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<Paper uid="E93-1017">
  <Title>The donkey strikes back Extending the dynamic interpretation &amp;quot;constructively&amp;quot;</Title>
  <Section position="3" start_page="0" end_page="130" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> The line If a farmer owns a donkey he beats it (1) from Geach \[6\] is often cited as one of the success stories of the so-called &amp;quot;dynamic&amp;quot; approach to natural language semantics (by which is meant Kamp \[12\], Heim \[9\], Sarwise \[1\], and Groenendijk and Stokhof \[7\], among others). But add the note It will kick back (2) and the picture turns sour: processing (1) may leave no beaten donkey active. Accordingly, providing a referent for the pronoun it in (2) would appear to call for some non-compositional surgery (that may upset many a squeamish linguist). The present paper offers, as a preventive, a &amp;quot;dynamic&amp;quot; form of implication =~ applied to (1). Based on a &amp;quot;constructive&amp;quot; conception of discourse analysis, an overhaul of Groenendijk and Stokhof \[7\]'s Dynamic Predicate Logic (DPI.) is suggested, although :=~ can also be introduced less destructively so as to extend DPL conservatively. Thus, the reader who prefers the old &amp;quot;static&amp;quot; interpretation of (1) can still make that choice, and declare the continuation (2) to be &amp;quot;semantically ill-formed.&amp;quot; On the other hand, Groenendijk and Stokhof \[7\] themselves concede that &amp;quot;at least in certain contexts, we need alternative externally dynamic interpretations of universal quantification, implication and negation; a both internally and externally dynamic treatment of disjunction.&amp;quot; A proposal for such connectives is made below, extending the dynamic interpretation in a manner analogous to the extension of classical logic by constructive logic (with its richer collection of primitive connectives), through a certain conjunctive notion of parallelism. null To put the problem in a somewhat general perspective, let us step back a bit and note that in assigning a natural language utterance a meaning, it is convenient to isolate an intermediate notion of (say) a formula. By taking for granted a translation of the utterance to a formula, certain complexities in natural language can be abstracted away, and semantics can be understood rigorously as a map from formulas to meanings. Characteristic of the dynamic approach mentioned above is the identification of the meaning of a formula A with a binary relation on states (or contexts) describing transitions A induces, rather than with a set of states validating A. In the present paper, formulas are given by first-order formulas, and the target binary relations given by programs. To provide an account of anaphora in natural language, DPL translates first-order formulas A r m DPL ro tiff 1 to p ogra s A f m (quan &amp;quot; ed) dynam'c logic (see, for example, Harel \[8\]) as follows A DPL - A? for atomic A</Paragraph>
    <Paragraph position="2"> The negation --,p of a program p is the dynamic logic test (\[p\] +-) ? with universal and static features (indicated respectively by \[p\] and ?),1 neither of which is intrinsic to the concept of negation. Whereas some notion of universality is essential to universal quantification and implication (which are formulated through negation</Paragraph>
    <Paragraph position="4"> and accordingly inherit some properties of negation), our treatment of (2) will be based on a dynamic (rather than static) form =~ of implication. Dynamic forms of negation ~, universal quantification and disjunction will also be proposed, but first we focus on implication.</Paragraph>
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