This paper discusses applications of a formal set-theoretic foundation for fuzzy set theory [23] to problems of approximate and uncertain reasoning, and also looks at other formalisms, such as the classical Alternative Set Theory 139) and its Intuitionistic version [21, 20] for uncertain reasoning. Formal definitions of these frameworks will be given. Many suggestions for semantic operations on values of formulae involving intermediate truth have been made; [41, 40] contain a survey of these. There are two main problems: how to reconcile the use of the linear order [0, 1] as the value space when in reality degrees of truth are often incomparable, and secondly, within this simplified range of values, to select operations which are both logically coherent with each other and intuitively reasonable. We will derive general semantic constraints on operations which will motivate the formal frameworks that we define.
