt. Everest is extended in space or Bob’s drunk driving is illegal. (∨2) Mt. Everest is extended in space or Bob’s drunk driving is legal. (∨3) (Mt. Everest is extended in space and Bob’s drunk driving is illegal) or 2 + 2 = 5. (∧1) Mt. Everest is extended in space and Bob’s drunk driving is illegal. (∧2) (Mt. Everest is extended in space or 2 + 2 = 5) and (Bob’s drunk driving is illegal or 3 + 8 = 12). Given (T∨*) we can say something about the metaphysical ground of the truth of (∨1) and (∨2). As earlier, suppose that correspondence and coherence with the body of law are the truth-reducing properties for
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atomic propositions belonging respectively to the empirical domain and the legal domain. Both of (∨1)’s disjuncts have their truth-reducing property, correspondence and coherence, respectively. The truth of (∨1) is thus doubly grounded. It is grounded in the first disjunct’s corresponding to reality and also in the second disjunct’s cohering with the body of law. (∨2), on the other hand, has just one disjunct that possesses its truth- reducing property. The truth of (∨2) is thus grounded solely in its first disjunct’s corresponding to reality. Given (T∧*) we can say something about the metaphysical ground of the conjunction-truth of (∧1). Both of (∧1)’s conjuncts have their truth-reducing property, correspondence and coherence, respectively. Jointly these facts ground the truth of (∧1). (∧1) is a conjunction whose conjuncts all have their truth-reducing property because the first conjunct corresponds to reality and the second conjunct coheres with the body of law. (∨1), (∨2), and (∧1) are less complex than (∨3) and (∧2). (∨3) and (∧2) are compounds with constituents that are themselves compounds. Transitivity helps us shed light on truth grounding for such compounds. The truth of (∨3), like the truth of any other disjunction, is identical to its being a disjunction with at least one disjunct that has its truth- reducing property. There is precisely one such disjunct, namely 〈Mt. Everest is extended in space and Bob’s drunk driving is illegal〉. This disjunct is itself a compound—a conjunction. This conjunction’s being true is identical to its being a conjunction with conjuncts that all possess their truth-reducing property. The truth of the conjuncts is identical, respectively, to correspondence to reality and coherence with the body of law. How about grounding? We have the following instances of grounding: (∨3)’s being a disjunction with at least one disjunct that has its truth-reducing property is grounded in its first disjunct, 〈Mt. Everest is extended in space and Bob’s drunk driving is illegal〉’s being a conjunction with conjuncts that all possess their truth-reducing property. In turn this conjunction’s being a conjunction with conjuncts that all possess their truth-reducing property is grounded jointly in the first conjunct, 〈Mt. Everest is extended in space〉’s corresponding to reality and
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the second conjunct, 〈Bob’s drunk driving is illegal〉’s cohering with the body of law. Applying transitivity to (∨3) we get that the truth of 〈(Mt. Everest is extended in space and Bob’s drunk driving is illegal) or 2 + 2 = 5〉 is grounded in 〈Mt. Everest is extended in space〉’s corresponding to reality and 〈Bob’s drunk driving is illegal〉’s cohering with the body of law. The truth of (∧2), like the truth of any other conjunction, is identical to its being a conjunction with conjuncts that all possess their truth- reducing property. (∧2) has two conjuncts, each of which is a disjunction. Their truth reduces to being a disjunction with at least one disjunct that has its truth-reducing property. For 〈Mt. Everest is extended in space or 2 + 2 = 5〉 there is exactly one such disjunct—the first one. This disjunct has its truth-reducing property, that is, 〈Mt. Everest is extended in space〉 corresponds to reality. For 〈Bob’s drunk driving is illegal or 3 + 8 = 12〉 there is also precisely one disjunct that has its truth-reducing property—again, the first one. This disjunct has its truth-reducing property, that is, 〈Bob’s drunk driving is illegal〉 coheres with the body of law. How about grounding? We have the following instances of grounding: (∧2)’s being a conjunction with conjuncts that all have their truth- reducing property is grounded jointly in 〈Mt. Everest is extended in space or 2 + 2 = 5〉’s being a disjunction with at least one disjunct that has its truth-reducing property and 〈Bob’s drunk driving is illegal or 3 + 8 = 12〉’s being a disjunction with at least one disjunct that has its truth-reducing property. 〈Mt. Everest is extended in space or 2 + 2 = 5〉’s being a disjunction with at least one disjunct that has its truth-reducing property is grounded in 〈Mt. Everest is extended in space〉’s corresponding to reality. 〈Bob’s drunk driving is illegal or 3 + 8 = 12〉’s being a disjunction with at least one disjunct that has its truth-reducing property is grounded in 〈Bob’s drunk driving is illegal〉’s cohering with the body of law. Applying transitivity we get that (∧2)’s being a conjunction with conjuncts that all have their truth-reducing property is grounded in 〈Mt. Everest is extended in space〉’s corresponding to reality and 〈Bob’s drunk driving is illegal〉’s cohering with the body of law. The resulting picture of truth grounding is this: the truth of any given atomic is identical to its truth-reducing property (correspondence, coherence, etc.). The truth of any compound is identical to its
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compound-specific, truth-reducing property. However, a compound’s having this property is ultimately grounded by facts about the instantiation of base-level truth properties.12
5
Strongly Pluralist Response A the Problem of Mixed Compounds
The problem of mixed compounds raised the following issues for the strong pluralist: 1. Something must be said about the nature of the truth property possessed by mixed compounds. In particular, while it is distinct from the truth properties possessed by its constituents, does its nature or instances somehow dependent on them—and if so, how? 2. Since the strong pluralist rejects the idea that there is a truth property that applies across all truth-apt domains, we need some assurance that the truth property possessed by neither disjunctions nor conjunctions is not a property of this kind. By using the strongly pluralist framework introduced above we can address both of these issues. Turning to the first issue, what we have is the following: a disjunction’s being true is simply identical to its being a disjunction with at least one disjunct that has its truth-reducing property. A conjunction’s being true is simply identical to its being a conjunction with conjuncts that all have their truth-reducing property. This account of the nature of disjunction-truth and conjunction-truth packs logical structure into both properties. This has the advantage of making clear how the truth of a compound depends on the semantic status of its constituents. Now, there is nothing extraordinary or mysterious about the property that reduces truth for mixed disjunctions. It is the same as the property that reduces truth for pure disjunctions—namely, disjunction-truth. Likewise, there is nothing extraordinary or mysterious about the property that reduces truth for mixed conjunctions. It is the same as the property that reduces truth for pure conjunctions—namely, con-
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junction-truth. In sum, from the point of view of truth reduction, mixed and pure compounds are on a par. Problem of mixed compounds solved. As for grounding, mixed disjunctions may have separate, sufficient grounds for their (disjunction-)truth pertaining to distinct domains. On the other hand, if pure disjunctions have separate sufficient grounds for their (disjunction-)truth, these grounds always belong to the same domain. Consider (PURE-∨) Mt. Everest is extended in space or London Bridge is extended in space. (MIX-∨)
Mt. Everest is extended in space or Bob’s drunk driving is illegal.
The truth of (PURE-∨) has two distinct, sufficient grounds pertaining to the empirical domain. 〈Mt. Everest is extended in space〉’s corresponding to reality is one, 〈London Bridge is extended in space〉’s corresponding to reality another. The truth of (MIX-∨), on the other hand, has two distinct, sufficient grounds that belong to different domains—the empirical domain and the legal domain, respectively.13 Mixed conjunctions have a sufficient ground that involves two distinct domains whereas pure conjunctions have a sufficient ground that involves just one domain. Consider: (PURE-∧) Mt. Everest is extended in space and London Bridge is extended in space. (MIX-∧)
Mt. Everest is extended in space and Bob’s drunk driving is illegal.
The (conjunction-)truth of (PURE-∧) has a sufficient ground involving just the empirical domain, namely 〈Mt. Everest is extended in space〉’s corresponding to reality and 〈London Bridge is extended in space〉’s doing so as well. The (conjunction-)truth of (MIX-∧), on the other hand, has a sufficient ground involving the empirical domain and the legal domain, namely 〈Mt. Everest is extended in space〉’s corresponding to reality and 〈Bob’s drunk driving is being illegal〉’s cohering with the body of law.
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Turn to the second issue, the issue of giving an assurance that the truth-reducing properties possessed by mixed disjunctions and mixed conjunctions do not apply across the board. This is straightforward. We just need to make two observations. The first observation is that the truth-reducing property for disjunctions, whether pure or mixed, is the property of being a disjunction with at least one disjunct that has its truth-reducing property. This is a property that only disjunctions can have, and so, the truth-reducing property for mixed disjunctions is not a generic truth property. The second observation is that the truth-reducing property for conjunctions, whether pure or mixed, is the property of being a conjunction with conjuncts that all have their truth-reducing property. This is a property that only conjunctions can have, and so, the truth-reducing property for mixed conjunctions is not a generic truth property.
6
Conclusion
We set ourselves two aims in this chapter. The first was to present and develop a version of strong truth pluralism. The second was to give a strongly pluralist solution to the problem of mixed compounds. We have achieved both aims. In Sects. 3 and 4 we introduced and developed a version of strong pluralism, an alethic incarnation of the (tokentoken) identity theory. The metaphysics of the view was presented in some detail, with a specification of its range of truth-reducing properties and an account of truth grounding. We introduced compoundspecific truth-reducing properties—identifying the truth of disjunctions with being a disjunction with at least one disjunct that has its truthreducing property and the truth of conjunctions with the being a conjunction with conjuncts that all have their truth-reducing property. The nature of these properties put us in a position to offer a straightforward solution to the problem of mixed compounds. We thus conclude that whatever insurmountable problems or challenges might seem to confront strong pluralism, the problem of mixed compounds is not one of them.
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Notes 1. Correspondence theorists include David 1994, Devitt 1984, Newman 2007, Rasmussen 2014, Russell 1912, Vision 2004, and Wittgenstein 1921. Coherence theorists include Bradley 1914, Rescher 1973, Walker 1989, and Young 2001. Pragmatists or neo-pragmatists include James 1907, 1909, Peirce 1878, and Putnam 1981. 2. For example, Beall 2000, 2013; Cook 2011; Cotnoir 2009, 2013a, b; Cotnoir and Edwards 2015; Edwards 2008, 2009, 2011, 2012, 2013, 2018; Gamester forthcoming; Kölbel 2008, 2013; Pedersen 2006, 2010, 2012a, b, 2014, ms-a, ms-b, ms-c; Pedersen and Edwards 2011; Pedersen and Wright 2013a, b, 2016; Wyatt 2013; Yu 2017. Gila Sher and Terence Horgan and various collaborators have developed a pluralist version of the correspondence theory that incorporates different ways of corresponding. See Sher 2005, 2013, 2016 and Horgan 2001; Barnard and Horgan 2006, 2013. Works by Lynch and Wright include Lynch 2001, 2004, 2006, 2009, and 2013 and Wright 1992, 1994, 1995, 1996, 1998, 2001, and 2013. 3. Versions of strong pluralism are presented in Cotnoir 2009, 2013a; Gamester forthcoming; Pedersen 2006, ms-a, ms-b, ms-c, Pedersen and Lynch 2018 (Sect. 20.6, due to Pedersen). 4. See, for example, Lynch 2009. For surveys that provide a systematic presentation of objections to pluralism, see Pedersen 2012b, Pedersen and Lynch 2018, Pedersen and Wright 2016. 5. The problem of mixed compounds is raised by Tappolet 2000, Sainsbury 1996, and Williamson 1994. Various pluralist options are discussed by Cook 2011, Cotnoir 2009, Edwards 2008, 2009, 2018, Gamester forthcoming, Lynch 2004, 2006, 2009, 2013, Pedersen 2012 b, Pedersen and Lynch 2018, Sher 2005, 2013, and Wright 2013. 6. It is quite tedious always to use formulations that make our endorsement of a token-token version of the identity theory explicit and distinguish it from its type-type counterparts. Sometimes we use formulations that are compatible with the type-type identity theory. However, in those cases it should be borne in mind that our reductionism kicks in at the level of tokens. One reason to opt for token identity is that it seems to integrate quite naturally with our account of truth grounding, that is the other component of our proposed metaphysics of truth. Grounding is usually regarded as a relation that obtains
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between particular facts or states of affairs rather than types of facts or states of affairs. 7. Wyatt (2013) and Edwards (2018), (this volume) are among the few sources that offer a systematic discussion of domains. Lynch (2009) talks about domains but only in passing. The formal aspects of what we say about domains here align with Yu (2017). For a more comprehensive treatment of domains and their relation to subject matter, see Pedersen (ms-e). 8. The idea that atomics belong to a unique domain has been debated in the literature. Discussion of this issue is often tied to the so-called problem of mixed atomics. Following Sher (2005), it is not clear, for example, which unique domain 〈Causing pain is bad〉 would belong to. It involves a mental concept (pain), a physical concept (causation), and a moral concept (badness). Our view is that 〈Causing pain is bad〉 belongs to the ethical domain. This is because the property figuring in an atomic proposition determines its domain membership. Since the property of being bad is an ethical property, 〈Causing pain is bad〉 belongs to the ethical domain. This kind of approach is suggested in Pedersen and Wright (2016, Sect. 4.5.1) and spelled out in more detail in Pedersen (ms-d). Edwards (2018) favors the same approach. Wyatt (2013) develops an account of domains according to which atomics can belong to more than one domain. He supplements this proposal with a story about how to determine a single truth-relevant property for propositions belonging to more than one domain. This is not the place to pursue an in-depth discussion of different approaches to mixed atomics and domains. For present purposes, we rest content with simply having noted some key differences. 9. Lewis 1986. 10. Thanks to Tim Fuller for raising the issue of hyperintensionality. Thanks to Jeremy Wyatt for extensive comments also. Substitutability salva veritate is often employed as a test for hyperintensionality. A notion or operator N on sentences is said to be hyperintensional if it does not allow intensionally equivalent sentences to be substituted salva veritate. The notion “propositional content of ” is hyperintensional in this sense. To see this consider sentences “p” and “p ∧ p” whose propositional content is respectively the proposition 〈p〉 and the proposition 〈p ∧ p〉. Although “p” and “p ∧ p” are intensionally equivalent, “p ∧ p” cannot be substituted salva veritate for “p” in “The propositional content of ‘p’ is 〈p〉”. This would clash with (p1), (p2), (p ∧ p1), and (p ∧ p2). Similarly, “p”
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cannot be substituted salva veritate for “p ∧ p” in “The propositional content of ‘p ∧ p’ is 〈p ∧ p〉”. 11. Rosen 2010. 12. The kind of grounding-theoretic account just provided is available to both strong and moderate pluralists. It should be noted, however, that there are significant differences with respect to the specifics of truth grounding within the frameworks of respectively strong and moderate pluralism. Crucially, within the framework of strong pluralism there is no grounding relationship between an atomic proposition’s corresponding (cohering, etc.) and its being true. For, remember an atomic proposition’s being true simply is its corresponding (cohering, etc.). Given this identity taking a proposition’s corresponding (etc.) to ground its own truth would violate irreflexitivity. However, grounding relations do obtain between atomic propositions’ having their base-level truth properties and instances of disjunction-truth and conjunction-truth. For details of a grounding-theoretic metaphysics for moderate pluralism, see Pedersen ms-a, ms-b, ms-c and Kim and Pedersen (ms). 13. We said, “if pure disjunctions have separate sufficient grounds for their (disjunction-)truth, these grounds always pertain to the same domain”. Strictly speaking, this claim needs to be qualified. It is only correct if we restrict attention to pure disjunctions with atomic constituents. To see this, note that 〈(Bob’s drunk driving is legal or Mt. Everest is extended in space) or (Bob’s drunk driving is illegal or there are no trees)〉 is a pure disjunction. Its disjuncts concern the same domain—namely, the legalempirical domain. However, (by transitivity) the pure disjunction in question has separate grounds for its truth in 〈Mt. Everest is extended in space〉’s corresponding to reality and in 〈Bob’s drunk driving is illegal〉’s cohering with the body of law. These grounds pertain to different domains.
References Barnard, R., and T. Horgan. 2006. Truth as Mediated Correspondence. The Monist 89: 31–50. ———. 2013. The Synthetic Unity of Truth. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 180–196. Oxford: Oxford University Press.
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Bradley, F. 1914. Essays on Truth and Reality. Oxford: Clarendon Press. Cook, R.T. 2011. Alethic Pluralism, Generic Truth and Mixed Conjunctions. The Philosophical Quarterly 61: 624–629. Cotnoir, A. 2009. Generic Truth and Mixed Conjunctions: Some Alternatives. Analysis 69: 473–479. ———. 2013a. Validity for Strong Pluralists. Philosophy and Phenomenological Research 83: 563–579. ———. 2013b. Pluralism and paradox. In Pedersen & Wright 2013a, 339–350. Cotnoir, A., and D. Edwards. 2015. From Truth Pluralism to Ontological Pluralism and Back. The Journal of Philosophy 112: 113–140. David, M. 1994. Correspondence and Disquotation: An Essay on the Nature of Truth. Oxford: Oxford University Press. Devitt, M. 1984. Realism and Truth. Oxford: Blackwell. Edwards, D. 2008. How to Solve the Problem of Mixed Conjunctions. Analysis 68: 142–149. ———. 2009. Truth-conditions and the Nature of Truth: Re-Solving Mixed Conjunctions. Analysis 69: 684–688. ———. 2011. Simplifying Alethic Pluralism. Southern Journal of Philosophy 49: 28–48. ———. 2012. On Alethic Disjunctivism. Dialectica 66: 200–214. ———. 2013. Truth, Winning, and Simple Determination Pluralism. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 113–122. Oxford: Oxford University Press. ———. 2018. The Metaphysics of Truth. Oxford: Oxford University Press. Gamester, W. forthcoming. Logic, Logical Form and the Disunity of Truth. To appear in Analysis. doi: https://doi.org/10.1093/analys/anx165. James, W. 1907. Pragmatism: A New Name for some Old Ways of Thinking. Cambridge, MA: Harvard University Press. ———. 1909. The Meaning of Truth. Cambridge, MA: Harvard University Press. Kim, S. & Nikolaj J.L.L. Pedersen. (ms). The Return of the Many: A Critical Appraisal of Moderate Truth Pluralism Through Metaphysics. Kölbel, M. 2008. “True” as Ambiguous. Philosophy and Phenomenological Research 77: 359–384. ———. 2013. Should We Be Pluralists About Truth? In Pedersen & Wright 2013a, 278–297. Lewis, D. 1986. On the Plurality of Worlds. Oxford: Blackwell.
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Lynch, M. 2001. A Functionalist Theory of Truth. In The Nature of Truth: Classic and Contemporary Perspectives, ed. M. Lynch, 723–749. Cambridge, MA: MIT Press. ———. 2004. Truth and Multiple Realizability. Australasian Journal of Philosophy 82: 384–408. ———. 2006. ReWrighting Pluralism. The Monist 89: 63–84. Lynch, M.P. 2009. Truth as One and Many. Oxford: Oxford University Press. ———. 2013. Three Questions for Alethic Pluralism. In Pedersen & Wright 2013a, 21–41. Newman, A. 2007. The Correspondence Theory of Truth: An Essay on the Metaphysics of Predication. Cambridge: Cambridge University Press. Pedersen, Nikolaj J.L.L. 2006. What Can the Problem of Mixed Inferences Teach Us About Alethic Pluralism? The Monist 89: 103–117. ———. 2010. Stabilizing Alethic Pluralism. The Philosophical Quarterly 60: 92–108. ———. 2012a. True Alethic Functionalism? International Journal of Philosophical Studies 20: 125–133. ———. 2012b. Recent Work on Alethic Pluralism. Analysis 72: 588–607. ———. 2014. Pluralism × 3: Truth, Logic, Metaphysics. Erkenntnis 79: 259–277. ———. (ms-a). Grounding Manifestation Pluralism. ———. (ms-b). Grounding Determination Pluralism. ———. (ms-c). Moderate Truth Pluralism and the Structure of Doxastic Normativity. ———. (ms-d). Moderate Pluralism About Truth and Logic: Truth and Logic as One, Quasi-truth and Quasi-logic as Many. ———. (ms-e). Subject Matter and Domains. Pedersen, Nikolaj J.L.L., and D. Edwards. 2011. Truth as One(s) and Many: On Lynch’s Alethic Functionalism. Analytic Philosophy 52: 213–230. Pedersen, Nikolaj J.L.L., and M.P. Lynch. 2018. Truth Pluralism. In The Oxford Handbook of Truth, ed. M. Glanzberg. Oxford: Oxford University Press. Pedersen, Nikolaj J.L.L., and C.D. Wright, eds. 2013a. Truth and Pluralism: Current Debates. New York: Oxford University Press. ———. 2013b. Pluralism About Truth as Alethic Disjunctivism. In Pedersen & Wright 2013a, 87–112. ———. 2016. Pluralist Theories of Truth. In The Stanford Encyclopedia of Philosophy, ed. E.N. Zalta, (Spring 2016 Edition). https://plato.stanford.edu/ archives/spr2016/entries/truth-pluralist/
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Peirce, C.S. 1878/1992. How to Make Our Ideas Clear. Reprinted in e Essential Peirce, ed. N. Houser and C. Kloesel, 124–141, Vol. 1. Bloomington: Indiana University Press. Putnam, H. 1981. Reason, Truth and History. Cambridge: Cambridge University Press. Rasmussen, J. 2014. Defending the Correspondence Theory of Truth. Cambridge: Cambridge University Press. Rescher, N. 1973. The Coherence Theory of Truth. Oxford: Oxford University Press. Russell, B. 1912. Problems of Philosophy. Reprinted by Oxford University Press, 1971. Rosen, G. 2010. Metaphysical Dependence: Grounding and Reduction. In Modality: Metaphysics, Logic, and Epistemology, ed. B. Hale and A. Hoffmann, 109–136. Oxford: Oxford University Press. Sainsbury, M. 1996. Crispin Wright: Truth and Objectivity. Philosophy and Phenomenological Research 56: 899–904. Sher, G. 2005. Functional Pluralism. Philosophical Books 46 (4): 311–330. ———. 2013. Forms of Correspondence: The Intricate Route from Thought to Reality. In Pedersen & Wright 2013a, 157–179. ———. 2016. Epistemic Friction: An Essay on Knowledge, Truth, and Logic. Oxford: Oxford University Press. Tappolet, C. 2000. Truth Pluralism and Many-Valued Logic: A Reply to Beall. The Philosophical Quarterly 50: 382–384. Vision, G. 2004. Veritas: The Correspondence Theory and Its Critics. Cambridge, MA: MIT Press. Walker, R.C.S. 1989. The Coherence Theory of Truth: Realism, Anti-realism, Idealism. London: Routledge. Williamson, T. 1994. A Critical Study of Truth and Objectivity. International Journal of Philosophical Studies 30: 130–144. Wittgenstein, L. 1921. Tractatus Logico-Philosophicus. In Annalen der Naturphilosophie. English translation by D. F. Pears & B. F. McGuinness (1961). London: Routledge & Kegan Paul. Wright, C. 1992. Truth and Objectivity. Cambridge, MA: Harvard University Press. ———. 1994. Realism, Pure and Simple: A Reply to Timothy Williamson. International Journal of Philosophical Studies 2: 327–341. ———. 1995. Truth in Ethics. Ratio 8: 209–226.
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———. 1996. Response to Commentators. Philosophy and Phenomenological Research 56: 911–941. ———. 1998. Truth: A Traditional Debate Reviewed. Canadian Journal of Philosophy, 24 (suppl. vol.): 31–74. ———. 2001. Minimalism, Deflationism, Pragmatism, Pluralism. In The Nature of Truth, ed. M. Lynch, 751–789. Cambridge, MA: MIT Press. ———. 2013. A Plurality of Pluralisms. In Pedersen & Wright 2013, 123–153. Wyatt, J. 2013. Domains, Plural Truth, and Mixed Atomic Propositions. Philosophical Studies 166: 225–236. Young, J.O. 2001. A Defence of the Coherence Theory of Truth. The Journal of Philosophical Research 26: 89–101. Yu, A. 2017. Logic for Pluralists. Journal of Philosophy 114: 277–302.
Methodological Pluralism About Truth Nathan Kellen
1
Introduction
Truth pluralism is the view that there are many ways of being true. Further, these ways of being true vary with the domain of discourse. While propositions about the physical world, for example SE There is an even number of the stars in the universe. may be true in virtue of some correspondence with the world, this does not seem plausible for other domains of discourse. Consider a proposition about morality, for example SC Sacrificing a younger child so an older child may live is morally permissible.
N. Kellen (*) University of Connecticut, Storrs, CT, USA © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_6
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If SC is true, then it seems true not in virtue of corresponding with some fact in the physical universe, but in virtue of something else, perhaps coherence with a set of other moral propositions.1 Let us call this feature domain-variability: DOMAIN-VARIABILITY: Ways of being true vary with domains.2 Nearly every truth pluralist account in the literature accepts domain- variability and treats it as essential to their pluralism.3 But why believe that ways of being true vary with domains? While we cannot rehearse here every truth pluralist’s motivation for accepting DOMAIN- VARIABILITY, I want to suggest that nearly all pluralists share a common motivation for accepting it. Consider what Crispin Wright notes as one of his key motivations for truth pluralism: A pluralistic conception of truth is also philosophically attractive insofar as an account which allows us to think of truth as constituted differently in different areas of thought might contribute to a sharp explanation of the differential appeal of realist and anti-realist intuitions. (Wright 1998, p. 58)
Or consider more recently (Pedersen and Lynch 2018): A principal reason for adopting truth pluralism is that the view provides a framework for understanding the intuitive appeal of respectively realism and antirealism with respect to different domains. The intuitive appeal stems in part from the observation that both traditional realist accounts of truth, such as the correspondence theory, and traditional antirealist accounts, such as the coherence theory, face a similar pattern of failure. Theories that seem plausible in some domains fail to seem as plausible in others. (Pedersen and Lynch 2018, p. 1)
Nearly every truth pluralist account in the literature explicitly includes at least one realist and one anti-realist domain.4 My claim is that DOMAIN-VARIABILITY is itself entailed by a more basic principle:
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(ANTI-)REALISM-VARIABILITY: The metaphysical account of domains varies; at least one domain is realist while at least one other is anti-realist. These two features are at the core of nearly every extant account of truth pluralism. Truth pluralism, properly understood, is a combination of realism and anti-realism into a single, coherent framework. It is this motivation which underwrites the more obvious aspect of truth pluralism—that there are many ways of being true which vary by domain. It is in this way that truth pluralism is the spiritual successor to the realism/ anti-realism debates which dominated analytic philosophy for decades after Michael Dummett first introduced his semantic anti-realism in (Dummett 1959). Truth pluralists are those who recognise that both camps got something right—that there were intuitions from both realism and anti-realism that should be captured by our theory of truth. In what follows, I will take this understanding of truth pluralism as a combination of realism and anti-realism and expand upon it by considering the extent to which various truth pluralisms are realist or anti-realist, in virtue of their methodological commitments. After drawing two distinctions which will allow us to categorise various pluralist views as realist or anti-realist I will introduce another form of truth pluralism, methodological truth pluralism, which does not privilege either its realist or anti- realist aspects.
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Truth Pluralism as (Anti-)realist?
I have argued that truth pluralism is best understood as an attempt to combine realism and anti-realism into a single framework. But this leads naturally to a further question: should the truth pluralist theory itself be categorised as a form of realism, anti-realism, or neither? That is: how should we characterise truth pluralism? Some surveys of the truth literature have lumped pluralist views in with anti-realist ones.5 There are a number of reasons one might do this. Realism has been the dominant position throughout history and thus
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there tends to be a strong presumption in its favour such that any deviation from it is characterised as a type of anti-realism (and of course pluralism is a deviation from global realism). On the other hand, there is some reason to consider truth pluralism as a form of realism. Anti-realism is generally characterised in terms of its epistemic constraint: if a domain is anti-realist then all of its truths are knowable in principle. But if a truth pluralism contains at least one realist domain, then there will be some unknowable truths, violating the anti- realist constraint, making it a form of realism. While each of these positions have merit, I want to suggest a more disciplined way of determining how a given truth pluralist theory should be characterised. To do so, we must examine the methodological accounts of the theories in question. As I’ve already noted, truth pluralists are pluralists precisely because they wish to account for competing realist and anti-realist intuitions. But some truth pluralists privilege one set of intuitions over the other, thus giving realist or anti-realist a privileged position in their theory. Consider, for example, Crispin Wright’s minimalism: A basic anti-realism – minimalism – about a discourse contends that nothing further is true of the local truth predicate which can serve somehow to fill out and substantiate an intuitively realist view of its subject matter… Because of its unassuming character, this minimalism, I suggested should always be viewed as the “default” stance, from which we have to be shown that we ought to move. (Wright 1992, p. 174)
Wright accepts a core Dummettian view that realism is a substantive doctrine, one oft-assumed but little argued for, which requires further justification over and above anti-realism. Nowhere is this clearer than in Wright’s discussion of his anti-realist account of truth, which he calls superassertibility, and its relation to other accounts of truth: If nothing bars the interpretation of a discourse’s truth predicate as superassertibility, then it is open to us to think of the truth of its statements as consisting merely in their durably meeting its standards of warranted assertion – a property for which all minimally assertoric sentences are eligible.
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But if truth and superassertibility can be prised apart – if it can be shown that superassertibility is a bad interpretation of the truth predicate in question – then the thought is at least strongly suggested that what confers truth on a statement is not a matter of its meeting standards internal to the language game, as it were, but its fit with an external reality. (Wright 1992, p. 142)
Here Wright repeatedly emphasises that superassertibility—his pluralism’s anti-realist aspect—has methodological priority over alternative accounts. For example, each sentence is couched in conditionals (“But if truth and superassertibility can be prised apart…”). For Wright, if a domain of discourse is eligible for truth, then it is first granted superassertibility as its truth predicate. If later it is shown that this will not do— that superassertibility fails as an interpretation of the truth predicate for the discourse—then we move to another, more realist account of truth. As Wright claims, “it is realism which must try to make good its case” by showing that anti-realism is unacceptable.6 While Wright’s pluralism is amenable to having realist aspects, its core is firmly anti-realist.7 Wright gives clear methodological priority to his anti-realism; anti-realism is the default from which we must move. This of course does not rule out his adopting realism in a particular domain, but it does make it more difficult to do so. It is in this way that Wright’s pluralism is best understood as a form of anti-realism. Not all truth pluralists follow Wright in privileging anti-realism. It is perfectly coherent to formulate a truth pluralism which not only does not privilege anti-realism but instead privileges realism; consider for example Edwards (2011): …although the notion of truth as correspondence to the facts might fit our domain of discourse about the material world, a different notion of truth – perhaps one with less metaphysical baggage, constructed out of coherence, or justification or warrant – may fit the domains in which the correspondence notion looks problematic. (Edwards 2011, pp. 31–2)
The principle here is the same as in the case of Wright and anti-realism. The realist account of truth (correspondence) is the default, and one from
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which we move if it cannot be made to work in some discourse, for example, in the moral domain. Edwards’ truth pluralism is, at its core, a realist account. When a pluralist account of truth sets either realism or anti-realism as the default for theory it is treating that aspect of the theory as being more methodologically fundamental. Wright holds anti-realism to be methodologically fundamental and realism to be secondary and vice versa for Edwards. This is a methodological (or epistemic) attitude that the pluralist has towards one aspect of their view: they believe that we more entitled to it, or that it’s prima facie more plausible, or simply that it is the default view. We can thus say that a view holds that (anti-)realism is methodologically fundamental if (anti-)realism is the default account of truth for a given domain. It may be helpful to introduce an analogy here. Imagine the truth pluralist constructing her theory as an assembly line worker. Down the line come various boxes, our domains, in a random order. The truth pluralist knows that each domain box will need to be stamped with a label: this one here is a realist box, that one is an anti-realist box, and so on. Like most pluralists, she has two stamps: one for her realist boxes and the one for her anti-realist boxes. The productive assembly line worker will put the stamp most likely to be used in her dominant hand so as to speed up the process. So which stamp goes in her dominant hand? Well it depends on whether she thinks the default status of the boxes is realist or anti- realist. The stamp she privileges will be the one which is what I am calling methodologically fundamental in her practice. We set out with the purpose of being able to determine whether truth pluralism was itself a realist or anti-realist view. With the notion of methodological fundamentality in hand we are able to distinguish pluralist views into two camps: those which hold realism to be methodologically fundamental and those which hold anti-realism to be methodologically fundamental. We can call the former views “realist” and the latter “anti- realist” for shorthand. This allows us to formulate an answer to the question of whether a given truth pluralist account is realist or anti-realist, depending on its methodological commitments. Now I wish to introduce a further distinction, built upon methodological fundamentality, which will allow us to sort truth pluralist views.
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Consider (Lynch 2009)’s manifestation functionalist pluralism. Lynch introduces a representationalist correspondence theory as his first account of being true. As he notes, this way of being true seems incompatible with the possibility of moral propositions being true, leading him to consider a coherence theory of truth for moral propositions. He proposes the property of supercoherence: SUPERCOHERENCE: A moral proposition P supercoheres with a moral framework F iff P coheres with F at some stage of inquiry in and continues to do so at any epistemically improved stage in + 1. (Lynch 2009, pp. 171–2) While it is easy to make sense of how supercoherence may be a way of being true, Lynch finds the account lacking, as it is “too permissive”, and leaves the possibility that even “craziest moral views” built on false empirical judgments may turn out to be supercoherent, and thus true.8 This leads him to conclude that …it is not enough for the fabric of our moral thought to be woven tightly – to be durably coherent – it must also be nailed down, or grounded on a firmer floor. (Lynch 2009, p. 173)
Lynch thus introduces his final, anti-realist account of truth, concordance: CONCORDANCE A moral proposition P is concordant iff P supercoheres with a moral framework F and F′s morally relevant non-moral judgments are true. (Lynch 2009, p. 176)
Note that “true” appears in the definition of concordance, which means that it can only be (non-circularly) understood when combined with another theory of truth, like his representational correspondence theory. Lynch defines concordance out of his representational correspondence theory, and thus the correspondence theory plays a crucial and ineliminable role in his second account of truth. Lynch’s concordance is
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thus essentially made up of two parts: an anti-realist theory (supercoherence) and a realist theory (representational correspondence). Obviously, this requires that Lynch treat his realist aspect as methodologically fundamental, but interestingly it also goes much further. For Lynch, realism is what I will call theoretically fundamental, insofar as another part of his theory is defined (partially) in virtue of the realist aspect of his pluralism. This is to be differentiated from those hold that realism is methodologically fundamental, for example (Edwards 2011). Edwards does not suggest adopting an anti-realist account built out of some sort of correspondence, but rather one built out of entirely separate, epistemic resources like coherence or warrant.9 Likewise, Wright’s pluralism, while methodologically anti-realist, does not attempt to build up a notion of correspondence from superassertibility. Let us take stock now. I have introduced two distinctions to help us determine whether a given pluralist account is realist or anti-realist. Methodological (anti-)realists treat (anti-)realism as the default, methodologically privileged account of truth. Theoretical (anti-)realists not only treat (anti-)realism as methodologically privileged, but they build their secondary account of truth out of the default one. Separating these into exclusive categories, we can sum the discussion up with the following table: Realist Anti-realist
Methodologically fundamental
Theoretically fundamental
Edwards (2011) Wright (1992)
Lynch (2009) ???
As of now, there is no truth pluralism in the literature which treats anti-realism as theoretically fundamental. It’s not clear how such an account would go or what would motivate it over a run-of-the-mill realist theory like a form of correspondence. Nonetheless, it seems to be a conceptually possible account of truth. It is worth stopping here to consider how this divvying up of the conceptual landscape compares with the other, major way of categorising truth pluralist views. All truth pluralists agree that there are multiple ways of being true, that is, there are multiple truth properties. Strong truth pluralists (e.g. Pedersen and Kim 2018) claim that there is no truth
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property had by all the true propositions, while moderate truth pluralists (e.g. Lynch 2009) believe that there is a generic truth property which all the true propositions have, in virtue of having one of the other, varying truth properties. While much of the literature up until this point has been dominated by moderate truth pluralisms, strong truth pluralism has recent defenders, including Pedersen and Kim (2018), Cotnoir (2013) and Ferrari, Morruzzi, Pedersen (ms.). We can also create a table for this distinction10: Strong truth pluralism
Moderate truth pluralism
Wright (1992), Cotnoir (2013) Pedersen and Kim (2018) (Ferrari, Morruzzi, Pedersen, ms.)
Lynch (2009), Wright (2013) (Edwards 2011)
Notice that this way of dividing truth pluralist views cuts across the fundamentality distinctions I introduced. The strong/moderate distinction does not track the realism/anti-realism distinction whatsoever. While Lynch (2009) and Wright (1992) differ according to the strong/moderate distinction, it is not because they fall on opposite sides of the realist/anti- realist debate, but rather because one advocates for a generic truth property and another does not.11 If we are concerned with distinguish types of truth pluralism based on their core commitments, I suggest that we do so in virtue of my fundamentality criteria.
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Methodological Pluralism About Truth
In the previous section I introduced two ways in which a truth pluralist can be (anti-)realist: by holding (anti-)realism as methodologically or theoretically fundamental. While these distinctions allow us to answer the question of whether a given truth pluralist theory is realist or anti- realist, and they allow us to distinguish various truth pluralist views in a new way according to their core, methodological commitments, the picture I have provided is not yet sufficient to capture the full breadth of truth pluralist views. So far when discussing realism and anti-realism I have only talked about pluralist views which privilege either one or the other theory. But must every truth pluralist pick a side?
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It may very well be that our language is so diverse that there is no reason to hold a privileged attitude towards realism or anti-realism with respect to arbitrary domains. Perhaps realism and anti-realism, contra (Wright 1992; Edwards 2011; Lynch 2009) are on a par. Neither realism nor anti-realism are the default, and neither require more evidence to establish themselves as the metaphysics of a domain than the other; instead, what they require is simply different evidence. Consider again the assembly line metaphor. Absent some knowledge of the general features of the boxes such that they are more likely to be say, realist than anti-realist, it makes little sense for the worker to assume that the next box coming down the line is realist. This of course does not mean that she ought to assume that it is anti-realist instead. What she ought to do is examine each box as it comes down the line and determine how to categorise it, based on its own particular features. So too for the truth pluralist, one might argue. Both realists and anti- realists have claimed methodological superiority for their views. Realists often note that realism better comports with common sense intuition, while anti-realists claim that anti-realism is more metaphysically and epistemically respectable. Perhaps what the pluralist ought to do is to step back from each of these claims and instead take a methodologically neutral stance, and examine each theory’s case with respect to each individual domain of discourse. Call such an approach to truth pluralism methodological pluralism about truth. Methodological pluralism about truth may have many virtues. It does not claim—nor need to provide support for such a claim—that our language and conceptual frameworks are structured in such a way that either realism or anti-realism are more likely to hold in a particular domain. The methodological pluralist about truth remains neutral (or silent) on that issue. Given the difficulty of establishing such conclusions, and the longstanding debates over their success, this should count as a point in favour of the methodological pluralist. Further, the methodological pluralist about truth is, in a sense, even more pluralist than its rival views. Pluralism was developed because monists failed to see that their theories of truth are implausible globally but plausible locally. The truth pluralist who treats realism or anti-realism as fundamental may commit the same sin as the monist, although on a
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smaller scale, in failing to recognise the true diversity of our language. They impose a type of monism—defeasible monism, but monism nonetheless—on our language; the methodological pluralist demurs from such a claim. This does not require that the methodological pluralist about truth cleanly divide up the language. It may simply turn out to be a fact of our language that most domains ought to be construed as realist. This does not undermine methodological pluralism. My methodological pluralism about truth is similar to Field’s (1994) methodological deflationism. It is a working hypothesis: do not assume that either realism or anti-realism will win out for a given domain absent any evidence one way or another. In this way, methodological pluralism about truth is actually compatible not only with pluralist accounts which, accept realism in all domains but one or vice versa, but in fact with monistic accounts of truth. While monistic accounts legislate that the same theory of truth holds globally, over all domains; the methodological pluralist does not, but nor does she assume that truth will in fact vary with domain.12 Methodological pluralism about truth, unlike methodological deflationism, does not come with it a commitment to a particular theory being the default for our working hypothesis, however.
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Conclusion
I have suggested that truth pluralism is best understood as the spiritual successor to the realism and anti-realism debate, insofar as it attempts to combine the best aspects of each theory. This raises a natural question: how ought we understand this new, mixed view? To answer this, I suggested that we again look at the core of truth pluralism: its realist and anti-realist motivations, and how it treats these views in theory construction. I suggested that some truth pluralists hold one of the two to be methodologically fundamental, that is, as the default stance from which we move. Others go further, in not only giving a privileged methodological status to one theory but to build the secondary account out of the first, thereby making the primary aspect theoretically fundamental. I then showed how these two distinctions can be used to categorise various types of truth pluralism in a way that cuts across the standard way of understanding the literature.
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I closed by considering one final view: methodological pluralism about truth, which does not privilege either its realist or anti-realist forebears. This methodological pluralism about truth may not be a completely new form of truth pluralism; many truth pluralists do not take a stand on whether realism or anti-realism comes first. They simply note that no account of truth works everywhere, and thus pluralism must be adopted. Perhaps if each pluralist spent more time reflecting on their methodology they would determine that they privilege one account over the other, or perhaps they would opt to remain methodologically neutral. What I have done here is made a plea to pay attention to the methodological commitments and motivations of various truth pluralisms, and to give a position—currently occupied or not—a name, and some arguments in its favour.13
Notes 1. Barring the truth of some fully reductive moral naturalism, that is. 2. This is similar to what (Beall 2013, p. 324) calls “language-relative truth pluralism”. 3. The only exception I am aware of is Beall (2013), if one accepts that his deflated truth pluralism is a pluralism about truth as opposed to a pluralism about truth-predicates which may be unrelated to the actual philosophically robust concept of truth. Note also that Beall’s pluralism does not accept DOMAIN-VARIABILITY, which again sets it far apart from other views in the literature. 4. This includes the most developed accounts of truth pluralism in Wright (1992) and Lynch (2009), as well as, for example Cotnoir and Edwards (2015), Edwards (2011), Pedersen (2006). Cotnoir (2013), who does not explicitly call for dividing domains by realism/anti-realism but provides a semantic framework for truth pluralisms which have certain domains which maintain classical logic and others which have paracomplete logics, a conclusion commonly held to follow from adopting realism and anti-realism respectively. Another potential outlier is Gamester (2017), who does divide up his truth pluralism into realist and antirealist parts, but the anti-realist parts are motivated by expressivism
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rather than the traditional metaphysical/semantic debates which the other pluralists are concerned with. 5. See, for example, Burgess and Burgess (2014) and Künne (2005). 6. Wright (1992, p. 174). 7. It is not even clear that Wright (1992) is truly a pluralist account, insofar as he never explicitly calls for more than one truth predicate. He notes ways in which we may work our way up from a superassertibility as the truth predicate of a given domain to a more realist truth predicate, for example, some form of correspondence, but never explicitly adopts a second truth predicate. Wright is a pluralist because he believes the concept of truth does not rule out the possibility of plurality of predicates; as he notes: “Minimalism is thus at least in principle open to the possibility of a pluralist view of truth” (Wright 1992, p. 25, emph. original). 8. Lynch (2009, p. 173). 9. Edwards does not suggest any particular theory of truth to be contrasted with correspondence, although he mentions some candidates which have previously been advocated as monistic views of truth (Edwards 2011, p. 32). 10. Cotnoir (2013) reads Wright (1992) as a strong truth pluralist. It is not clear that this is the only reading of the text, and Wright’s later work is explicitly a form of moderate pluralism. However, the reading is supported enough to worth including here. 11. As I note in the previous footnote, the placement of Wright (1992) is contentious. I prefer to read Wright as a moderate pluralist, who would thus end up in the same camp as Lynch (2009). This is despite their views being quite different when it comes down to methodology and motivations; nearly complete opposites in fact. On either way of interpreting Wright I do not believe that the strong/moderate distinction best captures the disagreement between these views. 12. This is analogous to the way in which Field’s (1994) methodological deflationism is compatible with inflationism about meaning or content. 13. Thanks to audiences at the University of Connecticut and Yonsei University for helpful feedback on this chapter. Thanks especially to Dorit Bar-On, Douglas Edwards, Filippo Ferrari, Will Gamester, Sebastiano Moruzzi, Nikolaj J. L. L. Pedersen, Joe Ulatowski, Cory Wright, Crispin Wright, Jeremy Wyatt and Andy Yu. Thanks most of all to Michael Lynch for countless discussion and feedback.
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References Beall, J. 2013. Deflated Truth Pluralism. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 323–338. Oxford: Oxford University Press. Burgess, A., and J. Burgess. 2014. Truth. Princeton: Princeton University Press. Cotnoir, A.J. 2013. Validity for Strong Pluralists. Philosophy and Phenomenological Research 86 (3): 563–579. Cotnoir, A.J., and D. Edwards. 2015. From Truth Pluralism to Ontological Pluralism and Back. Journal of Philosophy 112 (3): 113–140. Dummett, M. 1959. Truth. Proceedings of the Aristotelian Society 59 (1): 141–162. Edwards, D. 2011. Simplifying Alethic Pluralism. The Southern Journal of Philosophy 49 (1): 28–48. Field, H. 1994. Deflationist Views of Meaning and Content. Mind 103 (411): 249–285. Gamester, W. 2017. The Diversity of Truth: A Case Study in Pluralistic Metasemantics. PhD Dissertation, University of Leeds. Künne, W. 2005. Conceptions of Truth. Oxford: Clarendon Press. Lynch, M.P. 2009. Truth as One and Many. Oxford: Oxford University Press. Pedersen, Nikolaj J.L.L. 2006. What Can the Problem of Mixed Inferences Teach Us About Alethic Pluralism? Monist 89 (1): 102–117. Pedersen, Nikolaj J.L.L., and S. Kim. 2018. Strong Truth Pluralism. In Pluralisms in Truth and Logic, ed. J. Wyatt, Nikolaj J.L.L. Pedersen, and N. Kellen, 107–130. London: Palgrave Macmillan. Pedersen, Nikolaj J.L.L., and M.P. Lynch. 2018. Truth Pluralism. In The Oxford Handbook of Truth, ed. M. Glanzberg. Oxford: Oxford University Press. Wright, C. 1992. Truth and Objectivity. Cambridge, MA: Harvard University Press. ———. 1998. Truth: A Traditional Debate Reviewed. Canadian Journal of Philosophy 28 (sup1): 31–74. Wright, Crispin. 2013. A Plurality of Pluralisms. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 123–153. Oxford: Oxford University Press.
Normative Alethic Pluralism Filippo Ferrari
1
Introduction
Some philosophers have argued that truth is a norm of judgement.1 This thesis has been given in a variety of formulations—that true judgements are the correct ones; that it is better to judge truly than to judge falsely; and that the truth is what judges ought to pursue in enquiry. I will assume that truth somehow functions as a norm of judgement, and I will be focusing on two core questions concerning the judgement-truth norm—namely: (i) what are the normative relationships between truth and judgement? (ii) do these relationships vary or are they constant? I argue for a pluralist picture—what I call normative alethic pluralism2 (henceforth NAP)—according to which (i) there is more than one correct judgement-truth norm and (ii) the normative relationships between
F. Ferrari (*) Universität Bonn, Bonn, Germany © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_7
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truth and judgement vary in relation to the subject matter of the judgement. By means of a comparative analysis of disagreement in three core areas of the evaluative domain—refined aesthetics, basic taste, and morality—I show that there is an important variability in the normative significance that disagreement has in these areas—I call this the variability conjecture. By presenting a variation of Lynch’s scope problem for alethic monism, I argue that a monistic approach to truth’s normative function is unable to vindicate the normative variability conjecture. I then argue that NAP provides us with a very promising model to account for such a conjecture. I leave the discussion of both the metaphysical issue of what grounds these normative relationships and the epistemic issue of what justifies our beliefs about them aside. In particular, I take no stand on the question of whether truth’s normative function reflects a characteristic intrinsic to the nature of truth or whether it is grounded in some features external to truth’s nature. I intend the framework developed here to be compatible with rejecting the thesis that truth is an intrinsically normative notion.3
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The Truth-Norm
Advocates of the thesis that truth is a norm disagree about how the norm should be conceived, and many formulations of it have been given. The following list provides a sample of formulations that can be found in the recent debate (my emphases): Horwich We ought to want our beliefs to be true (and therefore not- want to have any false ones).4 James The true is the name of whatever proves itself to be good in the way of belief.5 Loewer True belief is valuable and, other things being equal, it is a rational doxastic policy to seek true beliefs.6 Lynch End of Inquiry: True propositions are those we should aim to believe when engaging in inquiry. Norm of Belief: True propositions are those that are correct to believe.7
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McHugh [T]he attitude of belief sets truth as the standard that a proposition must meet in order for it to be a fit object of that attitude.8 Wedgwood For any S, p: If S considers whether p, then S ought to believe that p if and only if p is true.9 Williams Beliefs aim at truth.10 While philosophers in the list take the truth norm to apply to belief, I take judgements to be the things that truth is primarily normative over. Following Shah and Velleman, I consider a judgement to be a cognitive mental act of affirming a proposition. It is a mental act because “it involves occurrently presenting a proposition, or putting it forward in the mind. It is a cognitive act because it involves presenting the proposition as true”.11 In this respect, judgements differ from beliefs, which are cognitive mental attitudes. The precise relationship between judgement and belief is a complex issue which won’t concern us in this chapter.12 Somewhat simplistically, I take it that the end product of a judgement is typically a belief whose content is a proposition. Since the examples I will consider involve an evaluative process, I take them to be fairly typical cases of beliefs formed through an act of judging. Thus, I use the expression ‘norms governing judgement’ as a catchall expression to indicate norms governing the formation, maintenance, and relinquishing of beliefs formed by means of an act of judging. Two things about this list and the current debate on the judgement- truth norm deserve some discussion. The first concerns the variability in the use of the normative vocabulary involved in the formulations above. Some formulations employ terms like ‘should’ or ‘ought’—thus using deontic terms; others involve notions such as ‘valuable’ and ‘good’—thus using an axiological vocabulary; others still talk in terms of truth’s being the aim of beliefs—using a teleological vocabulary; and, finally, others formulate the truth-norm in terms of correctness or fittingness—taking these notions to be normatively independent of any axiological or deontic element. I introduce the term criterial to indicate this latter category of normative notions. Mixed formulations—that is, formulations mixing different normative vocabularies—are also possible, as Lynch’s way of
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cashing out the normative role of truth both in terms of aim and in terms of correctness seems to suggest. Abstracting from the many issues concerning the precise formulation and interpretation of the truth norm, we can capture the highlighted variability in the ways in which the norm is conceived by distinguishing the following normative dimensions13: teleological Judgement aims at truth. criterial It is correct (fitting) to judge that p (if and) only if p is true. axiological It is valuable (good) to judge that p (if and) only if p is true. deontic One ought to judge that p (if and) only if p is true. These are four distinct dimensions of the normative constraint that truth can exert on judgement. In section “Minimal Normative Alethic Pluralism”, I show that the criterial, the axiological, and the deontic dimensions of the normative role of truth are normatively independent of each other. What about the teleological dimension? I take it to be (partly) constitutive of the act of judging. In judging, a subject performs an act that aims, constitutively, at the truth of the subject matter at issue. This means that all judgements, as I understand them here, deal with truth-apt discourse. In this respect, the teleological dimension is always present whenever a proper judgement is performed and for this reason I leave this dimension aside in the discussion that follows. With this in hand, I call a normative principle (NP) any principle expressing the normative constraint that truth exerts on judgement in terms of one or more of the three aforementioned dimensions—that is, criterial, axiological, and deontic. The notion of a NP so defined allows for some flexibility. In this sense, all the various norms in the list above count as NPs as I have defined the term. The second thing worth mentioning is that the philosophers in the list above assume—at least implicitly, in the context of their work—that their preferred NP applies uniformly to all judgements regardless of their
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subject matter. In other words, they all seem to share a commitment to a monistic conception of the normative function of truth—what I call Normative Alethic Monism (NAM). Such a conception encompasses the following two theses: Singularity: There is only one NP expressing truth’s normative function on judgement.14 Uniformity: NP applies uniformly to all judgements in all areas of truth-apt discourse. I take singularity and uniformity to provide a characterisation of NAM in terms of jointly sufficient and individually necessary conditions. Moreover, singularity and uniformity are independent theses: one might deny uniformity but endorse singularity by maintaining that with respect to a specific set of judgements the truth-norm is normatively silent15; or one might deny singularity but endorse uniformity by maintaining that truth uniformly exerts a plurality of normative functions for all judgements. However, neither of these ways of rejecting NAM amounts to what I take to be a genuine form of normative pluralism (more on this later). To make NAM more perspicuous, let us consider two specific applications of it. First, say that an alethic deontologist is a philosopher who thinks that the unique normative alethic principle governing all judgements is deontic—that is, it provides thinkers with some prescriptions about what to judge. An alethic deontologist, then, takes truth and falsity to always line up, respectively, with the obligatoriness and the impermissibility of the judgement in question. Second, those—alethic axiologists— who think of truth’s normative function in purely axiological terms—that is, in terms of what it is good or bad to believe—will take truth and falsity to always line up, respectively, with the good and the bad in the way of judgements. NAM imposes a structural rigidity in the normative function of truth which, I argue, makes it unable to account for differences in the normative significance of disagreement.
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isagreement and Its Normative D Significance
By the expression ‘normative significance of disagreement’ I mean the extent to which engaging in a disagreement licenses participants to negatively evaluate each other’s view. The kind of disagreement that matters for the purposes of this chapter is something analogous to what MacFarlane calls the simple view of disagreement (i.e. doxastic noncotenability)16: DIS To disagree with someone’s judgement that p is to issue judgements whose contents are jointly incompatible with p.17 Disagreement, in the DIS model, is thus minimally normatively significant in the following way: a commitment to judging that p is a commitment not only to the truth of p but also to the falsity of every content q which is incompatible with p, and consequently a commitment to assessing anybody endorsing q as judging falsely. However, what other kinds of negative normative assessment or reactive attitude18 to a contrary view are associated with an attribution of falsity and thus licensed by the presence of a disagreement varies in relation to the subject matter at issue. Call this the variability conjecture. With this in hand, let’s now turn to a discussion of some examples. The examples I will consider involve evaluative judgements—that is, judgements about evaluative matters such as basic taste, refined aesthetics, and, more controversially, fundamental morality.19 These are judgements for which a substantive kind of objectivity may be hard to sustain and, for this reason, they are taken to lack robustly representational content—in the sense of Wright (1992, Chap. 4). In this respect, these judgements are not subject to the possibility of a more fundamental kind of failure—a failure deriving from a misrepresentation of how things are objectively, that is, in mind-independent reality. Moreover, I should point out that these are just a few examples that I have chosen because I think they are particularly fit to show the kind of variability in the normative significance of disagreement I am interested
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in. Nothing hinges on which particular example we choose to highlight this variability. Also, I am not claiming that all examples of moral, aesthetic, and taste disagreement should be understood in the way suggested. Perhaps there is also substantive intra-domain variability. This, though, wouldn’t weaken my proposal—the more variability the better. However, in order to keep the discussion reasonably simple and smooth, I shall talk as if the three examples I discuss are paradigmatic of the three domains in question.
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Disagreement (1): Fundamental Moral
Consider the following disagreement about some fundamental moral value between Julie and Jill: Julie thinks that torturing people is always a morally deplorable practice. Jill disagrees, thinking that sometimes torturing people is after all morally acceptable. I take this to be an example of some deep incompatibility between moral values concerning fundamental human rights. Thus, we may ask: what kind of normative assessment of Jill’s contrary view would we typically expect from Julie in such a situation of disagreement? We would expect Julie to issue a strongly negative assessment of Jill’s view. Not only would Julie think that Jill is completely wrong, but she would also feel compelled to think that Jill ought to change her mind about that issue. In fact, she would find it quite deplorable that Jill holds such a morally bad view. What this suggests is that the kind of reaction we would expect from Julie amounts to a substantial criticism of Jill’s contrary view, which, in some cases, leads to a disposition to urge Jill to revise her view on torture. Even assuming that Julie and Jill consider each other equally knowledgeable on most of the relevant non-moral facts concerning torture, once the disagreement comes to light they will typically cease to consider each other as equally respectable moral judges, at least with respect to a range of topics closely connected to the subject matter of their disagreement.
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An appealing explanation of the appropriateness of this kind of substantive criticism of our opponent’s moral standing might be linked to a distinctive feature of moral value—namely, what we might call the higher- order heritability of moral value.20 The thought is that the moral condemnation of a certain action or practice in these fundamental cases engenders a commitment to judging the holding of any view favourable to that action or practice as alethically impermissible. In this respect, strong moral criticism of a certain practice carries with it a strong alethic criticism of the holding of any judgement supporting that practice. The criticism is typically conveyed by means of an attribution of falsity to such a judgement, which, in this case, goes hand in hand with assessing the judgement as impermissible. That said, it is important to keep in mind that this is a reconstruction of what we might typically expect in a situation of disagreement about fundamental moral issues. In other words, this is merely a conjecture concerning the typicality of a certain pattern of response to a situation of disagreement in some radical cases of incompatibility of fundamental moral values. There certainly may be a variety of contextually salient factors that would impact on the kind of critical assessment that we would deem appropriate—but for the sake of simplicity I will leave these atypical cases aside.
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Disagreement (2): Basic Taste
Consider now a situation in which Julie and Jill have a dispute about basic taste. I here draw an intuitive distinction between judgements about matters of basic taste and judgements about matters of refined aesthetics. Without endorsing any controversial thesis about where to draw the boundary between basic taste and refined aesthetics—there might be no sharp boundary—I will discuss two examples that might be taken as paradigmatic of each category. Let’s begin with the basic taste case: Julie thinks that oysters are delicious. Jill disagrees, thinking that they are tasteless.21
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In sharp contrast to the fundamental moral case, the kind of negative assessment of Jill’s contrary view we would typically expect from Julie would be paradigmatically weak—in fact nothing more than the realisation that they have divergent views on the taste of some particular oysters, and this alongside with an appreciation, in typical circumstances,22 of the legitimacy of Jill’s contrary view. If this approximates what people would typically expect in a situation of disagreement about basic taste, it shows that no substantive criticism of a contrary view should be expected. Absent pragmatic issues of coordination,23 a live and let live attitude is deemed appropriate. We may take as evidence of this that a rather natural follow-up that we might expect from Julie in coming to know about Jill’s contrary view would be a first-person qualification of the judgement— for example, “Oh, well, I just wanted to say that I like these oysters”—24 as a way of avoiding, in the dialectical context of the disagreement, the emergence of what might be considered an unjustified and futile quarrel. In fact, the mere presence of disagreement does not typically make either Jill or Julie less confident about her own opinion concerning oysters. What might happen sometimes is that, whenever it is plausible to assume a certain commonality of taste within the context of the disagreement, we experience a sense of surprise or curiosity in coming to know that someone we thought had a similar taste in fact has a radically different opinion about the food in question.
