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DUMMETT’S ARGUMENT FOR LOGICAL REVISION For Michael Dummett, classical inferences such as the law of excluded middle, double negation elimination, and reductio ad absurdum presuppose a reality in-principle unknowable. If this is right, then chagrin about the intelligibility of such an epistemically unfriendly reality entails chagrin about classical logic. Far better, argues Dummett, for our default logic to be intuitionistic, one that does not sanction use of strictly classical inferences. Most of the literature on this dialectic falls into two somewhat overlapping batches. The first consists of philosophical and mathematical treatises on intuitionistic logic and constructivism,1 and the second of discussions about the intelligibility of an unknowable reality.2 Only recently has a rigorous investigation been undertaken into Dummett’s arguments concerning the metaphysical and epistemic presuppositions of classical logic. As a result of this investigation, a number of philosophers have professed inability to discern even prima facie valid arguments for logical revision in Dummett’s oevre. For example, while discussing Dummett’s argument Neil Tennant says that one would be forgiven for viewing it as “a non-sequitur of numbing grossness” (Tennant 1997, p. 160). Like Tennant, Crispin Wright has recently offered his own new argument for intuitionism (Wright 1992). Unlike Tennant, he does not couple this argument with criticism of Dummett, though the very fact that he does not compare his argument with Dummett’s is telling. 1 Finally, in a recent article discussing Dummett and Wright’s arguments for logical revision, Joe Salerno states It seems that important attempts to make the [intuitionist] revisionist’s point fail. These are attempts made by Michael Dummett and Crispin Wright. The negative thesis is that, given the resources provided by either Dummett or Wright, choice of logic is not a realism-relevant feature—i.e, logical revision is not a consideration that is enjoined by one’s stance on the possibility of verification transcendent truth. (Salerno 2000, p. 212) If Salerno and Tennant are right about the nonexistence of an even prima facie valid argument for logical revision in Dummett’s oevre, then much of the literature on Dummettian anti-realism should be likened to the crowds’ discussion of the naked emperor’s fine garments. This would be completely outrageous, indeed scandalous. But Tennant and Salerno are wrong. While Tennant has done as much as anyone to advance Dummett’s anti-realist program (e.g. Tennant 1987, 1997), when explicating Dummett’s argument for logical revision, he (as well as Salerno) completely ignores important modal distinctions at the very heart of nearly all of Dummett’s writings on the theory of meaning. A close and careful reading of even a few of Dummett’s canonical works reveals an argument for logical revision that is prima facie valid and (once formalized) extraordinarily clear. So clear that it can be presented perspicuously in a Fitch style proof system.3 In what follows I begin by discussing the arguments that Salerno and Tennant mistakenly attribute to Dummett. I am able to use this discussion, as well as attention to some of Dummett’s canonical texts, to motivate and clarify Dummett’s actual argument. 2 My purpose is not to settle any issues between Dummett and his opponents, but rather to make clear what these issues are. Independent of considerations about “setting the record straight,” this clarifying of the issues is the most important accomplishment of my demonstration. By correctly formulating Dummett’s proof, I am able to precisely enumerate the four possible options for the defender of classical logic. I. SALERNO’S INTEPRETATION OF DUMMETT’S ARGUMENT “The Philosophical Basis of Intuitionistic Logic” is an early paper in which Dummett makes his case for verificationism . For example, he writes This [the realists’] conception violates the principle that use exhaustively determines meaning; or, at least, if it does not, a strong case can be put up that it does, and it is this case which constitutes the first type of ground which appears to exist for repudiating classical in favor of intuitionistic logic for mathematics. For, if the knowledge that constitutes a grasp of the meaning of a sentence has to be capable of being manifested in actual linguistic practice, it is quite obscure in what the knowledge of the condition under which a sentence