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Some Logics in the Vicinity of Interpretability Logics
Bulletin of the Section of Logic, Tome 53 (2024) no. 2, pp. 173-193 Cet article a éte moissonné depuis la source Library of Science

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In this paper we shall define semantically some families of propositional modal logics related to the interpretability logic 𝐈𝐋. We will introduce the logics 𝐁𝐈𝐋 and 𝐁𝐈𝐋^+ in the propositional language with a modal operator □ and a binary operator ⇒ such that 𝐁𝐈𝐋⊆𝐁𝐈𝐋^+⊆𝐈𝐋. The logic 𝐁𝐈𝐋 is generated by the relational structures lt;X,R,N gt;, called basic frames, where lt;X,R gt; is a Kripke frame and lt;X,N gt; is a neighborhood frame. We will prove that the logic 𝐁𝐈𝐋^+ is generated by the basic frames where the binary relation R is definable by the neighborhood relation N and, therefore, the neighborhood semantics is suitable to study the logic 𝐁𝐈𝐋^+ and its extensions. We shall also study some axiomatic extensions of 𝐁𝐈𝐋 and we will prove that these extensions are sound and complete with respect to a certain classes of basic frames. Finally, we prove that the logic 𝐁𝐈𝐋^+ and some of its extensions are complete respect with the class of neighborhood frames.
Keywords: interpretability logic, Kripke frames, neighbourhood frames, Veltman semantics 
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Celani, Sergio A. Some Logics in the Vicinity of Interpretability Logics. Bulletin of the Section of Logic, Tome 53 (2024) no. 2, pp. 173-193. http://geodesic.mathdoc.fr/item/BSL_2024_53_2_a2/

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