In many applications, logical reasoning is necessarily approximate in that an entailment “A implies B” only holds to a certain degree c. For such scenarios, a logic of approximate entailment (LAE) was developed, which allows reasoning about such entailments “A >c B”; however, it is not possible to combine “A >c1 B1” and “A >c2 B2” to some entailment “A >c3 (B1 and B2).”

This paper introduces an extended logic LAEC (LAE on a chain) that allows one to derive this combination for c3 = min(c1,c2), provided that the conclusions B1 and B2 are not contradictory. LAE interprets a formula in a set of worlds equipped with a function S(w1,w2) that describes the similarity of two worlds w1 and w2 with a number from 0 to 1. The core idea of LAEC is to extend this domain by a total order <= (that is compatible with S in that worlds that are farther apart along a chain w1 <= w2 <= ... are less similar) and the basic propositions by two modal operators that talk about this order. LAEC is subsequently generalized to the multisorted logic LAEPC (LAE on products of chains) that allows one to reason about different aspects of a domain.

The paper is self-contained and introduces LAE, LAEC, and LAEPC in a clear and systematic way; a small application example aims to demonstrate the practical relevance of the work. Further research will aim to generalize the logics and get rid of the necessity to prove the consistency of conclusions.