Abstract
I argue that the standard anti-realist argument from manifestability to intuitionistic logic is either unsound or invalid. Strong interpretations of the manifestability of understanding are falsified by the existence of blindspots for knowledge. Weaker interpretations are either too weak, or gerrymandered and ad hoc. Either way, they present no threat to classical logic.
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Notes
See e.g. Prawitz (1977, 4).
Notice that undecidability, unlike decidability is a tensed notion: what was once undecidable, e.g. Fermat’s Last Theorem, may become decidable. See also Shieh (1998).
Here I follow Williamson (2000) and unapologetically quantify over sentences, or propositions. Nothing in what follows hinges on this choice. The sensitive reader is invited to substitute universally quantified theses such as \({WVER}\) with the corresponding schemata, here and throughout.
I am here using ‘proof’ in a very broad sense, one according to which proofs need not be mathematical arguments. Dummett’s notion of verification, or the more neutral notion of correct argument, would probably be more appropriate. I’ll stick to talk of proofs here and throughout for the sake of simplicity.
See also Cozzo (2008, 126–132).
For a more detailed discussion, see Dummett (1979, 116), Prawitz (1980), Wright (1992, ch. 2), Salerno (2000), Wright (2001) and Incurvati and Murzi (2008). One might object that, even when he explicitly endorses the knowability of truth, Dummett would not thereby accept the Basic Revisionary Argument. For instance, Dummett (1976) takes semantic anti-realism to be “a regulative principle governing the notion of truth: if a statement is true, it must be in principle possible to know that it is true” (ibid., 99). In symbols:
$$ ({WVER_{Tr}})\, \forall \varphi( Tr(\ulcorner {\dot{\varphi}}\urcorner)\to\lozenge{\mathcal{K}} Tr( \ulcorner{\dot{\varphi}}\urcorner)). $$Dummett is here endorsing the claim that every truth is knowable—not quite \({WVER}. \) And \({WVER}_{Tr}\) only entails \({WVER}, \) if true sentences hold, i.e. if the following principle of semantic shift, as Dummett calls it, holds:
$$ {(Shift)} \; \forall\varphi Tr \; (\ulcorner {\dot{\varphi}}\urcorner) \to \varphi. $$However, one might insist, Dummett himself is prepared to question principles of semantic shift (see e.g. Dummett 2004, 14–39). Hence, it would be a mistake to attribute to Dummett a commitment to the Basic Revisionary Argument. One problem with this, is that we would still get an argument against the semantic Principle of Bivalence, that every sentence is either true or false, if we took as assumptions of a slightly modified version of the Basic Revisionary Argument the following three claims: that we’re justified in believing the Principle of Bivalence, that we’re justified in believing \({WVER_{Tr}}, \) and that we’re not presently justified in believing, of every sentence, that either it or its negation can be known to be true. To be sure, this latter argument would target classical semantics, as opposed to directly challenging classical logic. All the same, it would have the same two-step structure as the argument against \({LEM}\) we have just presented, and its relevance for the realism/anti-realism debate would be no less central than the Basic Revisionary Argument, at least insofar as one takes the Principle of Bivalence to be a “mark of realism”, as Dummett himself puts it. Thanks to Ian Rumfitt for raising this potential concern.
Proof: Let P be some forever-unknown truth, and assume that someone at some time knows that P is true but forever-unknown, i.e. assume \(\mathcal{K}(P \land \lnot\mathcal{K}P). \) If knowledge is factive and distributes over conjunction, \(\mathcal{K}P\land\lnot\mathcal{K}P\) follows. Contradiction. We must therefore negate, and discharge, one of our initial assumptions. By necessitation, and the modal principle \(\square \lnot A \vdash \lnot \lozenge A, \) we can conclude on no assumptions that truths of the form \(\varphi \land \lnot\mathcal{K}\varphi\) are unknowable. Now assume that there are forever-unknown truths. By existential generalization, \(Q \land \lnot\mathcal{K}Q\) follows. If \({WVER}\) holds, it is possible to know \(Q \land \lnot\mathcal{K}Q. \) But this contradicts our previous result, that truths of the form \(\mathcal{K}(\varphi \land \lnot\mathcal{K}\varphi)\) are unknowable. So there are no forever-unknown truths after all. By one step of arrow introduction, and assuming the full power of classical logic, we can conclude \(\forall\varphi(\varphi\to\lozenge\mathcal{K}\varphi)\to\forall \varphi(\varphi\to\mathcal{K}\varphi\)) follows. □
See Williamson (2000, ch. 12) for extensive, and persuasive, discussion.
The example is Wolfgang Künne’s. See Künne (2007).
See also Usberti (1995) for a treatment of the Church-Fitch proof along these lines.
Nothing in what follows hinges on this choice: the conditional here could equally be material, or intuitionistic.
Recall our quote in Sect. 1, that understanding a sentence “is to be able to recognize a verification of it if one is produced” (italics added).
Thus, the first layer of logical form of \((A \to B) \lor (C \land (D \lor \lnot Q))\) is simply \(\phi \lor \psi, \) and so on.
The second clause was suggested to me by Dag Prawitz (p.c.), in response to my observation that we cannot recognize proofs of \(P \land \lnot\mathcal{K}P. \)
See e.g. Wright (2003).
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Acknowledgments
I wish to thank the audience of the workshop “Anti-realistic Notions of Truth”, in particular Bernhard Weiss, Cesare Cozzo, Dag Prawitz and Gabriele Usberti, as well as the participants to two graduate lectures I gave at the University of Padova in November 2010, especially Massimiliano Carrara, Davide Fassio and Enrico Martino. Thanks are also due to Bob Hale and Dominic Gregory for helpful comments on previous drafts of some of this material, and to my former students in Sheffield, especially Joe Beswick and Jonathan Payne, for helping me think more clearly about manifestability. I gratefully acknowledge the generous financial support of the University of Sheffield, the Analysis Trust, and the Alexander von Humboldt foundation.
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Murzi, J. Manifestability and Epistemic Truth. Topoi 31, 17–26 (2012). https://doi.org/10.1007/s11245-011-9106-7
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DOI: https://doi.org/10.1007/s11245-011-9106-7