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<Paper uid="P89-1019">
  <Title>A Calculus for Semantic Composition and Scoping</Title>
  <Section position="6" start_page="158" end_page="158" type="concl">
    <SectionTitle>
5 Discussion
</SectionTitle>
    <Paragraph position="0"> The approach to semantic interpretation outlined above avoids the need for manipulations of logical forms in deriving the possible meanings of quantified sentences. It also avoids the need for such devices as distinguished variables (Gazdar, 1982; Cooper, 1983) to deal with trace abstraction. Instead, specialized versions of the basic rule of functional abstraction are used. To my knowledge, the only other approaches to these problems that do not depend on formal operations on logical forms are those based on specialized logics of type change, usually restrictions of the Curry or Lambek systems (van Benthem, 1986a; Hendriks, 1987; Moortgat, 1988). In those accounts, a phrase P with meaning p of type T is considered to have also alternative meaning tC/ of type T', with the corresponding combination possibilities, if p' : T' follows from p : T in the chosen logic. The central problem in this approach is to design a calculus that will cover all the actual semantic alternatives (for instance, all the possible quantifier scopings) without introducing spurious interpretations. For quantifier raising, the system of Hendriks (1987) seems the most promising so far, but it is at present too restrictive to support raising from noun-phrase complements.</Paragraph>
    <Paragraph position="1"> An important question I have finessed here is that of the compositionality of the proposed semantic calculus. It is clear that the application of semantic rules is governed only by the existence of appropriate syntactic licensing and by the availability of premises of the appropriate types. In other words, no rule is sensitive to the form of any of the meanings appearing in its premises. However, there may be some doubt as to the status of the basic abstraction rule and those derived from it. After all, the use of A-abstraction in the consequent of those rules seems to imply the constraint that the abstracted object should formally be a variable. However, this is only superficially the case. I have used the formal operation of A-abstraction to represent functional abstraction in this paper, but functional abstraction itself is independent of its formal representation in the Acalculus. This can be shown either by using other notations for functions and abstraction, such as that of de Bruijn's (Barendregt, 1984; Huet, 1986), or by expressing the semantic derivation rules in A-Prolog (Miller and Nadathur, 1986) following existing presentations of natural deduction systems (Felty and Miller, 1988).</Paragraph>
  </Section>
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