COLING 82, .1". Horecl~ \[ed.} 
North-Holland Publishing Company 
© Academitt 1982 
A LESNIEWSKIAN VERSION OF MONTAGUE GRAMMAR 
Arata ISHIMOTO 
Department of Industrial Administration 
Science University of Tokyo 
Noda, Chiba 
Japan 
We shall be concerned in this paper with the 
logical analysis of natural language on the basis 
of Lesniewski's ontology, which is a logical system 
without type-distinction between individuals and 
monadic predicates. This, it is believed, is also 
one of the features of natural language, and use 
will be made of this feature for developing a frag- 
ment of n atnral language. 
I Introduction According to Montague every simple sentence of 
natural language is of the form 
Np + Vp = (Det + N) + Vp, 
and, as emphasized by Barwise-Cooper \[1\] Det + N constitutes a gen- 
eralized quantifier to be applied to Vp or verb phrase as one-place 
predicate. (For Montague grammar consult among others Montague\[10\], 
Cresswell \[3\] and Jirku \[6\]<) 
Thus, simple sentences suck as: 
.1 the man walks, 
.2 every man walks, 
• 3 some (a) men (man) walk(s) 
.4 (at least) two men walk, 
• 5 (at least) three men walk, 
are respectively,of the following logical forms: 
1.11 (kPkQP1x man) walk, 
1.21 (~P~Q~x(P x O Q x) man) walk, 
1.31 (XP%Q3x(P x A Q x) man) walk, 
1.~l (kPkQBxBy(P x ^ P y A P x A Q y A xz~ y) man) Walk, 
1.51 (kPkQBx~y3z(P x A Q y ~ Q x A Q y A P z ~ Q z A 
x~y A y~z A z~x).man) walk, 
where QIxP x is the Russellian-type definite description to be de- 
fined as 
~x(P x ~ Q x) A ~x~ty(P x A P Y.~ X~ry) , 
with the scope restricted to Q. 
By means of k-con,~ersion, for example," from 1.11 and 1.21 we 
respectively obtain: 
walk ix man x, 
and 
Vx(man x O walk x), 
so that Det + N now becomes in both cases a generalized quantifier. 
139 
140 A. ISHIMOTO 
(Here as well as in what follows intension will not be taken into 
consideration. The proposed Lesniewskian theory will therefore 
be extensional.) 
Now it is to be remembered that 1.11-1.51 are the formulas of 
(second-order) predicate logic. In other words, the sentences of 
natural language are embedded in predicate logic although a large 
number of formulas of the logic do not have their counterparts in 
natural language. 
However, the embedding in traditional predicate logic is not 
necessarily the only possible way for us to understand the logical 
structure of natural language. In fact, natural language could also 
be embedded in Lesniewski's ontology augmented by a number of add- 
itional notions so that a fragment of natural language can be ac- 
comodated there, and ~n what follows we shall be concerned with a 
detailed construction of Lesniewskian-type logical grammar. 
2 Logical ~rammar based upon Lesniewski1~ ontology In Lesniew- 
skian-type Montague grammar we are all the same starting from the 
structural assumption of the simple sentences of natural language 
as mentioned at the beginning of the last section. Neverthel@ss, 
in Lesniewskian version of logical grammar, which will be abbre- 
viated as LMG in the sequel, Det, i.e. determiners, are represented 
not by generalized quantifiers but by the functors of Lesniewski's 
ontology with noun and intransitive verb phrases to be combined 
thereby as two arguments, which are now provided with the category 
of names in the sense of Lesniewski's ontology, not the one corres- 
ponding to monadic predicates of predicate logic. 
Without going into the details of the phrase structure and 
transformational rules necessary for generating a fragment of Eng- 
lish (which is by far smaller than that proposed by Cooper-Parsons 
\[2\] ) we shall present the deep or logical structures of a number of 
simple sentences (of English) as (well-formed) sentences belonging 
to the proposed Lesniewskian version of Montague grammar to be de-~ 
signated as LMG: 
2.11 ((the man) walk), 
2.12 ((every ((man or ) woman)) speak), 
2.13 ((some woman) not play)), 
2.14 ((every man) \[love (some woman)\] ) 
2.15 (the woman) \[(love and) admire) \] ' 
(every (boy ( or girl)\] l) 
2.16 (not ((every man) speak)), 
where \[ \]l is a combinator or operator which makes an intransitive 
verb phrase out of a transitive verb and a noun phrase. This com- 
binator is represented by a declension in inflectional languages 
such as Slavic ones, while in the case of uninflected languages such 
as English it is taken care of by word order. 
