Analysis of generalized square of opposition with intermediate quantifiers

Abstract

In this paper, we continue the development of a formal theory of intermediate quantifiers. These quantifiers are linguistic expressions such as “most”, “many”, “few”, and “almost all”, and they correspond to what are often called “fuzzy quantifiers” in the literature. In a previous study, we demonstrated that 105 generalized syllogisms with intermediate quantifiers are valid in our theory. In this paper, we turn our attention to another problem, which is the analysis of the generalized Aristotelian square of opposition, which, in addition to the classical quantifiers, can be extended by several selected intermediate quantifiers. We show that the expected relations can be well modeled in our theory. Our theory is developed within a special higher-order fuzzy logic, Łukasiewicz fuzzy type theory, and is very general, with a high potential for applications.