A Method to Single out Maximal Propositional Logics with the Disjunction Property II

This is the second part of a paper devoted to the study of the maximal intermediate propositional logics with the disjunction property (we will simply call maximal constructive logics), whose first part has appeared in this journal with the title “A method to single out maximal propositional logics with the disjunction property I”. In the first part we have explained the general results upon which a method to single out maximal constructive logics is based and have illustrated such a method by exhibiting the Kripke semantics of maximal constructive logics extending the logic ST of Scott, for which, in turn, a semantical characterization in terms of Kripke frames has been given. In the present part we complete the illustration of the method of the first part, having in mind some aspects which might be interesting for a classification of the maximal constructive logics, and an application of the heuristic content of the method to detect the nonmaximality of apparently maximal constructive logics. Thus, on the one hand we introduce the logic AST (“anti” ST), which is compared with ST and is seen as a logic “alternative” (or even “opposite”) to it, in a sense which will be precisely explained. We provide a Kripke semantics for AST and (without exhibiting them) show that (near the ones including ST and the ones including AST) there are maximal constructive logics which neither are extensions of ST nor are extensions of AST. Finally, we give a further application of the results of the first part by exhibiting the Kripke semantics of a maximal constructive logic extending AST. On the other hand, we compare the maximal constructive logics presented in both parts of the paper with a constructive logic introduced by Maksimova (1986), which has been conjectured to be maximal by Chagrov and Zacharyashchev (199 1); from this comparison a disproof of the conjecture arises.