The knower paradox in the light of provability interpretations of modal logic

Journal of Logic, Language and Information 14 (1):13-48 (2004)
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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.

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Citations of this work

Non‐Classical Knowledge.Ethan Jerzak - 2019 - Philosophy and Phenomenological Research 98 (1):190-220.
Paradoxes and contemporary logic.Andrea Cantini - 2008 - Stanford Encyclopedia of Philosophy.
Montague’s Paradox, Informal Provability, and Explicit Modal Logic.Walter Dean - 2014 - Notre Dame Journal of Formal Logic 55 (2):157-196.

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References found in this work

Modal Logic: An Introduction.Brian F. Chellas - 1980 - Cambridge University Press.
The Ways of Paradox, and Other Essays.Willard Van Orman Quine - 1966 - New York, NY, USA: Harvard University Press.
Mathematical Logic.Willard Van Orman Quine - 1940 - Cambridge: Harvard University Press.

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