Abstract
What makes necessary truths true? I argue that all truth supervenes on how things are, and that necessary truths are no exception. What makes them true are proofs. But if so, the notion of proof needs to be generalized to include verification-transcendent proofs, proofs whose correctness exceeds our ability to verify it. It is incumbent on me, therefore, to show that arguments, such as Dummett’s, that verification-truth is not compatible with the theory of meaning, are mistaken. The answer is that what we can conceive and construct far outstrips our actual abilities. I conclude by proposing a proof-theoretic account of modality, rejecting a claim of Armstrong’s that modality can reside in non-modal truthmakers.
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Notes
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- 4.
Clearly, in the presence of infinite proofs, the notion of Gödel number will need to be generalized.
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The assumption that languages are countable can be challenged. For example, infinite decimals are expressions (expressions using base-10 notation), though no (non-recurring) infinite decimal can ever be written down. Nonetheless, there are uncountably many such decimals, as the usual diagonalization argument shows.
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[8, p. 16].
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In symbols, Dummett expresses ‘Jones was brave or Jones was not brave’ as \((\alpha{\ {\square}\hspace{-1ex}\to} \beta)\vee(\alpha{\ {\square}\hspace{-1ex}\to}\neg\beta)\), where ‘\({\ {\square}\hspace{-1ex}\to}\)’ is the subjunctive conditional, ‘If it were α it would be β’. The correct analysis is \((\alpha {\ {\square}\hspace{-1ex}\to}\beta)\vee(\alpha {\ \lozenge\hspace{-1ex}\to}\neg\beta)\), where \(\alpha {\ \lozenge\hspace{-1ex}\to}\beta\) is equivalent to \(\neg(\alpha {{\square}\to}\neg\beta)\), that is, it is not the case that if it were α it would not be β, i.e., if it were α it might be β.
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For an elaboration of this argument, see [18].
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See the radical anti-realism of Jacques Dubucs: e.g., [7, p. 214].
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See, e.g., [19]. Note that a one-step inference is a function, from one proof to another.
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[3, p. 15].
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See [17].
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Note that nothing I have said above commits me to the S5 principle \(\lozenge p\to\Box\lozenge p\). If one were committed to it, one would need a proof that there is no proof of \(\neg p\). More plausibly, a theory of necessity as proof will reject \(\lozenge p\to\Box\lozenge p\) and endorse an S4 theory of modality.
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Acknowledgments
This paper was presented at the conference ‘(Anti-)Realisms, Logic and Metaphysics’: Nancy, 28 June–1 July 2006 and first published in Kriterion 50 (2010), 47–67.
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Read, S. (2012). Necessary Truth and Proof. In: Rahman, S., Primiero, G., Marion, M. (eds) The Realism-Antirealism Debate in the Age of Alternative Logics. Logic, Epistemology, and the Unity of Science, vol 23. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1923-1_13
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