MULTI-LEVEL PLURALS AND DISTRIBUTIVITY 
Remko Scha and David Stallard 
BBN Laboratories Inc. 
10 Moulton St. 
Cambridge, MA 02238 
U.S.A. 
ABSTRACT 
We present a computational treatment of the 
semantics of plural Noun Phrases which extends an 
earlier approach presented by Scha \[7\] to be able to 
deal with multiple-level plurals ("the boys and the 
girls", "the juries and the committees", etc.) 1 We ar- 
gue that the arbitrary depth to which such plural struc- 
tures can be nested creates a correspondingly ar- 
bitrary ambiguity in the possibilities for the distribution 
of verbs over such NPs. We present a recursive 
translation rule scheme which accounts for this am- 
biguity, and in particular show how it allows for the 
option of "partial distributivity" that collective verbs 
have when applied to such plural Noun Phrases. 
1 INTRODUCTION 
Syntactically parallel utterances which contain 
plural noun phrases often require entirely different 
semantic treatments depending upon the particular 
verbs (or adjectives or prepositions) that these plural 
NPs are combined with. For example, while the sen- 
tence "The boys walk" would have truth-conditions ex- 
pressed by: 2 
Vxe BOYS: WALK\[x\] 
the very similar sentence "The boys gather" could not 
be translated this way. Its truth-conditions would in* 
stead have to be expressed by something like: 
GATHER\[BOYS\] 
since only a group can "gather', not one person by 
himself. 
It is common to call a verb such as "walk" a 
"distributive" verb, while a verb such as "gather" (or 
"disperse ~ or intransitive "meet*) is called a 
~The wod( presented here was supported unOer DARPA contracts 
#N00014-85-C-0016 and #N00014-87.C-0085. The vmws and con- 
clusions contained in this document ere those of the authom and 
should not be intecpreted as neceeserily repr~tmg the official 
policies, e~ther expressed or implied, of the Defense Advanced 
Research Projects Agency or the United Statas Government. 
2We ¢jnore here the diecourse ~sues that bear on the inter- 
pretation of definite NPs 
"collective" verb. The collective/distributive distinction 
raises an important issue: how to treat the semantics 
of plural NPs uniformly. 
An eadiar paper by Scha ("Distributive, Collective 
and Cumulative Quantification" \[7\], hereinafter "DCC") 
presented a formal treatment of this issue which ex- 
ploits an idea about the semantics of plural NP's 
which is due to Bartsch \[1\]: plural NP's are always 
interpreted as quantifying over sets rather than in- 
dividuals; verbs are correspondingly always treated as 
collective predicates applying to sets. Distributive 
verbs are provided with meaning postulates which re- 
late such collective applications to applications on the 
constituent individuals. 
The present paper describes an improved and ex- 
tended version of this approach. Two important 
problems are addressed. First, there is the problem 
of ambiguity: the need to allow for more than one 
distribution pattern for the same verb. Second, there 
is the problem of "multi-level plurality': the con- 
sequences which arise for the distributive/collective 
distinction when one considers conjoined plural NPs 
such as "The boys and the girls". 
Both issues are addressed by a two-level system 
of semantic interpretation where the first level deals 
with the semantic consequences of syntactic structure 
and the second with the lexically specific details of 
distribution. 
The treatment of plural NPs described in this 
paper has been implemented in the Spoken Lan- 
guage System which is being developed at BBN. The 
system provides a natural language interface to a 
database/graphic display system which is used to ac- 
cess information about the capabilities and readiness 
conditions of the ships in the Pacific Reet of the US 
Navy. 
The remainder of the paper is organized as fol- 
lows: 
Section 2 discusses previous methods of handling 
the distributive/collective distinction, and shows their 
limitations in dealing with the problems mentioned 
above. 
Section 3 presents our two-level semantics ap- 
proach, and shows how it handles the problem of am- 
biguity. 
17 
Section 4 shows how a further addition to the two- 
level system - recursive enumeration of lexical mean- 
ings - handles the multi-level plural problem. 
Section 5 presents the algorithm that is used and 
Section 6 presents conclusions. 
