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<Paper uid="C69-2901">
  <Title>Celee, Marianne and Robert M. Sehwarcz, Nominal quantification in En@lish: a semantic description. System Development</Title>
  <Section position="2" start_page="2" end_page="2" type="metho">
    <SectionTitle>
T~ PROBLEM
</SectionTitle>
    <Paragraph position="0"> A. Requirements for a Semantic Theory 1. A Definition of Semantics  The term semantics is generally used to denote the system of relations between the expressions of a language and their meanings--in contrast to synts~x, which describes the acceptable structural forms of linguistic expression, and pragmatics, which concerns itself with the effects of communications in a language upon the communicants. The term &amp;quot;meaning&amp;quot; here is to be analyzed in terms of its particular relevance to the act of communication. An act of communication includes a sender who encodes a message into a signal (usually a string of phonetic or alphanumeric symbols), a channel along which the signal is sent, and a receiver who decodes the signal into the original message and interprets the message. Meaning is the functional import of the message, which calls forth a re sponse--cognitive~ affective, and/or conative--from the receiver in the partfcular communication situation; it is a relation of the message to the receiver. Let us now make the simplifying assumption that this meaning is determined by the functional form of the message under some &amp;quot;standard&amp;quot; functional interpretation (this is basically an oversimplification, since the rules of functional interpretation, especially on the affective response dimension, will vary from receiver to receiver and even for the same receiver over time); we may then define semantics as the explication of the relationship between the surface forms of linguistic utterances (as spatial or temporal arrangements of morphological units, which are the minimal linguistic units that have a direct functional relationship to the determination of meaning) and the functional forms of the messages they express. For a semantics to be formalized, this relationship must be sufficiently well defined to enable the processes of encoding and decoding to at least be formally explicable in terms of it, if not totally formalized within it.</Paragraph>
  </Section>
  <Section position="3" start_page="2" end_page="36" type="metho">
    <SectionTitle>
2. The Semantics of Formalized Languages
</SectionTitle>
    <Paragraph position="0"> The most successful attempts to formalize semantics have been for the formalized languages of the deductive sciences. Tarski, in his classic paper \[43\]3 has set down a systematic method for formalizing the semantics of a formalized language, which is any language for which the set of meaningful sentences is defined by a formal syntactic grammar, all sentences are unambiguous, and there is a set of (syntactically defined) axioms and rules of inference from which theorems in the language may be derived. The method involves construction of a metalan~uage containing expressions and axioms of a general logical kind, translations of the expressions and axioms of the language L to be characterized, and expressions and axioms which define the syntax of L. The notions of satisfaction and truth for the language L, which together constitute a semantic description of L, can be defined in the metalanguage M as follows: Given a domain D of individuals, a &amp;quot;semantic interpretation function&amp;quot; ~ is defined in M which assigns to each individual constant and individual variable of L an individual of the domain R, to each function letter of L of degree n m 1 a function from D n (the set of all ordered n-tuples of elements of D ) intoD, to each predicate letter of L of degree n ~ 1 an n-ary relation on D (defined as a subset of Dn), and to each phrase-forming rule of L a function Ti, which determines the semantic interpretation of the phrase formed in terms of the semantic interpretations of the constituent phrases. An n-tuple (al,...,an&gt; of individual constants then satisfies the sentential function F of n free variables whenever C/ (F) \[~5 (al),... ~ ~5 (an) \] = T (truth). The notion of truth is defined as a special case of the notion of satisfaction for sentential functions of zero free variables l (i.e., sentences of L). Examples of the application of the Tarski approach are to be found in any standard logic textbook in the truth-table interpretations (standard and non-standard) of the propositional calculus and the standard set of semantical rules for the first-order predicate calculus.</Paragraph>
    <Paragraph position="1"> The Tarski approach runs into difficulty, however, in that it establishes only a single notion of truth, without reference to the different ways in which truth may be epistemologically established. This becomes critical when it comes to establishing rules of substitutability for &amp;quot;oblique&amp;quot; or &amp;quot;nonextensional&amp;quot; contexts such as quotation, indirect discourse, modal sentences, and belief sentences, in which the unrestricted substitutivity of equivalence does not preserve truth-value--one must here define types of equivalence stronger than identity of reference, which is the type of semantic equivalence defined in the Tarski approach. It was mainly to deal with this problem that Carnap \[9\] proposed a method of semantic analysis called &amp;quot;the method of extension and intension.&amp;quot; In the framework of Tarski's formulation, the method can be stated as follows: Given a &amp;quot;model&amp;quot; of the language L, consisting of an individual domain D and a semantic interpretation function ~b for L over P, the extension of any well-formed expression E in L is defined as the set of values for E of all semantic interpretation functions #' over P which differ from # at most on their assignments to the free variables of E. Now if one considers the domain of possible models for L, the intension of any well-formed expression E in L may be defined as that function over models of L which yields as its value for any model the extension of E in that model.</Paragraph>
