NON-SINGULAR CONCEPTS IN NATURAL LANGUAGE DISCOURSE 
Tomek Strzalkowski 
Department of Computer Science 
Courant Institute of Mathematical Sciences 
New York University 
251 Mercer St. 
New York, NY 10012 U.S.A. 
Nick Cercone 
Centre for System Science 
School of Computing Science 
Simon Fraser University 
Burnaby, British Columbia VSA IS6 Canada 
We introduce a new approach to representing and manipulating various types of non-singular concepts 
in natural language discourse. The representation we describe is based on a partially ordered structure 
of levels in which the objects of the same relative singularity are assigned to the same level. Our choice 
of the representation has been motivated by the following main concerns: I. The representation should 
systematically distinguish between those language terms that are used to refer to objects of different 
singularity, that is, those classified within different but related levels of the model; 2. The representation 
should capture certain types of inter-sentential dependencies in discourse, most notably anaphoric-type 
cohesive links; 3. Finally, the representation should serve as a basis for defining a formal semantics of 
discourse paragraphs that would allow for capturing the exact truth conditions of sentences involving 
non-singular terms, and for computing interlevel inferences. In this paper we discuss (I) and (2) only. 
(3) is currently under investigation and will be the topic of a forthcoming article. We believe that our 
approach promotes computational feasibility, because we avoid the identification of general terms, like 
"temperature," "water," etc., with intensions, that is, functions over possible worlds. In our theory, 
the concept of non-singularity has a local (often subjective) character. 
1 INTRODUCTION 
Treating non-singular concepts is a difficult representa- 
tion problem in natural language research. Non-singular 
concepts, as we shall understand them here, are often 
abstract entities created to embrace a variety of smaller 
or larger collections of "instances" or "specimen." 
They are usually referred to using bare plural noun 
phrases (such as birds, alligators, presidents), or defi- 
nite singular noun phrases with "generic" interpreta- 
tion (the alligator, the president). The literature de- 
scribes numerous forms of non-singular concepts and 
corresponding to them non-singular terms that can be 
found in natural language discourse, including inten- 
sional (or functional), mass, generic, habitual, abstract, 
and more; see, for example, Montague (1974a), Lewis 
(1976), Barwise and Perry (1983), Quine (1960), Donnel- 
lan (1971), Vendler (1971), and Kripke (1972). In these 
and other writings, various treatments for the phenom- 
enon are suggested, but many of them do not properly 
capture the distinction between singular and non-sin- 
gular interpretation of linguistic descriptions. With the 
exception of intensional concepts, other forms of non- 
singularity have not been given satisfactory formal 
representations that would account for their role in 
natural language discourse. Perhaps the most successful 
treatment of non-singular terms in language thus far has 
been presented by Montague (1974a-d) with his formal- 
ization of intension, which can be traced back to the 
Fregean notion of sense. Unfortunately, the concept of 
intension does not capture all aspects of non-singu- 
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Computational Linguistics, Volume 15, Number 3, September 1989 171 
Tomek Strzaikowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse 
larity, for example, bare plurals denoting kinds cannot 
be adequately represented in Montague's intensional 
logic (IL) without some superficial, and often unplausi- 
ble, extensions (Carlson 1977, 1982). In addition, the 
enormous complexity of any non-trivial system of pos- 
sible worlds has proven to be disadvantageous for 
developing a computationally oriented application of 
Montague's theory (Strzalkowski and Cercone 1986). 1 
In this paper we do not attempt to improve on Mon- 
tague's grammar; in fact, our approach does, in part, 
rely on Montague's method of coupling the syntactic 
and semantic processing in order to assign to a sentence 
its meaning or at least a meaning-representing formula. 
Nonetheless, we also perceive the limitations of Mon- 
tague's method. It is not our intention here to build an 
alternative formal system for a fragment of English that 
would replace that of Montague's. Instead, we propose 
a computationally oriented approach to certain prob- 
lems with which Montague dealt only marginally or not 
at all. However informal our presentation may be, we 
make some effort to put things into a clear, semiformal 
setting. As various aspects of out theory crystalize, we 
expect a more formal version to emerge. 
We introduce a fragment of a new and, we believe, 
computationally feasible theory of names and descrip- 
tions, which offers a uniform treatment for many types 
of non-singular concepts found in natural language 
discourse. Although we limit our presentation to nomi- 
nal phrase constructions, the approach can be further 
extended to cover other types of phrases. We present 
the formal definition of non-singularity with respect to a 
particular discourse situation involving a discourse mes- 
sage; a number of individuals (parties); and their knowl- 
edge, beliefs, awareness, etc. We introduce a layered 
model of reality (the universe) as perceived by a dis- 
course participant, and define relative singularity of 
objects in this universe as an abstraction class of the 
layer-membership relation. Subsequently, linguistic de- 
scriptions and names are classified as singular, measur- 
ably singular, or non-singular depending upon what they 
are assumed to denote in the universe. The relationship 
between objects referred to in discourse and classified 
into different layers (levels) of the universe model has a 
particular significance for resolution of certain types of 
cohesive links in text. We call these links remote 
references because they cross level boundaries. In order 
to adequately explain the phenomenon of a remote 
reference, we propose a fairly general theory of names 
and descriptions in discourse. Although the theory lacks 
a complete formal specification at this time, we intro- 
duce a number of notions, such as a superohject and a 
coordinate, that, we believe, will prove helpful in further 
formalization of various intuitions presented here. 
In its present form, our theory focuses entirely on the 
problems of non-singularity and remote references in 
discourse that are created by the use of definite descrip- 
tions, bare plurals, and proper names. We do not 
describe in detail how one could actually compute 
remote references in discourse, except for two general 
rules introduced in Section 6. These rules, and perhaps 
some others yet to be developed, create a natural 
expansion of tlhe system for computing intersentential 
anaphoric references described in Strzalkowski and 
Cercone (1986). Except for a brief overview in Section 
2, we do not discuss many related phenomena, such as 
non-remote (or single level) references, opaque read- 
ings, non-referential interpretations, use of personal 
pronouns, and the like. The more general theory of 
stratified meaning representation (Strzalkowski 1986) 
addresses all of these concerns, as well as the problems 
of discourse coherence, selecting proper cohesive links 
in discourse, and building a discourse model. 
2 COMPUTING INTERSENTENTIAL DEPENDENCIES IN 
DISCOURSE 
A first step toward automating the process of discourse 
understanding is to grasp the meaning contents of the 
discourse message, at least the literal meaning. A dis- 
course normally consists of more than a single utter- 
ance, and although every utterance may be assumed to 
contribul:e something to the discourse meaning as a 
whole, this latter can only rarely be regarded as a simple 
sum of meanings of component utterances. Utterances, 
or sentences, making up a discourse are usually in- 
volved in complicated mutual dependencies, that often 
go beyond the text itself. A careful study of these 
extrasentential and intersentential dependencies in dis- 
course is necessary before a more successful attempt to 
design an automated discourse understanding system 
can be undertaken. 
In Strzalkowski (1986) and Strzalkowski and Cer- 
cone (1986) we introduced a rigorous method for han- 
dling certain cases of extrasentential dependencies in 
discourse, within a general framework, which we call 
the Stratified Model. The Stratified Model comprises a 
collection of disambiguating transformations that are 
applied to a discourse fragment before it can be assigned 
a final representation. These transformations include 
morphological analysis, lexical disambiguation, syntac- 
tic parsing, computing extrasentential dependencies, 
and pragmatic evaluation in discourse context. In order 
to compute anaphoric, and other, dependencies be- 
tween sentences in a discourse fragment, we first trans- 
late each sentence in the fragment into an intermediate, 
formal representation language. We call these transla- 
tions literal or "context-less" because they are derived 
based solely on sentences' syntactic structure. In fact, 
this transformation is done Montague-style (Montague 
1974d) except for the intermediate representation, 
which, in the Stratified Model, is a A-categorial language 
A, rather than an intensional logic. For this presentation 
it is enough to say that A is a typed predicate-calculus 
language: with A-operator. A formal definition of A is 
given in Strzalkowski and Cercone (1986). Next, we 
172 Computational Linguistics, Volume 15, Number 3, September 1989 
Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse 
proceed to compute various contextual dependencies of 
sentences, including extrasentential anaphoric links. 
Let L be a language of parse structures obtained in 
some grammar of a fragment of English. For the pur- 
pose of this presentation, L is identified with the set of 
phrase markers that can be generated from English 
sentences with a categorial grammar CAT similar to that 
of Montague (1974). Although some other syntactic 
system may be more suitable in practical application, 
we select CAT here for its relative simplicity and 
elegance. We concentrate on the translation of some 
example expressions, sentences, and paragraphs of L 
into a representation in a A-categorial language A that 
would capture both a sentence's logical form and its 
cohesive links to the surrounding discourse. In partic- 
ular, we shall look closely at the cohesive links created 
by inter-sentential anaphoric references appearing in 
different contextual situations. 
A possesses adequate expressive power to represent 
the meaning of a considerable spectrum of linguistic 
constructs found in a natural language discourse. What 
is of a particular interest to us, A provides a natural and 
uniform means for computing and representing extra- 
sentential dependencies. As we shall see in the next 
section, a meaning representation language so-defined 
is still inadequate for capturing some more difficult 
cases, which we call remote co-references. 
Our present effort is to describe a transformation ISD 
such that ISD _C L × A, and whenever a source 
expression in L consists of more than one sentence, a 
class of intersentential dependencies within this frag- 
ment is identified and resolved, if possible. It must be 
noted here that ISD represents a semantic process that 
is entirely independent, of any pragmatic or domain 
related factors. As a result a substantial amount of 
domain oriented ambiguity may be left unresolved. In 
any practical application, this transformation must be 
accompanied by a pragmatic process, as described in 
Strzalkowski (1986). ISD consists of a collection of 
translation rules {R1, R2 .... }, such that each rule is 
responsible for translating a specific type of depen- 
dency. Actually, only Rule 1 works directly on expres- 
sions of L, translating them into literal representations 
in A, independent of one another. Rules numbered 2 and 
up will take these literal translations and try to relate 
them pairwise looking, among other things, for unre- 
solved anaphoric references. Most of these rules can be 
written in terms of two distinguished expressions of A, 
S 1 and S 2, which we call the context-setting sentence and 
the current sentence, respectively. Expression S 1 is a 
A-representation of the linguistic context in which the 
sentence with translation S 2 is to be evaluated. Neither 
$1 nor S 2 must correspond to surface sentences, though. 
