A many-valued logic for approximate reasoning

A new system for many-valued logic, the Extended Post system of order p, p ≥ 2, is proposed as a system of logic supporting reasoning with facts and rules which are reliable to a specified extent. In an Extended Post system there are as many operations of logical disjunction and logical conjunction as there are truth values. The truth value associated with a particular operation of disjunction (conjunction) acts as a threshold value controlling the behavior of the operation. The availability of an extended set of logical operations provides improved flexibility in the symbolic translation of sentences from the ordinary word-language. Extended Post systems are equipped with a semantics in which graded rather than crisp sets correspond to predicates. The system exhibits a “rich” algebraic structure. The p operations of disjunction form a distributivity cycle. To each disjunction there corresponds a dual operation of conjunction, the two operations being distributive to one another. The p conjunctions form a dual distributivity cycle. Both propositional calculus and first-order predicate calculus of EP systems are developed. The application to approximate reasoning is described. It is shown that there exist distinct isomorphic copies of fuzzy logic, each corresponding to a distinct level of approximation and being complete to resolution.