    <Paragraph position="2"> The notion of intension may be formalized by considering the metalanguage translations of well-formed expressions of L to be intensional structures for these expressions, which, since intensions are funcZions, will take the form of function definitions. The translations may be defined by a translation function e which assigns to each individual constant, function letter, and predicate letter of L an appropriate function letter of the metalanguage M, to each variable of L a variable of M ranging over functions on models of L which map into the appropriate extensional range, and to each phrase-forming rule of L a function-definition operator ~i (which could be functional composition, c~nplement, union, intersection, iteration, transitive closure, summation, minimalization, etc. ). Given this definition, one can recreate Tarski's definition of satisfaction and truth by noting that for any model . of L the corresponding semantic interpretation i function C/i is given by C/i (E) = \[ e (E)\] (Mi). And one can define, along with the ordinary notion of (extensional) equivalence, the notions of L-equivalence and intensional isomorphism as equivalence of intension and intensional structure respectively--and show, as Carnap does in \[9\], how these stronger types of equivalence permit the establishment of suitable substitutability criteria for &amp;quot;oblique&amp;quot; contexts.</Paragraph>
    <Paragraph position="3"> To fit the semantics of formalized languages into our general definition of semantics, which presupposes use of the language for the purpose of communication, we must introduce one more thing into the metalanguage, namely, the performative operators of asserting, questioning, and ccmmmading. We posit that the language L is being used to communicate between two information and control systems A and B, both of which possess inccmplete and/or changing models of L 2 corresponding to knowledge of some environmental situation over which both A and B can exercise i 2~ote that the reference to disparate models of L (the only situation in which communication would ma/~e sense in this context) and changing models of L necessitates the use, at least PSmplicitly, of an intensional semantics.</Paragraph>
    <Paragraph position="4"> certain degrees of control. Then for any sentence $ of Lj an assertion '$.' from A to B carries the functional import of instructing B to modify its model so as to make S evaluate to truth, a ~uestion '$?' from A to B instructs B to evaluate S in its model and return the result to A, and a command 'S '' from A to B instructs B to modify its environment (if possible and if necessary) so that S ~-lll evaluate to truth in the model of the environment so changed. The metalanguage translations of the assertion, question, and command signs in L will be, of course, the corresponding performative operators. By identifying, now, the notion of message with metalanguage translation, the notion of arrangement of morphological units with syntactic description in the metalanguage, the notion of decoding with the translation function e, and the notion of encoding with the inverse of ~, we show how the semantics of formalized languages meets our general requirements for a formalization of semantics.</Paragraph>
    <Paragraph position="5"> 3- Natural Language Semantics We may now arrive at a set of specific requirements for a formal theory of natural language semantics by examining the crucial differences that are kno~m to exist between natural languages and formalized languages and noting the revisions and extensions of the formalized-language paradigm that are required to take these differences into account. This approach is indicated by the fact that the semantics of formalized languages represents the most highly-developed point of departure frc~ which to undertake a formal description of the real-world phenomenon of natural language semantics, and thus, if it indeed contains the potential of producing a description that fits the phenomenon, brings one much closer to that description than if one were to start with only the general definition of semantics given at the beginning of this section. What, then, does this approach indicate for the features of a revised paradigm under which the known properties of natural language as an instrument of communication may be subsumed? Natural languages, first of all, are used for a vastly wider variety of communication acts than are formalized languages. The messages that are communicated in natural language relate to virtually every area of human activity and extend to nearly every purpose involving some kind of human interaction. As a result, there is a large inventory of different types of messages that are expressed in natural language, each in its own particular way or ways. 3 The three basic types of performative operators-assertion, questioning, and commanding--are subject to modific~ion as to the manner of the request conveyed by the message (which indicates, among other things, the speaker' s perceived or intended relation to the hearer 4) and to functional combination, as in the case of threatening and warning, both of which combine commanding with asserting. A formal theory of natural language semantics must explicate these dimensions of the performative and also the relation of the performative to the notions of speaker, hearer, and context of utterance.</Paragraph>
