Deductive Systems of Fuzzy Logic

Lotfi Zadeh [23] is the author of the theory of fuzzy sets. A fuzzy subset A of a (crisp) set X is characterized by assigning to each element x of X the degree of membership of x in A In particular, if X is a set of propositions then its elements may be assigned their degree of truth, which may be “absolutely true”, “absolutely false” or some intermediate truth degree: a proposition may be more true than another proposition. This is obvious in the case of vague (imprecise) propositions like “this person is old” (beautiful, rich, etc.). And this leads to fuzzy logic. In the analogy to various definitions of operations on fuzzy sets (intersection, union, complement,…) one may ask how propositions can be combined by connectives (conjunction, disjunction, negation,…) and if the truth degree of a composed proposition is determined by the truth degrees of its components, i.e. if the connectives have their corresponding truth functions (like truth tables of classical logic). Saying “yes” (which is the mainstream of fuzzy logic) makes fuzzy logic to something principally different from probability theory since e.g. the probability of conjunction of two propositions is not determined by the probabilities of those propositions.