2BACKGROUND 
2.1 An Approach to Distributivity 
One possible way to generate the correct readings 
for "The boys walk" vs. "The boys gather" is due to 
Bennett \[2\]. Verbs are sub-categorized as either col- 
lective or distributive. Noun phrases consisting of 
"the" + plural then have two readings; a *sat" reading 
if they are combined with a collective verb and a 
universal quantification reading if they are combined 
with a distributive verb. 
Scha's "Distributive, Collective, and Cumulative 
Quantification" ('DCC') showed that this approach, 
while plausible for intransitive verbs, breaks down for 
the two-argument case of transitive verbs \[7\]. Con- 
sider the example below: 
"The squares contain the circles" 
\[3 
This sentence has a reading which can be ap- 
proximately paraphrased as "Every circle is contained 
in some square" so that in the world depicted above 
the sentence would be considered true. 
The truth-conditions which Bennett's approach 
would predict, however, are expressed by the formula: 
Vx e SQUARES: V.R CIRCLES: CONTAIN\[x,y\] 
which obviously does not correspond to the state of 
affairs pictured above. 
"DCC" avoids this problem by not generating a 
distributive translation directly. Noun phrases, regard- 
less of number, quantify over sets of individuals: a 
singular noun phrase simply quantifies over a 
singleton set. Nouns by themselves denote sets of 
such singleton sets. Thus, both "square" and 
"squares" are translated as: 
SQUARES* 
in which the asterisk operator "*" creates the set of 
singleton subsets of "SQUARES'. 
Verbs can now be uniformly typed to accept sets 
of individuals as their arguments. The 
collective/distributive distinction consists solely in 
whether a verb is applied to a large set or to a 
singleton set. 
Determiner translations are either distributive or 
collective depending upon whether they apply the 
predicate to ,the constituent singletons or to their 
union. Some determiners are unambiguously distribu- 
tive, for example the translation for "each': 
(;~X: (~.P. Vx E x: P(x))) 
Other determiners - "all', "some" and "three" - are 
ambiguous between translations which are distributive 
and translations which are collective. Plural "the', on 
the other hand, is unambiguously collective, and has 
the translation: 
(X,X: (~.~./:'(U(,X)))) 
where "U" takes a set of sets and delivers the set 
which is their union. 
The following is a list of sentences paired with 
their translations under this scheme: 
The boys walk 
WALK(BOYS) 
Each boy walks 
Vx e BOYS': WALK(x) 
The boys gather 
GATHER(BOYS) 
The squares contain the circles 
CONTAIN(SQUARES,CIRCLES) 
For "the" + plural NP's we thus obtain analyses which 
are, though not incorrect, perhaps more vague than 
one would desire. These analyses can be further 
spelled out by providing distributive predicates, such 
as "WALK" and "CONTAIN', with meaning postulates 
which control how that predicate is distributed over 
the constituents of its argument. For example, the 
meaning postulate associated with "WALK" could be: 
WALK\[x\] - \[#(x) • t3\] ^ \[rye x': WALK\[y\]\] 
which, when applied to the above translation 
"WALK\[BOYS\]', gives the result: 
\[#(BOYS) > 0\] ^ \[Vy ~ BOYS*: WALK\[y\]\] 
which represents the desired distributive truth- 
conditions. 
The meaning postulate for "CONTAIN" could be: 
CONTAIN\[u,v\] - Vy ~ v': 3xe u': CONTAIN\[x,y\] 
This meaning postulate may be thought of as ex- 
pressing a basic fact about the notion of containment; 
namely that one composite object is "contained" by 
another if every every part of the first is contained in 
some part of the second. Application of this meaning 
postulate to the translation 
CONTAIN\[SQUARES,CIRCLES\] 
gives the final result: 
Vy ~ SQUARES*: 3x E CIRCLES': CONTAIN\[x,y\] 
which expresses the truth-conditions we originally 
18 
desired; namely those paraphrasable by "Every circle 
is contained by some square'. 
In general, it is expected that different verbs will 
have different meaning postulates, corresponding to 
the different facts and beliefs about the world that 
pertain to them. 