S~ may represent a larger part of discourse, perhaps an 
entire paragraph; on the other hand, S 2 may constitute 
only a subclause of S l in which case we would talk of 
intra-sentential dependency. It should be noted here 
that the potentially explosive number of possibilities 
will be in fact limited by the actual structure of the 
discourse under consideration (see, among others, 
Grosz and Sidner 1985), as well as by the pragmatic and 
domain related information, not discussed here. 
Let us now consider a two sentence paragraph given 
below: 
$1: John interviewed a candidate. 
$2: The guy had impressive references. 
In the most natural reading of this paragraph, the 
anaphor of "the guy" is resolved against "a candidate" 
in the first sentence, so that the second sentence 
actually means: "the guy whom John interviewed had 
impressive references." When considered separately 
from one another, $1 and S 2 obtain the following trans- 
lations into A. 
S~ ~ 3x\[cand(x) & int(J,x)\] 
S 2 ~ 3x\[guy(x) & C(x) & Vy\[(guy(y) & C(y)) D 
(x=y)\] & had-imp-ref(x)\] 
In the translation of S 2, C is a free predicate variable 
that needs to be bound by the sentence's context. The 
context variable is introduced into the translation of a 
definite noun phrase containing a definite article, such 
as "the," by Rule 1 (Strzalkowski 1986, Strzalkowski 
and Cercone 1986). This rule generates literal transla- 
tions of sentences without considering their context, 
and does so in PTQ-style, that is, by assigning to each 
syntactic operation in CAT a formula formatting oper- 
ation in A (Montague 1974a). The next step is to resolve 
context references; we must find a binding for C occur- 
ring in S 2 in the context provided by $1 in order to 
obtain the final translation of the former. This intersen- 
tential dependency is captured by the translation Rule 2, 
which operates on the literal translations of both sen- 
tences delivered by Rule 1 (Strzalkowski and Cercone 
1986). In the example above, the second sentence 
obtains the desired translation as shown below. 
3x \[guy(x) & cand(x) & int(J, x) & had-imp-ref(x) 
& Vy \[{guy(y) & cand(y) & int(J,y)} D (x=y)\]\] 
Rule 2 (Perfect-Context Translation Rule): 
If the context-setting sentence S 1 has a referential 
interpretation in the form 
3u\[P(u) & F(u)\], 
and the current sentence S 2 contains an unresolved 
definite anaphor, that is, 
S 2 : :qu If(u) ¢~ Pl(u) & Fl(u ) & V x \[{Pt(x) & 
C(x)} ~ (x=u)\]\], 
then this anaphor can be resolved against S~, and the 
resulting translation of S 2 is obtained as 
AC\[S2\](Au\[P(u) & F(u)\]). 
A somewhat different problem arises when we consider 
a fragment with a possible non-referential interpreta- 
tion, as in 
I'/3 
m 
Computational Linguistics, Volume 15, Number 3, September 1989 
Tomek Strzalkowski and Nick Cercone Nou-Singular Concepts in Natural Language Discourse 
John wants to marry a princess. The girl must be rich 
and pretty. 
Now, Rule 2 can compute the anaphoric link between 
"the girl" and "a princess" only if both sentences 
receive their referential interpretations. In a case where 
both sentences are understood non-referentially, we 
have to use Rule 3, given below. No other combinations 
are possible. 
Rule 3 (Imperfect-Context Translation Rule): 
If the context-setting sentence S I has a non-referen- 
tial interpretation in the form 
imp (3u \[P(u) & F(u)\]), 
where imp is an imperfect operator, and the current 
sentence S 2, also in a non-referential interpretation, 
contains a definite anaphor which occurs in scope of 
an imperfect operator imp1 i.e., 
$2 = imp1 (3u \[C(u) & Pl(u) & Fl(u) & 
Vx \[{PI (x) & C(x)} 3 (x=u)\]\], 
then this anaphor can be resolved against St, with 
the resulting translation of S e derived as 
AC\[S2\](Au\[P(u ) & F(u)\]). 
Rule 3 encompasses a large class of non-referential 
contexts, which we call imperfect contexts, and which 
involve constructs including propositional attitudes 
(want, try, wish), intensional verbs (seek, conceive, 
think about), other complement-taking verbs (go, 
come), modal verbs (must, can, will), as well as pro- 
gressive tense forms. In Rule 3, all this is reduced to the 
formula with the imp operator which translates com- 
pound phrases, such as "John wants," or "John will." 
Thus, in the example given above, Rule 3 is applicable 
when both sentences contain a wide scope imp opera- 
tor. In this case, the second sentence of the fragment 
obtains the full translation with the following formula: 
must(3x \[gir/(x) & princess(x) & marries(J,x) & 
rich(x) & pretty(x) & Vy \[{girl(y) & princess(y) & 
marries(J,y)} D (x=y)\]\]) 
Other studied cases of intersentential anaphora (see 
Strzalkowski 1986a-c, Strzalkowski and Cercone 1986) 
include non-referential interpretation of discourse frag- 
ments involving attitude report verbs (believe, know, 
disagree). These cannot be translated with Rule 3, and a 
new rule, Rule 4, is developed to compute anaphoric 
links in texts similar to the one given below. 
John believes that a unicorn lives in the park. 
He thinks the creature has a long horn. 
Rules 5, 6, and 7 account for the pronominal anaphora, 
Rule 10 deals with certain instances of attributive use of 
definite noun phrases. Rules 8 and 9 are used when the 
antecedent of an anaphor is a proper name rather than a 
description. This is the situation where an interesting 
type of referential ambiguity occurs whose resolution 
may have far reaching consequences on the process of 
discourse understanding. 
Rule 9 (Names as Ultimate Referents): 
If the context-setting sentence S 1 has the form of 
FI(N) where N is an individual constant denoting a 
name, and the current sentence S 2 contains a definite 
anaphor, so that its literal translation has the form 
S 2 = 3x \[P(x) & C(x) & F 2 (x) & Vy \[{P(y) & 
C(y)) ~ (x=y)\]\], 
then the anaphor can be resolved against N as its 
ultimate referent with the following derivation: 
xplp( N) \]( Xx\[ :tC\[ S2\]( Xs\[ R(s) \]) \]) 
where \]V is the predicative use of name N. 
In the following fragment, 
Sylvester tries to catch a bird. The cat is clumsy. 
there are two possible ways of linking "the cat" with 
"Sylvester." In one reading, not very different from 
those processed with Rule 2, the definite anaphor refers 
primarily to the entity that can be described as "the one 
who tries to catch a bird," and only contingently to its 
name. In this case we acquire some new information 
about Sylvester, namely that it is a cat. In the other 
possible reading, the anaphor refers to the name only, 
and thu,; may draw on some context that is different 
from the first sentence in the fragment. This latter 
situation is handled by Rule 9. In the above fragment, 
Rule 9 would produce the following translation for "the 
cat is clumsy" (S is an individual constant denoting the 
individual named Sylvester, and Syl(x) means that x's 
name is Sylvester): 
cat(S) & Syl(S) & clumsy(S) & Vx \[{cat(x) & 
Syl(x)} 3 (x=S)\] 
There are more aspects of ISD transformation that merit 
attention. These include rules for dealing with other 
kinds of anaphora not discussed here, elliptical con- 
structions, enumerably singular (plural) terms, intrasen- 
tential anaphora, and non-anaphoric dependencies, as 
well as indirect and forward reference cases where 
access to the speaker/hearer knowledge base may be 
required. We also have to deal with the changing 
reference level. 
3 NON-SINGULAR TERMS IN DISCOURSE 
The rules discussed in Section 2 cover selected cases of 
intersentential anaphora where the reference level in 
discourse does not change from one sentence to an- 
other. There exists, however, a class of intersentential 
dependencies whereby a reference is made across 
boundm'ies of different reference levels in discourse. 
For example, in 
My new pet is an alligator. But the alligator cannot 
live in our climate. 
174 Computational Linguistics, Volume 15, Number 3, September 1989 
Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse 
"the alligator" in the second sentence most likely refers 
to a generic object of which the alligator in the first 
sentence is an instance or extension. Thus we can say 
that the second alligator is a non-singular superobject in 
which the first alligator somehow participates. The 
extent of such participation is not clear, but in general it 
can be observed that certain predications true of com- 
plexes of different kind are not preserved for their parts 
or elements, and vice versa. To represent this new kind 
of intersentential dependency we introduce a multilevel 
model for interpreting natural language expressions, 
such that the levels in the model would correspond 
(roughly) to the levels of reference in discourse. For 
instance, in the example above, the resulting represen- 
tation would have both alligators placed at different, 
though related, "object levels." Because of an inherent 
subjectivity of such classifications, the levels in the 
model may have fuzzy boundaries and are only partially 
ordered with the "lower than" (i.e., "more detailed 
than") relation with respect to some current level 
(corresponding to the level of reference at a present 
point in discourse). 
Consider now a somewhat larger fragment of text, 
excerpted from an article appearing in The New Yorker 
magazine. 
The two Ashanti kings are in somewhat different 
situations: the Ghanaian king is royally born, richly 
rewarded, divinely inspired and holds his office for 
life. The American Ashanti king is elected every two 
years from the ranks of an Ashanti social and cultural 
organization called the Asanteman Association of the 
United States of America, Inc. The first Stateside 
king, Kwadwo Tuffuor, was a plumber. The second, 
Kusi Appouh, repaired air-conditioners and refriger- 
ators. Kwabena Oppong is the third king; he drives a 
cab. 2 
The first two sentences in this fragment contain direct 
references to the higher-level entities, which are the two 
Ashanti kings, as if they were ordinary singular objects. 
