1 Fitch, F. ‘A Logical Analysis of Some Value Concepts’ Journal of Symbolic Logic 28 (1963) 135–42.CrossRefGoogle Scholar

2 See the excellent overview of such responses in Kvanving, J.L. ‘The Knowability Paradox and the Prospects for Anti-Realism’ Nous 29 (1995) 481–500.CrossRefGoogle Scholar

3 Tennant, Neil The Taming of the True (Oxford: Oxford University Press 1997), ch. 8.Google Scholar

4 Intuitionists might reject the transition from line 13 to 14, as P → 3i/ Ki/P only really follows from —(P A — 3y Ki/P) with the help of a classical negation rule such as the law of excluded middle or double negation elimination. In T. Williamson, ‘On the Paradox of Knowability’ Mind (1987) 256-61, the author suggests that this might be thought of as providing evidence for intuitionism, albeit not very much. The denial that any claim can be both true and unknown, as stated schematically in line 13, is problematic enough.

5 Note that a dialethist such as J.C. Beall might very well reject this assumption. For a dialethist response to Fitch's proof see Beall, J.C. ‘Fitch's Proof, Verificationism, and the Knower Paradox,’ Australasian Journal of Philosophy 78 (2000) 241–7.CrossRefGoogle Scholar

6 I would like to thank an anonymous reviewer for encouraging me to expand on this point.

7 Some of these rules can clearly be derived from simpler rules; for example T—i dist. follows from the intro and elimination rules for T and the normal rules for the logical operators.

8 This is O.K., since the conditional in 15 is schematic.

9 I owe this objection to an anonymous reviewer.

10 See the appendix for a restatement of the proof along these lines.

11 Dummett, Michael ‘What is a Theory of Meaning (II),’ in The Seas of Language, Dummett, M. (Oxford: Oxford University Press 1993), 61.Google Scholar

12 Nothing in Dummetf s point requires the existence of a God, as Dummetf s realist only need think that it is possible that God exists, and that this being's knowledge is relevant to our understanding of language. Thus, our discussion is orthogonal to Plantiga's argument's for theistic anti-realism in A. Plantiga, ‘How to Be an Anti- Realist,’ Proceedings of the American Philosophical Association (1982): 47-70.

13 For example, see Misak, C. Verificationism, (London: Routledge 1995),Google Scholar Jardine, N. The Fortunes of Inquiry, (Oxford: Oxford University Press 1986),Google Scholar and Hand, M. and Kvanvig, J.L. ‘Tennant on Knowability,’ Australasian Journal of Philosophy 77 (1999) 422–8.CrossRefGoogle Scholar

14 One should note that the inference K elim. and the conclusion of Fitch's proof together establish that the ‘is known’ predicate is coextensive with the truth predicate. Thus, we could have let Q just be ‘Q is not known’ and presented a notational variant of the standard liar paradox! Thus, it is the following philosophical discussion that justifies our use of knowability.

15 Montague, R. ‘Syntactical Treatments of Modality with Corollaries on Reflexion Principles and Finite Axiomatizability’ Acta Philosophical Fennica 16 (1963) 153–67,Google Scholar and Montague, R. and Kaplan, D. ‘A Paradox Regained’ Notre Dame Journal of Formal Logic 1 (1960) 79–90.Google Scholar

16 Beall, ibid.

17 On the assumption that P is true in the closest possible world in which God exists. This is no very big concession though, as the inference only fails then for any true sentences inconsistent with God's existence. Were the domain of quantification restricted to such sentences, the proof is clearly still inductively valid.

18 The theological relevance of Fitch's proof was first noted in Macintosh, J. ‘Fitch's Factives’ Analysis 44 (1984) 145–58,CrossRefGoogle Scholar and Macintosh, J. ‘Theological Question-Begging,’ Dialogue 30 (1991) 531–47.CrossRefGoogle Scholar Macintosh notes the similiarity of Fitch's proof and the ontological argument. However, Macintosh does not consider epistemic notions, nor does he use Fitch's proof in an argument against God's existence.

19 Montague (ibid.) took this to be the moral of his epistemic paradox. This might provide independent motivation for its employment in relation to this paradox. On the other hand, to the extent that one does not find it well motivated here, one might begin to question it in relation to Montague's paradox.

