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<Paper uid="P95-1051">
  <Title>Towards a Cognitively Plausible Model for Quantification</Title>
  <Section position="3" start_page="0" end_page="323" type="metho">
    <SectionTitle>
2 Disambiguation of Quantifiers
</SectionTitle>
    <Paragraph position="0"> Disambiguation of quantifiers, in our opinion, falls under the general problem of &amp;quot;lexical disambiguation', which is essentially an inferencing problem (Corriveau, 1995).</Paragraph>
    <Paragraph position="1"> 2 In recent years a number of suggestions have been made, such as discourse representation theory (DRT) (Kamp, 1981), and the use of what Cooper (1995) calls the &amp;quot;background situation ~. However, in beth approaches the available context is still &amp;quot;syntactic ~ in nature, and no suggestion is made on how relevant background knowledge can be made available for use in a model-theoretic model.</Paragraph>
    <Paragraph position="2">  Briefly, the disambiguation of &amp;quot;a&amp;quot; in (la) and (lb) is determined in an interactive manner by considering all possible knferences between the underlying concepts.</Paragraph>
    <Paragraph position="3"> What we suggest is that the inferencing involved in the disambiguation of &amp;quot;a&amp;quot; in (la) proceeds as follows: l. A path from grade and student, s, in addition to disambiguating grade, determines that grade, g, is a feature of student.</Paragraph>
    <Paragraph position="4">  2. Having established this relationship between students and grades, we assume the fact this relationship is many-to-many is known.</Paragraph>
    <Paragraph position="5"> 3. &amp;quot;a grade&amp;quot; now refers to &amp;quot;a student grade&amp;quot;, and thus  there is &amp;quot;a grade&amp;quot; for &amp;quot;every student&amp;quot;. What is important to note here is that, by discovering that grade is a feature of student, we essentially determined that &amp;quot;grade&amp;quot; is a (skolem) function of &amp;quot;student&amp;quot;, which is the effect of having &amp;quot;a&amp;quot; fall under the scope of &amp;quot;every'. However, in contrast to syntactic approaches that rely on devising ad hoc rules, such a relation is discovered here by performing inferences using the properties that hold between the underlying concepts, resulting in a truly context-sensitive account of scope ambiguities. The inferencing involved in the disambiguation of &amp;quot;a&amp;quot; in (lb), proceeds as follows:  1. A path from course and outline disambiguates outline, and determines outline to be a feature of course.</Paragraph>
    <Paragraph position="6"> 2. The relationship between course and outline is determined to be a one-to-one relationship.</Paragraph>
    <Paragraph position="7"> 3. A path from course to CS404 determines that CS404 is a course.</Paragraph>
    <Paragraph position="8"> 4. Since there is one course, namely CS404, &amp;quot;a course outline&amp;quot; refers to &amp;quot;the&amp;quot; course outline.</Paragraph>
  </Section>
  <Section position="4" start_page="323" end_page="323" type="metho">
    <SectionTitle>
3 Time and Memory Constraints
</SectionTitle>
    <Paragraph position="0"> In addition to the lingusitic context, we claim that the meaning of quantifiers is also dependent on time and memory constraints. For example, consider  (2a) Cubans prefer rum over vodka.</Paragraph>
    <Paragraph position="1"> (21)) Students in CS404 work in groups.</Paragraph>
    <Paragraph position="2">  Our intuitive reading of (2a) suggests that we have an implicit &amp;quot;most&amp;quot;, while in (2b) we have an implicit &amp;quot;all&amp;quot;. We argue that such inferences are dependent on time constraints and constraints on working memory. For example, since the set of students in CS404 is a much smaller set than the set of &amp;quot;Cubans&amp;quot;, it is conceivable that we are able to perform an exhaustive search over the set of all students in CS404 to verify the proposition in (2b) within some time and memory constraints. In (2a), however, we are most likely performing a &amp;quot;generalization&amp;quot; based on few examples that are currently activated in short-term memory (STlVi). Our suggestion of the role of time and memory constraints is based on our view of properties and their negation We suggest that there are three ways to conceive of properties and their negation, as shown in Figure 1  F'~gure I. Three models of negation.</Paragraph>