    <Paragraph position="6"> The explication of the non-performative parts of messages as intensional definitions requires some extension and elaboration</Paragraph>
    <Paragraph position="8"> in order to be applicable to natural language. First, people carry in their memories not one model but many, corresponding to the many different situations that they have knowledge of.</Paragraph>
    <Paragraph position="9"> Thus, a message must refer either to a specific model, to a specific range of models, or generically to all models in which the specified intensions have nonempty extensions. Restricting the range of applicable models is accomplished in natural language through presuppositions, which indicate prior conditions that must be satisfied in a model for a given message to be applicable to it. Indicating generic vs. model-specific information, as well as &amp;quot;given&amp;quot; (for locating the appropriate model) vs. &amp;quot;new&amp;quot; (for adding to the model) information in the model-specific case, is accomplished through the subject-predicate division and through an extensive assortment of quantifiers. Furthermore, as Morris \[31, 32\] has pointed out, natural-language expressions not only designate but also appraise and prescribe--thus, natural-language intensions may take on as extensions not only objects, sets, and relations, but also values and actions. Natural-language intensions may also take on as extensions other intensions, thus giving natural language a &amp;quot;recursiveness&amp;quot; of logical order and a self-referential capability (which leads, naturally, to the classical logical paradoxes).</Paragraph>
    <Paragraph position="10"> Intensional definitions are also more complex in their formal structure for natural languages than for formalized languages.</Paragraph>
    <Paragraph position="11"> Intensions may be defined by specifying the combination of tests and results that indicate which elements of any given model are to be included in their extensions--these tests may be on either &amp;quot;inherent&amp;quot; or &amp;quot;contextual&amp;quot; attributes of the element and the values of these attributes may be either countable sets or measurements on some continuous scale. For formalized languages, lO the identification function of an intension must distinguish clearly and unequivocally between exemplars and non-exemplars on the basis of a Boolean combination of the results fo the various tests. For natural languages, however, tests may be either criterial for identification of an exemplar or else have only a probabilistic bearing on identification; thus, the identification of exemplars of natural-language intensions is by no means clear-cut, but rather may resemble the assignment of degrees of confirmation to hypotheses (with a certain &amp;quot;level of confidence&amp;quot; being required for identification to take place). The use in English of generic determiners such as 'many', 'most', 'almost all', and 'few', and (corresponding) intensional adverbs such as 'co~only', 'usually', 'characteristically', and 'seldom', is indicative of the probabilistic nature of intensional definitionS +-n natural language. 5 The morphological structure of natural languages is also considerably more complex than that of formalized languages, as has been well recognized by contemporary linguists. The simple phrase-structure grammars that suffice to describe the syntax of formalized languages simply do not work for natural languages; to dascribe the surface syntactic structure of a natural language requires a system, such as a relational phrase-structure grammar (Bellert, \[4\]) or a complex-feature-symbol grammar, with the power of expressing the various relations of grammatical agreement among constituents. If the language to be analyzed is spoken language, the arrangements of morphological units are 5A full analysis of these generic determiners and adverbs, their logical interrelationships, and their relation to notions of probability is given in Celce and Schwarcz \[131. A capsule summa~j of this analysis is presented later in this paper.</Paragraph>
    <Paragraph position="12"> ll not simply linear strings of symbols (connected by whatever gra~natical relations) but are, rather, two-dimensional sequences consisting of both segmental and suprasegmental (stress, intonation, etc. ) morphemes. Furthermore, the exact correspondence between gra~aatical sentences and semantically-interpretable sentences that obtains in formalized languages does not hold for natural languages, which permit of both &amp;quot;grammatical nonsense&amp;quot; and syntactically deviant utterances that make perfect sense--the first phenomenon requires a semantic theory to posit nonsyntactic conditions for semantic acceptability; the second, a procedure for syntactic error correction in decoding.</Paragraph>
