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.
Keywords Believer Paradox  epistemic logic  hierarchy solutions to the semantic paradoxes  Knower Paradox  provability logic  self-reference  syntactical treatments of modalities
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DOI 10.1007/s10849-004-6406-y
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References found in this work BETA
Modal Logic: An Introduction.Brian F. Chellas - 2012 - Cambridge University Press.
The Ways of Paradox, and Other Essays.W. V. Quine - 1966 - Harvard University Press.

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Citations of this work BETA
Non‐Classical Knowledge.Ethan Jerzak - forthcoming - Philosophy and Phenomenological Research.
Paradoxes of Interaction?Johannes Stern & Martin Fischer - 2015 - Journal of Philosophical Logic 44 (3):287-308.

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