Fuzzy Truthoods Based on an Additive Semantic Measure with Break of Global Symmetry in Modal Logic

We propose an additive "semantic measure" theory based on the model of a perception based fuzzy truthoods in analogy to probability measure. Probability measure, which is the subject of the probability theory, is a special case of the fuzzy measure in the Dempster-Shafer theory when the focal sets are singletons. In the literature, the fuzzy measure is considered an extension of the probability measure. In the Dempster theory, the imprecise probabilities are defined with AND, OR and NOT operations to determine the lower and the upper probabilities. Currently, in the fuzzy set and the fuzzy logic theory, there is no analogous structure or connection to the probability theory. In this paper, we show that there exist a connection between the probability theory and the fuzzy set theory via its operations. We show that when we move from the context of the events to the context of possible worlds, in the modal logic, an isomorphism exist between the probability theory and the meta-theory based upon modal logic. With this isomorphism, it is possible to use the rules in the probability theory to formulate, in the context of possible worlds, the fuzzy logic rules with AND, OR and NOT operation with the t-norms, t-conorms and the fuzzy complement. When the formal theory of the probability is used for sets of worlds, the meaning of the probability changes dramatically. In the ordinary probability theory every event is separate, i.e., mutually exclusive, from the other events and is logically well defined. In the context of possible worlds, the events are semantic events that are not mutually exclusive, for which agents or experts assign different logic values in different worlds (contexts) to the same object. Since it is impossible to identify all the properties of an object, we evaluate hypothetical properties of the object from a partially open set of data or point of view. For this purpose the unknown property space of an entity or an object is divided into parts or granules, i.e., Corresponding Author: Germano Resconi. Dipartimento di Matematica, Università cattolica via Trieste 17. 25128 Brescia, Italy. E-mail: [email protected] www.mie.utoronto.ca/staff/profiles/turksen.html worlds, where hypothetical properties can be evaluated. However in the hypothetical nature of the local or context dependent evaluations, there are the possibilities of conflicting evaluations. That is the property evaluations of an object are substituted with a set of evaluations embedded in a conceptual space where granules are worlds. In this context, we propose an additive semantic measure theory that is based on the model of the fuzzy truthoods. Transformations in the logic values of a set of worlds generate asymmetry in the modal logic interpretation of uncertainty. At every transformation we associate a break of symmetry that we can compensate to generate new modus ponens and new tautology in the fuzzy logic. Similarities with the gauge theory in physics are possible.