It is remarked that any combinator could sometimes be applied 
from the left to the right as suggested by Cresswell \[5\] • This has 
already been practised in som@ of the above sample sentences. Thus 
such sentences like: 
((every man) \[(some (dog (or cat )) love\] 1 ), 
and 
((some woman) \[(some man)(love not)\] 1 ), 
are also well-formed, being close to the word order usual in Japa- 
nese. 
As is well known, 2.14 has another deep or logical structure 
in quantificational theory with 'some' having the wider scope than 
'every'. In this case, the given sentence is of the form: 
( \[(every man) love\] 2 (some woman)), 
A LESNIEWSKIAN VERSION OF MONTAGUE GRAMMAR 141 
where \[ !2 combines a noun phrase and a transitive verb phrase 
giving rlse to an expression corresponding to a monadic predicate. 
But, unlike \[ \]l, the doun phrase to be combined is in the nom- 
inative case. 
The use of \[ \]2 will be illustrated as follows: 
2.21 (\[(some man) hate\] 2 (every woman)), 
2.22 ((every woman) \[(some man) hate\]2), 
which are of the same structure with each other with 'some man' re- 
maining the subject of these sentences. 
As fs easily understood from the development up to the present, 
the (well-formed) expressions of the proposed LMG as a logic are de- 
fined in terms of the expressions both constant nnd variable having 
the category of names in the sense of Lesniewski's ontology and re- 
lations as well as of a number of logical operators not only senten- 
tial and quantificational but also name-for~ing and relation-#orm- 
ing. (For Lesniewski's ontology consult Iwanus \[5\], Luschei \[9\] 
and Slupecki \[11\]. ) 
3 Axiomatization of LMG as a logic ;w~ If we are to develop 
LMG as a logic,we have to axiomatize it as a logical system. 
Fortunately the axiomatization of Lesniewski's ontology has 
been intensively worked out ever since its single axiom was first 
proposed by Lesniewski himself in 1921. 
Thus, we are starting with the celebrated single axiom origi- 
nating from Lesniewski: 
3.01 e(a, b) ~ .(3x)e(x, a) 
~(x)(~(x, a)O ~(x, b)) 
A(x)(y)(C(x, a)A £(y, a).O E(x, y)), 
or its simplified version by Soboeinski \[19 : 
3.o2 ~(a, b) ~ (~x)(~(x, a) ^ E(x, b)) 
(x)(y)(~(x, a) A ~(y, a).D ~(x, y)), 
where e stands for 'the' and a, b,... and ~he like are (meta-) log- 
ical variables ranging over the expressions of the category of Les- 
niewskian names. On the other hand, e(a, b) stands for ((ea) b) or 
((a e) b) or (b (ea)) or (b (a e)), which are forthcoming as a re- 
sult of the liberalization due to Cresswell \[3\]. Analogously, 
A(a, b) (l(a, b)) represents ((A a) b) etc. ((I a) b~ etc.) with 
A (I) taking the place of 'every' ('some' ). (A and I are also 
known as syllogistic funetors corresponding to 'every' and 'some' 
respectively.) 
Nevertheless, 3.01 or 3.02 is not enough to develop LMG as a 
language. In fact, we need a number of additional axiom (schemata) 
for taking care of name- and relation-forming (logical)operators 
and the expressions involving \[ \]1 and \[ \]2. 
