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<Paper uid="P84-1064">
  <Title>A COMPUTATIONAL THEORY OF DISPOSITIONS</Title>
  <Section position="6" start_page="316" end_page="317" type="concl">
    <SectionTitle>
6. Negation of Dispositlona
</SectionTitle>
    <Paragraph position="0"> In dealing with dispositions, it is natural to raise the question: What happens when a disposition is acted upon with an operator, T, where T might be the operation of negation, active-to-passive transformation, etc. More generally, the same question may be asked when T is an operator which is defined on pairs or n-tuples of disp?sitions.</Paragraph>
    <Paragraph position="1"> As an illustration, we shall focus our attention on the operation of negation. More specifically, the question which we shall consider briefly is the following: Given a disposition, d, what can be said about the negaton of d, not d? For example, what can be said about not (birds can fly) or not (young men like young women).</Paragraph>
    <Paragraph position="2"> For simplicity, assume that, after restoration, d may be expressed in the form rd A Q A W s are BIs . (6.1) Then, not d = not (Q A ' s ore B ' s). (6.2) Now, using the semantic equivalence established in Zadeh (1978), we may write not (Q A's are B's)E(not Q)A's ore B'o , (6.3) where not Q is the complement of the fuzzy quantifier Q in the sense that the membership function of not Q is given by P,,ot Q(u).~- 1-pQ(u),0 &lt; u &lt; 1 . (6.4) Furthermore, the following inference rule can readily be established (gadeh, 1983a):  if Q is monotonic (e.g., Q A most).</Paragraph>
    <Paragraph position="3"> As an illustration, if d A birds can fly and Q A most, then (0.8) yields not (birds can fly) (ant (not most)) birds cannot fly. (o.g) It should be observed that if Q is an approximation to all, then ant(not Q) is an approximation to some. For the right-hand member of (0.9) to be a disposition, most must be  an approximation to at least a half. In this case ant \[not most\] will be an approximation to most, and consequently the right-hand member of (0.9) may be expressed -- upon the suppression of most -- as the disposition birds cannot fly.</Paragraph>
  </Section>
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