Modulated logics and flexible reasoning

Abstract

This paper studies a family of monotonic extensions of first-order logic which we call modulated logics, constructed by extending classical logic through generalized quantifiers called modulated quantifiers. This approach offers a new regard to what we call flexible reasoning. A uniform treatment of modulated logics is given here, obtaining some general results in model theory. Besides reviewing the “Logic of Ultrafilters”, which formalizes inductive assertions of the kind “almost all”, two new monotonic logical systems are proposed here, the “Logic of Many” and the “Logic of Plausibility”, that characterize assertions of the kind “many”, and “for a good number of”. Although the notion of simple majority (“more than half”) can be captured by means of a modulated quantifier semantically interpreted by cardinal measure on evidence sets, it is proven that this system, although sound, cannot be complete if checked against the intended model. This justifies the interest on a purely qualitative approach to this kind of quantification, what is guaranteed by interpreting the modulated quantifiers as notions of families of principal filters and reduced topologies, respectively. We prove that both systems are conservative extensions of classical logic that preserve important properties, such as soundness and completeness. Some additional perspectives connecting our approach to flexible reasoning through modulated logics to epistemology and social choice theory are also discussed.