    <Paragraph position="3"> In (a), we take the view that if we have no information regarding P(x), then, we cannot decide on -~P(x). In (b), we take the view that if P can not be confirmed of some entity x, then P(x) is assumed to be false 3. In (c), however, we take the view that if there is no evidence to negate P(x), then assume P(x). Note that model (c) essentially allows one to &amp;quot;generalize&amp;quot;, given no evidence to the contrary - or, given an overwhelming positive evidence. Of course, formally speaking, we are interested in defining the exact circumstances under which models (a) through (c) might be appropriate. We believe that the three models are used, depending on the context, time, and memory constraints. In model (c), we believe the truth (or falsity) of a certain property P(x) is a function of the following: np(P#) number of positive instances satisfying P(x) nn(P#) number of negative instances satisfying P(x) cf(P#) the degree to which P is ~gencrally&amp;quot; believed of x. It is assumed here that cfis a value v ~ {J.} u \[0,1\]. That is, a value that is either undefined, or a real value between 0 and 1. We also suggest that this value is constantly modified (re-enforced) through a feedback mechanism, as more examples are experienced 4.</Paragraph>
  </Section>
  <Section position="5" start_page="323" end_page="324" type="metho">
    <SectionTitle>
4 Role of Cognitive Constraints
</SectionTitle>
    <Paragraph position="0"> The basic problem is one of interpreting statements of the form every C P (the set-theoretic counterpart of the wff Vx(C(x)---)P(x)), where C has an indeterminate cardinality. Verifying every C P is depicted graphically in  cf(C,P), where cf(C,P)--O represents the fact that P is not generally assumed of objects in C. On the other hand, a value of cf near 1, represents a strong bias towards believing P of C at face value. In the former case, the processing will depend little, if at all, on our general belief, but more on the actual instances. In the latter case, and especially when faced with time and memory constraints, more weight might be given to prior stereotyped knowledge that we might have accumulated. More precisely:  1. An attempt at an exhaustive verification of all the elements in the set C is first made (this is the default meaning of &amp;quot;every&amp;quot;).</Paragraph>
    <Paragraph position="1"> 2. If time and memory capacity allow the processing of all the elements in C, then the result is &amp;quot;true&amp;quot; if np= ICI (that is, if every C P), and &amp;quot;false&amp;quot; otherwise. 3. If time and/or memory constraints do not allow an exhaustive verification, then we will attempt making a decision based on the evidence at hand, where the evidence is based on of, nn, np (a suggested function is given below).</Paragraph>
    <Paragraph position="2"> 4. In 3, ef is computed from C elements that are currently active in short-term memory (if any), otherwise cf is the current value associated with C the KB.</Paragraph>
    <Paragraph position="3"> 5. The result is used to update our certainty factor, ef, based on the current evidence ~.</Paragraph>
    <Paragraph position="4"> &amp;quot;c m np nn F'~ure 2. Quantification with time and memory constraints.  In the case of 3, the final output is determined as a function F, that could be defined as follows: (13) Frca,)(nn, np, e, cf, o9 =(np &gt; &amp;nn) ^ (cf(C,P) &gt;= co) where e and co are quantifier-specific parameters. In the case of &amp;quot;every&amp;quot;, the function in (13) states that, in the absence of time and memory resources to process every C P exhaustively, the result of the process is ~-ue&amp;quot; if there is an overwhelming positive evidence (high value for e), and if the there is some prior stereotyped belief supporting this inference (i.e., if cf &gt; co &gt; 0). This essentially amounts to processing every C P as most C P (example (2a)).</Paragraph>
    <Paragraph position="5"> ff &amp;quot;most&amp;quot; was the quantifier we started with, then the function in (13) and the above procedure can be applied, although smaller values for G and co will be assigned. At this point it should be noted that the above function is a generalization of the theory of generalized quantifiers, where quantifiers can be interpreted using this function as shown in the table below.</Paragraph>
    <Paragraph position="6"> 5 The nature of this feedback mechanism is quite involved, and will not be discussed be discussed here.</Paragraph>
    <Paragraph position="7">  We are currently in the process of formalizing our model, and hope to define a context-sensitive model for quantification that is also dependent on time and memory constraints. In addition to the &amp;quot;cognitive plausibility' requirement, we require that the model preserve formal properties that are generally attributed to quantifiers in natural language.</Paragraph>
  </Section>
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