Journal of Logic, Language and Information 14 (1):13-48 (2004)
AbstractThis 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.
Similar books and articles
The Unprovability of Consistency: An Essay in Modal Logic.George Boolos - 1979 - Cambridge, England: Cambridge University Press.
The paradox of the knower without epistemic closure.Charles B. Cross - 2001 - Mind 110 (438):319-333.
On strong provability predicates and the associated modal logics.Konstantin N. Ignatiev - 1993 - Journal of Symbolic Logic 58 (1):249-290.
The principle of uniform solution (of the paradoxes of self-reference).Nicholas J. J. Smith - 2000 - Mind 109 (433):117-122.
Finite Kripke models and predicate logics of provability.Sergei Artemov & Giorgie Dzhaparidze - 1990 - Journal of Symbolic Logic 55 (3):1090-1098.
Explicit provability and constructive semantics.Sergei N. Artemov - 2001 - Bulletin of Symbolic Logic 7 (1):1-36.
Fitch's proof, verificationism, and the knower paradox.J. C. Beall - 2000 - Australasian Journal of Philosophy 78 (2):241 – 247.
Added to PP
Historical graph of downloads
Citations of this work
Non‐Classical Knowledge.Ethan Jerzak - 2019 - Philosophy and Phenomenological Research 98 (1):190-220.
The Elimination of Self-Reference: Generalized Yablo-Series and the Theory of Truth.P. Schlenker - 2007 - Journal of Philosophical Logic 36 (3):251-307.
Some remarks on restricting the knowability principle.Martin Fischer - 2013 - Synthese 190 (1):63-88.
Montague’s Paradox, Informal Provability, and Explicit Modal Logic.Walter Dean - 2014 - Notre Dame Journal of Formal Logic 55 (2):157-196.
References found in this work
The Ways of Paradox, and Other Essays.Willard Van Orman Quine - 1966 - New York, NY, USA: Harvard University Press.
Formal Philosophy: Selected Papers of Richard Montague.Richard Montague - 1974 - New Haven: Yale University Press.
Propositional Attitudes: An Essay on Thoughts and How We Ascribe Them.Mark Richard - 1990 - Cambridge University Press.