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Disagreement (3): Refined Aesthetics
Last, consider the following dialogue about a matter of refined aesthetics: Julie thinks that Gould’s 1955 execution of Goldberg Variations is unequalled. Jill disagrees, thinking that the 1955 execution is too virtuosic. The 1981 version is preferable as a more mature interpretation. What kind of negative assessment of someone’s holding a contrary view should we expect in a disagreement about matters of refined aesthetics? Here it seems plausible to conjecture that the kind of critical reaction to
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Jill’s contrary view that people would deem appropriate for Julie to have is one that falls somewhere in the middle of the two extremes of the fundamental moral case and the basic taste case. Thus, we may typically expect Julie to be somehow surprised by Jill’s contrary judgement regarding Gould’s 1955 execution of Goldberg Variations. This is because she would think that Jill’s judgement is somehow off-colour. Thus, a certain degree of criticism would be regarded as appropriate. In particular, it would be natural for Julie to consider Jill’s aesthetic sensibility on this occasion to be not as good as her own. In this respect, contrary to the basic taste case, disagreement about refined aesthetics seems to give rise to an attribution of fault. Although it seems appropriate for Julie to continue to regard Jill as well-informed as she is about Glenn Gould’s musical production, by being committed to her own judgement about Gould and thus to her own scale of aesthetic value, she is committed to assessing Jill’s opinion as inferior—indeed, one that it would be better not to have. As a result, it might happen that in some cases the conversation continues by each subject trying to persuade the opposite party to change her mind. However, in contrast with the fundamental moral case described above, the degree of intensity of the reactive attitude that would be appropriate for Julie to have in response to Jill’s contrary view is significantly lower than in the fundamental moral case. We could certainly understand, if not justify, a high degree of heat in disputes about fundamental moral issues, but not in disputes about refined aesthetics. Regardless of how much aesthetically off-colour I consider your contrary view, granted that we are equally knowledgeable about the subject matter in question I would still feel some pressure to regard you as permitted to hold to the view you endorse.25
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ormative Alethic Monism N and the Normative Scope Problem
I will now argue that NAM falls prey to a serious objection that originates in a particular application of what is known in the truth pluralism debate as the scope problem for monistic (substantivist) conceptions of truth. The thought at the core of the objection is that no single substantive notion
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of truth—for example, correspondence, superassertibility, or coherence—seems to be fully adequate to play the truth role across all different domains.26 For instance, whereas correspondence might be the adequate notion of truth for our judgements about the physical constitution of the world, it might score less well as the notion of truth for the domain of basic taste and other evaluative domains. Similarly, a version of the coherentist conception of truth might be the adequate notion of truth in mathematics or in the moral domain, but it seems utterly inadequate for contingent judgements.27 Lastly, superassertibility might provide the right model for truth in some evaluative domains or in mathematics,28 but it is inadequate for judgements concerning matters that are epistemically inaccessible to us—for example, claims about the past for which no evidence has survived. Thus, although all these various notions of truth might provide a locally adequate model of truth, none of them gives us an adequate model of truth across all different domains. My contention is that the monistic conception of the normative role of truth faces an analogous problem. If something along the lines of the variability conjecture described in sections “Disagreement and Its Normative Significance”, “Disagreement (1): Fundamental Moral”, “Disagreement (2): Basic Taste”, and “Disagreement (3): Refined Aesthetics” is accurate, then it is easy to see why NAM is in trouble. It would in fact require thinkers to adopt a Procrustean attitude towards the negative assessment of a contrary view in the presence of a disagreement, regardless of the specific subject matter at issue. This would mean that, depending on which NP we take to characterise truth’s normative function, some of the normative assessments described in the examples would be deemed inappropriate. To illustrate: recall that alethic deontologism is the view according to which truth exerts only a deontic constraint over all judgements, cashed out in terms of ‘ought’, ‘permissible’, and ‘impermissible’. Then this view would have problems accounting for the kind of normative assessment that seems appropriate in the domain of basic taste—and, arguably, in the domain of refined aesthetic matters as well. This is because such a view would say that false judgements are always impermissible and thus that a reactive attitude of rational condemnation to a contrary judgement would be always legitimate.29 But this is not what we would typically
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expect in situations of disagreement about basic taste or, more controversially, refined aesthetics. Secondly, recall that alethic axiologism is the view according to which truth exerts only an axiological constraint over all judgements, cashed out in terms of ‘valuable’, ‘good’, and ‘bad’. This view would have problems in accounting for the normative assessment typical of disagreement about basic taste—and, arguably, about fundamental moral matters. This is because such a view would say that false judgements are always bad, but not impermissible. However, this prediction seems inadequate for the basic taste case—because no substantive criticism is forthcoming—and also for the fundamental moral case—because a more substantive negative reaction would be expected. Last, let us call alethic criterialism the view according to which truth exerts only a criterial constraint on judgement, cashed out in terms of ‘correctness’, ‘fittingness’, and ‘incorrectness’. Since the kind of normative assessment of a contrary view predicted by this view would be paradigmatically weak, it would be inadequate to account for the kind of normative assessment that occurs in disagreements about fundamental moral issues and refined aesthetic matters—where some (more or less) substantive criticism to a contrary view would typically be deemed appropriate. The core of the normative scope problem is thus that none of these monistic views is adequate as a general view about truth’s normative function. This puts considerable pressure on NAM.
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Normative Alethic Pluralism
In this and the following section “Introduction” introduce a pluralist framework for modelling the normative function(s) that truth exerts over judgements—Normative Alethic Pluralism—and show how this framework can be put to use to solve the scope problem. At the core of NAP are the following two theses: Plurality
There is more than one NP expressing truth’s normative function on judgement. Variability Truth exerts a variable normative function on judgements from different areas of truth-apt discourse.
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Plurality and Variability are meant to provide individually necessary and jointly sufficient conditions for NAP. They amount to the negation of Singularity and Uniformity respectively. Thus, according to NAP there is more than one correct NP expressing truth’s normative function and, moreover, different areas of discourse are governed by different NPs. NAP predicts that while the normative function of truth in one area of discourse might be expressed by a NP that encompasses the deontic dimension, in some other areas it might be expressed by a NP which encompasses the axiological dimension, and in other areas still it might be expressed by a NP which is exhausted by the criterial dimension— thus, without any deontic or axiological constraint. Interestingly, NAP leaves open the possibility of an area of discourse where truth is normatively inert, as it were, while still functioning as the aim of judgements in that area—and thus still exerting its teleological function. This means that it allows for a purely deflationary notion of truth (and falsity) that operates, locally, in some domains of discourse.30 In order to defend the cogency of NAP, it is important to show that the criterial, the axiological, and the deontic dimensions enjoy a certain degree of normative independence from each other. There are two ways in which these three dimensions can be normatively independent: (IND 1) Criterial⇏Axiological; Criterial⇏Deontic; Axiological⇏Deontic31 (IND 2) Deontic⇏Axiological; Deontic⇏Criterial; Axiological⇏Criterial Arguing for both (IND 1) and (IND 2) would get us full normative independence, which means the strongest available form of NAP—call it strong NAP. Otherwise we could argue for only one direction of normative independence, and we would get two possible moderate forms of NAP—the first endorses (IND 1) without endorsing (IND 2), while the second endorses (IND 2) without endorsing (IND 1).32 Since the relevant direction of normative independence that we need in order to deal with the variability illustrated in the examples above is that expressed by (IND 1), in what follows I will argue for (IND 1) only, and I will remain silent on (IND 2). Moreover, to provide a full defence of the relevant moderate version of NAP we need an argument to show that (IND 2) fails—that is, to show that we have at least one of the following directions of normative dependence:
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(DEP 1) deontic⇒axiological (DEP 2) deontic⇒criterial (DEP 3) axiological⇒criterial Even though I believe that these normative dependences are plausible, I won’t argue for them. What I will do in the next section is to defend a minimal version of NAP that is sufficient to solve the impasse generated by the normative version of Lynch’s scope problem. Such a minimal version can then be elaborated either in the direction of strong NAP or in that of moderate NAP.
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Minimal Normative Alethic Pluralism
To argue for minimal NAP we need to establish the following two claims: (i) that the presence of the criterial dimension by itself does not enforce either the axiological dimension or the deontic dimension and (ii) that the axiological dimension by itself does not enforce the deontic one. Let’s discuss the first claim first. The criterial dimension provides us with a standard for categorising judgements as correct or incorrect in accordance with the truth-value of their propositional content. There is an ongoing debate about how to interpret the correctness norm and in what sense it is normative33—whether in deontic terms, such as ‘ought’, or in what I call axiological terms, such as ‘good’, ‘bad’, and ‘valuable’. I reject both interpretations. More precisely, I do not think that we are forced to adopt either interpretation. I follow McHugh34 here in understanding correctness as a distinct normative property in its own right, as neither deontic nor axiological. This thesis presupposes that the category of the normative should be conceived—as Thomson (2008) suggests—as more variegated than philosophers have traditionally sustained. Taking this on board, it may be argued that for a judgement to be correct is for it to have a normative property. But it is not correct because one ought to hold it, or because it would be good if one held it. Correctness is distinct from these other normative properties. For one thing, quite intuitively, while tipping might be the correct thing to do in a certain restaurant, provided that you have received adequate service, it’s not something
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obligatory—thus refraining from tipping would be somehow incorrect, given the circumstances, but not impermissible. In this respect, the fact that it is correct to judge that p does not imply that the subject ought to judge that p. Moreover, something might be a correct thing to do, because, for example, it is within your rights to do so, without this implying that it is also a valuable or good thing. The correct way of setting the table is to put the knife to the right of the plate and the fork on the opposite side, but intuitively there’s nothing valuable in setting the table that way.35 In this respect, the criterial dimension does not entail the axiological one. In general, there should not be any deep difficulty in the thought that something could be correct without being valuable or obligatory, or incorrect without being disvaluable or impermissible. However, further inquiry is needed into what specific kind of sui generis normative notion correctness may be.36 Let us now turn to a discussion of the second normative independence claim—that is, that the presence of the axiological dimension by itself does not enforce the presence of the deontic one. The axiological dimension tells us that it is, at least pro tanto, good, or valuable, to judge according to what is true, and that it is, at least pro tanto, bad, or disvaluable, to judge falsely. Again, how to understand the axiological dimension is a much-discussed topic.37 However, for the purpose of this chapter, we can leave the many, undoubtedly important, issues raised by that debate aside, and focus on the question of whether the axiological dimension per se entails the deontic dimension. There are reasons for thinking that it does not. As McHugh points out Something may be bad without its badness being a matter of anyone’s having done anything they ought not have done, and without its being the case that there is anyone who ought to change it; some prospective state of affairs or object may be good without its being the case that there is anyone who ought to produce it or bring it about.38
Thus, a false judgement may be bad or disvaluable without its badness or disvalue being a matter of someone having done anything they ought not to have done in judging so. Analogously, a true judgement may be good or valuable without its being the case that anyone ought to judge that way.
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This thought can be substantiated by drawing a parallel between the moral case and the alethic case by means of the notions of supererogation and suberogation. In ethics, a supererogatory act is generally conceived as an act that goes beyond the call of duty and brings about a considerable moral good. As Chisholm puts it, supererogatory acts are “non-obligatory well-doing”.39 By contrast, following Chisholm’s characterisation, we can say that suberogatory acts are “non-forbidden ill-doing”, or, in Driver’s words “suberogatory acts are bad to do, but not forbidden”.40 Heroic acts are typical examples of the supererogatory: if I sacrifice my life in order to save a group of children who would otherwise die, this would be considered extremely good and valuable, but clearly non-obligatory—even if sacrificing my life is the only way to save those children. As an example of a suberogatory act, consider the following scenario. Julie has two perfectly healthy kidneys, but she refuses to donate one kidney to her sister Jill, who desperately needs one. Julie’s refusal would be clearly suberogatory. Since Julie is under no obligation to donate her kidney, her refusal is not an impermissible thing to do. However, we would certainly consider such a refusal morally disvaluable. This reflects the widespread thought that it would be better if Julia donated her kidney to Jill. What the categories of the supererogatory and suberogatory show is that there are acts that are intuitively morally optional but still open to an axiological assessment as to whether they are good or bad. In the alethic case, we may say that an alethically supererogatory judgement—understood, remember, as a cognitive mental act—is a good judgement to make, because true, but not obligatory, while an alethically suberogatory judgement would be a judgement that is bad in that it violates the axiological dimension of the normative function of truth without thereby being a judgement that ought not to be performed.41 Unless we can find an argument to show that the concept of alethic suberogation is incoherent we should remain open to the possibility that in some areas of discourse there are cases of suberogatory judgements. This means that we should be wary of assuming that compliance with or violation of the axiological dimension automatically engenders compliance with or violation of the deontic dimension. As a consequence, we have the idea that truth’s normative function in some domain may encompass the axiological dimension while at the same time being deontically silent.
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No doubt, more need to be said to have an exhaustive picture of how these dimensions of truth’s normative role are related to each other. However, I take the remarks that I have just offered as sufficient to establish the reasonableness of the partial normative independence claim (IND 1). This granted, we have at least the following four NPs governing truth’s normative function: (NP-1) Truth exerts a deontic constraint on judgements—perhaps together with both an axiological and a criterial constraint if we endorse (DEP 1 and DEP 2). (NP-2) Truth exerts an axiological but not a deontic constraint on judgements—perhaps together with a criterial constraint if we endorse (DEP 3). (NP-3) Truth exerts only a criterial constraint on judgements. (NP-4) Truth exerts no normative constraint—besides its teleological role.
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ormative Alethic Pluralism N and the Normative Scope Problem
How does NAP help in dealing with the normative version of Lynch’s scope problem as illustrated in section “Normative Alethic Monism and the Normative Scope Problem”? The specific kind of negative normative assessment that I have in mind here can be understood by means of the notion of normative fault. The conjecture concerning the variable normative significance of disagreement discussed above can be reformulated as follows: the kind of fault that is attributable to someone holding a contrary view in the context of a disagreement might vary in relation to the subject matter of the disagreement. What does this variation in normative fault amount to? How should we model it? Once we take normative alethic pluralism on board, we are open to the idea that there is more than one dimension in which truth governs judgement. Correspondingly, we have a plurality of ways in which someone holding a view that is judged false might be said to be at fault. Thus, we have the following four categories of fault-attribution:
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deontic fault
In judging not-p the subject is judging in a way she ought not to. axiological fault In judging not-p the subject is doing something disvaluable. criterial fault In judging not-p the subject is judging incorrectly. no fault It is perfectly fine for the subject to judge the way she does. As we saw in the previous section, the various dimensions of truth’s normative profile are to a certain extent independent of each other. The various kinds of attributions of normative fault I have just introduced enjoy an analogous kind of independence. In particular, in line with the minimal version of NAP I am endorsing here, the three main categories of fault-attribution listed above are independent of each other in the following way: (IND 3) c riterial fault⇏axiological fault; criterial fault⇏deontic fault; axiological fault⇏deontic fault. Whether the other direction of independence also stands depends on whether we argue for a moderate or a strong version of NAP. I intend to remain neutral on this issue. Thus, we might say that one’s recognition that one disagrees with another person about certain subject matters licenses an attribution of criterial fault without licensing an attribution of axiological or deontic fault. Reflecting on the kind of negative assessment that seems appropriate in the disagreement between Julie and Jill about oysters, this could be the right prediction. Thus, the (NP-3) model of the normativity of truth seems to be the adequate one for the domain of basic taste. By contrast, disagreements about other matters might license an attribution of both criterial and axiological fault without licensing an attribution of deontic fault. And this might be what happens in typical cases of disagreement about refined aesthetic matters. Although it would seem appropriate for Julie to assess Jill’s contrary but equally wellinformed view as somehow inferior—and thus to attribute axiological fault—it would seem inappropriate for Julie to assess Jill’s contrary view
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as impermissible. Thus, no deontic fault seems to be forthcoming. In this respect, the (NP-2) model of the normativity of truth would be the adequate one for the domain of refined aesthetics. Lastly, some disagreements might license an attribution of all these kinds of fault. This seems to be what happens in disagreements about fundamental moral issues where both parties would deem it appropriate to assess the opposite party’s contrary view not only as incorrect and morally less valuable but also as one that ought not to be had. If this is right, then (NP-1) would be the right model for the normativity of truth in the fundamental moral domain. An important question remains as to whether sense can be made of the (NP-4) model of the normativity of truth—that is, whether we can have sensible disagreements involving judgements teleologically constrained by truth where no fault is involved. This ultimately depends on whether we can have a normatively inert truth property operating locally, in some areas of discourse.42
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Conclusions
I have discussed what I take to be an interesting variability in the normative significance of disagreement and I have suggested that such variability requires some flexibility in the normative function that truth exerts in different areas of discourse. Because a monistic view on truth’s normative function—what I have called normative alethic monism—forces us to adopt a Procrustean attitude towards the normative significance of disagreement, it is inadequate as a general model of the normativity of truth. I have argued for this by presenting a variation of what is known in the truth pluralism literature as the scope problem. I have outlined a pluralist framework for understanding the normativity of truth—normative alethic pluralism—that promises to score better than NAM in addressing the variability in the normative significance of disagreement discussed above. The key point of my proposal is to understand variation in normative fault by looking at which dimensions of truth’s normative profile operate in a given domain of discourse. Once we adopt this pluralistic stance towards the normativity of truth, we obtain
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a framework which adequately models the variability in the normative significance of disagreement across different areas of discourse. In other words, NAP provides us with all the necessary tools for addressing the normative version of the scope problem.43
Notes 1. For example: Dummett 1959; Gibbard 2005; Horwich 2013; Shah and Velleman 2005; Wedgwood 2007; Wright 1992. 2. Since the label can be misleading in one important respect, I should clarify that with it I mean a pluralist account of the normative function that truth plays in relation to judgements. It is not part of the proposal to claim that this pluralist account requires or entails a pluralistic account of the nature of truth—although, the two views, taken together, gives a highly coherent and neat package. 3. See Ferrari (2016b) for an account of the normativity of truth—especially of what I call the axiological dimension of the normativity of truth (see below, section “The Truth-Norm”)—which is compatible with the minimalist conception of truth advocated by Horwich (in, e.g., Horwich 1998). For further discussion of value and Horwich’s minimalism, see Ferrari 2018. 4. Horwich 2013: 17. 5. James 1975: 42. 6. Loewer 1993: 266. 7. Lynch 2013: 24. 8. McHugh 2014: 177. 9. Wedgwood 2007. 10. Williams 1973: 136. 11. Shah and Velleman 2005: 503. 12. On this, see Chrisman 2016 and Sosa 2015, part III. 13. I focus on judgement rather than belief (as all the authors mentioned above do) to avoid intricate issues concerning the doxastic voluntarism versus non-voluntarism debate. 14. Singularity resembles Williamson’s simple account, but without the constitutivist element. See Williamson 2000: 240. 15. See Williams 2012 for an account of normative silence. 16. MacFarlane 2014: Chap. 6.
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17. Where ‘incompatibility’ here is understood in semantic terms, either as contrariness or contradictoriness of the propositional contents involved. 18. In the sense discussed by Strawson 1962. 19. NAP can be integrated into a realist conception of morality and aesthetics and help in getting a handle on the difference in normative significance between these kinds of disagreement and disagreement about some other factual matters. 20. This label has been suggested to me by Crispin Wright (personal conversation). 21. I assume, contra contextualists and expressivists, that even in the case of basic taste we can make sense of disagreement in terms of doxastic noncotenability. Defending this claim would take me too far away—but see MacFarlane 2014 and Ferrari and Wright 2017 for some arguments against a contextualist and expressivist treatment of basic taste judgements. 22. Here by typical circumstances I mean circumstances in which no appreciable defeater—for example, one of the parties being under anaesthetics or her gustatory sensibility being temporarily impaired because of the effect of a strong cough syrup that alters her taste—is in place. 23. E.g. a situation in which Julie and Jill have to decide whether to take their best friend to a French bistro. 24. A discussion of this point can be found in Ferrari and Wright 2017. Wyatt 2018, pp. 263–5 makes a similar prediction about first-person qualifications in cases of disagreement about basic taste. 25. See Ferrari 2016a for a more detailed discussion of the comparison between disagreement in basic taste and disagreement in refined aesthetics. 26. See Lynch 2009: 34–36. 27. A classical objection to coherentist accounts of truth can be found in Wright 1998. 28. If, for example, you endorse a constructivist account of mathematics. 29. It is important to highlight that the legitimacy of having a certain reactive attitude does not entail the legitimacy of expressing that attitude. There might be reasons (e.g. prudential, moral, or other kinds of contextual factors), that are independent of the norms governing judgements, that would make the expression of my reactive attitude inappropriate even though it would be legitimate for me to have such an attitude.
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30. The cogency of this option depends on whether we can make sense of a purely non-normative notion of truth. This is a debated issue among philosophers working on truth: see, for instance, Dummett 1959; Wright 1992; Lynch 2009; Wrenn 2015. For some replies, see, e.g., Horwich 1998; Ferrari 2016b; Ferrari and Moruzzi (2018). 31. ‘⇏‘should be read as: ‘does not enforce…’. 32. It is helpful to point out that the views that I call ‘strong NAP’ and ‘moderate NAP’ are quite different from the views that are often called ‘strong alethic pluralism’ and ‘moderate alethic pluralism’ in the alethic pluralism debate. Two remarks are especially relevant on this: first, that strong NAP doesn’t entail strong alethic pluralism and, second, that moderate NAP doesn’t entail moderate alethic pluralism. 33. See Thomson 2008 for a discussion of the various kinds of normativity in relation to judgements. 34. See McHugh 2014 and McHugh and Way 2015. 35. There is a similar, and familiar, contrast in the normative ethics debate: one might think that right actions are those that maximise utility but ask what is good about doing what is right. 36. McHugh (2014: 177) suggests that we should understand correctness in terms of fittingness: ‘the attitude of belief sets truth as the standard that a proposition must meet in order for it to be a fit object of that attitude […] For an attitude to be fitting is for it to have a normative property. But it is not fitting because you ought to hold it, or because you may hold it, or because it would be good if you held it. Fittingness, I maintain, is distinct from these other normative properties’. 37. See, for instance, David 2005; Hazlett 2013; Kvanvig 2003; Lynch 2005; McHugh 2012. 38. McHugh 2012: 10. 39. Chisholm 1963: 3. 40. Driver 1992: 286. 41. Turri first applied the category of the suberogatory to the case of the normativity of assertions, but differently, and with different aims; see Turri 2013. 42. Ferrari and Moruzzi (2018). 43. This paper has enormously benefitted from discussions with Elke Brendel, Matthew Chrisman, Massimo Dell’Utri, Douglas Edwards, Matti Eklund, Andreas Fjellstad, Patrick Greenough, Thomas Grundmann, Paul Horwich, Nathan Kellen, Michael Lynch, Giacomo Melis, Anne Meylan, Moritz Müller, Carol Rovane, Andrea Sereni, Erik
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Stei, Elena Tassoni, Joe Ulatowski, Giorgio Volpe, Jack Woods, Chase Wrenn, Cory Wright, Jeremy Wyatt, Luca Zanetti, Dan Zeman. Special thanks are due to Carrie Ichikawa Jenkins, Sebastiano Moruzzi, Nikolaj J. L. L. Pedersen, Eva Picardi and Crispin Wright. Moreover, I would like to acknowledge the generous support of the Deutsche Forschungsgemeinschaft (DFG—BR 1978/3–1) for sponsoring my postdoctoral fellowship at the University of Bonn. While working on this paper, I benefitted from participation in the Pluralisms Global Research Network (National Research Foundation of Korea grant no. 2013S1A2A2035514). This support is also gratefully acknowledged.
References Chisholm, R. 1963. Supererogation and Offense: A Conceptual Scheme for. Ethics. Ratio 5: 1–14. Chrisman, M. 2016. Epistemic Normativity and Cognitive Agency. Noûs. Doihttps://doi.org/10.1111/nous.12184. David, M. 2005. On Truth Is Good. Philosophical Books 46 (4): 292–301. Driver, J. 1992. The Suberogatory. Australasian Journal of Philosophy 70 (3): 286–295. Dummett, M. 1959. Truth. Proceeding of the Aristotelian Society 59: 141–162. Ferrari, F. 2016a. Disagreement About Taste and Alethic Suberogation. The Philosophical Quarterly 66 (264): 516–535. ———. 2016b. The Value of Minimalist Truth. Synthese. Doi https://doi. org/10.1007/s11229-016-1207-9. ———. 2018. The Value of Minimalist Truth. Synthese 195 (3): 1103–1125. https://doi.org/10.1007/s11229-016-1207-9. Ferrari, F., and S. Moruzzi. 2018. Ecumenical Alethic Pluralism. Canadian Journal of Philosophy. https://doi.org/10.1080/00455091.2018.1493880. Ferrari, F., and C. Wright. 2017. Talking with Vultures. Mind 126 (503): 911–936. Gibbard, A. 2005. Truth and Correct Belief. Philosophical Issues 15: 338–350. Hazlett, A. 2013. A Luxury of the Understanding: On the Value of True Belief. Oxford: Oxford University Press. Horwich, P. 1998. Truth. 2nd ed. Oxford: Oxford University Press. ———. 2013. Belief-Truth Norms. In The Aim of Belief, ed. T. Chan, 17–31. Oxford: Oxford University Press.
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James, W. 1975. Pragmatism and the Meaning of Truth. Cambridge MA: Harvard University Press. Kvanvig, J. 2003. The Value of Knowledge and the Pursuit of Understanding. Cambridge: Cambridge University Press. Loewer, B. 1993. The Value of Truth. Philosophical Issues 4: 265–280. Lynch, M. 2005. True to Life. Cambridge MA: The MIT Press. ———. 2009. Truth as One and Many. Oxford: Oxford University Press. ———. 2013. Three Questions for Truth Pluralism. In Truth and Pluralism, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 21–41. Oxford: Oxford University Press. MacFarlane, J. 2014. Assessment Sensitivity: Relative Truth and Its Applications. Oxford: Oxford University Press. McHugh, C. 2012. The Truth Norm of Belief. Pacific Philosophical Quarterly 93: 8–30. ———. 2014. Fitting Belief. Proceedings of the Aristotelian Society 114 (2): 167–187. McHugh, C., and J. Way. 2015. Fittingness First. Ethics 126 (3): 575–606. Shah, N., and D. Velleman. 2005. Doxastic Deliberation. Philosophical Review 114 (4): 497–534. Sosa, E. 2015. Judgement and Agency. Oxford: Oxford University Press. Strawson, P. 1962. Freedom and Resentment. Proceedings of the British Academy 48: 1–25. Thomson, J.J. 2008. Normativity. La Salle: Open Court. Turri, J. 2013. Knowledge and Suberogatory Assertion. Philosophical Studies 3: 1–11. Wedgwood, R. 2007. The Nature of Normativity. Oxford: Oxford University Press. Williams, B. 1973. Problems of the Self. Cambridge: Cambridge University Press. Williams, R. 2012. Indeterminacy and Normative Silence. Analysis 72 (2): 217–225. Williamson, T. 2000. Knowledge and Its Limits. Oxford: Oxford University Press. Wrenn, C. 2015. Truth. Cambridge: Polity Press. Wright, C. 1992. Truth and Objectivity. Cambridge, MA: Harvard University Press. ———. 1998. Truth: A Traditional Debate Reviewed. Canadian Journal of Philosophy Supplementary 28: 31–74. Wyatt, J. 2018. Absolutely Tasty: An Examination of Predicates of Personal Taste and Faultless Disagreement. Inquiry 61 (3): 252–280.
Truth in English and Elsewhere: An Empirically-Informed Functionalism Jeremy Wyatt
We should reconcile ourselves with the fact that we are confronted, not with one concept, but with several different concepts[.] [W]e should try to make these concepts as clear as possible (by means of definition, or of an axiomatic procedure, or in some other way); to avoid further confusions, we should agree to use different terms for different concepts; and then we may proceed to a quiet and systematic study of all concepts involved, which will exhibit their main properties and mutual relations. (Tarski 1944, p. 355)
This chapter explores the future of functionalism about truth. I’ll aim to defend functionalism while also explaining why and how functionalists should rely on empirical evidence regarding ordinary thought about truth. I’ll begin by sketching the functionalist framework and outlining some of its signature virtues. Next, I’ll raise what I take to be the most serious problem with functionalism—that although it entails empirical claims regarding ordinary thought about truth, its main proponent Michael Lynch has offered no empirical evidence in support of the view. J. Wyatt (*) Underwood International College, Yonsei University, Incheon, South Korea © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_8
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I’ll then venture to patch this lacuna by discussing some extant empirical data on ordinary thought about truth. These data, while inconclusive and in need of supplementation, support two underinvestigated pluralist views of ordinary thought about truth. Once we integrate these pluralist approaches with the functionalist framework, we get a much more nuanced view that is appropriately sensitive to relevant empirical findings. To close my defense, I’ll take a look at a second, pressing objection to functionalism that has been advanced by Cory Wright (2010) and explain why it doesn’t point to any serious flaws in the view. My overall conclusion, then, will be that functionalism is a distinctive and promising approach to the study of truth that can withstand philosophical scrutiny and cleanly integrate empirical inquiry into our ongoing search for the nature of truth.
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Truth: Concept and Property
The ur-question of truth theory is, of course, ‘What is truth?’ In contemporary work on truth, a distinction is often drawn between the ordinary, or folk concept truth and the property (or relation) truth that is allegedly represented by this concept.1 The folk concept truth is meant to be a concept that we employ prior to receiving any philosophical training and that professional philosophers tend to use when not thinking about philosophical issues pertaining to truth.2 William Alston (2002, p. 11) provides a paradigmatic motivation for drawing the concept-property distinction in truth theory, noting that3: [A] property might have various features not reflected in our concept of that property. To choose a well worn example, heat (the property of a physical object’s being more or less hot) is revealed by physics to be an average kinetic energy of constituent molecules, even though our ordinary concept of heat involves no such component (that’s not the way we ordinarily identify heat).
Alston’s insight is that since truth is a folk concept, it may fail in some respects to adequately represent truth. Analogies are especially helpful here, as Alston’s own analogy with heat shows. A similar and familiar
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case is that of jade. We ordinarily conceive of jade as being a single, typically green kind of mineral that is, for example, often used in Asian art. Yet while it is true that many familiar varieties of jade are green and that jade is often used in Asian art, chemists have discovered that there are actually two kinds of mineral—jadeite and nephrite—that we ordinarily call ‘jade’. This means that although our folk concept jade does provide a partially accurate representation of the nature of jade, there is more to the nature of jade than what it represents. As a result, when truth theorists set about their business, they actually confront not the single, traditional question ‘What is truth?’ but rather the following series of comparatively fine-grained questions: (i) What is the nature of the folk concept truth? (ii) What is the nature of the property truth that it allegedly represents? (iii) How might inquiry about truth inform inquiry about truth?
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Alethic Functionalism
The Framework The most developed response to these questions comes from Michael Lynch (2000, 2001, 2004a, b, 2005, 2009, 2013), who advocates functionalism about truth, or alethic functionalism. Alethic functionalism is inspired by the functionalist theories of mind and morality, respectively, championed by David Lewis (1966, 1970, 1972, 1994) and Frank Jackson (1998). Like these theories, alethic functionalism consists of two stages that involve the method of Ramsification.4 The first stage of alethic functionalism is the Conceptual Stage. The aim here is to offer an account of the folk concept truth. We begin by amassing a set S of sentences that contain ‘true’ or ‘truth’ (or both). The sentences in S must express propositions p1,…,pn that all and only the possessors of truth are disposed to believe upon reflection. Following Lynch, we can call p1,…,pn the folk theory FT of truth. It is worth emphasizing that according to the functionalist, possession of truth doesn’t require that one explicitly believe p1,…,pn. What is required, rather, is a
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disposition to believe p1,…,pn, provided that one has the concepts that one needs to understand these propositions and has also reflected on the propositions long enough to understand them. To illustrate, Lynch takes FT to include the propositions expressed by the following sentences5: (Objectivity) A belief is true iff with respect to that belief, things are as they are believed to be. (Norm of Belief ) It is prima facie correct to hold a belief iff that belief is true. (End of Inquiry) Other things being equal, holding true beliefs is a worthy goal of inquiry. The second stage of alethic functionalism is the Metaphysical Stage. The ambition here is to determine whether any actual properties have the features that are ascribed to truth in FT. To clarify what these features are, we construct the Ramsey sentence of FT. The first step is to regiment the grammar of the sentences that pertain to FT. In particular, we replace every instance of ‘true’ with ‘truth’ and adjust the sentences’ grammar appropriately. We then conjoin them to deliver the postulate PT of FT: (PT) A belief exemplifies truth iff with respect to that belief, things are as they are believed to be and it is prima facie correct to hold a belief iff that belief exemplifies truth and other things being equal, holding beliefs that exemplify truth is a worthy goal of inquiry. Next, we replace every instance of ‘truth’ in PT with an objectual variable y and we bind these variables with an existential quantifier. This delivers the Ramsey sentence RT of FT, which has the following form: (RT) There is a y (a belief exemplifies y iff with respect to that belief, things are as they are believed to be and it is prima facie correct to hold a belief iff that belief exemplifies y and other things being equal, holding beliefs that exemplify y is a worthy goal of inquiry).
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The final step is to determine whether any actual properties witness RT. If we discover that exactly one property does so, then we say that the folk theory FT is uniquely realized. In this case, we can simply call the realizing property truth. Accordingly, call this possibility monism. However, the functionalist also leaves room for two further possibilities. It may be that many properties witness RT; in this case, we say that FT is multiply realized, or that we’ve confirmed a certain sort of pluralism about truth. The third possibility is that upon investigation, we find that no property witnesses RT. In this case, we say that FT is unrealized, that is, that no actual property is represented by the folk concept truth. Call this result deflationism.6 Figure 1 summarizes the Conceptual and Metaphysical Stages of functionalism.7 One attractive feature of functionalism is that it incorporates Ramsification, which is a precise and well-understood method.8 It also comports with Alston’s insight by allowing for the possibility that the representation provided by the folk concept truth is partially accurate, even as there is more to say about the metaphysics of truth than what is articulated in this concept. If FT is uniquely or multiply realized, then the properties that realize it may have additional features that aren’t mentioned in FT. On the other hand, if FT is unrealized, then that, too, will probably come as a surprise to many possessors of the folk concept truth, insofar as they presumably take some property in the world to answer to their thought and talk about truth. This, then, would also be a sense in which the truth about the nature of truth diverges from the representation afforded by truth. Conceptual Stage
Metaphysical Stage
Identify the folk theory FT of truth
Construct the Ramsey sentence RT of FT Determine whether FT is uniquely realized, multiply realized, or unrealized
Monism
Fig. 1 Alethic functionalism
Pluralism
Deflationism
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In connection with this second virtue, I would emphasize a point that is sometimes overlooked in discussions of alethic functionalism. This is that functionalism itself is completely neutral regarding the metaphysics of truth, apart from the constraints imposed by FT. In principle, inquiry along functionalist lines could lead us to accept a monist, pluralist, or deflationary account of truth’s nature, and it’s clear that these metaphysics of truth are strikingly different. Rather than being regarded as a metaphysics of truth, then, functionalism is best described as a method for inquiring about truth that cleanly articulates a way in which conceptual inquiry about truth can inform metaphysical inquiry about truth.9
Functionalism’s Empirical Debt We’ve seen, then, that functionalism enjoys some significant virtues, and this is all to the good. However, I want now to describe what I take to be functionalism’s most serious shortcoming and to then explain how we should remedy it. The functionalist’s main ambition at the Conceptual Stage is to offer an account of the folk concept truth. Their driving idea is that this concept is underwritten by the folk theory FT of truth, and Lynch articulates a view as to which propositions comprise FT. We should now ask: what evidence do we have that FT in fact consists of the propositions expressed by (Objectivity), (Norm of Belief ), and (End of Inquiry)? Lynch offers two lines of evidence in support of his view on FT. One is that he suspects that the propositions at issue are among our most fundamental (implicit) beliefs about truth.10 The other is that certain influential philosophers such as Aristotle, James, and Peirce have taken these propositions to be central truths about truth.11 The basic problem is that these lines of evidence are inadequate in principle. Lynch’s view about FT is empirical—it is a view about how ordinary thinkers are, in fact, disposed to think about truth. However, we have little reason to believe that philosophers enjoy special insight into this issue. When we receive our professional training, we no doubt acquire a range of new abilities, but we don’t tend to spend much time in graduate school systematically examining how untrained, native speakers think
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about philosophically relevant issues. Accordingly, Lynch’s suspicion that we tend to believe the propositions expressed by (Objectivity), (Norm of Belief ), and (End of Inquiry) provides only very weak evidence that we actually tend to do so. Similarly, we have little reason to believe a priori that influential philosophers’ views about truth will closely align with those of ordinary thinkers. For this reason, the fact that Aristotle, James, and Peirce believed the propositions at issue provides only very weak evidence that ordinary thinkers are disposed to do the same. Functionalism, then, incurs an empirical debt that Lynch’s lines of evidence simply fail to settle. The good news, however, is that the functionalist can settle their empirical debt, provided that they are willing to do the proper sort of work. This work involves gathering empirical data on ordinary thought about truth and then using this data at the Conceptual Stage. In light of the results at the Conceptual Stage, the functionalist will then need to re-examine the options that are available at the Metaphysical Stage. In the remainder of the chapter, I want to outline some of the data that we currently have on ordinary thought about truth and to indicate what lessons they may hold. I’ll then explain how all of this affects the functionalist framework. To anticipate, I take the main upshot for functionalism to be that once we appreciate the trajectory along which our current data are leading us, we are able to glimpse some previously obscured options at the Metaphysical Stage.
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Data and Conceptual Pluralism
Interlinguistic and Intralinguistic Conceptual Pluralism As we’ll see, our current data point toward a notable degree of complexity in ordinary thought about truth. Accordingly, it will be useful to clearly articulate a pluralist view of ordinary thought about truth that has two main features. First, it should at least be supported by many of our data. Second, such a view should be able to serve as a guiding hypothesis in future studies of ordinary thought about truth.
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The basic thesis behind what we’ll call conceptual pluralism about truth is that there is more than one actual, folk truth concept. Notice, then, that conceptual pluralism isn’t the claim that there could be thinkers who use a truth concept that differs from the one used by, say, contemporary English speakers. Rather, it is a claim about the actual world to the effect that it contains thinkers some of whom use a different truth concept than do others. When we evaluate conceptual pluralism empirically, the following basic procedure should serve us well.12 We begin by examining ordinary speakers’ usage of ‘true’ and ‘truth’, as well as words in other languages that can be directly translated using ‘true’ or ‘truth’. Call any such word a piece of alethic vocabulary. If we find significant differences in speakers’ usage of alethic vocabulary, then we should go on to determine whether some speakers associate different concepts with these words than others do. Any concepts that we discover we’ll provisionally call truth concepts.13 We can distinguish between two kinds of conceptual pluralism. The first is what we’ll call interlinguistic conceptual pluralism: (Inter) There are at least two actual linguistic communities L1 and L2 such that some L1-members use a truth concept T1, whereas some L2- members use a distinct truth concept T2. Accordingly, if we discover two linguistic communities L1 and L2 such that all L1-members use a truth concept T1 and all L2-members use a distinct truth concept T2, this will verify interlinguistic conceptual pluralism. However, the thing to notice is that we needn’t arrive at precisely this discovery in order to show that interlinguistic conceptual pluralism is true. Rather, all that is needed to verify this view is that there be a linguistic community L1 some (but not necessarily all) of whose members use a truth concept T1 in addition to another linguistic community L2 some (but not necessarily all) of whose members use a distinct truth concept T2. A second variety of conceptual pluralism is what we can call intralinguistic conceptual pluralism:
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L1
T2
T2
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T3
L2
Fig. 2 Consistency of interlinguistic and intralinguistic pluralism
(Intra) There is at least one actual linguistic community L such that some L-members use a truth concept T1, whereas other L-members use a distinct truth concept T2. We can make one preliminary observation about the logical relationship between interlinguistic and intralinguistic pluralism—namely, that they are mutually consistent. Figure 2 brings this fact out by detailing two linguistic communities L1 and L2 that we might observe upon investigation. L1 and L2 individually confirm intralinguistic pluralism and they jointly confirm interlinguistic pluralism. In the rest of this section, I’ll be arguing that both intralinguistic and interlinguistic pluralism enjoy notable support from our existing data. As matters stand, the proper assessment is that these data are suggestive, though quite far from conclusive. This means that further study is needed before we can be very confident in any view of ordinary thought about truth. However, what I hope to bring out is that both intralinguistic and interlinguistic pluralism are very live theoretical options.
Intralinguistic Pluralism: Correspondence and Gender In discussing intralinguistic pluralism, I’ll be drawing on a recent study by Robert Barnard and Joseph Ulatowski (2013).14 Barnard and Ulatowski sought to investigate the extent to which their subjects agreed with what they call the correspondence root15:
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(CR) Truth involves agreement, copying, or correspondence between what (in the mind or in language) is true and what (in the mind or in the world) makes it true. They elicited responses from 200 English-speaking subjects to one of nine vignettes. To assess their subjects’ agreement with (CR), Barnard and Ulatowski presented them with a probe statement that is meant to be similar to (CR) but framed in non-technical language: (1) If a claim reports how the world is, then it is true. It is worth pausing at this stage to note that there is a significant bug in Barnard and Ulatowski’s experimental design. Insofar as it states that truth involves agreement, copying, or correspondence, (CR) suggests that correspondence is a necessary condition for truth. By contrast, the probe statement (1) states that ‘reporting how the world is’ is a sufficient condition for truth. As a result, the reactions of Barnard and Ulatowski’s subjects to (1) wouldn’t seem to provide evidence as to the attitudes that they would take toward (CR). Rather, Barnard and Ulatowski’s data look to provide insight into how their subjects would think about the converse of (CR), which we might call the modified correspondence root: (MCR) If a thought or claim copies, agrees with, or corresponds to what (in the mind or in the world) would make it true, then that thought or claim is true. When evaluating Barnard and Ulatowski’s findings in what follows, then, we will take their findings to concern (MCR), rather than (CR). Future studies could profitably investigate subjects’ attitudes toward (CR) by using the converse of (1), rather than (1), as a probe statement. Two of Barnard and Ulatowski’s findings involve the Bruno case and the Donna case16: Bruno case: Bruno has just finished painting his house. Bruno painted his house the same color as the sky on a clear summer day. Bruno claims his house is blue.
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Donna case: Donna is traveling in Germany, but does not speak German. She watches as a sailor asks for der Stadtplan and is handed what looks like a map of the city. Donna asks for der Stadtplan in a shop and is sold a city map. Donna still speaks no German, but believes that asking for der Stadtplan is a good way to obtain a city map from a German shopkeeper. In the Bruno case, Bruno makes an ordinary, observational claim. Similarly, in the Donna case, Donna forms a belief about an ordinary, practical issue. Accordingly, Barnard and Ulatowski hypothesized that these cases would elicit high degrees of agreement with (1) among their subjects.17 However, what they found was that when presented with these cases, male subjects expressed stronger agreement with (1) than female subjects. In particular, when presented with the Donna case, male subjects tended to agree with (1), whereas female subjects tended to disagree with (1). Using a 5-point Likert scale with a midpoint of 3, Barnard and Ulatowski found that the mean male and female responses to the Bruno case were 3.96 and 3.07, respectively. Likewise, the mean male and female responses to the Donna case were 3.56 and 2.60, respectively.18 These differences are statistically significant (Bruno: N = 41, t(39) = 2.305, p < 0.027; Donna: N = 40 (25 male, 15 female), t(37.701) = 2.521, p < 0.016). Pending further study of English speakers’ thought about truth, no single interpretation of Barnard and Ulatowski’s findings is clearly superior to its competitors. However, it is clear that their findings lend some support to intralinguistic pluralism. To explain their data along intralinguistic pluralist lines, we first hypothesize that there are at least two folk truth concepts TCL and TCA. To possess TCL, one must be disposed to accept (MCR) upon reflection (at least when thinking about certain topics). To possess TCA, by contrast, one need not be disposed to do so.19 Thus, we might call TCL a correspondence-leaning concept and TCA a correspondence-averse concept. The proffered explanation of Barnard and Ulatowski’s findings would then be that among English speakers, males are more likely than females to possess TCL, whereas females are more likely than males to possess TCA.
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Future studies should reevaluate this hypothesis by attempting to replicate Barnard and Ulatowski’s results among English speakers. They should also test similar hypotheses about other linguistic communities. Additionally, I would note that in evaluating the intralinguistic pluralist explanation of Barnard and Ulatowski’s findings, we should determine whether there are plausible explanations as to why male English speakers would be more likely to possess TCL, whereas female English speakers are more likely to possess TCA.
Interlinguistic Pluralism: English and Akan English Speakers on Correspondence Regarding interlinguistic pluralism, the first thing to say is that to my knowledge, there are no published, empirical studies that seek to evaluate this hypothesis.20 However, in a series of pioneering papers, Kwasi Wiredu (1985, 1987, 2004) brings to light two plausible conjectures that support interlinguistic pluralism. Wiredu’s conjectures concern English and the Ghanaian language Akan. Like Barnard and Ulatowski, Wiredu draws our attention to English and Akan speakers’ respective views on truth and correspondence. In doing so, he considers a claim that differs slightly from both (CR) and (MCR), which we can call the correspondence thesis: (CT) True propositions are those that correspond to facts. Wiredu’s first conjecture concerns English speakers’ attitudes toward (CT). Suppose that an ordinary English speaker were presented with the following attempt at a definition of truth: (2) True propositions are those that are true. It’s reasonable to hypothesize that most such speakers would recognize upon reflection that (2) is viciously circular, which would cause them to refrain from agreeing that (2) is an accurate definition of truth. By contrast, Wiredu would conjecture that adult English speakers will tend to
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evaluate (CT) differently, if asked to consider it as a definition of truth. His first conjecture is21: (WC1) Few (say, < 50% of ) native English speakers are disposed to believe upon reflection that (CT) is viciously circular. Notice, then, that (WC1) is not a conjecture about English speakers’ agreement or disagreement with (CT). Rather, it is a conjecture about the percentage of English speakers who are disposed to believe upon reflection that (CT) is viciously circular. Presumably, if one did believe that (CT) is viciously circular, then one would refrain from agreeing that it is an accurate definition of truth. However, what is worth observing is that an English speaker might refrain from believing that (CT) is viciously circular, even though they don’t accept it as an accurate definition of truth. This would be the position, for instance, of an English speaker who favors an alternative (e.g. relativist) definition of truth but grants that (CT) is a way that one might reasonably attempt to define truth. This shows that (WC1) doesn’t entail that most English speakers are disposed to believe upon reflection that (CT) is an accurate definition of truth (this, of course, is a virtue of (WC1), given the data from Barnard and Ulatowski (2013). (WC1) awaits direct empirical confirmation. However, it is weakly supported by two lines of empirical data, due respectively to David Bourget and David Chalmers and Barnard and Ulatowski. Bourget and Chalmers (2014) surveyed professional philosophers in 99 leading philosophy departments, most of which (89) are located in English-speaking countries.22 One of their survey questions was ‘Truth: correspondence, deflationary, or epistemic?’ 50.8% of their respondents (± 1.5%) indicated that they either accepted or leaned toward some sort of correspondence theory of truth (ibid. pp. 477, 498). This means that either a sizable minority or a slim majority of their respondents favored something like (CT) and would thus presumably take (CT) to be free from vicious circularity. In the present context, Bourget and Chalmers’ results are admittedly limited. One reason is that they didn’t explicitly use (CT) in their study. Another is that their target group consisted of professional philosophers,
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whereas (WC1) is a conjecture about native English speakers more generally. Despite these limitations, though, Bourget and Chalmers’ findings provide weak support for (WC1). Additionally, Barnard and Ulatowski sought in a recent study to examine the degree to which both professional philosophers and non- philosophers take truth to be objective. In doing so, they examined their subjects’ agreement and disagreement with various probe statements, one of which is: (3) When a claim is true, it expresses a fact. Barnard and Ulatowski (2017, Tables 3 and 5) found that 86% of their philosophers and 85.8% of their non-philosophers either agreed or strongly agreed with (3). It’s thus plausible that most of these subjects took (3) to be free from vicious circularity. These findings are also admittedly limited in the present context, given that (3)—in contrast to (CT)—doesn’t explicitly mention correspondence and wasn’t explicitly presented as a (partial) definition of truth. However, like Bourget and Chalmers’ findings, they also provide weak support for (WC1). In light of the limitations of the existing evidence that bears on (WC1), it would be especially helpful for future studies to directly examine the degree to which English speakers take (CT) to be viciously circular.
Akan Speakers on Correspondence When exploring English and Akan speakers’ views on truth and correspondence, Wiredu (1987, p. 29) also considers a direct translation of (CT) into Akan: (CTA) Asem no te saa kyerese ene nea ete saa di nsianim. In (CTA), ‘true’ and ‘fact’ are translated using the same Akan expression ‘ete saa’, which means it is so. Thus, as Wiredu points out, (CTA) can also be faithfully translated into English as ‘That a proposition is so amounts to its coinciding with what is so’. This observation leads Wiredu to offer the following insightful conjecture:
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This has the beauty of a tautology, but it teaches little wisdom. It seems to me unlikely that thinking in this language, one could be easily tempted into correspondence formulations of this sort. Indeed, in this language, it is pretty clear that the problem of truth must be the problem of clarifying the idea of something being so. (Ibid. Cp. 1985, p. 47; 2004, pp. 48–9)
We can thus render Wiredu’s second conjecture along the following lines: (WC2) Most (> 50% of ) native Akan speakers are disposed to believe upon reflection that (CTA) is viciously circular. Like (WC1), (WC2) awaits direct empirical confirmation. However, in their own investigations of Akan alethic vocabulary, J.T. Bedu-Addo (1985, pp. 70, 76–7) and Safro Kwame (2010) express sympathy with Wiredu’s contention that ‘true’ and ‘fact’ should be translated using a single Akan expression.23 We thus have strong preliminary evidence that (CTA)’s vicious circularity will be apparent to native Akan speakers. As a result, (WC2), like (WC1), is a plausible conjecture and should be empirically evaluated. (WC1) and (WC2) lend support to interlinguistic pluralism, which can explain them along the following lines. We first hypothesize that most English speakers possess some truth concept T1 as well as a distinct concept fact that is independent of T1 in the sense that it is definable without reference to T1. They deploy these concepts when interpreting (CT) and will accordingly take (CT) to be free from vicious circularity. By contrast, it seems that when they reflect on (CTA), most Akan speakers don’t deploy a truth concept as well as a distinct fact concept. Rather, they deploy a single concept being so that doubles as a truth concept and a fact concept. Since they deploy being so when interpreting (CTA), they will accordingly take (CTA) to be viciously circular. It then follows that T1 ≠ being so, which confirms interlinguistic pluralism. As a result, although additional empirical work is needed to fully assess (WC1) and (WC2), it is clear that interlinguistic pluralism promises a neat explanation of Wiredu’s conjectures.
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An Updated Functionalism
We’ve now seen that both intralinguistic and interlinguistic pluralism enjoy a good deal of plausibility. For this reason, it will be interesting to see how we can integrate them into the broader functionalist framework.24 In § 2, we saw that the standard functionalist approach consists of two stages—the Conceptual and Metaphysical Stages. In essence, the Conceptual and Metaphysical Stages now decompose into various sub- stages, which increases the number of options that are available at the Metaphysical Stage. Figure 3 offers a graphical depiction of this updated functionalist framework, which I’ll now briefly describe. Suppose that upon investigation, we discover that there are n actual, folk truth concepts T1,…,Tn, where n > 1. At conceptual sub-stage c1, we identify the folk theory FT1 of T1. We then move to the corresponding metaphysical sub-stage m1 and construct the Ramsey sentence RT1 for T1. Next, we determine whether FT1 is uniquely realized, multiply realized, or unrealized. We apply this procedure at all conceptual and metaphysical sub-stages c1,…,cn and m1,…,mn.
Conceptual Stages c1: Identify the folk theory FT1 of T1 . . . cn: Identify the folk theory FTn of Tn
Metaphysical Stages m1: Construct the Ramsey sentence RT1 of FT1 Determine whether FT1 is uniquely realized, multiply realized, or unrealized . . . mn: Construct the Ramsey sentence RTn of FTn Determine whether FTn is uniquely realized, multiply realized, or unrealized mf : Evaluate results at m1-mn
Monism
Fig. 3 Updated functionalism
Pluralism1 Pluralism2 Pluralism3 Deflationism
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Lastly, we want to evaluate the metaphysical results at m1,…,mn. Call the metaphysical sub-stage that we now occupy the final (pending further investigation) metaphysical sub-stage mf. Interestingly, at mf, we now have five basic possibilities: (a) Monism: All of the folk theories FT1,…,FTn are uniquely realized by the same property P; (b) Pluralism1: All of the folk theories are uniquely realized, but some are realized by a different property than others; (c) Pluralism2: Some, but not all, of the folk theories are multiply realized; (d) Pluralism3: All of the folk theories are multiply realized; and (e) Deflationism: All of the folk theories are unrealized. Options (a) and (e) are the most familiar, insofar as they respectively represent monism and a pure sort of deflationism. We can helpfully think of options (b)–(d) as representing increasingly complex grades of pluralism about the properties that realize the various folk theories. These grades of pluralism mark the basic degrees to which multiple realization can creep into our metaphysics of truth when we try to navigate the divide between monism and deflationism.25 What I would emphasize at this stage is that before we endorse any of these metaphysics of truth, we must do a great deal of further metaphysical—and empirical—work. My hope is that the updated functionalist framework will serve as a useful instrument for this work.
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Functionalism and Epistemic Circularity
Wright’s Criticism Thus far, my defense of alethic functionalism has mainly consisted of an effort to outline how the functionalist should pay off their considerable empirical debt. I want to wrap up this defense by looking at an objection to functionalism that comes from Cory Wright. This objection deserves extended reflection because it strikes at the conceptual bedrock of the
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functionalist program—namely, the functionalist’s use of Ramsification. Accordingly, if it succeeds, then all that I’ve said in defense of functionalism is essentially for naught. While Wright would presumably allow that theorists such as Lewis and Jackson can fruitfully use Ramsification to investigate the natures of mental states or moral rightness, he takes there to be a distinctive problem with using Ramsification to inquire about the nature of truth. Wright succinctly articulates his misgiving as follows26: Functionalists about truth implicitly define their (nominalized) τ-term true by using Ramsification to produce a sentence that indicates its denotation. But any implicit definition proceeds on the basis of explicit decisions that the principles constitutive of [the given theory] T are themselves true. Hence the circularity. In turn, making any explicit decisions that they are true requires already knowing in advance what truth is. Hence the epistemic circularity. If we suppose further that knowing in advance what truth is entails knowing what the τ-name truth denotes, then it becomes unclear why functionalists about truth ever required an implicit definition via Ramsification in the first place. Hence the problem.
To make our discussion of Wright’s challenge more fluid, we can simplify matters by supposing that the functionalist posits only one folk truth concept truth. Given the data from § 3, it is not obvious that this should be the functionalist’s actual stance. However, if the functionalist endorses conceptual pluralism, then a version of Wright’s objection will loom at each metaphysical sub-stage m1,…,mn, so the objection is pressing in either case. Wright takes functionalism to suffer from what he calls ‘epistemic circularity’. He points out that when the functionalist details the postulate PT at the Metaphysical Stage, it’s fair to think of them as attempting not only to clarify the folk concept truth, but also to implicitly define the word ‘truth’.27 The problem is supposed to arise from the alleged fact that in offering this implicit definition, the functionalist is claiming that they know that the conjuncts of PT are true. To know that the conjuncts of PT are true, Wright insists, the functionalist must have prior knowledge of the nature of the property truth.
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The question then becomes: how is the functionalist supposed to know what truth’s nature is? The functionalist aims to acquire knowledge of truth’s nature by constructing PT, using PT to construct the Ramsey sentence RT, and then determining which property witnesses RT. Yet if Wright is correct, then when they follow this procedure, the functionalist claims (implicitly) that they know that the conjuncts of PT are true. The core of the problem, then, is this: knowledge that PT’s conjuncts are true must, says Wright, be based on prior knowledge of truth’s nature—which, if the functionalist is correct, must be based on prior knowledge that PT’s conjuncts are true. It thus seems that in relying on Ramsification, the functionalist is claiming to have knowledge—that PT’s conjuncts are true—that they simply can’t acquire, given the strictures of their own view. To close, I’d like to mount a direct defense of functionalism against Wright’s worry. That is, I’ll aim to explain why functionalism isn’t viciously circular in the way that he suggests.28
Why Functionalism Isn’t Epistemically Circular For the sake of argument, I will suppose with Wright that the functionalist takes all of PT’s conjuncts to be true.29 We then pose Wright’s problem: to know that PT is true, mustn’t the functionalist already know the nature of the property that ‘truth’ denotes, that is, truth? But isn’t their view that our knowledge of truth’s nature is based on our prior knowledge of PT? If so, then it seems that to come to know that PT is true, they must already know that PT is true, which is impossible. The functionalist should dispel this impression of circularity by pointing out that they can know that PT is true without already knowing the nature of truth. This follows from a more general principle. Assuming that A is a competent user of the word ‘true:’30 (4) If A knows that p and they know that the content of sentence ‘S’ is that p, then A knows that ‘S’ is true. To see the plausibility of (4), suppose that a high school student Van looks at the whiteboard in his classroom and sees the sentence ‘The chemistry exam will begin at 1:30 p.m. today’. Van knows that the content of
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this sentence is that the chemistry exam will begin at 1:30 p.m. on that day. If he also knows that the exam will in fact begin at 1:30 p.m. on that day, then he knows that the sentence on the whiteboard is true. It’s simply not the case that to know that the sentence is true, Van must know what the nature of truth is. It’s enough that he’s a competent user of the word ‘true’ and thus understands how this word applies to English sentences. Let’s now think in the same vein about PT. Suppose that PT is the conjunction ‘S1 and S2 and S3’ and that the functionalist maintains that PT is true. We ask them: how do you know that PT is true? To explain this, the functionalist needs only point out (i) that the contents of its conjuncts are respectively that S1, that S2, and that S3 and (ii) that they know that that S1, that S2, and that S3. From (i), (ii), and (4), it follows that they know that PT is true. Consequently, it’s simply not the case that to acquire knowledge of PT, the functionalist must have antecedent metaphysical knowledge of truth’s nature. Like Van, what they need is an understanding of how ‘true’ applies to English sentences. They acquire this understanding simply in virtue of being a competent, mature English speaker—and thus prior to embarking on their analyses of truth and truth.31
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Conclusions
In this chapter, I hope to have conveyed why alethic functionalism promises to be a powerful tool for theorists studying the nature of truth. Functionalism delivers an insightful treatment of the concept-property distinction in truth theory and in doing so, sets our sights on the conspicuous value of empirically studying ordinary thought about truth. Despite considerable advances by experimental truth theorists, we’ve managed only to break ground on this avenue of inquiry. This means that we’ll need to expend a great deal of effort to see it to fruition—and in doing so, we should direct a sizable portion of our effort to the evaluation of conceptual pluralism. What awaits us as we pursue this line of inquiry is a sharper understanding of ordinary thought about truth, the nature of truth, and the interconnections between these topics. Whether the final
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metaphysics of truth is monist, pluralist, or deflationary, we will be able to confidently say that it is the product of clear-eyed metaphysical investigation disciplined by painstaking empirical research.32
Notes 1. In what follows, I’ll use italics to denote properties and relations and small caps to denote concepts. 2. Cf. Lynch (2009, p. 7). 3. For further illuminating reflections on this distinction, see Asay (2013, Chap. 1; 2018); Bar-On and Simmons (2007); Eklund (2017, § 2); and Lynch (2005, 2009, Chap. 1). 4. For a canonical account of Ramsification, see Lewis (1970). 5. (2009, Chap. 1). I’ve altered the wording of the sentences, though this won’t influence the concern that I’ll develop for functionalism in § 2. In making these alterations, I’ve been guided by the assiduous study of Lynch’s functionalism by Marian David (2013). 6. More specifically, this metaphysics of truth amounts to what Künne (2003, pp. 3–4, Chap. 2) calls “nihilism,” or what I prefer to call pure deflationism. Not all deflationists (e.g. Horwich 1998) are pure deflationists in this sense, but some (e.g. Quine (1948, 1970) and Strawson (1949, 1950)) have endorsed or would endorse the view. According to the taxonomy here, deflationists such as Horwich who commit to the existence of a single property truth should be classified as monists. Thus to fully flesh things out, we’d need to distinguish between deflationary monists and substantivist monists (and between deflationary and substantivist pluralists). Doing so requires a good bit of effort; for details, see Wyatt (2016). 7. As I’m using the expressions ‘pluralism about truth’ and ‘truth pluralism’, one may endorse what is often called ‘alethic pluralism’ without thereby endorsing truth pluralism. I take truth pluralism to be the view that more than one actual property realizes FT. In other words, truth pluralism is the view that more than one actual property fits the job description for truth—full stop, and hence not merely relative to some domain or other (on this distinction, see David (2013) and Edwards (2011, pp. 34–7)). Extant alethic pluralists—strong and moderate alike—thus stop short of endorsing truth pluralism. Strong alethic pluralists (e.g.
