An Approach to Non-Singular Terms in Discourse 
Tomek Strzalkowski 
School of Computing Science 
Simon Fraser University 
Burnaby. B.C.. CANADA 
V5A IS6 
Abstract 
A new Theory of Names and Descriptions that offers a uniform 
treatment for many types of non-singular concepts found in natural 
language discourse is presented. We introduce a layered model of the 
language denotational base (the universe) in which every world object 
Js assagned a layer (level) reflecting its relative singularity with respect 
tn other objects in the universe, We define the notion of relative 
singularity of world objects as an abstraction class of the layer- 
membership relation 
1. Introduction 
Linguistic (and related) literature describes numerous forms of 
non-singular concept~ that can be found in discourse including inten- 
sional (or functional) concepts, mass concepts, generic (or general) 
concepts attributive concepts, abstract concepts, etc. \[I\], \[2\]. 13\] \[4\] 16\] 
\[10\] Not all of these approaches could properly capture the 
distinction between singular and non-singular interpretation of linguis- 
tic descriptions, and some were originally devised to deal with singular 
terms only (such as Donnellan's attributive interpretation of definite 
descriptions \[2\]). With the exception of intensional concepts, these 
notions have not been given satisfactory formal representations that 
would account for their role in natural language discourse. Perhaps 
the most successful approach to non-singularity thus far has been 
presented by Montague \[4\] with his formalised concept of intension. 
Unfortunately, the concept of intension does not capture all aspects 
of non-singularity and the rigid translation system into intensional 
logic \[4\] seems to loose the smportant aspect of subjectivity in inter- 
preting natural language discourse. Also, the enormous complexity of 
any non trwial system of possible worlds proved to be a bar in 
developing a computationally-oriented application of Montague's 
theory 
In this paper we introduce a fragment of a new, and as we 
believe, computationally feasible Theory of Names and Descriptions 
that offers a uniform treatment for many types of non-singular con- 
cepts found in natural language discourse. Although we limit our 
presentation to nominal phrase constructions, the approach can be 
further extended to cover other types of phrases. In our theory we 
present the formalised definition of non-singularity with respect to a 
particular discourse situation involving a discourse message, a number 
of individuals (parties), and their knowledge, beliefs, awareness, etc, 
We introduce a layered model of reality (the universe) as perceived by 
a discourse participant, and define the notion of relative singularity of 
objects in this universe as an abstraction class of the layer- 
membership relation Subsequently, linguistic descriptions and names 
are classified as singular, measurably singular, or non-singular depend- 
ing on what they are assumed to denote in the universe The rela- 
tionship between objects addressed in discourse and classified into 
different layers (levels) of the universe 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, 
2. Non-singular terms in language 
Many philosophers and logicians, see \[1\]-\[4\], \[6\], \[10\]. appreciate 
that the u~age of the underlined nominal phrases in the following sen- 
tences has a "general" or "generic" character, except for "regular" 
singular interpretations which are possible only in some cases. 
E~ample j 
(~,d rh~ wears a crown. 
(lb) T~resident rs elected every four years. 
(lc) Gold is a yellow metal 
(ld) The temperature is a measure of molecular motion. 
:One can imagine hundreds of similar examples involving such non- 
singular objects as water, beat, the Pope, the number etc Unfor- 
tunately, there is no commonly accepted account of these species in 
362 
philosophical literature Some authors, see \[J\] and \[I0\], cautiously 
called them generic, or general (for example the king), or func 
tional (such as the number of students, ttre temperature) uses of 
(definite) descriptions. Others, like Kripke \[3\]. were quite close to 
consider them names (or at least some of them: heat gold). Yet 
others, see Quine \[6, 7\]. advocate the notion of abstract terms as 
being made of attributes, such as /being\] red (further abstracted as 
redness), or /being\] the man drinking tile martini (which cannot 
be so easily uominalized) which can predicate about "concrete" 
objects. 
There are numerous striking linguistic puzzles involving non- 
singular definite descriptions see \[I\]. \[4\], \[5\]. The following example 
illustrates the phenomenon 
Example2 
Consider the following inferences: 
(2a) The temperature is rising 
The temperature is ninety. 
so, Ninety is rising. 
(2b) The president met the Soviet leader many times 
The president is Reagan 
so. Reagan met the 5oviet leader many times 
(2c) The tiger lives in thejungle. 
