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<Paper uid="E95-1001">
  <Title>On Reasoning with Ambiguities</Title>
  <Section position="3" start_page="0" end_page="2" type="intro">
    <SectionTitle>
2 Consequence Relations
</SectionTitle>
    <Paragraph position="0"> In this section we will discuss some sample arguments containing ambiguous expressions in the data as well as in the goal. We consider three kinds of ambiguities: lexical ambiguities, quantifier scope ambiguities, and ambiguities with respect to distributive/collective readings of plural noun phrases. The discussion of the arguments will show that the meaning of ambiguous sentences not only depends on the set of its disambiguations. Their meanings also depend on the context, especially on other occurrences of ambiguities. Each disambiguation of an ambiguous sentence may be correlated to disambiguations of other ambiguous sentences such that the choice of the first disambiguation also determines the choice of the latter ones, and vice versa. Thus the representation of ambiguities requires some means to implement these correlations.</Paragraph>
    <Paragraph position="1"> To see that this is indeed the case let us start discussing some consequence relations that come to mind when dealing with ambiguous reasoning. The first one we will consider is the one that allows to derive a(n ambiguous) conclusion 7 from a set of (ambiguous) premisses F if some disambiguation of 7 follows from all readings of F. Assuming that 5 and 5~ are operators mapping a set of ambiguous representations a onto one of its disambiguations a ~ or a ~' we may represent this by.</Paragraph>
    <Paragraph position="2"> (1) v~3~'(r ~ p C/').</Paragraph>
    <Paragraph position="3"> Obviously (1) is the relation we get if we interpret ambiguities as being equivalent to the disjunctions of their readings. To interpret ambiguities in this way is, however, not correct. For ambiguities in the goal this is witnessed by (2).</Paragraph>
    <Paragraph position="4"> (2) ~ Everybody slept or everybody didnlt sleep.</Paragraph>
    <Paragraph position="5"> Intuitively (2) is contingent, but would - according to the relation in (1) - be classified as a tautology.</Paragraph>
    <Paragraph position="6"> In this case the consequence relation in (3) gives the correct result and therefore seems to be preferable.</Paragraph>
    <Paragraph position="7"> (3) v v l(r p C/') But there is another problem with (3). It does not fulfill Reflexivity, which (1) does.</Paragraph>
    <Paragraph position="8"> Reflexivity F ~ C/, if C/ e F To do justice to both, the examples in (2) and Reflexivity, we would have to interpret ambiguous sentences in the data also as conjunctions of their readings, i.e. accept (4) as consequence relation.</Paragraph>
    <Paragraph position="9">  (4) 35'3~(r ~ ~ 7 ~') But this again contradicts intuitions. (4) would support the inferences in (5), which are intuitively not correct.</Paragraph>
    <Paragraph position="10"> a. There is a big plant in front of my house.</Paragraph>
    <Paragraph position="11"> (5) ~ There is a big building in front of my house.</Paragraph>
    <Paragraph position="12"> b. Everybody didn't sleep. ~ Everybody was awake.</Paragraph>
    <Paragraph position="13"> c. Three boys got PS10. ~ Three boys got PS10 each.</Paragraph>
    <Paragraph position="14">  Given the examples in (5) we are back to (1) and may think that ambiguities in the data are interpreted as disjunctions of their readings. But irrespective of the incompatibility with Reflexivity this picture cannot be correct either, because it distroys the intuitively valid inference in (6).</Paragraph>
    <Paragraph position="15"> (6) If the students get PS10 then they buy books.</Paragraph>
    <Paragraph position="16"> The students get PS10. ~ They buy books.</Paragraph>
    <Paragraph position="17"> This example shows that disambiguation is not an operation 5 that takes (a set of) isolated sentences. Ambiguous sentences of the same type have to be disambiguated simultaneously. 1 Thus the meaning of 1We will not give a classification or definition of ambiguities of the same type here. Three major classes will consist of lexical ambiguities, ambiguities with respect to distributive/collective readings of plural noun phrases, and quantifier scope ambiguities. As regards the last type we assume on the one hand that only sentences with the same argument structure and the same set of readings can be of the same type. More precisely, if two sentences are of the same type with respect to quantifier scope ambiguities, then the labels of their UDRS's the premise of (6) is given by (7b) not by (7a), where al represents the first and a2 the second reading of the second sentence of (6).</Paragraph>
    <Paragraph position="18"> a. ((al b) V (a2 b)) ^ V (7) b. ((al -+ b) A el) V ((a2 --+ b) A a2) We will call sentence representations that have to be disambiguated simultaneously correlated ambiguities. The correlation may be expressed by coindexing. Any disambiguation ~ that simultaneously disambiguates a set of representations coindexed with i is a disambiguation that respects i, in symbols ~. A disambiguation ~i that respects all indices of a given set I is said to respect I, written ~. Let I be a set of indices, then the consequence relation we assume to underly ambiguous reasoning is given in (8) (s) p The general picture we will follow in this paper is the following. We assume that a set of representations F represents the mental state of a reasoning agent R.</Paragraph>
    <Paragraph position="19"> r contains underspecified representations. Correlations between elements of r indicate that they share possible ways of disambiguation. Suppose V is only implicitly contained in r. Then R may infer it from F and make it explicit by adding it to its mental state. This process determines the consequence relation relative to which we develop our inference patterns. That means we do not consider the case where R is asked some query 7 by another person B. The additional problem in this case consists in the array of possibilities to establish correlations between B's query and R's data, and must be adressed within a proper theory of dialogue.</Paragraph>
    <Paragraph position="20"> Consider the following examples. The data contains two clauses. The first one is ambiguous, but not in the context of the second.</Paragraph>
    <Paragraph position="21">  a. Every pitcher was broken. They had lost.</Paragraph>
    <Paragraph position="22"> Every pitcher was broken.</Paragraph>
    <Paragraph position="23"> b. Everybody didn't sleep. John was awake.</Paragraph>
    <Paragraph position="24"> (9) ~ Everybody didn't sleep.</Paragraph>
    <Paragraph position="25"> c. John and Mary bought a house.</Paragraph>
    <Paragraph position="26">  It was completely delapidated.</Paragraph>
    <Paragraph position="27"> John and Mary bought a house.</Paragraph>
    <Paragraph position="28"> If the inference is now seen as the result of R's task to make the first sentence explicit (which of course is trivial here), then the goal will not be ambiguous, because it simply is another occurrence of the representation in the data, and, therefore, will carry the same correlation index. In the second case, i.e. the case where the goal results from R's processing some external input, there is no guarantee for such a correlation. R might consider the goal as ambiguous, and hence will not accept it as a consequence. (B might after all have had in mind just that reading of the sentence that is not part of R's knowledge.) must be ordered isomorphically. On the other hand two sentences may carry an ambiguity of the same type if one results from the other by applying Detachment to a universally quantified NP (see Section 4).</Paragraph>
    <Paragraph position="29">  We will distinguish between these two situations by requiring the provability relation to respect indices. The rule of direct proof will then be an instance of Reflexivity: F t- 7i if ~'i E F.</Paragraph>
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