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Manifestability and Epistemic Truth

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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

  1. See e.g. Dummett (1973b, 1976 and 1978), Prawitz (1977), Tennant (1997).

  2. See e.g. Prawitz (1977, 4).

  3. See Brouwer (1908, 109), Dummett (1977, 8 and 1991b, 9) and (Prawitz 1980, 9). I’ll return to the question whether Dummett has really endorsed a version of the argument in Sect. 1.2 below.

  4. See Wright (1992 and 2001) and Salerno (2000). For a first fully explicit formulation of the argument, and a potential criticism, see Incurvati and Murzi (2008).

  5. 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).

  6. 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.

  7. An early version of the argument can be found in Dummett (1959). More mature versions are developed in Dummett (1973a, b and 1976).

  8. 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.

  9. See also Cozzo (2008, 126–132).

  10. 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.

  11. \({{\it {CT5}}}\) is the contrapositive of Frederic B. Fitch’s (1963) Theorem 5. Fitch credits the bulk of the proof to an anonymous referee, who was later identified as Alonzo Church. See Church (2009) and Salerno (2010).

  12. 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. □

  13. See Williamson (2000, ch. 12) for extensive, and persuasive, discussion.

  14. See Williamson (1982 and 1988).

  15. The example is Wolfgang Künne’s. See Künne (2007).

  16. See also Usberti (1995) for a treatment of the Church-Fitch proof along these lines.

  17. See Dummett (2009), Florio and Murzi (2009) and Murzi (2009) for further discussion.

  18. See Cozzo (1994), Tennant (1997, ch. 8), Prawitz (1998b, 302–303 and 329) and Hand (2003).

  19. Nothing in what follows hinges on this choice: the conditional here could equally be material, or intuitionistic.

  20. 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).

  21. 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.

  22. The second clause was suggested to me by Dag Prawitz (p.c.), in response to my observation that we cannot recognize proofs of \(P \land \lnot\mathcal{K}P. \)

  23. See e.g. Prawitz (1977), Dummett (1991b) and Tennant (1997).

  24. See e.g. Wright (2003).

  25. See e.g. Dummett (1991a and 1993).

  26. See also Tennant (2009) for a restriction that circumvents some of the criticisms advanced in Brogaard and Salerno (2006).

References

  • Appiah A (1985) Verification and the manifestation of meaning. Proc Aristotelian Soc Supp 59:17–31

    Google Scholar 

  • Bermudez J (2009) Truth, indefinite extensibility, and Fitch’s paradox. In: Salerno J (ed) New essays on the knowability paradox. Oxford University Press, Oxford, pp 76–90

    Chapter  Google Scholar 

  • Brogaard B, Salerno J (2006) Knowability and the closure principle. Am Philos Q 43(3):261–270

    Google Scholar 

  • Brouwer LEJ (1908) The unreliability of logical principles. In: Collected works I: philosophy and foundations of mathematics. Edited by A Heyting.

  • Byrne D (2005) Compositionality and the manifestation challenge. Synthese 144:101–136

    Article  Google Scholar 

  • Church A (2009) Anonymous referee reports. In: Salerno J (ed) New essays on the knowability paradox. Oxford University Press, Oxford, pp 13–20

    Chapter  Google Scholar 

  • Cozzo C (1994) What can we learn from the paradox of knowability? Topoi 13:71–78

    Article  Google Scholar 

  • Cozzo C (2008) Introduzione a Dummett. Laterza, Bari

    Google Scholar 

  • Dean W, Kurokawa H (2010) From the knowability paradox to the existence of proofs. Synthese 176:177–225

    Article  Google Scholar 

  • Dummett M (1959) Truth. Proc Aristotelian Soc 59:141–162. [Now in Dummett (1978), pp 1–25]

  • Dummett M (1973a) Frege: philosophy of language. Duckworth, London

    Google Scholar 

  • Dummett M (1973b) The philosophical basis of intuitionistic logic. In: Rose H, Shepherdson J (eds) Logic colloquium ‘73, North-Holland, Amsterdam. [Now in Dummett (1978), pp 215–247]

  • Dummett M (1976) What is a theory of meaning II. In: Evans G, McDowell J (eds) Truth and meaning. Oxford University Press, Oxford, pp 67–137

    Google Scholar 

  • Dummett M (1977) Elements of intuitionism. Oxford University Press, Oxford

    Google Scholar 

  • Dummett M (1978) Truth and other enigmas. Duckworth, London

    Google Scholar 

  • Dummett M (1979) What does the appeal to use do for the theory of meaning? In: Margalit A (ed) Meaning and use. Reidel, Dordrecht. [Now in Dummett (1993), pp 106–116]