2.2 Problems 
Conjuncbve Noun Phrases 
"DCC" only treated plural Noun Phrases (such as 
"the boys" and "some girls'), but did not deal with 
conjunctive Noun Phrases ('John, Peter and Bill', "the 
boys and the girls", or "the committees and the 
juries"). It is not immediately clear how a treatment of 
them would be added. Note that a PTQ-style 3 treat- 
ment of the NP "John and Peter": 
~.P: P(John' ) ^ P(Peter' ) 
would encounter serious difficulties with a sentence 
like "John and Peter carried a piano upstairs'. Here it 
would predict only the distributed reading, yet a col- 
lective reading is the desired one. 
It would be more in the spirit of the treatment in 
"DCC" to combine the denotations of the NPs that are 
conjoined by some form of union. For example, "John 
and Peter', "The boys and the girls" might be trans- 
lated as: 
;LP: P({John' ,Peter' )) 
~.P: P(BOYS U GIRLS) 
For a sentence like "The boys and the girls gather" 
this prevents what we call the "partially" distributive" 
reading - namely the reading in which the boys gather 
in one place and the girls in another. 
For this reason, it seems incorrect to assimilate all 
NP denota~ons to the type of sets of individuals. 
Noun phrases like "The boys and the girls" or "The 
juries and the committees', are what we call "multi- 
level plurals': they have internal structure which can- 
not be abolished by assimilation to a single seL 
Note that the plural NP "the committees" is a 
multi-level plural as well, even though it is not a con- 
junction. The sentence "The committees gather" has 
a partially distributive reading (each committe gathers 
separately) analogous with the partially distributive 
reading for "The boys and girls gather" above. 
Ambiguity and Discourse Effects 
The final problem for the treatment in "DCC" has 
to do with the meaning postulates themselves. These 
always dictate the same distribution pattam for any 
verb, yet it does not seem plausible that one could 
finally decide what this should be, since the beliefs 
and knowledge about the world from which they are 
derived are subject to variation from speaker to 
speaker. 
Variability in distribution might also be imposed by 
context. Consider the sentence "The children ate the 
pizzas" and a world depicted by the figure in 2.1 
where the squares represent children, and the circles, 
pizzas. Now there will be different quantificational 
readings of the sentence. The question "What did the 
children eat?" might be reasonably answered by "The 
pizzas'. If one were to instead ask "Who ate the 
pizzas?" (with a view, perhaps, to establishing in- 
dividual guilt) the answer "The children" would not be 
as felicitous, since the picture includes one square 
(child) not containing anything. 
It is to these problems with meaning postulates 
that we now turn in Section 3. The solution presented 
there is then used in Section 4, where we present our 
solution to the NP-conjunction/multi-level plural 
problem. 
3 THE AMBIGUITY PROBLEM 
3.1 The Problem with Meaning Postulates 
That certain predicates may have different dis- 
tributive expansions in different contexts cannot be 
captured by meaning postulates: since meaning pos- 
tulates are stipulated to be true in all models it is 
logically incoherent to have several, mutually incom- 
patible meaning postulates for the same constant. 4 
An alternative might be to retreat from the notion 
of meaning postulates per se, and view them instead 
as some form of conventional implicatures which are 
"usually" or "often" true. While it is impossible to have 
alternative meaning postulates, it is easier to imagine 
having alternative implicatures. 
For a semantics which aspires to state specific 
truth-conditions this is not a very attractive position. 
We prefer to view these as alternative readings of the 
sentence, stemming from an open-ended ambiguity of 
the lexicai items in question - an ambiguity which has 
to do with the specific details of distributions. 
Since this ambiguity is not one of syntactic type it 
does not make sense (in either explanatory or com- 
putational terms) to multiply lexical entries on its be- 
half. Rather, one wants a level of representation in 
which these distributional issues are left open, to be 
resolved by a later stage of processing. 
3We use the worn "style" because Montague's original paper 
\[6\] only conjoined term phrases with "or'. The extens~n to "and', 
however, is straJghtforward. 
4One might tfi to combine them into a single meening postulate by 
Iogr,,al disjunction. We have indicated Oefo~re \[9\] why this approach is 
not satisfactory. 
