On T-quantifiers and S-quantifiers

We show how the “classical” theory of T-norms and S-norms of fuzzy logic can be generalized to a theory of T-quantifiers and S-quantifiers, respectively. The key idea leading to this generalization is the fact that the (infinite) iteration of the two-valued conjunction and disjunction gives the two-valued all-quantifier and ex-quantifier, respectively. In the framework of fuzzy logic the same holds for min with respect to Inf and for max with respect to Sup. As a T-norm (S-norm) is commutative and associative, we can construct an all-τ-quantifier (an ex-σ-quantifier) from a given T-norm τ (S-norm σ). These quantifiers are characterized by axioms (T-quantifiers and S-quantifiers). Furthermore we show that the generating procedure is “complete” with respect to arbitrary T-quantifiers (S-quantifiers) and uniquely reversible