Towards a Cognitively Plausible Model for Quantification 
Walid S. Saba 
AT&T Bell Laboratories 
480 Red Hill Rd., Middletown, NJ 07748 USA 
and 
Carelton University, School of Computer Science 
Ottawa, Ontario, KIS-5B6 CANADA 
walid@eagle.hr.att.com 
Abstract 
The purpose of this paper is to suggest that 
quantifiers in natural languages do not have a 
fixed truth functional meaning as has long 
been held in logical semantics. Instead we 
suggest that quantifiers can best be modeled 
as complex inference procedures that are 
highly dynamic and sensitive to the linguistic 
context, as well as time and memory 
constraints 1. 
1 Introduction 
Virtually all computational models of quantification are 
based one some variation of the theory of generalized 
quantifiers (Barwise and Perry, 1981), and Montague's 
(1974) (henceforth, PTQ). 
Using the tools of intensional logic and possible- 
worlds semantics, PTQ models were able to cope with 
certain context-sensitive aspects of natural language by 
devising interpretation relative to a context, where the 
context was taken to be an "index" denoting a possible- 
world and a point in time. In this framework, the 
intension (meaning) of an expression is taken to be a 
function from contexts to extensions (denotations). 
In what later became known as "indexical semantics", 
Kaplan (1979) suggested adding other coordinates 
defining a speaker, a listener, a location, etc. As such, an 
utterance such as "I called you yesterday" expressed a 
different content whenever the speaker, the listener, or 
the time of the utterance changed. 
While model-theoretic semantics were able to cope 
with certain context-sensitive aspects of natural 
language, the intensions (meanings) of quantJfiers, 
however, as well as other functional words, such as 
sentential connectives, are taken to be constant. That is, 
such words have the same meaning regardless of the 
context (Forbes, 1989). In such a framework, all natural 
language quantifiers have their meaning grounded in 
terms of two logical operators: V (for all), and q (there 
exists). Consequently, all natural language quantifiers 
! The support and guidance of Dr. Jean-Pierre Corriveau of 
Carleton University is greatly appreciated. 
are, indirectly, modeled by two logical connectives: 
negation and either conjunction or disjunction. In such 
an oversimplified model, quantifier ambiguity has often 
been translated to scoping ambiguity, and elaborate 
models were developed to remedy the problem, by 
semanticists (Cooper, 1983; Le Pore et al, 1983; Partee, 
1984) as well as computational linguists (Harper, 1992; 
Alshawi, 1990; Pereira, 1990; Moran, 1988). The 
problem can be illustrated by the following examples: 
(la) Every student in CS404 received a grade. 
(lb) Every student in CS404 received a course outline. 
The syntactic structures of (la) and (lb) are identical, 
and thus according to Montague's PTQ would have the 
same translation. Hence, the translation of (lb) would 
incorrectly state that students in CS404 received 
different course outlines. Instead, the desired reading is 
one in which "a" has a wider scope than "every" stating 
that there is a single course outline for the course 
CS404, an outline that all students received. Clearly, 
such resolution depends on general knowledge of the 
domain: typically students in the same class receive the 
same course outline, but different grades. Due to the 
compositionality requirement, PTQ models can not cope 
with such inferences. Consequently a number of 
syntactically motivated rules that suggest an ad hoc 
semantic ordering between functional words are 
typically suggested. See, for example, (Moran, 1988) 2 . 
What we suggest, instead, is that quantifiers in natural 
language be treated as ambiguous words whose 
meaning is dependent on the linguistic context, as well 
as time and memory constraints. 
2 Disambiguation of Quantifiers 
Disambiguation of quantifiers, in our opinion, falls under 
the general problem of "lexical disambiguation', which 
is essentially an inferencing problem (Corriveau, 1995). 
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 
"background situation ~. However, in beth approaches the 
available context is still "syntactic ~ in nature, and no 
suggestion is made on how relevant background knowledge 
can be made available for use in a model-theoretic model. 
323 
Briefly, the disambiguation of "a" in (la) and (lb) is 
determined in an interactive manner by considering all 
possible knferences between the underlying concepts. 
What we suggest is that the inferencing involved in the 
disambiguation of "a" 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. 
2. Having established this relationship between students 
and grades, we assume the fact this relationship is 
many-to-many is known. 
3. "a grade" now refers to "a student grade", and thus 
there is "a grade" for "every student". 
What is important to note here is that, by discovering 
that grade is a feature of student, we essentially 
determined that "grade" is a (skolem) function of 
"student", which is the effect of having "a" fall under the 
scope of "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 "a" in (lb), proceeds as follows: 
1. A path from course and outline disambiguates outline, 
and determines outline to be a feature of course. 
2. The relationship between course and outline is 
determined to be a one-to-one relationship. 
3. A path from course to CS404 determines that CS404 is 
a course. 
4. Since there is one course, namely CS404, "a course 
outline" refers to "the" course outline. 
3 Time and Memory Constraints 
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. 
(21)) Students in CS404 work in groups. 
Our intuitive reading of (2a) suggests that we have an 
implicit "most", while in (2b) we have an implicit "all". 
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 "Cubans", 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 
"generalization" 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 
below. 
(a) (b) (c) 
F'~gure I. Three models of negation. 
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 "generalize", 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" 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. 
4 Role of Cognitive Constraints 
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 
Figure 2. It is assumed that the property P is generally 
attributed to members of the concept C with certainty 
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: 
3 This is the Closed World Assumption. 
4 ...... Thin Is similar to the dynamm reasoning process suggested by 
Wang (1995). 
324 
1. An attempt at an exhaustive verification of all the 
elements in the set C is first made (this is the default 
meaning of "every"). 
2. If time and memory capacity allow the processing of all 
the elements in C, then the result is "true" if np= ICI 
(that is, if every C P), and "false" 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). 
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. 
5. The result is used to update our certainty factor, ef, 
based on the current evidence ~. 
"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 > &nn) ^ (cf(C,P) >= co) 
where e and co are quantifier-specific parameters. In the 
case of "every", 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" 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 > co > 0). This 
essentially amounts to processing every C P as most C P 
(example (2a)). 
ff "most" 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. 
5 The nature of this feedback mechanism is quite involved, and 
will not be discussed be discussed here. 
quantifier np 
np- ICI 
nn 
np- 0 
every nn - 0 
some np> 0 nn < ICl 
no nn- ICI 
~>0 
s>O 
s<O 
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 "cognitive 
plausibility' requirement, we require that the model 
preserve formal properties that are generally attributed 
to quantifiers in natural language. 

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