  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.
