The last decade has seen an enormous development in infinite-valued systems and in particular in such systems which have become known as mathematical fuzzy logics. The paper discusses the mathematical background for the interest in such systems of mathematical fuzzy logics, as well as the most important ones of them. It concentrates on the propositional cases, and mentions the first-order systems more superficially. The main ideas, however, become clear already in this restricted setting.

There are several ways to extend the classic logical connectives for fuzzy truth degrees in such a way that their behavior for the values 0 and 1 work exactly as in the classical one. For each fuzzy semantics the formulas which are always true can change. But these sets are always a subset of the classical tautologies. The fuzzy semantics whose tautologies are identical to the classical tautologies are called here by “classic-like fuzzy semantics”. In this paper we will prove (...) that there are uncountable classes of classic-like fuzzy semantics and we will provide a characterization of them, i.e. we will determine a necessary and sufficient condition for a fuzzy semantic be classic-like. We also define a generic consequence semantic relation for fuzzy semantics and we will prove that for the classic-like fuzzy semantics, this notion coincides with the classical one. (shrink)