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date: 21 June 2021

Abstract and Keywords

Antinomies involving the assertibility modal arise when we consider what are called “syntactic interpretations” of the assertibility operator A, translation schemes in which the operator is interpreted in terms of a monadic predicate in the nonmodal fragment of the language. Such a translation scheme, of course, is nothing more than a systematic attempt to replace the assertibility modality by a concept, and trouble tends to arise precisely when the concept is too truthlike. In this chapter, a number of paradoxical syntactic interpretations of the assertibility modality are explored. In each case, the paradox consists in an inconsistency affecting the syntactic transcription of certain apparently unexceptionable combinations of modal principles concerning the assertibility operator. It is argued that the modal construal of the notion of assertibility is fundamental, and a standard model theoretic semantics is presented for it. I finally suggest that an assertibility predicate for a language L with the assertibility operator be recovered at the metalinguistic level: a sentence p will be said to be assertible in L if and only if the sentence Ap is true in L, with the result that the alternative forms for a theory of the assertibility property in L exactly mirror the alternative forms for a theory of truth for L. Some consequences of these observations for the philosophy of mathematics are explored.

Keywords: assertion, warranted assertibility, syntactic interpretations, semantic paradoxes, truth, Kripke semantics, experimental logics, Gôdel’s theorem, Lôb’s theorem

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