﻿Roles, Co-Descriptors, and the 
Formal Representation of Quantified English Expressions 
William A. Martin 1 
Laboratory for Computer Science 
Massachusetts Institute of Technology 
Cambridge, Massachusetts 02139 
A scheme is proposed for representing the logical form of English sentences, wherein 
the meaning of a network node depends on how it is related through "role-in" links to 
nodes representing more aggregate entities. It is argued that roles are a natural device for 
capturing many linguistic and philosophic distinctions, and that they are convenient for 
computational processing. In particular, it is shown how role-in links may be advantageous- 
ly used in lieu of quantifier scope to represent quantificational dependencies. 
1. Introduction 
In representing the semantics of English sentences, 
it is traditional to distinguish "logical form" from se- 
mantic content. Chomsky \[2\], for instance, introduces 
LF (his version of logical form) as a linguistic level of 
representation between syntax (phrase structure) and 
semantics. 
Representations of logical form, which are typically 
based on predicate calculus or lambda calculus, must 
be carefully chosen for empirical adequacy as well as 
computational convenience. There are at least three 
general ways in which predicate or lambda calculus 
might be carried over into representations of logical 
form: 
(la) directly (as by a theorem prover); 
(lb) by replacing quantification with dynami- 
cally scoped iteration procedures (as in 
Woods \[30\]); or 
1 This paper was substantially revised after Professor Martin's 
death by a group of his colleagues to satisfy length constraints and 
to take advantage of the reviewers' extensive and thoughtful com- 
ments. We wish to thank the editor and reviewers for their cooper- 
ation under these unfortunate circumstances. Prof. Martin's origi- 
nal acknowledgment follows. 
Ken Church, Lowell Hawkinson, Mitchell Marcus, Peter Szolo- 
vits, and Lucia Vaina read an earlier version of this manuscript and 
made many helpful comments. Ellen Lewis and Anne Schmitt did 
an excellent job of preparing the manuscript and figures. This 
research was supported by the Defense Advance Research Projects 
Agency and monitored by the Office of Naval Research under 
contracts N00014-75-C-0661 and N00014-80-C-0505. 
(lc) by translating into a semantic network 
formalism (as in Hendrix \[10\]). 
The second and third are more process-oriented; that 
is, they more closely relate elements of the representa- 
tion and states in the interpretation process. In my 
view, a process orientation is needed if issues of effi- 
ciency are to be discussed at all. 2 One key to efficien- 
cy is the delaying of decisions as long as possible. 
This is easier to achieve when differences between 
alternative interpretations are minimized. 
The two process-oriented approaches noted above 
differ in their treatment of scope dependencies. For 
example, we can model "each bottle" in 
(2) Each cork is fastened to each bottle by a 
small wire basket. 
with a quantifier like (¥x: bottle), and then translate 
that into an explicit FOR loop clause. Alternatively, 
we can employ Skolem functions, with "wire" and 
"basket" modeled as functions of a variable ranging 
over "bottle". These functions and variables can then 
be represented in a semantic network, along the lines 
discussed in "What's in a Link?" \[29\]. 
I choose to adopt a particular semantic network 
approach wherein it is more appropriate to view mean- 
ings of nodes decompositionally (i.e., as dependent 
upon what they are constituents of) rather than com- 
positionally (i.e., as fully determined by the meanings 
2 It is unclear how to pose efficiency questions in a non- 
process-oriented framework, where even effectiveness may be 
rejected as a desideratum. 
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American Journal of Computational Linguistics, Volume 7, Number 3, July-September 1981 137 
William A. Martin Roles, Co-Descriptors, and the Formal Representation of Quantified English Expressions 
of their constituents). More specifically, the meaning 
of a network node depends (decompositionally) on 
how it is related though "role-in links ''3 to nodes rep- 
resenting more aggregate entities. The decomposition- 
al view is preferable for explaining a number of lin- 
guistic phenomena. I will show, in this paper, that 
many important linguistic and philosophical distinc- 
tions can be analyzed as differences in role-in linkag- 
es. Among these are the intensional/extensional, 
referential/attributive, and distributive/collective dis- 
tinctions. 
My approach turns out to be more complex than 
some, but I will argue that it captures subtle cases in a 
more natural way and that it is superior for computa- 
tional purposes. Indeed it has been my experience (as 
in the development of MACSYMA \[13\]) that a repre- 
sentational formalism with more constructs often per- 
mits more efficient processing than a formalism based 
on a smaller set of more primitive operators. I suspect 
that this experience will strike home with many read- 
ers familiar with the difficulties encountered by 
predicate-calculus-based theorem provers. 
t~1 generic node 
• individual node 
may-be-described -as 
role-in 
inherits-from 
Figure la. Notation used in networks. 
2. Roles 
Figures la-lc illustrate the sort of semantic net- 
work I use. (In these and subsequent figures, only 
those nodes and links are shown that are relevant to 
the issues under discussion. By convention, links with- 
out arrowheads are to be taken as directed upwards.) 