The remaining three sentences directly refer to the 
entities that are instances of one of these kings, now 
seen as superobjects, at different time intervals. The 
discourse reference level has changed, and now we talk 
about lower-level entities. The superobjects can still be 
referred to, but only indirectly, through their instances; 
this is what we call the remote reference. 
The primitive notions of our theory are these of a 
singular object and a coordinate, a usually ordered set 
specifying a type of dimension that the object in ques- 
tion spans. A singular object is any entity to which we 
can directly refer using a nominal phrase of our lan- 
guage. The most common of the coordinates are time 
and space but other more abstract ones are also possi- 
ble. These two basic notions are then used to define the 
notion of the object's instance with respect to some 
coordinate. Thus the pet alligator in the example above 
is related to the generic concept of alligator by some 
species coordinate that somehow ties (or enumerates?) 
all alligators around the world. Similarly, if we use an 
appropriate coordinate consisting of two-year intervals 
we can decompose the American Ashanti king into its 
elected instances. If we reverse this process we can 
combine objects into complexes to which we can sub- 
sequently refer using collective terms, singular or plu- 
ral, such as, for example, "people" or "the man" 
(generic). The lower than relation between levels in the 
universe model derives from expanding the notion of 
instance over collections of objects. The relation intro- 
duces a partial ordering within the universe model and 
thus helps to trace changes in the reference level of 
discourse. The highly discrete approach taken here is 
favorably contrasted with other existing approaches to 
non-singular terms, including Quine (1960), Kripke 
(1972), Montague (1974), Carlson (1982), and others. 
While insights of Quine, Kripke and, perhaps even more 
so, Carlson are undoubtedly of great influence, they 
require reworking in more discrete terms. Finally, we 
may note that the research in artificial intelligence and 
computational linguistics has devoted relatively little 
attention to treatment of non-singular terms in natural 
language in general and in natural language discourse in 
particular; see, however, Sidner (1979) for some early 
attempts to recognize generics in discourse. One of the 
goals of the present research is to fill this gap. 
4 NON-SINGULAR TERMS IN LANGUAGE 
Many philosophers and logicians, Quine (1960), Kripke 
(1972), Donnellan (1971), Vendler (1971), Montague 
(1974), and Barwise and Perry (1983), note that the 
usage of the italicized nominal phrases in Example 1 has 
a general or generic character, except for regular singu- 
lar interpretations, which are only possible in some 
cases. 
Example 1. 
la. The king wears a crown. 
lb. The president is elected every four years. 
lc. Gold is a yellow metal. 
Id. Temperature is a measure of molecular motion. 
Hundreds of similar examples involving such non-sin- 
gular terms, and corresponding non-singular objects, as 
water, heat, the Pope, the number, etc. can be devised. 
Unfortunately, there is no generally accepted account 
of these non-singular terms in the philosophical litera- 
ture. Some authors, for example, Vendler (1971) and 
Barwise and Perry (1983), cautiously called them ge- 
neric, or general (for example, "the king"), or func- 
tional (such as "the number of students," "the temper- 
ature") uses of definite descriptions. Other authors, for 
example, Kripke (1972), were quite close to considering 
these kinds of non-singular terms as names (or at least 
some of them: heat, gold). Still other authors writing on 
the subject, for example, Quine (1960, 1973), advocated 
the notion of abstract terms as comprising of attributes, 
such as (being) red (further abstracted as "redness"), or 
Computational Linguistics, Volume 15, Number 3, September 1989 175 
Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse 
(being) the man drinking the martini (which cannot be 
so easily nominalized), which can be predicated of 
"concrete" objects. If we consider this discussion, the 
so-called "attributive use" of singular definite descrip- 
tions as identified by Donnellan (1971), may be consid- 
ered as addressing some abstract, higher-level, and 
therefore (in our interpretation) non-singular concepts. 
Carlson (1982) discusses the case of so-called "natural 
kinds," a specific type among generic terms. He advo- 
cates the view in which generic terms are taken as 
denoting entities in the same way that singular terms do. 
If we accept Carlson's position, then our ontology of 
objects becomes far richer than before, and we need to 
impose more structure on our representation to reflect 
this new situation accurately. Indeed, Carlson intro- 
duces a special R relation into Montague's IL which 
allows him to create individual objects out of generic 
objects, as well as stages out of ordinary singular 
objects. Thus, "Max believes that dogs are here" 
receives the following translation, where m and d are 
individual constants denoting Max and dog kind, re- 
spectively (Carlson 1977): 
Ax\[Bel(^3y\[R(y, x) & here'(y)\](m)\](d) 
This formula says that Max believes that some stages of 
dog-kind are here. One problem with this representation 
is that we have no idea how these stages are to be 
identified, in other words, how do we decompose a 
kind-level object into stage-level individuals. If other 
types of non-singular terms denote in similar fashion, 
then we may need an even more complex model in 
which various stages (or levels) of object aggregation 
can be reflected. 
Quine (1960) presents the most comprehensive ac- 
count of various categories of terms found among 
natural language expressions. Almost everything that 
one can say is made up of different kinds of terms, 
appropriately connected to yield meaningful utterances, 
which he classifies as singular, general, relative, ab- 
stract, attributive, etc. At present, we do not draw such 
fine distinctions in our classification, reserving the right 
to develop extensions along the lines of Quine as 
needed. We present a few examples to provide addi- 
tional insight and lay the foundation for our theory of 
names and descriptions. 
There are numerous linguistic puzzles involving non- 
singular definite descriptions, among them Partee's 
(1972) famous temperature problem. Example 2 illus- 
trates this phenomenon. 
Example 2 
2a. The temperature is rising. 
The temperature is ninety. 
Thus, Ninety is rising. 3 
2b. The president is elected every four years. 
The president is Reagan. 
Thus, Reagan is elected every four years. 
2c. The tiger lives in the jungle. 
My pet is a tiger. 
Thus, My pet lives in the jungle. 
2d. Americans drive big cars. 
John is an American. 
Thus, John drives big cars/a big car. 
The arguments in (2a-d) are normally considered in- 
valid. Various researchers agree that the definite de- 
scriptions "the temperature," "the president," and 
"the tiger" in the first sentences of (2a-c) should be 
interpreted functionally, that is, as intensions (Mon- 
tague 1974d), or functions over situations (Barwise and 
Perry 1983). Note that if the descriptions were to be 
interpreted singularly or as enumerating all instances of 
a non-singular object (that is, statements containing 
them were understood as making claims about each 
instance), the reasoning would be valid. In (2d) the 
situation is somewhat more complicated, because every 
extension of tile entity in the denotation of Americans is 
itself a compound entity, namely a generic entity. Thus, 
intensionality alone cannot explain why the first two 
sentences in (2d) appear connected. As a matter of fact, 
the reasoning displayed in (2d) is far more acceptable 
that those in (2a) to (2c). The reason, it seems, lies in 
our reluctance to apply singular interpretations to non- 
singular terms in first sentences of (2a-c), while this 
same move is more likely in (2d). In other words, while 
(2b') is an unlikely interpretation of (2b), (2d') can 
occasionally be accepted as a simplified version of (2d). 
2b'. Every president is elected every four years. 
\]Reagan is the president. 
Thus, Reagan is elected every four years. 
2d'. \]Every American drives big cars/a big car. 
John is an American. 
Thus, John drives big cars/a big car. 
Of course, (2a) and (2b), but not (2c) or (2d), could be 
valid if the definite noun phrases in the second sen- 
tences (the temperature, the president) were understood 
as co-referential with appropriate noun phrases in the 
first sentences. This avenue is also closed, however, 
because we do not regard these phrases as co-referen- 
tial, that is, they cannot be substituted for one another, 
nor cant be their denotations. Compare this with (2e) 
below, where the co-reference between "carnivorous 
animals" and "these beasts" is easily made. 
2e. Carnivorous animals live in these forests. 
These beasts are tigers. 
Thus, tigers live in these forests. 
We claim here that no two descriptions can be consid- 
ered co-referential unless they are used to refer to the 
objects that are at the comparable stages of aggregation. 
We call such object relatively singular and place them 
within the same level in our model. It is clear that the 
objects referred to by the corresponding descriptions in 
the firs~t two sentences of (2a) to (2d) are not relatively 
singular, and thus the conclusions in the third sentences 
are not forthcoming. Another type of co-reference, 
which we call a remote co-reference, can still occur, and 
we put this view forward in this paper. 
176 Computational Linguistics, Volume 15, Number 3, September 1989 
Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse 
5 A MULTILEVEL MODEL FOR INTERPRETING NATURAL 
LANGUAGE TERMS 
Initially, we note that our language tends to deal with 
singular objects only, no matter how complex their 
structure happens to be. A singular object is any entity 
that can be taken as a coherent whole, in other words, 
it can be referred to directly using a referring expression 
of language: a name, a definite description, a pronoun. 
Thus, at least as far as our ability to refer is concerned, 
all objects appear singular. Still, it is not the case that all 
objects are singular in the same way. Take, for example, 
two persons John and Mary. They are singular objects 
and they seem singular in the same way, in other words, 
singular relative to one another. Next take alligator, the 
species, and the alligator John owns. Although both are 
singular in their own right, they are not compatible 
when related to one another: the alligator John owns 
appears only a manifestation, or extension, of alligator 
the species at a certain space-time location. The indi- 
vidual alligator, which at some period of its life is owned 
by John is, therefore, singular in the same way John is, 
but this will not be the case when we consider the time 
slice of this alligator (an alligator stage, in Carlson's 
words) while it was owned by John. This latter appears 
only an instance of the individual alligator at some time 
interval. By the same token, if John owns many differ- 
ent alligators at different times (but, arguably, never 
more than one at a time) then we may risk to refer to 
John's alligator, an abstract object that generalizes over 
all alligator stages of all individual alligators ever owned 
by John (for example, "John always walks his alligator 
in the morning"). The new object, again, seems to 
belong in the same class of objects as John and Mary. 
