Saving the Truth Schema from Paradox
- Hartry Field
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
The paper shows how we can add a truth predicate to arithmetic (or formalized syntactic theory), and keep the usual truth schema Tr(〈A〉)↔A (understood as the conjunction of Tr(〈A〉)→A and A→Tr(〈A〉)). We also keep the full intersubstitutivity of Tr(〈A〉)) with A in all contexts, even inside of an →. Keeping these things requires a weakening of classical logic; I suggest a logic based on the strong Kleene truth tables, but with → as an additional connective, and where the effect of classical logic is preserved in the arithmetic or formal syntax itself. Section 1 is an introduction to the problem and some of the difficulties that must be faced, in particular as to the logic of the →; Section 2 gives a construction of an arithmetically standard model of a truth theory; Section 3 investigates the logical laws that result from this; and Section 4 provides some philosophical commentary.
- Anderson, A. R. and Belnap, N. D.: Entailment: The Logic of Relevance and Necessity, Vol. 1, Princeton University Press, Princeton, 1986.
- Beall, J. C.: Curry's paradox, in E. N. Zalta (ed.), (Online) Stanford Encyclopedia of Philosophy, CSLI, Stanford University, 2001.
- Boolos, G.: Provability, truth and modal logic, J. Philos. Logic 9 (1980), 1-7.
- Boolos, G.: The Logic of Provability, Cambridge University Press, Cambridge, 1993.
- Brady, R. T.: The non-triviality of dialectical set theory, in G. Priest, R. Routley and J. Norman (eds), Paraconsistent Logic: Essays on the Inconsistent, Philosophia Verlag, 1989, pp. 437-470.
- Feferman, S.: Toward useful type-free theories, I, J. Symbolic Logic 49 (1984), 75-111.
- Field, H.: Truth and the Absence of Fact, Oxford University Press, Oxford, 2001.
- Field, H.: Mathematical undecidables, metaphysical realism and equivalent descriptions, in L. Hahn (ed.), The Philosophy of Hilary Putnam, Open Court, 2003.
- Friedman, H. and Sheard, M.: An axiomatic approach to self-referential truth, Ann. Pure Appl. Logic 33 (1987), 1-21.
- Hajek, P., Paris, J. and Shepherdson, J.: The liar paradox and fuzzy logic, J. Symbolic Logic 65 (2000), 339-346.
- Kripke, S.: Outline of a theory of truth, J. Philos. 72 (1975), 690-716.
- McGee, V.: Truth, Vagueness, and Paradox, Hackett, Indianapolis, 1991.
- Priest, G.: Paraconsistent logic, in D. M. Gabbay and F. Günthner (eds), Handbook of Philosophical Logic, 2nd edn, Vol. 3, D. Reidel, Dordrecht, 2000, pp. 437-470.
- Priest, G.: An Introduction to Non-Classical Logic, Cambridge University Press, Cambridge, 2001.
- Restall, G.: Arithmetic and truth in Łukasiewicz's infinitely valued logic, Logique et Analyse 139-140 (1992), 303-312.
- Smorynski, C.: Self-Reference and Modal Logic, Springer-Verlag, New York, 1985.
- Saving the Truth Schema from Paradox
Journal of Philosophical Logic
Volume 31, Issue 1 , pp 1-27
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- law of excluded middle
- Hartry Field (1)
- Author Affiliations
- 1. New York University, NY, 10011, USA