Intuitionism and relevance (Q2782585)

Just like the logic of entailment is concerned with making classical implication relevant, the author of this monograph is concerned with making intuitionistic implication, and thus intuitionistic propositional logic, relevant. NEWLINENEWLINENEWLINEAfter introducing a semantic notion of intuionistic relevance \(\models_{\text{Irel}}\) (to be later made ``philosophically plausible'' by means of a modified Grzegorczyk semantics as in the author's [Log. Anal., Nouv. Sér. 41, No. 161-163, 167-188 (1998; Zbl 1009.03017)]) based on state descriptions, the author puts forward an axiom system for relevant intuitionistic implication \(A\rightarrow B\) between formulas \(A\) and \(B\) composed with \(\wedge\), \(\vee\), \(\sim\) alone, and proves a completeness theorem for these two notions. NEWLINENEWLINENEWLINEA completeness theorem is also proved for a hierachy of constructive systems of negations, after which the positive logic \({\mathbf E}+\) (in which no negation sign occurs) of \textit{R. Routley} and \textit{R. K. Meyer} [J. Philos. Logic 1, 192-208 (1972; Zbl 0317.02019)] is extended to a system containing a symbol for falsehood, on the basis of which negation is defined, giving rise to the minimal entailment system \textbf{ME} for which a completeness theorem is also proved. NEWLINENEWLINENEWLINEThe author takes great care to motivate and discuss philosophically the steps taken. NEWLINENEWLINENEWLINEErrata: In Theorem 2.6.1 and Definition 2.7.14, rel should be replaced by Irel.