A UNIFICATION-BASED SEMANTIC INTERPRETATION 
FOR COORDINATE CONSTRUCTS 
Jong C. Park 
University of Pennsylvania 
Computer and Information Science 
200 South 33rd Street 
Philadephia, PA 19104-6389 USA 
Internet: park@line, cis. upenn, edu 
Abstract 
This paper shows that a first-order unification- 
based semantic interpretation for various coordi- 
nate constructs is possible without an explicit use 
of lambda expressions if we slightly modify the 
standard Montagovian semantics of coordination. 
This modification, along with partial execution, 
completely eliminates the lambda reduction steps 
during semantic interpretation. 
1 Introduction 
Combinatory Categorial Grammar (CCG) has 
been offered as a theory of coordination in nat- 
ural language (Steedman \[1990\]). It has usually 
been implemented in languages based on first or- 
der unification. Moore \[1989\] however has pointed 
out that coordination presents problems for first- 
order unification-based semantic interpretation. 
We show that it is possible to get over the problem 
by compiling the lambda reduction steps that are 
associated with coordination in the lexicon. We 
show how our first-order unification handles the 
following examples of coordinate constructs. 
(1.1) Harry walks and every farmer walks. 
(1.2) A farmer walks and talks. 
(1.3) A farmer and every senator talk. 
(1.4) Harry finds and a woman cooks a mushroom. 
(1.5) Mary gives every dog a bone and some 
policeman a flower. 
We will first start with an illustration of why 
standard Montagovian semantics of coordination 
cannot be immediately rendered into a first-order 
209 
unification strategy. The lexicon must contain 
multiple entries for the single lexical item "and", 
since only like categories are supposed to conjoin. 
For example, the lexical entry for "and" in (1.1) 
specifies the constraint that the lexical item should 
expect on both sides sentences to give a sentence. 
Moore \[1989\] predicts that a unification-based 
semantic interpretation for sentences which in- 
volve for example noun phrase coordination won't 
be possible without an explicit use of lambda 
expressions, though there are cases where some 
lambda expressions can be eliminated by di- 
rectly assigning values to variables embedded in 
a logical-form expression. The problematic exam- 
ple is shown in (1.6), where proper noun subjects 
are conjoined. 
(1.6) john and bill walk. 
The argument is that if we do not change the se- 
mantics of "john" from j to AP.P(j), where P is a 
second order variable for property in the Montago- 
vian sense 1 , then the single predicate AX. walk(X) 
should accommodate two different constants j and 
b in a single variable X at the same time. Since 
the unification simply blocks in this case, the ar- 
gument goes, we need to use higher order lambda 
expressions such as AP.P(j) or AP.P(b), which 
when conjoined together, will yield semantics for 
e.g. "john and bill" as 
,~P.(P(j) ~ P(b)) . 
Combined finally with the predicate, this will re- 
sult in the semantics (1.7), after lambda reduction. 
(1.7) walk(j) & walk(b) 
1Montague \[1974\]. )~pVp(j) to be exact, taking in- 
tensionality into account. The semantics of the predi- 
cate "walks" will then be (^AX.walk(X)). 
Although Moore did not use quantified noun 
phrases to illustrate the point, his observation gen- 
eralizes straightforwardly to the sentence (1.3). In 
this case, the semantics of "and", "every" and 
"some" (or "a") will be (1.8) a, b, and c, respec- 
tively. 
(1.8) (a) AO.AR.AP.(Q(P) • R(P)) 
(b) AS. AP'. forall(X, S (X) =>P' (X)) 
(c) AS.AP". exists(X,S(X)~P' ' (X)) 
Thus, after four lambda reduction steps, one for 
each of Q, R, P' and P' ', the semantics of "a farmer 
and every senator" will be 
AP.(exists(X,faxmer(X)RP(X)) 
forall(X,senator(X)=>P(X))), 
as desired. 
Moore's paper showed how lambda reduction 
could be avoided by performing lambda reduction 
steps at compile time, by utilizing the lexicon, in- 
stead of doing them at run time. Consider again 
(1.8a). The reason why this formulation requires 
foursubsequent lambda reduction steps, not three, 
is that the property P should be applied to each 
of the conjuncts, requiring two separate lambda 
reduction steps. Suppose that we try to eliminate 
these two lambda reduction steps at compile time 
by making the argument of the property P explicit 
in the lexicon, following the semantics (1.9). 