Let us introduce, only intuitively at first, the relation 
Of relative singularity among objects. As suggested 
above, this relation will help us to break down the 
universe of objects into classes of relatively singular 
objects, which we call levels. The levels can be subse- 
quently partially ordered with lower than relation, i.e., 
L 1 < L2, indicating that level L~ consists of manifesta- 
tions (extensions, instances) of objects at level L 2. Let 
L 0 be an arbitrary level we select as our reference point; 
if our discourse operates at this level then L 0 defines the 
current level of reference of the discourse. Let L+ 1 and 
L_1 be two other levels different than Lo and such that 
L-I < Lo < L+~. At level L+I we place the objects we 
consider to be generalizations (or abstractions) of some 
measurable amount of objects from Lo. It is only from 
the perspective of L+~ that we are able interpret "The 
tiger lives in the jungle," or "The president is elected 
every four years," or "Birds can fly," or "Tourists 
start forest fires." The objects at L+~ are singular but 
only when related to one another within the same level; 
when viewed from L o they appear generic or functional 
or the like, in other words, non-singular. When we 
attempt to find a denotation for a nominal, such as "the 
tiger" or "tourists," within Lo, we attempt to give it 
either a singular or measurably singular interpretation. 
In a singular interpretation (if possible at all), we have a 
nominal refer to a specific object within the level 
(John's pet tiger), while in a measurably singular inter- 
pretation we use a quantification over a finite set of 
singular objects, also within the same level (every tiger, 
some tourists). Nominals denoting objects which are 
non-singular with respect to Lo, on the other hand, may 
have neither singular nor measurably singular interpre- 
tations within this level. In order to find proper deno- 
tations for them we must change the reference level 
from Lo to L+I. Thus, while the statement of "The 
President lives in the White House" when interpreted at 
level L÷~ can be argued to be equivalent to the state- 
ment "Every president lives in the White House" 
interpreted at L0, the same cannot be said of "The tiger 
lives in the jungle" and "Every tiger lives in the 
jungle." We must note that some objects found at L+~ 
could have been placed there by design rather than as a 
result of generalizing from Lo; an example of such 
higher-level object may be The President. 
If level L÷~ contains generalizations of objects from 
Lo, then level L_ l will contain their specializations or 
extensions. Descending upon L_~ we can see that what 
we previously considered to be the atom actually de- 
notes many different kinds of atoms (H, O, Ca, Fe, 
etc.), or that the mail is not the same every morning, or 
that Nicolas Bourbaki is the name of a group of 
mathematicians. 4 
A few definitions will help to put the above intuitions 
into a more formal setting. 
Def. 1. A use of a description is called singular if it 
refers to a singular object. A use of a description will 
be called measurably singular if it refers to some 
measurable quantity of a singular object. Otherwise 
we shall talk of non-singular use. 
Def. 2. An object level, or simply a level, is an 
arbitrary collection of relatively singular objects. On 
the language side, the corresponding reference level 
encompasses those singular and measurably singu- 
lar uses of descriptions that refer to the level's 
objects. 
Def. 3. For any level L, there are at least two distinct 
levels L_ I and L+t such that L+t contains these 
objects which are non-singular from the perspective 
of L, and L-I contains the objects for which the 
objects at L are non-singular. 
Def. 4. The level L o is an arbitrarily chosen level 
serving as a reference point. 
As described, the structure of levels is not yet adequate 
to capture the full complexity of the reference structure 
of discourse. A notion of coordinate has to be intro- 
duced along the following lines. We shall call T a 
coordinate, if T is a set of points or locations at which 
certain general (or abstract) objects, for example the 
president or the atom, are assigned more specific exten- 
sions or instances, such as President Reagan or H, Fe, 
Ca ..... A coordinate is usually an ordered set though 
Computational Linguistics, Volume 15, Number 3, September 1989 177 
L 
Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse 
the ordering may be partial only. Almost any object we 
can think of appears an instance of a more general 
concept, and often there will be more such concepts 
available, if we consider different coordinates. Water in 
a glass is an instance of some totality of water in the 
universe (space' coordinate), and also an instance of a 
concept of water as in "Water boils at I00 degrees 
Celsius." These examples suggest that a coordinate is 
usually a large set, often an infinite set, though perhaps 
no more than recursively enumerable. A non-singular 
object can be decomposed into instances in more than 
one way, depending which coordinate is used. An 
important observation is that when a higher-level object 
is decomposed with two different coordinates the result- 
ing sets of instances need not belong to the same level. 
The concept of the American president in the twentieth 
century is an instance of The President, but is still 
non-singular when compared with President Reagan, 
also an instance of The President, though with respect 
to a different coordinate. A somewhat finer structure of 
levels is required. 
Let L~'I T be the level where we place the instances of 
object N decomposed with coordinate T. By analogy, 
we define L~'I r to be the level such that for any object 
M, M ~ LN+'I r if N ~ L_ml r. In other words, L~'l r contains 
the superobject M generalizing over object N with the 
use of coordinate T. Suppose that we have an object N 
at level L0, to which we refer using a description N. 5 
Suppose further that coordinate T is selected so that for 
any x, y E Twe have that N-at-x :/: N-at-y. Let us use Nx 
to stand for N-at-x, where x is an element of T, and let 
(N x) be an expression (as translated into our meaning 
representation language) that refers to the object Nx, 
whenever the expression N refers to N, We obtain 
therefore that 
F1. Vx,y E T \[x --/: y ~ (N x) ~ (Ny)\] 
The new objects Nx'S cannot be placed at L 0 because, 
being instances of N, they are not singular relative to N 
(see Def. 2). Instead, we move them onto a new level 
LN_'I r leaving the original object N at L o. We say that the 
TN,T level rN,r is lower than the level L o, and write -~-1 L.~ \] 
L 0. Often we drop the superscripts N and T over the 
level symbol, assuming some lower level L_ 1, whenever 
it does not lead to ambiguity. Example 3 helps to 
illustrate the phenomenon just discussed. 
Example 3. Let us consider a rather naive concept of 
bird, as that of a winged creature that lay eggs and can 
fly. Let B be extension of this concept in our model and 
let Lo be set so that B ~ Lo. Using a genus coordinate, 
G, we can construct a level La_'~ containing such objects 
as eagle, hawk, and goose. Let's suppose that, initially, 
penguin can also be found at level La_'I c. Upon discovery 
that penguins cannot fly, however, one would wish to 
relax the characteristics of the concept B from Lo to 
contain both flying and non-flying birds. Nonetheless, 
two distinct concepts emerge, that of flying birds and 
that of non-flying birds, both of which become subordi- 
nates of the (now) more general concept B. These new 
concepts are placed at a new level La_'~ x, where K={kl, 
k2} is a class coordinate such that Bkl is the concept of 
flying bird, FB, and Ba~ is the concept of non-flying bird, 
NFB. The old level LB_'~ remains intact, though it is 
different from ra,x Moreover, rB,G < rB,X because Jr.,_ I . L,_ 1 z-,_ 1 
B,G ~a',cB, (- L_~ , where B' is either FB or NFB, and 
GB,C_G. There is another way of interpreting concept B 
as well: we introduce a specimen coordinate S that 
allows us to pick up specific birds, such as Opus, the 
penguin, at level ,-.-1 .rB's Note that this level is lower than 
LB'I c_ because it contains all levels "-.-lrx'sx, where X ranges 
za,o and SxC_S. Note that the structure over objects at ---1 , 
of levels has been created as described because of the 
following set of conditions. 
LFB,G~.~ < FB,K < Lo --i L~--I 
LNFB,GNFB ~ IB,K ~ Lo -! z-~-I 
LFB,6,.~ U TNFB,~,-B = L B_,~ < Lo --I J'~--I 
• rs,s with Speng,in C S Opus E L _pe~lguin,Sp ...... C L,_ 1 
Figure 1 further illustrates the concept of levels and 
coordinates. 6 For an easy interpretation of this drawing, 
note that gl ..... g5 E G, GFn O GNvB = G and gl, g2, 
g3 E GFB and g4, g5 E GNrn, and sl E Spengui n C S. 
Lo 
L~_.f 
L'_.? 
8x 
eagle 
FB,GFj N~,GNr s L_ I UL-1 
B 
83 
hawk goose 
L~_,~ 
~ 2 
g4 ~FB 
$1 ~ N 
penguin ostrich / 
Opus 
Figure 1. A structure of levels for a simple concept of bird. 
Now we can attempt to represent meanings of some 
simple statements about birds. For example, Birds can 
fly is represented at L0 as can-fly(B), while Opus is a 
bird would translate as 3sES \[(B s) = Opus\]. We cannot 
infer from these statements that Opus can fly because 
this would require the formula Vs \[can-fly((B s))\] to be 
true, which does not have to be the case considering 
that B may now contain instances of non-flying birds. 
Indeed, Opus cannot fly, which translates to 
--xan-fly(Opus), is not necessarily inconsistent with the 
above two. 7 The only thing we could say about each 
178 Computational Linguistics, Volume 15, Number 3, September 1989 
Tomek Strzalkowskiand Nick Cercone Non-Singular Concepts in Natural Language Discourse 
instance of B is, perhaps, that it is a bird and that every 
/'B,S bird is an instance of B. In other words, if x is a ,_,_ l 
variable then Vx \[bird(x) -= 3sES\[x=(B s)\]\] is true. This 
would lead to an equivalent L_ l translation of Opus is a 
bird as bird(Opus). Later, we will see that this equiva- 
lence is not generally valid. 
Let us now resume our general discussion on the 
characteristics of the structure of levels. At level L_ ~, 
we have an enumerable collection of different objects 
Nx's. Extending the description used for N (at Lo) over 
Nx's we refer to them as the N, a N, some N(s), every 
N, etc. It is possible, of course, that some other object 
N' found at L 0 is now disclosed to be N~, for some x E 
T. What that means is that we have placed N' incor- 
rectly at Lo, because it actually belonged to L_~. Con- 
sider the case of an ancient astronomer who believed 
that the Morning Star and the Evening Star were not 
only two different heavenly bodies, but also of the same 
genre as other stars. In our conventions, both the 
Morning Star and the Evening Star were placed at the 
level to which the planet Venus now belongs. When 
correcting this ancient misconception, we faced the 
problem of an instance of some object and the object 
itself were mistakenly assigned the same singularity 
level; that is, we had both Nx and N at Lo. Nevertheless, 
this situation represented the state of our knowledge of 
the world at the time. 
