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

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.