is true can consist, when that condition is not one which is always capable of being recognized as obtaining. (Dummett 1975a, 224) This is Dummett’s meaning-theoretic challenge to specify in what knowledge of meaning consists, in terms of practical abilities that can be correctly attributed to a competent language user. Dummett holds that a verificationist can provide such an account. The argument we are concerned with attempts to show that such verificationism entails intuitionist revision. This requires, in turn, understanding the logical structure of 3 Dummett’s verificationist claim. Then, where “[]” means “it is necessarily the case that,” we can formalize Dummett’s verificationist belief as Verificationism []X((X k1X) (~X k1~X)). Given some interpretations of the necessity operator, Dummett clearly intends this. A verificationist meaning theorist does not think that verificationism is a parochial truth about us, our language, and the world. Rather, it is supposed to reflect some deep truth about the nature of language, thought, and any possible world. Dummett holds that it is necessary that all truths are knowable. In “Revising the Logic of Logical Revision,” J. Salerno presents Dummett as holding that verificationism somehow renders classical logic inconsistent with the existence of sentences that can neither be proven nor refuted. This is an understandable interpretation of Dummett’s argument, as Dummett often seems to say that we all agree that there are such undecidable sentences in a language, and that the existence of such undecidable sentences, in conjunction with bivalence and verificationism, leads to a contradiction. For example, It is when the principle of bivalence is applied to undecidable statements that we find ourselves in the position of being unable to equate an ability to recognize when a statement has been established as true or as false with a knowledge of its truthcondition, since it may be true in cases when we lack the means to recognize it as true or false when we lack the means to recognize it as false. (Dummett 1976b, p. 63) 4 So the thought seems to be that since Verificationism, the existence of an undecidable sentence, and bivalence entail a contradiction, we should eschew bivalence. More formally, where the premises are, Verificationism []X((X k1X) (~X k1~X)) Undecidability Y(~k2Y ~k2~Y) Bivalence X(X ~X) and “” denotes absurdity, Dummett’s main claim seems to be: Salerno’s Version of Dummett’s Argument: Verificationism, Undecidability, Bivalence |- Using the law of excluded middle instead of Bivalence, and calling Verificationism “the knowability principle,” Salerno gives the argument for this in the following manner. Let us suppose that some indicative of the given class is undecidable. By accepting the law of excluded middle, one accepts the truth or falsity of every sentence, so the undecidable sentence is either true or false. First, suppose it is false. Then it follows from the knowability principle that we could prove it false. But we cannot prove it false, since it is undecidable. Second, suppose the undecidable sentence is true. Then it follows from the knowability principle that we could recognize it as true. Again, this is in contradiction with its undecidability! But then we have absurdity resting on excluded middle, the knowability principle, and the undecidability thesis. Something must go. (Salerno 2000, 214) If this is right then the debate between Dummett and his opponent is clear. The Dummettian anti-realist concludes that Bivalence caused the contradiction, and eschews Bivalence while asserting Verificationism and Undecidability. The Dummettian realist 5 concludes that Verificationism is the root of the evil and continues to assert Bivalence and Undecidability. Unfortunately (as Salerno points out), if this is the dialectic, then the Dummettian anti-realist loses the debate. Verificationism and Undecidability are themselves intuitionistically inconsistent! Reason why Salerno’s Version of Dummett’s Argument Fails: Verificationism, Undecidability |- As with the proof Salerno attributes to Dummett, the necessity operator in front of Verificationism isn’t needed to establish this. That is, one can intuitionistically prove the following. Important Lemma: |- X((X k1X) (~X k1~X)) ~Y(~k1Y ~k1~Y) 4 Given that “X((X k1X) (~X k1~X))” is just Verificationism without the modal, and that “~Y(~k1Y ~k1~Y)” is just the denial of Undecidability, our Important Lemma entails the Reason why Salerno’s