(1.9) AQ.AR.AP.(Q(AX.P(X)) • R(AX.P(X))) 
The first-order variable X ranges over the set of 
individuals, and the hope is that after lambda re- 
duction it will be bound by the quantifiers, such 
as forall, embedded in the expressions denoted 
by the variables Q and R. Since the same variable 
is used for both constructs, however, (1.9) works 
only for pairs of quantified noun phrases, which 
don't provide constants, but not for pairs involv- 
ing proper nouns, which do provide constants. In- 
cidentally, this problem is particular to a unifica- 
tion approach, and there is nothing wrong with the 
semantics (1.9), which is equivalent to (1.8a). This 
unification problem cannot be avoided by having 
two distinct variables Y and Z as in (1.10) either, 
since there is only one source for the predicate 
property for the coordinate noun phrases, thus 
there is no way to isolate the argument of the pred- 
icate and assign distinct variables for it at compile 
time. 
(1.10) AQ.AR.AP.(Q(XY.P(Y)) ~ R(XZ.P(Z))) 
210 
The way we propose to eliminate the gap be- 
tween (1.9) and (1.10) is to introduce some spuri- 
ous binding which can always be removed subse- 
quently. The suggestion then is to use (1.11) for 
the semantics of "and" for noun phrase conjunc- 
tion. 
(1.11) Semantics of"and"for NP Conjunction: 
AQ.AR.AP.(Q(AY.oxists(X,X=Y~P(X))) 
R(AZ.exists(X,X=Z~P(X)))) 
This satisfies, we believe, the two requirements, 
one that the predicate have the same form, the 
other that the variables for each conjunct be kept 
distinct, at the same time. The rest of the lambda 
expressions can be eliminated by using the notion 
of partial execution (Pereira & Shieber \[1987\]). 
Details will be shown in Section 3, along with some 
"more immediate but faulty" solutions. It is sur- 
prising that the same idea can be applied to some 
fairly complicated examples as (1.5), and we be- 
lieve that the solution proposed is quite general. 
In order to show how the idea works, we use a 
first-order Montagovian Intensional Logic (Jowsey 
\[1987\]; Jowsey \[1990\]) for a semantics. We apply 
the proposal to CCG, but it could equally well 
be applied to any lexicon based grammar formal- 
ism. We explain briefly how a CCG works in the 
first part of Section 2. As for the semantics, noth- 
ing hinges on a particular choice, and in fact the 
code we show is devoid of some crucial features 
of Jowsey's semantics, such as indices for situ- 
ations or sortal constraints for variable binding. 
We present the version of Jowsey's semantics that 
we adopt for our purposes in the second part of 
Section 2, mainly for completeness. In Section 3, 
each of the cases in (1.1) through (1.5), or varia- 
tions thereof, is accounted for by encoding lexical 
entries of "and", although only (1.3) and (1.5) de- 
pend crucially on the technique. 
We have a few words for the organization of 
a semantic interpretation system we are assum- 
ing in this paper. We imagine that it consists of 
two levels, where the second level takes a scope- 
neutral logical form to produce every possible, gen- 
uinely ambiguous, scoping possibilities in paral- 
lel and the first level produces this scope-neutral 
logical form from the source sentence. We as- 
sume that our second level, which we leave for 
future research, will not be very different from 
the one in Hobbs & Shieber \[1987\] or Pereira 
&: Shieber \[1987\]. The goal of this paper is to 
show how the scope-neutral logical forms are de- 
rived from natural language sentences with co- 
ordinate constructs. Our "scope-neutral" logical 
form, which we call "canonical" logical form (CLF), 
syntactically reflects derivation-dependent order of 
quantifiers since they are derived by a derivation- 
dependent sequence of combination. We empha- 
size that this derivation-dependence is an artifact 
of our illustrative example, and that it is not an 
inherent consequence of our technique. 
2 Background Formalisms 
A Combinatory Categorial Grammar 
The minimal version of CCG we need to process 
our examples contains four reduction rules, (2.1) 
through (2.4), and two type raising rules, (2.5) 
and (2.6), along with a lexicon where each lexical 
item is assigned one or more categories. For the 
reasons why we need these, the reader is referred 
to Steedman \[1990\]. 
