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<Paper uid="E99-1012">
  <Title>Ambiguous propositions typed</Title>
  <Section position="2" start_page="0" end_page="86" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> A widely held view expressed in (Carbonell and Hayes, 1987) is that &amp;quot;if there were one word to describe why natural language processing is hard, it is ambiguity.&amp;quot; For any given natural language utterance, a formal language such as predicate logic typically offers several non-equivalent (wellformed) formulas as possible translations. An obvious approach is to take the disjunction of all alternatives, assuming (for the sake of the argument) that the disjunction is a formula. Even if it were, however, various objections have been raised against this proposal (e.g. (Deemter, 1996)). For the purposes of the present paper, what is interesting about a word, phrase, sentence or discourse that is ambiguous in isolation is how it may get disambiguated when combined with other expressions (or, more generally, when placed in a wider context); the challenge for any theory of ambiguity is to throw light on that process of disambiguation. null *From June to mid-August 1999, I will be visiting IMS, Uni Stuttgart, Azenbergstr 12, 70174 Stuttgart, Germany. Where I might be after that is unclear.</Paragraph>
    <Paragraph position="1"> More concretely, suppose * were a binary connective on propositions A and B such that A * B is a proposition ambiguous between A and B. Under the &amp;quot;propositions-as-types&amp;quot; paradigm (e.g. (Girard et al., 1989)) identifying proofs of a proposition with programs of the corresponding type (so that &amp;quot;t: A&amp;quot; can be read as t is a proof of proposition A, or equivalently, t is a program of type A), disambiguation may take the form of type coercion. An instructive example with F as the con-</Paragraph>
    <Paragraph position="3"> where ap is function application (corresponding to modus ponens), while p. and qo are the first and</Paragraph>
    <Paragraph position="5"> Evidently, there is something conjunctive (never mind disjunctive) about o; but beyond the question as to whether the unambiguous propositions constituting the possible readings of an ambiguous proposition form a conjunctive or disjunctive set (whatever that may precisely mean), there is also the matter of the interconnected choices from such sets, mediated by terms such as pdeg(x) and qdeg(Y).</Paragraph>
    <Paragraph position="6"> To ground these abstract considerations in natural language processing, a few words about how to think of the terms t and types A are useful.</Paragraph>
    <Paragraph position="7"> For predicate logic formulas A, the terms t might be intuitionistic natural deduction proofs, related by the Curry-Howard isomorphism to a suitable typed A-calculus. A notable innovation made in Intuitionistic Type Theory (ITT, (Martin-LSf,  1984)) is to allow proofs to enter into judgments of well-formedness (propositionhood). This stands in sharp contrast to ordinary predicate logic (be it intuitionistic or classical), where well-formedness is a trivial matter taken for granted (rather than analyzed) by the Curry-Howard isomorphism. For a natural language, however, it is well-formedness that is addressed by building types A over sentences, nouns, etc (in categorial grammar; e.g. (Morrill, 1994)) or LFG f-structures (in the &amp;quot;glue&amp;quot; approach, (Dalrymple et al., 1993; Dalrymple et al., 1997)). Now, while ITT's rules for propositionhood hardly constitute an account of grammaticality in English, the combination (in ITT) of assertions of well-formedness (A type) and theoremhood (t: A) re-introduces matters of information content (over and above grammatical form), which have been applied in (Ranta, 1994) (among other places) to discourse semantics (in particular, anaphora). The present paper assumes the machinery of dependent functions and sums in ITT, without choosing between grammatical and semantic applications. In both cases, what ambiguity contributes to the pot is indeterminacy in typing, the intuition being that an expression is ambiguous to the extent that its typing is indeterminate. null That said, let us return to (1) and consider how to capture sequent inferences such as</Paragraph>
    <Paragraph position="9"> (i) and more complicated cases from iterated applications of., nested among other type constructs.</Paragraph>
    <Paragraph position="10"> The idea developed below is to set aside the con- and nective * (as well as notational clutter p., q.), (ii) and to step up from assertions t : A to (roughly)</Paragraph>
    <Paragraph position="12"> But what exactly could t ::A mean? The disjunctive conception t::A iff t:A for someAEA (3) would have as a consequence the implication t::-4 and .4 C B implies t::B.</Paragraph>
    <Paragraph position="13"> Now, if combinatorial explosion is a problem for ambiguity, then surely we ought to avoid feeding it with cases of spurious ambiguity. A complementary alternative is conjunction, t::A iff t:A for allAEA, (4) the object this time being to identify the C_-largest such set A, as (4) supports t::A and B C .4 implies t::B .</Paragraph>
    <Paragraph position="14"> But while (4) and (2) will do for Ax.y where y is a variable distinct from x, (4) suggests that (2) overgenerates for Ax.x. Spurious ambiguity may also arise to the left of ~- (not just to the right), if we are not careful to disambiguate the context.  (1) illustrates the point; compare</Paragraph>
    <Paragraph position="16"> where the context F is left untouched, to</Paragraph>
    <Paragraph position="18"> where the context gets trimmed. (5) and (2) yield</Paragraph>
    <Paragraph position="20"> To weed out spurious ambiguity, we will attach variables onto sets .4 of types, to form decorated expressions ct collect constraints on a's in sets C, hung as subscripts, }-c, on ~-.</Paragraph>
    <Paragraph position="21"> (3) and (4) are then sharpened by a contextual characterization, semantically interpreting judgments of the form t :: a and a typ by disambiguations respecting suitable constraints.</Paragraph>
  </Section>
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