Skip to Main Content
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.
Note: The Institute of Electrical and Electronics Engineers, Incorporated is distributing this Article with permission of the International Business Machines Corporation (IBM) who is the exclusive owner. The recipient of this Article may not assign, sublicense, lease, rent or otherwise transfer, reproduce, prepare derivative works, publicly display or perform, or distribute the Article.