This paper studies modal logics whose extensions all have the finite model property, those whose extensions are all recursively axiomatizable, and those whose extensions are all finitely axiomatizable. We call such logics FMP-ensuring, RA-ensuring and FA-ensuring respectively, and prove necessary and sufficient conditions of such logics in $$\mathsf {NExtK4.3}$$. Two infinite descending chains $$\{{\textbf{S}}_{k}\}_{k\in \omega }$$ and $$\{{\textbf{S}} _{k}^{*}\}_{k\in \omega }$$ of logics are presented, in terms of which the necessary and sufficient conditions are formulated as follows: A logic in (...) $$\mathsf {NExtK4.3}$$ is FMP-ensuring iff it extends $${\textbf{S}}_{k}$$ for some $$k\in \omega $$, it is RA-ensuring iff it extends $${\textbf{S}}_{k}^{*}$$ for some $$k\in \omega $$, and it is FA-ensuring iff it is finitely axiomatizable and extends $${\textbf{S}}_{k}^{*}$$ for some $$k\in \omega $$. (shrink)

We generalize the theory of canonical formulas for K4, the logic of transitive frames, to wK4, the logic of weakly transitive frames. Our main result establishes that each logic over wK4 is axiomatizable by canonical formulas, thus generalizing Zakharyaschev’s theorem for logics over K4. The key new ingredients include the concepts of transitive and strongly cofinal subframes of weakly transitive spaces. This yields, along with the standard notions of subframe and cofinal subframe logics, the new notions of transitive subframe and (...) strongly cofinal subframe logics over wK4. We obtain axiomatizations of all four kinds of subframe logics over wK4. We conclude by giving a number of examples of different kinds of subframe logics over wK4. (shrink)

We solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics as K4, S4, S4.1, K4.3, GL, or Grz are undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4,K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, if.

We solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics asK4,S4,S4.1,K4.3,GL, orGrzare undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4, K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, ifl1andl2are classes of transitive frames such that their depth cannot be (...) bounded by any fixedn< ω, then the logic of the class {5ℑ1× ℑ2∣ ℑ1∈l1, ℑ2, ∈l2} is undecidable. (On the contrary, the product of, say,K4and the logic of all transitive Kripke frames of depth ≤n, for some fixedn< ω, is decidable.) The complexity of these undecidable logics ranges from r.e. to co-r.e. and Π11-complete. As a consequence, we give the first known examples of Kripke incomplete commutators of Kripke complete logics. (shrink)