    <Paragraph position="13"> The above are only two of the phenomena that render explication of the process of encoding and decoding much more complex for natural languages than for formalized languages. In formalized languages all well-formed expressions are unambiguously interpretable in or out of context, their interpretations are determined in a straightforward compositional manner by function-definition operators in one-to-one correspondence With the syntactic formation rules of the language, and performatives are represented as single symbols preceding or following each sentence. For natural language none of these properties hold--indeed, semantic ambiguity and anomaly, discourse structure and other forms of context dependence, syntactic-semantic non-correspondence, idioms and figures of speech, and complex encodin~s of performatives are all common features of natural language. Their explication in a semantic theory requires, first, that the correspondence between syntactic form and semantic function be taken as many-to-many rather than as one-to-one; second, that intensional &amp;quot;wellformedness&amp;quot; relative to the particular domain of discourse and applicability to the model or range of models currently under  consideration be taken as criteria for semantic acceptability in a discourse context; and third, that the theory specify the various alternative encodings of a message rather than a single encoding. There is also a need to incorporate analogical processes into the explications of encoding and decoding in order to account for the metaphorical use of language.</Paragraph>
    <Paragraph position="14"> Finally, there are two inherent limitations that govern any attempt to formalize the semantics of natural languages: one formal, the other eplstemological. The formal limitation is a consequence of Tarski's theorem \[43\], which states that any consistent and complete semantic theory of a language must be formulated in a metalaugua~e of higher order than the lsmguage being described.</Paragraph>
    <Paragraph position="15"> But since the set of theorems of any deductive system must be recursively enumerable, and since there are subsets of natural languages sufficiently powerful to define any recursively enumerable set, any formalization of natural language semantics using a deductive logic (including the logic of computation) as a meta-language will be incomplete in the sense that there will be questions about the language 6 that are theoretically unanswerable in the metalanguage (one could, however, go to inductive logics and probabilistic metatheories as the basis for a metalanguage ).</Paragraph>
    <Paragraph position="16"> The epistemological limitation derives from the fact that, while formalized languages are uniquely defined, no two speakers of a natural language have quite the same idea of what their language is. It is clearly impossible, then, to formulate a semantic theory that describes all the speakers of a natural language.</Paragraph>
    <Paragraph position="17"> 6Including~ of course, any question as to the truth-value of a sentence expressing a logical paradox.</Paragraph>
    <Paragraph position="19"> Neither is it practicable to attempt an &amp;quot;ideal speaker-hearer&amp;quot; theory that purports to explain how native speakers of a language &amp;quot;generally&amp;quot; assign meanings to utterances and express meanings through utterances 3 since validation of such a theory would be next to impossible. A more appropriate goal, especially in light of the fact that the data for any semantic theory must ultimately derive from the use of the language for communication, is to construct a theory of a &amp;quot;typical speaker-hearer&amp;quot; of the language in question, whose validity would then derive from the ability of a physical realization of the theory (e.g., as a program running on a digital computer) to engage in successful purposive cCm~uni7 cation with native speakers of the language.</Paragraph>
    <Paragraph position="20"> Let us enumerate, then, the requirements for a formal theory of natural language semantics that have been indicated here:  1. The theory shall be couched in a formal metalanguage.</Paragraph>
    <Paragraph position="21"> 2. The metalanguage shall contain models of possible discourse contexts, expressions representing extensions, expressions representing intensions, and axioms defining the relation of extension to intension for any given model.</Paragraph>
    <Paragraph position="22"> 3. The metalanguage shall contain expressions representing the  messages ecmm~nicated in the natural language, which will contain performatives specified as to type and manner, intensional definitions of both fixed and recursive logical order with criterial and/or noncriterial components on the descriptive, appraisive, and prescriptive dimensions, presuppositions, and both generic and specific quantifiers.</Paragraph>
    <Paragraph position="23"> 7Further reasons for preferring the &amp;quot;typical speaker-hearer&amp;quot; model to the &amp;quot;ideal speaker-hearer&amp;quot; model are given in Schwarcz \[38\].  4. The metalanguage shall contain axioms characterizing the functional import of messages, sufficient to define both extensional and intensional equivalence, entailment, and contradiction among messages up to the limits of theoretical decidability.</Paragraph>
    <Paragraph position="24">  5. The metalsm~uage shall contain expressions and axioms defining a &amp;quot;standard&amp;quot; syntax of the language at the level of surface arrangements of morphological units, in terms of both phrase structure and relations of grammatical agreement.</Paragraph>
    <Paragraph position="25"> 6. The metalanguage shall contain axioms defining the possible encodings of any message in any discourse context to which it is applicable.</Paragraph>
