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≤s and D≥s, s ∈ Q+0. Our language allows making formulas such as D≤s(α, β) with the intended meaning ‘distance between formulas α and β 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.