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Cotnoir (2013) and Kim and Pedersen, this volume) take no actual property to realize FT simpliciter and moderate alethic pluralists (e.g. Edwards (2011) and Lynch (2009)) take exactly one such property to do so. This, I would stress, isn’t a criticism of these views, only an observation about their structure. It’s also worth noting that truth pluralism, in the present sense, is structurally similar to Beall and Restall’s logical pluralism, insofar as the properties that would realize FT would do so with respect to all domains. 8. In his most recent work (2009, 2013), Lynch has elected to not rely explicitly on Ramsification, thereby departing from the strategy in his earlier, pathbreaking work on functionalism (2000, 2001, 2004b, 2005). However, I would point out that in his recent work, Lynch still speaks of properties playing the ‘truth role’. The truth role is supplied by PT and playing it looks to be nothing over and above witnessing RT, so it seems that Lynch is still loyal at heart to Ramsification, even though he chooses not to wear the badge. 9. Lynch (2009, p. 84; 2013, p. 27) and Wright (2005, n. 14) also allude to this point (see also Devlin (2003)). However, other authors are a bit too quick in assimilating functionalism to pluralism. I have in mind here the otherwise illuminating critical discussions by Caputo (2012); Horton and Poston (2012); Newhard (2013, 2014, 2017) and Wright (2005, §§ 4.1, 4.2). What I would point out is that the criticisms advanced by these authors affect the functionalist only if they commit to some sort of alethic pluralism—a move that Lynch does make, but which is nevertheless entirely optional. 10. Lynch (2004a, Chap. 1; 2009, p. 8). 11. Lynch (2009, pp. 10–14; 2013, p. 24). 12. Cp. Barnard and Ulatowski (2013, 2017); Fisher, et al. (2017); Kölbel (2008); and Mizumoto (ms). See especially Ulatowski (2017, pp. viii, ix, 2, 8–9). 13. I say ‘provisionally’ because it is at this point that we should look for underlying similarities among the concepts that speakers deploy when they use alethic vocabulary. If we detect such similarities, we should then—and only then—advance a general account of what makes a concept a truth concept. To do otherwise would amount to gratuitous theory building in the absence of sufficient data. 14. Moltmann’s investigations (2015, 2018) into truth predicates, and related kinds of predicate, in various natural languages also look to bear significantly on intralinguistic pluralism. The same goes for the work of
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Dzobo (1992, pp. 79–83) and Wiredu (1985, 1987, 2004) on the Ghanaian languages Ewe and Akan. 15. 2013, p. 621. 16. Ibid. p. 631. 17. Ibid. p. 633. 18. Ibid. Figs. 2 and 3. 19. I include the parenthetical qualification to flag a further sort of variance that Barnard and Ulatowski (ibid. Fig. 1) found among their subjects. They found that their subjects (male and female alike) were more likely overall to agree with (1) when presented with the Bruno case than when presented with a case involving a simple arithmetical calculation. We might call this kind of variance topic-sensitivity. What I would point out is that this topic-sensitivity is independent of intralinguistic pluralism, insofar as the former may be present across the community of English speakers. For further investigation of topic-sensitivity, see Ulatowski (2017). 20. That said, Mizumoto (ms) has gathered interesting data pertaining to Japanese and English that do look to support interlinguistic pluralism. I should also note the fascinating collection of papers surveyed by Maffie (2001). By contrast, Matthewson and Glougie (forthcoming) investigate interesting cross-linguistic uniformities in the use of alethic vocabulary. 21. I take (WC1) to be a conjecture that Wiredu would be willing to make, although he doesn’t explicitly advance it. He comes extremely close to doing so at 2004, pp. 47–9. Cp. 1985, pp. 47–8, 49–50; 1987, p. 28. 22. Bourget and Chalmers (2014, § 2) note that in principle, anyone was allowed to take their survey, though the target group about which they mainly report are the professional philosophers from the mentioned departments. 23. Bedu-Addo argues that the best Akan expression to use is ‘nokware’, a view with which Kwame looks to be sympathetic. 24. My thinking here has been influenced by some suggestive remarks due to Wright (2005, pp. 18–21). 25. The grades of pluralism in (b)–(d) are notably different from the mainstream pluralist truth theories that have been developed thus far. One major difference is that the pluralisms in (b)–(d) make no reference to the notion of a ‘domain’, which figures prominently in mainstream pluralist theories. Rather, these pluralist views integrate Ramsification with conceptual pluralism, an approach that hasn’t been attempted by mainstream alethic pluralists.
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26. (2010, p. 8). Cp. (2005, pp. 21–2). 27. Cp. Lewis (1970, p. 429). 28. In this way, my reply to Wright differs from that of Lynch (2013, n. 12), who grants that functionalism is viciously circular, but insists (less than convincingly, to my mind) that every other theory of truth is viciously circular in the same respect. I should also note that my reply on behalf of functionalism differs from the reply that Wright (2010, § 6) finds most convincing. I lack the space to examine this reply, but my basic concern is that it rests on a mischaracterization of the functionalist’s ambitions. 29. Admittedly, it seems to me that the functionalist need not do so. By way of analogy, think of a non-racist psychologist studying the race-related concepts used by ordinary, racist subjects. It would be consistent for them to analyze these concepts along functionalist lines while refraining from commitment to the truth of their postulates. It would seem that the alethic functionalist enjoys a similar sort of freedom when analyzing ordinary thought about truth. 30. (4) is grounded in (one direction of ) what is sometimes called the transparency of the word ‘true’, as that word applies to sentences. It should also be helpful to remember the importance of the concept-property distinction in truth theory, as discussed in §1. 31. There is admittedly a further issue that must be explored—namely, how the functionalist could know that S1, that S2, and that S3. Though I can’t pursue this issue at length here, it’s plausible that the functionalist can acquire this knowledge in a rather familiar way. They can do so by reflecting on cases that pertain e.g. to the objectivity of truth or to the connections between A’s belief being true and its being correct for A to hold that belief. Put a bit more generally, the functionalist could come to know that PT is true by competently deploying the truth concept for which PT is the postulate in reflecting on cases that pertain to PT’s conjuncts. 32. I am grateful to audiences at the University of Bologna, Hong Kong University, Texas Christian University, and Yonsei University for their feedback on this chapter. Those who have helped me in developing the chapter include Wes Cray, Filippo Ferrari, Max Deutsch, Richard Galvin, Will Gamester, Patrick Greenough, John Harris, Blake Hestir, Ole Hjortland, Michael Lynch, Kelly McCormick, Sebastiano Moruzzi, Shyam Nair, Francesco Orilia, Giorgio Volpe, and Crispin Wright. I owe
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particular thanks to Jamin Asay, Teresa Kouri, Nikolaj J. L. L. Pedersen, Joe Ulatowski, and Cory Wright. Also, while working on this paper, I received support from the National Research Foundation of Korea (NRF 2013S1A2A2035514 and 2016S1A2A2911800). This support is gratefully acknowledged.
References Alston, W. 2002. Truth: Concept and Property. In What Is Truth? ed. Schantz, 11–26. New York/Berlin: de Gruyter. Asay, J. 2013. The Primitivist Theory of Truth. Cambridge: Cambridge University Press. ———. 2018. truth: A Concept Unlike Any Other. Synthese Special Issue Truth: Concept Meets Property, Wyatt, ed. https://doi.org/10.1007/ s11229-017-1661-z. Barnard, R., and J. Ulatowski. 2013. Truth, Correspondence, and Gender. Review of Philosophy and Psychology 4 (4): 621–638. ———. 2017. The Objectivity of Truth, a Core Truism? Synthese Special Issue Truth: Concept Meets Property. https://doi.org/10.1007/s11229-017-1605-7. Bar-On, D., and K. Simmons. 2007. The Use of Force Against Deflationism: Assertion and Truth. In Truth and Speech Acts: Studies in the Philosophy of Language, ed. Greimann and Siegwart, 61–89. London: Routledge. Bedu-Addo, J.T. 1985. Wiredu on Truth as Opinion and the Akan Language. In Philosophy in Africa: Trends and Perspectives, ed. Bodunrin, 68–90. Ile-Ife: University of Ife Press. Bourget, D., and D. Chalmers. 2014. What Do Philosophers Believe? Philosophical Studies 170: 465–500. Caputo, S. 2012. Three Dilemmas for Alethic Functionalism. The Philosophical Quarterly 62 (249): 853–861. Cotnoir, A. 2013. Validity for Strong Pluralists. Philosophy and Phenomenological Research 83: 563–579. David, M. 2013. Lynch’s Functionalist Theory of Truth. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 42–68. New York: Oxford University Press. Devlin, J. 2003. An Argument for an Error Theory of Truth. Philosophical Perspectives 17 (1): 51–82.
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Dzobo, N.K. 1992. Knowledge and Truth: Ewe and Akan Conceptions. In Person and Community: Ghanaian Philosophical Studies I, ed. Wiredu and Gyeke, 73–84. Washington, DC: The Council for Research in Values and Philosophy. Edwards, D. 2011. Simplifying Alethic Pluralism. Southern Journal of Philosophy 49 (1): 28–48. Eklund, M. 2017. What Is Deflationism About Truth? Synthese Special Issue Truth: Concept Meets Property, Wyatt, ed. https://doi.org/10.1007/ s11229-017-1557-y. Fisher, M., J. Knobe, B. Strickland, and F. Keil. 2017. The Influence of Social Interaction on Intuitions of Objectivity and Subjectivity. Cognitive Science 41: 1119–1134. Horton, M., and T. Poston. 2012. Functionalism About Truth and the Metaphysics of Reduction. Acta Analytica 27 (1): 13–27. Horwich, P. 1998. Truth. 2nd ed. Oxford: Oxford University Press. Jackson, F. 1998. From Metaphysics to Ethics: A Defence of Conceptual Analysis. Oxford: Oxford University Press. Kölbel, M. 2008. “True” as Ambiguous. Philosophy and Phenomenological Research 77 (2): 359–384. Künne, W. 2003. Conceptions of Truth. Oxford: Oxford University Press. Kwame, S. 2010. Nokware. In The Oxford Encyclopedia of African Thought, ed. F. Abiola Irele and B. Jeyifo. New York: Oxford University Press. Accessed at http://www.oxfordaasc.com/article/opr/t301/e275. Lewis, D. 1966. An Argument for the Identity Theory. The Journal of Philosophy 63 (1): 17–25. ———. 1970. How to Define Theoretical Terms. The Journal of Philosophy 67: 427–446. ———. 1972. Psychophysical and Theoretical Identifications. Australasian Journal of Philosophy 50: 249–258. ———. 1994. Reduction of Mind. In Companion to the Philosophy of Mind, ed. Guttenplan, 412–431. Malden: Blackwell. Lynch, M. 2000. Alethic Pluralism and the Functionalist Theory of Truth. Acta Analytica 24: 195–214. ———. 2001. A Functionalist Theory of Truth. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 42–68. New York: Oxford University Press. ———. 2004a. True to Life. Cambridge, MA: MIT Press. ———. 2004b. Truth and Multiple Realizability. Australasian Journal of Philosophy 82: 384–408.
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———. 2005. Alethic Functionalism and Our Folk Theory of Truth. Synthese 145: 29–43. ———. 2009. Truth As One and Many. Oxford: Oxford University Press. ———. 2013. Three Questions for Truth Pluralism. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 21–41. New York: Oxford University Press. Maffie, J. 2001. Truth from the Perspective of Comparative World Philosophy. Social Epistemology 15 (4): 263–273. Matthewson, L., and J. Glougie. Forthcoming. Justification and Truth: Evidence from Languages of the World. In Epistemology for the Rest of the World, ed. Stich, Mizumoto, and McCready. Oxford: Oxford University Press. Mizumoto, M. (ms) A Prolegomenon to the Cross-Linguistic Study of Truth. Moltmann, F. 2015. ‘Truth Predicates’ in Natural Language. In Unifying the Philosophy of Truth, ed. Achourioti et al., 57–83. Dordrecht: Springer. ———. 2018. Truth Predicates, Truth Bearers, and Their Variants. Synthese Special Issue Truth: Concept Meets Property. https://doi.org/10.1007/ s11229-018-1814-8. Newhard, J. 2013. Four Objections to Alethic Functionalism. Journal of Philosophical Research 38: 69–87. ———. 2014. Alethic Functionalism, Manifestation, and the Nature of Truth. Acta Analytica 29 (3): 349–361. ———. 2017. Plain Truth and the Incoherence of Alethic Functionalism. Synthese 194 (5): 1591–1611. Quine, W.V.O. 1948. On What There Is. Review of Metaphysics 2 (5): 21–36. Reprinted in From a Logical Point of View, 2nd Revised ed. Quine, 1–19. New York: Harper and Row. ———. 1970. Philosophy of Logic. 2nd ed. Englewood Cliffs: Prentice Hall. Strawson, P.F. 1949. Truth. Analysis 9 (6): 83–97. ———. 1950. Truth. Proceedings of the Aristotelian Society Supplementary 24: 129–156. Tarski, A. 1944. The Semantic Conception of Truth: And the Foundations of Semantics. Philosophy and Phenomenological Research 4 (3): 341–376. Ulatowski, J. 2017. Commonsense Pluralism About Truth: An Empirical Defence. Cham: Palgrave Macmillan. Wiredu, K. 1985. The Concept of Truth in the Akan Language. In Philosophy in Africa: Trends and Perspectives, ed. Bodunrin, 43–54. Ile-Ife: University of Ife Press. ———. 1987. Truth: The Correspondence Theory of Judgment. African Philosophical Inquiry 1 (1): 19–30.
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———. 2004. Truth and an African Language. In African Philosophy: New and Traditional Perspectives, ed. Brown, 35–50. Oxford: Oxford University Press. Wright, C.D. 2005. On the Functionalization of Pluralist Approaches to Truth. Synthese 145 (1): 1–28. Wright, C. 2010. Truth, Ramsification, and the Pluralist’s Revenge. Australasian Journal of Philosophy 88 (2): 265–283. Wyatt, J. 2016. The Many (Yet Few) Faces of Deflationism. The Philosophical Quarterly 66 (263): 362–382.
Part II Logic
Core Logic: A Conspectus Neil Tennant
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Introduction
As the title of this study implies, there is a much longer and more detailed treatment of Core Logic, to be found in Tennant (2017). Here we simply try to explain and summarize the main points and their relation to debates about monism and pluralism. The title of the Storrs conference in 2015 of which this volume is the proceedings was ‘Pluralism in Truth and Pluralism in Logic’. But the prevailing intellectual climate is so pluralist that the singulars in the conference title have metastasized so as to occasion, in the title of this volume, the plural noun ‘Pluralisms’. In this heady atmosphere, the present author presents a more conservative, but still accommodating, ‘absolutist pluralism’ about logic—albeit a reformist one. He believes there is such a thing as ‘core logical’ deductive reasoning, and that it is relevant, in a sense heretofore not satisfactorily explicated. It comes, however, in two
N. Tennant (*) Department of Philosophy, Ohio State University, Columbus, OH, USA e-mail: [email protected] © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_9
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varieties: constructive and non-constructive, as typified most importantly in mathematics, where mathematical theorems are deduced from mathematical axioms.
Locating the Core Systems ℂ and ℂ+ In the present author’s view, the decision tree for choice of logic, for an unmitigated absolutist, looks like this1: Is there one correct logic? Pluralist: No
Absolutist: Yes Is it Classical Logic? Quietist: Yes
Reformist: No Is it relevant? Yes
No Is it constructive?
Yes C
Core
No
Yes
No
Classical Core
I
Other
The formal-logical reforms that need to be carried out on Classical Logic C and Intuitionistic Logic I have to do with relevantizing formal proofs. The ‘absolutist pluralism’ adverted to above differs from this unmitigated absolutism by making the following concessive recommendations: So you’re a constructive mathematician? Then the right logic for you to use is Core Logic ℂ; or … So you’re a non-constructive, i.e. classical mathematician? Then the right logic for you to use is Classical Core Logic ℂ+.
The relevantizing effected within these two core systems is non-negotiable. Classical Core Logic ℂ+ is simply the ‘classicized extension’ of Core Logic ℂ.
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From now on, when we write ‘Core logician’, we intend this to be understood disjunctively, as ‘user of Core Logic for constructive reasoning, or user of Classical Core Logic for non-constructive reasoning’. This disjunctiveness is our concession to pluralism; while the commitment to relevance remains absolute.
On Being Conservative in One’s Aim This author is concerned (as Gentzen was) to complete satisfactorily Frege’s original task of regimenting the expert deductive reasoning of mathematicians in as natural a way as possible. And all such reasoning is relevant. Gentzen’s aim with his systems of natural deduction was clear: Wir wollen einen Formalismus aufstellen, der möglichst genau das wirk liche logische Schliessen bei mathematischen Beweisen wiedergibt. (Gentzen 1934, p. 183)
This was translated by M. E. Szabo as We wish to set up a formalism that reflects as accurately as possible the actual logical reasoning involved in mathematical proofs.
The difference between the Core logician’s approach and that of Gentzen is that the Core logician takes seriously the fact that such reasoning (whether or not it is constructive) is always fully relevant, in the way it proceeds from premises (say, mathematical axioms) to conclusions (say, mathematical theorems).2 One’s logical system should furnish formal proofs that fill in all the gaps that occur in the typically informal proofs that mathematicians provide, even when they are being (by their own disciplinary standards) very rigorous. Formal logicians have more demanding disciplinary standards. Yet their formal proofs should just be appropriately more detailed homologues of mathematicians’ informal proofs. Note that this observation applies both to the constructive mathematical reasoner and to the non-constructive, or classical one. So much for the conservativeness of the Core Logician’s formalizing project.
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On Being Accommodating in One’s Aim The Core Logician is accommodating in the following important regard. Since the time of Frege, intuitionistic or constructive mathematical reasoning has achieved a special systemic status, through the work of Heyting following Brouwer, resulting in the system I of Intuitionistic Logic. So there should be a formal-logical account of constructive mathematical reasoning conducted relevantly, and a more expansive formal-logical account of non-constructive (classical) mathematical reasoning, likewise conducted relevantly. These accounts are distilled, respectively, as the systems of Core Logic ℂ and Classical Core Logic ℂ+. The core systems ℂ and ℂ+ respectively relevantize the two more familiar and traditional systems I and C. And they do so in the same way, ‘at the level of the turnstile’.
What Is Wrong with R and Its Ilk By contrast, the relevance logics (such as R) in the Anderson-Belnap tradition (see Anderson and Nuel D. Belnap, Jr. (1975)) seek to relevantize at the level of the object-language conditional, thereby changing its Fregean sense. In relevant logics such as R, the following inferences fail, despite the fact that they directly render the Fregean stipulation that a conditional is true when its antecedent is false, and is also true when its consequent is true:
Øj j ®y
y j ®y
In these relevant logics, one is also deprived of Disjunctive Syllogism:
j Ú y Øj y
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despite its ubiquity in mathematical reasoning (e.g. about ordering relations). There is no known metatheorem to the effect that ordinary mathematical reasoning can be fully formalized in any system like R. Actually, there have been negative results in this regard—see Friedman and Meyer (1992). This is in stark contrast to the situation with ℂ (for constructive mathematical reasoning) and ℂ+ (for non-constructive mathematical reasoning).
The Uniformity of Relevantizing The right logic for constructivist reasoning must bear to the right logic for non-constructivist reasoning the same sort of relation (represented by the solid horizontal arrows below) that Intuitionistic Logic I bears to Classical Logic C. Likewise, the right logic for constructivist reasoning should bear to Intuitionistic Logic I the same sort of relation (represented by the dashed vertical arrows below) that the right logic for non-constructivist reasoning bears to Classical Logic C. Our claim is that ℂ is the right logic for constructivist reasoning, and ℂ+ is the right logic for non-constructivist reasoning. Note that Core Logic contains the aforementioned Disjunctive Syllogism (φ ∨ ψ, ¬φ : ψ) and Fregean truth-table respecting inferences (¬φ : φ → ψ and ψ : φ → ψ); but it does not contain either one of the two closely related Lewis paradoxes φ, ¬φ : ψ and φ, ¬φ : ¬ψ (see Figs. 1 and 2).
I
C
Fig. 1 From classical logic to core logic
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Fig. 2 Important system containments
What Is Wrong with M In addition to the four systems already mentioned (C, I, ℂ, and ℂ+), the reader might wonder how Johansson’s system M of Minimal Logic fits into the picture. Our contention is that M was intended by Johansson to be a relevantized system of constructive reasoning. And in this project, it failed, because, although it avoids the positive form of the first Lewis paradox (φ, ¬φ : ψ), it nevertheless contains the negative form (φ, ¬φ : ¬ψ). Moreover, M also fails to validate Disjunctive Syllogism. But the relevant validity of Disjunctive Syllogism is unimpeachable; and it is indispensable to mathematical reasoning (consider: b such that W is a set of nodes, and ≤ is an ordering on those nodes. ≤ is reflexive and transitive. They give as clauses (for w, u ∈ W) w ⊨ A ∧ B if and only if w ⊨ A and w ⊨ B w ⊨ ¬A if and only if, for all u such that w ≤ u, u ⊭ A. Lastly, situations are “simply parts of the world” (Beall and Restall 2006, p. 49), except that situations can be inconsistent, and parts of possible worlds cannot be. There is an important relationship between situations, called compatibility. Roughly, two situations are compatible if they do not “contradict” one another.4 Compatibility, according to Restall (1999), is probably non-reflexive, symmetric and such that every situation is compatible with something. Some examples of clauses on this semantics, for s and t situations and C the compatibility relation, are s ⊨ ¬A if and only if, ∀t such that sCt, t ⊭ A s ⊨ A ∧ B if and only if s ⊨ A and s ⊨ B. What we need now, to answer the question of whether the connectives in these logics do indeed share meanings, is to know what they take the identity conditions of the meanings of the connectives to be. Unfortunately, they do not say much on this front. What they do say is The clauses can… [all] be true of one and the same [connective] simply in virtue of being incomplete claims about… [the connective]. What is required is that such incomplete claims do not conflict, but the clauses governing negation do not conflict. The classical clause gives us an account of when a negation is true in a [model]…, and the constructive clause gives us an account of when a negation is true in a construction. Each clause picks out a different feature of negation. (Beall and Restall 2006, p. 98)
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This means that, somehow, there is one thing which is (for example) negation, and the various negation clauses given above all express different aspects of that one thing. To make sense of this claim, we need to fill in more details. I take it that Beall and Restall (or, at least Restall) want to take the meanings of the connectives on this picture to be maximal or minimal truth conditions, as described in Restall (2002). The maximal truth condition for a connective is something like the disjunction of all the clauses for the connectives in the admissible cases. That is, the maximal truth condition for negation (assuming the only admissible logics are classical, intuitionistic and relevant) would be something like (M is a Tarski model and M, s ⊨ ¬A if and only if it is not the case that M, s ⊨ A) OR (w is a node in a Kripke structure and w ⊨ ¬A if and only if, for all u such that w ≤ u, u ⊭ A) OR (s is a situation and s ⊨ ¬A if and only if, ∀t such that sCt, t ⊭ A)
The meaning of negation is then the disjunction of all three negation clauses given above.5 A minimal truth condition, on the other hand, is a bare biconditional between a sentence and an attribution of truth. So, for example, the minimal truth condition for negation would be something like ¬A is true in x if and only if A is not true in y for y compatible with x
The minimal truth condition straightforwardly captures the relevant negation clause. The classical negation clause is also straightforwardly captured: since classical situations are complete consistent situations, any situation which is compatible with a classical situation will not make anything true that the classical situation does not, so “compatible with” in this restriction will be reducible to “equal to”. Finally, we can capture an intuitionistic clause by again restricting ourselves to explicitly consistent situations. In this case, we have to assume that two explicitly consistent situations are compatible only in the cases where one can be “built” out of the other. As I will claim later, in order to be “added to” the maximal truth condition, a clause for a connective must have certain features,
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and I will argue that at least one of those features is being able to be made to fit a certain type of minimal truth condition. If this is acceptable, then we need to ensure that there is no conflict among clauses where they overlap. This is because Restall (2002) holds that “the two accounts [classical and relevant] agree where they overlap” (p. 438), which seems to accord with the claim above that incomplete claims do not conflict (from Beall and Restall 2006, p. 98). There are two questions we need to ask now: what, exactly, does it mean for two clauses to overlap? And do the intuitionistic and relevant clauses agree where they overlap? I think the answer to the second question is no, but it depends in large part on our answer to the first. It is to this which I now turn.
3
Overlapping Clauses
The method that Beall and Restall use to explain overlap involves describing all three consequence relations in a situation semantics (Restall (1999) and Beall and Restall (2001)). They state We… provide a model of how these things could be… to show how our three different claims about the behavior of negation can be true together… A world [classical model] is a complete, consistent situation… [We] take all constructions to be situations… [that are] not only consistent, but explicitly so. (Beall and Restall 2001, pp. 10–11)
Though this is ultimately the downfall of the view, we need the details to explain the problem. This method of explaining overlap involves restricting the clauses for the connectives in the situation semantics in certain ways to generate the clauses for the connectives in the other logics. At the very least, the road is simple enough for finding a classical consequence relation in such a semantics. We simply restrict ourselves to considering complete consistent situations, and restrict compatibility to be the identity relation.
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The case for finding an intuitionistic consequence relation in such a semantics is much harder. Beall and Restall (2001) claim we need to restrict ourselves to considering explicitly consistent situations, where each explicitly consistent situation makes ¬(A ∧ ¬A) true, for all A (Beall and Restall 2001, p. 11). Complete consistent situations (those which generate classical logic) are also explicitly consistent. Presumably, Beall and Restall hold that this generates the intuitionistic clauses for the connectives. In Sect. 4, I will show that this is not correct. To figure out whether two clauses for a connective overlap, we first, we need to assess what all the negation clauses have in common; what is it that makes them disjuncts in the maximal truth condition?6 As noticed in Hjortland (2013), all three negation clauses can be made to “fit” the following clause (*) for x ∈ D, x ⊨ ¬A if and only if, ∀y such that xRy, y ⊭ A This is something like the minimal truth condition alluded to above. Let’s look at precisely how this will work. First, we take the default to be that D is the entire situation space, and that R is compatibility (we will be forced to loosen these requirements later). This means that the default of (*) generates a relevant negation clause. Classical negation can then be recovered by letting D range over complete, consistent situations, and letting R be the identity relation. So far so good. Recall that Beall and Restall (2001) claim that the intuitionistic clause can be recovered by restricting our attending to explicitly consistent situations. In the language of (*), this means letting D range over explicitly consistent situations, and letting R be such that, if c is a stage, and cRs, then sRs, so that stages only “see” consistent situations. We are now in a position to articulate a definition for when two clauses overlap. Two clauses overlap when the intersection of their domains is non-empty. I think Beall and Restall would agree up to this point. We can also see now that all three clauses overlap, since the intersection of situations with complete consistent situations and/or with explicitly consistent situations will not be empty. This means what is left to determine whether all three negations have the same meaning is to assess whether or not they conflict where such overlap occurs.7
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The Problem
I will focus on showing that the intuitionistic clause conflicts with the relevant clause for negation. I will demonstrate this by showing the addition of the intuitionistic clause to a maximal truth condition which already has the relevant clause as one of its disjuncts makes negation arbitrary. If we add the intuitionistic clause to the maximal truth condition which already has the relevant clause, then we will be able to add clauses to the maximal truth condition which we ought not to consider part of the meaning of negation. I take it, showing that the relevant negation becomes arbitrary upon the addition of the intuitionistic clause demonstrates that the clauses conflict. So, both clauses cannot be part of the meaning of negation. In order to assess whether the intuitionistic clause for negation conflicts with the relevant clause, we need to settle just what stages-as- situations are. Beall and Restall claim that they are merely explicitly consistent situations. If stages are merely explicitly consistent situations, then changes in the relationship R in (*) need to be made. I call this horn “Option 1”. If they are not explicitly consistent situations, then there will be no overlap between the intuitionistic negation clause and the relevant negation clause. I call this horn “Option 2”. I will consider each option in turn, and show that both options make negation arbitrary, and thus no matter how Beall and Restall go, they will not be able to succeed in their connective meaning project. Suppose first that stages are merely explicitly consistent situations. To do this, we need to make sure that the intuitionistic clause for negation can appropriately fit (*) when D ranges over explicitly consistent situations. This means that, in addition to R being restricted in such a way that stages are only compatible with consistent situations, we also need to insist that R not be symmetric. Were the parallel relation to be symmetric, we would lose any notion of constructability. The ≤ relation is supposed to be something like a “constructed-from” or “built from” relation, so that we have w ≤ v just in case v is “built from” w. Suppose that the relationship between explicitly consistent situations (the “built-from” relationship) was indeed symmetric. Then anytime v was “built from” w,
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w would also be “built from” v. We could never construct anything new, so to speak, because each explicitly consistent situation would be built from those before and after it. Everything would be true at the first situation in the ordering. So, for (*) to properly capture the intuitionistic clause, R cannot be symmetric. If we want the negation clause for intuitionistic cases to properly fit (*), we need to restrict D to explicitly consistent situations, and R to some subrelation of compatibility which is not symmetric. If we allow this restriction, though, we have a problem. If all it takes to be part of the meaning of negation is to be “moldable” into (*) (where R can be any subrelation of C), then lots of things will fit this mold.8 Not all of them will be things we want to include as part of the meaning of negation. For example, let D be anything we like, and let R be empty.9 The empty relation is certainly a subrelation of compatibility. The clause here is ∀d ∈ D, d ⊨ ¬A if and only if, ∀t such that dRt, t ⊭ A. Since R is empty, this means the right-hand side of this conditional is always vacuously satisfied. The “empty-R” negation clause becomes d ⊨ ¬A for any A and any d ∈ D. This means that negated sentences are always true. Once again, if Beall and Restall are right, then this operator, which when appended to anything produces a true sentence, must be part of the meaning of negation. This makes negation arbitrary. An operator which, when appended to any sentence, makes that sentence true ought not be part of the meaning of negation. If intuitionistic stages are explicitly consistent situations ordered by a subrelation of the compatibility relation, and we can restrict R in this way, then we can see that intuitionistic negation and relevant negation cannot share a meaning. This suggests quite strongly that intuitionistic stages simply cannot be situations in the way Beall and Restall describe, at least not if we want the relevant and intuitionistic negations to mean the same thing. If they were, they would need to be ordered by a proper subrelation of compatibility, and that would allow us to add arbitrary clauses to negation. This rules out Option 1, and so we must now turn to Option 2. Even though intuitionistic stages are not situations, we can still make the intuitionistic clause for negation “fit” the (*) mold. We simply let D be nodes in Kripke structures, and let R be the ≤ relation. Here, we avoid
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the previous problem: nodes will only be related to the nodes in the future of their own Kripke structure, and the relationship will not be symmetric. There will be no overlap between the clause for negation in situations and the clause for negation in nodes of Kripke structures. The intersection of the domains will be empty, since one contains only situations, and the other contains no situations. Again, though, we have a similar issue. If all it takes to be part of the meaning of negation is to be “moldable” into (*) (with no requirements on the domain) then there is a problem. Again, there are lots of clauses which would fit this mold, and it is not the case that we would like to allow all of them to be part of the meaning of negation. Consider the following. Let us assume that the situation semantics is rich enough such that every atomic sentence is not true at some situation. Then, for any list of negated atomic sentences, we can construct a situation-in-a-model at which they will all be true.10 We do so simply by restricting our domain in certain ways. Suppose, for example, that the list we were given contained ¬A1,…,¬An. Then, we simply restrict our domain to the set/class of situations which make none of A1,…,An true. This will be non-empty, since because of our assumption, there is at least one situation which does not make true the conjunction of A1,…,An.11 Further, at any situation in this domain, the negations of A1,…,An are true, since the compatibility relationship will be restricted to our domain, and no situation in the domain makes A1,…,An true. In this case, negation is arbitrary. We can make the negation of any sentences we would like true, just by restricting the domain in certain ways. Thus, allowing restrictions to the domain in order to get the intuitionistic clause for negation requires allowing restrictions that gets arbitrary clauses for negation. I claim that Beall and Restall cannot allow this and still claim that they have succeeded in pinning down the meaning of negation. Summing up, we see that if stages are either explicitly consistent situations with the accessibility relation being a subrelation of compatibility, or if they cannot be situations at all, negation as described by Beall and Restall will be arbitrary if it includes both the intuitionistic and relevant clause. Given the way Beall and Restall discuss stages, and intuitionistic cases, these seem to be the only options. So, either negation is arbitrary
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or intuitionistic negation cannot be part of the “disjunction of all negation clauses” meaning. This means that either negation is arbitrary or Beall and Restall have not produced a pluralism without language change after all.12
5
A Different Situation Semantics?
There is another option here, and that is to go the way of Dunn (1993).13 There, he presents a method by which we might “fit” both relevant and intuitionistic negations into one model. This system has two relations, C and ⊑, so there is no need to make one “fit” the other, and we do not have the problem we saw above with trying to make the Kripke structure “in-the-future-of ” relation fit with our compatibility relation. Each relation corresponds to one “type” of negation: ¬∗A (defined by x ⊨ ¬∗A if and only if x∗ ⊭ A) and ¬⊥A (defined by x ⊨ ¬⊥A if and only if ∀y(y ⊨ A implies y⊥x)) (p. 331). Roughly, ∗ can be thought of as the operator from the Routley-Meyer relevant logic semantics, and so we use it to generate ⊑, which can be thought of as A ⊑ B ↔ everything A makes true, B makes true. ⊥ can be thought of as an orthogonality relation (see, e.g., Goldblatt’s semantics for orthologic), which we can use to interpret this C as a compatibility relation. By varying restrictions on the relations C and ⊑, and we can get classical, intuitionistic and relevant negation in this model. Moreover, it works even when we add ∧ and ∨. One might think now that this is all well and good, but that we still have two negations in play: ¬∗ and ¬⊥. Here is where Dunn’s system shines: one can show, under suitable circumstances, that these two negations really come to the same thing. Thomas (2015) shows how one could extend such a system to include a conditional connective as well. So Beall and Restall may be saved after all! Thomas’s (2015) expansion is subtle and elegant, but it unfortunately will not do for Beall and Restall’s position. In effect, it “lets” the quantifiers influence the meaning of negation. We can construct, for example, a model with two domains (Thomas’s term for something like our sets of situations above), such that φ ⊨ ¬¬φ holds when φ is quantifier free, but does not hold when it is not.14 We take two domains, call them ω and α,
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such that Thomas’s N relationship (the equivalent of our “R”, above) contains exactly < ω, α > and < α, α >, and then let the other relationships in the Thomas system (the range of each quantifier, and the tripartite relationship which is used to make sense of the arrow) be identity relations. Then, supposing all objects are P at ω, but that we introduce some new object, a, at α such that ¬Pa, then we will have both ω ⊨ ∀xPx and ω ⊨ ¬∀xPx. This in effect means that negation will not be compositional, since φ ⊨ ¬¬φ holds when φ is quantifier free, but does not hold when it is not. This seems to be allowing the quantifiers to influence the meaning of negation. Though this is not an issue for Thomas, it poses a problem for Beall and Restall. A non-compositional negation would not match natural language negation, and since Beall and Restall motivate their position in part by the applicability of their various logical systems, the connectives in their system must at least partly match the natural language connectives. Since natural language negation is arguably compositional, a non-compositional negation will not do the trick, and should not be admitted as part of the meaning of negation. Moreover, Thomas proves that all logics obtained by this system are subclassical, so this logic will be necessary, normative and formal as desired.15 Again, we see that if we try to use (*) to capture the meaning of negation, even in this new system, we will wind up letting things in which ought not to count as negation.
6
Merely Technical Meanings
It is open to Beall and Restall at this point to simply state that their definition of the connectives is merely technical, and that it does not matter that the meaning of negation really is arbitrary. This would allow them to say that it does not matter that negation is arbitrary, and maintain that their pluralism really is one in which the connectives across logics are pairwise synonymous, where synonymy amounts to sameness of meaning. However, on top of being unsatisfying for more general reasons, this move is not open to Beall and Restall because they motivate their pluralism in part by the ability of each logic to capture deductive reasoning in various applications. In particular, this means that the connective meanings have to have something in common with what the connective
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s peakers actually use. If they want their theory to be applicable, then the connectives cannot be merely stipulative. So they cannot just define negation technically, it has to match up with something in the world.
7
A Possible Solution
There is an interesting moral to draw here. There does seem to be something fundamentally different about intuitionistic negation and relevant negation. I strongly suspect this has something to do with the fact that intuitionistic and relevant consequences are both restrictions of classical consequence in different ways. Intuitionistic logic allows explosion but not double negation elimination, while relevant logic allows double negation elimination but not explosion. However, Beall and Restall have provided a meaning for the connectives on which classical and relevant negations mean the same thing. We can also do this for classical and intuitionistic negations. Here, we would use nodes in Kripke models as intuitionistic cases, and recover classical consequence, and the classical connective clauses, by restricting ourselves to one-node Kripke structures (see Beall and Restall 2006, p. 98). If both of these systems are legitimate, then we have a pluralism which is pluralistic in both logic and language. We have two model theories, that of the situations semantics and that of the Kripke structures, and two logics in each, classical and relevant in the first, and classical and intuitionistic in the second. On the one hand, we have what Beall and Restall want: two consequence relations with connectives which share a meaning, if sameness of meaning of the connectives in two logics amounts to being able to use the same model theory for both logics. On the other hand, we have what our earlier characterization of Carnap wants: sometimes, logic change does require connective meaning change. In a sense, this type of system might be thought of as a synthesis between Carnap and Beall and Restall. As a sketch of how it might go, consider the following.16 We might think that the meanings of the connectives are context sensitive in a certain way; for our purposes, the meanings are sensitive to the goals of certain uses of logic. That is, the meanings are sensitive to what the logic in question is being used to do (e.g., formalize a mathematical system,
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analyze a pattern of deductive inferences in natural language, etc.), or what we are trying to figure out when we are talking about logic. Consider two particular uses of logic: using classical logic to do classical analysis, and using intuitionistic logic to do smooth infinitesimal analysis. Interestingly, with smooth infinitesimal analysis, we can prove some negations of classical theorems, so that, for a certain group of elements, x, ¬∀x(x = 0 ∨ x ≠ 0) is true. Now, we might ask, what is the meaning of the connectives for each of these logics on the occasion of each use of logic? Well, that will be dependent on the contexts in which these uses take place. Depending on the goals of the pursuers of those goals, sometimes it might make sense to treat the logical connectives as meaning the same thing. On the other hand, sometimes it might make sense to treat the connectives as not sharing a meaning. Kouri Kissel (2018) develops a theory of how this might work based on a specific pragmatic linguistic framework, and Shapiro (2014) develops this on the basis of what type of mathematical goal each context involves. In this way, the conversational goals might be said to affect how many uses of each connective get to count as the same negation. Thus, sometimes the connectives will mean the same thing, and sometimes they will not.
8
Conclusion
Beall and Restall (2006) propose a logical pluralism where the connectives for each logic are pairwise synonymous. Due to the manner in which connectives are given their meaning, relevant negation and intuitionistic negation cannot mean the same thing. Thus, their pluralism is a pluralism of languages and logics, not just logics as desired.17
Notes 1. I argue elsewhere (see Kouri (2018)) that this is not the best way to understand the Carnapian picture. For the purposes of this chapter, though, it suffices. 2. It is not clear that Restall is right in such an assessment of Carnap. At the very least, Restall ought to grant that Carnap’s pluralism is a pluralism of
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logics and languages. Restall’s position, on the other hand, is just a pluralism of logics. Additionally, it is the case that Carnap could claim “A and ¬A together, classically entail B, but A and ¬A together do not relevantly entail B”. He would claim this is the case precisely because the connectives mean something different in each logic. In some sense, what this quote does for Restall is to show that Restall disagrees with the second half (“A together with its classical negation entail B, but A together with its relevant negation need not entail B”.) and not that Carnap disagrees with the first. Thanks to Hannes Leitgeb for pushing me on this issue. 3. There is a reason to think that the metalanguage Beall and Restall are using is also classical. Though they never explicitly make a claim about which metalanguage they are working with, it seems their description of stages in constructions and situations as subclassical suggests that classical logic is the strongest logic admissible on their picture, and their rejection of contra-classical logics suggests the same. If this is the case, then it seems that it would make a good choice for a metalanguage. For a criticism of their (probable) choice in metalanguage, see Read (2006). It should be pointed out here that Beall and Restall never explicitly make the claim that their metalanguage is classical. A referee has suggested to me that in personal correspondence, Restall has gone so far as to claim that there is no metalanguage associated with logical pluralism. I am not entirely sure what to make of this, as presumably, we need some language in which to discuss the logics which are admissible. Whether the metalanguage is classical, though, or even if there is no metalanguage, most of what follows can be re-stated accordingly, and so I will not pursue this complication further here. 4. Unfortunately, very little about the compatibility relation is committed to by Restall. This poses some problems for Beall and Restall’s project. In particular, it will turn out that it is important whether compatibility is primitive, or somehow determined by negation. 5. There is another way we can view this (if Beall and Restall’s metalanguage is classical, these will be equivalent). It might be rather than a disjunction, we have a conjunction of conditionals. The maximal truth condition for negation would then be (if M is a Tarski model then M, s ⊨ ¬A if and only if it is not the case that M, s ⊨ A) AND (if w is a node in a Kripke structure then w ⊨ ¬A if and only if, for all u such that w ≤ u, u ¬ A) AND (if s is a situation then s ⊨ ¬A if and only if, ∀t such that sCt, t ¬ A)
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Since I take it these are equivalent in their classical metalanguage (see endnote 3), I will focus on the disjunctive form. 6. This is in part necessary to rule out “junky” connectives, formed by disjoining the “wrong” clauses to the maximal truth condition. For example, it is required to prevent having to claim that the disjunction of the clause for conjunction in Tarski models with the clause for the arrow in situations is a legitimate connective meaning. Without this, the maximal truth condition is not a good candidate condition for the meaning of any connective, and we can dismiss Beall and Restall on those grounds. 7. There is an immediate response available on behalf of Beall and Restall here (thanks to Alexandru Radulescu for the suggestion). Beall and Restall claim that what they are doing is providing a precisification of the vague notion of logical consequence. So why not also assume that they are giving a precisification of a vague account of negation? Then, we would not need such a precise definition of negation, as I have suggested with maximal truth conditions and (*), and the examples I give in Sect. 4 can all be dismissed as borderline cases or outliers. However, I think there will still be a problem here, though it is different from any I will address here. There will be a tension between their firm claim that the connectives must mean the same thing, and the vague meaning of those connectives. In order to make the first claim, we must assume that there is something precise about the nature of the connectives, while to make the claim about vagueness, we must dismiss this preciseness. In effect, I think that if we want to pursue the claim that the connective meanings are vague, we are better to adopt something like the tentative conclusion I give in Sect. 7. 8. Additionally, of course, the logical consequence relationship will have to satisfy GTT and be necessary, normative and formal. For Beall and Restall, necessity is truth preservation in all cases, normativity is the ability to go wrong if you accept the premises and reject the conclusion of a valid argument, and formality is either schematicity or one of: providing norms for thought as such, indifference to identities of objects, or contentlessness (Beall and Restall 2006, Chap. 2). 9. This logical consequence relationship is also necessary, normative and formal, and satisfies the GTT. It satisfies the GTT since the cases in question are just whichever situations are “blind”. The empty-R relation is just a sublogic of what Beall and Restall refer to as relevant logic, which we had already assumed was necessary and formal, and so these characteristics are preserved. Additionally, it is normative, since we “go wrong” by assuming that the situations are compatible with something.
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10. Thanks to Graham Leach-Krouse for suggested an example of this type. 11. These restricted domains of situations still abide by the necessary- normative-formal requirements on being an admissible logic (see definition in endnote 9), and satisfy the GTT since the cases are simply situations of a specific type. These situations produce a consequence relation which is formal (since it is in a model) and normative, since we “go wrong” by assuming that the situations we are considering are not restricted to not making true a certain set of sentences. Finally, it is necessary. Beall and Restall define necessity as “the truth of the premises necessitates the truth of the conclusion” (p. 14), in other words, as long as it is not possible for the premises to be true and the conclusion to be false (p. 40). But in this case, this is not possible, since the possibilities in play are a subset of the original domain, so there are no possible cases where things are different. 12. One might think at this point that we might be able to accomplish the Beall and Restall project if we framed everything in terms of nodes in Kripke structures rather than situations. After all, terminal nodes in Kripke structures are classical models (see Beall and Restall 2006, p. 98). However, we find we have the same problem once again. The default instantiation of (*) would then be D=nodes in Kripke structures, R =≤. We would have to expand ≤ so that it could be symmetric, since compatibility is symmetric. However, Kripke structures give us no clues as to how to expand ≤ to appropriately capture compatibility, and so we can proceed as we wish, and expand it to any relationship. Then we simply take R to be the universal relation, and re-run the previous counterexample. Another option would be to consider cases as classes of pointed frames (thanks to Shay Logan for this suggestion). Then, we could capture the various logics by restricting what gets admitted to the class of pointed frames in question. Moreover, the two counter examples presented above would be less odd, since one ought to expect that odd frames come equipped with odd negations. However, we will still have a problem here (thanks to Beau Madison Mount for this suggestion). It is still the case that we will have “overlap” (e.g., pointed Kripke frames for propositional logic with the null signature will overlap with classical frames with a unary relation), and thus we will still be able to develop sentences which are true when the frame is considered as one kind of model, and not true when considered as the other. 13. Thanks to Aaron Cotnoir for suggesting this option. 14. Thanks to Ethan Brauer, Steven Dalglish and Giorgio Sbardolini for help in constructing this example.
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15. They will be necessary, normative and formal on the basis of Beall and Restall’s definitions. Formality here seems relatively simple. Necessity and normativity, on the other hand, are a bit odd, and this is the result of Beall and Restall’s definitions of them. For Beall and Restall, any sublogic of classical logic will be necessary, since they hold that relevant logic is necessary because worlds are a special type of situation. But here, since, on Thomas’s system, classical logic can be recovered as a special type of model, and since no model we are concerned with proves anything contra-classical, all models ought to be truth preserving, and so necessary on Beall and Restall’s picture. This reasoning mimics their discussion of relevant logic, where they say “The question is, in other words, whether truth-preservation over all situations guarantees truthpreservation over all possible worlds. The answer is ‘yes’ if possible worlds count as (perhaps special) situations” (Beall and Restall 2006, p. 54). Here, we have swapped “models” for situations, and “classical models” for possible worlds, but the reasoning is similar. Normativity, for Beall and Restall, comes down to there being some sort of mistake associated with reasoning invalidly on this type of logic. But they are not terribly specific about what types of things count as mistakes. The mistake associated with relevant logic is irrelevant reasoning, and the mistakes associated with constructive logic are “mistakes of constructivity” (Beall and Restall 2006, 70). In this case, then, I claim that not reasoning properly according to this new logic is reasoning compositionally when one shouldn’t. As one referee has suggested, there is a good reason for denying that all subclassical logics are normative and necessary. Though this seems plausible (and many commentators agree that Beall and Restall’s definitions of necessary and normative are problematic, see, e.g., Bueno and Shalkowski (2009) and Keefe (2013)), strictly speaking here all we only need is the Thomas system to match Beall and Restall’s definitions, and it seems to. On the other hand, if the Thomas system does not produce necessary, normative and formal logics, then it is not an option for Beall and Restall to begin with. 16. For more details, see Kouri Kissel (2018) and Shapiro (2014). 17. Acknowledgments: I would like to thank Stewart Shapiro, Shay Logan, Chris Pincock, Kevin Scharp, Neil Tennant and a referee for this volume for feedback on earlier drafts. Additionally, audiences at the 2016 Central APA, the UConn Conference on Logical Pluralism and Truth Pluralism, the Munich Center for Mathematical Philosophy and the 2017 Canadian Philosophical Association meeting provided incredibly helpful comments.
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References Barwise, J., and J. Perry. 1983. Situations and Attitudes. Vol. 78. Cambridge, MA: MIT Press. Beall, J.C., and Greg Restall. 2001. Defending Logical Pluralism. In Logical Consequence: Rival Approaches. Proceedings of the 1999 Conference of the Society of Exact Philosophy, ed. John Woods and Bryson Brown, 1–22. Stanmore: Hermes. ———. 2006. Logical Pluralism. Oxford: Clarendon Press. Bueno, O., and S.A. Shalkowski. 2009. Modalism and Logical Pluralism. Mind 118 (470): 295–321. Dunn, J.M. 1993. Star and perp: Two treatments of negation. Philosophical Perspectives 7 (May): 331–357. Hjortland, O.T. 2013. Logical Pluralism, Meaning-Variance and Verbal Disputes. Australasian Journal of Philosophy 91: 355–373. Keefe, R. 2013. What Logical Pluralism Cannot Be. Synthese 191 (7): 1375–1390. Kouri, T. 2018. A New Interpretation of Carnap’s Logical Pluralism. Topoi: 1–10. Online first. Kouri Kissel, T. 2018. Logical Pluralism from a Pragmatic Perspective. Australasian Journal of Philosophy: 96 (3): 578–591. Read, S. 2006. Review of Logical Pluralism Notre Dame Philosophical Reviews. https://ndpr.nd.edu/news/logical-pluralism/. Restall, G. 1999. Negation in Relevant Logics (How I Stopped Worrying and Learned to Love the Routley Star). In What Is Negation?, ed. H. Wansing and D. Gabbay, Volume 13 of Applied Logic Series, 53–76. Dordrecht: Kluwer Academic Publishers. ———. 2002. Carnap’s Tolerance, Language Change and Logical Pluralism. Journal of Philosophy 99: 426–443. Shapiro, S. 2014. Varieties of Logic. Oxford: OUP. Thomas, M. 2015. A Generalization of the Routley-Meyer Semantic Framework. Journal of Philosophical Logic 44 (4): 411–427.
Generalised Tarski’s Thesis Hits Substructure Elia Zardini
Earlier versions of the material in this chapter have been presented in 2015 at the Veritas Pluralism, Language and Logic Workshop (Yonsei University); in 2018, at the LanCog Seminar (University of Lisbon), at the LOGOS Workshop Pluralism and Substructural Logics (University of Barcelona), at the fifth SBFA Conference (Federal University of Bahia) and at the Workshop Disagreement within Philosophy (Rhine Friedrich-Wilhelm University of Bonn). I’d like to thank all these audiences for the very stimulating comments and discussions. Special thanks go to Agustín Rayo, Colin Caret, Bogdan Dicher, Catarina Dutilh Novaes, Luís Estevinha, Filippo Ferrari, Ole Hjortland, Luca Incurvati, José Martínez, Ricardo Miguel, Sergi Oms, Nikolaj J. L. L. Pedersen, Hili Razinsky, Lucas Rosenblatt, Sven Rosenkranz, Diogo Santos, Ricardo Santos, Erik Stei, Célia Teixeira, Pilar Terrés, Zach Weber, Jack Woods and Jeremy Wyatt. I’m also grateful to the editors Nathan Kellen, Nikolaj J. L. L. Pedersen and Jeremy Wyatt for inviting me to contribute to this volume and for their support and patience throughout the process. The study has been funded by the FCT Research Fellowship IF/01202/2013 Tolerance and Instability: The Substructure of Cognitions, Transitions and Collections. Additionally, the study has been funded by the Russian Academic Excellence Project 5-100. I’ve also benefited from support from the Project FFI2012-35026 of the Spanish Ministry of Economy and Competition The Makings of Truth: Nature, Extent, and Applications of Truthmaking, from the Project FFI2015-70707-P of the Spanish Ministry of Economy, Industry and Competitiveness Localism and Globalism in Logic and Semantics and from the FCT Project PTDC/FER-FIL/28442/2017 Companion to Analytic Philosophy 2.
E. Zardini (*) LanCog Research Group, Philosophy Centre, University of Lisbon, Lisbon, Portugal International Laboratory for Logic, Linguistics and Formal Philosophy, School of Philosophy, National Research University Higher School of Economics, Moscow, Russian Federation © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_11
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Starting Off
Generalised Tarski’s Thesis Beall and Restall (2006) (henceforth, ‘BR’) have influentially defended a specific version of logical pluralism, a general view according to which, roughly stated, there is more than one legitimate relation of logical consequence. More in detail, BR endorse (p. 35)1 the rather convoluted thesis (henceforth, ‘BRT’) that legitimate relations of logical consequence are all and only those that: BRT1 Drop out of Generalised Tarski’s Thesis:
GTT An argument is validx iff, in every casex where every premise is true, so is the conclusion2
(given a suitable assignment of a range of cases to ‘x’);
BRT2 Additionally satisfy some other conditions traditionally associated with logical consequence, i.e. necessity, normativity and formality. By way of some initial skirmishes, there are actually considerations I regard as decisive against all the three conditions of BRT2. As for the necessity condition,3 the argument from ‘Snow is white’ to ‘Actually, snow is white’ is valid, but, for no context C, ‘If ‘‘Snow is white’’ is true- as-uttered-in-C, ‘‘Actually, snow is white’’ is true-as-uttered-in-C’ is necessary4 [the point is due essentially to Kaplan (1989), pp. 507–510; see Zardini (2012) for elaboration and variations].5 As for the normativity condition, the argument from ‘P0’, ‘P1’, ‘P2’ …, ‘Pi’ and ‘All my scientific beliefs are that P0, that P1, that P2 …, that Pi’ to ‘All my scientific beliefs are true’ is valid, but it is not the case that one ought to believe that all one’s scientific beliefs are true [the point is due essentially to Makinson (1965); see Zardini (2018a) for elaboration and variations]. BR do mention the preface paradox (pp. 16–18) and apparently reject, on the basis of the considerations marshalled by the paradox, that all their scientific beliefs are true, but, in spite of their insisting that one also ought to believe that all one’s scientific beliefs are true, those
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considerations would seem to show not only that one ought to reject that all one’s scientific beliefs are true, but also that one is under no pressure at all to believe the extremely strong, unlikely, unreliable, arrogant—and indeed rather silly—proposition that all one’s scientific beliefs are true— there is no dilemma about believing that not all one’s scientific beliefs are true, that is just the unambiguously right thing to believe.6 As for the formality condition, there would seem to be a logic of, say, metaphysical necessity as a particular necessity distinct from, say, epistemic necessity and governed by principles different from those governing other necessities (there is no interesting logic of a thin, formal ‘necessity’; it is only thick, material necessities that have interesting logics). However, I know of no reasonable sense in which the notion of metaphysical necessity is formal [see Zardini (2018c) for elaboration and variations]. It’s as simple as that. Having noted all this about BRT2, in the following (really, from the last paragraph of this section onwards), I’ll rather focus on BRT1. I’m very sympathetic to logical pluralism as the general view roughly stated in the fourth last paragraph. To take a natural example, I think that classical analysis is legitimate (and true)—and so, a fortiori, that its accompanying classical logic is legitimate—and that intuitionist analysis [in the style of e.g. Brouwer (1927)] is also legitimate (and true), and so, a fortiori, that its accompanying intuitionist logic is legitimate. Their apparent incompatibility can quite satisfactorily be explained away by the very plausible hypothesis that they describe different mathematical structures.7 It is true that some members of either party make extremely bold claims to the effect that, as a matter of principle, the mathematical structures postulated by the other party cannot exist [e.g. Dummett (2000), 250–269]. But I find such claims very implausible and only backed up by very weak arguments, so that I think that it can safely be concluded that such members are plain wrong (there’s room for everyone in Plato’s heaven!). Here as elsewhere, first-order tolerance requires a modicum of second-order intolerance. In effect, I thus accept the ‘regionalist’ thesis [already suggested by Putnam (1968)] that different logics apply to different domains. I submit that the opposite idea that the same logic applies to every logical domain gets whatever spurious plausibility it has from comparison with the much more plausible idea that the same physics applies to every physical domain. But, while physical domains would seem connected and homoge-
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neous enough to make the latter thesis rather likely, the whole range of logical domains—going from the most exotic physical particles through baldness and truth to the most esoteric mathematical structures—are so completely disconnected and highly heterogeneous as to make the former thesis rather unlikely. Given the bewildering variety of logical domains, it is just plausible to expect that a single logical operation can obey a certain principle on a certain domain and not obey it on another domain. I’m not quite sympathetic to logical pluralism as typically understood in the contemporary debate—where BR have been a major player—for, on that understanding, even when taking a particular argument and even when choosing a particular ‘level of formality’, as when, for example, it is debated in philosophy of logic [e.g. Beall (2007)] whether the particular argument from ‘The Liar sentence is true iff it is not’ to ‘The Earth is flat’ is valid at the level of the logic of truth, it is supposed to be legitimate to hold that the argument is valid and also legitimate to hold that it is not valid, and so, given that in this case presumably it is legitimate to accept the premise of the argument if (and only if ) it is legitimate to hold that the argument is not valid,8 it is in effect supposed to be legit to accept the problematic instances of the T-schema and also legit to reject them! In addition to undermining the significance of what is in fact one of the most significant debates in philosophy of logic, pending first arguments such specific claim is in itself so helplessly relativist that it would seem it ought to lead to the rejection of any general view entailing9 it. I’m even less sympathetic to BRT (in particular, to its ‘only’-component). For, quite generally, BRT is yet another example of trying to impose a format to which every instance of a kind has to conform (in this case, every instance of the kind Legitimate Relation of Logical Consequence). Besides the telling considerations of Wittgenstein (1953), §66 (which would not seem to have adequately been heeded in much subsequent analytic philosophy), I think that the track record of such attempts in the history of thought in general and in philosophy in particular (and even more in particular about such philosophically central kinds as Truth, Knowledge, Cause, etc.) is extraordinarily negative. And I don’t see any reason for thinking that the specific case of the kind Legitimate Relation of Logical Consequence will deviate from this general tendency. Indeed, in the following I’ll corroborate this point by discussing a particular area that systematically violates GTT. The discussion is ultimately meant to achieve something more interesting
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than simply shooting a sundry series of cheap counterexamples at GTT, and even more interesting than pointing to a relatively unified area that systematically violates it; to wit, it is meant to uncover a fundamental divergence between logical consequence and truth preservation, and, relatedly, to invert the nowadays dominant order of priority between logical consequence and logical truth.10
Substructure The area that, in section “Generalised Tarski’s Thesis”, has been referred to as systematically violating GTT is the one of substructural logics [see Restall (2000); Paoli (2002) for a couple of general, philosophically sensitive surveys]. To recall, operational properties are those properties of a logic which concern particular logical operations (for example, reductio ad absurdum is an operational property in that it concerns negation). By contrast, structural properties are those properties of a logic which concern general features that only depend on considering premises and conclusions as a manipulable structure of unstructured objects (for example, reflexivity is a structural property in that it only depends on the premise and conclusion being the same object). Classical logic and many non-classical logics (for example, intuitionist logic) have a series of noteworthy structural properties: reflexivity, monotonicity, transitivity, contraction, commutativity and others. A logic is substructural iff it lacks one of the properties in that series. I’ll argue then that GTT systematically rules out all kinds of philosophically interesting substructural logics as legitimate relations of logical consequence. Given some of my work, I actually have a particular emotional attachment to this exclusion from the realm of ‘logical consequence’ :-( But, as in many other cases, it’s really not that interesting to determine exactly to which logics the label ‘legitimate relation of logical consequence’ applies (either in virtue of the vagaries of the use of that expression or in virtue of a sheer stipulation).11 What’s interesting is what kind is singled out by GTT, and how central it is for philosophical theorising about logic. In that respect, my cumulative argument will show that the kind singled out by GTT is not central for philosophical theorising about logic. And, as has already been foreshadowed, there will also be other consequences for broadly Tarski-inspired theses.
