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The paradox of the knower revisited. (English) Zbl 1348.03057
Summary: The Paradox of the Knower was originally presented by D. Kaplan and R. Montague [Notre Dame J. Formal Logic 1, 79–90 (1960; Zbl 0112.00409)] as a puzzle about the everyday notion of knowledge in the face of self-reference. The paradox shows that any theory extending Robinson arithmetic with a predicate \(K(x)\) satisfying the factivity axiom \(K(\overline A)\to A\) as well as a few other epistemically plausible principles is inconsistent. After surveying the background of the paradox, we will focus on a recent debate about the role of epistemic closure principles in the Knower. We will suggest this debate sheds new light on the concept of knowledge which is at issue in the paradox – i.e. is it a “thin” notion divorced from concepts such as evidence or justification, or is it a “thick” notion more closely resembling mathematical provability? We will argue that a number of features of the paradox suggest that the latter option is more plausible. Along the way, we will provide a reconstruction of the paradox using a quantified extension of S. N. Artemov’s Logic of Proofs [Bull. Symb. Log. 7, No. 1, 1–36 (2001; Zbl 0980.03059)], as well as a series of results linking the original formulation of the paradox to reflection principles for formal arithmetic. On this basis, we will argue that while the Knower can be understood to motivate a distinction between levels of knowledge, it does not provide a rationale for recognizing a uniform hierarchy of knowledge predicates in the manner suggested by C. A. Anderson [“The paradox of the knower ”, J. Philos. 80, No. 6, 338–355 (1984; doi:10.2307/2026335)].

MSC:
03F30 First-order arithmetic and fragments
03B42 Logics of knowledge and belief (including belief change)
03F40 Gödel numberings and issues of incompleteness
03F45 Provability logics and related algebras (e.g., diagonalizable algebras)
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