The networks shown in this figure have two kinds 
of nodes, generic and individual, and three kinds of 
links, inherits-from, may-be-described-as, and role-in. 
Generic nodes (e.g. ARCH) represent generic objects; 
individual nodes (e.g. ARCH-1 and ARCH-2) repre- 
sent instances of generic objects. Inherits-from links 
are used to say that the individuals ARCH-1 and 
ARCH-2 inherit properties from the generic ARCH. 
May-be-described-as links are used to indicate that an 
ARCH is a STRUCTURE. 4 Finally, role-in links are 
used to indicate that R1 and R2 are parts of the gen- 
eric ARCH. The "meaning" of a node depends prima- 
rily upon role-in links that connect it to nodes repre- 
senting more aggregate entities. 
My representation is similar to several in the litera- 
ture. Generic nodes correspond to Minsky frames \[16\] 
and roles (parts of generic objects) correspond to slots 
(parts of frames). One can also draw analogies with 
set theory. May-be-described-as links and inherits- 
from are similar to set inclusion and set membership. 
3 Role-in links could be viewed logically as Skolem dependen- 
cies. 
4 May-be-described-as and inherits-from are similar to "is-a" 
links. 
Figure lb. Simplified structural description of an arch. 
ARCH-1 
T STRUCTURE 
I~, ARCH 
R2 ARC,-2 
\ / 
\ # t / \ I t / 
# ~, / 
• R2-1 g R2-2 
Figure lc. An example of individual nodes. 
That is, 
(3a) ARCH may-be-described-as STRUCTURE 
(3b) ARCH-1 inherits-from ARCH 
are analogous to 
(4a) the set of all ARCHes is a subset of the 
set of all STRUCTUREs 
(4b) ARCH-1 is a member of the set of all 
ARCHes 
However, there is an important difference in the se- 
mantic interpretation: may-be-described-as links are 
interpreted as relating intensions expressed by generic 
nodes, whereas subset relations operate on sets of 
138 American Journal of Computational Linguistics, Volume 7, Number 3, July-September 1981 
William A. Martin Roles, Co-Descriptors, and the Formal Representation of Quantified English Expressions 
individuals. I consider the intension/extension distinc- 
tion extremely important, because intensional repre- 
sentations appear to be more natural for certain sen- 
tenees. 5 
My representation does not deal with contextual 
problems by assuming a set of objects called contexts 
(or partitions) and then somehow stipulating a context 
link for each node. Instead, I argue that roles allow a 
more natural modeling of the context phenomena. For 
example, Figure 2 shows how Hendrix \[10\] represents 
the fact that legal persons can own physical objects. 
Rectangles (and squares) represent partitions 
(contexts). An individual or set is in a partition if its 
representing circle is in the rectangle representing that 
partition. In this figure, the individual I is an implica- 
tion that any individual owning X has an individual 
agent Y, which is a legal person; an individual object 
Z, which is a physical object; an individual start-time 
t 1, which is a time; and an individual end-time t 2, 
which is also a time. 
This same information is shown in Figure 3, using 
the notation presented here. Figure 3 eliminates the 
individuals I and X, the implication node, the context 
surrounding I, and the context surrounding Y, Z, t 1, 
and t 2. Some of these are replaced in Figure 3 with a 
richer variety of node types; I believe these types have 
better linguistic and philosophical motivations than 
Hendrix's abstract individuals, contexts, and implica- 
tions. 
3. Reference and Definite Descriptions 
Russell's \[22\] analysis of a singular definite descrip- 
tion required the existence of a unique object satisfy- 
ing the description in order for the expression to de- 
note anything, and hence it fails to account for the 
successful reference of a noun phrase like "the clock" 
in 
(5) Did you wind the clock? 
Strawson \[25\] views these definite descriptions in 
relativistic terms: even when the referent cannot be 
determined in an absolute sense, as a particular clock 
for example, the description still has some meaning 
relative to whatever the context turns out to be. 
(6) When shall we say that a hearer knows what 
particular is being referred to by a speaker? 
Consider first the following case. A speaker 
tells a story which he claims to be factual. It 
begins, "A man and a boy were standing by 
a fountain," and it continues: "The man had 
a drink". Shall we say that the hearer knows 
5 1 choose to avoid the question of formal adequacy because 
almost all systems, intensional as well as extensional, are formally 
adequate in the sense that they are capable of representing all 
(effective) functions. A much more interesting question is whether 
they are capable of representing all functions in a natural way. 
which or what particular is being referred to 
by the subject expression in the second sen- 
tence? We might say so. For, of a certain 
_ range of two particulars the words "the man" 
serve to distinguish the one being referred to, 
by means of a description which applies only 
to him. But though this is, in a weak sense, 
a case of identification, I shall call it only a 
story-relative or, for short, a relative identifi- 
cation. For it is identification only relative 
to a range of particulars (a range of two 
members) which is itself identified only as 
the range of particulars being talked about 
by the speaker. That is to say, the hearer, 
hearing the second sentence, knows which 
particular creature is being referred to of the 
two particular creatures being talked about by 
the speaker; but he does not, without this 
qualification, know what particular creature 
is being referred to. The identification is 
within a certain story told by a certain 
speaker. It is identification within his story 
but not identification within history. 