20 See Tarski, A. ‘The Semantic Conception of Truth and the Foundations of Semantics’ Philosophy and Phenomenological Research 4 (1944): 341–76.CrossRefGoogle Scholar

21 Assuming that P is level one clearly involves no loss of generality. Also, one might implement semantic ascent slightly differently. Perhaps K only applies to sentences one level down. Perhaps only sentences of the same level can be conjoined. In these cases the premise assumed for reductio would be 3xK₃x(P₂ A -3yK₂yp₁) (P₂being the canonical metalinguistic translate of PJ). In these cases, our point would still hold. Line 6 would still not be a contradiction.

22 I assume, perhaps wrongly, that the burden is on the theist to show that 3xK_?fxP₁ and —3xK₂xP₁ do contradict each other. Perhaps the burden of proof is upon the theisf s enemy to show how 3xK₃xPi and —αxKˆxPi can be consistent. Unfortunately for the anti-theist though, success in this endeavor will almost certainly undermine the use of semantic ascent to block the paradox too. This strikes me as an extremely important issue for defenders of meta-linguistic resolutions of self-referential paradoxes, and one interesting in its own right for those who find such approaches ill-motivated. Such a discussion does not lie within the scope of this paper, since I am concerned with defending a different strategy for Fitch's proof and the new paradox.

23 Mavrodes, G. ‘Some Puzzles Concerning Omnipotence,’ The Philosophical Review 72 (1963) 221–3.CrossRefGoogle Scholar

24 See Williamson, Timothy ‘Tennant on Knowable Truth’ Ratio 13 (2000) 99–114CrossRefGoogle Scholar and Hand and Kvanvig, ibid.

25 Tennant, 275.

26 Hand and Kvanvig, 427.

27 Note that there is also the principle even more anti-realist than ARV.’ to the effect that if it is consistent to suppose God knows P (and P is true), then it is possible for us to know P. The moral of this, and Hand and Kvanvig's discussion (though clearly not one they endorse) is that the principle that all true claims are knowable might be completely irrelevant to the substantive debates between Dummettian anti-realists and their opponents! It seems that all along Dummettians and their opponents should have been arguing over the principle that if a claim is knowable at all then it is knowable to creatures like us. Fitch's proof does not undermine this principle. Whether this principle is felicitous for arguments for intuitionistic revision is, I think, an extremely important open question.

28 ibid., 424.

29 Depending upon one's philosophy of logic, one might object that failure to derive a contradiction, or consistency, is not the same as logical possibility. But the principle that if it is consistent to assume that P is known (and P is true), then it is possible in some weaker modality for P to be known is still non-trivial. How non-trivial depends upon the characterization of the weaker modality.

30 I owe this counter-argument to an anonymous reviewer.

31 Much of Tennant's discussion is relevant to characterizing this form of possibility. See for example, Tennant, 153.

32 Hand and Kvanvig, 426.

33 ibid., p. 425.

34 My earlier discussion of self-referential paradoxes shows that a better example might have involved an analog of Tennant's solution when considering the liar paradox. I don't, however, think this would work for precisely the reason that Tennant's solution is non-trivial, the role of different kinds of modality, which don't occur at least in standard versions of the liar paradox.

35 ibid., 426.

36 ibid.

37 I must thank Roy Cook. This paper began as a joint paper with him that used the self-referential paradox to defend a meta-linguistic solution to Fitch's proof (Roy proved a fixed point theorem to get the sentence that says ‘I am not knowable’ in the appropriate kind of language, and helped interpret the levels such that Fitch's proof was intuitively invalid). While the strategy seemed promising for Fitch's proof, an ever increasing Priestian disenchantment with semantic ascent as a solution to self-referential paradoxes in general (and the liar paradox in particular) led us to abandon the project. In this vein, one must admit that showing Tennant's solution to be free from ad hocness is not the same as showing it true. In particular, one would need to see how J.C. Beall's dialethist solution to Fitch's proof (ibid.) applies to the paradox of the stone and interacts with other facets of Dummett's program, such as Dummett's arguments for intuitionism. I hope that it is clear how the above demonstrations motivate such a project. I am also grateful to an anonymous reviewer, Andrew Arlig, Emily Beck Cogburn, Stewart Shapiro, and Mark Silcox for helpful suggestions.