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Knowability and bivalence: intuitionistic solutions to the Paradox of Knowability

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Abstract

In this paper, I focus on some intuitionistic solutions to the Paradox of Knowability. I first consider the relatively little discussed idea that, on an intuitionistic interpretation of the conditional, there is no paradox to start with. I show that this proposal only works if proofs are thought of as tokens, and suggest that anti-realists themselves have good reasons for thinking of proofs as types. In then turn to more standard intuitionistic treatments, as proposed by Timothy Williamson and, most recently, Michael Dummett. Intuitionists can either point out the intuitionistc invalidity of the inference from the claim that all truths are knowable to the insane conclusion that all truths are known, or they can outright demur from asserting the existence of forever-unknown truths, perhaps questioning—as Dummett now suggests—the applicability of the Principle of Bivalence to a certain class of empirical statements. I argue that if intuitionists reject strict finitism—the view that all truths are knowable by beings just like us—the prospects for either proposal look bleak.

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Notes

  1. See Church (in press) and Theorem 5 : in Fitch (1963).

  2. Assuming the full power of classical logic, this is the contrapositive of Fitch’s original Theorem 5 : \(\exists \varphi(\varphi\land\lnot{\mathcal{K}}p)\to\exists \varphi(\varphi\land\lnot\lozenge{\mathcal{K}}\varphi).\)

  3. Proof Assume that somebody at some time knows that p is true and forever-unknown. If knowledge is factive and distributes over conjunction, \({\mathcal{K}}p\land\lnot{\mathcal{K}}p\) follows. Contradiction. By negation introduction, \(\lnot{\mathcal{K}}(p\land\lnot{\mathcal{K}}p)\). By necessitation, \(\lnot\lozenge{\mathcal{K}}(p\land\lnot{\mathcal{K}}p)\). Now assume that all truths are knowable and that there is a forever-unknown truth. Then, \(p\land\lnot{\mathcal{K}}p\) must be knowable too. Contradiction. We must therefore negate, and discharge, one of our intitial assumptions. Anti-realists will discharge the second, thereby committing themselves, if classical logic is in place, to the claim that all truths will be known at some time. By one step of arrow introduction, \(\forall\varphi(\varphi\to\lozenge{\mathcal{K}}\varphi)\to\forall \varphi(\varphi\to{\mathcal{K}}\varphi)\) follows. □

  4. See Hart and McGinn (1976) and Hart (1979).

  5. See Williamson (1982) and Dummett (2007c, 2008).

  6. The terminology is Timothy Williamson’s (2000).

  7. Many thanks to an anonymous referee for providing stimulus for writing Sect. 2, 3 and 4.

  8. See also Williamson (1988, p. 429).

  9. See also Hand (2003, in press).

  10. See e.g. Dummett (1987, p. 285). I for one don’t think this a very serious problem. As Cesare Cozzo (1994, p. 77) observes, the standard intuitionistic argument for rejecting Bivalence holds even if proofs are conceived of as Platonic objects—after all, we have no guarantee that there is either a Platonic proof, or a Platonic disproof, of Goldbach’s Conjecture. If Bivalence is necessary for semantic realism, then a conception of truth as the existence of a Platonistic proof counts as an anti-realist one. See also Prawitz (1998b, p. 289) for a response to an argument by Dummett (1987, 1998) to the effect that Platonism about proofs enjoys commitment to Bivalence.

  11. See Dummett (1973, pp. 239–243).

  12. See Dummett (2004, 2006).

  13. Besides (2), Philip Percival points out two more untoward intuitionistic consequences of weak verificationism: \(\forall\varphi(\lnot{\mathcal{K}}\varphi\leftrightarrow\lnot\varphi)\) and \(\forall\varphi\lnot(\lnot{\mathcal{K}}\varphi\land\lnot{\mathcal{K}}\lnot\varphi).\) See Percival (1990).

  14. The example is Wolfgang Künne’s (2007).

  15. Proof Assume \(\lnot{\mathcal{K}}p,\)\(\lnot{\mathcal{K}}\lnot p\) and \(p\lor\lnot p.\) Then, \((p \land \lnot{\mathcal{K}}p) \lor (\lnot p \land\lnot{\mathcal{K}}\lnot p)\). Now assume \((p \land \lnot{\mathcal{K}}p).\) By existential introduction, derive \(\exists \varphi(\varphi\land\lnot{\mathcal{K}}\varphi).\) By arrow introduction, it follows that \((p \land \lnot{\mathcal{K}}p) \to \exists \varphi(\varphi\land\lnot{\mathcal{K}}\varphi).\) By similar reasoning, show that \((\lnot p \land \lnot{\mathcal{K}}\lnot p) \to \exists \varphi(\varphi\land\lnot{\mathcal{K}}\varphi).\) By disjunction elimination, conclude \(\exists \varphi(\varphi\land\lnot{\mathcal{K}}\varphi).\)

  16. See Dummett (2007c, p. 349).

  17. See also Dummett (2007b, pp. 303–304).

  18. See Florio and Murzi (in press) for a detailed presentation of the Paradox.

  19. See Dummett (1975) and, especially, Tennant (1997, Chap. 5).

  20. I am here assuming that our cognitive capacities can be parametrized, and that the cognitive capacities of actual agents have an upper bound. Let this upper bound be n. Then, any agent whose cognitive capacities exceed n to a significant degree will count as ideal.

  21. Proof Assume that \(q\land\lnot\exists xIx.\) Then, \(\lozenge{\mathcal{K}}(q\land\lnot\exists xIx)\) follows by weak verificationism. By the Paradox of Idealisation, however, \(\lnot\lozenge{\mathcal{K}}(q\land\lnot\exists xIx)\) holds too. We thus have a contradiction resting on (7), (8) and weak verificationism. A parallel reasoning shows that the Paradox of Idealisation and Dummett’s (1) give us the intuitionistically inconsistent \(\lnot{\mathcal{K}}(q \land \lnot \exists xIx)\) and \(\lnot\lnot{\mathcal{K}}(q \land \lnot \exists xIx).\)

  22. Proof Assume that some agent knows q. Call this agent a. By (7), a is an ideal agent, which contradicts our assumption that there are no ideal agents. Hence, nobody knows q. □

  23. See e.g. Tennant (1997, in press).

  24. See e.g. Wansing (2002).

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Acknowledgements

This is the sequel of a paper I wrote with my colleague and friend Salvatore Florio, to whom I am very much indebted. Many thanks to Dominic Gregory, Bob Hale, Stephen Read, Joe Salerno, Fredrik Stjernberg, Gabriele Usberti, Tim Williamson, and an anonymous referee for valuable comments and discussion on some of the topics discussed herein. An earlier version of this material was presented at the University of Cambridge and at the Eastern Division of the American Philosophical Association in Baltimore. I am grateful to the members of these audiences, especially to Luca Incurvati, for their valuable feedback. I wish to thank the University of Sheffield and the Royal Institute of Philosophy for their generous financial support.

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Murzi, J. Knowability and bivalence: intuitionistic solutions to the Paradox of Knowability. Philos Stud 149, 269–281 (2010). https://doi.org/10.1007/s11098-009-9349-y

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