Rules of inference in fuzzy sentential logic (Q1267518)

The aim of this study is to introduce some fuzzy rules of inference. Each of them has a counterpart in classical logic. We consider injective MV-algebra valued fuzzy sentential logics as these fuzzy inference systems are proved to be semantically complete [cf. \textit{E. Turunen}, Math. Log. Q. 41, No. 2, 236-248 (1995; Zbl 0829.03011)]. It is of special importance to realize that in the unit interval the only semantically complete fuzzy sentential logics are the injective MV-algebra valued fuzzy logics. It is well known that any nilpotent Archimedean T-norm generates an injective MV-algebra; moreover, injective MV-algebras in the unit interval are exactly the structures isomorphic with the Łukasiewicz MV-algebra. We follow \textit{J. Pavelka} [Z. Math. Logik Grundlagen Math. 25, 45-52, 119-134, 447-464 (1979; Zbl 0435.03020, Zbl 0446.03015, Zbl 0446.03016)] and define a fuzzy rule of inference as consisting of two components. The first component operates on formulae and is, in fact, a rule of inference in the usual sense; the second component operates on truth values and says how the truth value of the conclusion is to be computed from the truth values of the premises such that the degree of truth is preserved. After recollecting and proving some necessary algebraic definitions and results, we introduce several new fuzzy rules of inference and give a practical example of how to use them. The advantage of our approach is the semantical completeness of our logic; the degree of validity of any formula coincides with the degree of deduction of the formula.