19 
3.2 Two Levels of Semantic Interpretation 
To accommodate this our system employs two 
stages of semantic interpretation, using a technique 
for coping with lexical ambiguity which was originally 
developed for the Question-Answering System 
PHLIQA \[3\] \[8\]. The first stage uses a context-free 
grammar with associated semantic rules to produce 
an expression of the logical language EFL (for 
English-Oriented Formal Language). EFL includes a 
descriptive constant for each word in the lexicon, 
however many senses that word may have. Hence 
EFL is an ambiguous logical language; in technical 
terms this means either that the language has a 
model-theory that assigns multiple denotations to a 
single expression \[5\], or that its expressions are 
viewed as schemata which abbreviate sets of possible 
instance-expressions. \[g\] 
The second stage translates the EFL expression 
into one or more expressions of WML (for World 
Model Language). WML, while differing syntactically 
from EFL only in its descriptive constants, is un- 
ambiguous, and includes a descriptive constant for 
each primitive concept of the application domain in 
question. A set of translation rules relates each am- 
biguous constant of EFL to a set of WML expressions 
representing its possible meanings. Translation of 
EFL expressions to WM/expressions is effected by 
producing all possible combinations of constant sub- 
stitutions and removing those which are "semantically 
anomalous", in a sense which we will shortly define. 
EFL and WML are instantiations of a higher-order 
logic with a recursive type system. In particular, if (x 
and I~ are types, then: 
sets(.) 
sets(sets(=)) 
sets(sets(sets(.))) 
fun(~ 13) 
fun(sets(c¢),~) 
fun(sets(.),sets(13)) 
...o 
are all types. The type "sets(,)" is the type of sets 
whose elements are of type eL The type =FUN((x,~)" 
is the type of functions from type o~ to type 13. 
Every expression has a type. which is computed 
from the type of its sub-expressions. Types have 
domains which are sets; whatever denotation an ex- 
pression can take on must be an element of the 
domain of its type. Some expressions, being con- 
structed from combinations of sub-expressions of in- 
appropriate types, are not meaningful and are said to 
be "semantically anomalous". These are assigned a 
special type, called NULL-SET, whose domain is the 
empty set. 
For example, if =F" is an expression of type 
fun(o¢,~) and "a" is an expression of type 7. whose 
domain is disjoint from the domain of., then the ex- 
pression "F(a)" representing the application of "F" to 
"a" is anomalous and has the type NULL-SET. 
For more details on these formal languages and 
their associated type system, see the paper by 
Landsbergen and Scha \[5\]. 
3.3 Translation Rules Instead of Meaning 
Postulates 
We are now in a position to replace the meaning 
postulates of the "DCC" system with their equivalent 
EFt. to WML translation rules. For example, the 
original treatment of "contain" would now be 
represented by the translation rule: 
CONTAIN ->Zu, v: Vy E v': 3x E u': CONTAIN' Ix.Y\] 
Note that the constant "CONTAIN'" on the right-hand 
side is a constant of WML. and is notationally 
separated from its EFL counterpart by a prime-mark. 
The device of translation rules can now be 
brought to bear on the problem mentioned in section 
22. namely the distributional ambiguity (in context) of 
the transitive verb "eat*. The reading which allows an 
exception in the first argument would be generated by 
the translation rule: 
EAT -> ~.u, v: Vy ett: :ix E u*: EAT' \[x,y\] 
while the reading which allows no such exception 
would be: 
EAT ..> 
ZU.V: \[VX E V': :ly e u': EAT' \[y,x\]\] ^ 
\[Vx E U': :lye I/': EAT' Ix,Y\]\] 
We call this a "leave none out* tran~ation. When 
applied to the sentence "The children ate the pizzas" 
this generates the reading where all children are 
guilty. 
By using this device of translation rules a verb 
may be provided with any desired (finite) number of 
alternative distribution patterns. 
The next section, which presents this paper's 
treatment of the multiple plurals problem, will make 
use of a slight modification of the foregoing in which 
the translation rules are allowed to contain EFL con- 
stants on their right-hand sides as well as their left, 
thus making the process recursive. 
