The modal status of antinomies (Q1103609)

The paper presents two propositional paraconsistent modal logics, each based on the underlying propositional logic of Asenjo [\textit{F. G. Asenjo} and \textit{J. Tamburino}, ``Logic of antinomies'', Notre Dame J. Formal Logic 16, 17-44 (1975; Zbl 0246.02023)]. The logics (M and MD) are the analogues of the classical modal systems K and D, respectively. Semantics and axiom systems are given; soundness and completeness are proved (the latter by the canonical model construction). If B is any antinomic formula then \(\square B\) and \(\neg \diamond B\) are both valid in M, and these together with \(\neg \square B\) and \(\diamond B\) are valid in MD. These facts merely reflect the assumptions made that: i) any antinomic sentence is antinomic in all worlds; ii) there are no non-normal worlds. Each of these assumptions might be changed to produce different results. Hence, the formal considerations of this paper do not, on their own, solve the problem of the modal status of antinomic sentences.