On the Mechanization of Presuppositions and Partiality

It is well-known that many relevant aspects of everyday reasoning based on natural language cannot be adequately expressed in classical rst-order logic. In this paper we address two of the problems, rstly that of so-called presuppositions, expressions from which it is possible to draw implicit conclusion, which classical logic normally does not warrant, and secondly the related problem of partiality and the adequate treatment of undeened expressions. In natural language, presup-positions are quite common, they can, however, only insuuciently be modeled in classical rst-order logic. For instance, in the case of universal quantiication one normally uses restrictions in natural language 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. Similarly, undeened expressions can be handled in natural language discourses and utterances are not only classiied into the two categories`true' and`false'. This has led to the development of various better-suited many-valued logics. By combining diierent approaches we can now give a static description of presuppositions and undeenedness within the same framework. Additionally, we have developed an eecient mechanization of the induced consequence relation (which has been missing in the literature) by combining methods from many-valued truth-functional logics and sort techniques developed for search control in automated theorem proving.