4 MULTIPLE LEVELS OF PLURALITY 
4.1 Overview 
As we have seen in Section 2.2. utterances which 
contain multi-level plurals sometimes give rise to 
mixed collective/distributive readings which cannot be 
accounted for without retaining the separate semantic 
identity of the constituents. 
20 
Consider, for instance, the sentence "The juries 
and the committees gather". This has three readings: 
one in which each of the juries gathers alone and 
each of the committees gathers alone as well 
(distribution over two levels), another in which all per- 
sons who are committee members gather in one 
place and all persons who are jurors gather in another 
place (distribution" over one level), and finally a third in 
which all jurors and committee members unite in one 
large convention (completely collective). It seems in- 
escapable, therefore, that the internal multi-level 
structure of NPs has to be preserved. 
Indeed. it can be argued that the number of levels 
necessary is not two or three but arbitrary. As 
Landman \[4\] has pointed out. conjunctions can be ar- 
bitrarily nested (consider all the groupings that are 
possible in the NP "Bob and Carol and Ted and 
Alice"!). Therefore. the sets which represent collec- 
tive entities must, in principle, be allowed to be of 
arbitrary complexity. This is the view we adopt. 
Allowing arbitrary complexity in the structure of 
collective en~ties creates a problem for specifying the 
distributive interpretations of collective predicates: 
they can no longer be enumerated by finite lists of 
translation rules. An arbitrary number of levels of 
structure means an arbitrary number of ways to dis- 
tribute, and these cannot be finitely enumerated. 
In order to handle these issues it is necessary to 
extend the ambiguity treatment of the previous sub- 
section so that. as is advocated in \[9\], it recutsively 
enumerates this infinite set of alternatives. In order to 
do this we must allow EFL constants to also appear 
on the right-hand side of translation rules as well as 
on the left. 
In the next sub-section we present such a recur- 
sive EFL constant. Its role in the system is to deal 
with distributions over arbitrarily complex plural struc- 
tures. 
4.2 The PARTS Function 
For any complex structure there is generally more 
than one way to decompose it into parts. For ex- 
ample, the structure 
{ {John,Peter,Bill},{Mary,Jane,Lucy) } 
can be viewed as either having two parts - the sets 
'{John,Peter.Bill)' and '{Mary,Jane,Lucy}' - or six - the 
six people John,Peter,Bill,Mary,Jane, and Lucy. 
These multiple perspectives on a complex entity 
are accommodated in our system by the EFL function 
PARTS. This function takes a term, simple or com- 
plex, and returns the set of "parts" (that is, mathemati- 
cal "parts") making it up. Because there is in general 
more than one way to decompose a composite entitity 
into parts, this is an ambiguous term which can be 
expanded in more than one way. In addition, because 
the set-theoretic structures corresponding to plural en- 
titles can be arbitrarily complex, some expansions 
must be recursive, containing PARTS itself on the 
right-hand side. 
The expansions of PARTS are: 
1. PARTS\[x\] -> x (where x an individual) 
2. PARTS\[s\] => (for: s, collect: PARTS) 
(where s a set) 
3. PARTS\[s\] -> U(for: s. collect: PARTS) 
(where s a set) 
4. PARTS\[x\] ,,> F\[x\] 
Rule (1) asserts that any atomic entity is indivisible, 
that is, is its own sole part (remember, we are talking 
about mathematical, not physical parts here). Rules 
(2) and (3) range over sets and collect together the 
set of values of PARTS for each member; rule (3) 
differs in that it merges these into a single set with the 
operator 'U'. 'U' takes a set of sets and returns their 
union. In rule (4) "F" is a descriptive function. This 
rule is included to handle notions like membership of 
a committee, etc. 
