Remarks about the unification types of some locally tabular normal modal logics

Abstract

It is already known that unifiable formulas in normal modal logic $\textbf {K}+\square ^{2}\bot $ are either finitary or unitary and unifiable formulas in normal modal logic $\textbf {Alt}_{1}+\square ^{2}\bot $ are unitary. In this paper, we prove that for all $d{\geq }3$, unifiable formulas in normal modal logic $\textbf {K}+\square ^{d}\bot $ are either finitary or unitary and unifiable formulas in normal modal logic $\textbf {Alt}_{1}+\square ^{d}\bot $ are unitary.

Unification types in Euclidean modal logics

“…Recently, Balbiani et al [10,11] have proved that for all positive integers d≥2, the modal logic Alt 1 ⊕ d ⊥ is of unification type 1 and the modal logic K ⊕ d ⊥ is of unification type ω 16 . K5, KD5, the modal logics K5⊕χ 1 l and KD5⊕χ 1 l for each positive integer l≥2 and the modal logics Alt 1 ⊕ d ⊥ and K ⊕ d ⊥ for each positive integer d≥2 are locally tabular.…”

Decidability for $$\mathsf S4$$ Gödel Modal Logics

“…We have not provided an axiomatisation for the logic of reflexive transitive ♦-regular frames, as our completeness proofs rely on either Axiom FS2 or Axiom CD being available. Note that this class does not validate p → p. First results on logics of frames satisfying only forth-up confluence were recently presented in [34], where the K version was axiomatised and shown to be decidable.…”

KD45 with Propositional Quantifiers

“…Which of the four we are thinking of should always be clear from the letter that we use for the argument * . In (Balbiani and Tinchev, 2018), modulo notational differences, subsets of {p 0 , . .…”