The axiom (schemata) stipulating these operators are well- 
known, being of the forms: 
3.11 e(a, b and c) 5 .E(a, b) A e(a, c), 
3.12 e(a, b or e) ~ .c(a, b) V e(a, c), 
3.21 e(a, not b) ~.e(a, a)A~e(a , b), 
3.22 (R and S)(a, b) ~ .R(a, b) A S(a, b), 
3.23 (R or S)(a, b) ~ .R(a, b) V S(a, b), 
3.24 not R(a, b) ~ .e(a, a) A e(b, b)A~ R(a, b), 
3.25 en R(a, b) ~ R(b, a) 
3.26 R(a, b) ~ .e(a, a) A g(b, b), 
where in 3.26 R is atomic. We are also abbreviating such expres- 
142 A. ISHIMOTO 
sions as ((R and)S)and the like as (R and S) for the purpose of per- 
spicuity. 
Lemma 3.3 3.25 holds of any relation R. 
This is easily proved on the basis of 3.23-3.24 by induction on 
the length of the given R. 
3.~1 A(a, h) ~ (x)( c(x, a) D ~(x, h), 
3.h2 I(a, b) ~ ~x)( e(x, a) ~ e(x, b)). 
We are now presenting some of the axiom (schemata), which take 
care of the expressions containing \[ \] 1 and \[ \] 2. 
3.51 e(a, \[(e b) R\] 1 
.(Bx)( g(x, a)A (3z)( £(z, b) A R(x, z)) 
A(x)(y)( ~(x, a) ^ E(y, a). O ~(x, y)) 
^(x)(y)( E(x, b) ^ ~(y, b). O ~(x, Y) 
3.52 A(a, \[(I b) R\] 1 
(x)( s(x, a)O (BY)( e(y, b) % R(x, y)) 
3.53 l(b, \[(A a) R\] 2 ) (3y)( ~(y, b) A(x)( ~(x, a) O R(x, y)) 
It is noticed that some of these axioms are not well-formed as sen- 
tences of LMG as a language although they belong to LMG qua logic. 
Theorem 3.7 Every simple sentence of LMG (as a language) is 
equivalent to a sentence (of LMG as a logic), and this sentence in- 
volve only g and atomic relations besides logical operators with 
quantifiers binding only such name variables x and y as occur there 
in the context e(x, a) or R(x, y). 
The proof is carried out by induction on the number of symbols 
other than those mentioned in the theorem on the basis of axioms. 
It is again observed that the formulas to which these sentences 
of natural language are transformed are not necessarily those be- 
longing to LMG as a language. 
4 Translation of LMG into predicate logic It will be shown 
in this section that LMG as a language is embedded in first-order 
predicate logic (with equality) via a translation T to be defined 
presently. (The proposed translation dates from Prior \[8\] and has 
been elaborated by Ishimoto \[4\] and Kobayashi-Ishimoto \[7\].) 
The translation T is defined by induction on the number of the 
words contained in the given expression of LMG. 
In the first place, the basis is taken care of by: 
T a = Fa, 
T b = Fb, 
where a, b,... are (atomic) names constant and variable, and Fa, Fb, 
... are monadic predicates again constant or variable corresponding 
to a, b,... not necessarily exhausting all of them. 
T e = IPIQQlxP x, 
T A = IPIQ~x(P x O Q x), 
T I = IPIQBx(P x A Q x). 
We are now proceeding to the induction steps: 
T aA8 = T a A T 8, 
T av8 = T a V T 8, 
T aDS = T e D T 8, 
T ~ =~T ~, 
T (x)a =VF x T G, 
T (Bx)a =BF x T a, 
where e, ~ .... are meta-logical variables ranging over the 
A LESNIEWSKIAN VERSION OF MONTAGUE GRAMMAR 143 
sentences of LMG. 
Before" taking up the translation of relations, i.e. transitive 
verbs we have to introduce in advance another translation T 1 which 
transforms every relation in LMG into a (binary) relation of pre- 
dicate logic. The translation T 1 is defined inductively on the num- 
ber of the relation-forming operators employed for defining the 
given one. 
Starting with the basis: 
T 1 R = IxlYGR(X , y), 
where G R is the (binary) relation (of the Dredicate logic) corres- 
ponding to the give atomic relation of LMG, induction steps are: 
T 1 H and S = Ixly(T l R(x, y) A T, S(x, y)), 
T I E or S = Ixly(Ti R(x, y) ~ T, S(x, y)), 
T 1 not R = Ixly~Tl R(x , y), 
T 1 en E = IxlyTi R(y , x) 
(T1 R and S etc. will be abbreviated as Go . ~,etc.) 