Version of Dummett’s Argument Fails. Independently of undermining what one might take to be Dummett’s argument, this result is deeply problematic for the Dummettian. It seems to show that Verificationism and Undecidability are already jointly contradictory, without the assumption of Bivalence. One might think that, since no one would deny the existence of undecidable sentences, we have shown Dummett’s verificationism to be false. The result should not be interpreted in this manner. The proof just shows us that the modal “k2” in the statement of Undecidability (Y(~ k2Y ~ k2~Y)) needs to be sufficiently idealized in some manner. For example, one might interpret it as saying that 6 there exists a sentence such that it is not possible, given any finite extension of our conceptual resources, to know whether the sentence is true or false. Thus, depending upon the strength of the modal “k1” in Verificationism ([]X((X k1X) (~X k1~X))), a verificationist like Dummett need not be committed to anything clearly absurd. While this shows that the Dummettian anti-realist’s position is not obviously absurd, we still have no argument for logical revision. II. TENNANT’S INTERPRETATION OF DUMMETT’S ARGUMENT Strangely, the very argument that Salerno takes to be Dummett’s undoing is a key lemma in the argument Tennant attributes to Dummett. An immediate consequence of the Reason why Salerno’s Version of Dummett’s Argument Fails, is the 1st Lemma Tennant Attributes to Dummett:5 Verificationism |- ~(Undecidability) It should be noted that Tennant seems aware of the fact that one might interpret Dummett in the manner of Salerno. Where “” is the name of the 1st Lemma Tennant Attributes to Dummett, Tennant explicitly states the following. Note that , on Dummett’s own understanding of the course of argument, would not itself, rely on the principle of bivalence as a premise. . . [Dummett’s] argument must incorporate the principle of bivalence as a premise of its other subargument. . . (Tennant 1997, p. 181) One might think this is a more charitable reading of Dummett. However, Tennant’s reading is no more charitable, as the entire argument hinges on the “other subargument,” 7 2nd Lemma Tennant Attributes to Dummett: Bivalence |- Undecidability Were the 2nd lemma plausible, it and the 1st together would establish the inconsistency of Verificationism and Bivalence. (Tennnant 1997, pp. 180-181) Even more so than the argument Salerno attributes to Dummett, this forces a stark choice, one must either (with Dummett) give up Bivalence, or (with the classicist) give up Verificationism. The bulk of Chapter 6 of Tennant’s The Taming of the True (“The Manifestation Argument is Dead”) concerns itself with attempting to discern an argument from Bivalence to Undecidability. After a discussion involving, among other things, provably undecidable discourses such as Peano Arithmetic and new, unpublished, independence results attributed to Harvey Friedman, Tennant concludes the chapter by stating All that, however, still fails to make the desired logical transition available to the Dummettian: the transition, that is, from bivalence to the existence of recognition transcendent truths. (Tennant 1997, p. 194) Thus, like Salerno, Tennant concludes that Dummett never provided a prima facie valid argument for logical revision. III. DUMMETT’S ARGUMENT A careful study of Dummett’s texts on the theory of meaning reveals that, pace Salerno and Tennant, the principle of Bivalence by itself plays no role in Dummett’s argument. Rather, Dummett takes classical model-theoretic semantics, when used as part of a semantics for natural language or mathematics, to imply the possibility of the kind of undecidable sentences intuitionistically inconsistent with verificationism. As central as 8 this point is in all of Dummett’s writings on the “theory of meaning,” its dialectical relevance for Dummet’s revisionary arguments has been consistently overlooked. I formalize the principle thus. Dummett’s Insight TCS 2X(~k2X ~k2~X) where “TCS” means, roughly, “Classical model-theoretic semantics is the correct semantics for the logical operators, and is a part of the correct semantics for natural language and mathematics.”6 What is constitutive of TCS for Dummett’s purposes is that the truth predicate attaches correctly to a sentence in virtue of the referential relations of the subsentential units of the sentence. Thus, Dummett writes, To have a realistic view, it is not enough to suppose that statements of the