Intuitionistic modal logic and set theory

Abstract

The mathematical treatment of the concepts of vagueness and approximation is of increasing importance in artificial intelligence and related research. The theory of fuzzy sets was created by Zadeh [Z] to allow representation and mathematical manipulation of situations of partial truth, and proceeding from this a large amount of theoretical and applied development of this concept has occurred. The aim of this paper is to develop a natural logic and set theory that is a candidate for the formalisation of the theory of fuzzy sets. In these theories the underlying logic of properties and sets is intuitionistic, but there is a subset of formulae that are ‘crisp’, classical and two-valued, which represent the certain information. Quantum logic or logics weaker than intuitionistic can also be adopted as the basis, as described in [L]. The relationship of this theory to the intensional set theory MZF of [Gd] and the global intuitionistic set theory GIZF of Takeuti and Titani [TT] is also treated.