The finite frame property of some extensions of the pure logic of necessitation
Abstract
We study the finite frame property of some extensions of Fitting, Marek, and Truszczyński's pure logic of necessitation $\mathbf{N}$. For any natural numbers $m, n$, we introduce the logic $\mathbf{N}^+\mathbf{A}_{m, n}$ by adding the single axiom scheme $\Box^n \varphi \to \Box^m \varphi$ and the rule $\dfrac{\neg \Box \varphi}{\neg \Box \Box \varphi}$ (Ros$^\Box$) into $\mathbf{N}$. We prove the finite frame property of $\mathbf{N}^+\mathbf{A}_{m, n}$ with respect to Fitting, Marek, and Truszczyński's relational semantics. We also prove that for $n \ge 2$, the logic obtained by removing the rule Ros$^\Box$ from $\mathbf{N}^+\mathbf{A}_{0, n}$ is incomplete with respect to that semantics.
 Publication:

arXiv eprints
 Pub Date:
 May 2023
 DOI:
 10.48550/arXiv.2305.14762
 arXiv:
 arXiv:2305.14762
 Bibcode:
 2023arXiv230514762K
 Keywords:

 Mathematics  Logic
 EPrint:
 24 pages