Logic of approximate entailment in quasimetric spaces

• We propose an axiomatisation of Approximate Reasoning in the sense of E. Ruspini's paper “On the semantics of fuzzy logic”.
• The Logic LAEqLAEq deals with statements of the form that a property implies another one within a specified limit of tolerance.
• We present a proof calculus for LAEqLAEq and show its soundness and completeness.

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