A Resolution Calculus for Presuppositions

The semantics of everyday language and the semantics of its naive translation into classical first-order language considerably differ. An important discrepancy that is addressed in this paper is about the implicit assumption what exists. For instance, in the case of universal quantification natural language uses restrictions and presupposes that these restrictions are non-empty, while in classical logic it is only assumed that the whole universe is non-empty. On the other hand, all constants mentioned in classical logic are presupposed to exist, while it makes no problems to speak about hypothetical objects in everyday language. These problems have been discussed in philosophical logic and some adequate many-valued logics were developed to model these phenomena much better than classical first-order logic can do. An adequate calculus, however, has not yet been given. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. Unfortunately, restricted quantifications are not truth-functional, hence they do not fit the framework directly. We solve this problem by applying recent methods from sorted logics.