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<Paper uid="C86-1019">
  <Title>aodology and Verifiability in Rontague Grammar</Title>
  <Section position="1" start_page="0" end_page="90" type="abstr">
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Abstract
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    <Paragraph position="0"> 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.</Paragraph>
    <Paragraph position="1"> 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.</Paragraph>
    <Paragraph position="2"> Ne take the purpose of semantics to be as follows.</Paragraph>
    <Paragraph position="4"> (b) To define a 'meaning' of certain linguist-it expressions.</Paragraph>
    <Paragraph position="5"> (c) To generalize a 'meaning' referred as (h) in connection with internal world (human) and external world.</Paragraph>
    <Paragraph position="6"> 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.</Paragraph>
    <Paragraph position="7"> 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.</Paragraph>
    <Paragraph position="8">  (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.  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.</Paragraph>
    <Paragraph position="9"> 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 '&amp;quot; and '~' work efficiently for analysis, For example, consider the following sentences.</Paragraph>
    <Paragraph position="10"> 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,</Paragraph>
    <Paragraph position="12"> 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.</Paragraph>
    <Paragraph position="13"> 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.</Paragraph>
    <Paragraph position="14"> 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.</Paragraph>
    <Paragraph position="15"> 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.</Paragraph>
    <Paragraph position="16"> 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.</Paragraph>
    <Paragraph position="17"> 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.</Paragraph>
    <Paragraph position="18">  (I) The temperature is ninety.</Paragraph>
    <Paragraph position="19"> (2) The temperarure rises.</Paragraph>
    <Paragraph position="20"> (3) Ninety rises.</Paragraph>
    <Paragraph position="21"> The~e are translated into formulas in intensional logic as : (I) ~y(Vx(temerature' (x) &lt;--&gt; x=y) ^&amp;quot;y=n) (2) ~y(Vx(temerature' (x) &lt;--&gt; 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.</Paragraph>
    <Paragraph position="23"> 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.</Paragraph>
    <Paragraph position="24"> de dlcto: for al I (~I man) ((~I loves 112) (for some (112 woman))) ==&gt; YX(X man) =-&gt; (xlovesA-~y(y woman))) (i-9) de re: for som.(l;~ woman)(Ill loves ltg)(for all(ill man)) &amp;quot;=&gt;~y((y woman) A(Vx(x man) --&gt; 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.</Paragraph>
    <Paragraph position="25"> 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.  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.</Paragraph>
    <Paragraph position="26"> 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.</Paragraph>
    <Paragraph position="28"> The~e formulas are proved by means of the transitlvlty of &lt;. 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.</Paragraph>
    <Paragraph position="29"> That is, we can regard past tense as direct reference to some particular past time, not universal quantification.</Paragraph>
    <Paragraph position="30"> 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.</Paragraph>
    <Paragraph position="31"> 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).</Paragraph>
    <Paragraph position="32"> ~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&amp;quot;, 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.</Paragraph>
    <Paragraph position="33"> 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.</Paragraph>
    <Paragraph position="34"> 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.</Paragraph>
    <Paragraph position="35"> Since ~G is based on Fr~, the truth value  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.</Paragraph>
    <Paragraph position="36"> 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.</Paragraph>
    <Paragraph position="37"> 2. ~ and tln~uistiC/ 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.</Paragraph>
    <Paragraph position="38"> 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.</Paragraph>
    <Paragraph position="39"> 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.</Paragraph>
    <Paragraph position="40"> llere we recognize many problems in her insistencedeg 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.</Paragraph>
    <Paragraph position="41"> We believe human can never recounize such models in his  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.</Paragraph>
    <Paragraph position="42"> From above considerat\[ons, we Nlll conclude the following. Lingulstic~ is a part of philosophy rathe:&amp;quot; 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.</Paragraph>
    <Paragraph position="43"> 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.</Paragraph>
    <Paragraph position="44"> 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.</Paragraph>
    <Paragraph position="45"> 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.</Paragraph>
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