Logic of Approximate Entailment in quasimetric spaces

TL;DR

This paper introduces LAE, a logic for approximate entailment in quasimetric spaces, enabling reasoning with quantified imprecision and providing a sound and complete proof calculus for finite theories.

Contribution

It develops a formal logical calculus for LAE based on quasimetric spaces, including a proof of soundness and completeness using graph representations.

Findings

LAE generalizes classical propositional logic to handle approximate implications.

A proof calculus for LAE is established, proven sound and complete for finite theories.

Representation of proofs via weighted directed graphs is demonstrated.

Abstract

The logic LAE discussed in this paper is based on an approximate entailment relation. LAE 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 is based on the assumption that the distance function is a quasimetric. We provide a proof calculus for LAE 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.