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The Crash
Reflexivity A first structural property is reflexivity: (I) φ ⊢ φ holds. There are philosophical reasons to doubt (I) [Moruzzi and Zardini (2007), pp. 180–182].12 Firstly, (I) refers to circular arguments—so circular that they might be thought to violate epistemic and metaphysical principles governing logical consequence. Epistemically, one might try to forge some connection between validity and transmission [Wright (2000)] and think that a valid argument must be such that one can use it in at least some context to acquire a new justification for believing its conclusion13 [see Martin and Meyer (1982) for very broadly similar considerations].14 But in no context can one acquire a new justification for believing φ by inferring it by (I) from φ.15 Metaphysically, one might try to forge some connection between Ableitbarkeit and Abfolge [Bolzano (1837)] and think that a valid argument must be such that, if its premises are true in virtue of non-logical facts, its conclusion must be true (also) simply in virtue of the truth of the premises.16 But x-is-true-in-virtue-of-the-truth-of-y is non- reflexive (i.e. such that, for some x, it is not the case that x is true in virtue of the truth of x), even restricting to sentences that are true in virtue of non-logical facts.17 Secondly, notice that (I) is an atypical structural property, in that it licences the validity of rules (things of the form ‘Γ ⊢ φ holds’) rather than metarules (things of the form ‘If Γ0 ⊢ φ0, Γ1 ⊢ φ1, Γ2 ⊢ φ2 … hold, ∆ ⊢ ψ holds’). That would seem already in itself problematic: if at least almost all allegedly valid structural properties licence metarules rather than rules, that is good evidence that all allegedly valid structural properties licence metarules rather than rules. And it becomes even more problematic once it is realised that, for this reason, (I) violates the plausible principle that an argument is valid in virtue of some of the semantic properties of some expressions occurring in its premises or conclusion—principle that is entailed by some understandings of the plausible principle that argument validity is a matter of analyticity, for example by the understanding that an argument’s validity consists in the fact that its
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conclusion preserves the truth of its premises in virtue of the meaning of some expressions occurring in its premises or conclusion.18 That is all exciting stuff, but enter now GTT. In every casex where φ is true, so is φ. That holds no matter what x is, and so non-reflexive logics cannot drop out of GTT, and hence, by BRT1, non-reflexive logics are not legitimate relations of logical consequence. That’s an unhelpful ruling! The objection is sometimes raised that logical consequence is only concerned with truth preservation—i.e. with the fact that the conclusion is true except the premises are not—and variations thereof, whereas the reasons given in the second last paragraph in favour of non-reflexive logics clearly advert to quite different styles of considerations. Whatever its other merits [Zardini (2012)], the objection relies on an untenable separation of truth preservation from other features (such as those adverted to in the second last paragraph). For at least some of the same styles of considerations given in favour of non-reflexive logics also speak in favour, for at least some kind of implication, of the claim that, for some P, ‘If P, then P’ is not true. But that implication will also presumably satisfy the deduction theorem of its home logic, and so its home logic had better be non- reflexive on pain of validating ‘If P, then P’, and so on pain of failing to be truth preserving. Failure of desiderata for implication different from truth preservation thus amplifies, via the deduction theorem, into failure of the desideratum for logical consequence of truth preservation. Say that an argument immediately fails to be truth preserving iff its premises are true and its conclusion is not (which is presumably tantamount to the argument not being [truth preserving as per the idea presented at the beginning of this paragraph]), and say that an argument ultimately fails to be truth preserving iff its validity entails the validity of an argument that immediately fails to be truth preserving. Then, the point of this paragraph can be summed up by saying that, while (I) does not immediately fail to be truth preserving, it does ultimately fail to be truth preserving.
Monotonicity A second structural property is monotonicity: (K) If Γ ⊢ φ holds, Γ, ∆ ⊢ φ holds.
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(K) is uncontroversially not valid for many relations of non-deductive consequence (i.e. relations where the truth of the premises supports but does not guarantee the truth of the conclusion). That is not in itself particularly relevant, since BRT is only about logical consequence, which is a relation of deductive consequence (i.e. a relation where the truth of the premises does guarantee the truth of the conclusion). However, even in the case of logical consequence, there are philosophical reasons to doubt (K), as it licences arguments with irrelevant premises [a comment analogous to the one in section “Reflexivity” applies concerning the connection with truth preservation, cf Moruzzi and Zardini (2007), p. 184; Zardini (2015a), pp. 228–229].19 That is all exciting stuff, but enter now GTT. In every casex where every element of Γ, ∆ is true, so is every element of Γ. Therefore, if, in every casex where every element of Γ is true, so is φ, then, in every casex where every element of Γ, ∆ is true, so is φ. That holds no matter what x is, and so non-monotonic logics cannot drop out of GTT, and hence, by BRT1, non-monotonic logics are not legitimate relations of logical consequence. That’s an unhelpful ruling! It is an especially unhelpful ruling for anyone with an interest in relevant logics [e.g. Anderson and Belnap (1975)]: for, while these are traditionally more concerned with implication rather than with logical consequence and consequently presented axiomatically (pace the [logical- consequence]-first doctrine I’ll criticise in section “From Logical Consequence Back to Logical Truth”!), they can also naturally be presented in such a way that (K) is not valid in them (a presentation which philosophically makes a lot of sense, since as I’ve in effect noted in the second last paragraph, irrelevance is not more benign for logical consequence than it is for implication). Given that relevant logics are—together with intuitionist logic—BR’s main witness for logical pluralism, it is actually somewhat strange that they do not directly thematise this issue. The issue does obliquely creep up in their discussion of the status of LEM in their own version of relevant logics (pp. 53, 100–101): in that version, because they want to deny that φ ⊢ ψ ∨¬ψ holds, but implicitly assume (K), they must deny that ∅ ⊢ ψ ∨¬ψ holds, and so, I take it, deny that LEM is a logical truth20 (both of which they actually do). They sweeten
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the pill by saying that LEM is a necessary truth. But that’s not good enough. One problem with BR’s position is that many relevant logics do count LEM as a logical truth: for example, the relevant logic BX is defined in terms of adding LEM as a logical truth21 to the basic relevant logic B (analogously, a few other relevant logics …X are defined in terms of adding LEM as a logical truth to the relevant logic …), and the even more influential relevant logic R can be—and often is—partially characterised in terms of having LEM as a logical truth. Be that as it may with LEM in particular, the same issue arises with almost any alleged logical truth (save for absolutely trivial things like ‘For some P, P’). Since the whole hierarchy of relevant logics as standardly conceived relies on such alleged logical truths being logical truths in the pertinent logics, and since anyways I’d like to think that there are relevant logical truths in addition to absolutely trivial things like ‘For some P, P ’, I deem BR’s position untenable (even without going into the fact that it does not even speak to the case where (K) is applied to an argument with premise and conclusion). They should either, pace philosophical common sense (and their own choice of witness), deny that relevant logics are legitimate relations of logical consequence or, pace GTT, deny (K).
Transitivity A third structural property is transitivity: (S) If Γ ⊢ φ and ∆, φ ⊢ ψ hold, ∆, Γ ⊢ ψ holds. (S) is uncontroversially not valid for some relations of non-deductive consequence [Zardini (2015a), pp. 231–233]. Again, for the reason seen in section “Monotonicity”, that is not in itself particularly relevant. However, even in the case of logical consequence, there are philosophical reasons to doubt (S), as, in the presence of extremely compelling principles, it gives rise to the paradoxes of vagueness [e.g. Zardini (2008)]. For example, (S) gives rise to the Sorites paradox in the presence of the extremely compelling principle of tolerance (if something has a vague property, so does something relevantly similar to it).
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That is all very exciting stuff, but enter now GTT. If, in every casex where every element of Γ is true, so is φ, then, in every casex where every element of ∆, Γ is true, so is every element of ∆, φ. Therefore, if, in every casex where every element of ∆, φ is true, so is ψ, then, in every casex where every element of ∆, Γ is true, so is ψ. That holds no matter what x is, and so non-transitive logics cannot drop out of GTT, and hence, by BRT1, non-transitive logics are not legitimate relations of logical consequence. That’s a very unhelpful ruling!
Contraction Up to now, we’ve been implicitly assuming that premises are combined in a very coarse-grained way, suitably modelled by sets of sentences. But one of the hallmarks of a few substructural logics is precisely the rejection of that assumption. To open up the possibility of a more fine-grained way of combining premises, let’s henceforth replace in our formal modelling sets of sentences with series of sentences. A fourth structural property is then contraction: (W) If Γ, φ, φ ⊢ ψ holds, Γ, φ ⊢ ψ holds. There are philosophical reasons to doubt (W). Firstly, in the presence of extremely compelling principles, (W) gives rise to the semantic paradoxes [e.g. Zardini (2011)]. For example, (W) gives rise to the Liar paradox in the presence of the extremely compelling principle of correspondence (‘P’ is true iff P). Secondly, (W) forecloses an attractive account of what it is for an object to be changing from one state to its opposite [Zardini (2018d)]. That is all very exciting stuff, but enter now GTT. In every casex where every element of Γ, φ is true, so is every element of Γ, φ, φ.22 Therefore, if, in every casex where every element of Γ, φ, φ is true, so is ψ, then, in every casex where every element of Γ, φ is true, so is ψ. That holds no matter what x is, and so non-contractive logics cannot drop out of GTT, and hence, by BRT1, non-contractive logics are not legitimate relations of logical consequence. That’s a very unhelpful ruling!
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Commutativity A fifth structural property is commutativity: (C) If Γ, φ, ψ, ∆ ⊢ χ holds, Γ, ψ, φ, ∆ ⊢ χ holds. There are philosophical reasons to doubt (C). Firstly, there might be context change between an occurrence of a premise and another occurrence of a premise [e.g. Zardini (2014a)]. For example, if each next occurrence occurs on the day after the day on which the previous occurrence occurs, 〈‘Tomorrow it’ll rain’, ‘It’s not raining’〉 is naturally regarded as inconsistent (and so, let’s assume, as entailing everything), but 〈‘It’s not raining’, ‘Tomorrow it’ll rain’〉 isn’t.23 Secondly, (C) forecloses an attractive account of what it is for an object to be changing from one state to its opposite [Zardini (2018d)]. That is all very exciting stuff, but enter now GTT. In every casex where every element of Γ, ψ, φ, ∆ is true, so is every element of Γ, φ, ψ, ∆. Therefore, if, in every casex where every element of Γ, φ, ψ, ∆ is true, so is χ, then, in every casex where every element of Γ, ψ, φ, ∆ is true, so is χ. That holds no matter what x is, and so non-commutative logics cannot drop out of GTT, and hence, by BRT1, non-commutative logics are not legitimate relations of logical consequence. That’s a very unhelpful ruling!
3
Aftermath
Unireliabilism In all the cases we’ve considered in Sect. 2, GTT fails in that its satisfaction by a logic in effect requires that the logic capture the fact that, for every model in a certain class, it is not the case that the premises are true in the model and the conclusion is not. Ascending to a more abstract perspective, GTT fails in those cases arguably at bottom because it is thus a species of a more general kind of occurrence-insensitive, single-value, single-model reliabilism: ‘reliabilism’ because it only considers whether the
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conclusion has the relevant value when the series of premises do (rather than considering stronger connections between the conclusion having the relevant value and the series of premises doing); ‘occurrence-insensitive’ because it only considers which sentences the premises are (rather than considering, for example, how many times and in what order they occur); ‘single-value’ because it only considers one value relevant for all the series of premises and the conclusion (rather than considering different values relevant for different occurrences of the premises or the conclusion); ‘single-model’ because, on each test,24 it only considers one model for all the series of premises and the conclusion (rather than considering different models for different occurrences of the premises or the conclusion). Since such reliabilism is hostile to plurality in occurrences, values and models in testing for logical consequence, it may well be labelled for short ‘unireliabilism’. The issue surfaces slightly when BR note (p. 91) that, because of its ‘preservationism’, GTT rules out non-reflexive and non-transitive logics. But that’s just the tip of the substructural iceberg—as we’ve seen in Sect. 2, GTT systematically rules out all kinds of philosophically interesting substructural logics! Quite generally, GTT clashes with substructurality. And the culprit is not some vague, benign and acceptable ‘preservationism’—as we’ve seen in the last paragraph, GTT clashes with substructurality because it is a species of the much more precise, malign and questionable unireliabilism, which, obliterating as it does plurality in occurrences, values and models, as well as intensional connections between premises and conclusion, makes any version of ‘pluralism’ based on it barely worth its name. Indeed, once GTT’s unireliabilism has been identified as the culprit for its clash with substructurality, it becomes clear that, just as GTT generalises in a very natural way the classical idea that an argument is valid iff, for every classical model, it is not the case that the premises are true in the model and the conclusion is not (by relaxing its ‘classical- model’-component), one can generalise further in equally natural ways unireliabilism itself (by relaxing some of its components). And at least the specific substructural logics alluded to in Sect. 2—rather than representing conceptions of logical consequence totally alien to GTT—can actually be thought of as instances of such further generalisation. To wit, non-reflexive logics of the kinds alluded to in Sect. 2 can be thought of as requiring that the value of the conclusion be better than the values of the
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premises (thus clashing with the ‘single-value’-component of unireliabilism). Non-monotonic logics require that the value of the conclusion be due to the values of the premises (thus clashing with the ‘reliabilism’component of unireliabilism). Non-transitive logics of the kind alluded to in Sect. 2 can be thought of as allowing that the value of the conclusion be worse than the value of the premises (thus clashing with the ‘single-value’-component of unireliabilism). Non-contractive and non- commutative logics are sensitive to the number and order of occurrences of a premise (thus clashing with the ‘occurrence-insensitive’-component of unireliabilism). Non-commutative logics of the first kind alluded to in Sect. 2 can be thought of as testing different occurrences of the premises or the conclusion at different models (thus clashing with the ‘single- model’-component of unireliabilism).25 I see no reason for stopping at unireliabilism and not embracing these further dimensions of plurality which give rise to substructurality.26
Logic-Relative Version of GTT; an Absolute-Truth A Formulation of a Logic-Relative Version of GTT Although it concerns a variety of logics (most of which non-classical), BRT itself is a theory in classical logic [in this, it follows a well-trodden path: for a variety of reasons—some better than others according to e.g. Zardini (2018b)—the usual metatheory of many non-classical theories is indeed classical]. Classical (more accurately, non-substructural) reasoning was in fact crucial at several places in my arguments in Sect. 2. For example, when I argued that ‘in every casex where φ is true, so is φ’, I was simply helping myself to the classical logical truth ‘Every F is F’. Thus, taking our cue from a theme already emerged in fn 19, can’t we simply fix the problem by adopting instead a logic-relative version of GTT, requiring only that GTT hold for a logic once GTT is interpreted27 and evaluated using that very same logic (so that, for example, in discussing GTT for non-reflexive logics, ‘Every F is F’ is no longer available as a logical truth)? No, we can’t. To begin with, we’d need to supplement the logic with theories of syntax, of cases and of truth-in-a-case, and not every interesting logic will be able adequately to sustain such supplementations, since the required expressive resources might be lacking from the logic or they
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might be governed by too weak principles of the logic. Not every interesting logic has been developed with the purposes of doing syntax or semantics. Moreover, picking up from the second disjunct of the second last sentence of the last paragraph, it’s completely up for grabs which logics would satisfy the logic-relative version of GTT. Indeed, it is likely that quite a few interesting logics will not satisfy its logic-relative version, and we in fact know that even some of those that did not create problems for GTT will no longer satisfy its logic-relative version. For example, the Bočvar logic B3 [Bočvar (1938)] and the Kleene logic K3 [Kleene (1938)] will not satisfy the logic-relative version of GTT, since they lack a suitably weak implication vindicating claims—so crucial to GTT—along the lines of ‘Every F is G’ (for example, ‘In every casex where φ is true, so is φ’). In fact, we also know that some interesting logics that did create problems for GTT will still not satisfy its logic-relative version. For example, non- contractive logics will not satisfy the logic-relative version of GTT, since, letting ⊗ be a multiplicative conjunction, in typical non-contractive logics φ ⊢ φ ⊗ φ does not hold, but, in the lattice-theoretic semantics associated with typical non-contractive logics [e.g. Ono (2003)], ⊗ is idempotent on designated value (i.e., if φ gets designated value, φ ⊗ φ gets the same value), so that ‘If φ gets designated value in a certain casex, so does φ ⊗ φ’ and hence—since, presumably, φ is true in a case iff φ gets designated value in that case—‘If φ is true in a certain casex, so is φ ⊗ φ’ hold (and presumably do so even when using a non-contractive logic in the metatheory). GTT’s rulings were definitely unhelpful, but those of its logic-relative version are likely to be helpless. Furthermore, while the logic-relative version of GTT is designed to preserve its letter, it arguably abandons its spirit. For GTT’s spirit arguably incorporated unireliabilism, but the new substructural interpretations of its universal quantification and implication deprive GTT of one component or other of unireliabilism. Indeed, there is barely anything substantial in common at all, unireliabilist or otherwise, between, say, the sense in which a non-reflexive logic thinks it satisfies GTT and the sense in which a non-transitive logic thinks it does. The fact that the logic-relative version of GTT holds for the logics for which it does would seem simply to show that such logics are in the relevant respects expressively adequate
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rather than show that they all conform to the same format. GTT is paid lip service, but wildly different doctrines are actually pursued behind the chants of truth preservation. Finally, it is hard to see what roles cases are playing anyways in the logic- relative version of GTT. Taking a conditional of the form ‘If every premise is true, so is the conclusion’, say that its interpretation in terms of classical [universal quantification and implication] is an ‘[individual- truth]-preservation conditional’ (in the sense that the individual truth of all the premises is preserved by the conclusion), whereas its interpretation in terms of non-classical [universal quantification and implication] (as these are available in the logic in question) is a ‘[collective-truth]-generation conditional’ (in the sense that the collective truth of the series of premises generates the truth of the conclusion).28 (Make the same distinction if universal quantification is replaced by conjunction.)29 Then, for the non-classical logics that satisfy the logic-relative version of GTT, the non-classicality of plain [collective-truth]-generation conditionals (which are typically much stronger than the corresponding [individual- truth]-preservation conditionals) will presumably suffice to rule out the undesired classical arguments, without quantification over cases and with absolute truth instead of truth (or designated value) relative to cases.30 True, that might still rule in a lot of other undesired supra-classical arguments (i.e. arguments whose form is not valid even in classical logic), as, for example, the argument from ‘Italy will play against Sweden’ to ‘Italy will lose’ (since the [collective-truth]-generation conditional ‘If “Italy will play against Sweden” is true, so is “Italy will lose”’ sadly holds). But, for better or worse, such arguments will be ruled out by BRT2 (especially, by the necessity and formality conditions). Cases are supposed to be nothing less than the cornerstone of GTT (pp. 23–24), but they are nothing more than a frill of its logic-relative version.
From Logical Consequence Back to Logical Truth The logic-relative version of GTT cannot thus rescue BRT-style pluralism. Still, I think that, especially in its final, absolute-truth formulation (without cases or designated values, as per the last paragraph of section “A
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Logic-Relative Version of GTT; an Absolute-Truth Formulation of a Logic-Relative Version of GTT”), the logic-relative version of GTT affords an important insight into some of those logics that satisfy it31 (which include the specific substructural logics alluded to in Sect. 2).32 But, before I can adequately explain this in the ninth next paragraph, I first need to lay out some background. A mantra in contemporary philosophy of logic [e.g. Etchemendy (1988), pp. 74–78] is represented by the [logical-consequence]-first doctrine already mentioned in section “Monotonicity”, according to which the central property in logic is logical consequence, and logical truth is grounded33 in it as being merely a degenerate limit case of logical consequence: those valid arguments that have no premises and one conclusion (where a logical truth is taken to be the conclusion of any such valid argument)34 [from a historiographical point of view, it is usual to attribute to Tarski the shift in focus from logical truth to logical consequence against the evil Frege-Russell tradition, see e.g. Tarski (1930)]. The great insight is supposed to be that logic is not a body of truths but a body of rules, and so, in this respect, more akin to etiquette than to mathematics. As I’ve already mentioned and referenced in fn 20, BR themselves seem actually wary of the [logical-consequence]-first doctrine, apparently for the reason that one could equally well ground logical consequence in logical truth by saying that, given a logic L, the fact that φ0, φ1, φ2 …, φi ⊢L ψ holds is grounded in the fact that φ0 & φ1 & φ2 … & φi → ψ is a logical truth in L. While my main aim in this section is to argue against the [logical-consequence]-first doctrine, I should emphasise that by no means do I intend to defend an opposite universalist view along the lines of a [logical-truth]-first doctrine, and, in particular, that I regard the grounding just mentioned as untenable. Firstly, and less importantly, the grounding in question does not cover cases of logical consequence with infinitely many premises. My two cents is that this is probably best dealt with by extending the logic with infinitary conjunction—after all, logics without conjunction would already have had to be extended with (normal) conjunction to deal with cases of logical consequence with at least two premises. In for a penny, in for a pound. Of course, there might be issues as to whether every logic with
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valid arguments with at least two premises supports some such extension, and, if it does, whether it supports a privileged one that can be relied on in vindicating the grounding in question.35 Secondly, and more importantly, the grounding in question runs afoul of logics with non-standard behaviour of conjunction or implication. As for conjunction, a good example is given by Sb (fn 6), which is a philosophically interesting and technically well-behaved logic, but in which φ & ψ → χ can be a logical truth without φ, ψ ⊢ χ holding (essentially because φ, ψ ⊢Sb φ & ψ does not hold). As for implication, good examples are given by B3 and K3, which are philosophically interesting and technically well-behaved logics, but in which φ ⊢ ψ can hold without φ → ψ being a logical truth (essentially because φ → ψ is indeterminate as soon as either (B3) or both (K3) of φ and ψ are). (A dual, equally good example is given by the logic of paradox LP [Asenjo (1966)], which is a philosophically interesting and technically well-behaved logic, but in which φ → ψ can be a logical truth without φ ⊢ ψ holding (essentially because, in LP, φ → ψ is true as soon as φ is false, even if it is also true).) Notice that all these examples depend on taking as operative conjunction and implication those immediately available in the logics. It is certainly technically possible to extend the logics with further operations with a more standard behaviour; the problem is that there would typically seem to be no privileged way of doing so [witness the plethora of non-material implications proposed for extending K3 or LP; see e.g. Field (2008); Priest (2006) respectively, and Zardini (2016) for some critical discussion], which undermines the grounding of logical consequence in logical truth, since it leaves one with no determinate grounding basis. To elaborate on a particularly resilient version of the problem, suppose that, for example, a particular implication ⇒0 does at least the extensional part of the job for some such logic L, so that, even without restricting to the ⇒0-free language, φ0, φ1, φ2 …, φi ⊢L ψ holds iff φ0 & φ1 & φ2 … & φi ⇒0 ψ is a logical truth in L. A particularly resilient version of the problem is that there typically is at least one other, equally natural implication ⇒1 whose logic and so whose meaning differ, respectively, from the logic and so from the meaning of ⇒0 but which also does at least the extensional part of the job, so that, even without restricting to the ⇒1-free (or ⇒0-free) language, φ0, φ1, φ2 …, φi ⊢L ψ holds iff φ0 & φ1 & φ2 … & φi ⇒1 ψ is a
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logical truth in L. One would then be faced with the riveting question of whether logical consequence in L is grounded in the ⇒0-implicational logical truths of L or in the ⇒1-implicational logical truths of L (or in both, or in either, or partly in one partly in the other, etc.).36 Thirdly, and even more importantly, the grounding in question is shattered by logics with a standard behaviour of the determinacy operator [see Zardini (2014c) for some contrast with a non-standard behaviour of that operator]. A good example is given by the supervaluationist logic Sp [Fine (1975)], which is a philosophically interesting and technically well- behaved logic, but in which φ ⊢ ψ can hold without φ → ψ being a logical truth (essentially because φ ⊢Sp 𝒟 φ holds but φ → 𝒟 φ is not a logical truth in Sp). Points analogous to those made in the last paragraph hold concerning the possibility of extending the logic with a further implication satisfying the deduction theorem. Does the [logical-consequence]-first doctrine fare any better than the [logical-truth]-first one? Well, the Tarskian shift from logical truth to logical consequence is certainly beneficial insofar as one should also look at logical consequence, which, for the reasons mentioned in the last three paragraphs, in a few logics includes features that genuinely go over and beyond those of logical truth.37 This observation, however, falls dramatically short of vindicating the [logical-consequence]-first doctrine in its full strength, since, for example, the doctrine insists that, even for those logics where there is no such difference between logical consequence and logical truth, logical truth is grounded in logical consequence. I’m going to attack the [logical-consequence]-first doctrine precisely by arguing that, for some of those logics (in particular, for the specific substructural logics alluded to in Sect. 2), it is logical consequence that is grounded in logical truth rather than vice versa. My argument starts with the plausible if somewhat controversial assumption that logical consequence is itself grounded in facts about truth, rather than being primitive [e.g. Field (2015)] or grounded in other kinds of facts like, for example, facts concerning proof [e.g. Prawitz (2005)]. In turn, this standard, general semantic conception of logical consequence is typically accepted in a specific version, in classical logic, according to which logical consequence is grounded in guaranteed preservation of the truth of the premises by the conclusion [following in this Tarski again, especially Tarski (1936)].
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What for our purposes is interesting to observe is that such truth- preservation account of logical consequence is very close to the one got by reading (as it is in the spirit of BRT, see pp. 23–24) a dependence claim in the left-to-right direction of GTT. Let’s make this observation more accurate. The truth-preservation account cannot plausibly relax GTT and unireliabilism’s ‘reliabilism’-component, since the account (‘guaranteed preservation of the truth of the premises by the conclusion’) is all about the conclusion being true except the premises are not, a condition which, as I’ve already in effect observed in section “Reflexivity” (and reinforced in fn 19), is presumably tantamount to the condition that it is not the case that the premises are true and the conclusion is not.38,39 The truthpreservation account cannot plausibly relax GTT and unireliabilism’s ‘occurrence-insensitive’-component, since the account (‘guaranteed preservation of the truth of the premises by the conclusion’) is all about the truth of the premises (and the conclusion), and that is a question on which, for example, φ and φ, φ necessarily agree.40 They might disagree on some novel understanding of premises as tokens (fn 22), but such understanding is, at the present stage of inquiry, sheer speculation. The truth-preservation account cannot plausibly relax GTT and unireliabilism’s ‘single-value’-component, since the account (‘guaranteed truth preservation from the premises to the conclusion’) is all about identity in value between the premises and the conclusion.41 The truth-preservation account can plausibly relax GTT and unireliabilism’s ‘single-model’-component, since nothing in the account (‘guaranteed preservation of the truth of the premises by the conclusion’) prevents that the guarantee of truth preservation consists in its holding over a certain class of tests that might look at different models for different occurrences of the premises or the conclusion. Summing up, the truth-preservation account can plausibly [diverge from GTT and unireliabilism while remaining faithful to itself ] only with respect to their ‘single-model’-component: it is still essentially a species of occurrence-insensitive, single-value reliabilism. Therefore, the truth-preservation account clashes with substructurality for the same reasons for which GTT does (minus the ‘single-model’-problem with the first kind of non-commutative logic alluded to in section “Commutativity”). Substructural logical consequence is not grounded in truth preservation.42
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In what, pray, is it then grounded? It is at this juncture that the [collective-truth]-generation conditionals can play a role, at least for those substructural logics (extendable) with conjunction and implication behaving in the way mentioned in fn 31 (which include the specific substructural logics alluded to in Sect. 2; the proposal I’m about to develop is supposed to apply mainly to those logics, and possibly to some more substructural logics (extendable) with conjunction and implication behaving in the same way). For, although, for those logics, the truth- preservation account of logical consequence must be given up, we can still uphold the more general semantic conception of logical consequence by appealing to the idea that logical consequence is grounded in guaranteed generation of the truth of the conclusion by the collective truth of the series of premises (i.e. by replacing in the truth-preservation account an [individual-truth]-preservation conditional with a [collective-truth]-generation conditional). But how are we to spell out this idea more precisely? As a first stab, we could say that, in every specific substructural logic L alluded to in Sect. 2, the fact that φ0, φ1, φ2 …, φi ⊢L ψ holds is grounded in the fact that the [collective-truth]-generation conditional T ⌜φ0⌝ & T ⌜φ1⌝ & T ⌜φ2⌝ … & T ⌜φi⌝ → T ⌜ψ⌝, as interpreted by L, holds over the class of tests relevant for L. To say this would probably be the most direct modification of the most standard way of spelling out the truth- preservation account of logical consequence, but it is only a first stab as it would have a couple of serious problems. Firstly, even when L does have a ‘class of relevant tests’, these often involve as models rather exotic mathematical structures with no direct informal interpretation, and which can therefore hardly serve as a credible ground for logical consequence (contrary to the class of tests relevant for, say, classical logical consequence, which involve as models structures that can directly be informally interpreted as mundane possible situations). Secondly, as I’ve stressed in fn 30, the only notion of truth straightforwardly available in L is the one of absolute truth, whose interaction with the models involved in the tests relevant for L is however ill-defined (what is well-defined is the notion of truth-in-a-model, but it is well-defined in the classical metatheory of L, not in L itself, and, even if it could be defined in L itself, it would give the wrong results, for example, as per the argument in section “A Logic- Relative Version of GTT; an Absolute-Truth Formulation of a Logic- Relative Version of GTT”, for typical non-contractive logics).
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What is causing both problems is reliance on a class of tests, and the solution to both problems is therefore to get rid of it. But just how? Notice first that the only point of quantification over such class in the most standard way of spelling out the truth-preservation account of logical consequence is to ensure that logical consequence has the desired modal force. Observe then that, for the purposes of grounding, there is arguably no way of defining the class in purely non-logical terms (for one thing, as we’ve seen in fn 5, it won’t do to define it in terms of metaphysically possible worlds).43 If so, the most standard way of spelling out the truth-preservation account in effect amounts to grounding logical consequence in (among other things) logical possibility (of the models involved by the relevant tests). It does not offer a grounding of logical facts in non-logical facts; it only offers a grounding of logical facts in general in a privileged class of facts concerning logical possibility and individual-[truthin-a-model] preservation. But then it would seem that nothing of philosophical value is lost by taking as more fundamental not logical possibility as applied to models but logical necessity (aka logical truth) as applied to sentences,44 and say instead that the fact that φ0, φ1, φ2 …, φi ⊢L ψ holds is grounded in the fact that the [collective-truth]-generation conditional T ⌜φ0⌝ & T ⌜φ1⌝ & T ⌜φ2⌝ … & T ⌜φi⌝ → T ⌜ψ⌝ is a logical truth in L [cf Zardini (2014d), pp. 371–382 for the development of a congenial proposal in the specific case of the non-contractive approach to the semantic paradoxes mentioned in section “Contraction”],45 thereby grounding logical facts in general in a privileged class of facts concerning logical necessity and collective-[absolute-truth] generation (and, along the way, getting rid of classes of tests).46 If you’ve read fn 46, you might now be tempted by the thought that semantic ascent is dispensable, and so that we should rather say that the fact that φ0, φ1, φ2 …, φi ⊢L ψ holds is grounded in the fact that the object-language conditional φ0 & φ1 & φ2 … & φi → ψ is a logical truth in L, thereby grounding logical facts in general in a privileged class of facts concerning logical necessity and conjunction-cum-implication (and so jettisoning the semantic conception of logical consequence altogether). The temptation should be resisted, as the proposed alternative grounding arguably associates some facts about logical consequence with the wrong facts about logical truth. Let’s take as example a typical non-contractive logic LW (with ‘and⊗’ and ‘if→ …, then→ …’ as multiplicative conjunction
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and implication, respectively) and the fact that, in LW, by adjunction, ‘Snow is white’, ‘Grass is green’ entail ‘Snow is white and⊗ grass is green’. According to my proposal, that is grounded in (the logical necessity of ) the fact that, if→ ‘Snow is white’ is true and⊗ ‘Grass is green’ is true, then→ ‘Snow is white and⊗ grass is green’ is true, whereas, according to the alternative proposal, that is grounded in (the logical necessity of ) the fact that, if→ snow is white and⊗ grass is green, then→ snow is white and⊗ grass is green. The former account of the matter is clearly superior, since it respects the difference in the argument between the series of premises and the conclusion, and in particular it respects the point of the argument which is to the effect that the simple ‘Snow is white’ and ‘Grass is green’ together suffice for the complex ‘Snow is white and⊗ grass is green’. As in the classical case, it just is illuminating to use in the metalanguage the notions of conjunction and truth for combining two premises and then, mentioning a certain single sentence of the object language which involves an operator expressing the very same notion of conjunction, explaining why that conclusion follows from the two premises—it just is illuminating to use the idea of two sentences being both true to make a point about a single sentence involving ‘and’.47 All this is completely missed by the latter account, which flattens such wealth of structure concerning the interaction between premise combination and conjunction into a tautology that has nothing to do with either.48 Semantic ascent allows us to discern the fine structure of logical consequence: its combination of premises and its entailment from premises to conclusion.49 Having resisted the temptation, it only remains to state with due solemnity a consequence of the approach we’ve been pursuing which has been looming large for a while (and which, for that matter, is also a consequence of the alternative proposal discussed in the last paragraph). Namely, and quite literally, on this revolutionary approach, the [logical- consequence]-first doctrine is turned upside down: in the specific substructural logics alluded to in Sect. 2, not only logical truth is not grounded in logical consequence, but, vice versa, it is logical consequence that is grounded in logical truth.50 In those logics, the central property is logical truth, and logical consequence is grounded in it as being merely a degenerate limit case of logical truth: those logical truths that are [collective- truth]-generation conditionals (where the series of premises and the con-
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clusion of a valid argument are taken to be, respectively (and modulo semantic descent), the series of conjuncts in the antecedent and the consequent of any such logical truth).51 In those logics just as elsewhere, rules are only legitimate insofar as they are based on truths.
Notes 1. Throughout, page references are to BR unless otherwise stated. 2. Throughout, I follow BR in assuming, merely for simplicity, a multiple- premise, single-conclusion framework. The extension of my points to the (superior but) more complex multiple-premise, multiple-conclusion framework is straightforward. 3. BR claim that necessity is important for logical consequence at least partly because it guarantees that logic applies unrestrictedly in hypothetical reasoning (pp. 15–16). But it arguably doesn’t in the first place, since, say, the law of excluded middle (henceforth, ‘LEM’) is a logical truth but it is arguably wrong to reason that, if Brouwer were right, it would be the case that either Goldbach’s conjecture holds or it doesn’t. Notice that the hypothesis is perfectly possible (Brouwer might have held correct views), so it is not even the case that logic applies unrestrictedly to possible hypotheses (which would anyways raise the issue of why it should be more important unrestrictedly to apply to possible hypotheses rather than to, say, interesting hypotheses). 4. Throughout, I work with a standard framework for context-dependent languages where the truth of a sentence is relative both to a context of utterance and to a circumstance of evaluation, and I make such relativities explicit with the relevant parts of the construction ‘φ is true-as-uttered-in-C-as-evaluated-at-E’. 5. I’d like to forestall two likely reactions to this point. Don’t say that what is true is ‘For every context C, if “Snow is white” is true-as-uttered-inC-as-evaluated-at-C, “Actually, snow is white” is true-as-uttered-in-C-asevaluated-at-C’. The proposed Ersatz has the embarrassing problem (emerging in the opening ‘what is true [my emphasis, EZ]’) that neither it nor its embedded conditional is non-trivially necessary (so that the spirit if not the letter of the necessity condition would seem violated). What the proposed Ersatz indeed offers is universality over contexts, but that is a far cry from any interesting necessity: for every context C, ‘I exist’
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is true-as-uttered-in-C-as-evaluated-at-C, but there is no interesting sense in which the existence of a first person is necessary [Zardini (2012), pp. 266–268 will be excused if he perhaps got too carried away by his ‘semantic modality’]. Along these lines, the proposed Ersatz is not even of the right logical form: for the necessity condition is presumably to the effect that there is a certain unique truth-theoretic property (truth-asuttered-in-such-and-such-context) which is preserved at every circumstance from premises to conclusion (cf fn 3), whereas the proposed Ersatz is to the effect that every circumstance is such that at it some truth-theoretic property or other (truth-as-uttered-in-a-context-containing-that-circumstance) is preserved from premises to conclusion, with different such properties being preserved at different circumstances. That’s like proposing to understand the idea of universal access to healthcare not to the effect that that there is a certain unique health system that is accessed by everyone, but to the effect that everyone accesses some health system or other, with different such systems being accessed by different people. Also, don’t say that, while, for no context C, ‘If “Snow is white” is true-as-uttered-in-C, “Actually, snow is white” is true-as-uttered-in-C’ is metaphysically necessary, for every context C it is logically necessary. For, without further ado (which, as far as I know, has never been made), the claim that logical consequence is logically necessary is totally vacuous, as it is tantamount to the claim that logical consequence holds with the very special force of … logical consequence (in exactly the same way, also defeasible arguments could be claimed to be ‘defeasibly necessary’!). 6. Such failure of normativity spells disaster for normative constraints on logical consequence of the kind ‘If φ0, φ1, φ2 … ⊢ ψ holds, one should not accept φ0, φ1, φ2 … and reject ψ’, which are endorsed, for example, by Restall (2005); Beall (2015) [see Zardini (2016), pp. 313–314, fn 9 for some further discussion]. Notice that, if one proposed (what BR don’t) that, actually, ‘one should not accept φ0 & φ1 & φ2 … and reject ψ’, even setting aside various issues related to the possible infinity of φ0, φ1, φ2 … it would be hard to see how such proposal does not in effect amount to a wholesale rejection of the normativity of multiple-premise arguments (since the source of the normative fact that one should not accept φ0 & φ1 & φ2 … and reject ψ is presumably the fact that the single-premise argument φ0 & φ1 & φ2 … ⊢ ψ holds), which would be quite sad since the preface paradox can do nothing to undermine the incontrovertible fact that quite a few times multiple-premise arguments do have normative force (in particular, are such that one should not
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accept each of their premises and reject their conclusion, independently of whether one de facto accepts the conjunction of their premises). Indeed, while, as far as the issues raised by the preface paradox are concerned, in classical logic multiple-premise arguments can be taken to have normative force (when and) only when one should [accept the conjunction of their premises if one accepts each of them] (so that, if a multiple-premise argument has normative force and one accepts each of its premises, since one should then accept the conjunction of its premises, the proposal in question—at least when crucially supplemented by some sort of account of when one should [accept the conjunction of the premises of an argument if one accepts each of them]—by in effect appealing to the normativity of the corresponding single-premise argument can vindicate the claim that one should not reject its conclusion), that is no longer so in some logics with non-standard behaviour of conjunction such as the subvaluationist logic Sb [Jaśkowski (1948)]. In Sb, it is possible, for example, that φ, ψ entail χ with normative force, that one accepts each of φ and ψ but that φ & ψ is inconsistent (let alone entailed by φ, ψ), so that, for non-epistemic reasons (and so for reasons not concerned by the issues raised by the preface paradox), it is not the case that one should accept it. In such a situation, since the argument from φ, ψ to χ has normative force and one accepts each of φ and ψ, one should not reject χ, but, since it is not the case that one should accept φ & ψ, the proposal in question can no longer in effect appeal to the normativity of any φ & ψ-[single- premised] argument to vindicate the claim that one should not reject χ. 7. To be clear, the idea is that the objects are different, but the logical operations on them are the same: for example, there is a single operation of disjunction, which obeys LEM on the real numbers of classical analysis, whereas it does not obey it on the real numbers of intuitionist analysis. An alternative idea would be that the objects are the same, but the logical operations on them are different: for example, classical analysis talks about classical disjunction which obeys LEM on the real numbers, whereas intuitionist analysis talks about intuitionist disjunction which does not obey LEM on the same numbers. Such alternative is not only in itself unnatural, it is also incoherent in view of the well-known fact [Popper (1948)] that, on a shared domain of objects and properties, intuitionist logic would seem to collapse on classical logic: letting ‘orC’, ‘orI’, ‘identicalC’, ‘identicalI’, ‘notC’ and ‘notI’ express classical and intuitionist disjunction, identity and negation respectively, by using the deductive rules appropriate for each logical expression we can derive ‘r is identicalI with
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0 orI r is notI identicalI with 0’ from ‘r is identicalC with 0 orC r is notC identicalC with 0’, thereby making a hash of the intuitionist continuum. While the classical-analysis/intuitionist-analysis case is only one example of logical pluralism, I take the overall thrust of this discussion to provide some evidence to the effect that, in those cases where a logic is motivated by the aim of accounting for the specific behaviour of objects and properties in a certain domain (as the specific logics alluded to in sections “Transitivity” and “Contraction” are), plurality of legitimate relations of logical consequence is accommodated by distinguishing domains rather than by multiplying logical operations. In other cases, a logic is motivated instead by the aim of accounting for certain general, domain-unspecific features of logical consequence (as the specific logics alluded to in sections “Reflexivity”, “Monotonicity” and “Commutativity” are)—in those cases, plurality of legitimate relations of logical consequence is indeed most plausibly accommodated by multiplying logical operations. In both kinds of cases, contrary to logical pluralism as typically understood in the contemporary debate (as per the next paragraph in the text), plurality of legitimate relations of logical consequence is ultimately accommodated by distinguishing which fully interpreted sentences they apply to. 8. While there is usually a slide in going from its being legitimate to hold that φ does not entail an absurdity to its being legitimate to accept φ (for example, if φ is ‘The number of stars in the universe is odd’), there is no slide here in going from its being legitimate to hold that ‘The Liar sentence is true iff it is not’ does not entail ‘The Earth is flat’ to its being legitimate to accept ‘The Liar sentence is true iff it is not’, since the possibility that doing the former is not legitimate is essentially the only reason for doubting that doing the latter is legitimate. 9. Throughout, by ‘entail’ and its relatives I mean the converse of the relation of logical consequence. By ‘imply’ and its relatives I mean instead an operation expressed by a conditional operator. 10. Given this ultimate aim (and the fact that the abundant literature on GTT has already done a good job in this respect), in this chapter I won’t delve into other very plausible counterexamples, nor into the very obvious fact that, just as GTT is a very natural generalisation of an unduly restrictive notion of logical consequence, there are very natural generalisations of GTT itself in several directions (a fact to be handled with some caution, see fn 26).
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11. Although, should a substructural logic prove to be part of the best solution to, for example, the semantic paradoxes, I myself would find it terminologically misguided not to label it ‘legitimate relation of logical consequence’ (surely, whatever logic governs truth deserves to be labelled ‘legitimate relation of logical consequence’!). 12. In the text, I’ll briefly mention what I regard as the best such reasons not to commit to denial of (I)—an issue which obviously lies well beyond the scope of this chapter—but simply to make more vivid how interesting philosophical positions denying (I) would unhelpfully be outlawed by GTT. Analogous comments apply for the other structural properties to be considered below in the text. 13. That is, by relying in a non-deviant way on one’s belief in the premises of the argument and on one’s inference from them to the conclusion: one can perhaps acquire a new justification for believing, say, ‘There are circular arguments whose conclusion I’ve inferred from their premises’ by ‘going through’ the relevant instance of (I), but one would thereby be relying in a clearly deviant way on one’s belief in the premise of the argument and on one’s inference from it to the conclusion. As elsewhere [see e.g. Chisholm (1966), p. 30 for the much discussed case of deviant causation, in whose debate it would ironically seem presupposed that deviance only affects causation and not also rational connection], it is a tricky issue, lying beyond the scope of this chapter, to spell out exactly what such deviance is. 14. Even the crooked simplification argument, say, φ & ψ ⊢ φ passes muster, since one can be told that Al met with Bob and Cate and acquire a new justification for believing that Al met with Bob by inferring it by simplification from what one has been told. 15. This uncontroversial fact would easily be accounted for if, when transmission occurs, the justification for believing the conclusion were simply identical with the justification for believing the premises, for then it would be obvious that the justification for believing the conclusion of (I) cannot be new with respect to the justification for believing its premise. Pace Moruzzi and Zardini (2007), p. 181, that would seem however a simplistic view of what happens when transmission occurs (a mistake for which I assume the sole responsibility!), since, presumably, also the justification one has for inferring the conclusion from the premises is part of the justification one acquires for believing the conclusion. While the ‘mereology of justification’ is still in its infant days, a natural speculation is that
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nothing having as a part a justification j for believing φ can be a new justification with respect to j for believing φ, given which it still follows that the justification for believing the conclusion of (I) cannot be new with respect to the justification for believing its premise (assuming throughout, extremely plausibly, that one can only acquire a justification for believing the conclusion if one has a justification for believing the premises, and that the latter is then part of the former). Notice that, if the necessary condition for validity in question is the stronger one to the effect that one can use a valid argument in at least some context to acquire a first justification for believing its conclusion, the relevant uncontroversial fact is instead that in no context can one acquire a first justification for believing φ by inferring it by (I) from φ, which is easily accounted for, since it is obvious that the justification for believing the conclusion of (I) cannot be first with respect to the justification for believing its premise [see e.g. Zardini (2014b) for a recent discussion of transmission]. 16. Again, even simplification (fn 14) passes muster (if vacuously so), since, arguably, φ & ψ is always true in virtue of the (sometimes non-logical) fact described by φ, the (sometimes non-logical) fact described by ψ and the (always logical) fact that φ and ψ entail φ & ψ. 17. I’m not saying that x-is-true-in-virtue-of-the-truth-of-y is irreflexive (i.e. such that, for every x, it is not the case that x is true in virtue of the truth of x), since it isn’t [Zardini (2018b)]. 18. To belabour the point, φ does not preserve the truth of φ in virtue of the meaning of some expressions occurring in it, even if it is crucial for that preservation that, given the specific meanings involved in φ, the very same meanings are involved in φ. Compare: I don’t have the same age as my mother’s only son in virtue of being 38-year old, even if it is crucial for that sameness that, given my specific age, that very same age is had by my mother’s only son. 19. Read (1981,2003) offers a very different defence of the connection between non-monotonic logics and truth preservation by reinterpreting truth preservation in terms of a relevant implication (‘If every premise is true, so is the conclusion’). Such reinterpretation would seem however miles away from the original idea of truth preservation as presented in section “Reflexivity” (whose intelligibility Read does not contest), which is simply about the conclusion being true except the premises are not: that requires a ‘relevance connection’ between the premises being true and the conclusion being true just as little as the fact that an object
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weighs 1,171,979 grams except it is not now in my right pocket (which is presumably tantamount to every object that is now in my right pocket weighing 1,171,979 grams) requires a ‘relevance connection’ between an object being now in my right pocket and its weighing 1,171,979 grams. Having noted this, Read’s idea of reinterpreting truth preservation in terms of non-classical notions is in my view insightful in the respects I myself will exploit in sections “A Logic-Relative Version of GTT; an Absolute-Truth Formulation of a Logic-Relative Version of GTT” and “From Logical Consequence Back to Logical Truth”. 20. That follows, for example, under the [logical-consequence]-first doctrine, according to which logical truth is grounded in logical consequence as being merely a limit case of logical consequence with no premises and one conclusion. BR seem actually wary of the doctrine (pp. 12–13, despite p. 3), but they do accept at least that logical truths are co-extensional with the single conclusions that follow from no premises, which amply suffices for the implication at issue in the text. 21. And not simply as a necessary truth, for it is crucial for what is the logic of BX (i.e. what follows from what in BX, what is a logical truth in BX, what is a logical falsity in BX, etc.) that LEM have the same status as the other axioms of BX, which it is again crucial for what is the logic of BX that it be the one of logical truth. Analogous comments apply to the next two occurrences in the text of ‘as a logical truth’. 22. Throughout, I understand the elements of a series 〈x0, x1, x2 …〉 to be the members of the set {x0, x1, x2 …}, and, crucially, I understand the premises 〈φ0, φ1, φ2 …〉 to be the members of the set {φ0, φ1, φ2 …}, so that, for example, the only premise in 〈φ, φ〉 is φ (as a consequence of both understandings, I identify the premises 〈φ0, φ1, φ2 …〉 with the elements of that series). The understanding of premises just flagged as crucial would seem mandated by the traditional idea that premises are sentences [which, in general, I myself have defended in Zardini (2018e); BR, pp. 8–12, also consider judgement types and propositions, which however mandate the same understanding of premises]. In connection with noncontractive logics, sometimes [e.g. Girard (1995), p. 2] a different idea is aired to the effect that premises are ‘tokens’ (of some kind or other). I find extant proposals in this direction deeply problematic [see Zardini (2018b) for details], and I am much more attracted to seeing non- contractive logics as being sensitive to the number of occurrences of the same premise (i.e. sentence), so that, for example, the series 〈φ, φ〉 does not represent two tokens of φ, say φ19 and φ79 (whatever these may
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be!), but the double occurrence of φ. I’ll henceforth presuppose such broad understanding of non-contractive logics. The understanding of course invites the question how, when testing for logical consequence, there can be sensitivity to the number of occurrences if one only considers whether the premises (i.e. certain sentences) are true. My reply begins with the observation that, when, in the metatheory, one considers whether ‘the premises’ φ0, φ1, φ2 … are true, one is not simply considering whether φ0 is true, whether φ1 is true, whether φ2 is true …; one is combining φ0, φ1, φ2 … (to be sure, in that order) and considering whether they are all true—i.e. whether φ0 is true together with φ1 together with φ2 … It is then the mode of such combination that, if one’s metatheory is in a substructural logic, can be sensitive to the number of occurrences (and, as for the logics in section “Commutativity”, to their order): for example, if φ occurs twice as a premise as in 〈φ, φ〉, when testing for logical consequence one should not consider whether simply φ is true, but whether φ is true together with φ, and one might thereby be using a mode of combination under which the latter question may receive an answer different from the one received by the former question. I’ll make important use of this reply in the broader context of sections “A Logic-Relative Version of GTT; an Absolute-Truth Formulation of a Logic-Relative Version of GTT” and “From Logical Consequence Back to Logical Truth”. 23. Clearly, on this kind of logic of context change, also all the other structural properties considered in this chapter are not valid, but I think that it is more insightful to place it in the non-commutative camp rather than in any other substructural camp. 24. For our purposes, it is important to distinguish between models and tests. Roughly, models are single evaluations of the sentences of the whole language, whereas tests are partial checks for logical consequence looking whether, if the series of premises have the relevant value in the relevant model(s), the conclusion has the relevant value in the relevant model. Importantly, one test might look at different models for different occurrences of the premises or the conclusion. As a baby example, a language might have only two different models, and logical consequence for that language be determined by the tests checking for the fact that it is not the case that the series of premises are true in one model and the conclusion is not true in the other model [see Zardini (2014a) for a more adult example].
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25. I should note that the generalisations required by non-monotonic and non-contractive logics would seem in an importantly different ballpark, since their implementation is much less straightforward, and typically requires the adoption, at some level or other, of the operations of, respectively, intensional implication and multiplicative conjunction characteristic of those logics (I’ll myself follow this strategy in sections “A Logic-Relative Version of GTT; an Absolute-Truth Formulation of a Logic-Relative Version of GTT” and “From Logical Consequence Back to Logical Truth”). 26. All this of course does not mean that we should simply replace GTT with some sort of GGTT that allows for all possible generalisations in these directions (or others), since a few such generalisations will yield relations that are obviously not instances of the kind Legitimate Relation of Logical Consequence (but are not ruled out by BRT2 either). In a familiar oscillation, in the attempt at weakening too strong a condition that undergenerates with respect to a target kind, we end up with too weak a condition that overgenerates with respect to the kind. There is no (non-trivial) essence of Legitimate Relation of Logical Consequence. 27. Throughout, bear in mind that, as per regionalism (section “Generalised Tarski’s Thesis”), in some cases such ‘interpretation’ does not involve different logical operations and is more akin to the way in which, say, playing football and playing chess represent two different ‘interpretations’ of playing even though these do not involve different properties playing0 and playing1 exemplified (on the one hand by people and on the other hand) by football and chess, respectively. 28. That is only a very rough gloss and you shouldn’t read too much into it, not the least because the sense in which ‘the collective truth of the series of premises generates the truth of the conclusion’ varies dramatically from logic to logic (recall the last paragraph in the text). Still, the gloss is evocative at least for the specific substructural logics alluded to in Sect. 2, constituting a programmatic slogan congenial to the foundational role I’ll argue in section “From Logical Consequence Back to Logical Truth” these conditionals have. Thanks to Luís Estevinha for feedback on this matter. 29. If the logic in question has more than one operation of universal quantification, implication or conjunction, I assume that there is a most appropriate one for the role these conditionals are supposed to play, and I focus on that one (the assumption is arguably satisfied for all the specific substructural logics alluded to in Sect. 2).
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30. To set aside distracting issues concerning the opacity of absolute truth [e.g. Zardini (2015b)], in this discussion I assume that it makes sense to extend the target logics with a quoting singular term ⌜φ⌝ for each sentence φ of their original language and with a truth predicate T such that, for every sentence φ of their original language, T⌜φ⌝ is intersubstitutable with φ. Such extensions are straightforward (indeed, on my view, just as logical as, say, the extension of the conjunction-free fragment of intuitionist logic with conjunction), contrary to those that would be needed to develop a theory of truth-in-a-case (as per the third last paragraph in the text). (Such extensions are also harmless, since they only licence the intersubstitutability of T ⌜φ⌝ with φ if φ is T-free.) Moreover, focusing on languages for which such extensions make sense is justified since, in this discussion, my aim is to defend anti-universalist claims rather than universalist ones. Thanks to José Martínez and Ricardo Santos for feedback on some of these issues. 31. I should really be a bit more precise about what it is for a logic to satisfy the absolute-truth formulation of the logic-relative version of GTT. For our purposes, anticipating a bit, a natural way of making that notion precise is to say that a logic L satisfies the absolute-truth formulation of the logic-relative version of GTT iff [φ0, φ1, φ2 …, φi ⊢L ψ holds iff T ⌜φ0⌝ & T ⌜φ1⌝ & T ⌜φ2⌝ … & T ⌜φi⌝ → T ⌜ψ⌝, as interpreted by L, is a logical truth in L] (where, by fn 30, the last claim is equivalent with the claim that φ0 & φ1 & φ2 … & φi → ψ, as interpreted by L, is a logical truth in L, which makes satisfaction of the absolute-truth formulation of the logic-relative version of GTT by L a matter of L’s conjunction and implication correlating in the familiar ways to premise combination and entailment in L, respectively). Thanks to Ricardo Santos for urging this clarification. 32. Yes, including non-contractive logics, even given what I’ve said in section “A Logic-Relative Version of GTT; an Absolute-Truth Formulation of a Logic-Relative Version of GTT”, since we’re considering a formulation of the logic-relative version of GTT in terms of plain [collective- truth]-generation conditionals, without cases or designated values: while it is the case that, if φ gets a designated value, so does φ ⊗ φ, we do not have that, if φ is true, so is φ ⊗ φ [Zardini (2011)]. 33. Throughout, I understand ‘ground’ and its relatives in a suitably broad fashion, so as to include also the case of reduction (as is particularly plausible in the case of the [logical-consequence]-first doctrine).
