aodology and Verifiability in Rontague Grammar 
Seiki Akama 
Fujitsu Ltd. 
2-4-19,Sin-Yokohama, Yokohama, 222,Japan. 
Abstract 
Nethodological problems in Men\[ague Grammar are 
discussed. Our observations show that a 
model-theoretic approach to natural language semantics 
is inadequate with respect to its verifiability from 
logical point of view. But, the formal attitudes seem 
to be of use for the development in computational 
linguistics. 
O..introductlon 
In this paper we discuss the methodology of 
verifiability taken by researchers on model-theoretic 
semantics for natural language such as ~ontague ~rammar. 
Though Montague grammar (hereafter MG) has been 
developing since the publication of Montague\[lO\], there 
has been few serious studies of its 'sense' and 
methodology. 
Ne take the purpose of semantics to be as follows. 
I • , (a) To define a meanLng . 
(b) To define a 'meaning' of certain linguist-it 
expressions. 
(c) To generalize a 'meaning' referred as (h) in 
connection with internal world (human) and 
external world. 
ltere (a) is so abstract that it must he dicussed in 
general linguistic terms rather than in MG. But it is 
no doubt that the methodologies in ~G are based on the 
assumption (c). The problem (c) is central to MG. In 
MG semantic structure corresponding to syntactic 
structure of natural language is realized by means of 
its methodologies. 
The problem (c) is closely related with pragmatlcs 
and epistemology thus HG includes parts of them. As 
Chomsky's early transformational grammar was obliged to 
changes of the system for the sake of autonomous syntax 
hypothesis, the problem is important in MG. lntensional 
and possible-world semantic~ could solve parts of 
the problems. But it is difficult to ~ay that MG is a 
system facilitating (c). And methodological problems of 
MG including (c) are mainly ascribed to model theorj 
underlying MG. Ne shall focus on the point and discuss 
~G's methodology. Ezpecially following problems are 
investigated. 
(1) Is in\[arts\[anal logic necessary? 
(2) Can modal (tense) logic express a modality 
(tense) in natural language? 
(3) Is first-order logic necessary? 
(4) Is there a possibility of natural logic? 
(bO Are there appropriate methods for the 
interpretation of logical form? 
(6) Is there a distinction between logical words 
and content ~rds in natural language? 
~, MG and ~ode\[ Thepr2 
The purpose of model theory is to investigate the 
relationships between a true statement in some formal 
language and its model. Namely, it is to define a 
concept of 'truth' in some mathematical structures. In 
mathematical logic Tarski\[14\] first attempted to study 
the idea of model. In his paper Tarski mainly concerned 
himself with the definition of truth (the correct 
definition of a true sentence), lie confined his 
discussions to the object in the framework of predicate 
logic in the sense modern logic, lie despaired to define 
a true sentence in natural langufige. Since we are 
obliged to face to paradoxes for the sake of 
universality of natural language. But he suggested that 
there exists a possibility of application of the resutt~ 
for model theory, which he gave to the language he 
called 'formalized language', to natural language. 
88 
About forty years after the publication, Hontague, who 
is a disciple of his, could give a model theory for 
natural language. Moatague regarded intensional logic 
as a basis of his theory so as to overcome complexities 
of natural language. He was able to solve paradoxes, 
that Frege and others faced, by means of intensional 
logic. 
First we consider the problems of intensional 
logic. The model of intenslonal logo comes to be more 
complicated because it has a greater descriptive power 
than predicate logic in general. As Gall\[hi3\] pointed 
out, valid formulas in intensiona\[ logic fail to 
constitute a recurs\[rely enumerable set since it is 
essentially based on type theory. Thus we have no 
axiomatization for this logic. For this reason, we must 
restrict the scope of sentences in natural language 
capable of being treated by intensional logic. But the 
notation of intens~onal logic used in PTfi such as '" 
and '~' work efficiently for analysis, For example, 
consider the following sentences. 
Every man loves a man. (1-1) 
We have two interpretations for the sentences, namely, 
(every man)(loves a man.) (I-2) 
(every man loves)(a man.) (i-3) 
In general we call (i-2) de dicte reading, (I-3) de re 
reading, and obtain the following translations 
respectively, 
Vx(man' (x) --> love' (x,'~Q~y(~oman' (y) A Q{y})). 
(1-4) 
By(woman' (5,) AVx(man' (x) --> love' (x, *~PP{y}))). 
(I-5) 
Seen from the above formulas, in (1-2) that every 
man loves is not an individual 'woman' bat a property of 
property of a individual 'woman'. Ihat is, the meaning 
of individuals (inteesion) is considered as a mapping 
from possible-worlds to a reference (extension). If t.e 
define a possible-world as a set of indices, and 
determine the value for each index, then some extension 
is defined. But we doubt that an intenslon defined in 
intensional logic properly represents a meaning. 
