Fuzzy Quantifiers, Multiple Variable Binding, and Branching Quantification

Abstract

Lindström [1] introduced a very powerful notion of quantifiers, which permits multi-place quantification and the simultaneous binding of several variables. ‘Branching’ quantifification was found to be useful by linguists e.g. for modelling reciprocal constructions like “Most men and most women admire each other”. Westerståhl [2] showed how to compute the three-place Lindström quantifier for “Q ₁ A’s and Q ₂ B’sR each other” from the binary quantifiersQ ₁ andQ ₂, assuming crisp quantifiers and arguments. In the paper, I generalize his method to approximate quantifiers like “many” and fuzzy arguments like “young”. A consistent interpretation is achieved by extending the DFS theory of fuzzy quantification [3], [4], which rests on a system of formal adequacy criteria. The new analysis is important to linguistic data summarization because the full meaning of reciprocal summarizers (e.g. describing factors which are “correlated” or “associated” with each other), can only be captured by branching quantification.¹

The proofs of all theorems cited here are listed in [4].