An Axiomatic Theory of Fuzzy Quantifiers in Natural Languages

Many applications e.g. in approximate reasoning, data summarisation, information retrieval etc. can profit from the use of fuzzy quantifiers like “almost all” or “many”, which provide flexible means of information aggregation, and are capable of extracting meaningful linguistic summaries from large amounts of raw data. However, as will be shown by a number of counterexamples, existing approaches fail to provide a convincing interpretation of fuzzy quantifiers in the important case of two-place quantification (e.g. “about half of the blondes are tall”). The interpretation of fuzzy quantifiers should hence be based on a solid axiomatic foundation in order to guarantee predictable and linguistically wellmotivated results. In the report, an independent axiom system for “reasonable” approaches to fuzzy quantification is introduced, that are consistent with the use of quantifiers in NL. A number of linguistic adequacy criteria are formalized and it is shown that every model of the axiom system exhibits these essential properties. However, some principled adequacy bounds for approaches to fuzzy quantification are also established, which in most cases result from the known conflict between idempotence/distributivity and the law of contradiction in the presence of fuzziness. In addition, a broad class of models of the axiomatic framework is introduced. One of these models, which generalises the Sugeno integral (and hence the FG-count approach) can be shown to possess unique adequacy properties. Its analysis unveals the first definition of fuzzy cardinality which achieves adequate results with arbitrary quantitative one-place quantifiers. It is also shown how the Choquet integral (and hence the OWA approach) can be generalized to a model of the axiomatic framework. The resulting models not only represent a significant theoretical advance in fuzzy quantification; they are also practical. Efficient histogram-based algorithms for evaluating the resulting fuzzy quantifiers are described at the end of the report.