Incidence Calculus on Łukasiewicz's Three-valued Logic

Incidence calculus is a probabilistic logic which possesses both numerical and symbolic approaches. However, Liu in [5] pointed out that the original incidence calculus had some drawbacks and she established a generalized incidence calculus theory (GICT) based on Łukasiewicz's three-valued logic to improve it. In a GICT, an incidence function is defined to relate each proposition ϕ in the axioms of the theory to a set of possible worlds in which ϕ has truth value true. But the incidence function only represents those absolute true states of propositions, so it can not deal with the uncertain states. In this paper, we use two incidence functions i _* and i ^* to relate the axioms to the sets of possible worlds. For an axiom ϕ, i _* (ϕ) is to be thought of as the set of possible worlds in which ϕ has truth value true, while i ^* (ϕ) is the set of possible worlds in which ϕ is true or undeterminable. Since i ^* can represent the undeterminable state, our newly defined theory is more efficient to handle vague information than GICT.