We may now give names to some Nx'S and N may 
very well happen to be among them. This time, how- 
ever, N will not denote the old object from L0; this will 
be quite a different name referring the selected N~ and 
can be replaced by a definite description (N x). To 
illustrate this phenomenon, we may compare the con- 
cept of a programming language, such as Lisp, with its 
various implementations or versions available at differ- 
ent sites or times, and which are locally called Lisps as 
well. A more common transition of a name is, however, 
from an instantiation to a concept, which, ultimately, 
can create a similar effect. Consider for the moment the 
concept of a sun as a center (or one of several centers) 
of a solar system, and our own Sun as a specific 
instantiation of this concept. A schematic illustration of 
the descent process is shown in Figure 2. 
A process reverse to decomposition is that of ascend- 
ing to a higher level within the level hierarchy. Suppose 
that for some objects N ~, N 2, • •., considered distinct at 
L 0, we discover they share a certain property, such as 
being an N, so that we need a generalizing concept to 
talk about them. We pick up a coordinate T, and climb 
onto some higher level L~, that is, Lo N,r = t_, < = 
L 1, and establish a new object N there, a "superobject." 
Now, as viewed from L\], all Ni's are just the occur- 
rences of N at different values of coordinate T. In other 
words, the following equation holds: 
F2. Vi \[3xE/\[(N x) = Ni\]\] 
Note that all Ni's now belong to the level rN,r 1-'-l, which is 
a part of Lo. As before, we shall drop superscripts N and 
L~_7 
N: ~Lqv~ oMwa ~iun: t~ ~ttt d t mk rtmm) 
• o o S t o o 
~ tt~ ot N m t (oug bn) 
Figure 2. Descent from the current level onto a lower level. 
T for simplicity. No matter how we name N at L+~, the 
following Formula of Discovery summarizes our action: 
F3. Yx,y E T \[(N x) = (N y)\] 
It must be noted that (F3) is valid only when stated from 
the perspective of L+~. At Lo, Ni's remain distinct 
traditionally, so they remain distinguished in the lan- 
guage as well; cf. Formula F1. The generalization of 
other objects from L o onto L+~ may follow but, as in the 
case of decomposition discussed earlier, the process 
will largely remain implicit. These observations are 
illustrated in Example 4. 
Example 4. At level L0, we have object TP, named The 
President. Let T be the time coordinate (which is 
different than T in the last two examples). At Lo, we 
have, according to (F3), that Vx,y ~ T \[(TP x) = (TP y)\]. 
Later, we may discover that for some tl, tz E T, 
(TP t0=N and (TP t2)=R, and that at some level rTP,r ~L-J__ l 
where N and R belong, they are considered distinct and 
named Nixon and Reagan, respectively. But at Lo, 
R=N is true. The last observation can be made clearer 
if one imagines that TP is some abstract individual who, 
when observed in the early 1970s, is named Nixon, and 
who, when observed in the 1980s, is named Reagan. 
As described, the case of generalization (or abstrac- 
tion) is extremely common in natural language, and the 
best tangible manifestation of this phenomenon is com- 
mon nouns. Common nouns should be considered 
names of generalized concepts, which may be either of 
a physical or otherwise measurable nature (mass con- 
cepts such as water, snow, gold, temperature .... ) or 
of a non-physical, abstract nature (usually based on 
enumerable quantities of specific instances: tree, man, 
president .... ), or both (like fish, people .... ). 
Once the superobject N has been created, it begins to 
live a life of its own. Some new objects from Lo, 
different than Ni's, may now become instances of N at 
some, as yet, unutilized values of coordinate T. Also, 
we may use descriptions (N x) without caring whether 
they actually refer to any objects at Lo. In other words, 
the fact that a superobject N has no instance at certain 
location x ET at level L~'~ T does not preclude the use of 
Computational Linguistics, Volume 15, Number 3, September 1989 179 
Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse 
the description (N x) in the language. The resulting 
expression may not always be well defined, though, as 
is in the case of"the present king of France," where the 
adjective "present" specifies the value of time coordi- 
nate decomposing the superobject the king of France. 
This problem of non-denoting expressions created out 
of denoting general terms, which is widely discussed by 
Quine (1960, 1973), receives an elegant explanation in 
our theory. One remaining problem is the relationship 
between a superobject and its instances in some given 
decomposition. It is important that we do not equate a 
superobject at L+I with the set S of its instances at Lo, 
even if they may have actually given birth to this 
superobject. A superobject cannot be understood as a 
set of appropriate lower-level instances, since we would 
obtain only a measurable collection of singular objects. 
Instead, a superobject N can be identified with a family 
of functions {qb~ IT is a coordinate} such that each ~ is 
a function from coordinate T into an appropriate lower 
level, L~'I ~r. In particular, a superobject N at LP+'~, where 
P E L 0, can be considered (from Lo perspective) as a 
function ~ from T into L0 such that, whenever 
s ESC_Lo, then there is t E T such that ~ (t) = N t = S. 
The function qb~. is then arbitrarily extended beyond the 
set S. The following definition may be suggested. 
Def. 5. Let L and M be any two distinct levels of 
relatively singular objects. We say that level L is 
lower than level M, L < M, if there exists an object 
P at level M and a coordinate T such that L D ~'te'r ~vJt_ l . 
In order to avoid any misunderstanding, we add the 
following remark. It is possible that some Nj from 
among the Ni's was already recognized properly at Lo as 
our goal object N from L÷l, although its other occur- 
rences Ni for i :k j were not identified with it (cf. the 
Morning Star, the Evening Star, and Venus). In some 
sense, therefore, the previous concept of N was incom- 
plete, since it did not contain these other instances 
which, in turn, allowed this former concept to coexist 
with some of its would-be instances at the same level. 
This fact can be further reinforced in our language when 
we choose to name N after N j. This should not suggest 
that N and N j are one and the same object. The former 
is in a sense more mature, although, when referenced by 
name, one can hardly tell which one of the two concepts 
is being referred to, unless, of course, some additional 
clarifying context is present. Examples 5 and 6 below 
further illustrate this point. 
Example 5 We have the following distinct objects at 
level Lo: V, called Venus; MS, called the Morning Star; 
and ES, called the Evening Star. Upon discovery that 
they all represent occurrences of the same planet, we 
create a new object V', named Venus, at the level LI = 
LV'l T, and such that for some x,y,z E T, where Tis a time 
coordinate, (V'x) = V, (V'y) = MS, (V'z) = ES, where 
V', V, MS and ES are individual constants denoting V', 
V, MS and ES, respectively. Using the formula (F3), we 
conclude, from the perspective of L~, that V = MS = 
ES, while the same conclusion made at L o is false. 
Example 6. Let the level L o be as in Example 5, except 
that the object V is discovered not to be uniform. In 
fact, it contains occurrences of three different objects: 
planet Venus and some two heavenly bodies assumed to 
be Venus :in the mornings and the evenings. Now we 
cannot use our time coordinate T from the previous 
example to get the desired result of the object V' at L+ 1. 
Instead, we first descend to LV'l s over a coordinate S to 
differentiate the objects Vs,, Vs 2, V~ 3 for some st, $2, $3 E 
S. Let the V~, be a part of the ultimate object V'. In the 
same way, we create instances of MS and ES over the 
coordinate S at levels tMS,S and fES,S respectively. z.~_ 1 z.-_ I , 
Because V, MS and ES are relatively singular, their 
instances in decomposition with respect to the same 
coordinate yield objects that are also relatively singular. 
?V,S In other words, there is a level L 2 such that L 2 ~ ,-,-i t..J 
LMS,S tES,S Let MS~ be called the Morning Star and --1 \[J 1"--1 " 
ES~3 be called the Evening Star at L2. Now we can 
construct the ultimate object V' out of Vsl, MS~2, and 
ES~3 using a coordinate T such that {s~,sE,s3} C_ T and 
V" I = V~t, V" 2 = MS~2, and V's3 = ES~. We place V' at 
a new level L 3 L v d I"T. Note that L 3 C_L o, and thus that 
V' is singular at L0. 
Example 6 is more "realistic" than Example 5, but 
both examples have equal linguistic significance. Note, 
however, that having L 2 as L 0, and L 3 as L t, we could 
reconstruct Example 5. 
6 REMOTE REFERENCES, SUPERCONTEXTS~ AND 
SUBCONTEXTS 
Let us examine how the foregoing theory of non- 
singular terms could be utilized in assigning meaning 
representation to natural language discourse. In partic- 
ular, we, are interested in the problem of computing 
extrasentential dependencies in text. 
The translation rules described briefly in Section 2 
are applicable only when anaphoric references arise 
between relatively singular descriptions or, as we would 
say now, between descriptions denoting objects classi- 
fied within the same single level. We show that a similar 
technique can be used to compute another type of 
extrasentential dependencies, which we call the remote 
co-reference. Let us start with an example. The arrow 
symbol-~ is used here, and elsewhere, to mean "trans- 
lates to". 
ExamplE; 7. Consider the following discourse fragment. 
7a. The president 1 is elected every four years. 
7b. The president 2 is Reagan. 
There i,; nothing to prevent us from interpreting the 
president I and the president 2 at levels L t and L2, 
respectively, so that one of the following takes place. 
Either Lq = L 2, or L 1 < L 2, or L 2 < L 1, or simply L1 
L2, where < stands for the extended lower-level relation 
introduced in definition 5. The latter case does not 
interest us, since, in such an interpretation, both sen- 
tences were uttered at different occasions with no 
connection between them. 
