Fuzzy Empiristic Implication, A New Approach

Abstract

The present paper is a brief introduction to logical fuzzy implication operators, the basic properties of a fuzzy implication function, and ways to construct new fuzzy implication functions. It is also argued that logical implication functions are defined in a rather rationalistic manner. Thus a new, empiristic approach is proposed, defining implication relations that are derived from data observation and with no regard to any preexisting constrains. A number of axioms are introduced to define a fuzzy empiristic implication relation, and a method of computing such a relation is proposed. It is argued that the proposed method is easy and with small time requirement even for very large data sets. Finally an application of the empiristic fuzzy implication relation is presented, the choice of a suitable logical fuzzy implication function to describe an “If…then…” fuzzy rule, when observed data exists. An empiristic fuzzy implication relation is computed according to the data, and through schemas of approximate reasoning, the difference of it to any logical fuzzy implication function is measured. The fuzzy implication function that is closer to the empiristic best resembles the observed “If…then…” fuzzy rule.