(2.1) Function Application (>): X/Y ¥ => X 
(2.2) Function Application (<): Y X\Y => X 
(2.3) Function Composition (>B): X/Y Y/Z => X/Z 2 
(2.4) Function Composition (<B): Y\Z X\Y => XXZ 
(2.5) Type Raising, Subject (>T): np => s/(sknp) 
(2.6) Type Raising, Backward (<T): np => X\(X/np) 
The present fragment is restricted to the basic 
categories n, np and s. 3 Derived categories, or 
categories, are recursively defined to be basic cat- 
egories combined by directional symbols (/or \). 
Given a category X/Y or X\Y, we call X the range 
category and Y the domain category. Parentheses 
may be used to change the left-associative default. 
The semantics part to be explained shortly, (2.7a) 
through (2.7e) show examples of a common noun, 
a proper noun, a quantifier, an intransitive verb, 
a sentential conjunction, respectively. 
(2.7) Sample Lexicon 
(a) cat(farmer, n:X'farmer(X)). 
(b) cat(harry, np:AI'(h'B)'B). 
(c) cat(every, np: (X'A)'(X'B)'forall(X,A=>B) 
/n:X'A). 
2In Steedman \[1990\], this rule is conditioned by Z 
s\np in order to prevent such constructs as "*\[Harry\] 
but \[I doubt whether Fred\] went home" or "*\[I think 
that Fred\] and \[Harry\] went home." 
3For simplicity, we do not show variables for gen- 
der, case, tense, and number. Larger fragment would 
include pp, etc. 
211 
(d) cat (walks, s : S\np: (X'A)" (X'walk(X)) "S). 
(e) cat(and, (s: (St ~ S2)\s:S1)/s:S2),4 
A First-Order Montague Semantics 
In this section, we will focus on describing how 
Jowsey has arrived at the first-order formalism 
that we adopt for our purposes, and for further 
details, the reader is referred to Jowsey \[1987\] and 
Jowsey \[1990\]. The reader can safely skip this sec- 
tion on a first reading since the semantics we use 
for presentation in Section 3 lacks many of the new 
features in this section. 
Montague's PTQ analysis (Dowty, Wall & Pe- 
ters \[1981\]) defines an intensional logic with the 
basic types e, t and s, where e is the type of en- 
tities, t the type of truth values and s the type 
of indices. Derived types <a,b> and <s,a> are re- 
cursively defined over the basic types. A name, 
which is of type e, denotes an individual; individ- 
ual concepts are names relativized over indices, or 
functions from indices to the set of individuals. In- 
dividual concepts are of type <s, e>. A predicate 
denotes a set of individuals, or a (characteristic) 
function from the set of individuals to truth val- 
ues. Properties are intensional predicates, or func- 
tions from indices to the characteristic functions. 
Properties are of type <s,<e,t>>, or <e,<s,t>>. 
A formula denotes a truth value, and propositions 
are intensional formulas, thus of type <s,t>. 
By excluding individual concepts, we can en- 
sure that only truth values are relativized over in- 
dices, and thus a modal (omega-order) logic will 
suffice to capture the semantics. For this purpose, 
Jowsey defines two basic types e and o, where o 
corresponds to the type <s,t>, and then he de- 
fines derived types <a,b>, where a and b range 
over basic types and derived types. The logic is 
then made into first-order by relying on a fixed 
number of sorts and eliminating recursively de- 
fined types. These sorts include e, s, o, p and 
q, which correspond to the types e, s, <s,t>, 
<e,<s,t>> and <<e,<s,t>>,<s,t>> respectively 
in an omega-order logic. 
For a full exposition of the logic, the reader is 
referred to Jowsey \[1990\]. For our presentation, we 
4The category (s\s)/s has the potential danger 
of allowing the following construct, if combined with 
the rule <B: "*Mary finds a man who \[walks\]s\n p 
\[and he taIks\]s\s." The suggestion in Steedman \[1990\] 
is to add a new pair of reduction rules, X \[X\]~ ffi> X and 
conj X => \[X\]~, together with the category of "and" 
as conj. Thus, the category of "and harry talks" is 
now \[s\]t~, blocking the unwanted combination. 
will simplify the semantics and drop intensional- 
ity altogether. We also drop the sortal constraint, 
since our examples do not include belief operators 
and hence the only variables left are of sort e. 