  • Dummett M (1991a) Frege: philosophy of mathematics. Duckworth, London

    Google Scholar 

  • Dummett M (1991b) The logical basis of metaphysics. Harvard University Press, Harvard

    Google Scholar 

  • Dummett M (1993) The seas of language. Oxford University Press, Oxford

    Google Scholar 

  • Dummett M (2001) Victor’s error. Analysis 61(269):1–2

    Google Scholar 

  • Dummett M (2004) Truth and the past. Columbia University Press, New York

    Google Scholar 

  • Dummett M (2007) Reply to Wolfgang Künne. In: Auxier RE, Hahn LE (eds) The philosophy of Michael Dummett. Open Court, Chicago, pp 345–350

  • Dummett M (2009) Fitch’s paradox of knowability. In: Salerno J (ed) New essays on the knowability paradox. Oxford University Press, Oxford

    Google Scholar 

  • Fitch F (1963) A logical analysis of some value concepts. J Symb Log 28(2):135–142

    Google Scholar 

  • Florio S, Murzi J (2009) The paradox of idealisation. Analysis 69:461–469

    Article  Google Scholar 

  • Hand M (2003) Knowability and epistemic truth. Australas J Philos 81(2):216–228

    Article  Google Scholar 

  • Hand M (2009) Performance and paradox. In: Salerno J (ed) New essays on the knowability paradox. Oxford University Press, Oxford, pp 283–301

    Chapter  Google Scholar 

  • Hart W, McGinn C (1976) Knowledge and necessity. J Philos Log 5:205–208

    Article  Google Scholar 

  • Incurvati L, Murzi J (2008) How basic is the basic revisionary argument? Analysis 68(4):303–309

    Article  Google Scholar 

  • Künne W (2007) Two principles concerning truth. In: Auxier RE, Hahn LE (eds) The philosophy of Michael Dummett. Open Court, Chicago, pp 315–324

    Google Scholar 

  • Martin-Löf P (1995) Verificationism then and now, Vienna Circle Institute Yearbook 3. Kluwer, Dordrecht, pp 187–196

    Google Scholar 

  • Murzi J (2009) Knowability and bivalence: intuitionistic solutions to the paradox of knowability. Philos Stud 149(2):269–281

    Article  Google Scholar 

  • Prawitz D (1977) Meaning and proofs: on the conflict between classical and intuitionistic logic. Theoria 43:1–40

    Google Scholar 

  • Prawitz D (1980) Intuitionistic logic: a philosophical challenge. In: Wright GH (ed) Logic and philosophy. Martinus Nijhoff Publishers, The Hague, pp 1–10

    Google Scholar 

  • Prawitz D (1998a) Comments on Lars Bergström’s paper ‘Prawitz’s version of verificationism’. Theoria 64:293–303

    Google Scholar 

  • Prawitz D (1998b) Comments on Michael Dummett’s paper ‘Truth from the constructive standpoint’. Theoria 64:283–292

    Google Scholar 

  • Salerno J (2000) Revising the logic of logical revision. Philos Stud 99:211–227

    Article  Google Scholar 

  • Salerno J (2010) Knowability noir: 1945–1963. In: Salerno J (ed) New essays on the knowability paradox. Oxford University Press, Oxford, pp 29–49

    Google Scholar 

  • Shieh S (1998) Undecidability in anti-realism. Philos Math 6:324–333

    Article  Google Scholar 

  • Tennant N (1997) The taming of the true. Oxford University Press, Oxford

    Google Scholar 

  • Tennant N (2002) Victor vanquished. Analysis 62:135–142

    Article  Google Scholar 

  • Tennant N (2009) Revamping the restriction strategy. In: Salerno J (ed) New essays on the knowability paradox. Oxford University Press, Oxford, pp 223–239

    Chapter  Google Scholar 

  • Usberti G (1995) Significato e conoscenza: per una critica del neoverificazionismo. Guerini Scientifica, Milan

  • Williamson T (1982) Intuitionism disproved? Analysis 42:203–207

    Article  Google Scholar 

  • Williamson T (1988) Knowability and constructivism. Philos Q 38:422–432

    Article  Google Scholar 

  • Williamson T (2000) Knowledge and its limits. Oxford University Press, Oxford

    Google Scholar 

  • Williamson T (2009) Tennant’s troubles. In: Salerno J (ed) New essays on the knowability paradox. Oxford University Press, Oxford

    Google Scholar 

  • Wright C (1992) Truth and objectivity. Harvard University Press, Cambridge

    Google Scholar 

  • Wright C (2001) On being in a quandary. Mind 110:45–98

    Article  Google Scholar 

  • Wright C (2003) Vagueness: a fifth column approach. In: Beall JC (ed) Liars and heaps. Oxford University Press, Oxford, pp 84–105

    Google Scholar 

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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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