My pet is a tiger. 
so My pet lives in the j ungle 
The conclusions in (2a) to (2c) are wrong in general case The expla- 
nation given by numerous researchers chiefly amounted to the corro- 
boration that the definite descriptions the temperature, the 
president and the tiger in the first sentences of (2a). (2b) and (2c) 
respectively should be interpreted functionally, i.e., as iutensions \[4\], 
or functions over situations \[1\] Observe that if the descriptions were 
to be interpreted singularly or as enumerating all instances of a non- 
singular object (i.e., statements containing them were understood as 
making claims about each instance), the reasoning would be sound. 
We claim that unless some two descriptions (or names) are used 
singularly or measurably singularly at the same level no simple 
,eference can be made between them. In fact. another type of refer- 
ence that we call remote reference can still take place and we shall 
put this view forward in this paper 
3. Tire Theory of Names and Descriptions 
Initially let us observe that our language deals with singular 
objects only, no matter how complex their structure happens to be 
Suppose somebody is being put into posihon of the Observer who 
perceives all these objects and has to use his language to describe 
them Some objects are sharply distinguished from others so he 
chooses to give them names as John, Mary, Fatsy. Sun, .. The 
others have no clearly perceivable boundaries but he still may name 
them: tea. water, grass, snow ..... and then refer to some measur- 
able quantities of them as some tea, little snow. etc, Yet others 
appear to be numerous, though enumerable, displaying strong similari- 
ties to one another. It would be pointless for Observer to give them 
each a name Instead. he decides to refer to them as a cow, the man. 
this tree etc Still. he prefers to say the sun or the lake rather than 
to invent new names if he is not sure how many of them are there, 
even if he is aware of just one specimen LaLer tie may find out that 
some objects were gwen identical names, so having encountered them 
together he must refer to one as the John. the Sun, or a Fatsy. 
Having completed his job Observer, who is also a part of this world, 
may name h~mself Observer or the Observer, and happily sit down 
under a tree on the grass 
Let us call the whole collection of objects he has just described 
as the Observer level and use the symbol L 0 for it Suppose then 
we ask Observer to tell us as much as he can about L 0 Soon he finds 
out that his naming has its limits As he discovers new facts about 
his world it becomes more and more cumbersome for him to com- 
municate in terms of every man. some cats. several trees, each 
president, etc He discovers that some things be originally considered 
distinct appear to be instances of some single object. Also he must 
admit that the identity of some other objects has to be put into ques- 
tion Being smart enough. Observer invents two new levels. L+1 and 
L_j. which augment his world. At level L+I he places the new objects 
he discovered to be generalisations (or abstractions, if you like) of 
some measurable amount of objects from L 0 which displayed a 
striking similarity or even identity. From the perspective of L+l he is 
able to tell us that Tire tiger lives in the .jungle. that The 
president is elected every four years, and that The Morning Star 
and The Evening Star are actually two appearances of tire plar~et 
Venus. The objects at L+I are singular there, bnt they appear gen- 
eric" or "functional" or whatever of thai sort as seen from L 0. 
Observe that these objects may not have straightforward measurably 
singular descriptions at L 0 (like every tiger, some president, etc). 
and often it will not be possible to refer to them in the terms of the 
language available at L(~. In either case one may expect that some 
undescribable aspect of an L+I object can emerge at L n. even if they 
all have been derived from L 0 (which does not have to be the case). 
Next Observer invents a new generation of names at L+l. the 
president and the tiber may be among those names. On the other 
hand Observer might prefer to use definite descriptions here. for the 
similar reason he frequently decided so at L\[). In fact. we have no 
means to distinguish between names and definite descriptions in 
discourse We (.an only stick to linguistic conveutions. 
It probably would not take a long time before a new augmenta- 
tion for L+( becomes necessary. Two new levels L+1+1 and L+I_ t carl 
be added in a much the same fashion The level L+~_ 1 does not neces- 
sarily have to be L0 although it probably will. More or less the same 
happens at the level /._1 where Observer can now say that what he 
previously considered to be the atom actually denotes marry different 
kinds of atoms (H. O Ca. Fe etc.), that tea is not so uniform and 
many different teas can be found, and that under the name Joe 
Smith was actually hidden a group of crime story writers. Subse- 
quently the level L-1 will expand by L_I+ ( and L-a-( with the former 
often different than L 0. Let us now formalize our intuition 
Definition :1 
A use of a description will be called singular if it denotes or refers to 
a singular object. A use of a description will be called measurably 
singular if it denotes or refers to some measurable quantity of a 
singular object. Otherwise we shall talk of non-singular use. 
