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<Paper uid="W98-0309">
  <Title>Similarity and contrast relations and inductive rules</Title>
  <Section position="4" start_page="54" end_page="54" type="metho">
    <SectionTitle>
PATTERN OF INSTANTIATION
</SectionTitle>
    <Paragraph position="0"> UNILATERAL 1&amp;quot; = P.</Paragraph>
    <Paragraph position="1"> BILATERAL }&amp;quot; = C.</Paragraph>
    <Paragraph position="2"> Figure 3: The PATTERN OF INSTANTIATION parameter null Example 4 is then analysed as NEGATIVE POLARITY UNILATERAL, while Example 3 is analysed as NEGATIVE POLARITY BILATERAL. See (Knott, 1996;  Knott and Mellish, 1996) for a more detailed presentation of this framework, along with definitions of several more independent parameters.</Paragraph>
  </Section>
  <Section position="5" start_page="54" end_page="54" type="metho">
    <SectionTitle>
3 Comparisons and expectations
</SectionTitle>
    <Paragraph position="0"> Why might we assume that there is a defeasible rule underlying relations of SIMILARITY and CONTRAST? An initial piece of evidence comes from contexts such as the following, in which a similarity between two objects in one respect apparently gives rise to an expectation of similarities in other respects.</Paragraph>
    <Paragraph position="1"> (5) \[J0 was made by Richard Page.\] ...This brooch was also made by Richard Page. But whereas J0 was made in 1985, this brooch was made in 1990.</Paragraph>
    <Paragraph position="2"> This compare-and-contrast pattern is common in descriptive texts. We seem to have here a CONCESSION relation (signalled by but), whose satellite is a complex span comprising a SIMILARITY relation, and whose nucleus is a complex span comprising a CONTRAST relation: see Figure 4. The question is, why  do we have a CONCESSION relation between these two complex spans? According to the analysis of CONCESSION relations in Section 2.1. the relation must presuppose the existence of a defensible rule which is being defeated at this point. What could the rule be in this case? We would not want to suppose a rule that if an object is made by Page, it is likely to be made in 1990. Rather, we need a rule that states that if two objects are similar in some respect, they are also similar in other respects.</Paragraph>
  </Section>
  <Section position="6" start_page="54" end_page="55" type="metho">
    <SectionTitle>
4 Comparisons and inductive rules
</SectionTitle>
    <Paragraph position="0"> \Ve propose that a similarity relation should be represented as triggering such a rule. We suggest that similarity, like other relations, presupposes a defeasible rule: but that the rule in question is inductive in form. In brief, we suggest that a similarity relation between two propositions be thought of as permitting an inductive rule to fire, while a contrast relation be thought of as preventing an inductive rule from firing.</Paragraph>
    <Paragraph position="1"> In Section 4.1 we outline a tire rules, and in Section 4.2 for comparison relations with anism.</Paragraph>
    <Paragraph position="2"> mechanism for inducwe frame a definition respect to this mech-</Paragraph>
    <Section position="1" start_page="54" end_page="55" type="sub_section">
      <SectionTitle>
4.1 A recta-level system of inductive rules
</SectionTitle>
      <Paragraph position="0"> We will represent inductive rules as operating at a meta-level on a defeasible first-order logic based on commonsense entailment. We envisage inductive rules applying as second-order rules, whose conclusions are first-order defensible rules; when an inductive rule fires, it results in the addition or alteration of defensible rules within the first-order system. &amp;quot;  For the moment, we do not want so describe the workings of this second-order system in any detail; our concern here is really just to consider whether inductive rules might have a role in explaining certain structural properties of comparisons. Nevertheless, we will propose a rudimentary model of meta-level rules.</Paragraph>
      <Paragraph position="1"> The first-order system we propose is identical to commonsense entailment, except that each first-order defensible rule is associated with a strength, represented by a pair of values s/t, where t is the number of times the rule has been triggered and s is the number of times it has succeeded. A rule only becomes part of the set used in first-order defeasible reasoning if its values for t and s/t each reach a certain threshold. However, rules with values below the threshold can still be used as the preconditions for discourse relations.</Paragraph>
      <Paragraph position="2"> We then define a second-order defeasible connective &gt;&gt; to model inductive rules. An inductive rule has the general form given below:  (6) VP, a,b P(a) n P(b)(A...) &gt;&gt; C/  where C/ evaluates to one or more first-order defensible rules. When a rule is triggered, the t and s values of each of the right-hand side rules are incremented. We will be using just two inductive rules, which are given in Figure 5. Rule 7 says that given two objects in class c which are both P, we can increment the strength of the rule that asserts that objects in class c are typically P. Rule 8 says that if two objects have one property P in common, we should increment the strength of all rules that allow us to infer properties of one object on the basis of knowing properties of the other.</Paragraph>
    </Section>
    <Section position="2" start_page="55" end_page="55" type="sub_section">
      <SectionTitle>
4.2 Inductive rules in relation definitions
</SectionTitle>
      <Paragraph position="0"> How might we analyse also as presupposing an inductive rule? The first thing we need is a new parameter for specifying what sort of rule is being presupposed; this is given in Figure 6.</Paragraph>
    </Section>
  </Section>
  <Section position="7" start_page="55" end_page="55" type="metho">
    <SectionTitle>
RULE TYPE
</SectionTitle>
    <Paragraph position="0"> We can then define also as signalling a POSITIVE POLARITY, UNILATERAL and INDUCTIVE relation between the propositions it links. Here is an example: (9) Brooch B 1 is ornate. (...) Brooch B2 is also ornate.</Paragraph>