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34. The second main conjunct in its full strength is not guaranteed by the first one, and so, if at all desired, it must be added separately. Still, it represents the arguably most plausible grounding of logical truth in logical consequence, and that’s why, throughout, we’re focussing on it. Notice that ‘most plausible’ doesn’t imply ‘plausible’: following, but from no premises, would seem to make just as little sense as arriving, but at no places! Sometimes, the problem is fudged by invoking the empty set and saying that logical truths are those sentences that follow from the empty set; however, while it is useful formally to model logical consequence as a relation between sets (or, in view of substructurality, series), only an empty-set mystic would think that logical truths are characterised by a distinctive logical relation to the empty set (as opposed to many other more relevant objects). Some other times, the problem is fudged instead by invoking the 0ary operator t and saying that logical truths are those sentences that follow from t; however, while it is useful to introduce t in the formal study of logical consequence (especially in view of substructurality), its informal understanding is as of the ‘conjunction’ of all logical truths, which arguably prevents grounding a sentence being a logical truth in its following from t. Notice that an analogous problem affects the related idea that logical falsity is grounded in logical consequence as being merely a limit case of logical consequence with one premise and no conclusions (whereas, FWIW, the more general notion of inconsistency of a series of premises can still be grounded in logical consequence and logical falsity in terms of the series entailing the ‘disjunction’ of all logical falsities). Thanks to Ricardo Miguel for pressing me on the formulation of the [logical-consequence]-first doctrine. 35. Having noted all this, I’ll (pick up from fn 31 and) henceforth focus on the finite case myself. 36. Even setting aside the problem raised in the text, the extension of these logics with the desired operations typically requires complications going so far beyond the basic, natural framework of the logics as to make the [logical- truth]-first doctrine hardly credible for them. Moreover, the resulting operations are typically so tailor-made to fit the target relation of logical consequence that, given on the one hand the valid arguments of one of these logics and on the other hand its putatively grounding logical truths, the by far most plausible account is that the putatively grounding logical truths are what they are because the valid arguments are what they are rather than vice versa.
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37. Since, in arguing against the [logical-truth]-first doctrine, we’ve also touched on some dramatic cases (B3 and K3) which, at least on their standard sentential fragment, have no logical truths whatsoever but still have a wealth of valid rules, it’s just fair to mention that similar dramatic cases of logics [like TS of Cobreros et al. (2012)] which, at least on their standard sentential fragment, have no valid rules whatsoever but still have a wealth of valid metarules argue against the [logical-consequence]-first doctrine [as does the fact that the valid rules of ST of Cobreros et al. (2012) coincide with those of classical logic in spite of ST quite clearly being different from classical logic!]. The point iterates at higher orders. A thoroughly diverse picture of logics thus emerges, on which, for some logics, what is most fundamental are their logical truths (as I’ll argue starting from the next paragraph in the text); for some other logics, their valid rules; for yet some other logics, their valid metarules … Thanks to Bogdan Dicher, Sergi Oms and Lucas Rosenblatt for insisting on these points. 38. A further reinforcement: the condition that the conclusion is true except the premises are not would seem to be all about getting things right—no matter how inelegantly—in going from the premises to the conclusion; that is, just not getting them wrong; that is, not going from premises that are true to a conclusion that is not. 39. Notice that both the truth-preservation account of logical consequence and GTT constitute a very specific version of reliabilism, where the relevant property a valid argument is reliable about is truth (rather than knowledge, unfalsity, truth together with unfalsity, etc.). 40. That is, φ and φ, φ agree that their only premise, φ, is true. The additional strength of φ, φ over φ consists in representing φ as being true also as occurring twice. While that entails φ’s truth, it is not entailed by it. That’s not enough to make the additional strength of φ, φ relevant for the truth-preservation account of logical consequence: for example, the property of being known by me is also such that, while my knowledge of φ entails φ’s truth, it is not entailed by it, but that property is clearly irrelevant for the account. Having made this point, there is indeed an important question as to how to understand the idea that a premise is true ‘as occurring twice’, which I’ve addressed in fn 22. 41. It’s worth noting that, even if preservation is dropped in favour of a relation relating possibly distinct values (say, connection), the immediately resulting account (‘guaranteed connection between the truth-theoretic values of the premises and of the conclusion’) is still all about ‘truth-
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theoretic’ values such as truth, falsity, unfalsity, etc., whereas the distinction among designated values in the model theory of some non-reflexive and non-transitive logics does not sustain any such interpretation [for example, think of the non-[truth-theoretic] interpretation offered by Zardini (2008), pp. 347–349 of the distinction between two different kinds of designated values relevant for a wide family of non-transitive logics]. 42. We can further argue not only that substructural logical consequence is not grounded in truth preservation, and not only that substructural logical consequence is not co-extensional with truth preservation, but also that substructural logical consequence does not even require truth preservation. The best example for this is arguably offered by the non-contractive approach to the semantic paradoxes mentioned in section “Contraction” [especially as further developed in Zardini (2018f )], on which one can warrantedly accept, say, that the Liar sentence λ is true while holding that the argument from λ, λ to ‘Snow is black’ is valid, and so while holding that the premise (which occurs twice, see fn 22) of the argument is true and its conclusion is not. Substructural logical consequence does not require truth preservation. Thanks to Sven Rosenkranz for prompting this expansion of the argument. 43. To be clear, the claim is not that the intended class is not co-extensional with any class defined in purely non-logical terms—not the least because, under very plausible mathematical assumptions, such co-extensionality does obtain in the case of many logics! Especially for such logics, the claim is not that logical consequence is not correlated with individualtruth preservation over a class of tests defined in purely non-logical terms; it is rather that logical consequence is not grounded in such truth preservation, as it is only correlated with it because the class of tests defined in purely non-logical terms just so happens to be co-extensional with the intended class, which is in turn only characterisable as such partly in logical terms [for example, as the class of all and only those tests involving logically possible models, cf Etchemendy (1990), pp. 107–124]. 44. This is not to deny that something of heuristic value can be gained by working with the notion of a sentence being true in every logically possible model rather than with the notion of a sentence being logically necessary, just like something of heuristic value can be gained by working with the notion of a sentence being true at every metaphysically possible world rather than with the notion of a sentence being metaphysically necessary: in both cases, quantificational reasoning, based on the well-understood
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notion of truth-in-a-model or truth-at-a-world, can be easier than modal reasoning. But in neither case is the heuristics plausibly taken as evidence of the grounding of modal facts in quantificational ones concerning models or worlds. 45. Notice that this proposal does not require us to use L (in particular, its tricky conjunction and implication) in our own theory: it’s sentences that are logical truths, and so, on this proposal, we only need to mention— rather than use—the relevant sentences and hence mention, rather than use, the relevant operators and the logic governing them. Instead of employing ourselves the straightjacket of [individual-truth] preservation, we refer to L’s own interpretation and evaluation of [collectivetruth] generation. (Don’t complain that this might give us less of a clue as to which arguments are valid in L: the proposal is about what grounds the fact that an argument is valid in L, not how we get to know it.) Having noted this, as will emerge in particular in the next paragraph in the text, the proposal does rely on a canonical specification of the relevant sentences (grounding is sensitive to how the sentences are presented!), so that its understanding does require appreciation of some general semantic features of such sentences. 46. That this philosophical approach to grounding logical consequence for the specific substructural logics alluded to in Sect. 2 is viable is confirmed by technical facts concerning the formal definitions of some of these logics. To give one glaring example (whose obvious import for the foundational issues we’re investigating seems to me to have hitherto been grossly overlooked), as I’ve already in effect remarked in section “Monotonicity”, relevant logics are traditionally presented basically in terms of their logical truths, and, when they are taken to be non-monotonic, their relation of logical consequence is typically defined in terms of φ0, φ1, φ2 …, φi ⊢L ψ holding iff φ0 → (φ1 → (φ2 … → (φi → ψ))) …) is a logical truth in L. To give another glaring example (whose obvious import …), typical non-contractive logics have a semantic presentation employing a certain family of lattices (section “A Logic-Relative Version of GTT; an AbsoluteTruth Formulation of a Logic-Relative Version of GTT”): in such presentation, their relation of logical consequence is not defined, as is usual in broadly algebraic treatments of logics, in terms of preservation of designated value from φ0, φ1, φ2 … and φi to ψ in every model (for, as per the argument in section “A Logic-Relative Version of GTT; an AbsoluteTruth Formulation of a Logic-Relative Version of GTT”, that definition would give the wrong results), but in terms of φ0 ⊗ φ1 ⊗ φ2 … ⊗ φi → ψ getting designated value in every model (that is, being a logical truth).
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47. While this deft movement from use to mention is illuminating, the light it sheds is admittedly somewhat faint. But then there is only so much light to be had at these depths. 48. Don’t say that the fact that φ0, φ1, φ2 …, φi ⊢L ψ holds is grounded instead in the fact that the object-language, conjunction-free conditional φ0 → (φ1 → (φ2 … → (φi → ψ))) …) is a logical truth in L (cf fn 46), so that, in particular, the fact that, in LW, ‘Snow is white’, ‘Grass is green’ entail ‘Snow is white and⊗ grass is green’ is grounded in (the logical necessity of ) the fact that, if→ snow is white, then→, if→ grass is green, then→ snow is white and⊗ grass is green. Setting aside the adhocness of such deviation given the route we’ve followed starting from GTT, the deviation stumbles on exactly the same problem raised in the text when grounding the fact that, in LW, by modus ponens, ‘If→ snow is white, then→ grass is green’, ‘Snow is white’ entail ‘Grass is green’, since it grounds it in (the logical necessity of ) the tautologous fact that, if→, if→ snow is white, then→ grass is green, then→, if→ snow is white, then→ grass is green. Thanks to José Martínez for comments on some of these issues. 49. Semantic ascent could probably not fulfil this function if alethic deflationism held. Ergo, by modus tollens … 50. In turn, it is plausible that logical truth, even in those logics, is not primitive, and we may expect that its account will appeal to properties of the logical operations used or mentioned in a logical truth (the latter disjunct being particularly relevant when the logical truth involves [collective- truth]-generation conditionals). If so, whether structural properties are valid is grounded in whether certain sentences involving [collective- truth]-generation conditionals are logical truths (see fn 51 for a specific proposal), which is in turn grounded in the properties of logical operations. Substructurality is a logically interesting but philosophically shallow phenomenon caused by logically hidden but philosophically active underlying logical operations. Structure is grounded in operations. Thanks to Ricardo Miguel for pushing me on this. 51. Although a full treatment of the status of metarules (and metametarules, and metametametarules …) lies beyond the scope of this chapter, a natural way of extending the approach we’ve been pursuing to metarules for the specific substructural logics alluded to in Sect. 2 is to say that the fact that a metarule is valid (over and above its being admissible) in L is grounded in the fact that the conditional having as antecedent the conjunction of the series of [collective-truth]-generation conditionals corresponding to the ‘premise rules’ of the metarule and as consequent the [collective-truth]-generation conditional corresponding to the ‘conclusion rule’ of the metarule is
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a logical truth in L (and then iterate this strategy for metametarules, metametametarules, metametametametarules …). Thanks to Bogdan Dicher and Lucas Rosenblatt for their questions about the status of metarules.
References Anderson, A., and N. Belnap. 1975. Entailment. Vol. I. Princeton: Princeton University Press. Asenjo, F. 1966. A Calculus of Antinomies. Notre Dame Journal of Formal Logic 7: 103–105. Beall, J., ed. 2007. Revenge of the Liar. Oxford: Oxford University Press. ———. 2015. Free of Detachment: Logic, Rationality, and Gluts. Noûs 49: 410–423. Beall, J., and G. Restall. 2006. Logical Pluralism. Oxford: Oxford University Press. Bočvar, D. 1938. Ob odnom trexznačnom isčislenii i ego primenenii k analizu paradoksov klassičeskogo rasširennogo funkcional’nogo isčislenija. Matematičeskij sbornik 4: 287–308. Bolzano, B. 1837. Wissenschaftslehre. Vol. II. Sulzbach: Seidel. Brouwer, L. 1927. Über Definitionsbereiche von Funktionen. Mathematische Annalen 97: 60–75. Chisholm, R. 1966. Freedom and Action. In Freedom and Determinism, ed. Keith Lehrer, 11–44. New York: Random House. Cobreros, P., P. Égré, D. Ripley, and R. van Rooij. 2012. Tolerant, Classical, Strict. Journal of Philosophical Logic 41: 347–385. Dummett, M. 2000. Elements of Intuitionism. 2nd ed. Oxford: Oxford University Press. Etchemendy, J. 1988. Tarski on Truth and Logical Consequence. The Journal of Symbolic Logic 53: 51–79. ———. 1990. The Concept of Logical Consequence. Cambridge, MA: Harvard University Press. Field, H. 2008. Saving Truth from Paradox. Oxford: Oxford University Press. ———. 2015. What Is Logical Validity? In Foundations of Logical Consequence, ed. Colin Caret and Ole Hjortland, 33–69. Oxford: Oxford University Press. Fine, K. 1975. Vagueness, Truth and Logic. Synthese 30: 265–300. Girard, J. 1995. Linear Logic: Its Syntax and Semantics. In Advances in Linear Logic, ed. Jean-Yves Girard, Yves Lafont, and Laurent Regnier, 1–42. Cambridge: Cambridge University Press.
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Jaśkowski, S. 1948. Rachunek zdań dla systemów dedukcyjnych sprzecznych. Studia Societatis Scientiarum Torunensis 1: 55–77. Kaplan, D. 1989. Demonstratives. In Themes from Kaplan, ed. Joseph Almog, John Perry, and Howard Wettstein, 481–563. Oxford: Oxford University Press. Kleene, S. 1938. On a Notation for Ordinal Numbers. The Journal of Symbolic Logic 3: 150–155. Makinson, D. 1965. The Paradox of the Preface. Analysis 25: 205–207. Martin, E., and R. Meyer. 1982. S (for Syllogism). ms. Moruzzi, S., and E. Zardini. 2007. Conseguenza logica. In Filosofia analitica, ed. Annalisa Coliva, 157–194. Rome: Carocci. Ono, H. 2003. Substructural Logics and Residuated Lattices—An Introduction. In Trends in Logic. 50 Years of Studia Logica, ed. Vincent Hendricks and Jacek Malinowski, 193–228. Dordrecht: Springer. Paoli, F. 2002. Substructural Logics: A Primer. Dordrecht: Kluwer. Popper, K. 1948. On the Theory of Deduction, Part II. The Definitions of Classical and Intuitionist Negation. Indagationes Mathematicae 10: 44–54. Prawitz, D. 2005. Logical Consequence from a Constructivist Point of View. In The Oxford Handbook of Philosophy of Mathematics and Logic, ed. Stewart Shapiro, 671–695. Oxford: Oxford University Press. Priest, G. 2006. In Contradiction. 2nd ed. Oxford: Oxford University Press. Putnam, H. 1968. Is Logic Empirical? In Proceedings of the Boston Colloquium for the Philosophy of Science 1966/1968, Vol. V of Boston Studies in the Philosophy of Science, ed. Robert Cohen and Marx Wartofsky, 216–241. Dordrecht: Reidel. Read, S. 1981. Validity and the Intensional Sense of ‘and’. Australasian Journal of Philosophy 59: 301–307. ———. 2003. Logical Consequence as Truth-Preservation. Logique et Analyse 183: 479–493. Restall, G. 2000. An Introduction to Substructural Logics. London: Routledge. ———. 2005. Multiple Conclusions. In Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress, ed. Petr Hájek, Luis Valdés Villanueva, and Dag Westerståhl, 189–205. London: College Publications. Tarski, A. 1930. Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften. I. Monatshefte für Mathematik und Physik 37: 361–404. ———. 1936. O pojęciu wynikania logicznego. Przegląd filozoficzny 39: 58–68. Wittgenstein, L. 1953. Philosophische Untersuchungen. Oxford: Blackwell.
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Wright, C. 2000. Cogency and Question-Begging: Some Reflections on McKinsey’s Paradox and Putnam’s Proof. Philosophical Issues 10: 140–163. Zardini, E. 2008. A Model of Tolerance. Studia Logica 90: 337–368. ———. 2011. Truth without Contra(di)ction. The Review of Symbolic Logic 4: 498–535. ———. 2012. Truth Preservation in Context and in Its Place. In Insolubles and Consequences, ed. Catarina Dutilh-Novaes and Ole Hjortland, 249–271. London: College Publications. ———. 2014a. Context and Consequence. An Intercontextual Substructural Logic. Synthese 191: 3473–3500. ———. 2014b. Confirming the Less Likely, Discovering the Unknown. Dogmatisms: Surd and Doubly Surd, Natural, Flat and Doubly Flat. In Scepticism and Perceptual Justification, ed. Dylan Dodd and Elia Zardini, 33–70. Oxford: Oxford University Press. ———. 2014c. Evans Tolerated. In Vague Objects and Vague Identity, ed. Kensuke Akiba and Ali Abasnezhad, 327–352. Dordrecht: Springer. ———. 2014d. Naive Truth and Naive Logical Properties. The Review of Symbolic Logic 7: 351–384. ———. 2015a. Breaking the Chains. Following-from and Transitivity. In Foundations of Logical Consequence, ed. Colin Caret and Ole Hjortland, 221–275. Oxford: Oxford University Press. ———. 2015b. The Opacity of Truth. Topoi 34: 37–54. ———. 2016. Restriction by Noncontraction. Notre Dame Journal of Formal Logic 57: 287–327. ———. 2018a. Forthcoming. Closed without Boundaries. Synthese. ———. 2018b. Forthcoming. Instability and Contraction. Journal of Philosophical Logic. ———. 2018c. Forthcoming. The Underdetermination of the Meaning of Logical Words by Rules of Inference. In The A Priori: Its Significance, Grounds, and Extent, ed. Dylan Dodd and Elia Zardini. Oxford: Oxford University Press. ———. 2018d. Changing without Contra(di)ction. ms. ———. 2018e. The Bearers of Logical Consequence. ms. ———. 2018f. Unstable Knowledge. ms.
Logical Particularism Gillman Payette and Nicole Wyatt
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Introduction
Methodology in logic has, on the whole, conceived of a logic as a system of rules or principles. A correct logic is one which includes correct principles, and logical reasoning is reasoning in accord with such correct principles. The contemporary versions of this approach use a mathematically precise formal language to express these principles, which are understood similarly to the mathematical expression of laws of nature. We call this orthodox view ‘logical generalism’. The assumption of generalism is so broadly made that it is difficult to make sense of giving it up. Generalism might seem to be constitutive of G. Payette Department of Philosophy, University of British Columbia, Vancouver, BC, Canada e-mail: [email protected] N. Wyatt (*) Department of Philosophy, University of Calgary, Calgary, AB, Canada e-mail: [email protected] © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_12
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logic, and the idea of a non-generalist approach to logic incoherent. In this chapter, we aim to undermine the idea that logic and generalism go hand in hand by exploring in detail an approach to logic that is particularist rather than generalist. We subscribe to five central theses. First, we are anti-exceptionalists about logic—that is, we take the study of logic to be contiguous with other sciences. Second, we reject the idea that logical theories aim at correctness. Third, we endorse a view of explanation on which scientific theories need not be correct in order to be explanatory. Fourth, we deny that natural language inferences have logical forms independently of the use of a particular formal logic to model them. Fifth, we take logical validity to be a property of particular inferences/arguments, rather than a property of logical forms or schema. We have discussed the first three in Payette and Wyatt (2018) and the fourth in Wyatt and Payette (2018). Our focus in this chapter is on five. What is the particularist understanding of validity, and in what way does formal logic contribute to our understanding of valid inference? We also compare our view to that of the local logical pluralist.
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Validity
Our concern is with the validity of natural language arguments. This is not to deny that formal logics have many other important applications, nor that logical systems which have no relevance to natural language may be of considerable interest. But it is only when logics are understood as theories of a consequence relation in natural language that the debates between logical monists and logical pluralists make sense: for any logics to be correct there has to be something that they are correct with respect to. Following Priest (2014), we call this the canonical application for logic. Monists and pluralists agree that at least one logic is a correct theory of natural language validity. In contrast, logical nihilists and logical particularists share the view that no logic provides such a theory. For nihilists like Curtis Franks, this is a reason to give up the canonical application entirely:
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fixing our sights on [natural language validity] saddles logic with a burden that it cannot comfortably bear, and that logic, in the vigor and profundity that it displays nowadays, does and ought to command our interest precisely because of its disregard for norms of correctness. (Franks 2015, 148)
From Franks’ point of view, focus on the canonical application is both futile and pernicious. With respect to the latter, he argues that the pursuit of a correct logic obscures the fact that much of the real interest in logic lies in the interrelations between logics. For example, consider the enormous, indeed uncountable, variety of logics that appear by removing from classical propositional logic the law of excluded middle, thus generating intuitionistic propositional logic and those logics in between the two. Logicians can focus on particular ways of presenting these logics in terms of proof systems, and indeed on particular kinds of proof systems, for example, sequent calculi. They can show that the effect of removing a logical law can be reproduced by changing the structural parameters of one system that has all of the same proof rules for the connectives: allowing multiple formulas on the right-hand side of the sequents gives classical logic, while restricting the right-hand side to just one formula gives intuitionistic logic.1 One of the crucial observations logicians can make from this perspective is that classical logic collapses many distinctions that one may want to make. The most famous example: ¬¬p is logically distinct from p in intuitionistic logic, but not in classical logic. These results, and many like them, are part of our logical knowledge, but, Franks argues, they cannot be understood as part of some correct logic for natural language (Franks 2015, 154–5). With respect to the futility of pursuing the canonical application, Franks argues that even with respect to our most common and successful patterns of reasoning, we have no grounds for maintaining that these patterns are anything other than useful: it is a preconception that science is made possible by ahistorical norms of right reasoning. Once one considers the possibility that logic may be studied with patterns of thought adapted to what we learn along the way, it becomes hard to understand what special status the rudimentary principles we find ourselves reflexively appealing to are supposed to have. (Franks 2015, 159)
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Franks’ position, then, is that no logical system is likely to be correct. First, because they are put together partly by accident, and largely by efficacy. The dominant systems do not have—or cannot be seen to have, at least—some special metaphysical status. And second, because there is no evidence that any of the classes of inference rules we gather together have some property that makes them the valid rules: “the evidence suggests that the arrangement of our [logical] toolkit is a highly contingent matter” (Franks 2015, 159). In contrast, as particularists we are realists about the existence of a connection between premises and conclusions of valid arguments, and we think that logic is meant to help us study that connection. That is, we take it that there is a fact of the matter as to whether, for the utterance of a specific argument, that argument is valid or not. However, contra the standard view, we maintain that validity is a property of argument tokens, not argument types. Further, we agree with Franks that there is no feature or group of features that are sufficient or necessary for validity and also agree that there are no correct logical systems. We do not however share his view that the only way in which logic could contribute to our understanding of natural language is by offering a correct theory of true logical principles. On our view, logic studies the connections between premises and conclusions of valid arguments, which we understand in preservationist terms. While logical orthodoxy takes validity to be a matter of truth preservation between premises and conclusions, preservationists take a broader perspective on the sorts of features which a consequence relation can preserve.2 For the particularist, valid arguments are those in which there is some feature which obtains of the conclusion in virtue of obtaining of the premises. However, we treat that feature as variable. That is to say, some particular concrete arguments are valid because they preserve truth, some because they preserve relevance, and so on. Moreover, when one makes a syntactically identical argument on a subsequent occasion, it may fail to be valid, because on that particular concrete occasion, preserving truth (or relevance) may not be important to validity, whereas on the prior occasion it was. Logic can study both the wide variety of features that valid arguments preserve and the connections between premises and conclusions without needing to meet the burden of correctness Franks rightly rejects.3
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Logics
Our view is that formal logics are mathematical models of validity; admittedly, that is not an uncommon view. But there are important distinctions between our view and other versions of the logic-as-modelling view. Consider, as a representative example, the view of Roy Cook (2002). On his account logic aims to model natural language and in particular arguments constructed in natural language—including technical languages like that of mathematics.4 According to Cook, logical models—proof theory, model theory, formal language—contain certain mathematical objects that accurately represent some aspect of language/reality. These accurate representations are the representors.5 Other components of the models are merely mathematical ‘artifacts’ not intended to represent any part of reality. The standard metaphor is that of a model ship: the model may have certain components which do not exist in the real ship, and may lack certain components of the real ship. Which elements are the representors and which the artifacts in a model depends on what we want a model to do. For example, a model ship may not have engines. If we are looking for a model that represents the exterior of a ship, engines are unnecessary. But another model that is meant for hydrodynamic ‘tank tests’ of the ship may also lack engines, but include a lead block where the engines are in the actual ship to represent the position and mass of the engines (though not the moving parts of the engines since those don’t matter for this purpose). The real ship does not have a lead block where its engines are; but the lead block is a representation of a particular, relevant aspect of the real boat. But now suppose we want to put a model of the ship on a shelf, and we want to make sure that it doesn’t fall over, so we put in a lead block which happens to be where the engines are. The representors in this case are again the details on the exterior of the model. Before, the block was a representor; now it is an artifact. Given this understanding of modelling, in order for a logic to correctly represent validity in natural language, the elements of the model that stand in relations of consequence must be representors rather than artifacts. This means the formulas of the formal languages must represent the sentences of natural language. On the standard picture, they do so by
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representing the logical forms of these sentences. Among other things, this means that natural language sentences must have logical forms; that every natural language sentence represented by a given formula must have the same logical form; and that the formulas must also be composed of parts which themselves accurately represent the parts of the logical forms.6 There are a number of difficulties with this picture of logics as models. First, the distinction between representors and artifacts assumes that models represent, to the extent that they do, via their parts. It is true that in the model ship on the shelf the lead block doesn’t represent the engines. But it does combine with rest of the model ship structure to represent the usual (and preferred!) orientation of the actual ship. The lead block is not by itself a representor, but classifying it as an artifact seems a mistake: it does play some role in how the model functions as a model of the ship. The distinction rests on an overly simple picture of modelling; importantly, it may be the outcome of interactions between components that represents instead of a single component. Even in the case of hydrodynamic tank testing of model ships, it is really the interaction between the shape of the ship and the distribution of mass in the ship that is playing the key role. Representation can be an emergent property—A, B, and C can be combined to represent something without being used individually to represent. Second, in the specific case of logics, the picture advocated by Cook takes for granted that the formulas of formal logics are representors, not artifacts, and that what they represent are the logical forms of natural language sentences. However, as we have argued in detail elsewhere (Wyatt and Payette 2018), there is no basis for ascribing logical forms to natural language sentences independently of the use of a particular logical system. Ascriptions of logical form are theory-laden. From the point of view of the particularist, the idea that models, or parts of models, explain by representing is a mistake. We think the idea that the model and its components must represent is driven by the assumption that only correct theories can be explanatory. Our view is that logical theories cannot explain validity on theories of explanation in which correctness is a prerequisite.7 Once we have diverged from thinking of explanatory logical theories as correct, we can more easily question the requirement that models represent.
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Modelling
We have denied that logics represent features of natural language. Contra the nihilist, we maintain that logics do shed explanatory light on natural language consequence. To understand how non-representational models can explain, we turn to modelling in the sciences. We focus on three key features of explanatory but inaccurate or non-representative models: abstraction, imagination, and scaffolding. As Godfrey-Smith observes (2009), scientists from a variety of disciplines spend time considering things that don’t exist: examples include ideal gases, frictionless planes, infinitely large biological populations, wholly rational agents, and biologically implausible neural networks. Theoretical investigation of these apparently increases our understanding of the behavior of actual gases, the movement of physical objects, evolution in finite populations, human behavior, and human and animal cognition. All of these are part of model-based science, in which one system is explored as a basis for understanding another system. This seems puzzling—how can studying things that do not exist lead to knowledge of things that do? Unsurprisingly, philosophers of science disagree on exactly how ‘false’ models—models that rest on known false assumptions—function in scientific practice. But there are a number of salient themes in the literature. Firstly, many false models rest upon abstraction in various ways. When we posit ideal gases and wholly rational agents, we are abstracting away from details which matter in the actual world. Abstraction depends on making false assumptions and ignoring details, but increases the tractability of the problem. For example, in ecology scientists face two obstacles for studying real systems: (a) the time scale on which they operate tends to exceed the time available for study, and (b) their complexity makes it difficult to manipulate them systematically (Odenbaugh 2005, 233). Abstract mathematical models in ecology allow both problems to be overcome. Moreover, even when a more detailed and perhaps accurate model is available, an abstracted model may be more cognitively and computationally tractable, and thus more useful, as Woody (2013, 1574–5) argues with respect to the ideal gas law.8
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Secondly, false models support contrastive explanations. In many cases, scientists seek to answer not ‘why this?’ but ‘why this rather than that?’. Consider Galileo’s theory of falling bodies, which predicts that all bodies will fall at the same speed. Of course that is not what we observe in the actual world. But Galileo’s theory allows us to explain why in our world anvils fall faster than feathers precisely via attention to the unrealistic assumption that falling happens in a vacuum. Galileo’s false model points us to the correct explanation—air resistance—and away from the incorrect one—mass (Hindriks 2008, 342–3).9 Another aspect of this feature of false but productive models is that they support counterfactual reasoning of various sorts. Bokulich (2011), discussing Bohr’s model of the hydrogen atom, points out that while the model is fictional, it allows us to answer a wide range of ‘what-if-things- had-been-different’ questions. There is, she suggests, a pattern of counterfactual dependence between the emission spectrum of hydrogen and the elements of Bohr’s model (Bokulich 2011, 43–44). Odenbaugh, discussing ecology, calls this how-possibly modelling (Odenbaugh 2005, 236–7). Both contrastive explanations and counterfactual reasoning are made possible by the support models provide for imagination. Finally, false models may provide explanatory scaffolding for scientists. In ecology, mathematical models of communities led to the articulation of a number of concepts of community stability and complexity, and thus to alternative hypotheses concerning the relationships between them, all of which have gone on to provide an improved conceptual framework for studying actual populations (Odenbaugh 2005, 250). False models can also guide scientists in the construction of explanations by exhibiting the shape that good explanations should take. In chemistry, the ideal gas law provides, among other things, inferential scaffolding, in that it allows chemists to conceptualize actual gas properties as deviations from the ideal, and thus unifies the treatment of all gases under a single false model (Woody 2013, 2015). Scaffolding can be seen at work in graphical models as well: Woody (2014) shows how the visual organization of the periodic table functions like a map. It helps chemists navigate an aspect of their discipline “not by presenting the most detailed representation possible but by highlighting features of the landscape that are taken to be significant” (Woody 2014,
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142). And the table can function in this way despite the fact that the so- called periodic law that underlies it does not seem to reveal any causal structure or identify any mechanisms, nor is it precise enough to support scientific predictions (Woody 2014, 133–4, 143).10 Some arguments for logical nihilism are motivated by the failure of formal logic(s) to offer accurate predictions for natural language consequence. But this assumes that the only roles that can be played by logics with respect to natural language validity are prediction and description. Franks’ argument for nihilism is, in part, that no system can capture the variety of logical properties that we might be interested in. But he still thinks that were a logic useful for explaining natural language validity, it would have to be correct, that is, as a theory it would be true or as a model it would have to be accurate. In contrast, the literature on scientific modelling suggests that false models and theories can play a role in scientific explanation in a variety of ways.
5
Logicians
One common thread in our rapid survey of false models in science is the attention paid to the way in which scientists make use of these models, and the roles that abstraction, imagination, and scaffolding play. This echoes Ronald Giere’s views about modelling in general, which characterize representation as something scientists do with models for specific purposes, rather than as a timeless property of a model (Giere 2004, 743).11 When it comes to logics, the question is not whether the formal systems correctly represent some feature of natural languages. The right question is how these models are used to explain. Our claim is that logicians interested in natural language use logics to represent in order to accomplish their explanatory goals.12 Meeting those goals involves abstraction, imagination, and scaffolding. It turns out that this perspective makes sense of three practices of logicians which at first may not make sense on ‘representationalist’ accounts of modelling and veridical explanatory theories. Models in logic can be any or all of formal languages, proof theories, or model theories. Given the target phenomena of logical study we are
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considering, formal languages are rather important. Those languages carry the relation of consequence. But we must not forget the wide variety of models in use by logicians. There are various versions of formal language: for example, first-order versus propositional. There are many kinds of model theory: for example, algebraic versus possible worlds. And of course there are various kinds of proof theories: for example, Fitch versus axiomatic. That variety brings us to our first practice: generating many kinds of models used for specialized purposes. For example, modal logicians use multiple separate propositional languages to capture fragments of natural language: for example, temporal and deontic. These idealizations allow logicians to focus on certain phenomena rather than needing to express the modal connections in terms of first-order formulas, or deal with interactions between the various modal operators. Abstraction is essential to tractability, just as in other scientific contexts. Of course there is considerable investigation of the relationship between modal logics and first-order logic. This manifests in the practice of translating modal formulas into first-order formulas that express conditions on Kripke frames.13 But logicians also understand the limitations of that connection: certain first-order conditions on frames—like all worlds are related—cannot be represented via a modal formula. That brings us to another of the roles of modelling: imagination. Logicians use models to understand or imagine the variety of ways things might be. The natural language arguments are the only things that are observable in the science of logic: we do not see the semantics, we do not see logical forms, we do not see generalized proof rules. All of those are models that represent a way of thinking about validity. But we try to draw connections between different arguments; we try to extend our understanding in one case to other cases. Logicians also ask how it is that things could be different between arguments. These differences could be in the way the world is, but they could also be differences in what matters in the connection or disconnection between premises and conclusions. The model and proof theories are used to make sense of those questions. It is here that the second practice of logicians becomes particularly salient: representation theorems. Representation theorems are important in mathematics. They allow one to draw connections between different
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structures. In social choice theory, one lays out a set of axioms for how, for example, a choice function behaves, and then shows that those axioms correspond to a certain kind of preference relation.14 In logic, we see representation theorems in a variety of forms. One form we have already mentioned: modal correspondence theory.15 Some object language formulas, when true at every world in every model on a frame, mean that the relation in that frame has a certain property: transitivity, reflexivity, and so on. But the most common representation theorem one sees in logic is completeness.16 Generally completeness involves the representation of a certain semantic notion of consequence in terms of a proof-theoretic form of consequence—or semantic truth in terms of theoremhood.17 Another form of representation theorem is when one looks for a set of proof rules, usually in a sequent calculus, that can be characterized by a certain set of properties. Such characterizations were what Michael Dummett used to argue for the supremacy of intuitionistic logic in (Dummett 1991). How do these connect to imagination? Representation theorems show the range of variation for a given use of a model. A completeness theorem shows which logical relationships between formulas are in and which are out. If we want to include different forms, we know we have to change the model theory in some way. Similarly for sets of sequent proof rules. If we want the set of proof rules to have an extra property, we know that the consequence relation will have to change. Correspondence theory shows a closer connection between model theory and formal language than completeness theorems. It shows that sometimes general facts about the model theory can be brought into the object language: properties about the frame relation can be expressed by the formulas. If we want to change the model theory, then, we know the logically true formulas will change. The final role of models, scaffolding, explains why logicians haven’t discarded classical propositional logic (CPL) despite its limitations. They add to it and subtract from it in various ways. Sometimes they add propositional connectives as in modal logics. Sometimes they model sub- propositional structure as in first-order logic or branching quantifier logics. Subclassical logicians may weaken it by rejecting, for example, excluded middle. Logicians give systems which differ from CPL, and
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philosophers of logic tend to debate about the virtues and vices of rival logics in their propositional forms. We know the fit between propositional languages and natural language arguments isn’t perfect, but we keep them around. More importantly, logicians don’t get rid of CPL: it is what every logician cuts her teeth on. It also acts as a logica franca for metalogical investigations into logics. CPL acts as a conceptual scaffolding for understanding other logics. CPL is really big: you can prove a lot using it. Most of the propositional logics that are studied are sub-logics of CPL. That is largely because it is post-complete18 and it is structurally complete.19 Often, logics are studied by how their proof theory or semantics must be different from those for classical logic; it is our logical jumping-off point. In the end, we see all of those modelling practices from other sciences, which also use inaccurate models, at work in the practices of logicians. But all of our talk about applications and modelling may be reminiscent of another perspective in the philosophy of logic that we want to distinguish from our view: pluralism.
6
Pluralism
Let us take stock. Like the nihilist, we are convinced by the existence of counterexamples to every putative logical law that no logic describes a consequence relation in natural language. Unlike the nihilist, we do not conclude that logic contributes nothing to our understanding of natural language validity. According to the particularist, logics can and do reveal something about the genuine connection between premises and conclusions in natural language. Logics are best understood as explanatory but inaccurate/non-representative/fictional models; they function like other ‘false’ models in the sciences. Before we turn to pluralism we have to highlight a couple of important features of particularism. The common understanding of logics as models assumes, one, natural language utterances have logical forms, and two, what matters to consequence is stable across contexts and arguments. We think both assumptions are false. The former is argued against elsewhere.20 The latter is a
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common ground for the varied particularist projects. As Dancy puts it, “[t]he core of particularism is its insistence on variability” (2013).21 Whether moral or logical, particularism carries this commitment to the variable relevance of features. In the logical context, having the form of modus ponens, under some system of logical analysis, is a relevant feature to the assessment of some arguments, but not in the assessment of others. Faced with counterexamples, the particularist feels no need to offer increasingly recondite analyses of the arguments in order to escape the conclusion that modus ponens is invalid.22 As Dancy emphasizes, the particularist takes the view that moral—or in our case logical—reasons function in ways “not noticeably different from the way in which other reasons function—more ordinary reasons for action, say, or reasons for belief rather than for action” (Dancy 2013). We now want to say a little more about what separates our view from that of the pluralist. Following Haack (1978), we distinguish between global pluralism—the view that there are multiple logics which are correct come what may—and local pluralism, the view that logics are only correct relative to a particular domain of inquiry. Much of the more recent work on pluralism takes the view that global pluralism is the only pluralism worthy of the name, on the grounds that it is only the global pluralist who claims that a specific argument could be simultaneously valid and invalid.23 But Shapiro (2014) offers a robust defense of a particular version of local pluralism. Global pluralists and monists share the common assumption that there is a general-purpose consequence relation for natural language that logic(s) aim to describe. They differ on how many such relations there are, but agree that any such relation applies in any context of reasoning. In contrast, the local pluralist maintains that there are different consequence relations that apply in different contexts of reasoning. This might, on first glance, look very much like the particularist view. After all, we have said that validity is a matter of preserving some property or other in the move from premises and conclusion, and that what property must be preserved changes from case to case. The point of fundamental disagreement between us and the local pluralist lies with the notion of correctness. While the local pluralist allows what matters to consequence to vary from context to context, they think
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the aim of logic is accurate prediction and adjudication for natural language validity. To avoid the pitfalls of the global pluralist, the local pluralist needs some way of demarcating the area where each correct logic is in force. For example, a local pluralist might maintain that there are different logics for different areas of inquiry (which might be epistemologically or ontologically distinguished); or different logics for different sets of logical connectives (and perhaps that a single natural language connective can represent different logical connectives); or different logics for different sets of possible truth values (e.g. one might think that claims about the natural world must either be true or false, but that vagueness in social facts requires a plurality of truth values). The need for demarcation makes local pluralist vulnerable to the arguments of the nihilist. As Russell (2017) argues, we can ‘validate’ any purported logical law by restricting the substitution class into that law. For example, if we allow only true sentences to be substituted for φ and ψ in the form φ → ψ, ψ ⊨ φ, then this ‘law’—affirming the consequent—would seem to be correct. That isn’t a convincing justification of affirming the consequent precisely because the restriction on substitution is ad hoc. But then the question becomes, what restrictions are legitimate? The local pluralist needs a method of demarcation independent of any judgments about validity. For example, the local pluralist cannot restrict substitution to sentences where the laws of classical logic apply, and use that as an argument that classical logic is appropriate for that domain. Such a demarcation may be possible, but we are skeptical because any such demarcation would depend upon a logic neutral account of logical form.24 In contrast, the particularist has no need of an independent means of determining which logic applies to a particular argument; from our perspective the role of logic is not to adjudicate or predict validity but to explain it. Features and properties have variable relevance to validity, and we cannot determine in advance of considering a specific argument which properties are relevant. But, once we have decided what is relevant, logics can explain how the relevant validity-making properties could be preserved between premises and conclusions.25
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Particularist Logic
One natural objection to our view rests on the idea that logics are not just explanatory but also action guiding. Certainly logicians, qua scientists, may act as we have described, but non-logicians use logic to give arguments. Moreover, they use formal logic to give and check their arguments. That means they use a logic’s rules to ensure the validity of their arguments by following the rules of that logic. Now for what might seem to be a U-turn: this practice can be acceptable. We take logic to be the study of connections between premises and conclusions. Formal systems show how connections between premises and conclusions can be made. In classical logic, for example, we use a conception of ‘connection’ as truth preservation. Classical logic also provides a precise account of what ‘truth preservation’ means.26 This model sets a narrow boundary for what can count when evaluating an argument for validity. The flaw in the generalist way of thinking is the assumption that model must be applied everywhere, always. What about premises that can’t all be true simultaneously? Something other than truth must matter in that case. Here we see logic not just as a science, but as engineering. We look for some property that might suit our purposes, and then we see what we can do to model it as a connection. A favorite example is the notion of level of consistency or just ‘level’—from Jennings and Schotch (1984). Level is the answer to the question: what is the size of the smallest partition of the set in which all the sets in the partition are consistent? It measures inconsistency. The level of {p, ¬p} is two, because it takes at least two sets in a partition to get consistent sets. The level of {q ∧ p, p ∧ ¬ q ∧ r, p ⊃ (¬q ∧ ¬ r)} is three. One relation that preserves level is defined by saying t follows from a set of premises Γ when t follows classically from some set in every partition of Γ that is the same size as the one used to determine the level of the set, and each element of the partition is consistent.27 The level of the set of consequences on this definition will be the same as the level of the premises. This level-preserving approach to inconsistent consequence builds one model atop another as scientists are wont to do. It gives an altogether
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different interpretation to ‘preserving’ since the property that is preserved is between sets, not sentences, but why not? The truth property in the classical model stops working in the case of inconsistency. This is not the only place where something other than truth is used, but for different reasons. In deontic logic it has long been recognized that rules are not really true, and so some have tried to interpret satisfaction between models and formulas, not as truth, but as rule legitimacy—whatever that may mean.28 Others have added speech act satisfaction to model theories for speech acts, in addition to truth.29 With yet another focus, there are logics concerned with metaphysical grounding rather truth, see Fine (2012). These logics use the scaffolding provided by classical logic, just in different ways. In each of these cases, a way of connecting premises to conclusions is chosen. A model is engineered to understand the way that connection behaves across a range of applications. But each model focuses only on that one connection. The particularist doesn’t change the methodology of formal logic, she changes the interpretation of the end results to recognize the limitations of the model. But when is it legitimate to use a logic to construct an argument? There is a wonderful property that some logics have: conservative extension. Start with a language L, with a relation of consequence ▷ between the sentences of L. Then extend L to L* with some new bits.30 An extension of ▷ is a relation over L*, represented by ⊢, that contains ▷. We say that ⊢ is a conservative extension of ▷ when for each argument (A, b) in L, A ⊢ b iff A ▷ b. The new relation doesn’t make any arguments that were not already valid in ▷, valid. The particularist methodology is to offer an explanation of consequence for (a fragment of ) natural language, and is basically the standard methodology in logic. Let’s say ▷ is consequence in natural language. The logician focuses attention on some subset of L, L′, and gives a model that is extensionally adequate for the particular arguments in ▷ restricted to L′. The elements of L′ are replaced by their mathematical models constructed from the new bits in L*. What we then get is a language made from the union of L without the bits in L′, and L*. The arguments from ▷ restricted to L′ are replaced by their counterparts in L* that are related by ⊢.31
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The language of L* is formal, so it will have substitution instances from outside of L′. ⊢ is an extensionally adequate explanation of the arguments from L′; it may extend ▷, and that is why conservative extension is important. Building arguments using the new parts from L* allows her to be certain that she won’t offend the un-targeted portion of ▷. The explanation of natural language argument is fragmentary; ⊢ has limited application. Any arguments that extend ▷ are given or derived using rules which are chosen to characterize the abstract model, not reality per se. Argument construction proceeds, on the particularist account, by first deciding what connection matters between premises and conclusions. Then one deploys a logic developed to model that connection. Arguments constructed with this process in mind aren’t guaranteed validity—particularism doesn’t assume logics are correct. The goal of the particularist approach to argument construction is epistemological transparency: it wears its commitments on its sleeve.32
8
Conclusion
Reasons have variable relevance. The reason that a glass dropped on a cement floor broke is that it is fragile and the cement is hard. In another case, I might drop an identical glass and it might not break. In this second case, the fact that it is fragile and I dropped it on hard cement is not a reason that the glass broke, for the simple reason that the glass is intact. Dancy (2013) has pointed out that reasons can have this variability in the moral context, and we are pointing out that they have that variability in the logical context. The logical particularist maintains that an argument analyzed as having the form of modus ponens when using classical logic might well be the reason it is valid in one case. In another case, an argument with that form in classical logic is not valid, and the classical analysis provides no reason for taking it to be. Nonetheless, we think that the standard methodology of logic is productive and contributes to the historical goal of explaining validity in natural language reasoning. Logical particularists use logics as explanatory
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tools and take the explanations to be on par with those offered by other sciences. Logics do not explain validity on their own. In this chapter, we’ve sketched an account of the functioning of logics as models which is continuous with the use of such models in other scientific endeavors. Specifically, we have argued that models in logic serve our explanatory goals without aiming at, or succeeding at, representation.
Notes 1. Although we have argued elsewhere that giving the same rules to connectives does not mean that the connectives have the same meaning qua natural language connectives (Wyatt and Payette 2018), we do not doubt the mathematical significance of the construction and the purely mathematical conception of proof-theoretic meaning used in the construction. It is the mathematical significance of the constructions that interests Franks, if we have understood him correctly. We are not at odds on this point. 2. See Payette and Schotch (2007), Jennings et al. (2009). 3. As will become clear from the subsequent discussion, we do reject the view on which formal logic sheds light on the validity of arguments by directly representing the semantics of natural language sentences. See Stokhof (2007) for a discussion of the relationship between formal languages and natural language semantics that is somewhat parallel to the view of the relationship between formal logics and natural language argument advocated here. 4. Cook’s account of how logics model validity is very similar to that of Shapiro (2014): the primary difference is that Shapiro takes logics to be only in the business of modelling mathematical argument, while for Cook mathematical argument is a special case of a more general goal. 5. Cook refers to models as ‘mock-ups’ (Cook 2002). 6. Notice that this picture is absolutely neutral on the nature of the relationship between the phonological realization or the syntax of natural language sentences and their logical forms. 7. See Payette and Wyatt (2018) for more discussion of this point. 8. Abstraction here covers both idealization and fictionalization in science, where fictional models are those in which no amount of de-idealization—
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that is, no amount of adding in details left out or correcting false assumptions—would recover a correct description of the actual world (Bokulich 2011). 9. Hindriks (2008) also argues that models in the economics context can be better when containing a larger number of unrealistic assumptions. 10. The periodic table does not, contra the usual historical story, indicate directly the presence of missing elements, nor does it generate any predictions of specific quantitative properties. See Woody (2014) for further discussion. 11. Giere proposes that the representative features of models fall under the general schema: S uses X to represent W for purposes P. Some of this purpose relativity is already present in Cook’s view since which components are representors and which are artifacts may depend on the goal of the model. That goal is, presumably, imposed on the model. It is possible that Cook would be in closer agreement than we have suggested here, but note that critics, for example, Smith (2011), describe Cook’s view as one where representation is a relation between (parts of ) models and reality, rather than the 4-ary relation above. We won’t enter any further into debates about how to interpret Cook here since our goal is to advance our own view. 12. This is not a descriptive claim; that is, we are not claiming that this is how logicians understand what they are doing, whether collectively or individually. Rather this is a claim about how we should understand logical practice. In line with our anti-exceptionalism, we are approaching the epistemological and metaphysical questions in logic in the same way in which the philosopher of biology approaches those questions in biology. In both cases one must attend to the messy business of how science actually gets done. For further discussion of our methodology, see Payette and Wyatt (2018). 13. For example, □p ⊃ p is true at every world in every model on a Kripke frame (W, R) iff ∀w, wRw. 14. See Sen (1970). One can also consider Cozic and Hill (2015) to understand the larger role of representation theorems in science. 15. See van Benthem (1997) for an overview. 16. Here we are using ‘completeness’ in the (common) sense which includes both completeness and soundness theorems. 17. See Franks (2010) for more on interpretations of completeness.
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18. You cannot add a formula that isn’t a theorem to CPL as a new theorem without being able to then prove everything. 19. All of the admissible rules are derivable rules. For more on that distinction, see Iemhoff and Metcalfe (2009). 20. See Wyatt and Payette (2018). 21. See, for example, the discussions in McKeever and Ridge (2005), Cullity and Holton (2002) for moral particularisms, and Lakatos (1976) and Larvor (2001, 2008) for particularism in mathematics. John Norton’s material theory of induction is, we think, a particularist approach to inductive argument (2003, 2010). 22. See Kolodny and MacFarlane (2010), Dreier (2009), Cantwell (2008), or McGee (1985) for some possible counterexamples to modus ponens. 23. See, for example, Cook (2010), Beall and Restall (2006). 24. We argue in Wyatt and Payette (2018) that attributions of logical form are only possible from within a logic. 25. Logical particularism can be seen as a form of instrumentalism, but not at the level of theory justification. That is, we don’t take it that a ‘correct’ logic is the most useful one. Rather we take it that the practices and methods of formal logic are instrumentally justified because of their effectiveness in explanation. 26. Of course one could give other interpretations of the connection modeled by classical logic, and some people disagree with the classical interpretation of truth preservation, see, for example, Read (1994). 27. The examples above are not very interesting sets. Each has only its partition into unit sets, and so something follows when it follows classically from at least one of the formulas. 28. This is at least true for certain parts of the models in Castañeda (1981). 29. This is done in Searle and Vanderveken (1985). 30. a.k.a. mathematically defined logical connectives. 31. Note that we did not say that ⊢ explicates the arguments from L′. We agree with Woody, if we have interpreted her correctly, that what is often called ‘explication’ is simply a stage in explanation. Models, particularly mathematical ones, become part of the understanding of phenomena. 32. This is similar, we think, to the project in Field (2015) where commitments to relationships between degrees of belief are translated into logical rules. Particularism assumes that logical connections can be about things other than degrees of belief.
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References Beall, J., and G. Restall. 2006. Logical Pluralism. Oxford: Oxford University Press. Bokulich, A. 2011. How Scientific Models Can Explain. Synthese 180 (1): 33–45. Cantwell, J. 2008. Changing the Modal Context. Theoria 74 (4): 331–351. Castañeda, H.N. 1981. The Paradoxes of Deontic Logic: The Simplest Solution to All of Them in One Fell Swoop. In New Studies in Deontic Logic, ed. R. Hilpinen, 37–85. Chichester: Wiley-Blackwell. Cook, R.T. 2002. Vagueness and Mathematical Precision. Mind 111 (442): 225–247. ———. 2010. Let a Thousand Flowers Bloom: A Tour of Logical Pluralism. Philosophy Compass 5 (6): 492–504. Cozic, M., and B. Hill. 2015. Representation Theorems and the Semantics of Decision-Theoretic Concepts. Journal of Economic Methodology 22 (3): 292–311. Cullity, G., and R. Holton. 2002. Particularism and Moral Theory. Aristotelian Society Supplementary Volume 76: 169–209. Dancy, Jonathan. 2013. Moral Particularism. In The Stanford Encyclopedia of Philosophy (Fall 2013 Edn.), ed. E.N. Zalta. Metaphysics Research Lab, Stanford University. Dreier, J. 2009. Practical Conditionals. In Reasons for Action, ed. D. Sobel and S. Wall, 116–133. Cambridge: Cambridge University Press. Dummett, M. 1991. The Logical Basis of Metaphysics. Harvard: Harvard University Press. Field, H. 2015. What Is Logical Validity? In Foundations of Logical Consequence, ed. C.R. Caret and O.T. Hjortland, 33–70. Oxford: Oxford University Press. Fine, K. 2012. The Pure Logic of Ground. Review of Symbolic Logic 5 (1): 1–25. Franks, C. 2010. Cut as Consequence. History and Philosophy of Logic 31 (4): 349–379. ———. 2015. Logical Nihilism. In Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, ed. A. Villaveces, R. Kossak, J. Kontinen, and A. Hirvonen, 147–166. Berlin: De Gruyter. Giere, R.N. 2004. How Models Are Used to Represent Reality. Philosophy of Science 71 (5): 742–752.
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Godfrey-Smith, P. 2009. Models and Fictions in Science. Philosophical Studies 143 (1): 101–116. Haack, S. 1978. Philosophy of Logics. Cambridge: Cambridge University Press. Hindriks, F. 2008. False Models as Explanatory Engines. Philosophy of the Social Sciences 38 (3): 334–360. Iemhoff, R., and G. Metcalfe. 2009. Proof Theory for Admissible Rules. Annals of Pure and Applied Logic 159 (1–2): 171–186. Jennings, R.E., and P.K. Schotch. 1984. The Preservation of Coherence. Studia Logica 43 (1–2): 89–106. Jennings, R.E., B. Brown, and P. Schotch, eds. 2009. On Preserving: Essays on Preservationism and Paraconsistency. Toronto: University of Toronto Press. Kolodny, N., and J. MacFarlane. 2010. Ifs and Oughts. Journal of Philosophy 107 (3): 115–143. Lakatos, I. 1976. Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge: Cambridge University Press. Larvor, B. 2001. What Is Dialectical Philosophy of Mathematics? Philosophia Mathematica 9 (2): 212–229. ———. 2008. Moral Particularism and Scientific Practice. Metaphilosophy 39 (4–5): 492–507. McGee, V. 1985. A Counter Example to Modus Ponens. Journal of Philosophy 82 (9): 462–471. McKeever, S., and M. Ridge. 2005. The Many Moral Particularisms. Canadian Journal of Philosophy 35 (1): 83–106. Norton, J.D. 2003. A Material Theory of Induction. Philosophy of Science 70 (4): 647–670. ———. 2010. There Are No Universal Rules for Induction. Philosophy of Science 77 (5): 765–777. Odenbaugh, J. 2005. Idealized, Inaccurate but Successful: A Pragmatic Approach to Evaluating Models in Theoretical Ecology. Biology and Philosophy 20 (2–3): 231–255. Payette, G., and P.K. Schotch. 2007. On Preserving. Logica Universalis 1 (2): 295–310. Payette, G., and N. Wyatt. 2018. How Do Logics Explain? Australasian Journal of Philosophy 96 (1): 157–167. Priest, G. 2014. Logical Pluralism: Another Application for Chunk and Permeate. Erkenntnis 79 (SUPPL.2): 331–338. Read, S. 1994. Formal and Material Consequence. Journal of Philosophical Logic 23 (2): 247–265.
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Russell, G. 2017. An Introduction to Logical Nihilism. In Logic, Methodology, and the Philosophy of Science: Proceedings of the Fifteenth International Congress, ed. H. Leitgeb, I. Niiniluoto, P. Seppäla, and E. Sober, 1–10. London: College Publications. Searle, J.R., and D. Vanderveken. 1985. Foundations of Illocutionary Logic. Cambridge: Cambridge University Press. Sen, A. 1970. Collective Choice and Social Welfare. San Francisco: Holden Day. Shapiro, S. 2014. Varieties of Logic. Oxford: Oxford University Press. Smith, Peter. 2011. Squeezing Arguments. Analysis 71 (1): 22–30. Stokhof, M. 2007. Hand or Hammer? On Formal and Natural Languages in Semantics. Journal of Indian Philosophy 35 (5–6): 597–626. van Benthem, J. 1997. Correspondence Theory. In Handbook of Philosophical Logic, ed. D.M. Gabay and F. Guenthner, vol. 2, 2nd ed., 325–408. Dordrecht: Klewer. Woody, A.I. 2013. How Is the Ideal Gas Law Explanatory? Science & Education 22 (7): 1563–1580. ———. 2014. Chemistry’s Periodic Law: Rethinking Representation and Explanation After the Turn to Practice. In Science After the Practice Turn in the Philosophy, History, and Social Studies of Science, ed. L. Soler, S. Zwart, M. Lynch, and V. Israel-Jost, 123–150. London: Routledge. ———. 2015. Re-Orienting Discussions of Scientific Explanation: A Functional Perspective. Studies in History and Philosophy of Science Part A 52: 79–87. Wyatt, N., and G. Payette. 2018. Logical Pluralism and Logical Form. Logique et Analyse 241: 25–42.
Logical Nihilism Aaron J. Cotnoir
Ordinary language has no exact logic. Strawson (1950)
1
Outlining the View
The philosophy of logic has been dominated by the view that there is One True Logic. What is meant by ‘One True Logic’ is sometimes not made entirely clear—what is a logic and what is it for one of them to be true? Since the study of logic involves giving a theory of logical consequence for formal languages, the view must be that there is one true theory of logical consequence. What it means for a logic to be true is, roughly, for it to correctly represent something or other.1 What do logics represent? It is clear from the various uses of applied logic, they can represent many different sorts of phenomena. But for the purposes of traditional logic,
A. J. Cotnoir (*) Department of Philosophy, University of St Andrews, St Andrews, UK e-mail: [email protected] © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_13
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though, theories of consequence are frequently taken to represent natural language inference. As Beall and Restall (2006) note: Logic, in the core tradition, involves the study of formal languages, of course, but the primary aim is to consider such languages as interpreted: languages which may be used either directly to make assertions or denials, or to analyse natural languages. Logic, whatever it is, must be a tool useful for the analysis of the inferential relationships between premises and conclusions expressed in arguments we actually employ. If a discipline does not manage this much, it cannot be logic in the traditional sense. (8)
If this is right, it means that logic is connected to inferential practice of natural language speakers.2 Now perhaps logical theorizing is not an entirely descriptive enterprise and contains some element of normativity. But we think it is clearly true that a logic would have claims to be ‘correct’ only if it is constrained in some way by actual inferential practice. For the purposes of this chapter, we will follow Beall and Restall (and others3) in holding this characterization of logic as traditionally conceived.4 Once we have pinned down our subject matter, we see a number of possible outcomes. Logical Monism There’s exactly one logical consequence relation that correctly represents natural language inference. Logical Pluralism There’s more than one logical consequence relation that correctly represents natural language inference. Logical Nihilism There’s no logical consequence relation that correctly represents natural language inference. Much of the discussion in the philosophy of logic over the last decade has been devoted to the debate between logical monism and logical pluralism. But logical nihilism hasn’t been given nearly as much attention, even though the view has historical roots and is philosophically defensible.