On the basis of the translatzon T 1 thus introduced, T is de- 
fined for any relation R of LMG as follows: 
T R = IPIQT 1 R( IxP x, IxQ x), 
with R( IxP x, IxQ x) being defined as: 
~xBy(P x ^ Q y ~ R(x, y)) ~ VxVy(P x ^ P y.O x = y) 
^ ~xVy(Q x ^ Q y. O x = y). 
Lastly the translation T is applied to the operators \[ \]i: and 
\[ \]2in the following vay~ 
T \[ \]i = ~VIWlx(VlyW(x, y)), 
T \[ \]2 = ~V~W~y(V~xW(x, y)), 
where V and W are respectively the variables of the type of noun 
phrases and (binary) relations in predicate logic. 
Availing ourselves of the translation T thus defined, some 
sample sentences (of LMG) will be translated into the corresponding 
sentences of predicate logic. 
4.11 T 2.11 = T ((the man) walk) = T ((e man) walk) 
= ((T e T man) T walk) 
= (IPIQQIxP x) Fman) Fwalk = FwalklXFmanX , 
4.14 T 2.14 = T ((every man) \[love (some woman)\]l) 
= (IPIQ~x(P x D Q x) Fma n) 
((~VIWIx(VIyW(x, y~(IPIQ~x(p x A Q x) Fwoman) Glove ) 
= IQ~(FmanX ~ Q x)Ix(IQ~y(FwomanY A Q y) lyGlove(X, y)) 
~X(FmanXO BY(FwomanY A Glove(X' Y)))" 
As has been exemplified by the above translations we easily obtain: 
Lemma ~.3 Every sentence of LMG as a language is translated 
by T into a formula of first-order predicate logic with equality. 
Lemma h.4 The translation of the theses of LMG as a logic are 
provable in predicate logic. 
The proof is carried out by induction on the length of the 
proof. 
The treatment of the basis will be illustrated by the following 
example: 
4.~i T 3.02 ~ :T ~(a, b) ~ .(Bx)( ~(x, a)~ ~(x, h)) 
(x)(y)( c(x, a) ^ ~(x, b).D E(x, y)): 
:FblXFaX ~ . ~ Fx(FalXFxx ~ FblXFxX)~ 
VFx~Fy(Fa!xFxx A FalxFyX. ~ FylxFxx): 
:FblXFa ~ • ~x(Fax ~ FbX) ~ ~xVy(Fax ~ Fay. ~ x = y). 
144 A. ISHIMOTO 
Here use is ~made of some theses o~ second-o~der predicate logic. 
All the other axioms, if translated by T, will turn our to be 
provable in (higher-order) predicate logic. The induction steps 
do not present any difficulties. 
In view of Lemmas ~.3 and h.h we obtain, 
Corollary h.5 If a sentence of LMG as language is a thesis of 
LMG as a logic, then its T-transform is provable in first-order pre- 
dicate logic with equality. 
It is remarked that the proof of the T-transform of a sentence 
belonging to LMG as a language might involve formulas not necessa- 
rily belonging to first-order predicate logic. 
Lastly we wish to state without proof a lemma of fundamental 
importance, namely, 
Lemma h.6 If the T-transform of a sentence belonging to LMG 
as a language is provable in first-order predicate logic with equal- 
ity, then the sentence is a thesis of LMG as a logic. 
This is proved syntactically as well as semantically by the 
method employed in Ishimoto \[4\]. 
Combining Corollary ~.5 and Lemma h.6 we obtain, 
~L Theorem ~.7 For every sentence ~ of LMC as a language 
MG e iff T ~ is a thesis of first-order predicate logic with equal- 
ty. 
In view of theorem h.7 as far as the logical derivability of 
some sentences of natural language as specified above is concerned, 
there is no difference between first-order predicate logic and Les- 
niewski's ontology. Use will be made , it is hoped, of this fact 
in the various fields ;elated to the logical analysis of natural 
language. 
*An earlier version of this paper was read before the Austra- 
lasian Logic Conference, Wellington, 1981. 

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