Definition 2 
A level will be an arbitrary collection of singular objects. A level 
language will contain these and only singular and measurably singular 
uses of descriptions communicating of the level objects. 
Definition 3 
For any level L~. all names appearing in tbe L, language have singular 
interpretations. 
Definition 4 
For any level L n there will be at least two distinct levels L~ l and L,+ 1 
such that L,,+l contains the non-singular objects as seen from L,. and 
L,-t contains the objects for which the objects at L, are non-singular 
Definition 5 
Tbe Observer level L o is an arbitrary chosen level serving as a refer- 
ence point 
Suppose that we have an object N called N at level L 0. Let T 
be an arbitrary set we shall refer to as a coordinate. Suppose further 
that. for the coordinate T. the Observer discovers that the identity of 
N along that dimension can no longer be accepted. That is. there are 
at least two x. y E T such that N at x ~: N at y. Without losing gen- 
erality we can assume that the coordinate T has been chosen so that 
the following non-equation holds: 
• Vx.y ~ T.x~y,(N~) ~(Ny) 
Let (N x) denote an object N x for some x E T. The Observer cannot 
• place Nx's at L0 without violating definitions 2 and 3 Instead he 
moves them onto a new level L_Nt 7 leaving the original object N at L0. 
N may be no longer a "real" object but the concept remains in 
language LNt 7 can be attached to any existing level provided that the 
definitions 1 to 4 will never be violated It can also give a beginning to 
a new level. Note that the distribution of N over the coordinate T 
forces otber objects from L 0 to be distributed over T as well. and 
their instances placed at LN_I T This process may remain mostly impli- 
cit until we mm(e an utterance relating (N x) to other objects at LN_i T. 
In general we shall say that the level L_Ni T is lower than the level L 0, 
and write L~_lf<Lo Often we shall drop the superscripts N and T 
over the level symbol assuming some lower level L_~ whenever it does 
riot lead to ambiguity Observe that with the above account the level 
structure of objects has a dynamic, ever-changing character. Ally new 
empirical fact to be added to our world knowledge bears a potential 
reveberation in the level structure involving creation of new levels and 
moving objects between levels. At probably non-frequent idle states 
the definitions 1 to 4 assure the structure balance 
Moving at level L_ 1 the Observer is aware of an enumerable col- 
lection of different objects N,'s Extending the description used for N 
over N,'s the Observer refers 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 an Nx for some x (~ T. What that 
means is that we have wrongly placed N" at L 0. because it actually 
belonged to L_ l 13ut this was right at the time N" was placed there. 
ie. it mirrored tbe state of onr knowledge of the world at the time. 
We may now give names to some Nx's and N can very well happen 
among them. This time however N will not denote the old object from 
L0: this will be actually quite a different name referring to one 
selected N,. and which may be replaced by a definite description of (N~) 
On the other hand suppose we tlave some objects N l, N 2, 
considered distim:t at L 0 Suppose then that we discover some 
resemblance between them along some dimension (coordinate) T. so 
that we need a generalizing concept to talk about them. We climb to 
some higher level L~I 7-. i.e L0<LNI 7. and establish a new object a 
superobject N there Now as seen from LN+j T all N/s are just the 
occurences of N at L 0 at different values of coordinate T In other 
words, the following equation holds: 
• Vi Ix.x~ T,(Nx)=Ni 
Observe also that all Ni's now belong to the level N T L+l--1 which is a 
part of Lo As before we shall drop superscripts N and T for simpli- 
city. No matter how we name N at L+I the following Formula of 
Discovery summarizes our action: 
(FD) VxVy. x.y e T.(Nx) =(Ny) 
Remember that the formula FD is valid only when observed from L+I 
At L 0. Ni's remain distinct traditionally - so they remain distinct in 
the language as well. The generalisation of other objects from Lo 
onto LN+\] T may follow but. as in tbe case of decomposition discussed 
above, the process will remain largely implicit. Once the superobject 
N has been created it begins to live its own life. Some new objects 
from L 0 different than N/s. may now become instances of N at some 
not yet utilized values of coordinate T. Also. we may use descrip- 
tions (N x) without caring whether they actually refer to any objects 
at L 0. The latter property of general terms which is widely discussed 
by Qnine \[6 7\] gets a formal explanation in our theory. It is impor- 
tant not to confuse a superobject with a set S of lower level instances 
over some coordinate T as we would obtain a measurably singular 
concept only. Instead. a superobject can be identified with the func- 
tion N from Tinto L 0such that whenever s£Sthen there is a teT 
such that (N t)=s. and then extended arbitrarily beyond the set S 
Exam Die 3 
We have the following distinct object at some level L0: V called 
Venus. MS called Morning Star. and ES called Evening Star. Upon 
discovery that they all are just occnrences of the same planet we 
create a new object V" named Venus at some level LVl "T and such 
that for some ;~.y.z ¢ T. where T is a time coordinate. 