Following this line of thought, sentence (5) might 
be represented as follows, where clock-1 describes a 
particular clock and the-clock-1 describes the clock 
relative to the discourse. 
(7) r ............. > ............... , clot:k-1 room-1 house-1 t 
;~ discourse-1 real-world 
I 
I 
I 
I 
I the-clock-1 
I ....... 
Clock and Clock-1 are co-descriptors because they 
both describe the same individual. (The term co- 
description is chosen over co-reference or co- 
denotation to avoid implications of intensionality and 
existence.) 
Co-description is appropriate in (7), but surely not 
every occurrence of the-clock will refer to clock-1. 
For example, Webber \[28\] points out that in 
(8a) Wendy bought a yellow T-shirt that Bruce 
had liked. 
(8b) It cost twenty dollars. 
an appropriate description of the entity referred to by 
"it" is not "the yellow T-shirt that Bruce had liked," 
since (8a) is true even if Bruce had liked several T- 
shirts (and both the speaker and the listener were 
aware of the fact). Nor is it "the yellow T-shirt that 
Bruce had liked and Wendy bought," since (8b) can 
be true even if Wendy had bought several such T- 
shirts. An appropriate description is something like 
"the yellow T-shirt that Bruce had liked and that 
American Journal of Computational Linguistics, Volume 7, Number 3, July-September 1981 139 
William A. Martin Roles, Co-Descriptors, and the Formal Representation of Quantified English Expressions 
OWNINGS 
delineation 
IMPLICATIONS 
TIMES 
Figure 2. The delineation theorem of ownings (Hendrix). 
F-- -- --'L~IE ............................. 
I 
I r-~ ..................... o 
I I r '~ .............. I i I. agt 
W- - -< - - --0 l oobj 
process I I 
I I I legal-person I physobj Z 'I 
! ! 
I ! 
! ! owning ~/ 
~-- w Z 
i~',~ 
I ~, ~ ~ .............. I 
I 
startl~ime • 
end-time 
time 
I I 
I I 
i ~ - ,,,C ..................... i a 
I I 
Figure 3. The information of Figure 2 in the notation of this paper. 
140 American Journal of Computational Linguistics, Volume 7, Number 3, July-September 1981 
William A. Martin Roles, Co-Descriptors, and the Formal Representation of Quantified English Expressions 
Wendy bought and that was mentioned in (8b)". Ac- 
cordingly, (7) needs to be amended so that the-clock-1 
is relative to the sentence "did you wind the clock". 
(9) r- .............. > ............... j clock-1 room-1 house.1 = 
~1 discourse. 1 real-world 
did-you-win6\[ 
the-clock-1 I the-clock-1 
e. 
3.1 Referential and Attributive Use of 
Definite Descriptions 
Donnellan \[3\] has suggested that definite descrip- 
tions like "the clock" have two uses: referential and 
attributive. Attributive use asserts something about 
"whoever or whatever is the so-and-so"; referential 
use is intended to allow the hearer to identify at once, 
in his own terms, who or what is being talked about. 
(10) I will call the two uses of definite descrip- 
tions I have in mind the attributive use and 
the referential use. A speaker who uses a 
definite description attributively in an asser- 
tion states something about whoever or 
whatever is the so-and-so. A speaker who 
uses a definite description referentially in an 
assertion, on the other hand, uses the de- 
scription to enable his audience to pick out 
whom or what he is talking about and states 
something about that person or thing. In the 
first case the definite description might be 
said to occur essentially, for the speaker 
wishes to assert something about whatever or 
whoever fits that description; but in the ref- 
erential use the definite description is merely 
one tool for doing a certain job-- calling at- 
tention to a person or thing-- and in general 
any other device for doing the same job, an- 
other description or a name, would do as 
well. In the attributive use, the attribute of 
being the so-and-so is all important, while it 
is not in the referential use. 
To illustrate this distinction, in the case of 
a single sentence, consider the sentence, 
"Smith's murderer is insane". Suppose first 
that we come upon poor Smith foully mur- 
dered. From the brutal manner of the killing 
and the fact that Smith was the most lovable 
person in the world, we might exclaim, 
"Smith's murderer is insane". I will assume, 
to make it a simpler case, that in a quite or- 
dinary sense we do not know who murdered 
Smith (though this is not in the end essential 
to the case). This I shall say, is an attrib- 
utive use of the definite description. 
