The Knower Paradox in the light of provability interpretations of modal logic

Author: Égré Paul

Source: Journal of Logic, Language and Information, Volume 14, Number 1, December 2004 , pp. 13-48(36)

Publisher: Springer

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Abstract:

This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework of first-order arithmetic and in that of modal logic with fixed point operators. It is shown that the notion of a syntactical treatment of modalities is ambiguous between a self-referential treatment and a metalinguistic treatment of modalities, and that these two notions are independent. I survey and compare the provability interpretations of modality respectively given by Skyrms, B. (1978, The Journal of Philosophy 75: 368–387) Anderson, C.A. (1983, The Journal of Philosophy 80: 338– 355) and Solovay, R. (1976, Israel Journal of Mathematics 25: 287–304). I examine how these interpretations enable us to bypass the limitations imposed by the Knower Paradox while preserving the laws of classical logic, each time by appeal to a distinct form of hierarchy.

Keywords: Believer Paradox; epistemic logic; hierarchy solutions to the semantic paradoxes; Knower Paradox; provability logic; self-reference; syntactical treatments of modalities

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10849-004-6406-y

Affiliations: 1: IHPST, 13, Rue du Four, 75006, Paris, France, Email: paulegre@magic.fr

Publication date: 2004-12-01