3 A First-Order Unification 
We will follow the standard technique of combin- 
ing the syntactic information and the semantic 
information as in (3.1), where up-arrow symbols 
(,-,)5 are used to give structures to the seman- 
tic information for partial execution (Pereira & 
Shieber \[1987\]), which has the effect of perform- 
ing some lambda reduction steps at compile time. 
(3.1) Basic Categories 
(a) n: (de'do) 
(b) rip: (de'do)" (de'ro) "So (c) 
The term do in (3.1a) and (3.1b) encodes domain 
constraint for the variable de. Likewise, the term 
ro in (3.1b) specifies range constraint for de. The 
term So in (3.1b) and (3.1c) encodes the sentential 
constraint associated with a sentence. In order to 
avoid possible confusion, we shall henceforth call 
categories without ~emantic information "syntac- 
tic" categories. 
In this section, we will develop lexical entries for 
those coordinate constructs in (1.1) through (1.5), 
or variations thereof. For each case, we will start 
with "more immediate but faulty" solutions and 
present what we believe to be the correct solution 
in the last. (For those who want to skip to the 
correct lexical entries for each of the cases, they 
are the ones not commented out with %.) We have 
seen the lexical entry for sentential conjunction in 
(2.7d). The lexical entry for predicate conjunction 
can be similarly encoded, as in (3.2). 
(3.2) Lexical Entry for Predicate Co~unct~n 
cat(and, ((s:S\np:A'(X*(B1 ~ B2))'S) 
\(s:Slknp:A'(X'BI)'SI)) 
/(s:S2knp:A'(X'B2)'S2)). 
When the conjoined predicates are combined with 
the subject noun phrase, the subject NP provides 
only the domain constraint, through A in the first 
line. The range constraints in the last two NP 
categories guarantee that B1 and B2 will bear the 
same variable X in them, so that they can be safely 
SNot to be confused with Montague's ha~ek sym- 
bol, '^' 
212 
put as the range constraint of the first NP cate- 
gory. The CLF for (1.2) from (3.2) is shown in 
(3.3). 
(3.3) exists(Xl, farmer(Xl)~(walk(Xl)~ 
talk(Xl))) 
Let us turn to noun phrase coordination, e.g., 
(1.3). The first try, on the model of predicate con- 
junction, would be: 
(3.4) Lexical Entry for NP Conjunction: 
%cat(and, (np:A'(X'D)'(B & C) 
% \rip :AI" (Y'D) "B) 
% /rip: A2" (Z'D) "C). 
The intention is to collect the two domain con- 
straints via A1 and A2, to get the range constraint 
from D in the first line, and then to combine them 
by joining the two sentential constraints B and C 
of the domain categories. This idea however does 
not work, since the variables ¥ and Z do not ap- 
pear in the range constraint D. As a result, (3.4) 
will give the following ill-formed CLF for (1.3). 
exists (Xl, farmer (X i) &talk (X3)) 
Rforall (X2, senator (X2) =>talk (X3)) 
We therefore need to use distinct variables in place 
of D for the two range constraints which will have 
the same predicate symbol for their range cate- 
gories. Using the Prolog predicate univ ('=.. '), 
we can correct (3.4) as follows: 6 
(3.5) Lexical Entry for NP Conjunction: 
%cat(and, (np:A'(X'D)'(B & C) 
% \np : AI" (Y'B1) -B) 
/rip: A2" (Z'C1)'C) :- 
D =.. \[Pred, X\], 
% B1 =.. \[Pred, Y\], 
C1 =.. \[Pred, Z\]. 
This is an explicit case of a first-order simulation 
of second order variables. Unfortunately, this does 
not work, for several reasons7 First, this handles 
predicates of arity 1 only, and we need to know 
the type of each argument if we want to provide 
a different category for each predicate of different 
arity. Second, this can not be combined with pred- 
icate coordination, for example, such as "john and 
6D  .. \[P,X\] succeeds if D is unifiable with P(X). 
7One implementation-dependent reason is that the 
Prolog requires at least one of the two variables V and 
Fred to be already instantiated for the univ to work. 
This can not be expected when the noun phrase con- 
junction is being processed, since we don't yet know 
what predicate(s) will follow. 
a woman walk and talk," or some complex verbs 
that may require several predicates, such as "be- 
lieves", since it assumes only one predicate for the 
range constraint. 
