STRATIFIED OPERATORS AND GRADED CONSEQUENCE RELATIONS by

1. GRADED DEDUCTIVE TOOLS FOR GRADED INFORMATION In accordance with Tarski’s point of view, we identify a crisp deductive apparatus with a “deduction operator”, i.e. a closure operator D : P(Å)→ P(Å) where Å is the set of formulas in a logic. Given a set X of formulas, we interpret D(X) as the set of logical consequences of X. Now, we can imagine a "stratified" deduction apparatus, i.e. the availability of various deductive instruments each with a related degree of validity. We can represent such a state of affairs assuming that, for every λ ∈ [0,1], a crisp deduction operator Dλ is defined. Given a set X of formulas, we interpret Dλ(X) as the set of formulas that we can derive from X by using arguments which are "reliable" to degree λ. More generally, it is possible that the available information and the deduction apparatus are both stratified. In this case, we represent the stratified information by a fuzzy set v : Å→[0,1]. Then, if we denote by C(v,λ) the closed cut {α ∈Å : v(α)≥λ}, in the case α ∈ Dλ(C(v,λ)) for a suitable λ ∈ [0,1], we say that α is a consequence of v to degree λ. Obviously, we must consider the lower-constraint for the truth degree of α which is the best we are able to get. Then, it is natural to consider the number D(v)(α) = Sup{λ ∈ [0,1] : α ∈ Dλ(C(v,λ))} (1.1) as the best lower-constraint for this truth degree. This suggests a way to define new fuzzy logics that will be useful to investigate the interesting notion of a graded consequence relation proposed in Chakraborty [1988]. In the sequel we denote by U the interval [0,1].