Situations in which disjunctive syllogism can lead from true premises to a false conclusion
Relevance logicians have attacked the validity of disjunctive syllogism \((p\vee q, \sim p, \therefore q)\) where the disjunction is merely extensional (i.e. does not support \(\ulcorner\) If it were the case that \(p\), then it would be the case that \(q\urcorner\)). Geach, in his review (Philosophy 52, 495 (1977)) of \textit{A. R. Anderson} and \textit{N. D. Belnap}, Entailment. The logic of relevance and necessity, Vol. 1 (1975; Zbl 0323.02030), pointed out that counterexamples to extensional disjunctive syllogism were not forthcoming. Considering (1) \(\ulcorner p\) is most probably false\(\urcorner\), the author argues that (2) \(\ulcorner p\) is not most probably false\(\urcorner\) is the negation of (1) but not the contradictory of (1), because ``not most probably false'' means the disjunction of all possible truth values of \(p\) with the exception of ``most probably false''. If ``There will be a naval battle tomorrow'' is either most probable, or probable to some degree (other than 0 or 1), then we can use this fact to construct a disjunctive syllogism of the form \(\ulcorner\)(2), (1) or \(q\), so \(q\urcorner\) where \(\ulcorner q\urcorner\) can only be either true or false and is in fact false; this syllogism will have true premises and a false conclusion.
scientific article; zbMATH DE number 1192481
scientific article