The contrast with such a use of the sen- 
tence is one of those situations in which we 
expect and intend our audience to realize 
whom we have in mind when we speak of 
Smith's murderer and, most importantly, to 
know that it is this person about whom we 
are going to say something. 
For example, suppose that Jones has been 
charged with Smith's murder and has been 
placed on trial. Imagine that there is a dis- 
cussion of Jones's odd behavior at his trial. 
We might sum up our impression of his be- 
havior by saying, "Smith's murderer is in- 
sane". If someone asks to whom we are ref- 
erring, by using this description, the answer 
here is "Jones". This, I shall say, is a refer- 
ential use of the definite description. 
The distinction is brought out nicely by the follow- 
ing pair of sentences from Moore \[17\] and their corre- 
sponding semantic net representations (Figures 4 and 
5). 
(1 la) The President has been married 
since 1945. referential 
(llb) The President has lived in the 
White House since 1800. attributive 
Sentence (lla) refers to the person who is currently 
President, while sentence (1 lb) refers to whoever was 
president at each point in time since 1800. The truth 
of (llb) is not dependent on who is filling the role of 
President. In fact, we can go even farther. In the 
past the President has been killed and it has taken a 
while to swear in the new President. During this time 
there is no President, yet (llb) is still true. In (lla), 
by contrast, the referent of "the President", president- 
1, is not a role in "the President is married", as shown 
in Figure 4. Therefore, we are not at liberty to have 
the referent change with time.6 
The referential/attributive distinction might have 
been ascribed to differences of scope in quantified 
expressions. 
(12a) (the p : president) (¥ t since 1945) p has 
been married at time t. referential 
(12b) (V t after 1800) (the p : president) p has 
lived in the white house at time t. attributive 
(13a) ((~ (p) (true-at-time (married p) 
(every (t) (after t 1945)))) 
(the (y) (pres y))) referential 
(13b) (true-at-time (live-in (the (y) 
(pres y)) white-house) 
(every (t) (after t 1800))) attributive 
6 A method for the sequential binding of quantified variables 
in determining the truth of an expression was introduced by Hintik- 
ka \[11\]. Although he was working with predicate calculus, he 
obviously had in mind the same general strategy proposed here. 
American Journal of Computational Linguistics, Volume 7, Number 3, July-September 1981 141 
William A. Martin Roles, Co-Descriptors, and the Formal Representation of Quantified English Expressions 
president-1 
the-president-1 
7\ real-world 
\ 
\ 
• discourse-1 
the-president.has-been- 
married-since-1945-1 
/0\ the-president-is-married- 1 
/ \ 
/ \ 
~ subject-of-'is-married-l' 
Figure 4. Referential use of the president. 
0~ real-world 
\ 
\ 
\ 
\ 
• discourse-1 
the-president-has-been. 
married-since-1945-1 
the-president-1 
/0\ the-president-is-married- 1 
/ \ 
/ \ 
/ \ 
/ \ 
11~::::. ~ subject-of-'is-married-l' 
Figure 5. Attributive use of the president. 
My representation, in contrast, makes it easy for the 
interpreter to delay the decision because it minimizes 
the difference between the two cases. It is also possi- 
ble for the interpreter never to decide which reading is 
intended. For example, in a sentence such as, "The 
president has owned a terrier since February," the 
listener will not be able to distinguish between the two 
readings. If a later sentence requires the listener to 
make a distinction, only a minimal alteration to the 
representation is required. Partee \[19\] makes an argu- 
ment that further supports this approach. 
(14) ... having a particular individual in mind 
(the "referential" case) and knowing nothing 
about an individual other than some descrip- 
tive phrase (the "attributive" case) may be 
just two extremes on a continuum of 
"vividness". One may consider, for instance, 
the case of a detective tracking down a crim- 
inal and obtaining more and more clues, in- 
cluding fingerprints, voice recordings, photo- 
graphs of varying clarity, etc. It is not at all 
clear at what point the detective, who may 
be described as "looking for the man who 
did so-and-so" stops looking for "whoever it 
is that did so-and-so" and starts looking for 
a particular individual. 
3.2 Subject Co-Descriptors (Verb Phrase Deletion) 
Both Figures 4 and 5 employ co-description be- 
tween the-President and subject-of-x, where x is the 
sentential predicate. Sentences (15a) and (15b) from 
Sag \[23\] offer further support for this convention. 
(15a) The chickens are ready to eat, and the 
children are too. 
(15b) John likes flying planes, and Bill does too. 
In these examples, the subject plays the same role in 
both conjuncts. That is, if the chickens are ready to 
be eaten, then the children are also ready to be eaten, 
and if the chickens are ready to do the eating, then the 
children are ready to do the eating. However, (15a) 
cannot mean that the chickens are ready to be eaten 
and the children are ready to do the eating. Sentence 
(15b) illustrates the same point. It is ambiguous 
whether John likes to fly planes himself or whether he 
would prefer someone else do the flying, but whichev- 
er way he likes the flying done, Bill will want it done 
the same way. 
