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<Paper uid="J82-1003">
  <Title>From English to Logic: Context-Free Computation of &amp;quot;Conventional&amp;quot; Logical Translation 1</Title>
  <Section position="4" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2. Intensional and 'Conventional' Translations
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    <Paragraph position="0"> We should emphasize at the outset that our objective is not to impugn Montague grammar, but merely to make the point that the choice between intensional and conventional translations is as yet unclear. Given that the conventional approach appears to have certain advantages, it is worth finding out where it leads; but we are not irrevocably committed to this approach.</Paragraph>
    <Paragraph position="1"> Fortunately, the translation component of a parser for a Gazdar-style grammar is easily replaced.</Paragraph>
    <Paragraph position="2"> Montague grammarians assume that natural languages closely resemble formal logical systems; more specifically, they postulate a strict homomorphism from the syntactic categories and rules of a natural language to the semantic categories and rules required for its formal interpretation. This postulate has led them to an analysis of the logical content of natural language sentences which differs in important respects from the sorts of analyses traditionally employed by philosophers of language (as well as linguists and AI researchers, when they have explicitly concerned themselves with logical content).</Paragraph>
    <Paragraph position="3"> The most obvious difference is that intensional logic translations of natural language sentences conform closely with the surface structure of those sentences, except for some re-ordering of phrases, the introduction of brackets, variables and certain logical operators, and (perhaps) the reduction of idioms. For example, since the constituent structure of &amp;quot;John loves Mary&amp;quot; is \[John \[loves Mary\]\], the intensional logic translation likewise isolates a component translating the VP &amp;quot;loves Mary&amp;quot;, composing this VP-translation with the translation of &amp;quot;John&amp;quot; to give the sentence formula. By contrast, a conventional translation will have the structure \[John loves Mary\], in which &amp;quot;John&amp;quot; and &amp;quot;Mary&amp;quot; combine with the verb at the same level of constituent structure.</Paragraph>
    <Paragraph position="4"> In itself, this difference is not important. It only becomes important when syntactic composition is assumed to correspond to function application in the semantic domain. This is done in Montague grammar 5 We consistently use infix form (with the predicate following its first argument) and square brackets for complete sentential formulas.</Paragraph>
    <Paragraph position="5"> by resort to the Schoenfinkel-Church treatment of many-place functions as one-place functions (Schoenfinkel 1924, Church 1941). For example, the predicate &amp;quot;loves&amp;quot; in the above sentence is interpreted as a one-place function that yields a one-place function when applied to its argument (in this instance, when applied to the semantic value of &amp;quot;Mary&amp;quot;, it yields the function that is the semantic value of &amp;quot;loves Mary&amp;quot;). The resultant function in turn yields a sentence value when applied to its argument (in this instance, when applied to the semantic value of &amp;quot;John&amp;quot;, it yields the proposition expressed by &amp;quot;John loves Mary&amp;quot;). Thus, a dyadic predicator like &amp;quot;loves&amp;quot; is no longer interpreted as a set of pairs of individuals (at each possible world or index), but rather as a function into functions.</Paragraph>
    <Paragraph position="6"> Similarly a triadic predicator like &amp;quot;gives&amp;quot; is interpreted as a function into functions into functions.</Paragraph>
    <Paragraph position="7"> Moreover, the arguments of these functions are not individuals, because NPs in general and names in particular are assumed to denote property sets (or truth functions over properties) rather than individuals. It is easy to see how the postulate of syntactic-semantic homomorphism leads to this further retreat from traditional semantics. Consider Gazdar's top-level rule of declarative sentence structure and meaning: &lt;10, \[(S) (NP) (VP)\], (VP' NP&amp;quot;)&gt;.</Paragraph>
    <Paragraph position="8"> The first element of this triple supplies the rule number (which we have set to 10 for consistency with the sample grammar of Section 4), the second the syntactic rule and the third the semantic rule. The semantic rule states that the intensional logic translation of the S-constituent is compounded of the VP-translation (as functor) and the NP-translation (as operand), where the latter is first to be prefixed with the intension operator A. In general, a primed syntactic symbol denotes the logical translation of the corresponding constituent, and a double-primed symbol the logical translation prefixed with the intension operator (thus, NP&amp;quot; stands for ANP').</Paragraph>
    <Paragraph position="9"> For example, if the NP dominates &amp;quot;John&amp;quot; and the VP dominates &amp;quot;loves Mary&amp;quot;, then S' (the translation</Paragraph>
    <Paragraph position="11"> Similarly the translation of &amp;quot;Every boy loves Mary&amp;quot; comes out as ((loves' AMary') A(every' boy')), given suitable rules of NP and VP formation. 6 Note the uniform treatment of NPs in the logical formulas, i.e., (every' boy') is treated as being of the same semantic category as John', namely the (unique) seman-</Paragraph>
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