Generalized fuzzy quantifiers and the modeling of fuzzy branching quantification
Ingo Glöckner · International Journal of Intelligent Systems · 2009
Lindström (Theoria, 1966; 32:186–195) introduced a very powerful notion of quantifiers, which permits multiplace quantification and the simultaneous binding of several variables. “Branching” quantification was found to be useful by linguists, e.g., for modeling reciprocal constructions like “Most men and most women admire each other.” Westerståhl (In Gärdenfors P., editor. Generalized Quantifiers. New York: Reidel; 1987. pp 269–298) showed how to compute the three-place Lindström quantifier for “Q1 As and Q2 Bs R each other” from the binary quantifiers Q1 and Q2, assuming crisp quantifiers and arguments. In the paper, this basic method will be generalized to vague quantifiers like “many” and fuzzy arguments like “young.” A consistent interpretation is achieved by extending the DFS theory of fuzzy quantification (Glöckner, TR97-06, 1997; TR2002-07, (Int J Approx Reason) 2003; 2004; 37(2):93–126), which introduces fuzzy generalized quantifiers in conformance with the original linguistic notion, and controls their interpretation by 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. The proofs of all theorems cited in the paper are listed in (Glöckner, TR2002-07, 2003). © 2009 Wiley Periodicals, Inc.