The goal of this paper is to generalize a notion of characteristic (or Jankov) formula by using finite partial Heyting algebras instead of the finite subdirectly irreducible algebras: with every finite partial Heyting algebra we associate a characteristic formula, and we study the properties of these formulas. We prove that any intermediate logic can be axiomatized by such formulas. We further discuss the correlations between characteristic formulas of finite partial algebras and canonical formulas. Then with every well-connected Heyting algebra we (...) associate a set of characteristic formulas that correspond to each finite relative subalgebra of this algebra. Finally, we demonstrate that in many respects these sets enjoy the same properties as regular characteristic formulas. In the last section we outline an approach how to generalize these obtained results to the broad classes of algebras. (shrink)

We study the finite frame property of some extensions of Fitting, Marek, and Truszczyński’s pure logic of necessitation $$\textbf{N}$$ N. For any natural numbers m, n, we introduce the logic $$\textbf{N}^+\textbf{A}_{m,n}$$ N + A m, n by adding the single axiom scheme $$\Box ^n \varphi \rightarrow \Box ^m \varphi $$ □ n φ → □ m φ and the rule $$\dfrac{\lnot \Box \varphi }{\lnot \Box \Box \varphi }$$ ¬ □ φ ¬ □ □ φ ($${\text {Ros}}^\Box $$ Ros □ ) (...) into $$\textbf{N}$$ N. We prove the finite frame property of $$\textbf{N}^+\textbf{A}_{m, n}$$ N + A m, n with respect to Fitting, Marek, and Truszczyński’s relational semantics. We also prove that for $$n \ge 2$$ n ≥ 2, the logic obtained by removing the rule $${\text {Ros}}^\Box $$ Ros □ from $$\textbf{N}^+\textbf{A}_{0, n}$$ N + A 0, n is incomplete with respect to that semantics. (shrink)

The finite model property (FMP) in weakly transitive tense logics is explored. Let \(\mathbb {S}=[\textsf{wK}_t\textsf{4}, \textsf{K}_t\textsf{4}]\) be the interval of tense logics between \(\textsf{wK}_t\textsf{4}\) and \(\textsf{K}_t\textsf{4}\). We introduce the modal formula \(\textrm{t}_0^n\) for each \(n\ge 1\). Within the class of all weakly transitive frames, \(\textrm{t}_0^n\) defines the class of all frames in which every cluster has at most _n_ irreflexive points. For each \(n\ge 1\), we define the interval \(\mathbb {S}_n=[\textsf{wK}_t\textsf{4T}_0^{n+1}, \textsf{wK}_t\textsf{4T}_0^{n}]\) which is a subset of \(\mathbb {S}\). There are (...) \(2^{\aleph _0}\) logics in \(\mathbb {S}_n\) lacking the FMP, and there are \(2^{\aleph _0}\) logics in \(\mathbb {S}_n\) having the FMP. Then we explore the FMP in finitely alternative tense logics \(L_{n,m}=L\oplus \{\textrm{Alt}_n^F, \textrm{Alt}_m^P\}\) with \(n,m\ge 0\) and \(L\in \mathbb {S}\). For all \(k\ge 0\) and \(n,m\ge 1\), we define intervals \(\mathbb {F}^k_{n,m}\), \(\mathbb {P}^k_{n,m}\) and \(\mathbb {S}^k_{n,m}\) of tense logics. The number of logics lacking the FMP in them is either 0 or \(2^{\aleph _0}\), and the number of logics having the FMP in them is either finite or \(2^{\aleph _0}\). (shrink)

This paper contains a brief overview of the area of admissible rules with an emphasis on results about intermediate and modal propositional logics. No proofs are given but many references to the literature are provided.