The Finite Frame Property of Some Extensions of the Pure Logic of Necessitation
Taishi Kurahashi, Y. Sato · Studia Logica · 2024
Abstract 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.