given class are determined, by the reality to which they relate, either as true or as false; one has also to have a certain conception of the manner in which they are so determined. This conception consists essentially in the classical two-valued semantics: and this, in turn, embodies an appeal to the notion of reference as indispensable. (Dummett 1982, p. 231)) On this conception a simple atomic sentence like “Fred is envious” is true if, and only if, the entity referred to by “Fred” is in the extension of the set of entities referred to by “envious.” Then, for Dummett, it is the way the truth of logically complex statements are determined in classical model theory that is important. He writes, 9 The truth value of a quantified statement is, on this conception, determined by the truth-values of its instances, so that the instances stand to the quantified statement just as the constituent subsentences of a complete sentence whose principal operator is a sentential connective stand to the complex sentence: the truth-value of the quantified statement is a truth-function of the truth-values of its instances, albeit an infinitary one if the domain is infinite. The truth-value of a universally quantified statement is the logical product of the truth-values of its instances, that of an existentially quantified statement the logical sum of the truth-values of its instances. These operations, these possibly infinitary truth-functions, are conceived of as being everywhere defined, that is, as having a value in every case: in other words, the application of the operation of universal or of existential quantification to any predicate that is determinately true or false of each object in the domain will always yield a sentence that is itself determinately either true or false, independently of whether we are able to come to know its truth-value or not. (Dummett 1982, pp. 231-232) To grasp Dummett’s point, consider a possible world containing an infinite sequence of red and green objects. Now consider the sentence, “There are infinitely many red objects.” If classical model theory gave the correct interpretation of this sentence, then the sentence would be made true or false in the way Dummett describes above. Yet there would be no way for the inhabitants of such a world to tell whether or not the sentence were true or false, since they could not survey the infinite domain. Thus in the described possible world, there does exist an in-principle undecidable sentence. In the case of decidable, yet contingent, empirical predicates such as “red” it is easy to generate such 10 scenarios given the quantificational apparatus of classical logic, and the attendant modeltheoretic interpretation of those quantifiers.7 Dummett’s Insight does allow one to deduce absurdity from Verificationism and TCS. This, then is the form of Dummett’s real argument for logical revision. Dummett’s Argument: Verificationism, Dummett’s Insight |- ~TCS Proving this will require greater clarity about the inferential roles of key modal notions occurring in the premises. Premises: Verificationism Dummett’s Insight []1X((X k1X) (~X k1~X)) TCS X(~k2X ~ k2~X) Then, our demonstration necessarily involves the following modal inferences. Transformation rules: K. |- P, therefore |- []L P. Rules of inference: []~ dist. []x~P, therefore ~xP [] dist. []x(P Q), therefore []xP []xQ [] >. []LP, therefore []yP (If P is logically necessary, then P is necessary in any other modality)} With these rules we can establish Dummett’s Argument. Since the modal reasoning is somewhat complicated, I present this as a formal proof. Demonstration of Dummett’s Argument: 1. X((X k1X) (~X k1~X)) 11 ~Y(~k1Y ~k1~Y) by Important Lemma 2. []L(X((X k1X) (~X k1~X)) ~Y(~k1Y ~k1~Y)) 1 K. 3. []1(X((X k1X) (~X k1~X)) ~Y(~k1Y ~k1~Y)) 2 [] >. 4. []1X((X k1X) (~X k1~X)) []1~Y(~k1Y ~k1~Y) 3 [] dist. 5. []1X((X k1X) (~X k1~X)) Verificationism 6. []1~Y(~k1Y ~k1~Y) 4, 5 elimination 7. ~1Y(~k1Y ~k1~Y) 6 []~ dist. 8. TCS X(~k2X ~ k2~X) Dummett’s Insight 9. | TCS For ~ introduction 10.| X(~k2X ~ k2~X) 8, 9 elimination 11. | X(~k2X ~ k2~X) ~1Y(~k1Y ~k1~Y) 7, 10 introduction 12.