The solution we propose is to use the revised 
semantics of "and" in (1.11) instead. That is, we 
expect (3.6) from (1.3): 
(3.6) Proposed Semantics of (1.3): 
exists (Xl, farmer(Xl) 
~(exists (X2, (X2=Xl)&talk (X2)) ) ) 
&f orall(X3, senat or (X3) 
=>(exists (X2, (X2=X3) ~tt a\]k (X2)) ) ) 
We need to distinguish the variable X2 in the 
second line from the variable X2 in the fourth 
line, via something like c~ conversion, since in the 
present form, the Prolog will consider them as the 
same, while they are under distinct quantifiers. 
In fact, since we are separating the semantic in- 
terpretation into two levels, we can further pro- 
cess the CLF at the second semantic interpretation 
level to eliminate those spurious bindings such as 
exists(X, (X=u)~tu) along with variable renaming 
to derive the logical form (3.7) from (3.6): 
(3.7) exists (Xl, farmer(Xl ) &talk(Xl) ) 
aforall (X3, senator (X3) =>talk (X3)) 
(3.8) produces the CLF (3.6) for (1.3). 
(3.8) Lexical Entry for NP Conjunction: 
cat (and, 
(np:A'(X'D)'(B ~t C) 
\np: A1" (Y" (exists (X, (X=Y) &D) ) ) "B) 
/np : A2" (Z" (exists (X, (X=Z) ~tD) ) ) "C). 
The reason why we are able to maintain in the 
two domain categories two different forms of range 
contraints is that the only place that will unify 
with the actual range constraint, i.e., the predi- 
cate, is the range constraint part of the range cat- 
egory only. We note in passing that Jowsey pro- 
vided yet another approach to noun phrase coordi- 
nation, a generalized version of his idea as shown 
below. 
(3.8a) Lexical Entry for NP Conjunction: 
cat(and, 
(np:(X*A)'(X'D)'B 
\np:(Y'A1)*(Y'C)'B) 
/np:(Z'A2)'(Z'forall(X,(X=Y v X=Z)=>D))'C). 
For example,(3.8a) will give the following seman- 
tics for (1.3). 
exists(Xl,farmer(Xl)&forall(X2,senator(X2) 
=>forall(X3,(X3=Xl v X3=X2)=>talk(X3)))) 
213 
This approach has its limits, however, as indicated 
in the footnote 8. 
We now turn to some of the non-standard con- 
stituent coordination. First, consider (1.4), which 
is an instance of Right Node Raising (RNR). The 
CCG syntactic category of the conjunction "and" 
in this case is (C\C)/C, where C is s/np. (3.9) 
shows one derivation, among others, for (1.4). The 
syntactic category of "finds" is (sknp)/np. 
(3.9) One derivation for (1.4). 
harry finds and a woman cooks a musMroom 
..... >T ....... >T - 
s/(s\np) .... s/(s\np) ..... np 
............. >B >B 
s/np s/np 
(s/np)k(s/np) 
< 
s/rip 
........... > 
s 
Assuming that the category of "finds" is as follows, 
(3.10) Lexical Entry for '~nds": 
cat(finds. ((s:S\np:AI'(X'A)'S) 
/np:A2"(Y'find(X,Y))'A)). 
here is the first try for the RNR "and." 
(3.11) Lexical Entry for RNR Conjunction: 
%cat(and, ((s:S/np:A'CX'(Bl~B2))'S1) 
% \(s:S/np:A'(X'B1)'Si) 
% /(s:S3/np:A'(X'B2)'S2). 
For example, (3.11) will produce the CLF (3.12) 
for the sentence "harry finds and mary cooks a 
mushroom." 
(3.12) exists(Xl,musbxoom(Xl)~find(h,Xl)& 
cook(m,Xl)) 
However, this works only for pairs of proper nouns. 
For example, for the sentence "every man finds 
and a woman cooks a mushroom," it will give the 
ill-formed CLF (3.13) where the domain constraint 
for the noun phrase "a woman" is gone and X3 
is therefore unbound. This happens because the 
sentential constraint S2 is not utilized for the final 
sentential constraint. 
(3.13)%forall(Xl,man(X1)=>exists(X2, 
%mushroom(X2)&find(XI,X2) 
%&cook(X3,X2))) 
Putting the two sentential constraints Sl and s2 
together as follows does not work at all, since the 
relation between S and SO is completely undefined, 
unlike the ones between S1 and B1 and between S2 
and B2. 
