Algebraic Semantics for Relative Truth, Awareness, and Possibility

TL;DR

This paper introduces relativized Boolean algebras (RBAs), algebraic structures that provide semantics for propositional logic with truth relative to local domains, extending to modal logics and awareness theories.

Contribution

It develops RBAs as a new algebraic framework for relative truth, integrating modal operators and offering semantics for awareness and possibility logics.

Findings

RBAs generalize Boolean algebras with local truth domains

Modal RBAs model various awareness and possibility logics

Behavior aligns with propositional logic laws locally

Abstract

This paper puts forth a class of algebraic structures, relativized Boolean algebras (RBAs), that provide semantics for propositional logic in which truth/validity is only defined relative to a local domain. In particular, the join of an event and its complement need not be the top element. Nonetheless, behavior is locally governed by the laws of propositional logic. By further endowing these structures with operators -- akin to the theory of modal Algebras -- RBAs serve as models of modal logics in which truth is relative. In particular, modal RBAs provide semantics for various well known awareness logics and an alternative view of possibility semantics.