Graded Consequence Relations

Abstract

Formula (6.1) in Chapter 3 for the canonical extension of a closure operator enables us to apply a crisp deduction apparatus to fuzzy information, i.e., information “stratified” at several levels of validity. Now, it is also possible to apply a “stratified” deduction apparatus (i.e., various deductive instruments each with a related degree of validity) to a crisp information. We can represent such a state of affairs assuming that, for every λ ∈ U, a crisp deduction operator D _λ is defined. Given a set X of formulas, we interpret D _λ (X) as the set of formulas that we can derive from Xby using arguments which are “reliable” to degree A.