180 Computational Linguistics, Volume 15, Number 3, September 1989 
Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse 
Consider first that L~ = L 2 = L0. If the two definite 
descriptions were to co-refer then we would be talking 
of the same object (individual) in both sentences. That 
interpretation, although possible, does not agree with 
our intuition. In this case the conclusion of 
7c. Reagan is elected every four years. 
follows immediately. 
Let us assume next that L 2 TP, T DL_ 1 < L 1 = L o, where 
TP is the object at LI referred to by the presidenh, and 
T is a time-based coordinate. If the presiden h is used as 
a name, we can expect to obtain the following transla- 
tions: 
7a ---> eefy(TP) 
7b ---> 3t \[SL(t) & ((TP t)=R)\] 
with "elected every four years" ---> eefy, "the pres- 
ident" ~ AP\[P(TP)\] 
where t E T and SL is a selector over T provided by the 
discourse situation (for example now, here, etc.). This 
selector could be obtained if we consider a possible 
translation of (7b) at L2 as pres(R) & loc(R, now), from 
which it follows that 3tET\[((TP t)=R & loc((TP t), 
now)\], with SL = At\[loc((TP t), now)\]. 
In a more general case, we would take the phrase the 
presidenh as an ordinary definite description. Assuming 
some external context C which allows for the use of the 
definite article, the following translation can be derived: 8 
7a---> 3r \[p(r) & C(r) & Vy \[(p(y) & C(y)) 
D (r=y)\] & eefy (r)\] 
7b ---> 3x \[p(x) & C(x) & Vy \[(p(y) & C(y)) 
D (x=y)\] & :lt\[SL(t) & p((x t)) & 
((x t)=R)\]\] 
where "president" ---> p. 
The first of these formulas can be interpreted, with 
respect to some implicit context C (which may be 
reasonably assumed to limit our attention to the U.S. 
government), as: there is exactly one object such that it 
is a president and has a property of being elected every 
four years. Clearly, the object we refer to in (7a) is not 
any particular person, but the office of the President of 
the U.S. The other formula says, in turn, that there is a 
value t of the time coordinate T at which the instance of 
the general object referred to in (7a) (the President of 
the U.S.) is identical with the object R (which stands for 
Reagan). Observe that R refers to the individual Reagan 
as he appears at t, but not necessarily beyond that. The 
latter formula gives the full translation of (7b) when the 
remote reference to level L~ has been resolved. Before 
this interlevel connection is established, however, we 
can assign (7b) only the a literal translation at L2, as 
shown below, where C2 is a local context at level L2. 
7b ----> :Ix \[p(x) & C2(X ) t~ Vy \[(p(y) & Cz(Y)) D 
(x=y)\] & (x=R)\] 
A point that is worth discussion is the role of the 
selector SL in the full translation of (7b). This selector 
may be interpreted as a local context at L z determined 
by a discourse situation, as in Barwise and Perry (1983). 
To understand its influence on the form of the refer- 
ence-making sentence, compare Example 7 with Exam- 
ple 8 below. 
Example 8 
8a. The tiger lives in the jungle. 
8b. My pet is a tiger. 
Unlike (8b), in the "story" of the president the local 
context allows (and requires) us to use "the" in (7b) 
because there may be at most one instance of the 
general term at a given discourse situation (in this case, 
the President of the U.S.). This is not the case when 
more than one value of the coordinate T satisfied the 
selector condition. The use of a definite description in 
such a context suggests, therefore, that a single instance 
of a general term is being picked up by the selector. This 
means, in turn, that except for a main "remote refer- 
ence" in cross-level referencing, another local refer- 
ence is being made at these levels where the definite 
description is used to denote some object. The remote 
reference itself does not need a definite description to 
be used for establishing a connection between an in- 
stance and the general term. The fact that we use a 
definite article when referring to an instance of a general 
concept, as in (7b), implies only that the local context 
SL contains (or is expected to contain) exactly one 
instance of the general object. Notice that the temporal 
aspect of this local context is extremely influential. 9 
When we decompose a general object with respect to a 
time coordinate, we are more likely to obtain a unique 
instantiation in a local context. However, when a coor- 
dinate does not contain a time element, as in (8a,b), we 
cannot, in general, exclude the possibility that instances 
other than the one we intend to refer to in subsequent 
statements may be present in a local context. Thus a 
definite description is not used until the local context is 
properly narrowed. In the case of (8a,b) above, for 
example, we may continue the discourse at the lower 
level, saying "The animal is quite friendly," and mean- 
ing "The animal which is my pet tiger is quite friendly." 
This discussion strongly supports our theory of 
names and descriptions, particularly the existence of 
levels and coordinates. Having established a higher- 
level object (or superobject), we can freely discuss its 
instances across various coordinates. The local refer- 
ences can be accounted for by singular translation rules, 
which we discuss in Strzalkowski and Cercone (1986). 
Finally, let us define the notion of remote reference 
for objects classified into different naming levels. 
Def. 6. An object N at a level L n is said to be remotely 
referenced by a description M if M refers to an 
object M at some level L m such that either L m D_ 
L~ 'r, or L m D_L~f, for some coordinate T. 
Let us summarize this discussion briefly now. In some 
part of a discourse, a certain (general) object X is 
addressed; that is, there is some part, $1, of the dis- 
course (presented as a single sentence in our examples, 
Computational Linguistics, Volume 15, Number 3, September 1989 181 
Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse 
for simplicity), such that S 1 predicates something of 
X--that is SI(X), where X is a description that refers to 
X. In a subsequent part of the discourse, however, the 
discourse changes the level of reference and only some 
instance(s) of X with respect to some coordinate T is 
addressed; that is, there is some t E T such that S2((X 
t)), where S 2 is this new part of the discourse. Appar- 
ently, the discourse internal cohesion would be compro- 
mised if we did not allow the higher level object X be a 
target of a remote reference by a description (X t) 
denoting one of its instances. In such a case we say that 
SI(X) creates a supercontext for (X t). We can further 
say that X and (X t) are remotely co-referential. By 
analogy, if S! contains a reference to an instance of an 
object X, that is, we have SI((X t)) for some coordinate 
Twhere t E T, and $2 contains a reference to X, that is, 
S2(X), then SI((X t)) is a subeontext for X. In this case 
too, X and (X t) are remotely co-referential. 
We have just arrived at a very important property of 
a natural language discourse, which influences both 
coherence and cohesion, and is absolutely essential in 
proper representation of discourse meaning content. 
The following translation rule specifies how such a 
representation should be derived.I° The approach pre- 
sented here is entirely original, although the problem of 
the "general term/an instance" connection has been 
known for a long time and other, less general, solutions 
have been suggested (see Partee 1972 and Montague 
1974). 
Rule 11 (Supercontext Translation Rule): 
If the context-setting sentence St with the translation 
L+1, where 3x \[Pl(x) & Fl(x)\] is interpreted at level g.r 
is an object satisfying sentence S 2 when interpreted 
at level L o, and $2 contains an unresolved remote 
reference P2, that is, 
$2 = 3y \[P2(Y) & F2(y)\], 
then the full translation of S 2 is obtained as 
AQ\[AC\[MQ,c\](C1)\](Au\[3t\[AY\[P2(y) & F2(Y)\]((u t))\]\]), 
where the supercontext C1 is hx\[Pi(x) & Fl(x)\], and 
MQ, c abbreviates the following expression 
MQ,C = 3x\[C(x) & Vy \[C(y) D (x=y)\] & Q(x)\]. 
Thus far we have considered two types of cross-level 
references: the trivial one where L~ = L2 = Lo, and the 
remote reference in supercontext, that is, with L~ < L2 
= L o. Let us now turn to the remaining type, for which 
L 0 = L 1 < L2. We call such situations remote references 
in subeontext. 
Example 9. Let TP and T be as in Example 7. This time 
suppose that we have the following situation: L 0 = L I < 
L 2 ~_Lt+p| r, where tp is TPt for some t E T. Our discourse 
may now look as follows. 
9a. A president1 sits in the first row. 
9b. The president 2 is elected every four years. 
182 
Sentence; 9b has the following translation with a remote 
reference to "the president" in (9a). 
9b -* 3x \[p(x) & 3t \[SL(t) & p((x t)) & sfr((x t))\] 
& Vy \[(p(y) & 3t \[SL(t) & sfr((y t))\]) D 
(x=y)\] & eefy(x)\] 
where "sits in the first row" ---> sfr. 
The formula translating (9b) says that the president, a 
unique superobject denoted by x, is elected every four 
years. The uniqueness of the president superobject is 
determined by the fact that one of its instances with 
respect to some space-time coordinate, identified in the 
formula by the term (x t) where t is an element of this 
coordinate, has the property of sitting in the first row. 
One may wonder whether the use of the definite 
description in (9b) is always necessary to maintain the 
remote reference, since we rejected such a necessity in 
supercontexts. It should be clear enough, considering 
the level structure introduced earlier, that we can and 
have to use definite descriptions in subcontextual ref- 
erences. We illustrate this point with another example. 
The problem is discussed at a great length in Strzal- 
kowski (1986). 
Example 10. In the following discourse, the descriptions 
a presidentl and a president 3 refer to objects which are 
not relatively singular and therefore do not belong to the 
same level. Although the discourse seems connected 
and coherent, there is no straightforward correspon- 
dence between the two sentences. 
10a. ,lohn wants to become a king rather than a 
presidenti. 
10b. That is because a president3 is elected every 
several years, while the king rules for a lifetime. 
Here a presidentl refers to an L0-instance PI of an 
rv,,r A president 3 L+ l-level object Pz specifically Pz E ~+1 • 
refers to some object P3, which is still non-singular from 
the perspective of level Lo. P3 is an instance of Pz but 
with respect to a coordinate T' different than T. In other 
words, P3 is placed on the level L~:i r, which is not a part 
of L0. There is no direct correspondence between a 
president~ and a president3 beyond the fact that P2 E 
f P3,r' It may well happen, however, that for LP'\[ r n ~+1 • 
some coordinate U and some u ~ U, we shall have (P3 
u) denoting Pv This is why John would rather be a king 
.... What relates these two "presidents" is a compo- 
sition of two remote references: the one made by a 
president I to the object P2 (we may wish to call this 
object the president2), and the other made by a 
president 3 to P2. This same effect will be produced if we 
use other measurably singular descriptions like "every 
president," "some presidents," or "most presidents" 
in (10b). If we had the defini~te "the president" instead 
of indefinite "a president" in (10b), however, we would 
obtain a clear instance of a remote reference in subcon- 
text, that is, with P2 = P3. The objects and their 
respective levels in the presidents case are illustrated in 
Figure 3. 