In MS individuals and truth values are assumed as 
semantic primitives. Using possible-world semantics we 
can extend predicate logic. This extension causes the 
atructure of model to be more complex, and produces lots 
of contradictions as natural language semantics. Above 
all the problems of logical enuivalence is serious. For 
example, assume a and h for logically equivalent 
formulas, that is, a and b are true in same indices. 
Then it is a valid inference from doubt(p,'a) to 
doubt(p,'b). If we doubt a, we would doubt b logically 
equivalent to a from the standpoint of logically 
equivalence thus for p, a and b have differernt 
meanings. To put it more correctly, the meaning of 
'doubt' in a and b is different unless p knows the 
correct sense of logically equivalence between a and b. 
Such a statement fails to be explained in tradltonal 
logic. This is nothing but a limitation of ordinary 
mode\[ theory. Researchers such as geenan\[8\], 
Thomason\[15\] and Turner\[1G\] tried to extend tntensionai 
logic from various viewpoints. Thomason added 
intensional logic to third type of propositions, which 
is a denotation of a sentence. Thus we clearly need a 
domain containing at least two propoaitiona of a model 
for intone\[anal logic. Eeenan introduced the concept of 
9ntple~ ca ly perfection, that is, the element of the 
ontology are poasible denotatona for extensions for 
expressions, by means of Boolean algebra. ~is 
motivation is to restrict a domain of intenslonal logic. 
Thus the set of possible world is defined in terms of 
~Oxlmally conslstent sot of propositions, sentence 
denotations. Turner\[16j extended intenslonai 1ogle in the sense 
of type-free theory in which a self-annlication is 
permitted for the treatment of nominalizations. We are 
very intere:;ting in such strategies since in Scott-type 
denotational semantics we have no intermediate language 
as in PTQ. Thus we can obtain semantic interpretation 
of a sentence directly. We have an idea for types of 
natural language, namely, polvmorohic types, which can 
have various types. These types are essentially 
considered as a subset of type-free theory. 
Above mentioned trials are restrictions to a mode\[ 
for intensional logic. But such perplexed constructions 
muct cause us more difficulties in reality. Hunt we 
give up thi.'~ logic? It is certain though intensionai 
logic has the sides against our intuitions, it can 
provide a powerful model for some phenomena. For 
example, consider the following sentences referred to as 
~sadox. 
(I) The temperature is ninety. 
(2) The temperarure rises. 
(3) Ninety rises. 
The~e are translated into formulas in intensional logic 
as : 
(I) ~y(Vx(temerature' (x) <--> x=y) ^"y=n) 
(2) ~y(Vx(temerature' (x) <--> x-y) .~rise' (y)) 
(3) rise' ('n) . (I-7) 
As seen from (?) Hontague dealt with noun phrases as 
objects which have intensions and extensions. In the 
examples, intensions are represented as functions that 
denote some number at each index, and extensions are 
rendered as particular number such as 90 at certain 
index. Namely, the truth value of sentence (2) in (1-6) 
depends not on extension but on intension. For this 
reason verbs such as 'rise' referred to as intensional 
verb...~. But such for~lisms seem to be recaputuiated in 
the framework of predicate logic. If so, it is 
effective from not only intuitive but also computational 
point of views. ~Such formalisms are divided into b#o 
approaches. One is an approach that is an extension of 
predicate logic to intensional logicusing some devices 
as in Schubert and Pelletier\[13\]. Another is an 
approach that intensionnI logic is interpreted as a 
programming language such as LISP as in Hobbs and 
Rosenschein\[G\]. Schubert and Pelletier stated that 
predicate logic is suitable from the viewpoint of A\[ 
systems. According to them, the expressions in 
intensional logic are not comprehensive to human being. 
For example, it is better understandable to capture 
definite noun phrases as individuals than a set of 
properties. Slot representations conquest gaps to 
intensional. In this formulation a proper name is 
represented as a constant, a common noun as a monadlc 
pedicate and a transitive verb as a binary predicate. 
'Hary' =:> Hary I 
'boy' "=> (Ili boy) (I-8) 
'loves' ~:> (lit loves tt2) 
ltere ~n is called argument slot that is filled from 
higher number in turn. The sentence (i-2) and (i-3) are 
translated as follows. 
de dlcto: 
for al I (~I man) ((~I loves 112) (for some (112 woman))) 
==> YX(X man) =-> (xlovesA-~y(y woman))) (i-9) 
de re: 
for som.(l;~ woman)(Ill loves ltg)(for all(ill man)) 
"=>~y((y woman) A(Vx(x man) --> x loves y)) (i-I0) 
These translations are similar to the formulas in 
predicate logic, ltere slot representations enable us to 
operate a scoplng of noun phre~es. This device seems to 
have some simulating with combinators in combinatory 
logic. 
