In Section 2 I give a criterion of decidability that can be applied to logics (i.e. Tarski consequence operators) without the finite model property. In Section 3 I study ukasiewicz-style refutation procedures as a method of obtaining decidability results.This method also proves to be more general than Harrop's criterion.

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.

This paper provides a semantic analysis of admissible rules and associated completeness conditions for arbitrary deductive systems, using the framework of abstract algebraic logic. Algebraizability is not assumed, so the meaning and significance of the principal notions vary with the level of the Leibniz hierarchy at which they are presented. As a case study of the resulting theory, the nonalgebraizable fragments of relevance logic are considered.

Hereditary structural completeness is established for a range of substructural logics, mainly without the weakening rule, including fragments of various relevant or many-valued logics. Also, structural completeness is disproved for a range of systems, settling some previously open questions.

tic sequenzen-kalkul of Gentzen, into rules for PCc, the classical sequenzenkalkul. We shall limit ourselves here to sequenzen or turnstile statements of the form A„A„..., A„ I- B, where A„A„..., A„(n ~ 0), and B are wffs consisting of propositional variables, zero or more of the connectives '5', "v', ' ', ')', and '=', and zero or more parentheses. One can pass from PCi to PCc by amending the intelim rules for ' a result of long standing, or by amending (...) the intelim rules for either one of.. (shrink)

Uniform infinite bases are defined for the single-conclusion and multiple-conclusion admissible rules of the implication–negation fragments of intuitionistic logic and its consistent axiomatic extensions . A Kripke semantics characterization is given for the structurally complete implication–negation fragments of intermediate logics, and it is shown that the admissible rules of this fragment of form a PSPACE-complete set and have no finite basis.