Operators in the paradox of the knower
type: Item
instance of: scholarly article
title: Operators in the paradox of the knower
zbMATH Open document ID: 0803.03003
DOI: 10.1007/BF01064487
author: Patrick Grim
published in: Synthese
publication date: 1995-01-03T00:00:00Z
review text: The Paradox of the Knower can be developed in a formal arithmetic equipped with the usual introspective machinery for the Gödel diagonalization theorem to obtain. This theorem says that with every well-formed formula \(Wx\) in the system, there is a sentence \(S\) such that \(S\equiv W\ulcorner S\urcorner\) (\(\ulcorner S\urcorner\) is the Gödel number of \(S\)). Now, if this system contains a predicate symbol, \(K\), with the premises: \(K\ulcorner A\urcorner\to A\), \(K\ulcorner(K\ulcorner A\urcorner\to A)\urcorner\), \(I(\ulcorner A\urcorner, \ulcorner B\urcorner)\to (K\ulcorner A\urcorner\to K\ulcorner B\urcorner)\), where \(I\) is the deducibility relation for the system, then it is inconsistent. This syntactic property becomes a paradox, ``the Paradox of the Knower'', if we believe that \(K\ulcorner A\urcorner\) can be interpreted as ``\(A\) is known''. In trying to resolve the paradox, some authors have suggested that the knowledge predicate \(K\ulcorner A\urcorner\) rather ought to be dealt with as an operator, or sentential connective. That is, not taking a term, like a Gödel number, as argument but rather a sentence. The author argues against such views. In particular, he deals with systems where all formulae, \(W\), are recoverably encoded, not by Gödel numbers but by closed formulae, ``counter-formulas'', \([W]\). He finds that the diagonalization theorem still holds, and that the corresponding premises are inconsistent whereby also the interpretation of \(K[A]\) as ``\(A\) is known'' becomes paradoxical. However, if \(K\) operates directly on non-encoded formulas, a corresponding diagonalization theorem need not hold, and \(KA\to A\), etc., need not give an inconsistency. The author concludes that the diagonalization theorem and the Paradox of the Knower are not merely a question of whether a sentence-generating function takes terms or sentences as arguments. Rather, it is the ``effective expressive power'', of the system, that matters.
Mathematics Subject Classification ID: 03A05
Mathematics Subject Classification ID: 03F30
Mathematics Subject Classification ID: 03F40
zbMATH DE Number: 490766
zbMATH Keywords: Paradox of the Knower
zbMATH Keywords: formal arithmetic
zbMATH Keywords: Gödel diagonalization theorem
zbMATH Keywords: Gödel number
zbMATH Keywords: knowledge predicate
zbMATH Keywords: operator
zbMATH Keywords: inconsistency
MaRDI profile type: Publication
cites work: On modal logic with propositional quantifiers
cites work: Q3235339
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cites work: A paradox regained
cites work: Q5726696
cites work: Necessity predicates and operators
full work available at URL: https://doi.org/10.1007/bf01064487
OpenAlex ID: W1991226376
description: scientific article; zbMATH DE number 490766