Ilobbs and Rosenschein tried to convert intensional 
logic to S-expressions in LISP. The lambda expressions 
are considered as the pure LISP thus the conversion is 
plausible. Such expressions are exemplified as follows. 
(constant) -:> (QUOTE ~() 
m(a variable of type <s,b> for any b) ==> (\[IUOTE~() 
"W "> (LIST(QUOTE QUOTE,S) (i-\[i) 
"~ ,-> EVAL o( 
The sentence (l) in (I-7) is translated in 
ninety ==> (L,~BDA(Q) ((g~) (INT* 90)) 
be ==> (L,1NBDA (P) (LA,~BDA ix) ((P*) (INT~(LA~BDA (y) 
(EOU~L (X*) (YD))))))) 
the temperature ==> (LAHBDA (P)(FOR SO,'IE 
ENTITY-COYCEPTS (LAY, BDA (Y) (FOR ALL 
E,~IT ITY-CONCEPTS (LA,'iBDA (X) 
(At~D(IFF(TEHP X)(EQUAL X Y))((P~)Y))))))) 
the temperature is ninety 
==> ((TIIE (FUNCTIO~I TEHP)) 
(INT~(BE (INT~(FUNCTION NUIETT)))) 
INT ~ (L,~HBDA (G) (LAHBDA (*) G)) (i-12) 
Here we may assume there is a variabIe named * to the 
value of which are applied to produce the corresponding 
extensions. Above two trials are for approximating the 
functions of lntensional logic by means of simpler 
~ystem in order to reduce inherited complexities in this 
logic. In any case deficiencies of intensionaI logic 
are ascribed to model theory, and even if we take it 
off, it is doubtful that intension formulated in 
intensional logic corresponds to the meaning of 
linguistic expressions. 
Next we consider tense logic and modal logic. As 
both logic.~ are based on possible-world semantics we 
come to face tbe name problems in genera1, tlere ~Je 
discuss the problems involved in direct app\[ications to 
natural language. In tense logic the operators P and T 
are able to apply infinitely in principle but in 
practlce the scope of tense has some boundary. Thus it 
is not easy to solve tense in natural language only by 
these t~o operators. Bauerle\[2~ introduced third 
operator T (it is the case on ... that ...) so as to 
overcome shortcomings of traditional tense logic as in 
the axiomatization by Priori13\]. In tense logic the 
following relations hold. 
FF ~ --> F~ (1-13) 
PP P--> P@ (1-14) 
The~e formulas are proved by means of the transitlvlty 
of <. Such relations assume all forms of the past 
(future) tense as quantification over times past 
(future). But to avoid the infinite application of 
tense operators we must take a strategy that tense can 
be considered as a point of reference by Reichenhach. 
That is, we can regard past tense as direct reference to 
some particular past time, not universal quantification. 
Similarly in modal logic it is doubt that the t~o 
operators enable us to explain the modality of natural 
language. First of all modalities are divided into the 
o~.ctive and the su___bb\]ec__tive. And modal logic can 
manage only objective modaliLy. Suppose the folloNing 
examples. 
John cannot be a Japanese. (1-I5) 
It is impossible John is a Japanese. (1-16) 
If we translate these sentences into formulas in 
~G we obtain the one in only (I-16). 
~QJapanese' (j) (= I:I~Japanese' (j)) (I-17) 
In other words the sentence in (1-15) belong to the 
category of snbjective modality thus it is impossible 
that the subject is a logical connection of the function 
to each constihmnt (namely content word) in the 
statement rather than some kind of operation to the 
statement (namely truth value). Unfortunately, most of 
the modalities in natural language belong to objective 
modality. We can state that semantic", in logic is not 
always linguistically valid. Chomsky\[3\] called HG a 
type of descriptive semantic~ except that he thlnk~ it 
is not semantics really, in the sense that it does not 
deal with the classical questions of semantics such as 
the relation between language and the world. 
The situations do not change even if we restrict 
logic to predicate logic. And if we want predicate 
logic to be psychologically real, though we will discuss 
thin in section 2 in detail, we will reply in negative 
due to Lowenheim-Skolem'n theorem. 
When we interpret the so-called logical forms, if 
we depend on the idea of intensional logic, it happens a 
lot of irrationalities. Namely, the interpretation is 
nothing but a decision procedure of truth condition. 
Since ~G is based on Fr~, the truth value 
89 
of a sentence is a function of the one of parts and it 
is difficult to add interpretation of linguistic 
constralnt~ to the system of formal logic. Thus Natural 
Logic was proposed. Lakoff\[9\] said that the semantic 
structure of natural language co~responds to the 
grammartlcal structure and that Natural Logic must be 
able to explain logical inferences in natural language. 
