Distances and Degrees of Uncertainty in Many-Valued Propositions of Experts and Application of These Concepts in Problems of Pattern Recognition and Clustering

摘要

This paper is concerned with formulas of the n-valued logic, which was introduced and studied by Lukasiewicz. These formulas have various applications, in particular, for representing many-valued propositions of experts. The methods of mathematical logic and of the models for the n-valued logic introduced here are used to find model "distances" on formulas (propositions) and uncertainty degrees (measures) of formulas as the characteristics of their falsity on the class of models under consideration (possible universes). The properties of the thus-introduced distances and uncertainty measures of formulas are examined. Ways for defining metrics and degrees of uncertainty on classes of equivalent formulas are put forward, and the useful properties thereof are established. They can be employed for clustering problems, construction of decision functions, and pattern recognition.