Suppose PARTS is applied to the structure: 
{ {John,Peter,Bill),{Mary~Jane,Lucy} ) 
corresponding, perhaps, to the denotation of the NP 
"The boys and the gids'. The alternative sets of parts 
of this structure are: 
(1) {John,Petar,BilI,Mary,Jane,LucY } 
(2) { {John,Peter,Bill},{Mary,Jane,Lucy} } 
Let us see how these ~-re produced by recursively 
expanding the funclion PARTS. Suppose we invoke 
rule (3) to begin with. This produces: 
U(for: { {John,Peter,Bill},{Mary,Jane,Lucy} }, 
collect: PARTS) 
Now suppose we invoke rule (2) on this, resulting in: 
U(for: { {John,Peter,Bill),{Mary,Jane,Lucy} }, 
collect: ~.x: (for: x, collect: PARTS)) 
In the final step, we invoke rule (1) to produce: 
U(for:{ {John,Peter.Bill},{Mary,Jane,Lucy) } 
collect: Zx:. (for: x, 
collect: ~.x: x) 
This expression simplifies to: 
{John,Peter,BUI.Mary,Jane,Lucy) 
which is just the expansion (1) above. 
Now suppose we had invoked rule (2) to start 
with, instead of rule (3). This would produce the ex- 
pansion: 
for: { {John.Petar,Bill},{Mary,Jane,Lucy) ), 
collect: PARTS 
The rest of the derivation is the same as in the first 
21 
example. We invoke rule (2) to produce the expan- 
sion 
for: { {John,Peter, Bill},{Mary,Jane,Lucy} }, 
collect: ~.x:. (for: x, collect: PARTS) 
Rule (1) is then invoked: 
for: { {John,Peter.Bill},{Mary,Jane,Lucy} }, 
collect: ~.x:. (for: x, 
collect: ~.x:. x) 
There are now no more occurrences of PARTS left. 
This expression reduces by logical equivalence to: 
{ {John, Peter,Bill},{Mary,Jane,Lucy} } 
which is just the expansion (2). 
We now proceed to the distributing translation 
rules for verbs, which make use of the PARTS func- 
tion in order to account for the multiple distributional 
readings economically. 
4.3 The Distributing Translation Rules 
The form below is an example of the new scheme 
for the translation rules, a translation which can cope 
with the problem originally posed in section 2.1, "The 
squares contain the circles'. "s 
CONTAIN -> 
~.u,v : Vx • PARTS\[{v}\]: 
3y • PARTS\[{u}\]: CONTAIN' \[y,x\] 
This revised system can now cope with multi-level 
plural arguments to the verb "contain". Suppose we 
are given "The squares contain the circles and 
triangled'. The initial translation is then: 
Vx • PARTS\[{{CIRCLES,TRIANGLES}}\]: 
3y • PARTS\[{SQUARES}\]: 
CONTAIN' \[y,x\] 
The ranges of the quantifiers each contain an occur- 
rence of the PARTS function, so it is ambiguous as to 
what they quantify over. Note, however, that the 
WML predicate CONTAIN' is typed as being ap- 
plicable to individuals only. Inappropriate expansions 
for the quantifier ranges therefore result in anomalous 
expressions which the translation algorithm filters out. 
The first range restriction: 
PARTS\[{{CIRCLES,TRIANGLES}}\] 
is expanded to: 
U(for: {{CIRCLES.TRIANGLES}}, 
collect: ~.x: U(for: x, 
collect: Zx (for: x, collect: ~.x. x))) 
by a sequence of expansion rule applications 
(3),(3),(2),(2), and (1). This final form is equivalent to: 
SNote one othe¢ modification with rescNmt to the tre~lent 
presented in section 2.1: predicates transiting verbs are now al- 
Iowe~ to operate on individuals instea¢l of sets only 
U(CIRCLES,TRIANGLES) 
The other restriction, 'PARTS\[{SQUARES}\]', is 
reduced by similar means to just 'SQUARES'. We 
have, finally: 
Vx E U(CIRCLES,TRIANGLES): 
3y • SQUARES: CONTAIN' \[y,x\] 
which expresses the desired truth-conditions. 
4.4 Partial Distribution of Collective Verbs 
Let us take up again the example "The juries and 
committees gather*, Recall that this has three read- 
ings: one in which each deliberative body gathers 
apart, another in which the various jurors combine in a 
gathering and the various committee members com- 
bine separately in another gather, and finally, one in 
which all persons concerned, be they jurors or com- 
mittee members, come together to form a single 
gathering. 
