Abstract
I will be concerned with the following question: are there compelling arguments for postulating a distinction between the truth of a statement and the recognition of its truth, when truth is conceived along the lines of a suitable generalization of the intuitionistic idea that it should be characterized as the existence of a proof? I will argue that the distinction is not necessary within the conceptual framework of intuitionism by replying to two arguments to the contrary, one based on the paradox of inference, the other on considerations concerning the content of a statement.
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Notes
An argument for the explicit nature of evidence is given by Williamson (2000, ch. 9). A discussion of this point is beyond the limits of the present paper.
“To be in a position to know p, it is neither necessary to know p nor sufficient to be physically and psychologically capable of knowing p. No obstacle must block one’s path to knowing p. If one is in a position to know p, and one has done what one is in a position to do to decide whether p is true, then one does know p. […] Thus being in a position to know […] is factive” (Williamson 2000, 95).
Dummett (1977, 394).
E.g. Prawitz (1987, 139): “The usual intuitionistic attempt to explain the logical constants in terms of what counts as proofs of sentences of different logical forms is quite misleading in that respect.”
For instance by Prawitz (1987, 140).
Ibid. A different construal would take it as an argument for the thesis that a neo-verificationist explanation of the logical constants requires such a distinction, where neo-verificationism is characterized by an integration of intuitionistic and Gentzenian ideas. But Dummett’s overall position encourages the former construal; see for instance Dummett (1975b, 31).
For instance, a proof of “Prime(n)∨¬Prime(n)” is the primality test as applied to n, i.e. the operations permitting to establish wheter n is prime or not. The test as applied to n should not be confused with the general method consisting in applying to every number x the test for x (this general method is a proof of “∀x(Prime(x)∨¬Prime(x))”).
Actually, in Prawitz’s definition the clauses for the constants different from ⇒ and ∀ require that the subproofs are canonical; but, as Prawitz remarks (footnote 9), this requirement can be left out.
Of course I am not speaking of hypothetical evidence, which is recognized by intuitionists, but of indirect evidence.
With “conceptual necessity” I mean a principle whose validity can be extracted from the sole analysis of the concepts involved.
This is not to say that a distinction on different grounds is not possible or even desirable, of course.
I am assuming that a proof of “A is true” is the introspective observation that the actual mental state of the subject is a proof of A.
References
Berker S (2008) Luminosity regained. Philos Imprint 8(2):1–22
Brandom R (1976) Truth and assertibility. J Philos 73(6):137–149
Casalegno P (2002) The problem of non-conclusiveness. Topoi 21(1–2):75–86
Dummett M (1973) Frege: philosophy of language. Duckworth, London
Dummett M (1975a) The justification of deduction. In: Proceedings of the British Academy, LIX, pp 201–231. [Now in Truth and other enigmas, Duckworth, London, pp 290–318]
Dummett M (1975b) The philosophical basis of intuitionistic logic. In: Rose HE, Sheperdson JC (eds) Logic Colloquium, vol 73. North Holland, Amsterdam, pp 5–40
Dummett M (1977) Elements of intuitionism. Clarendon Press, Oxford
Dummett M (1991) The logical basis of metaphysics. Duckworth, London
Frege G (1956) The thought: a logical inquiry. Mind 65(259):289–311
Heyting A (1931) Die intuitionistische Grundlegung der Mathematik. Erkenntnis 2: 106–15. [Eng trans The intuitionist foundations of mathematics. In: Benacerraf P, Putnam H (eds) Philosophy of mathematics. Prentice-Hall, Englewood Cliffs, 1964, pp 42–49
Heyting A (1958) On truth in mathematics. In: Verslag van de plechtige viering van het honderdvijftigjarig bestaan der Koninklijke Nederlandse Akademie van Wetenschappen. North Holland, Amsterdam, pp 277–279
Martin-Löf P (1985) On the meaning and justification of logical laws. In: Bernardi C, Pagli P (eds) Atti degli Incontri di Logica Matematica, vol II. Università di Siena, pp 291–340
Prawitz D (1987) Dummett on a theory of meaning and its impact on logic. In: Taylor B (ed) Michael Dummett—Contributions to philosophy. Nijhoff, Dordrecht, pp 117–165
Prawitz D (1998) Truth and objectivity from a verificationist point of view. In: Dales HG, Oliveri G (eds) Truth in mathematics. Oxford University Press, Oxford, pp 41–51
Prawitz D (2005) Logical consequence from a constructivist point of view. In: Shapiro S (ed) The Oxford handbook of philosophy of mathematics and logic. Oxford University Press, Oxford, pp 671–695
Tarski A (1935) Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica 1, pp. 261–405. [Eng trans The concept of truth in formalized languages. In: Tarski A (1983) Logic, semantics, metamathematics 2nd edn. Hackett Pub. Co., Indianapolis, pp 152–278]
Usberti G (1995) Significato e conoscenza. Guerini e Associati, Milano
Usberti G (2006) Towards a semantics based on the notion of justification. Synthese 149(3):675–699
Williamson T (2000) Knowledge and its limits. Oxford University Press, Oxford
Acknowledgment
I am much indebted to Luca Tranchini for helpful comments on this paper. This work was supported by the MIUR fund No. 2007H7X4YE_002.
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Usberti, G. Anti-Realist Truth and Truth-Recognition. Topoi 31, 37–45 (2012). https://doi.org/10.1007/s11245-011-9110-y
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DOI: https://doi.org/10.1007/s11245-011-9110-y