%cat (and, % 
% 
((s:S/np:A'(X'(SIaS2))'SO) 
\(s:SI/np:AI'(X'BI)'BI)) 
/(s:S2/np:A2"(X'B2)'B2)). 
This problem is corrected in (3.14), which will pro- 
duce the eLF (3.15) for (1.4): 
(3.14) Lexical Entry for RNR Co~unctmn. 
catCand, ((s:S/np:A'CX'CSl&S2))'S) 
\(s:SI/np:AI"(X'BI)'BI)) 
I(s:S2/np:A2"(X'B2)*B2)). 
(3.15) Semantics of (1.4) from (3.14): 
exists(Xl,mushroom(Xl)kfind(h,Xl)) 
kexists(X2,.oman(X2)kcook(X2,Xl))) 
(1.5) shows another case of non-standard con- 
stituent coordination, which we will call an in- 
stance of Left Node Raising (LNR). The syntactic 
category of "and" for LNR is (C\C)/C where C 
is (sknp)\(((sknp)/np)/np). (3.16) shows one 
syntactic derivation for (1.5). The syntactic cate- 
gory of "gives" is ((sknp)/np)/np. 
(3.16) One derivation for (1.5), fragment. 
every dog a bone 
<T ...... <T 
((sXnp)/np) \ ( ((sknp)/np)/np) 
(sknp) \ ((sknp)/np) 
<B 
(s\np)k(((sMap)/np)/np) 
Again, we assume that the category of "gives" is: 
(3.17) LexicM Entry for "gives": 
cat(gives, ((s:Slknp:AI'(X*S2)'S1) 
/np:A2"(Y'give(X,Z,Y))'B) 
/np:A3"(Z'B)'S2). 
(3.18) shows the first try for the lexical entry, s 
(3.18) Lexical Entry for LNR Conjunction. 
%cat(and, 
% (((s:_\np:_) 
% \(((s:S\np:(X'A)-(X'(S4 ~ S6))'S) 
/np:AI'(Y-B)'SI)/np:A2"(Z'SI)-S2)) 
Sin this case, we can no longer use the disjunctive 
technique such as forall(Xl, (Xl= v Xl= )=>give( 
,X1, )) for the CLF, since Xl is now a pair. The prob- 
lem gets worse when the conjoined pairs do not have 
the same type of quantifiers, as in (1.5). 
214 
% \((s:_\np:_)\(C(s:_knp:_) 
% Inp:A3"(Y'B)'S3) 
% /np:A4"(Z'S3)'S4))) 
% /((s:_\np:_)k(((s:_knp:_) 
% /np:AS"(Y'B)'SS) 
% /np:A6"(Z'SS)'S6))). 
It gives the eLF (3.19) for (1.5): 
(3.19) Semantics of (1.5) from (3.18): 
forall (Xl, dog (X 1 ) =>exist s (X2, bone (X2) 
~give (m, Xl ,X2) ) ) ~exist s (Xl, policeman(Xl) 
• exist s (X2, flo.er (X2) ~give (m, X I, X2) ) ) 
Unfortunately, (3.18) favors quantified nouns too 
much, so that when any proper noun is involved in 
the conjunction the constant for the proper noun 
will appear incorrectly in the two sentential con- 
straints at the same time. It seems that the only 
way to resolve this problem is to create four vari- 
ables, Y1, Y2, 7.1 and Z2, at the semantics level, 
similar to idea in (1.11). (3.20) implements this 
proposal. 
(3.20) Lexical Entry for LNR Conjunction. 
cat(and, 
(((s:_\np:_) 
\(((s:S 
knp:(X'A)'(X'(S4 ~ Se))'S) 
/np: At" (Y'B)'SI) 
/np:A2" (Z'S1) "$2)) 
\((s :_\np:_) 
\(((s :_knp:_) 
/np : A3" (Y 1" 
(exists (Y, (Y=Y1)~B)) ) "$3) 
/np: 
A4" (Zl" (exists (Z, (Z=Z I) ~$3) ) ) "S4) ) ) 
/((s :_\np:_) 
\(((s:_\np:_) 
/rip: 
A5" (Y2" (exists (Y, (Y=Y2) kB) ) ) "$5) 
/np: 
AS" (Z2" exists (Z, (Z=Z2) &SS) ) ) "S6) ) ). 
(3.20) will give the eLF (3.21) for (1.5). 
