Generalized Quantifiers and Natural Language

# Generalized Quantifiers and Natural Language - Generalized...

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Generalized Quantifiers and Natural Language Jon Barwise and Robin Cooper 0 Introduction In 1957, the Polish logician Andrej Mostowski pointed out that there are many mathematically interesting quantifiers that are not definable in terms of the first- order V, 3 and initiated study of so-called generalized quantifiers (cf. Mostowski 1957). Since then logicians have discovered and studied a large number of generalized quantifiers. At last count there were well over 200 research papers in this area. Most of this work has been directed toward cardinality quantifiers (e.g. Keisler 1969) and topological quantifiers (e.g. Sgro 1977) which are not particularly relevant to natural language, but even so, it has forced logicians to rethink the traditional theory of quantification. The quantifiers of standard first-order logic (as presented in elementary logic textbooks) are inadequate to treat the quantified sentences of natural languages in at least two respects. First, there are sentences which simply cannot be symbolized in a logic which is restricted to the first-order quantifiers V and 3. Second, the syntactic structure of quantified sentences in predicate calculus is completely different from the syntactic structure of quantified sentences in natural language. The work on general- ized quantifiers referred to above has led to new insights into the nature of quantifiers, insights which permit logical syntax to correspond more closely to natural language syntax. These insights, we argue, may also make a significant contribution to linguistic theory. Section 1 discusses the nature of generalized quantifiers and their relationship to the syntax of English in general terms. Section 2 develops a logic containing generalized quantifiers. Section 3 shows how this logic may be formally related to a fragment of a syntax for English. Section 4 is the main section of the paper. In it we discuss some of the general implications of the notion of generalized quantifier for a theory of natural language of the kind that is interesting to linguists. Our conclusion, in section 5, attempts to draw some general conclusions about the relationship between syntax, semantics and logic.

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76 Jon Barwise and Robin Cooper The paper has four appendices. Appendix A contains additions to the fragment in section 3 which are suggested by the results in 4 4. Appendix B contains some possible semantic postulates on the meaning of non-logical determiners. Appendix C contains the proofs of the facts about quantifiers asserted in the body of the paper. Appendix D consists of a chart classifying English determiners according to the semantic categories introduced in 3 4. Some (but not all) of the points made in section 1-3 of this paper are implicit or explicit in Montague (1974), especially in PTQ, "The Proper Treatment of Quantifica- tion in Ordinary English". (Some of the suggestions in 1-3 are also similar to suggestions in other papers: e.g. Fenstad (1978); Peacocke (1979)). Our hope is to develop Mon-
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• Fall '07
• Dever
• Semantics, Predicate logic, Quantification, natural language, First-order logic

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