<title>Fuzzy logical semantics of Bayesian decision making</title>

Abstract

An approach to uncertainty and decision making problems alternative to probabilistic one is studied. It is based on fuzzy sets technique and has deep relations with the underlying first order fuzzy logic. In this logic not only logical connectives but also quantifiers have fuzzy interpretation. It is shown that all fundamental concepts of probability and statistics such as joint distribution, conditional distribution, etc., have meaningful analogs in new context. Thus the classical concepts obtain fuzzy-logical semantics. As a result one may treat them as formulas of first order logic. It is shown, that this appraoch makes it possible to utilize rich conceptual experience of statistics. In particular it leads to fuzzy Bayesian approach in decision making, which plays the same role in fuzzy problems of optimal decision making as its probabilistic prototype in the theory of statistical games, and provides methods for construction of optimal strategies. Connection with underlying fuzzy logic provides the logical semantics for fuzzy decision making. In this approach a priori information is represented by a fuzzy predicate and an experiment--by fuzzy universal quantifier, etc. As a result the notion 'good decision strategy' is expressed by a first order formula in this logic.