Logic of approximate entailment in quasimetric spaces

Abstract

The logic LAE q discussed in this paper is based on an approximate entailment relation. LAE q generalises classical propositional logic to the effect that conclusions can be drawn with a quantified imprecision. To this end, properties are modelled by subsets of a distance space and statements are of the form that one property implies another property within a certain limit of tolerance. We adopt the conceptual framework defined by E. Ruspini; our work is towards a contribution to the investigation of suitable logical calculi. LAE q is based on the assumption that the distance function is a quasimetric. We provide a proof calculus for LAE q and we show its soundness and completeness for finite theories. As our main tool for showing completeness, we use a representation of proofs by means of weighted directed graphs. • We propose an axiomatisation of Approximate Reasoning in the sense of E. Ruspini's paper “On the semantics of fuzzy logic”. • The Logic LAE q deals with statements of the form that a property implies another one within a specified limit of tolerance. • We present a proof calculus for LAE q and show its soundness and completeness.