The following sentence from Partee \[20\] has the 
same sort of ambiguity as (15a) and (15b) above; I 
will use it to illustrate how such ambiguities are dealt 
with in my scheme of representation. 
(16) The prosecutor believed that he would 
win the case, and so did the defense 
attorney. 
The missing verb phrase can be understood in two 
ways, either that the defense attorney would win or 
that the prosecutor would win. This can be explained 
in terms of my representation, as shown in (17a) and 
(17b) below. Note that he-1 is co-descriptive with the 
subject role in (17a) and with the-prosecutor-1 in 
(17b). Thus (17a) represents the sloppy attributive 
interpretation where he-1 refers to whatever the sub- 
ject happens to be, whereas (17b) represents the non- 
sloppy referential interpretation where he~l refers to 
the-prosecutor-1 in both conjuncts. 
142 American Journal of Computational Linguistics, Volume 7, Number 3, July-September 1981 
William A. Martin Roles, Co-Descriptors, and the Formal Representation of Quantified English Expressions 
(17a) b\ discourse- 1 sloppy 
/ % 
/ \ 
prosecutoy-~ \ the-prosecutor-believed- 
t ~ that-he-would.win-1 I I 
/ I the-pro cutor-1 
t _~~ 
subject-of-'the-prosecutor. 
J believed-that-he-would.win'-1 
II he-1 
(17b) 
/ 
prosecutoy-~ 
the-prolcutor-1 
~/he-1 
t x discou rse-1 nonsloppy 
\ 
\ the-prosecutor-believed- 
~ that-he-would-win.1 
/ I 
/ I 
/ I 
/ ~ ~ subject-of-'the-prosecutor. 
/ believed-that-he-would-win'-1 / 
3.3 Restrictive and Non-Restrictive Modifiers 
Another opposition which can be captured with the 
attributive/referential distinction is that between re- 
strictive and non-restrictive modifiers. 
(18a) My uncle, who is 70, is bald. non-restrictive 
(18b) My uncle who,is 70 is bald. restrictive 
Sentence (18a) has a non-restrictive relative clause, 
"who is 70", modifying, "my uncle". That is, "who is 
70" is not used to pick out the uncle who is bald, but 
just to give extra information about him. This sen- 
tence is equivalent to the conjunction, "My uncle is 
bald & my uncle is 70." By contrast, in the restrictive 
reading (18b), "who is 70" picks out a particular un- 
cle. The distinction between (18a) and (18b) can be 
captured as follows. 
( 19 a ) n on- restrictive 
/IlK discourse-1 
/ t I %.~ 
I I I ~ 
/ i my-uncle- 
my-uncle-1 subject-of my-uncle- 
is-bald-1 
" Q ,, 'who-is-70'-1 
I 
I % % 
who-1 subject-of- 'who-is-70'-1 
(19b) ..~ discourse.1 restrictive 
/ I 1 
t I 
/ I 
t I uncle-1 / 
my-uncle- 
who-is-70-1 
my-uncle-1 
/ 
/ 
J S' 
/ 
my-uncle- 
J ~ is-bald-1 
I 
I i subject-of 
my-uncle- is.bald-1 
% 
,.Q who-is-70-1 
/ 
/ 
J 
/ 
t 
who-1 subject-of- 
'who-is-70'-1 
Note that the head of the relative clause must be 
either attributive or referential for both main and rela- 
tive clauses. It is not possible to have (20), for exam- 
ple. This argues that the relative pronoun "who" 
should be treated as a co-descriptor of a sentence role 
of the main clause, not a discourse role. 
(20) *The President, who has been married 
since 1945, has lived in the White House 
since 1800. 
4. Discourse Iteration 
We have seen that explicit nesting of structural 
descriptions can explain the referential/attributive 
distinction, sloppy binding, and restrictive/descriptive 
relative clauses. This section will show that quantifier 
interpretation can also be explained in a similar man- 
ner. Consider sentence (21), which is three ways 
ambiguous, as shown in (22a)-(22c). 
(21) Every boy wants a lion. 
(22a) They will take any lion. 
O\ discoursed 
\ 
\ 
\ 
\ every-boy-wants- 
a-tion-1 
1 
I 
I 
1 • a-lion-1 
American Journal of Computational Linguistics, Volume 7, Number 3, July-September 1981 143 
William A. Martin Roles, Co-Descriptors, and the Formal Representation of Quantified English Expressions 
(22b) 
iteration- 1 
lion-1 
/ 
I 
I 
I 
I 
Each wants his own specific lion. 
/O\ discourse-1 
\ 
N 
\ 
N 
a-lion.1 
every- boy-wants- 
I~ a-lion-1 t 
I I 
I object-of- 
every- boy-wants- 
a-lion-1 
(22c) 
/ 
/ 
lion-1 ~ 
They all want the same specific lion. 
