Algebraic semantics for hybrid logics

TL;DR

This paper develops algebraic semantics for hybrid logics, establishing dualities and completeness results for various classes of hybrid algebras, enhancing the theoretical understanding of hybrid logical systems.

Contribution

It introduces hybrid algebras as a new algebraic semantics, proves duality with descriptive frames, and shows completeness of axiomatic extensions with respect to these algebras.

Findings

Established duality between hybrid algebras and descriptive frames

Proved completeness of basic hybrid logics with respect to hybrid algebras

Extended completeness results to permeated hybrid algebras with additional rules

Abstract

We introduce hybrid algebras as algebraic semantics for hybrid languages with nominals and, possibly, the satisfaction operator. We establish a duality between hybrid algebras and the descriptive two-sorted general frames of Ten Cate. We show that all axiomatic extensions of the basic hybrid logics, with or without the satisfaction operator, are complete with respect to their classes of hybrid algebras. Moreover, we show that by adding the usual non-orthodox rules to these logics, they become complete with respect to their classes of permeated hybrid algebras, corresponding to strongly descriptive two-sorted general frames.