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
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
should hence be based on a solid
in order to guarantee predictable
and linguistically well-motivated
In the report,
an independent axiom system for
`reasonable' approaches to fuzzy quantification is introduced,
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.
An Axiomatic Theory of Fuzzy Quantifiers in Natural Languages.
Technical Report TR2000-03, University of Bielefeld, Technical Faculty, PO-Box 100131, 33501 Bielefeld, Germany, Feb. 2000, 53 pages.
Portable Document Format PDF, 53 pages, 700KB