The validity paradox in modal \(\text{S}_5\)
type: Item
instance of: scholarly article
title: The validity paradox in modal \(\text{S}_5\)
zbMATH Open document ID: 0935.03011
DOI: 10.1007/BF00413822
published in: Synthese
publication date: 2000-05-02T00:00:00Z
review text: Jacqette's paper shows that, first, the Pseudo-Scotus paradox is not derivable in non-modal logics and logics weaker than \(\text{S}^{-0.3ex}_5\); and, second, that the ordinary concept of validity is paradoxical in \(\text{S}^{-0.3ex}_ 5\) (and its extensions). A paradigmatic formulation of Pseudo-Scotus is: 1. This argument is valid. 2. This argument is invalid. Given the usual definition of validity, \textit{invalidity} of the argument follows from its \textit{validity}. Suppose (P) to be \textit{invalid}. Then, 1. can be true while 2. is false. In that case, both the premise and the conclusion assert the \textit{validity} of (P). Both horns of the dilemma bring about the result that argument (P) is valid if and only if it is invalid. Neither Tarskian semantic hierarchy nor exclusion of demonstratives can prevent the paradox from appearing. Instead, the author shows that the argument involves a modal fallacy: From \textit{invalidity} follows not \textit{validity}, but only \textit{possible validity} of (P). After a critical discussion of some attempts by Read and Sorensen to state the problem, Jacquette demonstrates that the characteristic \(\mathbf{P} p \supset \mathbf {NP} p\) in \(\text{S}^{-0.3ex}_ 5\) together with the assumption that logical validity is logical validity in all possible worlds allow to obtain the paradox.
reviewed by: Uwe Scheffler
Mathematics Subject Classification ID: 03A05
Mathematics Subject Classification ID: 03B45
zbMATH DE Number: 1322774
zbMATH Keywords: pseudo-Scotus
zbMATH Keywords: modal fallacy
zbMATH Keywords: validity
zbMATH Keywords: paradox
author: Dale Jacquette
MaRDI profile type: Publication
cites work: Q3281052
cites work: Q4692665
cites work: Q3731564
description: scientific article; zbMATH DE number 1322774