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A clarification: there is another view which can rightly be called ‘Logical Nihilism’ which is not the view under consideration here. Mortensen (1989) has argued that anything is possible based on a kind of thorough-going empiricism. On one reading, Mortensen believes everything is logically possible.5 And if everything is possible, then for any argument from premises X to conclusion A there will be some possible case according to which X is satisfied, but A not. Thus, the logical consequence relation is empty. This view plausibly deserves the title ‘Logical Nihilism’6 and in fact has been defended under that name by Estrada- González (2011) and Russell (2017). So, I think it is worth distinguishing two types of logical nihilism: Logical Nihilism 1 There’s no logical consequence relation that correctly represents natural language inference; formal logics are inadequate to capture informal inference. Logical Nihilism 2 There are no logical constraints on natural language inference; there are always counterexamples to any purportedly valid forms. On the second view, there is a consequence relation (namely the empty one) which gets natural language inference ‘right’. The focus of this chapter is on logical nihilism in the first sense. In what follows, I present and defend a number of arguments in favor of logical nihilism. The arguments are grouped into two main families: arguments from diversity (§2) and arguments from expressive limitations (§3). The arguments are often simple syllogisms, pointing to fundamental differences between natural languages and formal consequence relations. Many of the arguments involve familiar problems in the philosophy of logic. The arguments, taken individually, are interesting in their own right; they each highlight an important way in which the formal methods of logic can be seen to be inadequate to modeling natural language inference. But the arguments taken jointly are more significant; by presenting all the arguments together, we can build something of a cumulative case for logical nihilism. Of course, if any of these arguments are sound then logical nihilism is correct. But the arguments reinforce one another, such
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that logical nihilism presents us with a unified view across a broad range of issues in philosophy of logic. I conclude (§4) by considering related philosophical issues and sketching a general outlook on logic and formal methods that is nihilist-friendly. Before presenting these arguments, however, let me respond to an immediate worry. In presenting and endorsing arguments for a view, one ordinarily takes them as good arguments, where good arguments are (at the very least) valid arguments. But if there is no correct theory of logical consequence, then presumably there are no valid arguments either. Thus, one might suspect that any attempt to argue for logical nihilism undermines itself. This worry could be serious if logical nihilism entailed that there were no valid arguments (as in Logical Nihilism 2). But notice that logical nihilism (of both sorts) is consistent with their being standards governing good inference in natural language. And Logical Nihilism 1 merely claims that there is no formal theory that perfectly captures these standards. And that’s perfectly compatible with these arguments being formally valid in some regimented language that is adequate for more restricted purposes. It’s also compatible with arguments conforming to inherently informal standards on good inferential practice, or an abductive methodology in the philosophy of logic (e.g. Williamson 2017). So the worry is ill-founded.
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Arguments from Diversity
The first batch of arguments for logical nihilism follows a simple recipe. The first ingredient is an argument for logical pluralism; this establishes that no single logic can be an adequate theory of inference. The second ingredient is a constraint on logical consequence that rails against logical pluralism—that is, a constraint that requires there be at most one single adequate theory of inference. Combine these two ingredients and stir; the result is logical nihilism. For if there cannot be one single correct theory, and any correct theory must not be plural, then there cannot be any correct theory at all. Fortunately for us, such ingredients are not hard to come by.
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There are a number of types of logical pluralism that have been defended in the literature; I won’t consider all of them. I will focus on Beall and Restall’s (2000, 2006) case-based pluralism, and Lynch’s (2009) domain-based pluralism. I’ll also briefly consider Varzi’s (2002) logical relativism and Russell’s (2008) truth-bearer dependent version of logical pluralism.7
An Argument from Necessity Beall and Restall (2006) suggest that the notion of logical consequence may be analyzed by the Generalized Tarski Thesis (GTT). GTT An argument is validx if and only if in every casex in which the premises are true, the conclusion is true. Of course, we have yet to specify what sorts of cases are under consideration here. Logicians are frequently concerned with models, but the notion of consequence itself doesn’t determine that models are the only possible option. Once this is granted, there is a straightforward argument to pluralism. The idea is that settled core of logical consequence (‘the intuitive or pre-theoretic notion’ (Beall and Restall 2000)) is given by GTT. An instance of GTT is obtained by a specification of casesx in GTT, and a specification of the relation of being true in a case. An instance of GTT is admissible if it satisfies the settled role of consequence, and if its judgments about consequence are necessary, normative, and formal (in some sense or other). A logic, then, is an admissible instance of GTT. Beall and Restall contend that there are at least three admissible instances of GTT. They defend this claim by appeal to the diverse purposes logic may be put to, each of which corresponds to a different kind of case. There are the complete and consistent cases of classical logic (i.e. worlds), the incomplete cases of intuitionistic logic (i.e. constructions), and the inconsistent (and/or incomplete) cases of relevant logic (i.e. situations). As a result, there is no single correct consequence relation, but many correct consequence relations relative to which kind of cases we intend.
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Suppose we agree with Beall and Restall that there are a number of different ways of specifying the notion of a case, each with its own specification of truth-in-a-case, and hence a number of instances of GTT. We might still disagree that any one of these instances is admissible because the resulting consequence relations fail to be necessary. Clearly, it’s a plausible constraint on logic that it must be necessary; an argument is valid whenever it is necessarily truth preserving. But on the surface, the necessity constraint appears to require us to look at all kinds of cases, if they really are genuine cases. This is, in effect, a version of an objection to logical pluralism developed by Bueno and Shalkowski (2009). Thus, on Beall and Restall’s account, none of the major families of logics they consider satisfy the necessity constraint. By their own standard, none of these are logics at all. Only a very weak consequence relation survives this scrutiny, according to their accounting of the necessity constraint as quantification over all cases. Once the partisan spirit of logical monism is replaced with the open-minded embrace of cases suitable to alternative logics, no commonly promulgated consequence relation seems to satisfy the necessity constraint. Hence, according to their own accounting of the constraints on relations of logical consequence, there are no such relations. (p. 299f )
It is worth noting that there are actually two sorts of objections here, each resulting in a different form of logical nihilism. The first objection is that, on an absolutely general reading of necessity—necessity in the broadest possible sense—as quantifying over all cases of any type, no instance of GTT counts as admissible. And hence there are no logics; this is a form of Logical Nihilism 1. On the other hand, we might instead specify a general notion of case as follows: c is a case* iff c is a casex for some x. Similarly, we can give a general specification of truth-in-a-case as follows: A is true-in-a-case* iff c is a casex for some x and c represents A to be true-in-a-casex. This seems to suffice for an instance of GTT. And indeed, this will constitute an admissible instance especially with regard to necessity (read again in the broadest sense). But, if we take seriously the vast range of logics which have
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been put forward by various logicians, then given corresponding notions of casesx and truth-in-a-casex, it is doubtful that any substantive logical principles will survive.8 This would lead to a situation close to Logical Nihilism 2. But even if we simply stick with the cases Beall and Restall have already admitted, it seems clear that the logic resulting from quantifying over all such cases is still going to be far too weak to adequately account for natural language inference. I do not mean to suggest that Beall and Restall have nothing to say in response to this style of argument—indeed, they do (2006). Whether their response is ultimately successful I leave to the reader to decide. But I merely want to show that arguments for case-based pluralism might naturally be converted to an argument from diversity that attempts to establish logical nihilism.
An Argument from Topic-Neutrality A similar sort of strategy equally applies to domain-based pluralists. Some pluralists about truth have suggested that there are viable forms of logical pluralism that sit naturally with their views. Pluralism about truth is the view that different domains of inquiry (e.g. science, mathematics, ethics, history, etc.) have different truth properties associated with them; so being true in one domain might consist in corresponding to a fact, whereas being true in a different domain might correspond to a kind of generalized coherence. Lynch (2008, 2009) has recently argued that, because truth properties in domains of inquiry are so vastly different, which inferences are valid (i.e. which inferences preserve these properties) can vary from domain to domain.9 This is a domain-based logical pluralism. However, many have contended that logical consequence must be topic-neutral. Indeed, we might agree with Sher (1996) that ‘formality and necessity play the role of adequacy conditions: an adequate definition of logical consequence yields only consequences that are necessary and formal’ (p. 654). That is, what arguments are valid can depend only on their form, and cannot depend on what the arguments are about. How to cash out this topic-neutrality constraint is somewhat controversial (see MacFarlane 2002 for various proposals), but it seems clear that any
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domain-based logical pluralism will not meet such a constraint. Of course, one might accept that within a domain logical consequence does not depend on any further facts about the content of those propositions. But there is still the residual fact that which argument forms are valid and invalid will depend on the domain of inquiry one is in. The truth pluralist argues that any adequate theory of consequence must be domain- relative. By topic-neutrality, there can be no adequate theory of consequence. As a result, one can convert an argument for domain-based pluralism into an argument from diversity for logical nihilism.
Other Arguments from Diversity Another strategy along the same lines applies to Varzi’s (2002) brand of logical pluralism. According to logical relativism, the correct inferences depend crucially on what one takes to be logical—as opposed to the non- logical—vocabulary. Varzi advocates for a kind of skepticism about any objective criterion on where to draw the logical/non-logical divide.10 Of course, one might be perfectly happy to accept that there are natural language consequence relations for each way of drawing the divide. Indeed, one might rather appeal to instead to ‘meaning postulates’ or ‘semantic constraints’ as (cf. Sagi 2014) for all vocabulary to generate semantic entailment relations. Formal semanticists often deliver such theories. But there are general reasons why formal semantics does not deliver the logic of natural language. Glanzberg (2015, §2.2) argues that such meaning postulates or semantic constraints will, at best, deliver analytic entailments which should not intuitively count as valid (e.g. ‘John cut the bread’ lexically entails ‘The bread was cut with an instrument’ because of constraints on the meaning of ‘cut’). This seems to dictate against counting such semantic ‘entailment’ relations to be logical consequence proper. Glanzberg (2015, §2.1) also argues that natural language semantics, whether in its neo-Davidsonian or type-theoretic guises, are primarily concerned with absolute semantics. That is, they give semantic clauses for sentences expressing the meanings that speakers actually understand those sentences to have. They do not give relative semantics, that is, semantic clauses that are generally specified over a range of models, which
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is precisely what is required in order to specify logical consequence.11 The end result is that, if logic must be formal, then there is no logical consequence relation that is adequate to natural language. At the risk of belaboring the point, allow me to give a final argument from diversity. Russell (2008) argues that different views on the relata of the consequence relation (e.g. sentences, propositions, etc.) yield different consequence relations. If one takes sentences to be the relata of consequence, then contextual aspects of language can validate inferences which are intuitively invalid (e.g. ‘I am here now’ is a logical truth in Kaplan’s LD).12 Alternatively, if one takes propositions to be the relata of consequence, then propositions can validate inferences which are intuitively invalid as well (e.g. ‘Hesperus is Hesperus’ entails ‘Hesperus is Phosphorous’). Russell draws logical pluralism as her conclusion, but I’d contend that the arguments better support logical nihilism: both of these consequence relations deliver intuitively invalid validities. Of course, a nihilist need not reject that such semantic or metaphysical necessitation relations exist; it is just that they do not appear adequate to the role of logic as traditionally conceived. The various available arguments from diversity are arguments on-the- cheap, as it were. They piggyback on arguments for logical pluralism, together with monist constraints about the concept of logical consequence. Monists may not be sympathetic to the underlying pluralistic motivations, and pluralists may not be sympathetic to the monistic reading of the constraints on logical consequence. Though I wouldn’t expect monists or pluralists to agree, I think it is striking that logical nihilism can make sense of the motivations behind both lines of thought.
3
Arguments from Expressive Limitations
The second class of arguments follows a different sort of recipe. It starts from a given phenomenon of natural language inferential practice, and argues that no formal language can exhibit that phenomena. Many arguments of this sort have been given in the past and recent literature. But again, I think it is worth putting them all in one place to display their cumulative weight. Every argument from expressive limitations needs to
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be refuted in order to defend against logical nihilism. And I hope to display just how difficult this task is.
The Argument from Semantic Closure The first argument from expressive limitations for logical nihilism appeals to considerations around semantic closure. 1 . Natural languages are semantically closed. 2. No formal language is semantically closed. 3. So, no formal language is adequate to natural language. The argument appeals to an alleged fundamental difference between formal languages (from which logical consequence relations are derived) and natural language. The idea is that natural languages are capable of representing their own semantics; we could express all the truths about English in English. But formal languages cannot generally be supplemented with semantic notions that apply to themselves on pain of triviality. Consider premise 1. Many find it straightforwardly obvious that natural languages like English are semantically closed in the sense that we can use natural languages like English to talk about the semantic properties and relations of sentences of English. Scharp (2013) argues that it is a fundamental presupposition of formal semantics that such a thing is possible. In attempting to give a complete theory of natural languages as a whole, couched in a given natural language, we must presuppose that that language can serve as a metalanguage for itself. Take the semantic property ‘is true’ as an example. Eklund (2002) has argued that in order to be competent with the use of the English truth predicate, one must be willing to apply it to all sentences of the English language. As for premise 2, there are a number of different ways of supporting this claim. Tarski showed clearly that the truth predicate could not be expressed in any formal language (of certain expressive richness) if the consequence relation was to be classical due to semantically ‘rogue’ sentences like the Liar sentence. This means that, at the very least, premise 2
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is true with respect to classical formal languages.13 This result has led many to reject that classical logic is the One True Logic. But the well-known phenomenon of revenge plagues non-classical formal languages as well. The literature is predictably populated with articles that find inexpressible concepts for non-classical logics that purport to be semantically closed. Even absent a general recipe for finding a revenge paradox for every formal language, the sheer volume of failed attempts at semantic closure would justify a plausible pessimistic meta-induction14: if premise 2 has proven true for every formal language so far, then there’s good reason to think the issue is systemic. There are somewhat more general recipes for finding revenge problems even for non-classical approaches to paradox. Here is a basic one: we begin with three desiderata for any semantically closed formal language 𝕷. Characterization There is a property φ s.t. all and only ‘rogue’ sentences of 𝕷 are φ. Semantic Closure φ is expressible in 𝕷; there is a name ⟨α⟩ for every sentence α in 𝕷. Revenge Immunity No sentence attributing φ to a ‘rogue’ sentence is itself φ. Let Γ be the class of ‘rogue’ sentences. Now, [by SEMANTIC CLOSURE] construct the following sentence λ ∶ φ⟨λ⟩—the sentence which says of itself that it is a rogue sentence. Now, it will typically be the case that λ is a paradigm case of a rogue sentence; λ ∈ Γ. So, [by characterization], φ⟨λ⟩ is true. But [by REVENGE IMMUNITY], φ⟨λ⟩ ∉ Γ. Contradiction. One might object that this is merely a proof that, contrary to appearances, λ is not rogue. But then consider the sentence γ ∶ φ⟨γ⟩ ∨ ¬T⟨γ⟩, where T is our truth predicate. If γ is not rogue, then it must fall under some other semantic category (i.e. true or false). But in either case, we get the standard Liar reasoning yielding a contradiction. So γ must be rogue. But since γ is an attribution of φ to a rogue sentence, we have a violation of REVENGE IMMUNITY. This is perhaps especially problematic since, if γ is rogue,
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that’s precisely what one disjunct of γ says, and thus it should intuitively be true. The characterization of paradoxical sentences seems destined to break down. There are, of course, a number of options for replying to a revenge charge like this. The first option is to reject that Characterization is a desiderata at all. Of course, the characterization constraint has two directions. One might reject the ‘all’ direction, and attempt to leave some paradoxical sentences uncharacterized. An immediate reply would be that one has thereby failed to give a complete account of the semantic paradoxes. On the other hand, one might reject the ‘only’ direction, and claim that we must throw out some babies (non-rogues) when throwing out the bathwater (rogues). Two immediate replies come to mind: first, this will result in a situation where being a ‘rogue’ sentence is not expressible in 𝕷 (even if φ is); after all φ will apply to some non-rogues. Second, being a ‘rogue’ sentence appears not to be doing any work in the resulting theory, the characterization of paradoxes happens using φ. One might opt for second response to revenge: reject Revenge Immunity—that is, accept that some attributions of φ are themselves ‘rogue’. An immediate reply to this approach, however, is that since the theory itself includes attributions of φ to the paradoxical sentences, one’s theory relies on exactly the problematic phenomenon. That is, one is forced to use rogue sentences in giving one’s theory. A third response to these revenge charges would be simply to reject Semantic Closure. This is Tarski’s solution (for formal languages) in that he rejects any such φ is expressible. But again linguistic appearances suggest the contrary for natural languages, and so this will lead us to a version of logical nihilism. Perhaps one might attempt to agree with Priest (1984) that this merely shows semantic closure can be had only on pain of inconsistency; an implicit argument for dialetheism. But Beall (2015) has outlined a general argument barring semantic closure for formal theories on pain of triviality. The rough idea is to classify ‘rogue’ sentences of a formal semantic theory (couched in a formal language together with its consequence relation) as trivializer-sentences: sentences that, relative to that language’s consequence relation, yield the theory containing all sentences of the language. Then there can be no trivializer predicate for that theory characterizing all and
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only the trivializing sentences of the language. That’s because a sentence which says of itself that it is a trivializer for that very theory will be uncharacterized by that predicate. The upshot is that any formal language is either semantically incomplete or absolutely inconsistent (i.e. trivial). Another related issue for semantic closure is that there are very general reasons for thinking that the validity predicate for a logical consequence relation of a formal language will not expressible in the language itself. One reason for thinking this has been put forward by Field (2008). Gödel’s second incompleteness theorem says that no sufficiently strong formal language can prove its own consistency. If we had semantic closure (including an adequate truth and validity predicates), we would be able to do just such a thing. After all, if we could show in a formal language that the validity relation was truth preserving, we would be able to assert all the axioms of a logical theory to be true, and then be confident that by closing the theory under logical consequence, we did not introduce any inconsistencies. Beall and Murzi (2013) have also argued that no validity predicate will be expressible in any formal language (satisfying certain structural rules like contraction and transitivity) that contains its own truth predicate.15 This is due to the fact that one can use such a validity predicate to construct variants of Curry’s paradox. Now, it may be that by rejecting c ertain structural rules might allow one to express a validity predicate.16 But even still, the resulting (non-contractive, or non-transitive) logical consequence relation will be highly non-standard, and one wonders whether such a logic could be adequate to natural language inference. The underlying worry is that by attempting to accommodate semantic closure in a formal language, we are required to weaken the logical consequence relation for that language to such a degree that it no longer resembles our patterns of natural language inference. It is no accident that the literature regarding semantic closure continues to grow, for these issues are directly tied up with the adequacy of formal methods for giving an adequate treatment of natural language inference. It is worth saying that quite a lot of brilliant technical work has gone into these problems, and that the debate is far from settled. But logical nihilism is not outside the bounds of reason even considering the best work on these problems.
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The Argument from Quantification The second argument from expressive limitations trades on another limitation of formal languages to do with absolute generality. 1 . Natural languages have absolutely unrestricted quantifiers. 2. No formal language has absolutely unrestricted quantifiers. 3. No formal language is adequate to natural language. Each premise is, of course, controversial so let’s look at the case for each.17 What evidence is there for premise 1? Certain uses of natural language quantification appear to require an unrestricted reading. For example, part of the main function of philosophical uses of quantification is to rule out the existence of certain objects: ‘The universal set does not exist’ should not be true if the universal set does exist but we just can’t quantify over it. Even the rejection of unrestricted quantification seems to presuppose the availability of unrestricted quantification; saying ‘one cannot quantify over absolutely everything’ appears to presuppose that there is something that one cannot quantify over.18 Additionally, McGee (2003) argues that natural language universal quantification would be unlearnable if it weren’t unrestricted. There are also general reasons for accepting premise 2. Here is an argument that has been put forward repeatedly in the literature, based on two key claims. All-in-One Formal quantification involves quantification over sets (or at least domains that are set-like). No Universe There is no universal set (or set-like domain). What reason is there for thinking that formal quantification involves quantification over sets? Grim (1991) and Priest (2002) have both provided arguments along these lines. In fact, Priest (2002) has argued for a stronger claim, what he calls the ‘Domain Principle’, according to which all quantification (whether formal or not) involves sets.19 Far and away,
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the best formal theory of natural language quantifiers is Generalized Quantifier Theory; and generalized quantifiers are relations between sets. However, whether this is required for any adequate theory of natural language quantification is controversial. There are a few available lines of response. One might reject All-In- One, and suggest that plural quantifiers provide a counterexample to this claim (Boolos 1985).20 It is, of course, relevant that plural quantification is isomorphic to second-order quantification over non-empty monadic predicates. And second-order quantification (at least on its standard semantics) intrinsically involves quantification over sets. Many have argued that plural quantifiers are no more innocent than higher-order quantifiers, or are at least ‘set-like’ in the relevant sense.21 To be sure these issues are complicated, and debate is ongoing. But there are many who simply do not see a way to preserve unrestricted quantification, and opt rather for an indefinitely extensible hierarchy of quantifiers.22 Perhaps one might suggest a proof- theoretic approach to quantification without domains. Of course, there is the worry about the failure of any proof-theory to be complete with respect to higher-order models. A potential suggestion would be to give rules (schematic or proof- theoretic) for quantifiers that are open-ended, in the sense they govern the behavior of all possible extensions of the language (e.g. higher-order quantifiers) all at once.23 But even on this proposal, it will still be true that no single formal language can achieve quantification over absolutely everything. At best, the view delivers a single concept of quantification which will be retained no matter how one’s formal language is expanded to ever more inclusive domains. Instead of rejecting All-in-One, one might instead reject No Universe and consider formal theories with set-like universal objects. In ordinary set theory, we reject a universal set because it would have to be a member of itself. And for Russell-paradox related reasons, no set can be a member of itself. Class theory can have a universe of sets, but not a universe of classes since proper classes are not members of anything including other classes. Similarly, mereology has a universal object. Could one use mereological models for quantification? Uzquiano (2006) shows that this method (supplemented with some natural assumptions) can only recover quantification over sets up to certain types of limit cardinals; but if there
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are strongly inaccessible cardinals, then this quantification is not truly unrestricted. Another option would be to appeal to alternative set theories. Paraconsistent set theories can non-trivially contain universal sets guaranteed to exist by Naïve Comprehension (Brady 2006; Weber 2010, 2012). Such sets will be inconsistent, but in a paraconsistent setting we can have inconsistent sets (including the Russell set) non-trivially. So, in this case it appears we can have unrestricted quantification (Priest 2007). But even if this proposal looks promising, it comes with a similar cost— we lose a plausible form of restricted quantification. Any adequate theory of restricted quantifiers requires the following: Modus Ponens All As are Bs, x is an A; hence x is a B. Weakening Everything’s a B; hence all As are Bs. Contraposition All As are Bs; hence all non-Bs are non-As. That is, any theory that rejects the above principles will be inadequate to natural language inference. But any theory that accepts them reintroduces a form of Ex Falso Quodlibet.
B A→B ¬B ¬B → ¬A ¬A
Here, on the assumption that B and ¬B, we can infer ¬A. But this is something that simply cannot be valid in a dialetheic setting, since it would entail the truth of (the negation of ) every sentence whatsoever.24 There are set theories based on classical logic with a universal set: Quine’s (1937) New Foundations, and descendants (see Forster 1992). But Linnebo (2006, p. 156f ) notes, they are technically unappealing and lack a unified intuitive conception. More importantly, they have a
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niversal set only by placing further restrictions on Comprehension, a u limitation which Williamson (2003, p. 425f ) argues would also make any account of ordinary restricted quantification impossible.25
Other Arguments from Expressive Limitations Another key argument from expressive limitations appeals to an obvious phenomenon of natural language: vagueness. The philosophical literature on vagueness is enormous, and the number of theoretical options are too numerous to outline here. I want to highlight some views of vagueness that are broadly in line with the nihilist outlook on logic. One key player is, of course, Dummett who argued that the use of vague predicates in natural language is intrinsically inconsistent. What is in error is not the principles of reasoning involved, nor, as on our earlier diagnosis, the induction step. The induction step is correct, according to the rules of use governing vague predicates such as ‘small’: but these rules are themselves inconsistent, and hence the paradox. Our earlier model for the logic of vague expressions thus becomes useless: there can be no coherent such logic. (Dummett 1975, p. 319f )
If there can be no coherent logic of vague expressions, but natural language contains vague expressions, then we have a straightforward argument to logical nihilism. More recently, Eklund (2002) argued that languages with vague expressions are inconsistent languages. Ludwig and Ray (2002) suggest that no sentence involving a vague term can be true. Aside from these strong claims about the inconsistency or incoherence of natural language, however, one might want to contend there is still a mismatch between formal logics and natural language with how they treat vagueness. It’s not outlandish to suggest no formal language, with its mathematical precision, could ever hope to perfectly represent natural language vagueness. Or one might agree with Sainsbury (1996) and Tye (1994) that any semantics for vague expressions would have to appeal to a similarly vague metalanguage.26 Hiding in the background to most formal approaches to vagueness is the ubiquitous revenge problem of higher-order vagueness. It may be that
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such problems inevitably render precise formal methods inadequate for the vague aspects of natural language inferences.27 There are probably other arguments from expressive limitations that can be made; I have simply pointed to three of the more difficult problems in philosophical logic: semantic closure, unrestricted quantification, and vagueness. And as emphasized, these problems are not new, and the debate over the correct treatment of such phenomena ongoing. Still the logical nihilist permits a unified perspective: the expressive resources of natural language and our best formal languages might not perfectly coincide, and that might be because there are important and deep differences between natural language inference and logical consequence.
4
Related Issues
An issue I’ve largely set aside is the fact that logic is taken to be normative. Perhaps this unreasonably privileges psycho-semantic facts about how we in fact infer, rather than how we ought to infer. On this conception, appeals to linguistic data and evidence for ‘semantic intuitions’ can appear irrelevant. The question of logic is: ‘what is best inferential practice?’ But I want to stress that even if logical consequence is conceived as a fundamentally normative relation, there might well be arguments for logical nihilism. For example, we might contend that an objective normative relation would be metaphysically and epistemically strange. Roughly, we might apply Mackie’s (1977) arguments to logical consequence itself. This case is made strongly by Field (2009): Quite independent of logic, I think there are strong reasons for a kind of an- tirealism about epistemic normativity: basically, the same reasons that mo- tivate antirealism about moral normativity, or about aesthetic goodness, ex- tend to the epistemic case. (For instance, (i) the usual metaphysical (Humean) worry, that there seems no room for ‘straightforward normative facts’ on a naturalistic world-view; (ii) the associated epistemological worry that access to such facts is impossible; (iii) the worry that such normative facts are not only nonnaturalistic, but ‘queer’ in the sense that awareness of them is supposed to somehow motivate one to reason in a certain way all by itself.) (p. 354)
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And if we have good reason to reject entities which are metaphysically or epistemically or motivationally strange, we would have good reason to reject logical consequence (qua objective and normative).28 If logical consequence is a subjective normative relation, then whether that relation holds would be relative either to our aims or our conventions. This view is called ‘logical pragmatism’ by Haack (1978). Here, the aim of formal logic is not so much correct representation, but rather utility. But this view is not opposed to logical nihilism: it may well be that the best inferential practice for natural language users (i.e. the most useful) has no formal analogue. After all, semantic closure, unrestricted quantification, and vagueness are useful features of natural language. But here again what counts as ‘best’ in the pragmatic sense will depend in large part on what natural language is used for.29 In defending logical nihilism, I do not intend to impugn any honorific status of natural language inference. This is over against so-called Inconsistency Theories (with which logical nihilism has much in common) according to which natural language is inconsistent because of, for example, the truth predicate or vague expressions.30 Logical nihilists are free to disagree with the claim that natural languages are inconsistent, perhaps because they agree with Tarski (1944) that ‘the problem of consistency has no exact meaning with respect to this [natural] language’ and only arises with the sufficiently precise formulation of a formal language. Some inconsistency theorists claim that natural language is incoherent or meaningless because it has no precise logic.31 Here again, logical nihilists are free to disagree. We might well agree with Wright (1975) that natural language inference can be unprincipled without being incoherent. Nor am I impugning formal methods in the study of natural language, in philosophy, or elsewhere. As Glanzberg (2015) rightly notes, the application of formal methods has been one of the greatest successes in the study of natural languages of the last half-decade. The application of formal techniques to, for example, epistemology, metaphysics, perhaps virtually every area of philosophy has likewise yielded significant progress. But we should be aware that formal languages serve a representational function, and—like modeling tools used in any science—their applications have limits.
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This is something the early analytic pioneers of logic, Frege and Tarski, understood well. Consider Frege: I believe I can make the relationship of my Begriffsschrift to ordinary language clearest if I compare it to that of the microscope to the eye. The latter, due to the range of its applicability, due to the exibility with which it is able to adapt to the most diverse circumstances, has a great superiority over the microscope. Considered as an optical instrument, it admittedly reveals many imperfections, which usually remain unnoticed only because of its intimate connection with mental life. But as soon as scientific purposes place great demands on sharpness of resolution, the eye turns out to be inadequate. The microscope, on the other hand, is perfectly suited for just such purposes, but precisely because of this is useless for all others. (Frege 2002, §V)
Formal languages require a significant degree of abstraction and idealization from natural language inference.32 Only certain aspects of the model are intended to represent the phenomena being modeled. This perspective might even help to handle some of revenge problems involving semantic closure, absolute generality, vagueness, and so on. Too-easy revenge appeals to artifacts of the model and argues they are inexpressible in the model. But once we see the clear separation, we can see that there’s no obligation for the theorist to take these burdens on.33 The view of logic-as-modeling is sometimes billed as a form of logical pluralism (e.g. Cook 2010). And while it is clear my sympathies lie with this view, a rather more drastic opinion of the limitations of formal methods inclines me to say that no formal language gets it right, rather than to say that many formal languages get it right more-or-less. The nihilist accepts that there are no correct or completely general formal theories. But that needn’t mean there aren’t a lot of reasonably good, useful, and explanatory ones. In certain areas of inquiry, some of the more useful features of natural language are unnecessary and may even be detrimental to the theoretical purposes at hand. Science and mathematics (both pure and applied), and even many areas of philosophy, require the kind of precision inherently informal languages cannot provide. Formal logic has its best application in areas where we can regiment our language and revise our practice. But we shouldn’t forget that not all of our inquiry
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is like this: sometimes regimentation leaves us with an expressive loss, and sometimes revising our practice is not possible or even desirable. Logical nihilism reminds us to respect the differences between model and reality.34
Notes 1. Of course, there are many ways one might think about truth; one needn’t think of truth in terms of correct representation. But it is a reasonable way of thinking about the issue, and I think it is what the intuitive slogan ‘One True Logic’ is getting at. 2. This quote strongly suggests that Beall and Restall would not disagree with the characterization of logics as being in the business of correct representation of natural language inferential practice. 3. Compare recent authors like Bueno and Colyvan (2004), “The aim of logic is taken to be to provide an account of logical consequence that captures the intuitive notion of consequence found in natural language” (p. 168). Or Resnik (2004), “As practitioners of inference we make specific inferences […] As logicians we try to formulate a systematic account of this practice by producing various rules of inference and laws of logic by which we presume the practice to proceed. This aspect of our work as logicians is like the work of grammarians” (p. 179). Or consider Cook (2010) “[A] logic is ‘correct’, or ‘acceptable’, etc., if and only if it is a correct (or acceptable, etc.) codification of logical consequence. The idea that the philosophically primary (but obviously not only) goal of logical theorizing is to provide a formal codification of logical consequence in natural language traces back (at least) to the work of Alfred Tarski” (p. 195). 4. Cook (2010, p. 495f ) gives a detailed account of what it means to say a logical consequence relation is adequate to natural language inference. I am assuming something like his definition is suitable for this purpose. 5. Another way of reading Mortensen is as arguing that real broad possibility outstrips pure logical possibility. In this case, then, there may be logically impossible scenarios that are not, broadly speaking, impossible. Mortensen would then not count as a logical nihilist in the sense above. 6. Parallel disputes over the metaphysics of composition. Here universalism states that composition always occurs, whereas nihilism claims that the
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composition relation is basically empty. ‘Emptyism’ just doesn’t have the same ring to it. And unfortunately, ‘Noneism’—the most natural name for the view defended in this chapter—is already taken. 7. There is also Eklund’s (2012) Carnapian language-relative approach; Shapiro’s (2011, 2014) contextual approach, and Cook’s (2010) logic-as- modeling approach. Later I’ll touch on some issues that directly relate to their motivations. 8. Similar arguments have been made by Read (2006, p. 208f ) and Priest (2006, Chap. 12). 9. In Cotnoir (2013), I outline general approach to validity motivated by this idea. See also Pedersen (2014) for more motivations. 10. This was Tarski’s (1936) early view. But I also consider it to be quite possible that investigations will bring no positive results in this direction, so that we shall be compelled to regard such concepts as logical consequence as relative concepts. The fluctuation in the common usage of the concept of consequence would in part at least be quite naturally reflected in such a compulsory situation [of a relatively-defined concept of consequence]. (p. 420) See also Etchemendy (1990) and Dutilh-Novaes (2012). 11. Glanzberg’s (2015) rich and carefully argued paper is concerned with rejecting the view that natural language (a structure with a syntax and a semantics) determines a logical consequence relation. His position is very similar to logical nihilism of the sort I’m defending here, and he is probably one of the view’s closest allies. But strictly speaking Glanzberg’s view is compatible with logical monism and logical pluralism, since there could be one (or more) correct theory of natural language inference, even if it isn’t possible to simply read such a thing off from natural language itself. 12. Compare Zardini (2010, 2014) who argues that mid-argument context shifts invalidate standard inferences (e.g. “I am sitting” can fail to entail itself ). 13. See also Bacon (2015) who argues that there can be no ‘linguistic’ theories of paradox based in a classical language due to revenge problems. 14. Thanks to Cory Wright for suggesting this way of framing the issue. 15. Related arguments first appeared in Whiǣle (2004), and Shapiro (2011). See also Murzi (2014). For dissenting voices see Cook (2014), Wansing and Priest (2015).
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16. See e.g. Zardini (2013), Weber (2014). 17. For good introductory discussions of these issues see Florio (2014) and Rayo and Uzquiano (2006, p. 1). The following discussion is indebted to them in various ways. 18. See Lewis (1991) and Williamson (2003). 19. Priest himself rejects ‘No Universe’ for his preferred set theory; for discussion see below. 20. See Uzquiano (2009), Rayo (2002). 21. For example, Jané (1993), Resnik (1988), and more recently Linnebo (2003). 22. For example, one might utilize hyperplural quantification (Rayo 2006), which under certain assumptions, isomorphic to the type hierarchy (Linnebo and Rayo 2012). There are also modal approaches (Linnebo 2010 and Studd 2013) which allow set-theoretic domains to be indefinitely extensible. 23. For some defenders of this view, see Lavine (2006), McGee (2006), and Williamson (2006). 24. This problem is discussed, and some possible lines of response explored in Beall et al. (2006). 25. See also Weir (2006). 26. See Cook (2002) for discussion. 27. Wright (2010) contends that higher-order vagueness worries are pseudoproblems; it is a revenge problem that only arises for views which misunderstand first-order vagueness. The logical nihilist can afford sympathy to such claims. 28. Field takes this to be an argument for pluralism (because of an underlying antirealist pluralism about epistemic norms), one might well think such considerations provide better reasons to be nihilist about logic. 29. The comparison with morality is instructive: consider the relevance of anti-theory views in ethics (e.g. Clarke 1987) to logical nihilism, or even the similarities between particularism (e.g. Dancy 1983) and Hofweber’s (2007) view that we should give up the ideal of deductive inference as exceptionless and monotonic. Natural language inferences need not be exceptionless or monotonic, and often are not. But they might still be generically valid, in the sense that generics like “Humans are bipeds” are true. 30. E.g. Chihara (1979), Eklund (2002), and Ludwig (2002).
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31. E.g. Scharp (2013) who thinks defective concepts like ‘truth’ need to be replaced, or Patterson (2009) who argues that we understand natural language using a false semantic theory, such that strictly speaking natural language sentences have no meanings. 32. See especially Glanzberg (2015, §IV), but also Cook (2002, 2010), Shapiro (2006), and Scharp’s (2013) metrological naturalism. 33. This point is made clearly and forcefully in Beall (2007) with respect to the semantic paradoxes. Cook (2002) argues for a similar perspective with respect to vagueness. 34. I’d like to thank audiences at the Truth Pluralism and Logical Pluralism Conference at the University of Connecticut, the Swiss Society for Logic and Philosophy of Science at the University of Neuchâtel, the Northern Institute of Philosophy 2011 Reading Party, the University of St Andrews Philosophy Society, and the students in my 2014 and 2017 Philosophy of Logic seminars. Their comments and questions led to many improvements and developments in the paper. Special thanks to Colin Caret, Roy Cook, Matti Eklund, Ole Hjortland, Michael Lynch, Julien Murzi, Stephen Read, Gillian Russell, Gil Sagi, Kevin Scharp, Stewart Shapiro, Keith Simmons, Crispin Wright, and Elia Zardini for discussions on these topics over a number of years. Thanks also to an anonymous referee for helpful comments on a previous version of the paper. The biggest debt of gratitude is owed to my PhD supervisor Jc Beall, who disagrees with many of the ideas in this paper. A reaction against one’s academic upbringing can be a sign of deep respect; and I hope this paper is taken in that spirit.
References Bacon, A. 2015. Can the Classical Logician Avoid the Revenge Paradoxes? Philosophical Review 124 (3): 299–352. Beall, J. 2007. Prolegomena to Future Revenge. In Revenge of the Liar, ed. J. Beall, 1–30. Oxford: Oxford University Press. ———. 2015. Trivializing Sentences and the Promise of Semantic Completeness. Analysis 75: 573–584. Beall, J., and J. Murzi. 2013. Two Flavors of Curry Paradox. Journal of Philosophy 110: 143–165. Beall, J., and G. Restall. 2000. Logical Pluralism. Australasian Journal of Philosophy 78 (4): 475–493.
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———. 2006. Logical Pluralism. Oxford: Oxford University Press. Beall, J., R.T. Brady, A.P. Hazen, G. Priest, and G. Restall. 2006. Relevant Restricted Quantification. Journal of Philosophical Logic 35 (6): 587–598. Boolos, G. 1985. Nominalist Platonism. The Philosophical Review 94: 327–344. Brady, R. 2006. Universal Logic. Stanford: CSLI Publications. Bueno, O., and M. Colyvan. 2004. Logical Non-apriorism and the Law of Non- contradiction. In The Law of Non-contradiction: New Philosophical Essays, ed. Graham Priest, Jc Beall, and Bradley P. Armour-Garb, 156–175. Oxford: Oxford University Press. Bueno, O., and S.A. Shalkowski. 2009. Modalism and Logical Pluralism. Mind 118 (470): 295–321. Chihara, C. 1979. The Semantic Paradoxes: A Diagnostic Investigation. Philosophical Review 88: 590–618. Clarke, S.G. 1987. Anti-theory in Ethics. American Philosophical Quarterly 24 (3): 237–244. Cook, R.T. 2002. Vagueness and Mathematical Precision. Mind 111: 227–247. ———. 2010. Let a Thousand Flowers Bloom: A Tour of Logical Pluralism. Philosophy Compass 5 (6): 492–504. ———. 2014. There Is No Paradox of Logical Validity. Logica Universalis 8 (3–4): 447–467. Cotnoir, A.J. 2013. Validity for Strong Pluralists. Philosophy and Phenomenological Research 86 (3): 563–579. Dancy, J. 1983. Ethical Particularism and Morally Relevant Properties. Mind 92 (368): 530–547. Dummett, M. 1975. Wang’s Paradox. Synthese 30 (3–4): 301–324. Dutilh-Novaes, C. 2012. Reassessing Logical Hylomorphism and the Demarcation of Logical Constants. Synthese 185 (3): 387–410. Eklund, M. 2002. Inconsistent Languages. Philosophy and Phenomenological Research 64 (2): 251–275. ———. 2012. The Multitude View on Logic. In New Waves in the Philosophy of Logic, ed. G. Restall and G. Russell, 217–240. Palgrave Macmillan. Estrada-González, L. 2011. On the Meaning of Connectives (Apropos of a Non-necessitarianist Challenge). Logica Universalis 5 (1): 115–126. Etchemendy, J. 1990. The Concept of Logical Consequence. Cambridge, MA: Harvard University Press. Field, H. 2008. Saving Truth from Paradox. Oxford: Oxford University Press. ———. 2009. Pluralism in Logic. Review of Symbolic Logic 2 (2): 342–359. Florio, S. 2014. Unrestricted Quantification. Philosophy Compass 9 (7): 441–454.
326
A. J. Cotnoir
Forster, T.M. 1992. Set Theory with a Universal Set. Oxford: Oxford University Press. Frege, G. 2002. Begriffsschrift, a Formula Language, Modeled upon that of Arithmetic, for Pure Thought. In From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, ed. J. Van Heijenoort, 1–82. Cambridge, MA: Harvard University Press. Glanzberg, M. 2015. Logical Consequence and Natural Language. In Foundations of Logical Consequence, ed. C.R. Caret and O.T. Hjortland, 71–120. Oxford: Oxford University Press. Grim, P. 1991. The Incomplete Universe: Totality, Knowledge, and Truth. Cambridge, MA: MIT Press. Haack, S. 1978. Philosophy of Logics. Cambridge: Cambridge University Press. Hofweber, T. 2007. Validity, Paradox, and the Ideal of Deductive Logic. In Revenge of the Liar, ed. J. Beall, 145–158. Oxford: Oxford University Press. Jané, I. 1993. A Critical Appraisal of Second-Order Logic. History and Philosophy of Logic 14: 67–86. Lavine, S. 2006. Something About Everything: Universal Quantification in the Universal Sense of Universal Quantification. In Absolute Generality, ed. A. Rayo and G. Uzquiano, 98–148. Oxford: Oxford University Press. Lewis, D.K. 1991. Parts of Classes. Oxford: Blackwell. Linnebo, Ø. 2003. Plural Quantification Exposed. Noûs 37 (1): 71–92. ———. 2006. Sets, Properties, Unrestricted Quantification. In Absolute Generality, ed. A. Rayo and G. Uzquiano, 149–178. Oxford: Oxford University Press. ———. 2010. Pluralities and Sets. Journal of Philosophy 107 (3): 144–164. Linnebo, Ø., and A. Rayo. 2012. Hierarchies Ontological and Ideological. Mind 121 (482): 269–308. Ludwig, K. 2002. What Is the Role of a Truth Theory in a Meaning Theory. In Meaning and Truth: Investigations in Philosophical Semantics, ed. O. Campbell, D. Shier, and J.K. Campbell. New York: Seven Bridges Press. Ludwig, K., and G. Ray. 2002. Vagueness and the Sorites Paradox. Philosophical Perspectives 16: 419–461. Lynch, M.P. 2008. Alethic Pluralism, Logical Consequence, and the Universality of Reason. Midwest Studies in Philosophy 32 (1): 122–140. ———. 2009. Truth as One and Many. Oxford: Oxford University Press. MacFarlane, J.G. 2002. What Does It Mean to Say that Logic Is Formal? PhD thesis, Citeseer. Mackie, J.L. 1977. Ethics: Inventing Right and Wrong. New York: Pelican Books.
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McGee, V. 2003. Universal Universal Quantification. In Liars and Heaps, ed. M. Glanzberg and J. Beall, 357–364. Oxford: Oxford University Press. ———. 2006. There Is a Rule for Everything. In Absolute Generality, ed. A. Rayo and G. Uzquiano, 179–202. Oxford: Oxford University Press. Mortensen, C. 1989. Anything is possible. Erkenntnis 30 (3): 319–337. Murzi, J. 2014. The Inexpressibility of Validity. Analysis 74 (1): 65–81. Patterson, D. 2009. Inconsistency Theories of Semantic Paradox. Philosophy and Phenomenological Research 79 (2): 387–422. Pedersen, Nikolaj J.L.L. 2014. Pluralism x 3: Truth, Logic, Metaphysics. Erkenntnis 79 (2): 259–277. Priest, G. 1984. Semantic Closure. Studia Logica: An International Journal for Symbolic Logic 43 (1/2): 117–129. ———. 2002. Beyond the Limits of Thought. Oxford: Oxford University Press. ———. 2006. Doubt Truth to Be a Liar. Oxford: Oxford University Press. ———. 2007. Review of Absolute Generality. Notre Dame Philosophical Reviews. https://ndpr.nd.edu/news/absolute-generality/. Quine, W.V.O. 1937. New Foundations for Mathematical Logic. The American Mathematical Monthly 44: 70–80. Rayo, A. 2002. Word and Objects. Noûs 36 (3): 436–464. ———. 2006. Beyond Plurals. In Absolute Generality, ed. A. Rayo and G. Uzquiano, 220–254. Oxford: Oxford University Press. Rayo, A., and G. Uzquiano. 2006. Introduction. In Absolute Generality, ed. A. Rayo and G. Uzquiano, 1–19. Oxford: Oxford University Press. Read, S. 2006. Monism: The One True Logic. In A Logical Approach to Philosophy, ed. D. DeVidi and T. Kenyon, 193–209. New York: Springer. Resnik, M.D. 1988. Second-Order Logic Still Wild. The Journal of Philosophy 85 (2): 75–87. ———. 2004. Revising Logic. In The Law of Non-contradiction: New Philosophical Essays, ed. Graham Priest, J.C. Beall, and Bradley P. Armour- Garb, 178–193. Oxford: Oxford University Press. Russell, G. 2008. One True Logic? Journal of Philosophical Logic 37 (6): 593–611. ———. 2017. An Introduction to Logical Nihilism. In Logic, Methodology and Philosophy of Science, ed. P.S. Hannes Leitgeb, Ilkka Niiniluoto, and E. Sober. London: College Publications. Sagi, G. 2014. Formality in Logic: From Logical Terms to Semantic Constraints. Logique et Analyse 56 (227): 259–276. Sainsbury, R.M. 1996. Concepts Without Boundaries. In Vagueness: A Reader, ed. R. Keefe and P. Smith, 186–205. Cambridge, MA: MIT Press. Scharp, K. 2013. Replacing Truth. Oxford: Oxford University Press.
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A. J. Cotnoir
Shapiro, S. 2006. Vagueness in Context. Oxford: Oxford University Press. ———. 2011. Varieties of Pluralism and Relativism for Logic. In A Companion to Relativism, ed. D. Hales, 526–552. Malden: Wiley-Blackwell. ———. 2014. Varieties of logic. Oxford: Oxford University Press. Sher, G. 1996. Did Tarski Commit ‘tarski’s Fallacy’? Journal of Symbolic Logic 61 (2): 653–686. Strawson, P.F. 1950. On Referring. Mind 59 (235): 320–344. Studd, J. 2013. The Iterative Conception of Set. Journal of Philosophical Logic 42: 697–725. Tarski, A. 1936. On the Concept of Logical Consequence. 2nd ed, 409–420. Indianapolis: Hackett. ———. 1944. The Semantic Conception of Truth: And the Foundations of Semantics. Philosophy and Phenomenological Research 4 (3): 341–376. Tye, M. 1994. Sorites Paradoxes and the Semantics of Vagueness. Philosophical Perspectives 8: 189–206. Uzquiano, G. 2006. The Price of Universality. Philosophical Studies 129: 137–169. ———. 2009. Quantification Without a Domain. In New Waves in the Philosophy of Mathematics, ed. O. Bueno and Ø. Linnebo, 300–323. London: Palgrave Macmillan. Varzi, A.C. 2002. On Logical Relativity. Noûs 36 (1): 197–219. Wansing, H., and G. Priest. 2015. External Curries. Journal of Philosophical Logic 44: 453–471. Weber, Z. 2010. Transfinite Numbers in Paraconsistent Set Theory. Review of Symbolic Logic 3 (1): 71–92. ———. 2012. Transfinite Cardinals in Paraconsistent Set Theory. Review of Symbolic Logic 5 (2): 269–293. ———. 2014. Naive Validity. The Philosophical Quarterly 64 (254): 99–114. Weir, A. 2006. Is It Too Much to Ask, to Ask for Everything? In Absolute Generality, ed. A. Rayo and G. Uzquiano, 333–368. Oxford: Oxford University Press. Whittle, B. 2004. Dialetheism, Logical Consequence, and Hierarchy. Analysis 64 (4): 318–326. Williamson, T. 2003. Everything. Philosophical Perspectives 17 (1): 415–465. ———. 2006. Absolute Identity and Absolute Generality. In Absolute Generality, ed. A. Rayo and G. Uzquiano, 369–390. Oxford: Oxford University Press. ———. 2017. Semantic Paradoxes and Abductive Methodology. In The Relevance of the Liar, ed. B. Armour-Garb, 325–346. Oxford: Oxford University Press.
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Wright, C. 1975. On the Coherence of Vague Predicates. Synthese 30 (3/4): 325–365. ———. 2010. The Illusion of Higher-Order Vagueness. In Cuts and Clouds: Vagueness, Its Nature, and Its Logic, ed. R. Dietz and S. Moruzzi. Oxford: Oxford University Press. Zardini, E. 2010. Truth Preservation in Context and in Its Place. In Insolubles and Consequences, ed. C. Dutilh-Novaes and O. Hjortland, 249–271. London: College Publications. ———. 2013. Naive Logical Properties and Structural Properties. The Journal of Philosophy 110: 633–644. ———. 2014. Context and Consequence: An Intercontextual Substructural Logic. Synthese 191: 3473–3500.
Varieties of Logical Consequence by Their Resistance to Logical Nihilism Gillian Russell
Recent work on logical pluralism has suggested that the view is in danger of collapsing into logical nihilism, the view on which there are no valid arguments at all.1 The goal of this chapter is to argue that the prospects for resisting such a collapse vary quite considerably with one’s account of logical consequence. The first section lays out four varieties of logical consequence, beginning with the approaches Etchemendy (1990) called interpretational and representational, and then adding a Quinean substitutional approach as well as the more recent universalist account given in Williamson (2013, 2017). The second section recounts how the threat of logical nihilism arises in the debate over logical pluralism. The third and final section looks at the ways the rival accounts of logical consequence are better or worse placed to resist the threat.2
G. Russell (*) Department of Philosophy, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA e-mail: [email protected] © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_14
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Varieties of Logical Consequence
In a volume on logical pluralism, it is not unusual to write or assume that there are different views of logical consequence; classical and relevant logicians disagree about the extension of the relation, and monists and pluralists disagree in a different way about whether there can be more than one consequence relation. Still, these varieties of logical consequence are not the main ones that I wish to draw your attention to in this section. Two monists who agree on a logic can have different views of logical consequence in the sense I have in mind. Essentially, that is because they might agree that an argument form—for example, modus ponens—is valid, but disagree about the features in virtue of which it is valid. We might say that they disagree about what logical consequence is or, to borrow a phrase from Etchemendy, that they disagree about the concept of logical consequence.3 So what is logical consequence? In informal philosophy it has often been said—in fact, I’ve said it quite emphatically myself—that an argument is valid just in case any possible world in which all of the premises are true is one in which the conclusion is true. I’ll call this characterisation the modal slogan. Once we get down to work, the modal slogan gets us into trouble. One problem is that there are sentences which express necessary truths which are not logically valid, such as 2 + 2 = 4, Hesperus is Phosphorus and Water is not CO2.4 This problem has a natural counterpart for consequence: there are no possible worlds where Hesperus is bright is true but Phosphorus is bright is not—and the latter is still not a logical consequence of the former. A second problem for the modal slogan is the converse of the first: indexical logics like Kaplan’s Logic of Demonstratives have logical truths which need not express necessary truths, such as Ap ↔ p, Np → p, NLocated(I, Here) and dthat[α] = α.5 This problem too has a counterpart for logical consequence: there are possible worlds in which I am not here now, but in which snow is white. Still, since I am here now is a logical truth (in LD), it is a logical consequence of any arbitrary set of sentences, including {snow is white}. This gives us an example of an argument where Α ⊨ Β even though there are possible worlds where Α is true but Β is not. Hence the modal slogan both under- and over-generates instances of logical consequence.6
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In practice, these problems present few difficulties when we are doing formal logic, and that is because the modal slogan is dropped in favour of a characterisation in terms of models. Exact definitions vary slightly from author to author, but they are generally acknowledged to be descendants of Tarski’s definition: The sentence X follows logically from the sentences of the class K if and only if every model of the class K is also a model of the sentence X. (Tarski 1936, p. 417)
Tarski uses ‘model of a sentence X’ to mean something (in his paper it was a sequence of objects, in contemporary formulations it is a model) which satisfies X. Where—as is usually the case in an argument—X is a sentence (a formula with no free variables), a model of X will be a model which makes X true. Hence we might say: The sentence X is a logical consequence of the set of sentences K just in case every model that makes K true also makes X true.
The structural similarity to the modal slogan is clear, but it would be dangerous, for the reasons given above, to assume that models represent possible worlds. Our first two varieties of logical consequence differ in their views of what models do represent. Roughly, on the interpretational account, models represent different assignments of meanings to the non-logical expressions in our language. On the representational account, models represent different arrangements of the world.7 The difference is perhaps best illustrated with an example, so suppose that we have a first-order language, L, with logical expressions (∧, ¬, ∀, =) as well as non-logical individual constants (a, b, c, …), non-logical predicates (P, Q, R, …), first-order variables (x, y, z, …) and punctuation ((, ), , ). We adopt a standard definition for wffs (well-formed formulas and sentences).8 A model M for L consists of a pair (D, I) with D a nonempty set (the model’s domain) and I an interpretation function which assigns appropriate extensions from D to all the non-logical expressions in the language.9
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Now consider two models, K and J. K = 〈DK, IK〉 and is such that the sentence Fa is true in K, because a I K (the denotation assigned to a in K) is a member of F I K (the extension assigned to the one-place predicate F in K). Model J differs on this: Fa is false in J, that is, IJ(Fa) = F, since IJ assigns an object to a which is not in the set F I J . On the representational conception of logical consequence, K and J don’t differ on the meaning of any of the expressions in Fa, rather they differ in representing situations in which the referent of a ‘is F’ or not. On this way of thinking about it, models represent something like states of the world (admittedly in a highly abstract fashion, but that is to be expected in logic) and so when a set of premises entails a conclusion, that is because no matter how the world is, if the premises are true, the conclusion is. On the interpretational conception—by contrast—K and J differ in assigning different meanings to the expressions a and F. With the meanings assigned to the expressions in K, the sentence is true, but with the meanings assigned to it in J, the sentence is false. On this view, if the conclusion of an argument is true in all the models in which the premises are all true, then that is because no matter how we interpret the non-logical expressions in the argument, if the premises are true, then the conclusion is. These two views of formal models fit naturally with two different intuitive conceptions of what logical consequence is. On one, ¬ (Snow is white ∨ Grass is green) entails ¬ snow is white because of the meanings of ∨ and ¬. Since the meanings of ∨ and ¬ are sufficient to guarantee that the argument is valid, it won’t matter what interpretation the sentences Snow is white and Grass is green receive. So on this intuitive conception, there are two natural ways to test for validity: (i) keep the logical constants the same but vary the interpretation of the non-logical expressions or (ii) keep the logical constants the same but substitute different expressions for the non-logical expressions. We might call these the meaning variation and word substitution tests, respectively. The latter is an approach that Quine endorsed: The relation of implication in one fairly natural sense of the term, viz. logical implication is readily described with the help of the auxiliary notion of logical truth. A statement is logically true if it is not only true but remains true when all but its logical skeleton is varied at will; in other words, if it is
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true and contains only logical expressions essentially, any others vacuously. Now one statement may be said to logically imply another when the truth- functional conditional which has the one statement as an antecedent the other as consequent is logically true. (Quine 1981, §5)10
There’s a natural problem with this substitutional approach though, identified already by Tarski11: it makes the relation of logical consequence depend on the richness of the available language. This condition may in fact be satisfied only because the language with which we are dealing does not possess a sufficient stock of extra-logical constants. The condition […] could be regarded as sufficient for the sentence X to follow from the class K only if the designations of all possible objects occurred in the language in question. This assumption, however, is fictitious and can never be realised. (Tarski 1936, pp. 415–416)
The meaning-variation test avoids this problem. Rather than substituting, for example, a name that already refers to the object which is not in the extension of ‘F,’ we reinterpret the original name, a, so that it refers to that object, giving us our counterexample. This was Tarski’s approach. He conceived of a valid argument as one that preserved truth no-matter how we reinterpreted the non-logical expressions. Though we usually regard contemporary definitions of logical consequence as descendants of Tarski’s 1936 definition, there are several striking differences between it and contemporary definitions. The first is that rather than assigning alternative interpretations to the non-logical expressions directly, Tarski first forms the sentential function corresponding to each sentence by replacing the non-logical expressions with syntactically appropriate variables (like constants with like variables, unlike with unlike). For example, we might start with the sentence ¬Odd(2) ∧ Prime(2) and form the sentential function:
ØXx Ù Yx
(1)
Here the non-logical predicates Odd and Prime have been replaced with the predicate variables X and Y, and the individual constant 2 has been replaced with the individual variable, x, to form a sentential function.
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Sentential functions are then satisfied (or not) by the closest thing in Tarski’s system to a modern-day model: a sequence of objects. A sequence of objects which would satisfy (1) above would be one which began (the set of planets, Ceres, the set of moons, …)—assuming the ‘objects’ in this sequence are assigned to the non-logical expressions in the order they appear in the sentence. A sequence of objects which would not satisfy (1) would be (the set of even numbers, 4, the set of prime numbers). Such sequences can easily be reconceptualised as functions from the non- logical expressions to suitable meanings, and hence as an analogue to our modern-day interpretations. Another difference between Tarski’s definition and contemporary ones is that Tarski’s models have no counterpart for the contemporary domain of quantification. As Etchemendy notes, domains are a very natural addition to a model if you are thinking of consequence representationally— we can represent worlds with fewer or more objects—but seem unmotivated on the interpretational account Tarski favoured12; we wouldn’t expect a reinterpretation of the non-logical expressions to affect the number of objects there are. Still, we normally think that variable domains are important for achieving the right extension for ‘logical consequence’—we wouldn’t want ∃x ∃ y(x ≠ y) (or any other sentence equivalent to the claim that there are at least n objects for some n ∈ ℕ) to be true in every model (and thus a logical truth) or false in every model (and thus a logical falsehood). If we think of models representationally, this naturally motivates the inclusion of a domain, which then allows us to have variable domain models, resulting in (what we normally think of as) a sensible extension for the consequence relation. On the other hand, the representational conception of a model looks less natural—and the interpretational conception more natural—when it comes to thinking about why two names, say a and b, might be assigned the same referent in one model and a different referent in another. On the interpretations account, this simply represents an alternative semantic value for at least one of the names, a reinterpretation like any other. Just as 2 could be a name for Julius Caesar, so Hesperus could be a name for Mars. On the representationalist account—where we assume that the semantic value of a name is an object—this looks much harder to explain. If a refers to Venus and b refers to Venus, it is hard to see how we can
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retain those interpretations (as the spirit of the account instructs us to do) and somehow—by changing the world—end up with a model in which I(a) ≠ (b). Here is where we are: we have three varieties of logical consequence: interpretational, representational, and substitutional. On the interpretational account, we vary the interpretations of the non-logical expressions in an argument, and if every way of doing so makes the conclusion true if the premises are all true, then the conclusion is a logical consequence of the premises. This view is the historical ancestor of contemporary model theory, and it makes some limited sense of that model theory, but it is hard to motivate contemporary models containing variable domains from this perspective. On the substitutional account, we keep the logical expressions fixed, and substitute syntactically appropriate alternative expressions uniformly throughout the argument. If doing so can result in true premises but a non-true conclusion, the argument is not valid. This account was endorsed by Quine, but makes the extension of the relation of logical consequence hostage to the richness of the language. On the representational account we retain the interpretation of all expressions in the argument, but consider various different states the world might be in. If every state that makes all the premises true makes the conclusion true, then the conclusion is a logical consequence of the premises. This motivates the variable domains of contemporary models quite naturally, but it is hard to see why different models should be allowed to assign different interpretations to expressions which actually share an interpretation (which they are permitted to do in contemporary model theory). Now it only remains to introduce the fourth and final conception of logical consequence that I will consider.
