A fuzzy logic with similarity

Fuzzy set and fuzzy logic are an important and a practical mathematical tool for the processing of uncertain and vague information. The similarity relation is a basic notion in fuzzy set and has been applied to almost every field where uncertainty and vagueness are concerned. In this paper, a fuzzy logic with similarity is proposed and resolution- and paramodulation-based fuzzy reasoning is discussed. In order to effectively deal with similarity in reasoning procedure, the inference rule paramodulation is extended to fuzzy reasoning. It is shown that resolution and paramodulation are complete and sound for fuzzy predicate calculus. That is, on one hand, if a set of clauses is S-unsatisfiable, then there is a refutation using the resolution and/or paramodulation from the set; on the other hand, if every clause in a set of clauses is something more than a "half-truth" and the most unreliable clause has truth-value a, then it is guaranteed that all the logical consequence obtained by repeatedly applying the resolution and/or paramodulation will have truth-value no less than a.