| 11 ~ elimination 13. ~TCS 9-12 ~ introduction Q.E.D. III. Q.E.D.? The main purpose of this paper is to show that Dummett himself provided the resources for a prima facie valid argument for logical revision. However the point of this is not 12 merely to give Dummett his due. Like all good philosophical demonstrations, Dummett’s carves the logical space of possible positions in terms of how one reacts to his proof. My presentation of Dummett’s argument sets in bold relief several morals about how debate about intuitionistic logical revision should continue. The proof shows there to be exactly four strategies for disagreeing with Dummett. One may: (1) argue that Verificationism is false, (2) argue that Dummett’s Insight is false, (3) argue that the proof equivocates on line 9 due to incomparable notions of possibility (1 and 2) and (4) argue that the proof equivocates on line 9 due to incomparable notions of knowability (k1 and k2). In closing I would like to suggest that these combine to produce a challenge to the Dummettian. One could argue that insofar as Verificationism and Dummett’s Insight are plausible they involve incomparable modal notions. Then the Dummettian would face a dilemma; either one of Verificationism or Dummett’s Insight is false, or the proof equivocates on line 11. To see how this strategy might work, assume that “k1” meant “is knowable by God” and “k2” means “is knowable by a person.” Then line 11 (X(~k2X ~ k2~X) ~1Y(~k1Y ~k1~Y)) would be the claim that: (1) it is possible there exists a sentence unknowable by people, and (2) it is not possible that there exists a sentence unknowable by God. Clearly these two claims do not contradict one another. But then the conclusion that truth conditional semantics is mistaken would not follow.8 Now what is really interesting is that Verificationism becomes much less controversial if one merely affirms that all truths are knowable by God, as long as this claim isn’t taken to presuppose God’s existence. Likewise, Dummett’s Insight becomes much less controversial if it merely affirms that the correctness of truth conditional 13 semantics implies that it is possible that some sentences are unknowable to us. Now, for the purpose of argument assume that any more controversial statements of Verificationism and Dummett’s Insight are false. Then our dilemma would undermine Dummett’s argument; either one of Verificationism or Dummett’s Insight are false, or the proof equivocates on line 9. Another strategy for establishing the same dilemma would be to argue that the truth of Verificationism forces “k1P” to mean something like “defeasible evidence is available for P,” and that the truth of Dummett’s Insight requires “k2” to mean “P can be known with certainty.” Again, line 11 would equivocate, merely stating that (1) it is possible that some sentence can’t be known with certainty to be true or to be false, and (2) it is not possible for a sentence to be such that there is no defeasible evidence available for its truth or falsity. Of course these possible objections do not even mention the possibility operators 1 and 2 occuring in Verificationism and Dummett’s Insight. Given the indisputable ambiguity of the modal “is possible” it would be surprising if one couldn’t argue for our dilemma by focusing on them. Finally, it should also be clear that the proper interpretations of the modals in Dummett’s key claims will be in some sense discourse dependent. It might be the case, for example that our dilemma undermines Dummett’s argument when applied to empirical claims, but fails to undermine it when applied to mathematical claims. Or that it fails for both kinds of claims, but for different reasons involving the modals. I will not further explore these possible refutations. I have come to praise Dummett’s argument, not bury it. The reason I enumerate the classicist’s strategies is to 14 establish that future anti-realist debate about logical revision should be a debate about the proper modal notions occurring in Dummett’s argument. I take heart in the fact that Dummett himself is aware of this. In “What is a Theory of Meaning (II)” he writes In this way, we try to convince ourselves that our understanding of what it is for undecidable sentences to be true consists in our grasp of what it would be to be able to use such sentences to give direct reports of observation. We cannot do this; but we know just what powers a superhuman observer would have to have in order to be able to do it- a hypothetical being for whom the sentences in question would not be undecidable. And we tacitly suppose that it is in our conception of the powers which such a superhuman observer would have to have, and how he would determine the truth-values of the sentences, that our understanding of their truth-conditions consists. This line of thought is related to a second regulative