    <Paragraph position="26"> 7. The metalanguage shall contain axioms defining the possible  decodings of arrangements of morphological units that are ~C/ell-formed in the &amp;quot;standard&amp;quot; syntax or deviate from it by at most a tolerable degree and determining the intensional well-formedness and applicability to a given discourse context of these decodings.</Paragraph>
    <Paragraph position="27"> 8. The theory, to be validated as a description of a &amp;quot;typical speaker-hearer&amp;quot; of the language under consideration, must support a physical embodiment that is capable of engagiD~ successfully in purposive corm~unication with native speakers of the lar~uage.</Paragraph>
    <Paragraph position="28"> B. Computational Avenues of Approach to a Semantic Theory Since language is an instrument of communication and communication is essentially purposive, any semantic theory that one develops for the computer will of necessity be based, unless one is simply engaged in an academic exercise, on the purpose for which one I wishes to communicate with the computer in natural language. In  this section several such purposes and the sorts of semantic theory they have led to or are likely to lead to will be described. The purpose of oldest vintage is, of course, translation by machine from one languaPS~ to another. The problem here is, given a discourse in one language 3 to produce a discourse in a second language that has the same functional import with respect to a model of the domain of discourse as the first. Perhaps the reason that no efforts in this direction have achieved notable success to date is that the model of the domain of discourse and its functional interaction with the language have generally been ignored in the design of translation systems. The direction that will lead to a breakthrough here is that of developing domainspecifi___.____~e (rather thau langllage-specific) translation systems for well-understood and formally structurable domains of discourse such as physics and mathematics--once a fomnal model of the subject matter ~id a canonical procedural language for communicating with that model are defined, efforts can be directed toward specifying the decodings of as much of the relevant natural-language subsets as possible into the procedural language aud reasonable encodings of the procedural langaago into each of the natural languages. Data management and infor,~ation retrieval is another purpose of -widespread interest. The domains of discourse to which these systems may apply may be arbitrarily broad or narrow; whatever the case, the requirements for formal structurability and a formal procedural language for storing and retrieving information in the data base are present. If the system is to do deductive question answering (or ~hat Travis \[~5\] has called &amp;quot;analytic information retrieval&amp;quot;), the system must be able to store and utilize the logical relationships among concepts and facts. The  problem of specifying encodings and decodings here is simpler than for machine translation 3 since the user may make do with a fairly restricted natural-language subset for input, and natural-language output may be generated in a canonical form if it is in fact necessary at all. Thus it is possible here to get by with an oversimplified semantic theory, but for that very reason it can be expected that more progress can be made sooner with this than with any other approach (and this has, in fact, turned out to be the case).</Paragraph>
    <Paragraph position="29"> Another purpose is the use of natural language to interface ~r'+-th pictorial information. Here the model is a set of logical statements describing the visual image, derived by the application of pattern-recognition operations to the visual image. The model, once derived, can then be either directly encoded into a set of natural-language sentences or else used as a data base for information retrieval. An alternative approach is to decode natural-language retrieval statements into search procedures on the visual image itself, performing the pattern-recognition operations, then, during the execution of these search procedures. If the visual image is what a robot sees in its environment, the robot may not or, Sy be asked about what it sees but also told to move about in its environment and to move parts of the environment about~ thus 3 the intensional structure of the robot's message language will include a prescriptive as well as a descriptive dimension. As in the case of data management and information retrieval systems, the input language can be restricted and the output language can be minimal, thus obviating the need for sophisticated fornmlations of decoding and encoding.</Paragraph>