(3.21) Semantics of (1.5) from (3.20): 
f orall (Xl, dog(Xl) =>exist s (X2, X2=Xl 
&exist s (X3, bone (X3) ~exist s (X4, X4=X3 
\give (m,X2, X4) ) ) ) ) 
\exist s (Xl, policeman(Xl) ~exist s (X2, X2=Xl 
Rexist s (X3, flower (X3)~exist s (X4, X4=X3 
agive (m, X2, X4) ) ) ) ) 
Using the technique of eliminating spurious bind- 
ings, (3.21) may be replaced by a logical form 
(3.22): 
(3.22) forall(Xl ,dog(Xl) 
=>exists (X3 ,bone (X3) ~give (m, Xl, X3) ) ) 
~exists (Xl, policeman(Xl) 
&exists (X3, flo.er (X3)~give (m, Xl, X3) ) ) 
In addition to this, (3.20) gives the CLF (3.23) for 
(3.24), 
(3.23) exists (Xl, Xl=j~exist s (X2, bone (X2) 
• exist s (X3, X3=X2 &give (m, X 1, X3) ) ) ) 
• exists (X 1, Xl=b~exist s (X2. flo.er (X2) 
Rexist s (X3, X3=X2 ~give (m, X1, X3) ) ) ) 
(3.24) mary gives john a bone and bill a flower. 
for which no CLF could be derived if we were using 
(3.18). This completes our demonstration for the 
technique. 
The natural question at this point is how many 
lexical entries we need for the conjunct "and". If 
natural language makes every possible category 
conjoinable, the number of entries should be in- 
finite, since function composition can grow cate- 
gories unboundedly, if it can grow them at all. We 
predict that in natural language we can limit the 
conjunction arity to n, where n is the maximum 
arity in the lexicon. 
4 Conclusion 
The system described in this paper is implemented 
in Quintus Prolog. We expect that the approach 
can be extended to any lexicon-based grammar of 
the same power as CCG if it provides means for 
term unification. 
The reason we choose to eliminate all the 
lambda expressions is that it allows uniform treat- 
ment within first-order unification, since Jowsey's 
results suggest that in other respects natural lan- 
guage semantics can be characterized in a first- 
order logic. As an alternative, we could choose 
to enforce uniform treatment within second-order 
unification, using the idea for example in Na- 
dathur & Miller \[1988\]. Although we leave this 
possibility for future research, we believe that this 
option might turn out to be more appropriate in 
terms of elegance of the approach. And the result- 
ing conceptual clarity might be exploited to design 
a schema for generating these entries for "and". 
the content. I am also very grateful to Dr. Mark 
Johnson, who suggested, and took pains of going 
over in detail, another way of presenting the thesis, 
that resulted in the material in the introduction 
section. All errors are however entirely mine. The 
author was supported by the ARO grant DAAL03- 
89-C-0031PRI. 

References 
David R. Dowty, Robert E. Wall & Stanley Peters 
\[1981\], Introduction to Montague Seman- 
tics, D. Reidel Publishing Company. 
Jerry R. Hobbs ~ Stuart M. Shieber\[January- 
June 1987\], "An Algorithm for Generat- 
ing Quantifier Scopings," Computational 
Linguistics 13, 47-63. 
Einar Jowsey\[1987\], "Montague Grammar and 
First Order Logic," Edinburgh Work- 
ing Papers in Cognitive Science: Catego- 
rim Grammar, Unification Grammar and 
Parsing 1, 143-194. 
Einar Jowsey 
\[1990\], Constraining Montague Grammar 
for Computational Applications, Doctoral 
Dissertation, Department of AI, Univer- 
sity of Edinburgh. 
Richard Montague \[1974\], in Forma/ Philosophy, 
Richmond H. Thomason, ed., Yale Uni- 
versity Press. 
Robert C. Moore\[1989\], "Unification-Based Se- 
mantic Interpretation," Proceedings of 
the ACL. 
Gopalan Nadathur & Dale Miller\[1988\], "An 
Overview of A-Prolog," Proceedings of the 
Fifth International Logic Programming 
Conference. 
Fernando C.N. Pereira & Stuart M. Shieber \[1987\], 
Prolog and NaturM-Language Ananlysis, 
CSLI Lecture Notes Number 10. 
Mark J. Steedman \[April 1990\], "Gapping as Con- 
stituent Coordination," Linguistics and 
Philosophy 13, 207-263. 