principle governing the notion of truth: If a statement is true, it must be in principle possible to know that it is true. This principle is closely connected with the first one [that if a statement is true, then there must be something which makes it true]: for, if it were in principle impossible to know the truth of some true statement, how could there be anything which made that statement true? (Dummett, 1976b, p. 61) Interestingly, Dummett takes this kind of verificationism to be one that he and the defender of classical logic should agree about. Clearly then, for Dummett the important question concerns how super the superhuman observer needs to be. But this is precisely the question of how to construe the modal notions occurring in the premises of Dummett’s Argument. Thus, if I am right (about Dummett’s argument), then Dummett is 15 right. Anti-realists and realists should conclude that significant progress in many of the debates begun by Dummett now requires attaining much greater clarity and insight into the relevant modal notions.9 REFERENCES Appiah, Anthony. (1986) For Truth in Semantics, Basil Blackwell Press. Dragalin, A.G. (1980) Mathematical Intuitionism, Introduction to Proof Theory, American Mathematical Society. Dummett, M. (1967) “Platonism”, in Dummett, Michael. (1978) Truth and other Enigmas, Harvard University Press, 202-214. Dummett, M. (1975a) “The Philosophical Basis of Intuitionistic Logic”, in Rose, H. and Shepherdson, J. (eds.), Logic Colloquium ’73, Amsterdam, Oxford and New York, reprinted in Dummett, Michael. (1978) Truth and other Enigmas, Harvard University Press, 215-247. Dummett, M. (1975b) “Wang’s Paradox”, Synthese, 30, 301-24 reprinted in Dummett, Michael. (1978) Truth and other Enigmas, Harvard University Press, 248-268. Dummett, Michael. (1976a) “Is Logic Empirical,” Lewis, H. (ed.) Contemporary British Philosophy, London, pp. 45-68, reprinted in Dummett, Michael. (1978) Truth and other Enigmas, Harvard University Press, 269-289. Dummett, M. (1976b) “What is a Theory of Meaning? (II)”, in G. Evans and J. McDowell 16 (eds.), Truth and Meaning, Oxford: Clarendon Press, reprinted in Dummett, Michael. (1993) The Seas of Language, Clarendon Press, 34-93. Dummett, Michael. (1982) “Realism,” Synthese, 52, 55-112, reprinted in Dummett, Michael. (1993) The Seas of Language, Clarendon Press, 230-276. Dummett, Michael. (1991) The Logical Basis of Metaphysics, Harvard University Press. Prawitz, Dag. (1965) Natural Deduction: A Proof-Theoretical Study, Almqvst and Wiksell. Salerno, Joseph. (2000) “Revising the Logic of Logical Revision,” Philosophical Studies, 99, 211-227. Shapiro, Stewart. (1993) “Anti-Realism and Modality,” in J. Czermak (ed.), Philosophy of Mathematics, Proceedings of the 15th International Wittgenstein-Symposium, Part 1, HolderpPichler-Tempsky, 269-287. Tennant, Neil. (1987) Anti-Realism and Logic: Truth as Eternal, Clarendon Press. Tennant, Neil. (1997) The Taming of the True, Clarendon Press. Wright, Crispin. (1992) Truth and Objectivity, Harvard University Press. Wright, Crispin. (1993) Realism, Meaning, and Truth, 2nd ed., Basil Blackwell Press. 17 ENDNOTES 1 For example, Prawitz’s Natural Deduction: A Proof Theoretical Study, and Dragalin’s Mathematical Intuitionism: An Introduction to Proof Theory. 2 For example, Appiah’s For Truth in Semantics and many of the articles in Wright’s Realism Meaning and Truth. Tennant Anti-realism and Logic: Truth as Eternal and The Taming of the True combine both strains. 3 Of course the fact that one can formalize a proof does not entail that one should do so. However, the dialectical situation we find ourselves in is special. First, since this argument has eluded those Dummett exegetes (e.g. Salerno, Tennant, and Wright) sympathetic to intuitionism and verificationism, I chose to pay the price of a kind of inelegance when what is purchased is logical unassailability. Second, it is very important that Dummett not commit himself to the validity of classical logic in any part of his argument. Unfortunately, it is much more difficult to discern whether a complicated nonformal argument involves resources unavailable to the intuitionist. For the reader moved by this issue, the time wasted by trudging through a formal proof is gained back tenfold by not going down the path of fruitlessly trying to discern a tu quoque argument against Dummett. Third, and most important, by explicitly presenting Dummett’s argument, I am able to show exactly where the defender of classical logic should demure. 