~\ discourse-1 
/ 
/ \ 
/ X 
\ every-boy-wants- / II 
/ ~ a-tion-1 
/ I 
/ I 
I 
i object-of- 
a-lion-1 every-boy-wants- 
a.lion-1 
The three readings (22a)-(22c) can be interpreted as 
follows. In each case, want-a-lion must be true for 
each boy. In (22a), a-lion-1 is attributive: it can be 
picked as needed for each wanting. In (22b), a-lion-1 
is referential, but it refers to a role in an iteration of 
wanting at the discourse level. (Not surprisingly, giv- 
en my account, this is a difficult reading for people to 
construct.) Finally, (22c) gives the by-now-familiar 
referential case where every boy is constrained to 
wanting the same lion. 7 Let us now turn to a discus- 
sion of the discourse iteration construct used in (22b). 
(23a) (FOR EVERY X / CLASS : (P X) ; (Q X)) 
Every X in CLASS that satisfies P 
also satisfies Q. 
(23b) (FOR SOME X / CLASS : (P X) ; (Q X)) 
Some X in CLASS that satisfies P 
also satisfies Q. 
(23c) (FOR GEN X / CLASS : (P X) ; (Q X)) 
A generic X in CLASS that satisfies P 
also satisfies Q. 
(23d) (FOR THE X / CLASS : (P X) ; (Q X)) 
The single X in CLASS that satisfies P 
also satisfies Q. 
The FOR statement applies the filter P(x) in turn to 
each element in the CLASS, and then applies Q(x) to 
those elements that pass the filter. The key words 
EVERY, SOME, GEN, THE, and so on, specify a 
particular enumeration function. 
I prefer to represent quantified phrases in terms of 
role-in links (interpretable as Skolem dependencies) 
rather than iteration procedures, and to use predicates 
on intensions and sets in addition to predicates on 
individuals. By converting Woods' FOR construct to 
my notation, the difference between the two referen- 
tial readings of "every boy wants a lion" can be 
spelled out in more detail as shown in (24a) and (24b) 
below. (Note that (24a) is a slight elaboration of 
(22b), and that (24b) is identical to (22c)). (24a) 
captures both the EVERY and GENeric options. In 
both cases, lions are individuated by the boys who 
want them. The individuating is represented by the 
role-in link (Skolem modification) from lion-1 to 
want-boy-lion- 1. 
(24a) Every boy wants his own lion. 
4.1 Procedural Representation of Knowledge 
By procedural representation of knowledge, I mean 
knowledge in the form of procedures. For example, 
the knowledge that every boy wants a lion could be 
cast in the form "if one were to check every boy in 
question and count those who want lions, then the 
count of those who want lions would be equal to the 
count of the boys checked". Woods \[30\] introduced a 
FOR iteration construct for representing knowledge of 
quantified propositions procedurally. 8 Examples of the 
use of this construct are: 
7 Partee \[19\] points out that this type of sentence and those 
of the previous section pose a problem for the analysis of the indef- 
inite article in terms of the features +specific \[7\]. Something more 
is needed to produce all the readings. Features can be used as 
follows: if a hearer has decided a node is definitely attributive it 
can be marked +attributive, if it is referential the co-descriptor is 
shown. Otherwise, it is unmarked and has no co-descriptor. 
/ 
FOR-EVERY-BOY / / / 
BOYS:T; / g 
WANT-BOY-LION-1 \ \ 
II boy 
/ 
/ 
/ 
want.boy-lion-1 
I 
I 
I 
I 
lion-1 v-...~ a-lion-1 
/0\ discourse-1 
\ 
\ 
\ every.boy-wants- 
a-lion-1 
object-of- 
every-boy-wants- 
a-lion-1 
8 Readers familiar with iteration macros and programming 
languages will see that Woods' FOR is quite limited, e.g. iteration is 
limited to a single variable. I use it here because it is simple, is well 
explained by Woods, and will facilitate comparison of my sugges- 
tions with Woods' earlier work. 
144 American Journal of Computational Linguistics, Volume 7, Number 3, July-September 1981 
William A. Martin Roles, Co-Descriptors, and the Formal Representation of Quantified English Expressions 
(24b) Every boy wants the same specific lion. 
~\ discourse-1 
/ X 
I N 
I % 
/ 
/ 
I 
I 
\ every-boy-wants- @ 
t a-lion-1 
I 
I I 
/ I 
/ i object-of- 
lion-1 ~ ~ a.lion- 1 every-boy-wants- 
a-lion-1 
In contrast to (24a), (24b) contains no iteration at all. 
In this case, all the boys want the same lion. Since 
lions are not individuated with respect to boys, there is 
no motivation to introduce an iteration (or Skolem 
dePendency). On this reading, "every boy wants a 
lion" can be conceptualized just like "a boy wants a 
lion". 