These readings are accounted for by the following 
translation rule for GATHER: 
GATHER => ;Lx:. Vy • PARTS\[{x}\]: GATHER' \[y\] 
Applying this rule to the initial translation: 
GATHER\[{{JURIES,COMMITrEES}}\] 
produces the expression: 
Vy • PARTS\[{{JURIES,COMMITTEES}}\]: 
GATHER' \[y\] 
The various readings of this now depend upon what 
the range of quantification is expanded to. This must 
be a set of sets of persons in order to fit the type of 
GATHER', which is a predicate on sets of persons. 
We will now show how the PARTS function 
derives the decompositions that allow each of these 
readings. Because of the collective nature of the 
terms "jury" and "committee" ,we will use rule (4), 
which uses an arbitrary descriptive function to decom- 
pose an element. 
Suppose that 'JURIES' has the extension '{Jl ,J2,J3}' 
and 'COMMITTEES' has the extension '{Cl,C2,C3}'. 
Suppose also that the the descriptive function 
'MEMBERS-OF' is available, taking an organization 
such as a jury or committee onto the set of people 
who are its members. Let it have an extension cor- 
responding to: 
Jl "-~ {a,b,c} 
J2 -'>' {d.e.f) 
J3 ~ {g,h,i} 
c 1 ~ {j,k,I} 
c 2 --+ {m,n,o} 
c 3 --+ {p,q,r} 
where the letters a,b,c, etc. represent persons. 
The derivation (3),(3),(2),(4) yields the first of the 
readings above, in which the verb is partially dis- 
22. 
tributed over two levels. The range of quantification 
has the extension: 
{ {a,b,c},{d,e,f},{g,h,i},{j,k,I},{m,n,o},{p,q,r} } 
This is the reading in which each jury and committee 
gathers by itself. 
The derivation (3),(2),(3),(4) yields the second 
reading, in which the verb is partially distributed over 
the outer level. The derivation produces a range of 
quantification whose extension is: 
{ {a,b,c,d,e,f,g,h,i},{j,k,l,m,n,o.p,q,r} } 
This is the reading in which the jurors gather in one 
place and the committee members in another. 
Finally, the derivation (2),(3),(3),(4) yields the third 
reading, which is completely collective. This deriva- 
tion produces a range of quantification whose exten- 
sion is: 
{ {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r} } 
This is the reading in which all persons who are either 
jurors or committee members gather. 
5 OTHER PARTS OF SPEECH 
In this section we discuss distributional considera- 
tions for argument-taking parts of speech other than 
verbs - specifically prepositions and adjectives. 
Prepositions in our system are translated as two-place 
predicates, adjectives as one-place predicates. The 
distributional issues they raise are therefore treatable 
by the same machinery we have developed for tran- 
sitive and intransitive verbs. 
5.1 Prepositions 
Prepositions are subject to distributional con- 
siderations that are analogous to those of transitive 
verbs. Consider: 
The books are on the shelves 
Given facts about physical objects and spatial loca- 
tion, the most plausible meaning for this sentence is 
that every book is on some shelf or other. This would 
be expressed by the translation rule: 
Zu, v : Vx• PARTS(u): :ly~ PARTS(v): ON' (x,y) 
Note the similarity with the translation rule for 
"CONTAIN". from which it differs in that the roles of 
the first and the second argument in the quantifica- 
tional structure are reversed. 
5.2 Adjectives 
The treatment of adjectives in regular form is ex- 
actly analogous with that given intransitive verbs such 
as "walk". Thus, for the adjective "red", we may have 
the translation rule: 
RED => Zu : Vxe PARTS(u): RED(x) 
A more interesting problem is seen in sentences con- 
taining the comparative form of an adjective, as in: 
The frigates are faster than the carders 
What are the truth-conditions of this sentence? One 
might construe it to mean that every frigate is faster 
than every carrier, but this seems unneccesarily 
strong. Intuitively, it seems to mean something a little 
weaker than that, allowing perhaps for a few excep- 
tions in which a particular carrier is not faster than a 
particular frigate. 