Universalist Logical Consequence On the accounts we have looked at so far, the relation of logical consequence is a metalinguistic one, concerning preservation of the property of truth between a set of premises and a conclusion. Timothy Williamson (2013) proposes an alternative, non-metalinguistic conception of logical consequence, one which is again a recognisable descendent of Tarski’s
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approach. The view is easier to understand if we start with Williamson’s technical definition, and this is in turn easier to grasp if we begin with logical truth, and then generalise to logical consequence afterwards. Call a sentence A a logical truth just in case a related sentence A′, generated by uniformly replacing all the non-logical vocabulary with appropriate new variables (distinct vocabulary with distinct variables) and closing the result under universal quantification, is true. For example, each of the sentences on the left below is a logical truth just in case its counterpart on the right is true:
a=a Fa Ú ØFa Af ® f
"x ( x = x ) "X"x ( Xx Ú ØXx ) "X ( AX ® X )
Among other things, these illustrations make it clear that we are to quantify into all non-logical positions, including predicate and sentential positions. Call the result of replacing all the non-logical expressions with appropriate variables the sentence’s shell. Then we can express the new definition succinctly: Definition 1 (Logical Truth) A is a logical truth just in case the universal closure of A’s shell is true.13 The universal closure of a sentence will not (in general) be a metalinguistic claim but merely a very general one, and hence the truth conditions for claims about logical truth do not require metalinguistic facts to hold. One might reasonably wonder what is required to hold so that ∀X ∀ x(Xx ∨ ¬ Xx) or ∀X(X ∨ ¬ X) are true. How is quantification into predicate position or sentence position to be understood? One view would be that we are quantifying over properties and propositions, respectively, and that, for example, ∀X(X ∨ ¬ X) is true just in case whatever proposition we assign to the variable X, X ∨ ¬ X is true. Similarly, whatever object we assign to x and property to X, Xx ∨ ¬ Xx is true. This raises questions about what properties and propositions are, but just as first-order logic can make a lot of progress while taking the idea of an object for granted, so we can pursue this approach further without
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endorsing particular views on these matters. All we need to say for now is that propositions are the values of sentential variables, much as objects are the values of first-order variables, and similarly properties are the values of predicate variables. Now we generalise to logical consequence. This must be done with some care since there is no obvious way to close an argument’s shell using quantifiers (we normally quantify into sentences, not arguments). So consider an argument with premises, such as modus tollens: P → Q, ¬Q ⊨ ¬ P. Call the result of uniformly replacing the non-logical vocabulary with variables throughout the argument the argument’s shell. The shell for modus tollens is X → Y, ¬Y ⊨ ¬ X. Then we say that the conclusion of the argument is a logical consequence of the premises just in case every assignment of appropriate values (objects to first-order variables, properties to predicates, etc.) that makes both X → Y and ¬Y true also makes ¬X true. Here is another illustration, this time with a first-order case: a=b Fa Fb
The shell for the argument is: x=y Xx
Xy
An assignment will be a function that assigns objects to x and y and a property to X. Our new definition of logical consequence tells us that the argument is valid just in case there is no such assignment that makes x = y and Xx true, and Xy not true. In general: Definition 2 (Logical Consequence) A sentence is a logical consequence of a set of premises just in case there is no assignment that makes all the premises of the argument’s shell true and the conclusion not true.
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This is our fourth variety of logical consequence. It is much newer than the other three varieties, and no doubt its merits will become clearer over time. In this chapter, I want to look at how it fares—in comparison with the other approaches—against the threat of logical nihilism.
2
What Is Logical Nihilism?
Logical nihilism—as we will understand it here—is the view that there are no laws of logic, or equivalently that the relation of logical consequence is empty. For any arbitrary set of premises and conclusion, the nihilist holds that the premises do not entail that conclusion.14 Putative laws of logic—laws endorsed by others, but not by the nihilist—include modus ponens, and the law of excluded middle, conjunction elimination and the principle of non-contradiction, and more generally, any well- formed atomic sentence in which ⊨ is the main predicate, flanked by a name for a set of sentences on the left, and a name for a sentence on the right, as in: A ® B, A
AÙB A, ØA A A ® B, B
B A Ú ØA A Ø(A Ù ØA B A A
( MP ) ( LEM ) ( ÙE ) ( PNC ) ( Explosion ) ( ID ) ( AC )
Call sentences of this form E-sentences (‘E’ for entailment). Many logics reject one or more of these putative laws. Classical logic rejects (AC), paraconsistent logics reject (Explosion), Strong Kleene and intuitionistic logics deny that (LEM) is a law. Logical nihilism rejects them all, including relatively uncontroversial principles, such as conjunction elimination
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and identity. More comprehensively, for any set of sentences K and any conclusion X, the nihilist is committed to:
K X
What this amounts to will depend on what logical consequence is, that is, on what ⊨ means. Different conceptions of consequence correspond to different versions of logical nihilism. Representational nihilism is the view that for any argument, there is always a case—roughly, a state of the world—in which the premises are true and the conclusion is not. Interpretationalist nihilism is the view that in any argument there is always an interpretation of the non-logical expressions which makes the premises true and the conclusion not. Substitutional nihilism is the view that there is always a way to substitute expressions for the non-logical expressions in an argument which makes the premises true and the conclusion not true. And universalist nihilism will be the view that there is always an assignment of values to the non-logical expressions in an argument which makes the premises of an argument’s shell true without making the conclusion true. Nihilism, then, is relatively simple to state and understand, even if, like logical consequence, it comes in several flavours. But one might wonder why anyone would entertain such an apparently wild view. My goal in this section will not be to convince you of nihilism itself, but rather to convince you that the view is a live option, so that you might be interested in safeguards against it.15 One of the reasons it is a live option stems from the extreme generality of logical laws. Claims that are very general are, in one sense, very strong, and hence take much to establish and little to refute. For example, it is not sufficient for the truth of modus ponens that the truth of
snow is white
and
snow is white ® grass isgreen
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guarantees that of
grass isgreen
—though I’m confident that it does—because this argument is only one instance of modus ponens. In order for modus ponens to be a logical law, every single instance would need to be truth-preserving.16 Similarly, it is not sufficient for the validity of modus ponens that every single argument you can think of that takes that form is truth-preserving. There is always a chance that the counterexample involves some case, or some interpretation, or some assignment that you haven’t thought of. Do we have reason to think that every instance of modus ponens is truth-preserving—including those instances that feature Sorites predicates, truth and other metalinguistic predicates, self-reference, sentences which generalise over the mathematical objects in infinite sets, context-sensitivity, fictional names, and so on? The history of the paradoxes—many of which have motivated the rejection of classical laws of logic—suggests any such faith could be misplaced. While we might feel confident that the instance above is truth-preserving (and I do feel confident about that), I am not similarly confident that every argument of the same form—no matter how rich the language in which it is formulated—is also so. To the extent one is not confident of that, one is not confident that modus ponens is a logical law.
Do We Have Proofs of the Logical Laws? In mathematics, one way to establish even very general claims is by giving a proof, and certainly we give proofs of logical laws. These proofs are of two kinds. The first is formal, that is, in a particular formal proof system. We might, for example, give the following sequent calculus proof of the Law of Excluded Middle:
cut
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Fig. 1 A truth-table proof of Modus tollens
But such a formal proof of the Law of Excluded middle serves as a proof (in the intuitive sense) of the Law of Excluded Middle only on the assumption that the proof rules are sound. Logicians who deny the law— like strong Kleene logicians, intuitionist logicians, and nihilists—reject this assumption, and so this kind of proof will not establish a law of logic. A second standard way to establish a law of logic is via informal (though still rigorous) model-theoretic proof. One example is to use a truth-table to display classical interpretations (one per row; see Fig. 1 above). A second example is to use classical first-order models, as in this model-theoretic proof of Fa ⊨ ∃xFx: Take an arbitrary model M = (D, I) which makes Fa true. Then |a|M ∈ |F|M. Consider an assignment v of elements from D to variables such that v(x) = |a|M. Then Fx is satisfied by to M on v, and hence M makes ∃xFx true. Generalising, any model which makes Fa true makes ∃xFx true. Still, paraconsistent logicians don’t take the truth-table above to show that modus tollens is a logical law, because they hold that it fails to display all the relevant interpretations. When the truth-table is expanded to include the interpretations on which sentence letters may receive the truth-value ‘Both’ (B), we find interpretations on which the premises may both be ‘Both’ while the conclusion is false—a counterexample to the law (Fig. 2). Similarly, with the model-theoretic proof, a free logician will claim that we have failed to consider relevant models. What about models in which a’s referent is not a member of the domain of quantification? So though we have proofs of general logical laws, both formal and informal, unless we know that one particular set of models, or one particular proof system, is the right one, we do not know that these are
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Fig. 2 A paraconsistent truth-table of Modus tollens
c orrect. And if we don’t have proofs of the logical laws, then we don’t have proofs that logical nihilism is false.
Logical Nihilism and Logical Pluralism A second reason to concern ourselves with nihilism is that an argument has been given for it in the literature on logical pluralism.17 On one much-discussed version of logical pluralism—the view due to Beall and Restall (2000)—validity is defined using what they call the Generalised Tarski Thesis (GTT): (GTT) An argument is valid x if and only if, in every case x in which the premises are true, so is the conclusion.
This is naturally understood as assuming a version of the representationalist approach to consequence. The (GTT) leads to pluralism on Beall and Restall’s view because the word case is an everyday, imprecise expression which can be made more precise in a variety of ways. We might
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consider truth-preservation over possible worlds, paraconsistent or paracomplete models and more. More than one of these precisifications of case is, they think, legitimate, and the different legitimate precisifications lead to different extensions for the word valid. Priest has pressed an objection to this view18: The obvious reply to this argument is that it is only truth-preservation over all situations that is, strictly speaking, validity. One of the points about deductive logic is that it will work come what may: we do not have to worry about anything except the premises. (Priest 2006, p. 202)
The objection draws on a view of logic that is both intuitive and has historical precedent, namely, that logic is supposed to be completely general; if an argument K ⊨ X preserves truth in (say) arithmetic but not in geometry, or in classical cases but not in dialethic ones—then it is not really a law of logic after all. Logic is supposed to work in absolutely all cases. Beall and Restall respond with a reductio ad absurdum on the assumption that logic is completely general: they say that this assumption will lead to absurdity—in particular to logical nihilism, or some similar absurdly weak logic: …we see no place to stop the process of generalisation and broadening of accounts of cases. For all we know, the only inference left in the intersection of (unrestricted) all logics might be the identity inference. (Beall and Restall 2000, p. 92)
Elsewhere I’ve argued that not even identity would remain (Russell 2017), but one might think that this is not of the greatest importance. Most philosophers and logicians regard both all-out logical nihilism and a view on which only identity is left (or some other very minimal set of E-sentences—call such a view logical minimalism) as absurd. They think that if your philosophy of logic leads you to either logical nihilism or logical minimalism, it must be wrong. Though discussions of pluralism have tended to assume a case-based (and so representationalist) approach to consequence, this is inessential to the debate. A structurally similar issue—nihilism begot by generality—
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arises for other varieties of consequence as well. It is easy to see this if we reformulate the GTT to get different conceptions of pluralism for each of the conceptions of consequence discussed above: An argument is validx if and only if on every uniform substitutionx (of appropriate non-logical expressions for the non- logical expressions in the argument) on which the premises come out true, so does the conclusion. (GTT-i) An argument is validx if and only if on every interpretationx of the non-logical expressions on which the premises are true, so is the conclusion. (GTT-r) An argument is validx if and only if in every casex in which the premises are all true, so is the conclusion. (GTT-u) An argument is validx if and only if on every assignmentx of values to the non-logical expressions on which the premises are true, so is the conclusion. (GTT-s)
To the extent that we think that the words substitution, interpretation, and assignment are—like case—still imprecise and tolerant of various reasonable precisifications, the way is open for those precisifications to result in different extensions for valid. Moreover, the monist may object, as above, that this does not take the every in the respective definitions seriously enough. It is only a logic which takes all of the different precisifications into account which is— strictly—logic. And so then the pluralist can suggest again that this commitment to absolute generality in logic will lead to nihilism.
3
Modes of Resistance
So much for the big picture: a commitment to generality in logic seems to push us towards nihilism, or at least towards weaker and weaker logics. Still, when we look at the details, the availability, plausibility, and persuasiveness of certain kinds of counterexample vary with our conception of consequence. We are now in a position to explore some of this variation in the third and final section of this chapter. In the coming subsections, I will look at the issues associated with empty names, vagueness, and self-
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reference, in that order. In each case I will begin by sketching a possible view (or views) of the topic that would lead to a rejection of a classical entailment principle. Since my goal is to look at safeguards that logics might have against such things, I won’t argue for these sketches of views on empty names, vagueness, and so on. Rather, my aim is to put them on the table so that we can look at the internal resources that each conception of logic has for resisting them. Things will get quite complicated, but I will finish by drawing out a general pattern: if our conception of consequence quantifies over elements that go ‘deeper’ than the phenomenon that is thought to cause trouble, then the phenomenon does not threaten our logic. Deeper logics, then—and especially Williamson’s universalist variety—have more resistance to pressures to weaken and, ultimately, against nihilism. But the cost of this depth is complimentary: it doesn’t allow us to study the effects of interesting surface phenomena on consequence.
Empty Names Classical logic does not model empty names. In classical model theory, an interpretation must assign an extension to every individual constant. Natural languages have empty names—Pegasus, Santa, and so on—and we might want to know about entailment relations on sentences which contain them. In particular, a commitment to generality in logic could lead us to ask what will happen to the logic if we consider all names, including the empty ones. What will happen depends on one’s conception of logical consequence. For example, on the substitution-conception, we are supposed to be able to substitute any name uniformly for another name throughout a valid argument without losing validity. But if we substitute Pegasus for a throughout this classical validity:
a = a $x ( x = a )
(2)
we appear to have a classically valid argument whose premise is a classical logical truth, but whose conclusion is false. Thus if we reject the classical restriction on which kinds of terms can be substituted—that is,
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if we are serious in our commitment to generality—then we come under pressure to reject this principle of entailment. Things look different if we are employing (GTT-u). Then the problem of empty names gets no traction, because (GTT-u) doesn’t quantify over linguistic expressions, but over assignments of referents to them. Whatever object we assign to Pegasus in ∃xx=Pegasus (Bismarck, Du Bois, the number 3, the Eiffel Tower, etc.), the sentence comes out true. Intuitively, the substitutional conception is saying that a = a is a logical truth if it is true whatever name we substitute for a, and hence it is making a claim about names, including empty names. But the universalist conception aims to make no claims about names, only about objects. It says that a = a is a logical truth if (roughly) every object is identical to itself. It bypasses the problem of empty names, and hence is resistant to the counterexample. What about the remaining two varieties, interpretational and representational? Let’s begin with interpretational, that is, let’s suppose that contemporary models represent different ways the non-logical parts of languages can be. It is fairly clear that the models do this by assigning different referents to names, that is, it is built into our standard conception of a model that an interpretation for a name is a referent. That makes space for the interpretationalist to say that when they consider only complete interpretation functions, they already are considering all (really all) the possible interpretations for the names. Still, this response on behalf of (GTT-i) rings a bit hollow, since the possibility under consideration is precisely that a name has not a referent. If the goal is to represent all the ways a language can be, having a part that doesn’t have an interpretation (in the sense of having a referent) is prima facie one of those ways. So it looks as if the interpretational conception of consequence ought to consider partial interpretation functions if they are really to consider all the ways language can be, and this would lead to a rejection of (2). The representational variety can seem more ‘worldly’ in spirit than the interpretational. It regards models as representing different states of the world—different cases. If Pegasus is an empty name, then there is no Pegasus. Is there still a case in which Pegasus flies and another in which Pegasus does not fly? My inclination is to say no. Making ‘Pegasus flies’ true
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is a not condition that a case can meet or fail to meet if ‘Pegasus’ has no meaning, any more than making the sky dfsndfnsf is a condition that something can meet or fail to meet. But perhaps you think it is too quick to say that the name Pegasus has no meaning. It could be associated with some descriptive condition—a sense—that, as it happens, isn’t satisfied, but could be in some cases. Perhaps you want to say that Pegasus doesn’t exist, but that there are still models on which Pegasus flies is true, namely, models in which something satisfies the condition associated with Pegasus, and that thing flies. Such a conception of names could still be associated with interpretation functions that assign objects to names, but we might expect that names would be permitted to lack a referent, that is, the requirement that the interpretation function be a complete function over the set of individual constants would be dropped. This is consistent with the representational approach to models. Just think of the possible changes to the states of the world as including the deletion or inclusion of the referents of names, and we keep the language the same by retaining the name’s descriptive sense (which is ignored in the model). This isn’t my view of names, for broadly Kripkean reasons. But if you like this view, then you might think that the representational approach is as threatened by the existence of empty names as the interpretational one. To sum up the results of this section, empty names motivate a decisive weakening of the logic in the case of the substitutional conception of consequence, but not on the worldly conception. Meanwhile defenders of the interpretational and representational conceptions can make moves to avoid admitting incomplete interpretation functions on names. The defender of the interpretational conception would deny that ‘empty’ is a legitimate interpretation of a name (because interpretations of names are referents), and the defender of the representational conception would deny that empty names can be used to place conditions on cases. It is perhaps worth noting, first, that both these ways of avoiding weakenings have a decidedly Millian flavour, and second, that on the conceptions of logical consequence that permit us to avoid weakening in the face of empty names, we do this by finding a way of thinking of logic and/or names that allows us to ignore the phenomenon. This is somewhat unsatisfying, and this is a point that I will return to at the end of this chapter. For now, let’s look at a new threat.
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Vagueness and Incomplete Predicates Predicates in classical models must have complete extensions, where this means that the extension (the set of things of which the predicate is true) and the anti-extension of the predicate (the set of things of which the negation of the predicate is true) exhaust the domain of the model: everything in the domain is in one set or the other.
But one might think that the extensions of vague predicates in natural languages are incomplete. On one version of the idea, the conventional meanings associated with our colour predicates determine a range of shades which are red, and a range of shades which are not red, but do not legislate on all of the shades which fall in between. We might imagine a series of colour cards progressing gradually from the brightest of reds at one end to the pinkest of pinks at the other. (Represented in black and white below.) Anything which is pink is not red, but while there are shades on one end which fall in the conventional extension of ‘red’, and shades at the other end which fall in the extension of ‘pink’ (and so in the anti-extension of ‘red’) there are shades in the middle about which our linguistic conventions say neither that the shade is red, nor that it is not red. red
not-red
This suggests that the interpretation function for a predicate P might assign an extension |P|+ (the set of things that ‘are P’) and an anti-extension
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|P|− (the set of things that ‘are not P’) though these two sets need not exhaust the domain of the model.19 Then if |a| ∈ |P|+, Pa is true, if |a| ∈ |P|−, then ¬Pa is true, but if |a| is in neither the extension nor the anti-extension of the predicate, then both Pa and ¬Pa are neither true nor false—a third truth status, which I will abbreviate to neither. To argue for a third truth-status is not yet to have made a case for a non-classical logic, because a philosopher who believes in three truth- statuses could still hold that all and only the classical entailments are valid. But there is a well-worn path from a third truth-status to a non- classical logic. For if Pa is neither on some interpretations, then what of super-sentences containing it, such as ¬Pa be or Pa ∨ ¬ Pa? One approach adopts the Strong Kleene truth-tables, on which ¬Pa is neither if Pa is and a disjunction takes neither if both disjuncts do. Assuming that a logical truth is true on all interpretations we have a counterexample to the classical law of excluded middle when Pa gets the neither truth status.20
A Ú ØA
(3)
Different varieties of consequence have different levels of resistance to this kind of counterexample. There are some similarities between the responses open to them here and the responses available in the previous empty names case. For example, the (GTT-s) is especially vulnerable; if there are predicates with incomplete extensions and logic is supposed to consider all uniform substitutions of predicates for predicates, it is hard to see how we can justify rejecting counterexamples to a logical law based on substituting incomplete predicates. The (GTT-i) account of logical consequence is subject to similar pressures. If there are natural language predicates with incomplete extensions, and we are thinking of models as representing different ways the language could be, it seems artificially restrictive to ban ways on which predicates get indeterminate extensions, that is, incomplete interpretation functions in models. However, just as in the case of empty names, there is an option here if the interpretationalist has the will to take it: if they take a Russellian view of propositions, they might wish to maintain that an interpretation for a predicate simply is a property. They can hold (as in Russell 1918) that just as the world contains objects, so it contains properties, which
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serve as the meanings of predicates. To interpret a predicate just is to assign it one of these worldly properties (much as to interpret a name is to assign it a referent) and, on the assumption that properties have determinate extensions, interpreted predicates must similarly have determinate extensions. Still, perhaps even more so than in the case of names, this rings a little hollow: we are considering modifying a logic to take account of predicates with incomplete extensions, since it seems that this is one of the ways language can be. Though a strict Russellian might simply redescribe this view of vagueness as one on which one of the ways language can be is uninterpreted (or improperly or incompletely interpreted), this is still, intuitively, one of the ways language can be. To insist on restricting the interpretations we look at to incomplete ones seems incompatible with a commitment to generality. On the representational account, the meanings of all the expressions in an argument we are testing remain as they are, and we consider whether there are any ways the world could be which make the premises true and the conclusion not. If the P in Pa ∨ ¬ Pa is a determinate predicate (say ‘is prime’), it would seem that no matter what worldly changes we make to the referent of a (or its surrounding world)21 Pa will be true or false (but not neither), and so Pa ∨ ¬ Pa would always be true. The problem is that still won’t mean that the law of excluded middle is a logical truth, given the way the representational approach is normally implemented. Because representationalists concede that logic is supposed to be formal, so that what we would need for the LEM to continue to hold is the stronger claim that all sentences of the form A ∨ ¬ A are true in all models, and a sentence Qa ∨ ¬ Qa, where Q is an indeterminate predicate, will provide us with a counterexample. Finally, let’s look at the universalist account, for here things are different again. On this conception, when we ask whether Qa ∨ ¬ Qa is a logical truth, we are asking whether there is any assignment of properties and objects to X and x, respectively, in the sentence’s shell— Xx ∨ ¬ Xx—such that the shell does not come out true. Here there is no direct reference to indeterminate predicates, and the question naturally arises, whether there is any such thing as an indeterminate property. No doubt there is much to be said either way, but there is clearly space for a fan of the worldly account to take the following view: while there
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are indeterminacies in the interpretations of predicates—for example, in the meanings of predicates—there is no such indeterminacy or vagueness in the world itself. Nothing about a cloud, or a rainbow, or a furry cat, is really indeterminate. The world is as it is, the various molecules and photons and hairs are where they are, and if the truth-value of a sentence about a mountain or a heap or the colour of a playing card is indeterminate, that is just because the rules for when to call something ‘red’ (or ‘a mountain’ or ‘a heap’) don’t tell us enough about what it is to be called in these worldly cases. This view has several interesting consequences (and perhaps for some they are too interesting), but one of them is that the existence of indeterminate predicates need have no impact on logic. Perhaps ‘red(a) ∨ ¬ red(a)’ contains an indeterminate predicate, but when we ask whether it has the form of a logical truth, we are asking whether there are any assignments on which Xx ∨ ¬ Xx fails to come out true, and if there are no assignments which have indeterminate properties in their ranges—because there are no indeterminate properties—the answer will be ‘no.’
Self-Reference and Overdeterminacy We will look at one more example before drawing some conclusions. It is well known that some metalinguistic sentences and expressions—such as ‘This sentence is not true’ and ‘heterological’—generate paradoxes. These paradoxes have led logicians to weaken logics, in two ways. On the way I will focus on here—dialetheism—the paradoxes are taken to show that there are true contradictions. Dialetheists hold that the contradictory conclusions of the paradoxes are true.22 We can model this using a similar idea to the one employed to handle indeterminacy: suppose that predicates like ‘true’ and ‘heterological’ are associated with conventional rules which give a sufficient condition for being ‘F’ and a sufficient condition for being ‘not F’ but—the paradox shows us—these conditions need not be exclusive. Some things might meet the condition for falling in the extension of ‘heterological’, making ‘x is heterological’ true of them. Some things fall in the anti-extension, making ‘x is not heterological’ true of them, and some things—like ‘heterological’ itself—fall in both the
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extension and the anti-extension of ‘F’, making both the sentence and its negation true. heterological palindrome
monosyllabic
long
heterological meaningful
heterological
polysyllabic
English
We might call such predicates inconsistent. A pressing worry about this view is that it leads to triviality via (Explosion):
A,Ø A B
(4)
So dialetheists generally reject (Explosion), and indeed, this seems well motivated given their view: if there are models in which A is true and ¬A is true, then there are models in which both A and ¬A are both, and these need not be models in which B is true. So once again, we have a view which suggests abandoning a principle of classical logic. The situation with respect to the various conceptions of logical consequence is in some ways similar here to that in the case of incomplete predicates. For example: • if the substitutionalist is committed to generality, and inconsistent predicates exist, it is hard to see how they can resist the pressure to include them. • the interpretationalist can either allow inconsistent interpretations (naturally leading to a weakening of the logic) or make the Russellian move of insisting that interpretations just are assignments of worldly properties and combine it with the metaphysical view that there are no inconsistent worldly properties.
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• the representationalist might hold that there are no cases in which explosion arguments employing consistent predicates have true premises and non-true conclusions, but since they allow that logic is formal, and there will be arguments of the same form with inconsistent predicates, this is not sufficient to save explosion. • the universalist again has a natural defence: argue that there are no inconsistent properties and then point out that since their conception of consequence quantifies over properties, the existence of inconsistent predicates does not provide any new counterexamples to explosion (just as the existence of empty names doesn’t show that it’s not true that every object is self-identical). The major difference, I think, between the incompleteness and inconsistency cases, is that philosophers are generally more adverse to the idea that there are inconsistent properties than they are to the idea that there are incomplete properties. That is, despite the parallels between the cases, people are more sympathetic to the idea that there might be an incompleteness in the world than they are to the idea that there might be inconsistency ‘out there.’
4
Conclusion
Things have become quite complicated, but this is only, I think, because there are four varieties of logical consequence on the table, and some of these offer more than one option when confronted with a linguistic expression which does not meet the standard assumptions of classical logic. The beginnings of a pattern are emerging from this complexity, so let me try to draw that out. You can take logical consequence to be a relation that quantifies over various things: expressions, meanings of various kinds, or over bits of the world (objects and properties and the like.)23 The substitutional view takes it to quantify over expressions, the universalist view takes it to quantify over bits of the world, and the interpretationalist view can either be understood as quantifying over meanings or—if you take the line that a meaning just is a bit of the world (this is what I’ve been calling the ‘Russellian’ take)—over bits of the world. The
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representationalist view might be thought of as quantifying over cases (which could be thought of—tolerating some metaphysical hair—as bits of the modal universe), but in practice the assumption that logic is formal means that it usually inherits the features of quantifying over expressions. Any view which takes consequence to quantify over meanings has the option of saying that it doesn’t quantify over the kinds of meaning that result in the problems. For example, if the issue is empty names, then the problem is arising at the level of reference. An interpretationalist view could hold that consequence quantifies over assignments of reference, and hence needn’t take into account interpretations which assign no referent to a name. If, on the other hand, they quantified over what Carnap called ‘individual concepts’ or senses of names (in either case an aspect of meaning which determines the referent) the question would naturally arise: what about individual concepts which fail to determine a referent in some circumstances? And finally, we also sometimes have the option of combing the view of logical consequence with a view of the metaphysics of the problem. Take incompleteness. On some views, there are incomplete predicates but no incomplete meanings—the predicates are incomplete precisely because they fail to link up to a unique meaning. On others, there are incomplete predicates, incomplete meanings, but no incompleteness in the world itself. And on others still there are incomplete predicates, incomplete meanings, and incomplete worldly properties, like being a mountain or being orange. We might speak of linguistic vagueness, semantic vagueness, and ontic vagueness. We can also talk of linguistic dialetheism, semantic dialetheism, and ontic dialetheism. Suppose you think that there are incomplete predicates, but no incomplete meanings or worldly properties. Then you can resist counterexamples based on incompleteness by adopting a conception of consequence that quantifies over meanings or worldly properties—not predicates. Suppose you think instead that there are incomplete predicate meanings, but not worldly properties. Then you will probably think that there are incomplete predicates (the ones that express those meanings) but you can avoid counterexamples by adopting a conception of consequence that quantifies over worldly properties. Finally suppose you think that there are incomplete worldly properties (as you might if you believe in ontic
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vagueness). Then you are likely to be stuck with any counterexamples (and hence stuck with a weaker logic) based on incompleteness, since there will be predicates which attribute those worldly properties. So here is a high-level trend (much complicated, as we have seen, in the details): you can define consequence by quantifying over things on a vertical spectrum that has linguistic expressions at the top, various layers of meanings in the middle, and worldly things at the bottom. Various putative counterexamples arise from phenomena like incompleteness, inconsistency, and empty names, and for some of these phenomena it is controversial how ‘deep’ they go. For example, it is controversial whether incompleteness is a phenomenon restricted to expressions, or whether there are also incomplete meanings and incomplete properties. Similarly, there are names that lack referents, and Frege thought there were meanings without referents (names with sense but no referent; though Millians deny this), but it is hard to see how there could be objects but no referent—the reference failure phenomenon goes at most as deep as meanings. But if you think the phenomenon which generates a counterexample goes at most as deep as level n, then you can avoid it by selecting a conception of logical consequence that quantifies over a level that is lower than level n. Say, level n − 1.24 Finally, let me note that there appears to be a cost to opting to define consequence in a way that avoids the problems of, for example, empty names and incomplete and inconsistent predicates, namely, that it merely avoids the problems. Avoidance is sensible when we are engaged in some projects, but more generally, if such things exist then they can feature in arguments, and we might want to know the extension of valid over such arguments. A conception of logical consequence which simply doesn’t apply doesn’t answer these questions— it merely declines to consider them.
Notes 1. For example, Beall and Restall (2006, p. 92) and Russell (2016). 2. A note on terminology: in this chapter I will follow the common practice of treating the expressions ‘(logical) consequence,’ ‘entails,’ and ‘valid’ as intertranslatable—an argument is valid just in case its conclusion is a
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(logical) consequence of its premises, and this is so just in case the premises together entail the argument’s conclusion. Hence the different views of logical consequence that I will be looking at are just as much different views of validity and different views of entailment. There are two other words that it might be tempting to use here: implication (perhaps being Quineanishly careful to use this for entailment rather than for a conditional) and inference. But following Harman (1986) I take inference to be a different topic altogether, and I’ll be cautious about using ‘implication’ because I think Quine’s suggested regimentation (i.e. restricting its use to talk of the entailment relation) is less entrenched than would be ideal for successful communication (Quine 1966, pp. 165–166; Quine 1981, §5). 3. For example, in the title of Etchemendy (1999). I recognise that Etchemendy’s word ‘concept’ is fighting talk in some philosophical circles. I mean to talk—as Etchemendy did—about different views of what features an argument has to have to be valid and have no special commitment to using the word ‘concept’ or to any particular construal of that phrase. 4. Sometimes people respond to this problem by distinguishing different kinds of necessity. 2 + 2 = 4 and Hesperus is Phosphorus, they might claim, are metaphysically necessary, but not logically necessary, and it is the logical modality in terms of which logical consequence is defined. There are two problems with this response. The first is that it replies on a controversial claim about the kind of necessity possessed by the propositions expressed by these sentences. In, for example, Kripke’s modal argument of the necessity of Hesperus is Phosphorus, the ‘□’ that that sentence inherits is the very same one applied to Hesperus is Hesperus (Kripke 1980). The second problem is that the account threatens to be circular. What is logically necessary if it is not the things which hold in virtue of logic alone—the logical truths. But this assumes that we already have an account of logical consequence. 5. Kaplan (1989). Here A is the actuality operator, N the now operator, and α a singular term, so that informal instances of these sentences might be Actually it is snowing if, and only if, it is snowing., if it is snowing now, then it is snowing, I am here now and dthat[the shortest spy]=the shortest spy. 6. An additional problem for the modal slogan is that: necessity and logical truth come apart in the model theory for modal logics. □p may be true at some points in a model and false at others. This would make no sense if logical truth were necessity—necessity might be world relative but logical truth is not.
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7. The terminology here is from Etchemendy (1999). I find this terminology somewhat difficult to remember, but one useful mnemonic is to note that the one view is interpretational because it speaks of different interpretations of the linguistic items, while the other is representational because it talks about the things that get represented. 8. Here is one: (i) if Π is an n-place predicate and t1,…,tn are n terms, the Πt1,…,tn is an atomic wff; (ii) if t1 and t2 are terms, then t1 = t1 is an atomic wff; (iii) if A is a wff, then ¬A is a wff; (iv) if A and B are wffs, then A∧B is a wff; (v) if A is a wff and x is a first-order variable, then ∀xA is a wff. If A contains no quantifiers, then all the variables in A are free. If ∀ is immediately followed by a variable ξ, then we say that it is a ξ-binding quantifier. In a wff ∀ξA, the ξ-binding quantifier ∀ is said to bind all free occurrences of ξ in A. A variable which is bound in A is not free in A. Any wff with no free variables is a sentence. 9. That is, nI will be an element of D, where n is a name, PI will be a subset of Dn, where P is an n-place predicate, and fI will be a complete function from Dn to D, where f is an n-place function. 10. See also “Their characteristic [logical truths] is that they not only are true but stay true even when we make substitutions upon their component words and phrases as we please, provided merely that the so-called ‘logical’ words ‘=’, ‘or’, ‘and’, ‘not’, ‘if-then’, ‘everything’, ‘something’, etc., stay undisturbed” (Quine 1950, p. 4). 11. Tarski too considers a variation of the word-test approach: “If, in the sentences of the class K and in the sentence X, the constants—apart from purely logical constants (like signs being everywhere replaced by like signs), and if we denote the class of sentences thus obtained from K by ‘K′’, and the sentence obtained from X by ‘X′’, then the sentence X′ must be true provided only that all the sentences of class K′ are true.” (Tarski 1936, p. 415) 12. Etchmendy (1999, pp. 68–69; 111–114). 13. It is a bit loose to speak of ‘the’ universal closure of ‘the’ shell, since there will be many trivial variants, for example, ∀Y(Y → Y) and ∀Z(Z → Z) are also universal closures of a shell of P → P. If desired, we might enumerate the variants and count only the first (or the fifth) as the universalisation. Since the issue won’t be of importance here, I won’t mention it again. 14. In his chapter in this volume, Aaron Cotnoir distinguishes two views called ‘logical nihilism’ and focuses on one. The view I have in mind here is the other one.
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15. I’ve looked in more depth at one argument for logical nihilism here: (Russell 2017) 16. All of our conceptions of logical consequence agree that this is necessary for logical consequence, though some may require more than this. 17. Admittedly, that argument was intended to be a reductio on someone else’s view. But one person’s reductio ad absurdum is another person’s argumentum ad absurdum. 18. There are similar lines of argument in Read (2006, p. 61) and Bueno and Shalkowski (2009, p. 11). 19. The problems of higher order vagueness make it clear that this is not sufficient to accommodate and explain vagueness, but there are still many views on which the move is regarded as necessary. 20. And in fact Strong Kleene logic provides counterexamples to all the classical logical truths. 21. We leave to the previous section discussion of what happens if a does not exist. 22. That is, The Liar sentence is true and the Liar sentence is not true or ‘Heterological’ is heterological and ‘heterological’ is not heterological. 23. And one of the reasons things are especially complicated is that the most familiar views can be understood as quantifying over either meanings or bits of the world. 24. It would be especially interesting in future work to examine this pattern against the phenomenon of context-sensitivity. Context-sensitivity is a paradigmatically linguistic, rather than worldly, issue, but there are still views (such as MacFarlane’s non-indexical context-sensitivity) which allow it to go deeper than others. Moreover, context-sensitivity is known to have interesting non-classical effects on logic.
References Beall, J., and G. Restall. 2000. Logical Pluralism. Australasian Journal of Philosophy 78: 475–493. ———. 2006. Logical Pluralism. Oxford: Oxford University Press. Bueno, O., and S. Shalkowski. 2009. Modalism and Logical Pluralism. Mind 118: 295–321. Etchemendy, J. 1990. The Concept of Logical Consequence. Cambridge: Harvard University Press.
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———. 1999. On the Concept of Logical Consequence. Stanford: CSLI. Harman, G. 1986. Change in View. Cambridge, MA: MIT Press. Kaplan, D. 1989. Demonstratives: An Essay on the Semantics, Logic, Metaphysics, and Epistemology of Demonstratives. In Themes from Kaplan, ed. J. Almog, J. Perry, and H. Wettstein. New York: Oxford University Press. Kripke, S.A. 1980. Naming and Necessity. Oxford: Blackwell. Priest, G. 2006. Doubt Truth to be a Liar. Oxford: Oxford University Press. Quine, W.V.O. 1950. Methods of Logic. Cambridge, MA: Harvard University Press. ———. 1966. Three Grades of Modal Involvement. In The Ways of Paradox, and Other Essays, 158–176. Cambridge, MA: Harvard University Press. ———. 1981. Mathematical Logic. 3rd ed. Cambridge: Harvard University Press. Read, S. 2006. Monism: The One True Logic. In A Logical Approach to Philosophy: Essays in Honour of Graham Solomon, ed. D. DeVidi and T. Kenyon, 193–209. Berlin: Springer. Russell, B. 1918. The Philosophy of Logical Atomism. Chicago/LaSalle: Open Court Classics. Russell, G. 2016. Logical Pluralism. In The Stanford Encyclopedia of Philosophy, ed. E.N. Zalta, CSLI, Summer 2013 edn. ———. 2017. An Introduction to Logical Nihilism. In Logic, Methodology and Philosophy of Science—Proceedings of the 15th International Congress. College Publications. Tarski, A. 1983/1936. On the Concept of Logical Consequence. In Logic, Semantics and Metamathematics, ed. J. Corcoran, 2nd ed., 409–420. Indianapolis: Hackett. Williamson, T. 2013. Modal Logic as Metaphysics. Oxford: Oxford University Press. ———. 2017. Semantic Paradoxes and Abductive Methodology. In Reflections on the Liar, ed. B. Armour-Garb, 325–346. Oxford: Oxford University Press.
Part III Connections
Pluralism About Pluralisms Roy T. Cook
1
Introduction
In recent years various sorts of pluralisms have come into vogue in the philosophy of logic. For present purposes, two in particular stand out: • Logical Pluralism: The claim that two (or more) distinct logics are correct (or legitimate, or best). • Alethic Pluralism: The claim that two (or more) distinct accounts of truth are correct (or legitimate, or best). Pluralisms, of either variety, come in two flavors1: • Domain-specific Pluralism: There are two (or more) distinct accounts of [insert the phenomenon in question], and each applies to a distinct domain (typically, one not overlapping the domains to which the other accounts apply). R. T. Cook (*) Department of Philosophy, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_15
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• Domain-independent Pluralism: There are two (or more) distinct accounts of [insert the phenomenon in question], and each applies everywhere. Interestingly, the most well-developed account of alethic pluralism, Michael Lynch’s account as developed in Lynch (2008, 2009) and elsewhere, is a domain-specific alethic pluralism, while the first (and perhaps still most well-developed) account of logical pluralism—JC Beall and Greg Restall’s account as developed in Beall and Restall (2000, 2001, 2006)—is a domain-independent logical pluralism. Stewart Shapiro’s logical relativism—presented in Shapiro (2015)—presents a rich, domain-specific logical pluralism as an alternative to the BeallRestall domain-independent approach. Domain-independent alethic pluralism remains the most under-examined of the four options, although Sher (2013) is a promising recent exploration in this direction.2 Here we shall focus on the domain-specific versions of both alethic pluralism and logical pluralism. The reason is not that these accounts are in some way superior to domain-independent pluralisms about the same subject matter.3 The reason that we will restrict our attention for the most part to domain-specific pluralisms is that we are interested in the connections, if any, between alethic pluralism and logical pluralism—and in particular, whether the former entails the latter or vice versa—and the only extant arguments that I am aware of for such connections focus on domain-specific accounts. I have no doubt that arguments for or against the existence of connections between domain-independent alethic pluralism and domain-independent logical pluralism (or between combinations of domain-specific and domain-independent pluralisms) can be formulated, and hopefully some of the work carried out here will prove helpful in such endeavors. But this chapter will focus primarily on the domain-specific case, although I will return briefly to other combinations, noting some apparent difficulties in formulating similar arguments for independence, in the concluding section. If domain-specific alethic pluralism entails domain-specific logical pluralism, and vice versa, then in some sense we really only have one
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pluralism, not two. If, however, the two sorts of pluralism are independent of each other, then we truly have two distinct kinds of pluralism— that is, we have a plurality of pluralisms. The purpose of this chapter is to argue that domain-specific alethic pluralism does not entail domain- specific logical pluralism (contrary to Lynch (2008, 2009) and Pedersen (2014)), nor does domain-specific logical pluralism entail domain- specific alethic pluralism, and hence we do have such a plurality of pluralisms.4 To accomplish this, in Sect. 2 I will present a simple model that shows both how one might be a domain-specific pluralist about logic while being a monist about truth and how one might be a domain-specific pluralist about truth while being a monist about logic. I will then, in Sect. 3, use this model to identify the fallacy in the arguments found in Lynch (2008, 2009) and Pedersen (2014).5 Section 4 will then further flesh out the model, distinguishing between different senses in which a domain might be epistemically constrained. Finally, in a short concluding section, I will tie up some loose ends and gesture in the direction of the aforementioned difficulties in extending the present account to domain-independent pluralisms (and combinations of domain-specific and domain-independent pluralisms). Before moving on, a methodological note is in order: Although this chapter is a response to arguments found in Lynch (2008, 2009) and Pedersen (2014), the arguments made below do not only apply to versions of domain-specific alethic pluralism that take truth to be a functional concept manifested by different lower-level properties in different domains. The arguments are much more general than this, applying to any view according to which truth obeys different principles in different domains, regardless of whether that is a result of their being different sui generis truth properties in different domains, or a single truth property that behaves differently in different domains, or anything in between (e.g. the various accounts of domain-specific alethic pluralism inspired by Wright (1992)). In short, the argument is a logical one, and is independent of much of the metaphysical theorizing that is a requisite of spelling out this-or-that particular brand of alethic pluralism in detail.
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Three Theories
Our argument for the independence of domain-specific alethic pluralism and domain-specific alethic pluralism (i.e. for a plurality of pluralisms) will proceed by constructing a philosophical model—one consisting of three distinct domains with differing logics and differing accounts of truth—and showing that one substructure of the model satisfies domain- specific alethic pluralism but not domain-specific logical pluralism, while another substructure of the model satisfies domain-specific logical pluralism but not domain-specific alethic pluralism. In order to construct such a model, we need to consider three ingredients: 1 . The domains in question 2. The logics that govern each domain 3. The accounts of truth that govern each domain Our model will require three distinct domains, which we shall simply call D1, D2, and D3. Once we have carried out a bit more of the formal work, there will be much more to say about what sorts of domain D1, D2, and D2 are. Now, for each domain, we need to decide on a logic that governs that domain, and on a theory of truth that governs that domain. First, for concreteness, let’s assume that we are working in some standard formal language LPA+T(x) for arithmetic that contains6:
∀, ∃, ¬, ∧, ∨, →
as primitive logical vocabulary (plus the usual countable infinity of variables, etc.), plus:
0, s ( x ) , +,×
as non-logical arithmetic vocabulary, plus a (non-logical) truth predicate T(x).7 We will use LPA to refer to the sublanguage containing all and only those formulas of LPA+T(x) not containing the truth predicate T(x).
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The non-truth-theoretic axioms are the standard axioms of Peano Arithmetic PA with induction restricted to formulas not containing T(x), and we assume that all three domains satisfy the axioms of PA (so understood). Second, we need to settle on some logics. Here we will only need to consider two familiar logics: First-order classical logic (C) and first-order intuitionistic logic (H). We will assume that the logic that applies to both D1 and D2 is H, and the logic that applies to D3 is C. Finally, we need two theories of truth. We will base these two theories on the familiar compositional axioms for truth8: [A1] (∀x)(AtomPA(x)→ (T(x) ↔ T0(x))) ↔ ¬T(x))) [A2] (∀x)(SentPA(x)→ (T(‘ ℜx’) [A3] (∀x)(∀y)(SentPA(x)⋀Sent(y)→ (T(‘x ∧ y’) ↔ (T(x)⋀T(y)))) [A4] (∀x)(∀y)(SentPA(x)⋀Sent(y)→ (T(‘x ∨ y’) ↔ (T(x)∨T(y)))) y’) ↔ (T(x)→T(y)))) [A5] (∀x)(∀y)(SentPA(x)⋀Sent(y)→(T(‘x → )x’) ↔ (∀x)T(sub(x, v))) [A6] (∀x)(FormPA(x)→T(‘( ∀x )x’) ↔ (∃x)T(sub(x, v))) [A7] (∀x)(FormPA(x)→T(‘( ∃x Following standard usage, we will call this theory CT↾.9 This is our first theory of truth. Our second theory of truth, which we shall call CT↾+, is simply CT↾ supplemented with the Arithmetic Law of Bivalence:
(
Biv PA : ( ∀x ) Sent PA ( x ) → (T ( x ) ∨ T ( ¬ x ))
)
We will assume that CT↾ is the theory of truth that applied to D1, and CT↾+ is the theory of truth that applies to D2 and D3. We can sum all of this up in the following table: D1 D2 D3
Logic
Truth theory
H H C
CT↾ CT↾+ CT↾+
Before moving on, it is worth making a technical point. The theory of D2 and the theory of D3 are not identical. In the context of CT↾ with
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intuitionistic logic H, BivPA entails that excluded middle holds for each sentence in LPA, but it does not entail that excluded middle holds for all sentences in LPA+T(x). Another way of putting this is as follows: We do not have: For all Φ ∈ LPA+T(x):
CT + H Φ iff CT + C Φ
CT + H Φ iff CT + C Φ
although we do have: For all Φ ∈ LPA:
(The latter result can be proven by a simple, albeit tedious induction which is left to the reader.) Thus, the theories of each domain are distinct. Now, what does all of this have to do with whether domain-specific alethic pluralism entails domain-specific logical pluralism (or vice versa)? The answer is simple: • If the only legitimate domains are D1 and D2, then domain-specific logical pluralism is false, since there is only one logic (H), yet domain- specific alethic pluralism is true, since there are two domain-specific theories of truth: CT↾ (for D1) and CT↾+ (for D2). • If the only legitimate domains are D2 and D3, then domain-specific alethic pluralism is false, since there is only one theory of truth (CT↾+), yet domain-specific logical pluralism is true, since there are two domain-specific logics: H (for D1) and C (for D2). Thus, domain-specific alethic pluralism does not entail domain- specific logical pluralism (since the first observation provide a counterexample to the validity of the inference), and domain-specific logical pluralism does not entail domain-specific alethic pluralism (since the second observation provides a counterexample). In short, we have shown that with respect to logic and truth, we have a plurality of pluralisms.
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Before moving on, a methodological clarification is in order. In the construction given above, we have equated a theory of truth, in the relevant sense, with a formal theory codified in the language LPA+T(x). As a result, one might worry that the argument just given has missed the mark, since most (if not all) extant defenses of alethic pluralism in the literature treat truth as a substantive concept and a theory of truth as more than merely a formal theory (e.g. Wright (1992), Lynch (2009)).10 This is of course right, but it is not a problem, since it is also the case that philosophically substantial accounts of the nature of truth “bring with them” formal theories that codify (the formal portions of ) that account, and sufficiently different such informal, philosophical accounts of truth will typically be accompanied by correspondingly different formal theories (such as CT↾ and CT↾+). In particular, although Lynch (2009)—our main target here—does not specify the formal theories of truth that correspond to his informal, substantial accounts in any detail, it is clear that the two types of truth theory on which he focuses the majority of his attention—accounts where truth is epistemically constrained and accounts where it is not— will correspond to formal theories that differ from one another in exactly the manner in which CT↾ and CT↾+ differ (i.e. in terms of whether they include some version of bivalence as a truth-theoretic principle). Thus, although the argument given above only applies to the accounts of truth developed by Lynch (and Wright) indirectly, via the link between informal philosophical accounts of truth and the formal theories corresponding to such informal accounts, it nevertheless does apply.11
3
What Went Wrong?
The observations of the previous section are enough to see that there is no direct connection, in either direction, between the two sorts of pluralism at issue—at least in a formal sense. But it is worth examining each of these three domains in a bit more detail to understand the independence just established a bit more deeply. Fleshing out the philosophical aspects of typical formulations of domain-specific pluralism (of both sorts) will help us in this task.
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Discussions of domain-specific alethic pluralism—and, in particular, the discussions of the connection between domain-specific alethic pluralism and domain-specific logical pluralism found in Lynch (2008, 2009) and Pedersen (2014)—typically focus on two kinds of truth: correspondence and superwarrant. I will assume that correspondence is familiar to the reader (or, at least, familiar enough for our purposes here), but superwarrant deserves a little bit more attention. A proposition being superwarranted is defined by Lynch as follows: The proposition that p is superwarranted just when it is warranted without defeat at some stage of inquiry and would remain so at every successive stage of inquiry. A stage of inquiry is a state of information: it is always open to extension, and potentially incomplete: At any particular stage of inquiry, we may have no warrant for a proposition and no warrant for its negation. A belief is warranted without defeat at a stage of inquiry as long as any defeater for the belief at a given stage is itself undermined by evidence available at a later stage. (Lynch 2008, 124–125)
We need not delve into these details too deeply, however, since for our purposes the crucial consequence of truth being superwarrant in a particular domain is that truth is epistemically constrained in that domain— that is, the following principle of epistemic constraint for truth holds for all Φ in the language appropriate for that domain: ECT: If T(‘Φ’) then it is feasible to have a warrant for Φ. (see Lynch (2008, 134))
In domains where truth is correspondence, however, epistemic constraint might fail to hold (but in another sense it might not—more on this anon!). Both Lynch (2008) and Pedersen (2014) argue that alethic pluralism entails logical pluralism (at least, if our alethic pluralism contains at least one domain where truth is correspondence and at least one domain where truth is superwarrant, see Lynch (2008, 133–135)) along the following lines12: 1. There is a domain D1 where truth is superwarrant, hence epistemically constrained.
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2. There is a domain D3 where truth is correspondence, hence not epistemically constrained. 3. Not every instance of excluded middle is true in D1. 4. Every instance of excluded middle is true in D3. (C) Hence, the logic of D1 is distinct from the logic of D3. Let’s call this the epistemic constraint argument. Before moving on to the real problem with the epistemic constraint argument, it is worth noting two smaller worries. First, epistemic constraint holding of a domain does not entail that excluded middle necessarily fails in that domain. After all, traditional intuitionists are logical monists—they do not think that excluded middle is logically true on any domain, even though there are many domains that are intuitionistically provably decidable and hence all instances of excluded middle are true (and, for mathematical domains, necessarily true) in that domain. Hence, there is some reason to doubt the quick move from (1) to (3). Along similar lines, the failure of excluded middle does not immediately entail that truth is not correspondence. Neil Tennant, for example, has developed an understanding of intuitionistic logic that seems compatible with the correspondence theory of truth. In particular, he argues that: We are maintaining that these instances of [excluded middle] are better thought of as being held to be (necessarily) true on a priori rather than on ‘logical’ grounds. Indeed, the holding true (as a matter of necessity) of every such instance [of excluded middle] … expresses an essentially metaphysical belief. This belief is to the effect that the world is determinate in every expressible regard. (Tennant 1996, 213)
If this is right, then truth could be correspondence, but the world might not be determinate enough (in the relevant sense) to guarantee that, for each sentence Φ, either Φ corresponds to the way the world is or Φ fails to correspond to the way the world is. As a result, the inference from
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truth being correspondence to all instances of excluded middle being true—that is, the inference from line (2) to line (4)—is also questionable. These worries being noted, let’s set them aside and, for the sake of argument, grant that: • If truth is correspondence in domain D, then all instances of excluded middle hold for sentences about D. • If truth is superwarrant in domain D, then some instances of excluded middle fail to hold for sentences about D. The key to seeing how these observations are compatible with the independence of domain-specific alethic pluralism and domain-specific logical pluralism lies in how, exactly, we understand “hold”. As the subscripts I use in the above reconstruction of the Lynch/ Pedersen argument indicate, the epistemic constraint argument presupposes that, with respect to correspondence and superwarrant, there are only two types of domain possible: • Those like D1, where the logic is H and (at least some) instances of excluded middle fail to hold. • Those like D3, where the logic is C and (all) instances of excluded middle hold. In short, the epistemic constraint argument assumes that we cannot have a domain like D2, where the logic is intuitionistic but all instances of excluded middle nevertheless hold (for sentences about D2—i.e. those sentences in LPA—although not necessarily for all sentences about the semantics of the language describing D2, i.e. not necessarily all sentences in LPA+T(x)). But why could there not be such a domain? What prevents there being a domain where instances of excluded middle are not logical truths, but are nevertheless guaranteed to be true in virtue of principles in our theory of truth? Michael Lynch attempts to rule out something very much like domains like D2. It is worth quoting the passage in question at some length:
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… why not say that [intuitionistic logic] is not only the default logic, it is the only logic, and that domains whose logic appears classical only do so because we are employing additional principles such as Bivalence which aren’t part of the one true logic. Alethic pluralism … is certainly consistent with this suggestion. But the suggestion comes at a price. According to the suggestion, [intuitionistic logic] holds in all domains. Bivalence is not recognized as a logical principle by [intuitionistic logic]. Therefore, in every domain, Bivalence is not recognized as a logical principle. Therefore in domains which, according to this suggestion, nonetheless appear classical – and therefore abide by bivalence – Bivalence must be true for some non-logical reason. And one might wonder what that reason might be. A full assessment of this suggestion, therefore, requires drawing the boundaries of logic, an issue well beyond the scope of the current essay. But one small point is worth making: it seems natural that if one domain allows some inferences as valid and another does not, they have different logics. And domains where Bivalence holds will allow some inferences as valid that other domains (which don’t sanction Bivalence) will not. So the natural thought is that they have different logics. (Lynch 2009, 103–104)
There is a lot that needs unpacking and clarifying here, and in particular, we are going to do some of the boundary-drawing with regard to logic that Lynch places outside the scope of his own project. As we shall see, this will allow us to easily identify the reason that Bivalence (or something like it) might be true of a domain without being a logical truth (with respect to that domain). Lynch’s argument against the truth-but-not-logical-truth of Bivalence can be reconstructed as follows13: 1 . Bivalence is a logical principle. 2. If a logical principle is true in domain D, then it is valid in D. (C) Hence, the logic of any domain where Bivalence is true is C. It turns out that both premises in this argument fail, however.
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First, Bivalence is not a logical principle in the relevant sense. The difference between classical logic and intuitionistic logic is not the validity of Bivalence, it is the validity of (all instances of ) excluded middle. Bivalence is a metalogical, semantic principle—one that utilizes the truth predicate—and hence Bivalence is not a logical truth (not even for the classical logician). I have argued at length for a similar claim about the Tarskian T-schema in Cook (2012)—the same argument applies here. To put this point bluntly, a logical principle is either (i) a sentence that contains only logical vocabulary, or (ii) a schema that contains only logical vocabulary and metavariables.14 The truth predicate T(x) is not a piece of logical vocabulary (nor, obviously, a metavariable)—hence Bivalence is not a logical principle (again, see Cook (2012) for more discussion). But perhaps the first premise in our reconstruction of Lynch’s argument, more carefully formulated, is that Bivalence entails a logical principle—namely, the logical schema known as excluded middle—and that if Bivalence holds of a domain D, then excluded middle is true in D.15 Hence, we could reformulate the argument as: 1 . Bivalence entails the truth of a logical principle: excluded middle. 2. If a logical principle is true in domain D, then it is valid in D. (C) Hence, the logic of any domain where Bivalence is true is C. This still won’t do, however. The reason is simple: Even if we reformulate (1) in this way, premise (2) is still false. We should remember that what is at issue here is logical validity—not merely truth (or even necessary truth). Tarski provided us with a useful first approximation of what we mean by logical consequence (and hence of logical validity as the special empty-premise case of logical consequence): Consider any class ∆ of sentences and a sentence Φ which follows from the sentences of this class. From an intuitive standpoint it can never happen that both the class ∆ consists only of true sentences and the sentence Φ is false. Moreover, since we are concerned here with the concept of logical, i.e. formal, consequence, and thus with a relation which is to be uniquely determined by the form of the sentences between which it holds … the
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consequence relation cannot be affected by replacing the designations of the objects referred to in these sentences by the designations of any other objects. (Tarski 1936, 414–415)
According to Tarski, logical consequence is a relation that holds between statements (and collections of statements) of our actual language and involves two crucial ideas: A (natural language) statement Φ is a logical consequence of a set of (natural language) statements ∆ if and only if: • Necessity: The simultaneous truth of every member of ∆ guarantees the truth of Φ. • Formality: This guarantee follows solely from the logical form of Φ and of the members of ∆.
