We prove that $\textbf {K}5$ and some of its extensions that do not contain $\textbf {K}4$ are of unification type $1$.

This paper proves the finite axiomatizability of transitive modal logics of finite depth and finite width w.r.t. proper-successor-equivalence. The frame condition of the latter requires, in a rooted transitive frame, a finite upper bound of cardinality for antichains of points with different sets of proper successors. The result generalizes Rybakov’s result of the finite axiomatizability of extensions of$\mathbf {S4}$of finite depth and finite width.

In this paper we use “generic submodels” to prove that each normal extension of G.3 (K4.3W) has the finite model property, by which we establish that each proper normal extension of G.3 is G.3Altn for some n≥0.

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)

Pietruszczak (Bull Sect Log 38(3/4):163–171, 2009) proved that the normal logics K45 , KB4 (=KB5), KD45 are determined by suitable classes of simplified Kripke frames of the form ⟨W,A⟩ , where A⊆W. In this paper, we extend this result. Firstly, we show that a modal logic is determined by a class composed of simplified frames if and only if it is a normal extension of K45. Furthermore, a modal logic is a normal extension of K45 (resp. KD45; KB4; S5) if (...) and only if it is determined by a set consisting of finite simplified frames (resp. such frames with A≠∅; such frames with A=W or A=∅; such frames with A=W). Secondly, for all normal extensions of K45, KB4, KD45 and S5, in particular for extensions obtained by adding the so-called “verum” axiom, Segerberg’s formulas and/or their T-versions, we prove certain versions of Nagle’s Fact (J Symbol Log 46(2):319–328, 1981) (which concerned normal extensions of K5). Thirdly, we show that these extensions are determined by certain classes of finite simplified frames generated by finite subsets of the set N of natural numbers. In the case of extensions with Segerberg’s formulas and/or their T-versions these classes are generated by certain finite subsets of N. (shrink)

In the basic modal language and in the basic modal language with the added universal modality, first-order definability of all formulas over the class of all frames is shown. Also, it is shown that the problems of modal definability of first-order sentences over the class of all frames in the languages and are both PSPACE-complete.

Hume’s famous character Cleanthes claims that there is no difficulty in explaining the existence of causal chains with no first cause since in them each item is causally explained by its predecessor. Relying on logico-mathematical resources, we argue for two theses: (1) if the existence of Cleanthes’ chain can be explained at all, it must be explained by the fact that the causal law ruling it is in force, and (2) the fact that such a causal law is in force (...) cannot explain the occurrence of the events in the chain. In order to perform (1), we manage to express in mathematical terms the intuitive idea that indefinitely delayed explanation is ultimately no explanation. In order to achieve (2), we identify a logical relation we can prove to be as strong as the causal relation at issue in the Cleanthes passage, according to a precise notion of strength of relations. (shrink)