We present a basis for the admissible rules of intuitionistic propositional logic. Thereby a conjecture by de Jongh and Visser is proved. We also present a proof system for the admissible rules, and give semantic criteria for admissibility.

An algorithm recognizing admissibility of inference rules in generalized form (rules of inference with parameters or metavariables) in the intuitionistic calculus H and, in particular, also in the usual form without parameters, is presented. This algorithm is obtained by means of special intuitionistic Kripke models, which are constructed for a given inference rule. Thus, in particular, the direct solution by intuitionistic techniques of Friedman's problem is found. As a corollary an algorithm for the recognition of the solvability of logical equations (...) in H and for constructing some solutions for solvable equations is obtained. A semantic criterion for admissibility in H is constructed. (shrink)

We show that the variety of Heyting algebras has finitary unification type. We also show that the subvariety obtained by adding it De Morgan law is the biggest variety of Heyting algebras having unitary unification type. Proofs make essential use of suitable characterizations (both from the semantic and the syntactic side) of finitely presented projective algebras.

In Iemhoff we gave a countable basis for the admissible rules of . Here, we show that there is no proper superintuitionistic logic with the disjunction property for which all rules in are admissible. This shows that, relative to the disjunction property, is maximal with respect to its set of admissible rules. This characterization of is optimal in the sense that no finite subset of suffices. In fact, it is shown that for any finite subset X of , for one (...) of the proper superintuitionistic logics Dn constructed by De Jongh and Gabbay ), all the rules in X are admissible. Moreover, the logic Dn in question is even characterized by X: it is the maximal superintuitionistic logic containing Dn with the disjunction property for which all rules in X are admissible. Finally, the characterization of is proved to be effective by showing that it is effectively reducible to an effective characterization of in terms of the Kleene slash by De Jongh. (shrink)

We study the problem of finding a basis for all rules admissible in the intuitionistic propositional logic IPC. The main result is Theorem 3.1 which gives a basis consisting of all rules in semi-reduced form satisfying certain specific additional requirements. Using developed technique we also find a basis for rules admissible in the logic of excluded middle law KC.

We investigate admissible inference rules for the multi-modal logic VSK+ extending the logic VSK – the logic of Visibility, Perception and Knowledge. The logic VSK has been introduced by M.Wooldridge and A. Lomuscio [21]. VSK was intended for reasoning about properties of computational agents situated in some environment. Admissible rules are important for modelling of logical consequence. We consider these rules for VSK+, the logic of a wise agent . The main result of our paper is the construction of an (...) algorithm which determines admissible inference rules in VSK+. The algorithm is based on the proof of existence of computable bounds on the size of special Kripke 3-frames refuting inadmissible rules. (shrink)

Our investigation is concerned with the finite model property with respect to admissible rules. We establish general sufficient conditions for absence of fmp w. r. t. admissibility which are applicable to modal logics containing K4: Theorem 3.1 says that no logic λ containing K4 with the co-cover property and of width > 2 has fmp w. r. t. admissibility. Surprisingly many, if not to say all, important modal logics of width > 2 are within the scope of this theorem–K4 itself, (...) S4, GL, K4.1, K4.2, S4.1, S4.2, GL.2, etc. Thus the situation is completely opposite to the case of the ordinary fmp–the absolute majority of important logics have fmp, but not with respect to admissibility. As regards logics of width ≤ 2, there exists a zone for fmp w. r. t. admissibility. It is shown that all modal logics A of width ≤ 2 extending S4 which are not sub-logics of three special tabular logics have fmp w.r.t. admissibility. (shrink)

We consider structural completeness in modal logics. The main result is the necessary and sufficient condition for modal logics over K4 to be hereditarily structurally complete: a modal logic λ is hereditarily structurally complete $\operatorname{iff} \lambda$ is not included in any logic from the list of twenty special tabular logics. Hence there are exactly twenty maximal structurally incomplete modal logics above K4 and they are all tabular.