4.2 Collectives, Distributives, and Pluralities 
In applying predicates to pluralities, we must be 
careful not to confuse a predicate on a plurality with a 
predicate on a collective. If a process, like wanting, is 
done by a plurality, then a plurality of wantings oc- 
curs. This contrasts with a collective, which acts as a 
single unit. Fauconnier \[5\] demonstrates this distinc- 
tion with (25). 
(25a) The men {gathered, united, quarreled}, collective 
(25b) The men took off their hats. distributive 
(25c) The men carried the couch, plurality 
As a further example, observe that in 
(26) Everybody gave $1000 to many of the men. 
we must decide whether the men receiving money 
receive $1000 apiece or just participate in a group 
receiving $1000. We must also decide whether 
"everybody" acted individually or collectively. Note 
that one cannot personally receive $1000 if $1000 is 
given to a group he is in, but one can be personally 
told the news if the news is told to a group he is in. 
From this we see that the distributive reading may or 
may not require a separate instance of some state or 
process for each individual. 
A plural noun group may be understood in any of 
the three ways. Vendler \[27\] gives examples intended 
to show that the choice between "each" and "every" 
influences our preferences. 
(27) Suppose I show you a basket of apples and I 
tell you, "Take all of them". If you started 
to pick them one by one I should be sur- 
prised. My offer was sweeping: you should 
take the apples, if possible, "en bloc". Had I 
said, "Take every one of them," I should not 
care how you took them, provided you do 
not leave any behind. If I say, "Take each of 
them," one feels that the sentence is unfin- 
ished. Something like, "Take each of them 
and examine them in turn," is expected. 
Thus I expect you to take them one after the 
other not missing any. 
"All" favors collective interpretation, but permits dis- 
tributive interpretation. "Every" favors distributive 
interpretation. "Each" strongly favors distributive 
interpretation and favors distinct actions for each indi- 
vidual. 
4.3 Multiple Iterations 
The previous sections suggested that an iteration 
need be constrUcted only when two descriptions in the 
sentence are related in a special way, e.g., where one 
is individuated by the other. Otherwise, predicates on 
collectives or pluralities provide an adequate descrip- 
tion. In this way, I avoid iteration, which "is relatively 
expensive from a computational perspective. There 
are other ways to eliminate iterations. For example, 
the apparent double iteration in sentence (28a) can be 
reduced to a single iteration. Formula (28b) repre- 
sents the natural reading where each cork is associated 
with one bottle. It might appear that it requires two 
nested FOR loops as in (28c), but in fact, it can be 
implemented with a single loop (28d). 
(28a) Each cork is fastened to each bottle by a 
small wire basket. 
(28b) (¥x) (Vy) if x is a cork & y is a bottle 
& x is the cork in bottle y 
then x is fastened to y by a small 
wire basket. 
(28c) Forall x in corks do two nested loops 
Forall y in bottles 
if x is the cork in bottle y then 
x is fastened to y by a small wire basket. 
(28d) Forall y in bottles do single loop 
let x = the cork in bottle y 
x is fastened to y by a small wire basket. 
Most people understand this sentence by iterating over 
bottles and individuating the corks by the bottles. The 
single loop implementation can be derived straightfor- 
wardly from my representation. In some other repre- 
sentation, it may require a very clever compiler optimi- 
zation, such as "loop jamming". 
Sentences (29a) and (29b) are similar to (28a) in 
this respect. 
(29a) All the boys kissed all the girls. 
(29b) Each man and each woman will be joined 
in marriage here tonight. 
These sentences provide evidence that a quantifier like 
"each" doesn't necessarily set up an iteration. It usu- 
ally does, and therefore these sentences are a bit odd. 
But these sentences are understood when each distrib- 
uted quantified expression has a discourse co- 
American Journal of Computational Linguistics, Volume 7, Number 3, July-September 1981 145 
William A. Martin Roles, Co-Descriptors, and the Formal Representation of Quantified English Expressions 
description which participates in an iteration. Setting 
up an iteration in (29a) and (29b) expresses the spe- 
cific dependence of boys on girls, and of each man 
and woman on each other. 
(34) 
sail-boat motor-boat 
4.4 Human Processing of Quantified Expressions 
In the previous example we could have had each 
man married to each woman! Indeed, it is not always 
obvious how an iteration should be constructed. 
When someone hears the statement 
(30) A requirement for the course is the carv- 
ing of a block of wood into each of the 
12 designs. 
his reasoning might be as follows: "Well, let's see. 
We take the wood and carve the first design. (He 
pursues the distributive referential reading with one 
block.) Oh! Oh! Now how do we carve a second de- 
sign, the block is used up. Well, maybe we could fit 
the twelve designs on one block, or we could cut the 
block into twelve pieces. Or maybe I should abandon 
the referential reading and use twelve blocks." There 
can be no doubt that world knowledge (pragmatics) is 
required to choose between such readings. Consider 
now the following pair of sentences. 
