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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

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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 
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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