Computatiomd Linguistics, Volume 15, Number 3, September 1989 
Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse 
P|.T PS,T' £+\] IIL+\] 
P2,T' L_! 
/'o * ° PI * 
P2 
al~nlaml ~ 
a pm~d~t (~ mla tD Imme ) 
Figure 3. The structure of levels in the president example 
(Example 10). 
The case of the remote reference in subcontext is 
summarized with the translation Rule 12. 
Rule 12 (Subcontext Translation Rule): 
If the context-setting sentence S t with the translation 
~,T 3x \[Pn(x) & Fl(x)\] is interpreted at some level L_ 1, 
where ~ is an object satisfying the current sentence 
S 2 when interpreted at L o, and S 2 contains an unre- 
solved remote reference P2, that is, 
S 2 = =Ix \[P2(x) & C(x) & Vy \[(P2(Y) & C(y)) 
D (x=y)\] & F2(x)\] 
then the full translation of $2 is obtained as 
S2(P 2) --~ AC\[LE\](Au\[::II \[CI((U t))\]\]) 
where the subcontext Cn = Ax \[Pn(x) & Fn(x)\] is 
derived from $1. 
7 SUPEROBJECTS VS. PLURAL TERMS 
We now examine the nature of superobjects, that is, the 
objects placed at level L÷ 1. In particular, we are inter- 
ested in what sets them apart from their instances. Let 
us again consider the term "The President" as referring 
to the superobject TP at some level L+~ (of. Example 7). 
We may say that such superterm has the property of 
collective referring to all of its instances at once, but 
without necessarily making a reference to any of these 
instances in particular. At level L0, if we want to refer to 
a set of objects, we use one of the enumerating quanti- 
tiers "every" or "each," or we use some sort of plural, 
such as in "These presidents were married" (each). We 
note, however, that most plurals, especially the so- 
called "bare" plurals, like presidents, tigers, or meet- 
ings can actually refer collectively, which makes them 
akin of superterms, though many of these plurals can be 
ambiguous between the collective and a non-collective 
interpretations. Just like in the case of singular super- 
objects, the objects denoted by plural terms, such as 
presidents, tigers, Americans, etc., cannot be always 
identified with the corresponding sets of lower-level 
instances. It turns out that the plural terms actually 
denote superobjects, 11 and therefore they should be 
interpreted at the same level as respective singular 
superterms. We will see that the generalization leads 
naturally to plural terms that may or may not induce 
equivalent singular superterms. Conversely, a plural 
equivalent to a singular term may suggest the most 
natural coordinate to decompose the superobject in its 
denotation into lower-level instances. When a singular 
term lacks a plural equivalent, however, we may admit 
that the object in its denotation is not naturally decom- 
posable and that we are now looking at the bottom-most 
level in some decomposition hierarchy. A further de- 
composition may be still possible, but it can only 
produce objects that will never assume an independent 
status and will remain recognized only as instances of 
some more general superobject scattered over that or 
another coordinate. This phenomenon is characteristic 
of so-called mass objects and their corresponding mass 
terms. As an example, consider such nouns as water, 
gold, or heat. They lack plural equivalents, and there is 
no obvious way to decompose them into lower-level 
instances, except with 4-dimensional space-time coor- 
dinate. Note that although we can occasionally use a 
morphologically plural term, like waters, these usually 
will not be equivalent to singular superterms, and they 
will often denote other mass objects as well, such as in 
"waters of the Nile." Space-time coordinates can also 
be used to obtain instances of otherwise non-decompos- 
able entities, such as John, into space-time slices, called 
"stages" (Carlson 1982). Quite naturally, the question 
of where one level ends and another begins arises. The 
following two examples provide some insight into the 
level-boundary problem. 
Example 11. Consider the following sentences. 
1 la. Mary brings (some) water every day. 
1 lb. John picks up the mail every morning. 
Let "water" in (1 la) be the name of some superobject 
W at level L+I. Presumably, Mary brings only a part of 
W, but we can say that W is being brought by Mary 
every day. This is the same W every day, although each 
time possibly a different part of it is in transit, which 
leads to the obvious translation (at L+t), 
i. 1 la ---> brings-every-day(M,W) 
where M and W are individual constants denoting L÷I 
objects M (Mary) and W (water). Alternatively, if we 
evaluate (lla) at level L o = ,-,-lrw'r with a space-time 
coordinate T, in which case (lla) should read "Mary 
brings some water every day," we obtain the following 
interpretation: 
ii. 1 la---> Vx \[day(x) D 3t \[brings(x, M, (W t))\]\] 
where W is as before, M is an individual constant 
denoting a space-time stage of Mary,12 and brings(x,y,z) 
should be read at x y brings z. Now (W t) denotes some 
instance of the superobject W at level L 0' of which we 
Computational Linguistics, Volume 15, Number 3, September 1989 
m _ 
183 
Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse 
can say that it is water as well; that is, water((W t)). At 
L 0' we replace (W t) by a singular variable u, obtaining 
ii'. 1 la---~ Vx \[d(x) D 3u \[water(u) & brings(x, M, u)\]\] 
The translations featured in (i), (ii), and (ii') may appear 
to be equivalent statements made from different per- 
spectives and at different levels of detail. This is not the 
case, though. In particular, (ii) and (ii') are not equiva- 
lent. This point is further illustrated with (1 lb). Let m be 
the mail that John is picking up at any one particular 
occasion. There is a space-time coordinate T such that 
the level L~I r contains a unique superobject for which 
the following holds. 
iii. 1 lb ~ 3x \[mail(x) & C(x) & Vy \[(mail(y) & C(y)) 
(x = y)\] & picks-ev-morn(J, x)\] 
The context C is used to determine the uniqueness of 
John's mail superobject. Note that the quantification m,T 
ranges over objects at L+~ . Because John's mail super- 
object is decomposable with T we obtain the following 
equivalent cross-level interpretation. 
iv. 1 lb ~ 3x \[mail(x) & C(x) & Vy \[(mail(y) & C(y)) 
3 (x = y)\] & Vu \[morn(u) D 3t \[picks(u, J, (x t))\]\]\] 
Here, t ranges over elements of coordinate T, and (x t) 
refers to an L 0 instance of John's mail superobject. The 
three arguments of picks(x,y,z) are understood as at x y 
picks up z. As it stands, (iv) still needs to be interpreted m,T 
at L+~ . To reduce this translation to the form that 
would be interpretable at Lo, we have to replace the 
definite reference in the first line of (iv), that is, 
3x \[mail(x) & C(x) & Vy\[(mail(y) & C(y)) 
D (x = y)\]\] 
with a predicate, interpretable at L o, that would 
uniquely indicate John's mail. Then replacing (x t) by a 
variable z, we obtain the following formula. 
iv'. Vu \[morn(u) D 3z \[John's-mail(z) & 
picks(u, J, z)\]\] 
Note that equivalence to (iv) requires the John's-mail 
predicate to be true of an empty z, which can happen if, 
at some occasion, John receives no mail at all. Other- 
wise (iv) and (iv') are not equivalent. In (iv) we can 
maintain truth of picks (u,J,(x t)) because we do not 
require (x t) to denote anything at Lo; we merely say 
that a t exists. This example actually shows the power 
of the multilevel representation. Observe that we have 
just delivered a very strong argument supporting the 
claim that superobjects are not merely sets of their 
instances. It must be noted here again (cf. footnote 7) 
that some limit needs to be set up for the amount of 
exceptions to a general statement like (iv) that we are 
willing to tolerate. 
There remains one more reading of (lib) that does 
not seem to require any reference to higher-level ob- 
jects. This reading could be paraphrased as: Every 
morning there exists the mail such that John picks it up, 
or more formally 
v. I lb ~ Vx\[morn(x) D 3u \[mail(u) & C(u) & 
Vy\[(mail(y) & C(y)) D (u = y)\] & picks(x,J,u)\] 
This time the variable u is bound at Lo. Is this transla- 
tion feasible? We can answer both "yes" and "no." A 
"yes" answer indicates that the transformation gives us 
a singu\]lar interpretation of (11b) at L o. Note that be- 
cause of the uniqueness clause in it, (v) says no more 
than that John keeps picking up the same thing every 
morning, since the context C does not depend on x and 
is the same each time. A "no" answer indicates that the 
latter interpretation most probably does not express our 
intention. The translation of (v), although possible, is 
not equivalent to either (iii), (iv), or (iv'). 
At the beginning of this section we stated that bare 
plurals, like presidents or meetings, behave much like 
singular superobjects. Let us look somewhat closer at 
this issue now. Consider the pairs of sentences (12a,b) 
and (12c,d) below. 
Example 12. 
12a. The faculty meeting is held every month. 
12b. Faculty meetings are held every month. 
12c. The rat can live in most countries. 
12d. Rats can live in most countries. 
It seems to us that the preferred interpretation of 
sentences in (12b) and (12d) is such that the plural noun 
phrase ,(faculty meetings, rats) is understood as denot- 
ing a higher-level concept. For the same reason as (12a), 
and (lib) before it, (12b) will remain a truthful L÷I 
statement in spite of the fact that, at some occasions, 
meetings can be canceled (due to holidays, for exam- 
ple). In the case of (12c) and (12d) the point is somewhat 
finer. "\['his time we may be less reluctant to say that 
(12d) means, in fact, that every rat can live in most 
countries, because we talk about possibilities. There is 
a catch in here, however; though every rat can live at 
many different places, not every one, or perhaps even 
none, car~ live at most of these places. 