(31a) Everybody at MIT knows a dialect of LISP. 
(31b) Everybody at IJCAI knows a dialect of LISP. 
Everyone at a university might conceivably know the 
same specific dialect, while everyone at an internation- 
al conference might not. 
Van Lehn \[26\] reports that when people are given 
a sentence like 
(32) A quick test confirmed that every drug 
was psychoactive. 
they claim they understand it, but are then unable to 
state whether there was one test per drug or only one 
test for all. This ambiguity can be expressed in predi- 
cate calculus using quantifiers. 
(33a) (\] test) (V drug) C(test, drug) 
(33b) (¥ drug) (3 test) C(test, drug) 
But in my representation, the ambiguity is represented 
by the presence or absence of a role-in link, which 
makes it easy to delay disambiguation. There is no 
shuffling of quantifiers to shift between the two inter- 
pretations; instead, disambiguation is achieved by sim- 
ply inserting a role-in link, when and if appropriate. 
4.5 Ambiguity and Generality 
The problem with (32) arises in part because of the 
difficulty of distinguishing between a general expres- 
sion and an ambiguous one. For example, 
illustrates that both a sail boat and a motor boat can 
be described by the general expression boat. This 
contrasts with 
(35) = = pronoun-mine mineral-mine 
where there is no generalization of the pronoun-mine 
and mineral-mine senses of "mine". Finally, 
(36) A quick test confirmed 
? th~t every drug 
/////%',,,,,,~,~ p sy c h o act i ve 
forall drugs forsome test 
forsome test forall drugs 
points out that it is uncertain whether "a quick test 
confirmed that every drug was psychoactive" has a 
general sense. The general sense of "boat" is resolved 
semantically -- by choosing between more specific 
concepts such as "sailboat" or "motorboat". If (35) 
does have a general sense it might be resolved seman- 
tically, but it might also be resolved pragmatically -- 
by binding the concept into the discourse structure in 
different ways. Philosophers have largely ignored this 
issue since they have been interested in formal lan- 
guages whose terms are logically unambiguous. Van 
Lehn's results would tend to indicate that people do 
have a general sense of this expression. 
The advantage of the representational scheme pro- 
posed here is that it allows the resolution of ambiguity 
by incrementally adding to what is already present. 
We have discussed many reasons why scoping deci- 
sions should be delayed, ranging from computational 
efficiency to Van Lehn's empirical observations. 
Role-in links provide a simple way to accomplish this. 
5. Conclusion: The Asymmetry of Roles 
Roles introduce a very interesting asymmetry. In 
my representation, the meaning of a node depends on 
its roles in more aggregate entities, not just on its 
constituents. For example, I use role-in links as Sko- 
lem modifiers where others have employed quantifier 
scope. In this way, I can represent quantificational 
dependencies a8 role-in links to nodes representing 
more aggregate entities. I have argued that roles are a 
natural device for explaining many linguistic and philo- 
sophical distinctions, and they are convenient for com- 
putational processing. 
Section 3 showed how roles can help represent 
referring expressions. The referential/attributive dis- 
146 American Journal of Computational Linguistics, Volume 7, Number 3, July-September 1981 
William A. Martin Roles, Co-Descriptors, and the Formal Representation of Quantified English Expressions 
tinction was formulated in terms of roles so that the 
attributive use asserts something as a role in such-and- 
such whereas the referential use asserts something as 
being the such-and-such without a role dependency. I 
choose to use roles where others (e.g. Moore) have 
employed scope dependencies, because the ambiguity 
persists even when the scoping context is unavailable, 
as in "The president has owned a terrier since Febru- 
ary". In this way, I can delay the binding decision 
because I can represent the dependencies (or lack 
thereof) without hypothesizing an outer context. 
Furthermore, I have argued that verb phrase dele- 
tion provides additional support for the claim that role 
dependencies are associated with references, not with 
contexts. That is, in a sentence like "The prosecutor 
believed that he would win the case, and so did the 
defense attorney", there are two ways to interpret 
"he": as a co-descriptor of the subject or as a co- 
descriptor of the prosecutor. Note that whichever way 
it is taken in the first conjunct, it will be taken the 
same way in the second. There is a natural explana- 
tion for this if the co-description dependency is associ- 
ated with the reference "he", whereas in a system 
where this dependency was associated with the con- 
text, an ad hoc stipulation would be required. 
Section 4 argued that quantifier dependencies 
should be associated with references as opposed to 
contexts. My approach leads to a natural representa- 
tion for collectives, distributives, and pluralities. By 
minimizing the differences in the representation of 
these three cases, an interpreter is in a better position 
to delay binding decisions. This is consistent with Van 
Lehn's empirical observations. Furthermore collec- 
tives and pluralities can be interpreted without itera- 
tion, saving considerable processing effort. 

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