    <Paragraph position="1"> According to our model, one of the effects of this relation is to cause the inductive rule 7 to fire, which in turn has the effect of increasing the strength of the generalisation that brooches of the class to which B1 and B2 belong are typically ornate. This seems a plausible effect, particularly in a descriptive context where a reader/hearer is being informed about objects in an unfamiliar domain.</Paragraph>
    <Paragraph position="2"> Now consider a contrast relation, of the kind signalled by whereas. On our model, this relation would be NEGATIVE POLARITY, UNILATERAL and INDUC-TIVE. Here is an example of such a relation: (10) Brooch B1 is ornate, whereas Brooch B2 is simple.</Paragraph>
    <Paragraph position="3"> According to the model, the effect of this relation is simply to leave the s and t values of the generalisation that 'all brooches of the relevant class are ornate' unchanged. This is not particularly satisfactory; we might want to make it have a more significant effect, perhaps by increasing only the t value (thereby reducing s/t): but as things are currently formulated, changes to the strengths of first-order rules are only possible when an inductive rule fires.</Paragraph>
    <Paragraph position="4"> PATTERN OF INSTANTIATION for INDUCTIVE relations We have seen that the POLARITY parameter appears to do useful work for INDUCTIVE relations.</Paragraph>
    <Paragraph position="5"> We should now consider whether the PATTE~-N OF INSTANTIATION parameter is productive for tNDUC-TIVE relations. Those we have seen so far have all been UNILATERAL. What might an INDUCTIVE BI-LATERAL relation look like': Such a relation would have to hold between two propositions, one being an individual proposition, and the other being a generalisation for which the individual proposition provided inductive support. One possibility is that this class of relations are those which can be signalled by the connective indeed. Consider the following example: null (11) This jewel is elaborate. Indeed, most Art-Deco jewels are elaborate.</Paragraph>
    <Paragraph position="6"> If we assume that the proposition most Art-Deco jewels are elaborate takes as its semantic value a first-order defeasible rule of the form Vx isa(x, art_deco) &gt; elaborate(x), then it seems that we can describe the relation signalled by indeed as POSITIVE POLARITY. INDUCTIVE and BILA.T-ERAL. I Z Defeasible rules in commonsense entailment are actually intended to represent the semantics of generic sentences. However, the generalisa.tion introduced by indeed does not have to be a generic; the proposed account of indeed undergenerates in this regard.</Paragraph>
  </Section>
  <Section position="8" start_page="55" end_page="55" type="metho">
    <SectionTitle>
5 Structural consequences of
</SectionTitle>
    <Paragraph position="0"> inductive rules We will now consider whether the proposed account of comparison relations can help us in accounting for some of their unusual structural characteristics.</Paragraph>
    <Section position="1" start_page="55" end_page="55" type="sub_section">
      <SectionTitle>
5.1 Compare-and-contrast structures
</SectionTitle>
      <Paragraph position="0"> Firstly, consider again Example 5. To recap: what we have here is a pair of comparison relations, apparently linked by a CONCESSIOn' relation, and the difficulty is to explain why the CONCESSION relation applies. We can begin to account for this effect by noting that the initial similarity relation between the first two sentences causes Rule 8 to fire as well as Rule 7. The effect of Rule 8 firing is to add/increment the strength of a whole set of first-order rules allowing inference from J0's possession of a given property to the brooch's possession of that property (and vice versa). One of these rules allows an inference from the date of manufacture of one Object to that of the other. The contrast relation in the second sentence provides information that explicitly states that this inference is not permitted, and must result in the newly-added rule being defeated if consistency is to be preserved. If we can take this to be a case where Rule 8 is defeated, which seems plausible, then we can consider the high-level relation signalled by but to be NEGATIVE POLARITY. BILAT-ERAL and INDUCTIVE. thereby subsuming it within a very general account of the contexts where this connective is applicable.</Paragraph>
    </Section>
    <Section position="2" start_page="55" end_page="55" type="sub_section">
      <SectionTitle>
5.2 Violations of adjacency
</SectionTitle>
      <Paragraph position="0"> Finally, we can consider whether the account of comparisons as presupposing inductive rules prorides any way of explaining the violation of adjacency which the similarity relation signalled by also appears to permit. Our suggestion here is that since the rules presupposed by comparison relations are of a different sort from those presupposed by causal/inferential relations, it is possible that the theorem-proving systems which search for the inferences that can be drawn from the incoming facts in a discourse are different for the two kinds of rules. It is uncontroversial that there should be methods for constraining the search for inferences to be drawn.</Paragraph>
      <Paragraph position="1"> for both types of rule; in any large system of facts and rules there is an explosion of possible inferences to make. For causal/inferential relations, we could postulate that the search for inferences is constrained by the compositional structure of the discourse, and thus influenced by the nucleus-satellite structure of its relations; and that it is this which leads to the criterion of adjacency being enforced.</Paragraph>
      <Paragraph position="2"> For comparison relations, on the other hand. we could imagine different criteria for constraining the search space: for instance, we could suggest that the search does not take structural prominence into account, but is simply limited to the previous n propositions. There is no space here to explore this possibility in an3- detail', however, it seems an interesting one to consider.</Paragraph>
    </Section>
  </Section>
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