One of this section's key observations is that plural 
terms are, in many respects, equivalent to singular 
superterms. We can assume, for now, that plural terms, 
cautiously named prototypes of superterms, actually 
denote superobjects as well. For some plural terms, we 
can find equivalent singular versions like tigersmtiger, 
presidentsmpresident, etc. Others do not have this 
property. Alternatively, mass terms will usually lack 
plural equivalents, for example, "water" cannot be 
identified with "waters" in general. Still others may 
expose a surprising mixture of the properties, like 
"people," which is a plural term with morphologically 
singular form and which may also be used as mass 
superterm. 
8 CONCLUSION 
We presented an approach to representing various kinds 
of non-singular concepts in natural language dis- 
course. 13 The major observation of our theory is that 
184 Computationan Linguistics, Volume 15, Number 3, September 1989 
Tomek Strzalkowski and Nick Cercone 
reality, as perceived by an intelligent individual, can be 
regarded as a partially ordered structure of levels such 
that each level contains only those objects that are 
considered relatively singular. Observe that there are 
virtually no restrictions imposed upon the notion of 
relative singularity, so that the distribution of objects 
between levels of the world model may differ among 
different individuals. Non-singular objects, called su- 
perobjects, are placed at a number of higher levels, 
which are related to the current level with various 
coordinates. Conversely, a singular object may be de- 
composed along a coordinate, and new objects, so 
obtained, will be placed at some lower level. This same 
coordinate can be used then to obtain instances of other 
objects at this lower level, so that the relative singular- 
ity of objects within each level is maintained. This 
theory also contributes to a better understanding of 
discourse internal cohesion by introducing the notion of 
remote reference in text. 
We believe that our approach promotes computa- 
tional tractability of some more difficult properties of 
natural language discourse. Among these, the notion of 
intension as formalized by Montague (1974a-d) has long 
been considered difficult for a practical realization, and 
a more tractable alternative has been sought ever since. 
Our theory of names and descriptions takes us a step in 
this direction, although an implementation or a formal 
complexity evaluation have yet to be attempted. Ob- 
serve that with the concept of superobject and coordi- 
nate, we no longer have to identify general terms like 
"temperature," "president," or "water" with inten- 
sions, that is, functions over possible worlds. Superob- 
jects are not some mysterious, extra-world entities, but 
they acquire a concrete status which makes them as 
comprehensible as ordinary objects. In a simplified 
account, our notion of coordinate can be loosely related 
to the possible world theory's concept of index. The 
coordinate is, however, far more selective than the 
index. In a particular discourse situation, we can pick 
up that aspect of intensionality of some concept that is, 
at present, relevant to our understanding of the dis- 
course. Our freedom in selecting that or another coor- 
dinate is all important. Note also that structures of 
coordinates may vary considerably. For example, a 
pure time coordinate may consist of time points as well 
as consists of time periods, and time periods are of 
much greater significance in practice (examine "the 
president" example). Coordinates connote more than 
indices. They can be used to define a non-singular status 
of an object which is otherwise purely extensional (the 
examples of "water" from the last section). In this 
sense, the concept of non-singularity has a local, and 
often subjective, character. 
Non-Singular Concepts in Natural Language Discourse 
ACKNOWLEDGMENTS 
The authors would like to thank Jim Delgrande and the anonymous 
referees whose many comments have helped to make this paper into 
a more readable and comprehensive article. The research reported 
here was supported in part by the Natural Science and Engineering 
Research Council of Canada under Operating Grant number A4309 
and Operating Grant number A9219, by the Office of the Academic 
Vice-President, Simon Fraser University, and by SFU's Open Grad- 
uate Scholarship. We also thank the LCCR for use of facilities. 
NOTES 
I. For Moutague's system to work it is necessary that a possible 
world is understood as a completely specified alternative reality, 
and cannot be thought of as merely a different model, or a partial 
situation (Montague 1974a, Barwise and Perry 1985). Although 
various "Montaguc-style" or "Montague-inspired" approaches 
to natural language processing have been proposed in computa- 
tional linguistics (see, for example, Landsbergcn and Scha 1979, 
Bronncnberg et al. 1980) none of them ever attempted to tackle 
a full-scale semantics for IL. 
2. "Profiles: A gentle reign," by Susan Orlean. The New Yorker, 
Dec. 12, 1988, p. 50. 
3. is in "is rising" and in "is ninety" are of course different uses of 
the verb to be. This is the identity is in the second sentence that 
we are concerned with here. 
4. This example was suggested by one of the reviewers. 
5. We use boldface to denote objects, and romanface to indicate 
their names and other descriptions referring to them. 
6. There is no connection between this and other drawings pre- 
sented in this paper, and any semantic network formalism. 
7. This is not to say that we can have just any amount of negative 
evidence in the form of literals '-lfly((B t)) and still keep on 
believing that fly(B). An empirical verification or refutation of 
the latter will depend upon the number of instances seen and the 
quantitative relationship between the positive and the negative 
evidence (Strzalkowski 1989). Nonetheless, we have to be 
prepared to accommodate exceptions, up to a point. The discus- 
sion of these issues is beyond the scope of the present article. 
8. A detailed discussion on how to produce such translations can be 
found in Strzalkowski and Cercone (1986) and Strzalkowski 
(1986). 
9. This is, quite clearly, due to the temporal aspect of the majority 
of sentences produced in natural language discourse (use of verb 
tenses and temporal adverbs). 
10. Before a translation rule like one of these presented in this paper 
can be used, a sentence must undergo numerous transformations 
within the Stratified Model (Strzalkowski 1986). These transfor- 
mations include, but are not limited to, morphological analysis, 
lexical disambiguation, and syntactic parsing. 
11. Quine (1960) seems to suggest this conclusion; refer, for example 
to his double interpretation of mass terms (pp. 120-121). 
12. To be precise, we should represent Mary-stage as (M t), that is, 
as a term referring to an instance of the object M. However, our 
naming convention discussed earlier allows for replacing the 
definite description (M t) by the name of the Lo object in its 
denotation. We utilize this option here. 
13. This approach to representing non-singular concepts contributes 
to the transformation Fn_ 1 of the Stratified Model; see Strzal- 
kowski (1986). In particular, rules 11 and 12 add greatly to the 
explanation of text cohesion, about which transformation. Fn-~ 
is concerned, although they introduce some ambiguity into the 
transformation. 

REFERENCES 
Barwise, J. and Perry, J. 1983 Situations and Attitudes. The MIT 
Press, Cambridge, MA. 
Barwise, J. and Perry, J. 1885 "Shifting situations and shaken 
attitudes." Linguistics and Philosophy 8;(1): 105-161. 
Bronnenberg, W.; Bunt, H.; Landsbergen, J.; Scha, R.; Schoenmak- 
ers, W.; van Utteren, E. 1980 "The Question Answering System 
PHLIQAI." In L. Bolc (ed.) Natural Language Question Answer- 
ing Systems. Macmillan. 
Carlson, G. 1977 "A unified analysis of English bare plural." Linguis- 
tics and Philosophy 1: 416-456. 
Carlson, G. 1982 Generic Terms and Generic Sentences. Journal of 
Philosophical Logic 11: 145-181. 
Donnellan, K. 1971 Reference and Definite Descriptions. In D. D. 
Steinberg, L. A. Jakobovits (eds.) Semantics. Cambridge Univer- 
sity Press, Cambridge, England; 100-114. 
Kripke, S. 1972 Naming and Necessity. In D. Davison, G. Harman 
(eds.) Semantics of Natural Language. Reidel, Dordrecht; 253- 
355. 
Landsbergen, J. and Scha, R. 1979 "Formal languages for semantic 
representation." In Allen and Petofi (eds.)Aspects of Automatized 
Text Processing: papers in text linguistics. Buske, Hamburg. 
Lewis, D. 1976 General Semantics. In B. H. Partee (ed.) Montague 
Grammar. Academic Press; 1-50. 
Montague, R. 1974a On the Nature of Certain Philosophical Entities. 
In Thomason 1974 Selected Papers of Richard Montague. Yale 
University Press, New Haven, CT. 
Montague, R. 1974b English as a Formal Language. In Thomason 
1974 Selected Papers of Richard Montague. Yale University 
Press, New Haven, CT. 
Montague, R. 1974c Universal Grammar. In Thomason 1974 Selected 
Papers of Richard Montague. Yale University Press, New Haven, 
CT. 
Montague, R. 1974d The Proper Treatment of Quantification in 
Ordinary English. In Thomason 1974 Selected Papers of Richard 
Montague. Yale University Press, New Haven, CT. 
Partee, B. H. 1972 Opacity, Coreference, and Pronouns. In D. 
Davison, G. Harman (eds.) Semantics of Natural Language. 
Reidel, Dordrecht; 415--441. 
Quine, W. V. 1960 Word and Object. The MIT Press, Cambridge, MA. 
Quine, W. V. 1973 The Roots of Reference. Open Court. La Salle, IL. 
Strzalkowski, T. 1986. A Theory of Stratified Meaning Representa- 
tion. Doctoral dissertation, School of Computing Science, Simon 
Fraser University, Burnaby, B.C., Canada. 
Strzalkowski, T. 1986a An Approach to Non-Singular Terms in 
Discourse. In Proceedings of the l l th International Conference on 
Computational Linguistics (COLING-86). Bonn, West Germany. 
Strzalkowski, T. 1989 "A meaning representation for generic sen- 
tences." (unpublished manuscript). 
Strzalkowski, T. and Cercone, N. 1985 A Framework for Computing 
Extra-Sentential References. In Proceedings of the Theoretical 
Approaches to Natural Language Understanding. Halifax, Nova 
Scotia; 107-116. 
Strzalkowski, T. and Cercone, N. 1986 A Framework for Computing 
Extra-Sentential References. Computational Intelligence 2(4): 
159--180. 
Thomason, R. (ed.) 1974 Selected Papers of Richard Montague. Yale 
University Press, New Haven, CT. 
Vendler, Z. 1971 Singular Terms. In D. D. Steinberg, L. A. Jakobo- 
vits (eds.) Semantics. Cambridge University Press, Cambridge, 
England; 115-133. 