Thus it is possible to consider that Natural Logic 
possesses similar framework to TG rather than HG. From 
the standpoint of Gg theory in TG, IIornstain\[?\] 
pur:ueted logical forms, lie claimed that semantics 
should also he exp\[ained from the same hypotheses 
(hmateness) as syntax. We think that his approach is 
more realistic and rational theory if such theories are 
to be formalized in view of psycho\[egg. We can find a 
similar approach, though it may be more ambitious, in 
Johnson-Laird\[8!. Necessity of Natural Logic seems to 
be derived from the drawbacks of formal logic owing to 
its artificallty. As we take up the sixth problem 
before, there is a clear distinction between logical 
words and content words, and we faced strict type 
constraints. ~ost inferences in natural language are 
executed by means of logical words. In an extreme terms, 
we can infer only if ~e know inference rules. But our 
daiIy inferences seem to depend on the property of 
content words. 
We therefore need the counterpart of inference 
rules in logic for inferences depended on content ~ords. 
The abuse of mean{as postulates at lexicaI level 
provide no solution. Since Natural Logic is based on 
the principle of universal grammar in grammartical 
theory. But if Natural Logic adopts predicate logic as a 
device for logical forms, it is impossible that the 
logic overcome its difficulties. 
2. ~ and tln~uisti¢ Theor', 
Finally we shall investigate into philosophlcal 
aspects of ~g. We can find fen research involved in the 
issues of methodology and philosophy in HG. • The 
exception is Partee\[ll\]. She tried to justify 
theoretical validity of MG in connection with 
psychological reality. Hen\[ague himself apprared to 
reconstruct linguistics oa the basis of the same 
methodo\[ogy in mathematics, thus there ex\[sta no 
psychological factor here. Dowty\[~\] also stands in the 
position that semantics is a field handling the 
reIationships between linguistic exprssioas and external 
worlds. Are there hypotheses in ~G in different place 
from our mind? We hard to receive such radical 
opinions. Even if we discover reality in ~G, it is 
doubtful whether theoretical validity of HG is verified. 
For example, we have the assumption that individuals 
and truth values are semantic primitives in ~G. What is 
an individual? At a first glance individuals are 
grasped at ease, but we can never clarify what it is. 
The assumption of model theory says that a set of 
individuals is never empty in some structure. Suppose a 
possible-~orld that consists of only humans as its 
elements. Even if this set has countably infinite 
power, i t will be empty someday because humans are 
mortal. This contradict: the assumption. Hare doubtful 
fact is tm~ individuals corresponding to dead humans are 
represented in a model. And, by Lowenheim-Skolem's 
thereto there exists a countable model if a model exists. 
This impties that we have difficulties to identify a 
set of individuals in its model. Can ~e find 
verifiabill ty and reality in such concepts? 
Now we cannot deny a human semantic competence. 
Partee derided level of semantics into t~o parts and 
insisted that semantics in lexica\[ level ia a mental 
par~. The claim sho~s that it is improper to advance 
model-theoretic approaches in ~g to linguistic lever. 
llere we recognize many problems in her insistence° 
According to her argument, it is realistic to choose 
appropriate individuals and possible-worlds in models of 
intensional logic and Hontague's attempt is to define 
not a unique intensional modot but a family of models. 
We believe human can never recounize such models in his 
90 
mind. She said that human need not know all 
possible-worlds and choose opt\[mai world by means of the 
mechanisms as \[nductiou. This idea'is very suspicious 
but we do not know how to verify it now. That is, the 
specification of a particular actual model, $dlich she 
called, cannot be 'realistic' if we use model theoretic 
semantics as intensional (or predicate) logic. 
From above considerat\[ons, we Nlll conclude the 
following. Lingulstic~ is a part of philosophy rathe:" 
than psychology. Since psychology has not complete 
systems, we do not intend to say psychology i~ an 
incomplete study, the object of semantics is bo~h humaa~ 
ourselves and external worlds. Of course we can mention 
that methodology in ~G is a small part of our internal 
world. ~e want to insist that we ought to unify 
pragmatics as ~G provided the ~ay unifying syntax and 
semantics. ~ethodology in ~G must be a foothold of it. 
At that time it does not matter whether there exists a 
reality in the methodology. The important thing is that 
such a methodology can constitute a part of realistic 
linguistic theory. \[n other words, logical forms may be 
interpreted both more logically and psychologically. 
After all we can oniy see the worlds through tinted 
glasses, namely our language. To make matters worse, we 
never take off our glasses. Living things such as bees 
and birds may look the ~orlds in more effective ways. 
And we want to know abner the worlds more. To do so, we 
come to set down our tinted lense. In the case of ~G 
its settings are performed by model theory. If the 
degree of lense slip down we will look at the world in 
strayed eyes. If we fall into the case, we should 
reflect on ourselves again. This reflection MII cause 
us to find the way hew to know natural language better. 
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