4 Here is the proof. Claim: |- X((X k1X) (~X k1~X)) ~Y(~k1Y ~k1~Y) Proof: 1. | X((X k1X) (~X k1~X)) Assumption for introduction 18 2. | | Y(~k1Y ~k1~Y) Assumption for ~ introduction 3. | | | (~k1P ~k1~P) Assumption for elimination 4. | | | (P k1P) (~P k1~P) 1 elimination 5. | | | (P k1P) 4 elimination 6. | | | ~k1P 3 elimination 7. | | | | P Assumption for ~ introduction 8. | | | | k1P 5, 7 elimination 9. | | | | k1P ~k1P 6, 8 introduction 10.| | | | 9 ~ elimination 11.| | | ~P 7-10 ~ introduction 12.| | | ~P k1~P 4 elimination 13.| | | k1~P 11, 12 elimination 14.| | | ~k1~P 3 elimination 15.| | | k1~P ~k1~P 13, 14 introduction 16.| | | 15 ~ elimination 17.| | 3-16 elimination 18.| ~Y(~k1Y ~k1~Y) 2-17 ~ introduction 19. X((X k1X) (~X k1~X)) ~Y(~k1Y ~k1~Y) 1-18 introduction 5 In his discussion Tennant does not modalize Verificationism either. As with Salerno, presenting his argument with the modal causes no problem, as []P logically entails P. 19 6 For our purposes, we can assume that TCS itself entails bivalence, though this should not be completely without controversy. As far as I know, nowhere in the secondary literature on Dummett is this entailment questioned, though it does not follow without further argumentation. Classical model-theoretic semantics of the kind utilized by mathematical logicians as well as natural language semanticists defines truth in a model, not truth simpliciter. Thus, classical semantics simply tells us that a sentence is either true or false in any interpretation of the sentence. It is only if one takes truth in an intended interpretation to be a good model of truth simpliciter, that one gets bivalence from classical semantics. While Dummett has extraordinarily insightful things to say about this issue in, for example, The Logical Basis of Metaphysics and “Wang’s Paradox,” no one has yet systematically explored (from within the framework of antirealism) the prospects of using classical semantics to justify classical inference but not to utilize truth in an intended interpretation as a notion of truth simpliciter. However, in what follows I assume that TCS includes the assumption that, to borrow a phrase from Stewart Shapiro, truth in a model is a good model of truth. 7 Dummett takes the structures in which the classical mathematician trades as being exactly like the red-green world I describe. For Dummett, this is why the classical mathematician is committed to the view that it could be the case that the natural number structure is such that for example, every even number is the sum of two primes, while no finite proof of such a claim exists (e.g. see Dummett’s “Platonism”). This is the key to understanding why Dummett sees intutionism as the main competing position to Platonism in the philosophy of mathematics. It is also key to understanding why one who 20 restricted his attention to Dummett’s “The Philosophical Basis of Intuitionistic Logic” might read the argument in the way Tennant does. Modal talk in mathematics is somewhat dicey, since most of us accept both that (in some sense) mathematical truth entails mathematical necessity and (in some sense) that an undecided mathematical claim is possibly true and possibly false. So one who ignored the second modal intuition (perhaps due to allegiance to the first) while restricting her attention to Dummett’s discussion of mathematics (e.g. in “The Philosophical Basis. . .”) might collapse the modal in Dummett’s Insight. On this view Dummett would believe that truth conditional semantics entailed the actual existence of an undecidable sentence. Then, if one ignored Dummett’s voluminous writings on the role of logical semantics in theory of meaning, one might (like Tennant) foist upon Dummett the view that bivalence itself entails such a sentence. 8 This is exactly why, in the concluding chapter of The Logical Basis of Metaphysics, Dummett is at great pains to show that even if God exists, he does not have knowledge of a completed infinity. 9 Stewart Shapiro first encouraged me to render explicit the modal commitments of Dummettian anti-realists. It is a consequence of my discussion that Shapiro’s “Antirealism and Modality,” as well as Anthony Appiah’s For Truth and Semantics deserve much more critical attention from Dummettians than they have received. I would also like to thank Emily Beck Cogburn, Joseph Salerno, William Taschek, and Neil Tennant for encouragement and helpful comments on an earlier draft of this material. 21