(V';~) = V (V" v) =MS.(V'z) = ES. According to the FD for- 
mula we conclude from L+1 that V=MS=ES. while the same conclu- 
sion made at L n is false 
Examp~ 
At level Lo the OI)server is aware of the object TP named The 
President. Let T be the time coordinate (different than in the last 
363 
example). At L 0we have according to the FD formula that 
° VxVy. x,y E T (TPx) =(TPy) 
Later the Observer may dicover that for some tl. t2 E T, (TP tl)=N 
and (TP t2)=R, aud that at some level LT_~ 'T where N and R belong, 
they are considered distinct and named Nixon and Reagan respec- 
tively. But at L o. R=N is true. The last observation can be made 
clearer if one imagines that TP is some abstract individual which (like 
Venus) when observed in early 70's is named Nixon, while when 
observed in 80's is called Reagan, \[\] 
Definition 6 
An object N at a level L, is said to be remotely referenced if the 
reference comes from some level Lnj such that either L,<L~ or 
Lm< L, 
Typical cases of remote references in discourse have been listed 
in Example 2 
4. Superobjects 
Let us now examine the nature of superobjects i.e.. the objects 
placed at level L+l. It turns out that the plural terms, e.g. 
presidents, tigers, etc., are actually prototypes of superobJects, see 
\[6\]. and they should therefore be placed at the same level as 
respective superobjects. We will see that the generalization leads 
naturally to plural terms which may or may not induce equivalent 
singular superobjects. Conversely. a plural equivalent to a superobject 
may suggest the most natural coordinate to decompose the latter 
onto some lower level. When a superobject lacks a plural equivalent, 
however, we may admit that this object's origin has been traced 
down. A further decomposition is still possible but this process may 
often produce objects that will never assume an independent status 
and will remain recognized only as instances of- this general concept 
scattered over that or another coordinate This phenomenon is 
characteristic of the so-called mass objects and their corresponding 
mass terms Quite naturally the question of- where one level ends 
and another begins arises The following example gives some insight 
into the problem of level boundaries. 
Exampl£5 
Consider the following sentences. 
(Sa) Mary brings (some) water every day. 
Let water in (5a) be the name of some superobject w at the 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+I ) 
(i) 5a --, (br-e-d M w) 
where br-e-d stands for brings every day. 
'On the other hand. suppose that Mary brings some water every day. 
Except for the above interpretation, we also have the measurably 
singular reading at L0 where w is scattered over some coordinate T 
so that ~tE T such that (W t) is being brought by Mary. i.e.. 
(~t (br M (w t))). This clause is, of course, relative to every day so 
at L 0 we could have 
(ii) 5a -~ (Vx(dx) D (~\]t(brM(wt))))t 
where brines -~ hr. day ~d 
Both translations are essentially equivalent, and this equivalence is by 
no means accidental. It lends a strong support for our Theory of 
Names and Descriptions. and explains the intuition underlying its for- 
mulation. 
5. Conclusion 
In this paper we presented a new approach to representing vari 
ous kinds of non-singular concepts in natural language as the Theory 
of Names and Descriptions. -I-he major observation of the Theory is 
that 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 which are considered relatively singular Observe 
t To be precise we should represent Mary as (M l) here. i.e.. as an instance of 
the L 41 object M at some I E T. However. our nanling convention discussed in section 
3 allows for replacing lhe definite description by a new name at the level L 0. We shall 
utilize this option here. 
364 
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 superobjects, are placed at a number of higher 
levels which are related to the current level with various coordinates 
Conversely. a singular object may be decomposed 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 singularity of objects within 
each level is maintained For more details concerning various aspects 
of tile theory the reader is referred to \[8\] and \[g\] 
Acknowledgements 
Tile author would like to thank Dr. Nick Cercone and Dr. Jim 
Delgrande for their comments and suggestions that helped to intprove 
the quality of this paper This research was supported in part by the 
Natural Science and Engineering Research Council of Canada under 
Operating Grant number A4309, by the Office of the Academic Vice- 
President Simon Fraser University, and by the SFU's Open Graduate 
Scholarship Thank the LCCR for use of facilities, 

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