On the other hand, another requirement 
eliminates truth-conditions which are too weak. For if 
"The  gates are faster than the carders" is true, it 
must surely be the case that "The carriers are faster 
than the frigates" is false. This requirement holds not 
only for "faster", but for the comparative form of any 
adjective. 
The treatment of comparative forms in the Spoken 
Language System can be illustrated by the following 
schema: 
(~.x,y: larger(<uf>(x),<uf>(y))) 
in which '<uf>' is filled in by an "underlying function" 
particular to the adjective in question. For the adjec- 
tive "fast", this underlying function is "speed". 
The requirement of anti-symmetry for the distribu- 
tions of comparatives is now reduced to a requirement 
of anti-symmetry for the distributional translation of 
the EFL constant "larger'. In this way, the anti- 
symmet~/ requirement is expressed for all compara- 
tives at once. 
Obviously anti-symmetry is fufilled for the 
universal-universal translation, but, as we have 
pointed out, this is a very strong condition. There is 
another, weaker condition which fufills anti-symmeW: 
larger, -> 
~.u,v:. Vx ~ PARTS\[u\]~y • PARTS\[v\]: larger' \[x,y\] ^ 
Vx • PARTS\[v\]: 3y • PARTS\[u\]: larger, \[y,x\] 
When applied to the sentence above, this condition 
simply states that for every frigate there exists a car- 
tier that is slower than it. and conversely, for every 
carrier there exists a frigate that is faster than it. 
This is anti-symmetric as required. For if there is 
some frigate that is faster than every carrier, there 
cannot be some carrier that is faster than every 
frigate. 
6 THE ALGORITHM 
The algorithm which applies this method is an ex- 
tension of the previously-mentioned procedure of 
generating all possible WML expansions from an EFL 
expression and weeding out semantically anomalous 
ones. The two steps of generate and test are now 
embedded in a loop that simply iterates until all EFL- 
level constants, including 'PARTS', are expanded 
23 
away. This gives us a breadth-first search of the 
possible recursive expansions of 'PARTS', one which 
nevertheless does not fail to halt because seman- 
tically anomalous versions, such as those attempting 
to quantify over expressions which are not sets, or 
those applying descnptive relations to arguments of 
the wrong type, are weeded out and are not pursued 
any further in the next iteration. 
We can now define the function TO-WML, which 
takes an EFL expression and produces a set of WML 
expressions without EFL constants. It is: 
TO-WML(exp) "clef 
expansions <- (exp} 
until ~(3e e expansions: EFL?(e)) 
do 
becjin 
expansions <- U(for: expansions, 
collect: ~.e for: AMBIG-TRANS(e) 
collect: SIMPLIFY) 
expansions <- {e e expansions: TYPEOF(e)= 
NULL-SET) 
end 
The function AMBIG-TRANS expands the EFL-level 
constants in its input, and returns a set of expres- 
sions. The function EFL? returns true if any EFL 
constants are present in its argument. The function 
TYPEOF takes an expression and returns its type; it 
returns the symbol NULL-SET if the expression is 
semantically anomalous. Note that if a particular ex- 
pansion is found to be semantically anomalous it is 
removed from consideration. If no non-anomalous 
expansion can be found the procedure halts and the 
empty set of expansions {\] is returned. In this case 
the entire EFL expression is viewed as anomalous 
and the interpretation which gave rise to it can be 
rejected. 
7 CONCLUSIONS 
We have shown how treatments of the 
collectJve/dis~butive distinction must take into ac- 
count the phenomenon of "partial distributivity', in 
which a collective verb optionally distributes over the 
outer levels of structure in what we call a "multi-level" 
plural. Multiple levels of structure must be allowed in 
the semantics of such plural NPs as "the boys and the 
girls", "the committees", etc. 
We have presented a computational mechanism 
which accounts for these phenomena through a 
framework of recursive translation rules. This 
.'ramework generates quantifications over alternative 
levels of plural structure in an NP, and can handle 
NPs of arbitrarily complex plural structure, It is 
economical in its means of producing arbitrary num- 
bers of readings: the multiple readings of the sen- 
tence such as "The juries and the committees 
gathered" are expressed with just one translation rule. 
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