Abstract
The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema “True(A)↔A”, and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in “ordinary” contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True(A) with A within the language.
The logic is also shown to have the resources required to represent the way in which sentences (like the Liar sentence and the Curry sentence) that lead to paradox in classical logic are “defective”. We can in fact define a hierarchy of “defectiveness” predicates within the language. Contrary to claims that any solution to the paradoxes just breeds further paradoxes (“revenge problems”) involving defectiveness predicates, there is a general consistency/conservativeness proof that shows that talk of truth and the various “levels of defectiveness” can all be made coherent together within a single object language.
Similar content being viewed by others
REFERENCES
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.
Field, H.: On conservativeness and incompleteness, J. Philos. 81 (1985), 239–60.
Field, H.: Deflating the conservativeness argument, J. Philos. 96 (1999), 533–40.
Field, H.: Saving the truth schema from paradox, J. Philos. Logic 31 (2002), 1–27.
Gupta, A. and Belnap, N.: The Revision Theory of Truth, MIT Press, Cambridge, MA, 1993.
Hajek, P., Paris, J. and Shepherdson, J.: The liar paradox and fuzzy logic, J. Symbolic Logic 65 (2000), 339–346.
Ketland, J.: Deflation and Tarski's paradox, Mind 108 (1999), 69–94.
Kripke, S.: Outline of a theory of truth, J. Philos. 72 (1975), 690–716.
Restall, G.: Arithmetic and truth in ?ukasiewicz's infinitely valued logic, Logique et Analyse 139-140 (1992), 303–312.
Rogers, H.: Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967.
Shapiro, S.: Conservativeness and incompleteness, J. Philos. 80 (1983), 521–31.
Shapiro, S.: Proof and truth: Through thick and thin, J. Philos. 95 (1998), 493–521.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Field, H. A Revenge-Immune Solution to the Semantic Paradoxes. Journal of Philosophical Logic 32, 139–177 (2003). https://doi.org/10.1023/A:1023027808400
Issue Date:
DOI: https://doi.org/10.1023/A:1023027808400