The validity paradox in modal \(\text{S}_5\)
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.
scientific article; zbMATH DE number 1322774