Abstract
In Sect. 11.2, we argue that the adoption of a tenseless notion of truth entails a realistic view of propositions and provability. This view, in turn, opens the way to the intelligibility of the classical meaning of the logical constants and consequently is incompatible with the antirealism of orthodox Intuitionism. In Sect. 11.3, we show how what we call the “potential” intuitionistic meaning of the logical constants can be defined, on the one hand, by means of the notion of atemporal provability and, on the other hand, by means of the operator K of epistemic logic. Intuitionistic logic, as reconstructed within this perspective, turns out to be a part of epistemic logic, so that it loses its traditional foundational role, antithetic to that of classical logic . In Sect. 11.4, we uphold the view that certain consequences of the adoption of a temporal notion of truth, despite their apparent oddity, are quite acceptable from an antirealist point of view.
with G. Usberti
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Notes
- 1.
In some respects, this seems to be the move implicit in Martin-Löf’s type theory, in particular in his distinction between propositions and judgements. According to it, “A is potentially true” is a judgement, and a general feature of judgements is that the logical operations cannot be applied to them; as a consequence, a judgement expressing the content of PEM does not exist. However, this answer is acceptable only to someone who subscribes to (1) the distinction between judgements and propositions, and (2) the reasonability of the interdiction to apply negation to a judgement. In Martino and Usberti (1991), we stated some reasons not to accept either (1) or (2). Observe that if PEM were meaningless for the reason at issue, so would be Martin-Löf’s claim that “If A is actually true, then it is potentially true”, where the inapplicability of the logical constants to judgements is violated too.
- 2.
A similar interpretation of the intuitionistic logical constants has been proposed by Carlo Dalla Pozza (1991).
- 3.
For a nice introduction to this topic see (Shapiro 1985).
- 4.
For a formulation of this paradox and the problems it raises see also the introduction to the present issue.
- 5.
Observe that, even if our epistemic operator K may be analysed as \(K= \Diamond K_{0}\), it is not to be regarded as empirical, because of the presence of the modal ingredient: the mere knowability of a proposition holds quite independently of any empirical event of actual knowledge.
References
Dalla Pozza, C. (1991). Un’interpretazione pragmatica della logica proposizionale intuizionistica. In G. Usberti (Ed.), Problemi fondazionali nella teoria del significato (pp. 49–75). Firenze: Leo Olschki Editore.
Dummett, M. (1977). Elements of intuitionism. Oxford: Clarendon Press.
Dummett, M. (1987). Reply to dag prawitz. In B. Taylor (Ed.), Michael Dummett: Contributions to philosophy (pp. 281–286). The Hague: Nijhoff.
Martin-Löf, P. (1991). A path from logic to metaphysics. In G. Sambin & G. Corsi (Eds.), Atti del Congresso Nuovi Problemi della Logica e della Filosofia della Scienza (pp. 141–149), Vol. II. Bologna: CLUEB.
Martino, E., & Usberti, G. (1991). Propositions and judgements in Martin-Löf. In G. Usberti (Ed.), Problemi fondazionali nella teoria del significato (pp. 125–136). Firenze: Leo Olschki Editore. Reprinted here as chapter 9.
Prawitz, D. (1987). Dummett on a theory of meaning and its impact on logic. In B. M. Taylor (Ed.), Michael Dummett: Contributions to philosophy (pp. 117–165). Dordrecht: Springer.
Shapiro, S. (1985). Epistemic and intuitionist arithmetic. In S. Shapiro (Ed.), Intensional mathematics (pp. 11–46). Amsterdam: North Holland Publishing Company.
Williamson, T. (1988). Knowability and constructivism. Philosophical Quarterly, 38, 422–432.
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Martino, E. (2018). Temporal and Atemporal Truth in Intuitionistic Mathematics. In: Intuitionistic Proof Versus Classical Truth. Logic, Epistemology, and the Unity of Science, vol 42. Springer, Cham. https://doi.org/10.1007/978-3-319-74357-8_11
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