Some Logics in the Vicinity of Interpretability Logics

Abstract

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 ⟨X, R, N⟩, called basic frames, where ⟨X, R⟩ is a Kripke frame and ⟨X, N⟩ 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.