A propositional logic with binary metric operators

Abstract

The aim of this paper is to combine distance functions and Boolean propositions by developing a formalism suitable for speaking about distances between Boolean formulas. We introduce and investigate a formal language that is an extension of classical propositional language obtained by adding new binary (modal-like) operators of the form $D_{leqslant s} $ and $D_{geqslant s}$, $sinmathds{Q}_0^+$. Our language all-ows making formulas such as $D_{leqslant s}(alpha,beta)$ with the intended meaning `distance between formulas $alpha$ and $beta$ is less than or equal to $s$'. The semantics of the proposed language consists of possible worlds with a distance function defined between sets of worlds. Our main concern is a complete axiomatization that is sound and strongly complete with respect to the given semantics.