An immediate corollary is that a principle Φ (e.g. excluded middle) is a logical truth if and only if (i) Φ is necessarily true, and (ii) its (necessary) truth is somehow guaranteed by its logical form. But there are plenty of cases where logical principles are true—even necessarily true—in some domain without being logically true. We have already mentioned one such case: All instances of excluded middle are true in intuitionistic domains that are decidable, but this in no way entails that the logically monist intuitionist accepts that excluded middle is logically true in such domains. Another example is provided by anyone who accepts a platonist account of standard mathematics and also accepts that the linguistic resources of standard second-order logic are, properly speaking, logical resources (not everyone accepts this, of course, e.g. Quine (1970)). I myself am such a person (as are many others—e.g. Shapiro (1991, Shapiro 1997)). Any such view will entail the truth—in fact, the necessary truth—of the following logical principle:
( ∃f ) ( ∀x ) ( ∃y ) ( f ( x ) = f ( y ) → x = y ) ∧ ( ∃x ) ( ∀y ) ( ¬f ( y ) = x )
expressing the fact that the mathematical universe is Dedekind infinite. But the vast majority of people holding such views will reject that this claim is logically true.16
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Now, why does the intuitionist deny the logical truth of excluded middle even in cases when the domain in question is decidable? Why does the (second-order) platonist deny the logical truth of the claim that the universe is Dedekind infinite? Tarski has already given us the answer: In each case, the principle in question is true, and even necessarily true, but the principle is not guaranteed to be true merely in virtue of its form. On the contrary, in the first case we need to carry out a proof of the decidability of the theory of the domain in question in order to see that all instances of excluded middle are true, and in the latter case we need to argue that platonism is, in fact, true (and, in addition, that at least one true mathematical theory has an infinite domain). Hence, anyone who wants to accept something along the lines of domain D2 needs to only argue that Bivalence is true of the domain in question, but is not true (on that domain) in virtue of its logical form. Perhaps it is true because we have a (non-logical) warrant for the domain being decidable, or perhaps it is true because we have some sort of metaphysical guarantee that the domain in question is determinate in certain respects (along the lines of (Tennant 1996)). Or perhaps we have some other sort of non-logical guarantee that Bivalence, and hence all instances of excluded middle, is true of the domain. None of these would in any way imply that excluded middle is a logical truth, and none of these are particularly implausible. Hence, the epistemic constraint argument fails, and we do indeed have a plurality of pluralisms.
4
Pluralism and Epistemic Constraint
In the previous two sections, we formulated an argument that domain- specific alethic pluralism and domain-specific logical pluralism are independent, and critiqued an argument due to Lynch and Pedersen for the opposite conclusion. If all we cared about was whether the two kinds of pluralism were independent, then we could stop here. But presumably the reason we are interested in the connections between domain-specific alethic pluralism and domain-specific logical pluralism is that we find one or the other of these views interesting.17 And the three toy domains constructed in Sect. 2 and used in the argument for independence (and
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the debunking of the epistemic constraint argument) also illustrate some interesting, and as far as I can tell novel, observations about the role that epistemic constraint can and should play in accounts of domain-specific pluralism of either sort. As we have already noted, the alethic pluralist believes that truth comes in at least two varieties—correspondence and superwarrant—and argues that epistemic constraint for truth: ECT: If T(‘Φ’) then it is feasible to have a warrant for Φ.
holds of domains where truth is superwarrant, but not necessarily for domains where truth is correspondence (and we agreed, for the sake of argument and Tennant (1996) notwithstanding, to accept that epistemic constraint does not hold in domains where truth is correspondence). Much of the point of the previous section was to carefully drive a wedge between logical principles that are merely true, or even necessarily true, and those that are logically true (i.e. necessarily true in virtue of logical form). Given this, it makes sense to ask whether logical truth is similarly epistemically constrained in particular discourses. In other words, we can also ask whether the following principle—we shall call epistemic constraint for logical truth—holds in a particular discourse: ECLT: If Φ is a logical truth, then it is feasible to have a warrant for Φ.
The first thing to notice is that ECT entails ECLT: If Φ is a logical truth, then it is a truth, and so by ECT it is feasible to have a warrant for Φ. But there seems to be no reason to think that ECLT entails ECT. Let’s assume that, if ECT holds, then the right logic for the domain in question is H. The reason is simple and follows the (correct core of ) reasoning found in Lynch (2008, 2009) and Pedersen (2014): If logical truth is epistemically constrained in a domain D, then it would seem that there is no guarantee that we can know, of every sentence Φ about D, on purely logical grounds, that either Φ is true or ¬Φ is true. But, of course, as discussed in the previous section, there might be other non-logical grounds on the basis of which we can conclude that Bivalence holds, and hence (if our theory of truth is powerful enough otherwise, as CT↾+ is)
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that all instances of excluded middle (for sentences about the domain, at least—although not necessarily for sentences involving the truth predicate T(x)) are true. This leaves three remaining combinations, corresponding to each of the domains constructed in Sect. 218: 1. Both ECLT and ECT hold of D. Then the logic of D is H, and the theory of truth should not include BivLD. Hence, the domain is (in relevant respects) like D1. 2. ECLT holds but ECT does not hold of D. Then the logic of D is H, and the theory of truth should include BivLD. Hence, the domain is (in relevant respects) like D2. 3. Neither ECLT nor ECT holds of D. Then the logic of D is C, and the theory of truth should include BivLD. Hence, the domain is (in relevant respects) like D3. Thus, once we understand that logical truth being epistemically constrained does not imply that truth is epistemically constrained (although the converse implication does hold), we can see that the notion of epistemic constraint, generally speaking, divides up domains into three sorts, not two, and this in turn explains how domain-specific alethic pluralism might hold while domain-specific logical pluralism might fail, or vice versa.
5
Concluding Remarks
In the previous section, we have seen that, in principle, at least, domain- specific alethic pluralism is independent of domain-specific logical pluralism, and we also noted that the one simple way to understand this phenomenon is by noting that logical truth being epistemically constrained does not entail that truth is likewise epistemically constrained. In short, with respect to the domain-specific versions of the two types of pluralism at issue, we do have a pluralism about pluralisms. We’ll conclude the chapter by examining whether similar stories might be told
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about various other combinations of domain-specific pluralism, domain- independent pluralism, and monism. Since we can choose any of these three options for each of logic and truth, this gives us nine distinct possible combinations: DS Logical Plur. DI Logical Plur. Logical Mon.
DS Alethic Plur.
DI Alethic Plur.
Alethic Mon.
1 4 7
2 5 8
3 6 9
We have already shown that domain-specific logical pluralism is compatible with alethic monism (option 3), and domain-specific alethic pluralism is compatible with logical monism (option 7). In addition, although the arguments of Lynch (2008, 2009) and Pedersen (2014) fail to show that domain-specific alethic pluralism covaries with domainspecific logical pluralism, the discussion found there makes a strong case for their compatibility. More simply, we can just consider a view that accepts all of D1, D2, and D3 (or relevantly similar domains) as legitimate (option 1). And no one doubts that the combination of alethic monism and logical monism (option 9) is coherent, even if many readers of this chapter (and this volume) will no doubt think such a view is likely incorrect. In short, we’ve covered the corners of this chart, but have addressed none of the options in the “+”-shaped region remaining. It appears, however, that if we restrict our attention to the two kinds of truth (epistemically constrained or not) and the two kinds of logic (epistemically constrained or not) that have concerned us thus far, then only two of the remaining options are coherent. If one is a domain-independent logical pluralist, then this implies that there are at least two logics that apply everywhere. Assume that one is epistemically constrained and the other not. Then this means that, according to at least one of these everywhere applicable logics, logical truth is not epistemically constrained (everywhere). But this means truth must (everywhere) fail to be epistemically constrained (since truth being epistemically constrained implies logical truth being epistemically constrained). And if truth must fail to be epistemically constrained, we must be alethic monists. This rules out options (4) and (5), but leaves (6) intact.
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Along similar lines, if one is a domain-independent alethic pluralist, then there are two accounts of truth that apply everywhere. Assume one is epistemically constrained and the other not. Then, according to at least one of these everywhere applicable accounts of truth, truth is epistemically constrained (everywhere). But this means that logic must (everywhere) be epistemically constrained (since, again, truth being epistemically constrained implies logical truth being epistemically constrained). And if logic must be epistemically constrained, we must be logical monists. This rules out options (2) and (5) (again), but leaves (8) intact. I do not, in fact, think these brief arguments actually show that domain-independent logical pluralism entails truth monism, or that domain-independent alethic pluralism entails logical monism.19 Rather, what I think that these brief, somewhat un-careful arguments show is that, if we are to develop a view that combines a domain-specific pluralism of one sort with a domain-independent pluralism of the other sort (i.e. option 2 or 4), or one that combines domain-independent pluralism of both sorts (i.e. option 5), then defending that combination as a coherent position on the 3 × 3 grid of possible positions above will require additional theoretical tools not mobilized here. For example, if we are domain-independent alethic pluralists, so truth both is epistemically constrained and is not epistemically constrained (depending on which notion we use), perhaps there is a way to keep the former notion from “bleeding” into logical truth, entailing that logical truth must be epistemically constrained everywhere (for any legitimate logic). I am optimistic that such theoretical tools can be found, and that accounts of logic, truth, and their interrelations corresponding to options (2), (4), and (5) can (and likely will be) carefully formulated and defended as coherent. But for the present I will be content to have shown that, contrary to the suggestion found in Lynch (2008, 2009) and Pedersen (2014), options (3) and (7) are coherent, leaving examination of the other options for another day.20
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Notes 1. Further, other sorts of pluralisms—such as metaphysical pluralisms— also fall on one or the other side of the domain-specific/domain-independent divide. 2. See Edwards (2012) and Pedersen and Wright (2013a, b) for good surveys of alethic pluralism and Cook (2010) and Russell (2016) for good surveys of logical pluralism. 3. On the contrary, I am rather unsympathetic to domain-specific logical pluralism, and rather sympathetic to domain-independent logical pluralism, despite the good efforts of (Shapiro 2015). And I am wholly unsympathetic to either sort of alethic pluralism. For some of that story, see (Cook 2014). 4. It is important to note that I am not arguing that we have a plurality of true pluralisms (see the previous footnote). The point is that the accounts in question—domain-specific alethic pluralism and domain-specific logical pluralism—are distinct positions, regardless of whether they are correct or not. 5. I am, in what follows, going to concentrate on the argument as given in Lynch (2008, 2009). Pedersen (2014) characterizes itself as presenting a slightly more detailed and cleaned up version of Lynch’s argument (and extending the conclusions to metaphysical pluralism, which is interesting but orthogonal to our concerns here). I would like to note that Pedersen’s essay was extremely helpful in sorting out how, exactly, the argument in question is supposed to work. 6. We need both quantifiers and all four propositional connectives in LPA+T(x) if we are to allow the logic in question to be full intuitionistic logic, since the familiar equivalences that allow for various (classical) definitions of one operator in terms of another fail intuitionistically. 7. For an argument that the truth predicate is non-logical, see (Cook 2012). We will return to discuss the relevance of this fact in detail below. 8. A brief explanation of the notation in these axioms: • T0(x) is the arithmetically definable truth predicate for ∅0 sentences • AtomPA(x) is the (purely arithmetic) predicate expressing “x is the code 0
of an atomic sentence of PA”.
• SentPA(x) is the (purely arithmetic) predicate expressing “x is the code
of a sentence of PA”.
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• FormPA(x) is the (purely arithmetic) predicate expressing “x is the code
of a formula of PA with exactly one free variable”. • ℜ is the recursive function that maps the code of a formula to the code ). , ∨ , and → of its negation (and similarly for ∧ • ∀v is the recursive function that maps the code of a formula with one free variable to the code of the universal quantification of that formula (and similarly for ∃v ). • sub(x, y) is the recursive function that maps a number v and the code of a formula with one free variable to the code of the formula obtained by replacing the free variable with the numeral for that number. Note that our presentation of CT↾ contains more axioms than in many standard presentations, since we are not treating the conditional as defined in terms of negation and disjunction. 9. Nothing hinges on which particular theory of truth we chose, so long as combining that theory with Heyting Arithmetic HA (i.e. the axioms of PA plus the logic H) and combining that theory with classical Peano Arithmetic (i.e. the axioms of PA plus classical logic C) give extensionally distinct theories. The simplicity (and conservativeness) of CT↾ makes it a convenient choice, however. 10. Thanks are owed to an anonymous referee for pressing this point. 11. Further, the discussion of the next two sections serves to provide an explanation of the formal construction just given in terms of philosophical notions mobilized in Lynch’s account—in particular, in terms of the role played by epistemic constraint. 12. We shall see the reason for the odd numbering of the domains shortly. 13. This reconstruction involves a simplification—one harmless in the present context—namely that whatever logics we are taking to be candidate logics for our domains, if we supplement any of those logics with the law of excluded middle, we obtain C. This is, of course, not true for many logics (i.e. substructural logics), but it is true of H. 14. Of course, non-logical vocabulary can be substituted for the metavariables in instances of the latter sort. 15. In fact, this won’t work simply because the version of Bivalence we added to CT↾—that is, BivPA—does not entail excluded middle for all sentences in LPA+T(x). Let’s set aside this technical observation, however, since we have more important, more philosophical fish to fry.
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16. A slightly snarkier way of making the point: If the truth, or even the necessary truth, of a logical principle entailed its logical truth, and if second-order logic is logic, then any platonist would presumably also be some sort of logicist. 17. I certainly find them interesting, even if, as I noted in endnote 3, I ultimately think neither of them is correct. 18. Here LD is the language about domain D—that is, all sentences about D not including the truth predicate T(x), hence BivLD is Bivalence restricted to the sentences in LD. 19. This, despite the fact that I think that the combination of domain- independent logical pluralism and alethic monism (i.e. option 6) is the right option. The point is that I don’t think the other options are incoherent—I just think that they are wrong. 20. Thanks are owed to Nikolaj J. L. L. Pedersen, Nathan Kellen, and an anonymous referee for helpful comments on early versions of this material.
References Beall, J., and G. Restall. 2000. Logical Pluralism. Australasian Journal of Philosophy 78: 475–493. ———. 2001. Defending Logical Pluralism. In Logical Consequence: Rival Approaches. Proceedings of the 1999 Conference of the Society of Exact Philosophy, ed. J. Woods and B. Brown, 1–22. Stanmore: Hermes. ———. 2006. Logical Pluralism. Oxford: Oxford University Press. Cook, R. 2010. Let a Thousand Flowers Bloom: A Tour of Logical Pluralism. Philosophy Compass 5 (6): 492–504. ———. 2012. The T-schema Is Not a Logical Truth. Analysis 72 (2): 231–239. ———. 2014. Should Anti-realists Be Anti-Realists About Anti-realism? Erkenntnis 79: 233–258. Edwards, D. 2012. Pluralist Theories of Truth. In Internet Encyclopedia of Philosophy, ed. J. Fieser and B. Dowden. http://www.iep.utm.edu/plur-tru/ Lynch, M. 2008. Alethic Pluralism, Logical Consequence, and the Universality of Reason. Midwest Studies in Philosophy 32: 122–140. ———. 2009. Truth as One and Many. Oxford: Oxford University Press. Pedersen, Nikolaj J.L.L. 2014. Pluralism × 3: Truth, Logic, Metaphysics. Erkenntnis 79: 259–277.
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Pedersen, Nikolaj J.L.L., and C.D. Wright. 2013a. Pluralist Theories of Truth. In The Stanford Encyclopedia of Philosophy, ed. E.N. Zalta. https://plato.stanford.edu/entries/truth-pluralist/. ———. 2013b. Truth and Pluralism: Current Debates. Oxford: Oxford University Press. Quine, W. 1970. Philosophy of Logic. Cambridge, MA: Harvard University Press. Russell, G. 2016. Logical Pluralism. In The Stanford Encyclopedia of Philosophy (Winter 2016 edition), ed. E.N. Zalta. https://plato.stanford.edu/archives/ win2016/entries/logical-pluralism/. Shapiro, S. 1991. Foundations Without Foundationalism: The Case for Second- Order Logic. Oxford: Oxford University Press. ———. 1997. Philosophy of Mathematics: Structure and Ontology. Oxford: Oxford University Press. ———. 2015. Varieties of Logic. Oxford: Oxford University Press. Sher, G. 2013. Forms of Correspondence: The Intricate Route from Thought to Reality. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 157–179. Oxford: Oxford University Press. Tarski, A. 1936. On the Concept of Logical Consequence. In Logic, Semantics, Metamathematics, ed. A. Tarski, 2nd ed., 409–420. Indianapolis: Hackett. ———. 1983. Logic, Semantics, Metamathematics. Ed. J. Corcoran, 2nd ed. Indianapolis: Hackett. Tennant, N. 1996. The Law of Excluded Middle Is Synthetic A Priori, If Valid. Philosophical Topics 24 (1): 205–229. Woods, J., and B. Brown, eds. 2001. Logical Consequence: Rival Approaches. Proceedings of the 1999 Conference of the Society of Exact Philosophy, Hermes Stanmore. Wright, C. 1992. Truth and Objectivity. Cambridge, MA: Harvard University Press.
A Plea for Immodesty: Alethic Pluralism, Logical Pluralism, and Mixed Inferences Chase B. Wrenn
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Introduction
There is a natural, if inchoate, intuition that truth might have a variable nature, depending on what the subject matter is. Given the great differences between natural science, mathematics, ethics, and aesthetics, does it not stand to reason that ‘Electrons have negative charge,’ ‘1 + 1 = 2,’ ‘Torture is wrong,’ and ‘Duck Soup is funny’ are all true, but the truth of each is a different property from the truth of the others? The arch-deflationist W. V. Quine had the intuition,1 but it was Crispin Wright (1992) who brought the prospects of pluralism about truth into contemporary discussion. These days, a variety of different ways of working out the details of a pluralist conception of truth are on offer. By alethic pluralism (AP), I mean the view that there are different, but substantial, truth-properties in different discourses.
C. B. Wrenn (*) Department of Philosophy, The University of Alabama, Tuscaloosa, AL, USA e-mail: [email protected] © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_16
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There is also a natural, if inchoate, intuition that logic is variable, depending on what you are talking about. As alternative logics have proliferated, often motivated by the concerns of specific domains of inquiry, one might guess that the logic appropriate for some inquiries just isn’t the same as the logic appropriate for others. The distinctive feature of a logic is the relation of logical consequence it describes. Equivalently, logics can be distinguished in terms of which inferences qualify as valid in them—provided we understand ‘inferences’ not as psychological events but as sets of premises and conclusions. By logical pluralism, I mean the view that there are multiple, legitimate relations of logical consequence. Domain-specific logical pluralism (DLP) is the view that different domains of inquiry are governed by different relations of logical consequence. Both AP and DLP face the problem of mixed inferences. This problem arises when we try to explain the validity of inferences whose premises and conclusion are not uniformly drawn from the same domain of inquiry. Some versions of alethic pluralism, such as Michael Lynch’s manifestation functionalism, are specifically designed with this problem in mind. However, there is a natural route from AP to DLP and, as Lynch is aware, the addition of DLP to AP threatens to reintroduce the problem of mixed inferences, even for manifestation functionalism. Nevertheless, Lynch contends, the threat can be avoided through modesty: when there are multiple candidate logics on the table, we should be cautious and employ only the weakest one of them or, if none is weakest, we should endorse as valid only those inferences validated by all candidates (Lynch 2008, 2009, Ch. 5). The modest approach, I will argue, is unsatisfactory on two counts. (1) It is bound to invalidate certain intuitively valid inferences. (2) It threatens to collapse into a view I call virtual logical pluralism, which isn’t really a form of logical pluralism at all. On the other hand, there appears to be no good alternative to the modest approach for the advocate of AP + DLP. If that is so, then it seems alethic pluralists should steer clear of domain-specific logical pluralism—or vice versa.
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Alethic Pluralism
The distinctive claim of alethic pluralism is that truth is (in some sense of ‘is’) a different property for claims in different discourses or domains of inquiry—but truth is also more substantial than deflationists claim it to be. Pluralists differ in their understanding of the ‘is’ in their distinctive claim. Some take it to be a matter of identity: the truth predicate designates different properties when applied to claims from different discourses (Wright 1992, 2013). Some take it to be a matter of realization or manifestation: the property playing the role of truth in one discourse need not be the same property as what plays the truth-role in another (Lynch 2001, 2009). Not only can alethic pluralists differ on what it means to be truth for a discourse, but they can differ on which properties are truth in which discourses. One pluralist might think correspondence is truth in mathematics, while another thinks mathematical truth is something epistemic, for example. A main motivation for pluralism is the idea that realism seems appropriate for some discourses (such as physical science) but not others (such as aesthetics). Let us call a property a truth-candidate if it is the sort of property pluralists think might be truth for some discourse or other. Two prominent truth-candidates are robust correspondence and superwarrant. A claim robustly corresponds to the world when it represents the world as being a certain way, the world is that way, and the world’s being that way is independent of our actual or possible possession of evidence that that is how it is. (Henceforth, I will omit ‘robust’ and refer to this as ‘correspondence.’) The notion of superwarrant comes from Wright’s notion of superassertability (1992). A claim is superwarranted when there is a stage of inquiry in which it is warranted without defeat and it would remain that way in all successive stages of inquiry (Lynch 2009, p. 38; see also Pedersen 2014). Plausibly, something like correspondence is truth in an area such as physics, chemistry, or everyday discourse about middle-sized dry goods and their determinate properties. Superwarrant is a plausible contender in areas where truth seems evidentially constrained. For example, no joke
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could be secretly funny—funny despite the fact that no one could know. Truth and knowability walk together in discourse about what is funny, and one might suppose the truth about humor must coincide with whatever our most considered, unimprovable judgments would say about it.
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Mixed Inferences and Logical Pluralism
The following argument illustrates the problem of mixed inferences (Tappolet 1997):
Mix ( 1) The one about the nun at the dude ranch is funny. (2) Either snow is white or the one about the nun at the dude ranch isn’t funny. (3) Therefore, snow is white. Mix’s conclusion and its first premise come from difference discourses. Suppose truth is something epistemic, such as superwarrant, in discourse about what is funny, and it is something realist, such as correspondence, in discourse about the color of snow. Then the argument appears valid, but it also appears not to transmit truth from its premises to its conclusion, given pluralism. Truth for (1) is superwarrant, but truth for (3) is not. So, alethic pluralists have to find a way to explain the validity of such inferences. An especially promising approach is functionalism about truth (Lynch 2001, 2008, 2009). In Lynch’s version, a number of ‘truisms’ serve to define the role or job description of truth. They include, among others: Objectivity: The belief that p is true if, and only if, with respect to the belief that p, things are as they are believed to be. and End of Inquiry: Other things being equal, true beliefs are a worthy goal of inquiry.
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Whatever property best satisfies the truisms for a given discourse’s atomic claims is said to ‘manifest’ truth for those claims.2 Correspondence might manifest truth in some discourses, while superwarrant manifests it in others. Truth itself is the property that essentially satisfies the truisms. In a discourse where correspondence satisfies the truisms, propositions have the property of truth by having the property of correspondence, not very differently from the way that objects can have the property of redness by having the property of being some particular shade of red.3 Every true proposition has the property of truth, and the property of truth plays the truth-role essentially. For mixed compounds such as (2), and in mixed inferences such as Mix, there may be no property other than truth itself that manifests truth. Premise (2) is the sort of compound that is true whenever it has a true component. The fact that different properties manifest truth for each of its parts doesn’t change the fact that, if either component has a truth-manifesting property, then the compound is true. The truth of the component, however it might be grounded, grounds and determines the truth of (2). Likewise, even if different properties manifest truth for (1), (2), and (3), the inference from (1) and (2) to (3) has the feature crucial to deductive validity: If (1) and (2) each have the property of truth, then (3) must also have the property of truth. That is how the functionalist tries to solve the problem of mixed inferences. Just as alethic pluralists think there are multiple truth-properties, logical pluralists believe there are multiple relations of logical consequence. Nevertheless, any relation of logical consequence will have these core features (Beall and Restall 2006): GTT: An argument is valid iff every casex in which its premises are true is a casex in which its conclusion is true.4 NNF: The logical consequence relation is normative, necessary, and formal. The ‘x’ subscript in GTT indicates a parameter for different logical systems. Each system brings with it a different conception of which ‘cases’ are relevant. Different systems count different arguments as valid and deliver different logical consequence relations.
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As Beall and Restall’s discussion illustrates, it can be difficult to state NNF’s notions of normativity, necessity, and formality precisely. The general ideas, though, are clear enough. Logical consequence is ‘normative’ in the sense that it is somehow wrong to accept the premises of a valid argument while rejecting its conclusion (ibid., p. 16). It is ‘necessary’ in the sense that valid reasoning tells us not only that if a set of premises are true, then so is a conclusion, but also that if those premises were true, then the conclusion would be true as well (ibid., pp. 15–6). Following MacFarlane (2000), Beall and Restall identify three different, and non-equivalent, ways in which we might plausibly clarify the idea that logic is ‘formal’ (ibid., pp. 18–23). What will matter for my purposes, though, is the idea that logic abstracts away from the particular semantic content of thought, though of course there remains room to debate just where to draw the line between logical form and semantic content. Despite the possibility of such debate, I will make some assumptions about which features of a claim are ‘formal’ and which involve its ‘semantic content.’ According to pluralists, what discourse a claim belongs to, and so what truth-candidate is relevant to the claim, depends on what the claim is about. For example, here is Lynch: What makes a proposition a member of a particular domain? The obvious answer: the subject matter it is about … That said, it would be nice to say at least something more about what, at least in general, distinguishes propositions of one domain from another. A natural answer is this: the kind of concepts (moral, legal, mathematical) that compose the proposition in question. (2009, pp. 79–80)
I will assume that the question of what domain a claim belongs to is a question about its semantic content, not its logical form. For a relation between premises and a conclusion to be formal, it must be independent of the actual content of those claims. Two arguments alike in logical form will be alike in terms of whether that relation holds between their premises and their conclusion. Both classical and intuitionistic entailment are formal in this way.
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Logical pluralism is just the view that there is more than one genuine relation of logical consequence. Domain-specific logical pluralism (DLP) holds that different domains of inquiry or discourses have different logics. The logical consequence relation in ethics, for example, might not be the same as the logical consequence relation in physics. It might be intuitionistic entailment in the former case, classical entailment in the latter. Such a view could be attractive to alethic pluralists on purely intuitive grounds: if logic tells us about the preservation of truth in inference, and different discourses have different truth-properties, why should we expect different discourses to have the same logic? Lynch describes a more direct path from AP to DLP, provided we assume that truth is sometimes non-epistemic and sometimes superwarrant (2008, pp. 134–5; 2009, pp. 94–6; see also Pedersen 2014). In domains where truth is non-epistemic, we might expect the classical logical laws to hold, and so logical consequence in those domains would be classical. For example, the classical Law of Excluded Middle (LEM) holds for those discourses. But in domains where superwarrant is truth, logical consequence amounts to the following (given that ‘cases’ amount to possible worlds): SWLC: P is a logical consequence of Q iff every possible world in which Q is superwarranted is a possible world in which P is superwarranted. There is, however, no general guarantee that either a proposition or its negation must be superwarranted. So, LEM does not hold for discourses in which superwarrant is truth, and SWLC’s relation of logical consequence is not the same as classical logical consequence. Though DLP may be tempting to alethic pluralists, it also threatens to reintroduce the problem of mixed inferences. Functionalism aimed to explain the validity of Mix in a way that didn’t require assigning it to a discourse. But given DLP, there are multiple relations of logical consequence that vary from discourse to discourse. So, we need to know which relation of logical consequence is relevant to evaluating a mixed inference such as Mix. Its premises and conclusion come from different discourses, with different, proprietary relations of logical consequence. DLP does not provide a discourse-neutral relation of logical consequence, and so we
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seem not to have a relation of logical consequence appropriate for the mixed inference itself. And with no such relation on the table, we have no way of accounting for the inference’s validity. What is to be done?
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Modesty
One might be tempted to try for DLP the same functionalist move Lynch uses to resolve AP’s problem of mixed inferences. In that case, one would consider ‘logical consequence’ to be a higher-order relation. One proposition is a logical consequence of another just in case they bear a relation to one another that plays the role of (or manifests) logical consequence for them. Such a move is unhelpful in this case. We need to know what relation plays the role of logical consequence for Mix. It is hard to see how an answer to that question could be given while respecting the formality requirement. There are two, related worries here. First, the functionalizing move does not shed much light on the question of whether (3) is, in fact, a logical consequence of (1) and (2). On the functionalizing move, it is a consequence if it bears the relation to them that plays the role of logical consequence appropriate for the content of the argument. But that is merely to avoid the real question: What logical consequence relation is appropriate for the content of that inference? The second worry concerns the formality of the higher-order relation of consequence. Classical and intuitionistic consequence are formal; what P and Q are about is irrelevant to whether P is a classical consequence of Q and to whether it is an intuitionistic consequence of Q. However, whether P and Q are related by a relation that plays the role of logical consequence for them does depend on their content. If P has a form that would make it a classical consequence of Q, but not an intuitionistic consequence of Q, then the question whether P bears a relation to Q that plays the role of logical consequence for them is one whose answer depends on what P and Q are about. Two inferences could be identical in terms of their logical forms, while one of them has a conclusion that bears the functionalized consequence relation to its premises and the other
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does not. So, the functionalized relation fails the formality condition on being a relation of logical consequence. Rather than functionalizing logical consequence, Lynch’s own proposal is to be as modest as possible: “[W]hen reasoning across domains, logical caution is in order: We want to limit the number of logical truths that we endorse, so as to respect those domains which, by virtue of the property that plays the truth-role within them, enforce less [sic] logical laws than others” (2008, p. 138; 2009, p. 101). He codifies that idea in the following way: MODEST*: Where a compound proposition or inference contains propositions from distinct domains, the default governing logic is that comprised by the intersection of the domain-specific logics in play. (Lynch 2008, p. 139; 2009, p. 102)5
We can identify a logic with the inferences it validates. A logic is ‘in play’ for an inference when it governs the domain of at least one of the inference’s components (i.e., premises, conclusion, or a component of a premise or conclusion). The intersection of the logics in play would be a logic that validates all and only those inferences validated by all the logics in play. A mixed inference, then, qualifies as valid if and only if all the logics in play would count it as valid. That, we should note, is enough to validate Mix, assuming Disjunctive Syllogism and Double Negation Introduction are valid in all the logics in play for it. It also avoids the formality problem, because each logic’s proprietary relation of logical consequence can be formal, while the question of which one is relevant to a given argument will depend on the argument’s content. Since DLP itself already says that the governing logic of an unmixed inference depends on its content, MODEST* doesn’t violate the formality condition in any way that an advocate of DLP should find troubling. This way of tackling the problem is certainly modest and cautious. But, I think, it is too modest and cautious. Its zeal not to validate any invalid inferences leads it to invalidate certain intuitively valid inferences. It also risks collapsing DLP into what I will call virtual logical pluralism.
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MODEST* Is Too Modest
There are two kinds of valid mixed inferences that the modest approach invalidates. For illustrative purposes, let us suppose ‘Jack went up the hill’ is governed by classical logic and correspondence truth, and let us suppose that ‘1 + 1 = 2’ is from a domain governed by intuitionistic logic, in which Double Negation Elimination (DNE) is not a valid form of inference. Consider this argument:
Inessential Mix ( 4) 1 + 1 = 2 and it’s not the case that Jack did not go up the hill. (5) Therefore, Jack went up the hill. Getting from (4) to (5) requires an application of DNE, and so the argument is not intuitionistically valid. Since intuitionistic logic governs one of (4)’s conjuncts, MODEST* tells us the inference is valid if and only if it is both intuitionistically and classically valid. So, we have to count it as invalid, which seems wrong. I call this a case of ‘inessential mixing,’ since the proposition that 1 + 1 = 2 appears to play no essential role in the inference, apart from compelling a logic too weak to validate it. The recipe for inessential mixing is just this: Take an inference that is valid in a given logic L, and then add a (non-contradictory) proposition from a domain with a logic that does not validate that inference. The proposition can be added as a new premise, or it can be conjoined with one of the old premises. Either way, the addition of the new proposition requires, according to MODEST*, a weakening of the logic so that the inference is no longer valid. There are also cases of ‘essential mixing.’ Here is one:
Essential Mix ( 6) Either it’s not the case that Jack did not go up the hill or 1 + 1 ≠ 2. (7) 1 + 1 = 2. (8) Therefore, Jack went up the hill.
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In this case, we again have an inference that is not intuitionistically valid, but now the intuitionistic proposition that 1 + 1 = 2 is essential to the argument. By MODEST*’s lights, the argument is invalid, and that seems wrong too. The recipe for essential mixing is also straightforward. Start with an inference that is valid in a given logic, L1, whose components all come from a domain governed by L1, and that has an essential premise that is not a component of the conclusion. Uniformly replace all occurrences of that premise in the argument (including as a component of another claim) with a (non-contradictory) proposition from a discourse governed by a logic, L2, that does not validate the inference. This alteration, by MODEST*, requires the new inference to be governed by L2 rather than L1; it makes the new inference invalid. But as Essential Mix illustrates, such inferences can be valid after all. Lynch’s MODEST* proposal is too modest to validate all the mixed inferences that, intuitively, we should want validated. Indeed, it invalidates even Mix, on the assumption that propositions about what is funny are governed by a logic that does not validate Disjunctive Syllogism.6
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Virtual Logical Pluralism
A second problem for MODEST* is one Lynch himself anticipates. If we are retreating to the weakest logic in play anyway, why not give up on logical pluralism altogether and hold that the One True Logic is something extremely weak, such as Neil Tennant’s ‘core logic’ (1987, 2012a, b), while “domains whose logic appears to be classical only do so because we are employing the classical portion” of a weaker-than-classical logic with classical logic as a proper part (Lynch 2008, p. 139)? Lynch’s reply is that, if we made such a move, we would be adopting a One True Logic in which classical principles such as Bivalence do not hold. As logical principles, they are incorrect. So, in domains where Bivalence does hold, it must hold “for some non-logical reason. And one might wonder what that reason might be” (Lynch 2008, p. 139). Virtual logical pluralism (VLP) is the view that there is only one relation of logical consequence, but some discourses have features that make
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them appear to be governed by stronger logics, as special cases of the One True Logic. Here are some examples to illustrate how VLP might work: Semantic Spandrels: The overall logic of a language containing a transparent truth predicate is paraconsistent, but such that the segment of the language lacking the predicate behaves perfectly classically. Here, the lack of a transparent truth predicate makes non-semantic discourse appear classical, as a special case of the non-classical One True Logic. (Beall 2009) Quasi-Quinean Intuitionism: Quine was no intuitionist, but suppose the One True Logic is intuitionistic, while Bivalence and the Law of Excluded Middle have proven indispensable to the practice of science. Then scientific discourse can be seen as including them as axiom schemata, or seen as presupposing their instances. The classical logic of science is a special case of the intuitionistic One True Logic, justified on the grounds of its indispensability.
For present purposes, it isn’t necessary for us to evaluate VLP’s merits and demerits, but only to note that endorsing VLP involves giving up DLP. MODEST* invites a retreat to VLP from DLP, in the face of the problem of mixed inferences. Is Lynch’s reply sufficient to block that retreat? It appears not to be. First, as Semantic Spandrels indicates, there can be logical reasons for the Law of Non-Contradiction to hold in some discourses but not others, even when the One True Logic is paraconsistent. In this case, the logical reason is precisely what Beall (2009) goes to such great pains to show: truth-value gluts emerge in semantic discourse only because that discourse includes a logical device, ‘transparent truth,’ characterized by the intersubstitutability of Tp and p in all nonopaque contexts. The Law of Non-Contradiction holds in non-semantic discourses, not for a substantive, non-logical reason, but rather for the logical reason that their languages don’t include that logical device.7 So, Lynch’s claim that we would need non-logical reasons for classical logical principles to hold in some discourses rather than others appears to be incorrect.
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Second, Quasi-Quinean Intuitionism reminds us that a generally Quinean outlook toward logic is always available. If we want to explain why classical principles hold in a given area of inquiry, the answer could be that we have found those principles to be so fruitful in that area that the principles have become central theoretical commitments. From the Quinean perspective, logical principles are distinctive only in their centrality to our web of belief. If the One True Logic isn’t classical, there is no in-principle reason our web couldn’t still have pockets of classicism. The classical principles in those areas would have the same status as central and very-unlikely-to-be-revised structuring theories in science. A chemist is no more likely to give up the belief that hydrogen has atomic number 1 than to give up DNE. If DNE isn’t part of the one true logic, but it is important to chemistry, then let it be an important part of chemical theory that, when we’re talking about chemistry, we accept every instance of~~p ⊢ p. For the Quasi-Quinean, even the principles of the One True Logic are justified by way of their centrality and importance to our web of belief. On this view, the distinction between ‘logical’ and ‘non-logical’ reasons for accepting an inference is too tenuous to do much serious theoretical work. A Quasi-Quinean would thus find Lynch’s way of blocking the retreat to VLP unsatisfying. There is no compulsion for an advocate of DLP to retreat to VLP, but the move is hard to resist. It is especially hard to resist if the advocate of DLP also endorses MODEST*. In the end, it is hard to see the difference between accepting DLP and MODEST* or, instead, just embracing VLP. It is thus hard to see why an alethic pluralist wouldn’t just give up on logical pluralism.
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A striking feature of Inessential Mix and Essential Mix is that there are chains of inferences from the premises to the conclusion that are valid, even by the lights of MODEST*. The premises of both arguments intuitionistically entail: (9) It’s not the case that Jack did not go up the hill.
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And that claim, from a domain whose governing logic is classical, classically entails the conclusion of both arguments. An advocate of AP + DLP might hope to find a solution to the problem of mixed inferences in that fact. The result would be a modification of MODEST* in the following way: MODEST**: Where a compound proposition or inference contains propositions from distinct domains, the default governing logic is that comprised by the intersection of the domain-specific logics in play. An inference Γ ⊢ p is valid if and only if either (i) it is valid in its default governing logic, or (ii) there is a proposition q such that the inference Γ ⊢ q is valid in its default governing logic and q ⊢ p is valid.8
MODEST** gives a recursive characterization of validity. In the base case, Γ ⊢ p is valid in the inference’s default governing logic, as in MODEST*. The recursive clause (ii) allows for the case that there is a chain of individually valid inferences from the premises to the conclusion, which may not be valid in the default governing logic of Γ ⊢ p, as there is for both Inessential Mix and Essential Mix by way of (9). Clause (ii) requires only that q ⊢ p be valid, not that it be valid in its default governing logic. That allows for the chain of inferences to include more than one intermediate step. For example, let I, R, and C be claims from domains governed by intuitionistic, relevantistic, and classical logic, respectively, and let S be a classical but not relevantistic consequence of C. We should want I & ~~(R&C) ⊢S to be valid. Although the premise entails ~~(R&C) in all the logics in play, it does not entail C in all the logics in play. Further, ~~(R&C) ⊢S is not valid in its (relevantistic) default governing logic. To get from ~~(R&C) to S, we must go by way of the relevantistically valid ~~(R&C) ⊢C and then the classically valid C⊢S. Requiring q ⊢ p to be valid in its default governing logic would not deliver the desired result. MODEST** ensures that p is a logical consequence of Γ so long as there is a chain of individual inferences, each valid in its default governing logic, from Γ to p. That likewise ensures that logical consequence for mixed inferences is transitive.
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There are at least two disadvantages to such an approach. First, an advocate of DLP might want to allow for the possibility that logical consequence in some discourses is not transitive. For example, one might think that doing so is the best way to avoid sorites paradoxes in vague discourse.9 In such a logic, the following argument would be invalid
Pure Bald ( 10) A person with 0 hairs is bald. (11) For all x, if a person with x hairs is bald, then so is a person with x+1 hairs. (12) Therefore, a person with 100,000 hairs is bald. Pure Bald is invalid by the lights of a non-transitive logic, because getting from (10) and (11) to (12) requires intermediate steps, such as: ( 13) If a person with 0 hairs is bald, then a person with 1 hair is bald. (14) A person with 1 hair is bald. (15) If a person with 1 hair is bald, then a person with 2 hairs is bald. (16) A person with 2 hairs is bald. and so on to: (17) A person with 999,999 hairs is bald. The point of having a non-transitive logic is to stop such chains of inferences. Given MODEST**, though, we are guaranteed transitivity of logical consequence when we deal with propositions from multiple domains. So, consider this inference, where ‘P(x)’ stands for ‘A person with x hairs is bald,’ ‘M’ stands for ‘Murder is wrong’ (or some other claim from a non-vague discourse), and ‘…’ abbreviates the obvious conjuncts:
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Mixed Bald (18) P(0) &M (19) (P(0) &M→P(1) &M) & … & (P(99,999) &M→P(100,000) &M) (20) P(100,000) In the non-transitive default governing logic, (18) and (19) can entail10: (21) (P(1) & M) & (P(1) & M→P(2) & M) & … & (P(99,999)& M → P(100,000) & M) The inference from (21) to (22) is valid in its default governing logic: (22) (P(2) & M) & (P(2) & M → P(3) & M) & … & (P(99,999)& M → P(100,000) & M) By similar steps, each valid in its default governing logic, we ultimately reach the conclusion P(100,000) & M, and then P(100,000). Since there is such a chain of individual inferences, each valid in its default governing logic, MODEST** counts Mixed Bald as valid. But that just means MODEST** undoes the very work that having a non-transitive logic for vague discourses was supposed to do. This disadvantage might not be very worrisome to some pluralists. Beall and Restall claim that non-transitive logics cannot possibly satisfy GTT, and so non-transitive logics are not really logics at all (2006, p. 91). A fan of AP + DLP could say the same. A second, more serious disadvantage is that this approach still invalidates some valid mixed inferences, such as this one:
Disjunctive Mix ( 23) Either it’s not the case that Jack didn’t go up the hill or 1 + 1 = 2. (24) Therefore, either Jack went up the hill or 1 + 1 = 2. The inference from (23) to (24) is not intuitionistically valid, and (23) doesn’t have any intuitionistic consequences from which (24) can be validly inferred. Both MODEST* and MODEST**, then, count Disjunctive Mix as invalid. But that seems wrong; the argument is valid.
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The problems discussed here arise for DLP if we try to solve the problem of mixed inferences by way of MODEST* or a modification of it. The advocate of AP + DLP might then try to find a different solution to the problem. Unfortunately, such a solution looks hard to come by. The trouble is that mixed inferences can have unmixed conclusions. So, we want to guarantee that no sound mixed inference has a conclusion that is untrue in its home discourse. Short of MODEST* and its relatives, there may be no way to guarantee that. The problem can be seen by considering an argument such as this (Lynch 2008, pp. 136–7): NIX: If it is not the case that Sophie’s choice is morally right, then grass is not green. But grass is green; so Sophie’s choice is morally right.
Suppose (a) that Sophie’s choice is to decide which of her children to give over to the Nazis (as in the novel), (b) that ‘Sophie’s choice is morally right’ is neither true nor false, and (c) that conditionals with antecedents that are neither true nor false are true.11 Then the argument’s premises are true, but its conclusion is not true. In evaluating NIX, we need to use a logic that does not count it valid. The argument for MODEST*, then, is that if we evaluate inferences according to any logic stronger than the logic that applies to any of their components, then we will wind up endorsing certain conclusions that are not true in their home discourses. So, to maintain AP + DLP without losing the ability to evaluate any mixed inferences as valid at all, we need to take the cautious route embodied by MODEST*. AP + DLP thus faces an unfortunate dilemma. On the one hand, it needs MODEST* in order to validate any mixed inferences at all without giving up on its AP component. On the other hand, MODEST* invalidates some inferences that, intuitively, the advocate of AP + DLP should want to count as valid, and it invites rejecting DLP in favor of VLP. There may be a way out of this dilemma. Maybe a creative modification of AP + DLP can validate the various valid mixed inferences discussed here without validating NIX. MODEST* does not require, but
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only invites, the retreat from DLP to VLP, and maybe that is an invitation easily declined. Nevertheless, it does appear to be a serious problem for AP + DLP, and there is one obvious way to avoid it: just don’t combine AP and DLP in the first place.
Notes 1. “Science, thanks to its links with observation, retains some title to a correspondence theory of truth; but a coherence theory is evidently the lot of ethics” (Quine 1981, p. 63). 2. Lynch (2013) downplays the role of discourses in pluralist theories, but most pluralists (including earlier time-slices of Lynch) suppose that, at least for atomic propositions, what truth-property is relevant to a proposition is a function of what discourse the proposition belongs to. 3. Different versions of functionalism characterize the relationship between truth itself and the various other properties that “realize,” “manifest,” or “play the role of ” truth differently. For discussion see Wright (2013). 4. GTT gives a generic characterization of logical consequence because an argument is valid if and only if its conclusion is a logical consequence of its premises. 5. Lynch also considers a solution on which the governing logic is the weakest logic among those governing the various domains involved in the compound proposition or inference. That solution, however, is just a special case of MODEST*, where the logics in question are ordered in such a way that one of them is weakest. 6. Some paraconsistent logics reject Disjunctive Syllogism. Maybe some jokes both are and aren’t funny. Then such a logic might well be the right one for discourse about comedy. 7. There is room for some debate as to what does or does not constitute a “logical reason” for a logical law to hold. Here I assume that law holds for a logical reason when its instances are logical truths. On Beall’s approach, the instances of ~(p& ~p) are logical truths, so long as they don’t involve the transparent truth device. 8. I owe thanks to Jack Lyons, Stewart Shapiro, and Sarah Wright for pointing out the need to consider versions of this move on behalf of AP + DLP.
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9. See Zardini (2008) and Cobreros et al. (2012). 10. Whether they do entail it might depend on the details of the non- transitive logic in question. The entailment holds for a logic whose non- transitive consequence relation is the ⊨ct of Cobreros et al. (2012). 11. These are the suppositions Lynch makes in presenting the case.
References Beall, J.C. 2009. Spandrels of Truth. New York: Oxford University Press. Beall, J.C., and G. Restall. 2006. Logical Pluralism. Oxford: Clarendon Press. Cobreros, P., P. Egré, D. Ripley, and R. van Rooij. 2012. Tolerant, Classical, Strict. Journal of Philosophical Logic 41 (2): 347–385. Lynch, M.P. 2001. A Functionalist Theory of Truth. In The Nature of Truth, ed. M.P. Lynch, 723–749. Cambridge, MA: MIT Press. ———. 2008. Alethic Pluralism, Logical Consequence and the Universality of Reason. Midwest Studies In Philosophy 32 (1): 122–140. ———. 2009. Truth as One and Many. Oxford/New York: Oxford University Press/Clarendon Press. ———. 2013. Three Questions for Truth Pluralism. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 21–41. New York: Oxford University Press. Pedersen, Nikolaj J.L.L. 2014. Pluralism x 3: Truth, Logic, Metaphysics. Erkenntnis 79: 259–277. Quine, W.V. 1981. On the Nature of Moral Values. In Theories and Things. Cambridge, MA: Harvard University Press. Tappolet, C. 1997. Mixed Inferences: A Problem for Pluralism About Truth Predicates. Analysis 57 (3): 209–210. Tennant, N. 1987. Natural Deduction and Sequent Calculus for Intuitionistic Relevant Logic. Journal of Symbolic Logic 52: 665–680. ———. 2012a. Changes of Mind: An Essay on Rational Belief Revision. Oxford: Oxford University Press. ———. 2012b. Cut for Core Logic. The Review of Symbolic Logic 5 (3): 450–479. Wright, C. 1992. Truth and Objectivity. Cambridge, MA: Harvard University Press.
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———. 2013. A Plurality of Pluralisms. In Truth and Pluralism: Current Debates, ed. Nikolaj J.L.L. Pedersen and C.D. Wright, 123–153. New York: Oxford University Press. Zardini, E. 2008. A Model of Tolerance. Studia Logica 90 (3): 337–368.
Logic for Alethic, Logical, and Ontological Pluralists Andy D. Yu
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Introduction
Following Wright (1992), one might maintain that the debate between realists and antirealists about a given domain of subject matter—whether the domain is about scientific, mathematical, or ethical subject matter— is best formulated as a debate about the nature of the truth property associated with the domain. This motivates the standard understanding, due to Wright (1992), of alethic pluralism, according to which there are distinct, domain-specific truth properties, where domains are associated with subject matter.1 To illustrate, the correspondence truth property is associated with the scientific domain, the coherence truth property is associated with the mathematical domain, and the superassertible truth property (which takes truth to be epistemically constrained) is associated with the ethical domain. Before continuing, let me clarify two simplifying assumptions. First, pluralists reject truth as it is represented in the familiar account of logic in favor of an alternative representation that captures its distinctness and A. D. Yu (*) Faculty of Law, University of Toronto, Toronto, ON, Canada © The Author(s) 2018 J. Wyatt et al. (eds.), Pluralisms in Truth and Logic, Palgrave Innovations in Philosophy, https://doi.org/10.1007/978-3-319-98346-2_17
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domain specificity. I set aside the suggestion that pluralists can unqualifiedly grant truth as it is represented in the familiar account, as just T (“truth”). Not least, pluralists themselves concede that they should say something more. Crucially, the suggestion does not do justice to pluralists’ commitment that there are distinct domain-specific truth properties. For the commitment suggests that something is missing from the familiar representation of truth, so that the familiar account stands in need of revision. Given the intertwining of truth with logic, alethic commitments motivate corresponding logical commitments. Second, for simplicity, truth bearers are sentences. That truth bearers are sentences is compatible with propositions and other entities also being truth bearers. My focus in this chapter is on how pluralists of various stripes might go about doing logic. As I explain in what follows, alethic pluralists face challenges to provide accounts of atomic sentences, quantified sentences, complex sentences, and logical consequence that capture their view of truth. But the challenges might not be limited to these. For alethic pluralists might also be logical pluralists, in taking distinctions between truth properties to induce corresponding distinctions between logical consequence relations. Similarly, alethic pluralists might also be ontological pluralists, in taking distinctions between truth properties to induce corresponding distinctions between quantifiers. Alethic pluralists who are also logical pluralists, or ontological pluralists, should provide a unified account of logical consequence, or of quantified sentences, that captures their view of truth and logical consequence, or of truth and quantifiers. My plan for this chapter is as follows. In Sect. 2, I review the account I propose on behalf of alethic pluralists.2 In Sect. 3, I extend the account to accommodate logical pluralism as well as ontological pluralism.
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Alethic Pluralism
In this section, I outline logical challenges for alethic pluralists and review my answers to them on behalf of alethic pluralists. In the first subsection, I outline four challenges, involving atomic sentences, quantified sentences, complex sentences, and logical consequence. In the second subsection, I review my answers to them.
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Logical Challenges for Alethic Pluralists Alethic pluralists face four challenges, which arise from considering mixtures of sentences about distinct subject matters. The first challenge, to maintain a unified account of atomic sentences, arises from considering atomic sentences, which can be about distinct subject matters.3 Pluralism is often illustrated using straightforward sentences. Consider the following sentences: (1) Water is H2O. ( 2) 1 + 1 = 2. (3) Murder is wrong. These sentences are about exactly one subject matter, and so—given that they are true—are true in exactly one way. Given that (1) is a scientific sentence, it is correspondence true; given that (2) is a mathematical sentence, it is coherence true; and given that (3) is an ethical sentence, it is superassertible true. But what about when matters are less straightforward? Not all sentences are about exactly one subject matter. Atomic sentences can be mixed, in that the sentences are about distinct subject matters; these contrast with sentences that are pure, which are not about distinct subject matters. But if mixed sentences are true, it is unclear in what way or ways they are true. Consider the following mixed atomic sentence: (4) The number π is beautiful. Apparently, (4) is about distinct subject matters, namely, mathematical subject matter and aesthetic subject matter. It is unclear in what way or ways true mixed sentences, such as (4), are true. Given that (4) is true, in what way or ways is it true? Perhaps it is both correspondence true and superassertible true. Alternatively, perhaps it is just superassertible true. Finally, perhaps it is true in neither of these ways, but if so, then in what way or ways is it true?
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The challenge is for pluralists to maintain an account of mixed atomic sentences, and more generally, a unified account of atomic sentences, whether pure or mixed. We can put the challenge thus: First logical challenge. Maintain a unified account of atomic sentences.
The second challenge, to maintain a unified account of quantified sentences, arises from considering quantified sentences, which can be about distinct subject matters.4 Consider the following mixed quantified sentence: (5) Everything is self-identical. As with (4), (5) also seems to be about distinct subject matters. Straightforwardly, (5) seems to be about everything. But if (5) is about everything, and so is associated with a generic domain, then the domain- specific property associated with the generic domain seems to be a generic truth property. While even pluralists might grant the existence of a generic truth property, it remains unclear to what extent they should grant such a property theoretical significance.5 The challenge is for pluralists to maintain an account of mixed quantified sentences, and more generally, a unified account of quantified sentences, whether pure or mixed. We can put the challenge thus: Second logical challenge. Maintain a unified account of quantified sentences.
The third challenge, to maintain a standard account of complex sentences, arises from considering logical combinations of sentences, which can be about distinct subject matters.6 Consider the following mixed complex sentences: (6) Water is H2O and 1 + 1 = 2. Given that the scientific sentence (1) is correspondence true and the mathematical sentence (2) is coherence true, their conjunction (6) is also
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true. But in what way or ways is (6) true? The options available here are similar to the options available when considering mixed atomic sentences. Perhaps (6) is both correspondence true and coherence true. Alternatively, perhaps it is just correspondence true. Finally, perhaps it is true in neither of these ways, but if so, then in what way or ways is it true? Here, one version of this last option is salient. Given that a conjunction is true iff both conjuncts are true, perhaps there is a truth property common to (1), (2), and (6). But such a property seems to be a generic truth property. Again, it is unclear to what extent pluralists should grant such a property significance. The challenge is for pluralists to maintain an account of mixed complex sentences, and more generally, a standard account of complex sentences, whether pure or mixed. We can put the challenge thus: Third logical challenge. Maintain a standard account of complex sentences, according to which the following obtains:
(i) a negation is true iff the negand is false; (ii) a conjunction is true iff each conjunct is true; (iii) a disjunction is true iff either disjunct is true.
Plausibly, what counts as a standard account should be understood in a looser rather than a stricter sense. To permit a reasonable amount of flexibility, the standard account of complex sentences sought need not be identical to the account from the familiar account of logic. Let me elaborate. Let us assume a sentential language ℒ, with atomic sentences Atomic(ℒ) and sentences Sent(ℒ). Also, let us adopt the standard convention of using the metavariables α, β, γ, … (with or without subscripts) to range over atomic sentences, ϕ, ψ, χ, … (with or without subscripts) to range over all sentences, and Γ to range over sets of sentences. For simplicity, we focus on the logical operators that are negation, conjunction, and disjunction. The material conditional and the material biconditional are defined in terms of these operators in the usual way, as follows: ϕ → ψ: = ¬ϕ ψ and ϕ ↔ ψ: = (ϕ → ψ) (ψ → ϕ).
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On the familiar account of complex sentences, a model M: Atomic(ℒ) → {T, F } is a function from atomic sentences to truth values, where T represents the generic truth property and F represents the generic falsity property.7 Given a model M, we get the semantic value function |·|M : Sent(ℒ) → {T, F} from sentences to truth values based on the model M. We define | · |M by |α|M = M(α) (where α is atomic) and the following: (LO) (i) |¬ϕ|M = T iff |ϕ|M = F; (ii) |ϕ ∧ ψ|M = T iff |ϕ|M = T and |ψ|M = T; (iii) |ϕ ∨ ψ|M = T iff |ϕ|M = T or |ψ|M = T. Recalling the point about permitting a reasonable amount of flexibility, it is enough that the account of complex sentences sought is similar enough to the familiar account to count as standard. Granted, whether an account is similar enough to the familiar one is vague. Still, examples show that we have enough of a grip of what counts as standard for the notion to be fruitfully employed. The strong Kleene account of complex sentences is a standard account, but a noncompositional account probably is not. The fourth challenge, to maintain a standard account of logical consequence, arises from considering mixed arguments, where the premises and conclusion can be about distinct subject matters.8 Consider the following argument, which has premises (1) and (2) and conclusion (6): ( 1) Water is H2O. (2) 1 + 1 = 2. (6) Water is H2O and 1 + 1 = 2. Clearly, the argument from (1) and (2) to (6) is valid: (6) is a logical consequence of (1) and (2). On the standard account of logical consequence, logical consequence necessarily preserves truth. So perhaps some single truth property is preserved from (1) and (2) to (6). But what property is that? Perhaps the truth property is common to (1), (2), and (6). If so, then the property seems to be a generic truth property. Once again, it is unclear to what extent pluralists should grant a generic truth property
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significance. The challenge is for pluralists to maintain an account of mixed arguments, and more generally, a standard account of logical consequence. We can put the challenge thus: Fourth challenge. Maintain a standard account of logical consequence, according to which logical consequence necessarily preserves truth.
To permit a reasonable amount of flexibility, the standard account of logical consequence sought need not be identical to the account from the familiar account of logic. On the familiar account of logical consequence, we have the following: (LC) Γ ⊨ ϕ iff for every model M, if |γi|M = T for every γi in Γ, then |ϕ|M = T. Logical consequence preserves truth across all models, and is classical. It is enough that the account sought is similar enough to the familiar account to count as standard. Examples show that we have enough of a grip of what counts as standard for the notion to be fruitfully employed. A many-valued account of logical consequence in terms of designation is standard, but not an account that validates the inference from ϕ to ¬ϕ. In sum, the four logical challenges for pluralism are to maintain unified accounts of atomic sentences and quantified sentences, and standard accounts of complex sentences and of logical consequence.
Logic for Alethic Pluralists I now review the account that I propose on behalf of alethic pluralists.9
For a Sentential Language I begin by reviewing the account for a sentential language. On the account I propose on behalf of pluralists, there is a one-to-one correspondence between domains, domain-specific truth properties, and domain-specific falsity properties. Pure domains are associated with
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exactly one subject matter, while impure domains are associated with two or more subject matters. Where domains are either pure or impure, pure domains generate all domains. Each atomic sentence is assigned to exactly one domain. Negations are always assigned to the same domain as the negand, while conjunctions and disjunctions may or may not be assigned to the same domain as each operand, depending on whether or not the operands are assigned to the same domain. Each atomic sentence is then assigned a domain-specific truth value, where the relevant domain is the one it is assigned. The domain-specific truth values of negations, conjunctions, and disjunctions are determined by the domain-specific truth values of the operands. Logical consequence necessarily preserves domain- specific truth. To incorporate talk of domains, we assume a structure 𝕯 = ⟨𝒟, ⊙⟩, where D is a nonempty set and ⊙ is a binary operation on 𝒟. Intuitively, 𝒟 is the set of domains, and ⊙ is a mixing operation on domains. To capture the idea that there are distinct domains (and associated domain- specific truth properties, as well as domain-specific falsity properties), we require that there are at least two members in 𝒟: |𝒟| ≥ 2. Let Di denote a member of 𝒟. Intuitively, Di ⊙ Dj is the mixed domain for Di and Dj. Since we want the mixed domain for identical domains to be the same domain and do not want the order in which we mix domains to matter, we require that 𝕯 is a semilattice: (DS) (i) Di ⊙ Di = Di (idempotency); (ii) Di ⊙ Dj = Dj ⊙ Di (commutativity); (iii) (Di ⊙ Dj) ⊙ Dk = Di ⊙ (Dj ⊙ Dk) (associativity). As usual, a partial order ≤ on 𝒟 is induced by setting the following: (PO) Di ≤ Dj iff Di ⊙ Dj = Dj. Note that ≤ in turn induces a strict partial order
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