    <Paragraph position="30">  The use of the computer to develop models of human thought processes is a purpose that can lend revealing insights into the nature of a semantic theory. Here one starts with hypotheses about the structure of human memory and the information processes that take place there, embeds these hypotheses in a computer program, and runs the program to determine the consequences of these hypotheses in terms of predictions of observable behavior. In terms of a semantic theory, the emphasis here is * likely to focus on models, messages, and the pragmatic functions of messages on models; only limited attention is likely to be paid to the syntactic structure of the language, and encoding and decoding are likely to be formulated in a rough-and-ready heuristic fashion rather than in a way motivated by linguistic considerations. 8 Nevertheless, such models are an excellent way to test the workability of semantic ideas, for the models' linguistic poverty is compensated for in experimentation by their designers' linguistic flexibility--and once the innards are working right, they may serve as a basis for a more linguistically sophisticated formulation of decoding and encoding. A purpose incorporating both analytic information retrieval and psychological modeling is computer-aided instruction with natural language. 9 The capabilities required here are to semantically analyze a student's natural-language response or question, to compare an analyzed response to a standard &amp;quot;correct response&amp;quot; to determine the logical difference if any, to generate remedial feedback in natural language by application of &amp;quot;tutorial decision rules&amp;quot; to the structure representing this difference,  and to answer a student's analyzed question and generate a naturallar~age reply. For natural-language CAT all the components of a semantic theory, except perhaps for encoding, must be developed to their full extent with respect to the subject areas to be taught. The linguistic requirements are not quite so severe as for machine translation, since the capability of dynamic interaction enables students to put up with a certain amount of rigidity on the machine's part and since the machine will not be required to analyze or generate long coherent discourses, but the requirement of thorough and complete logical analysis is more demanding here than in any other application of a semantic theory.</Paragraph>
    <Paragraph position="31"> Finally, there is the purpose of enabling people to program the computer in natural language. Messages here are statements in a general-purpose progrsm~ning language which includes capabilities for defining both macros and closed subroutines; they will thus have both descriptive and prescriptive dimensions. Nouns, verbs, and adjectives will be decoded into either data items (if proper names, numbers, or truth values), primitive functions, macros, or closed subroutines, conjunctions and prepositions will decode into operators for combining program steps, adverbs will decode into designations of program sequencing, and quantifiers will decode into specifications for iterative loops. The decoding process will likely be some form of syntax-directed compiling, which exactly fits the decoding paradigm for formalized-language semantics, except in that the procedure may allow for a small degree of ambiguity. Encoding will either be completely standardized or else be defined in terms of a sublanguage of output specifications that may be associated in an arbitrary manner with computational procedures. All this assumes, however, that natural language is being used to program the computer in  for small western cities?' produced a ten-item request 3 and the question 'For the smoggy high-income cities what is the ageincome value-range?' produced twenty separate procedural requests. Although most of Kellogg's e~perimentation has been performed on a data base of census information, his system has also been successfully tier prostrated with airline-schedule and educational data bases.</Paragraph>
    <Paragraph position="32"> If Kellogg's system can be f~Atlted as a semantic theory~ other than in its lack of a nontrivial formulation of encoding, it is principally in its failure to deal with certain of the requirements specific to the semantics of natural languages. Chief in importance among these are noneriterial attributes of intensions (except those quantified by 'some'), recursiveness of logical order, the appraisive dimension of language, discourse structure recognition, disambiguation by discourse context, and deviations from standard syntax.. 14 The logic of equivalence, entailment, and contradiction among messages, particularly on the intensional side, has also not been formalized to the extent that it could be. In all fairness, however, it must be pointed out that few if any of the other current approaches to semantics have dealt with any of these requirements (except the last 3 for predicate-calculus-based systems) in a formally satisfying way. Kellogg has succeeded in putting together the best of current knowledge in linguistics, fomal semantics, and systems programming to develop an eminently usable formalization of English semantics for the computer.</Paragraph>
    <Paragraph position="33"> Both the linguistic and the computational formalizations of natural language semantics, when looked at individually, can be seen to fall considerably short of the requirements for a i Thish last item# as well as undefined words and zemantic anomalies, is handled by Kellogg through appropriate feed-back messages to the user.</Paragraph>
    <Paragraph position="34">  semantic theory that is adequate for natural languages. When taken collectively, however, they contribute an enormous reservoir of ideas and experience upon which one may draw in undertaking the formulation of an adequate semantic theory. With the addition of recent advances in linguistic theory, programming languages, and artificial intelligence to this reservoir, ~ may draw from it the elements that will combine to make up an adequate approach. Let us now look at one possible such approach.</Paragraph>
  </Section>
  <Section position="4" start_page="36" end_page="47" type="metho">
    <SectionTitle>
AN OFERI~IONAL-MEANING APPROACH TO SEMANTICS
A. Methodological Basis
</SectionTitle>
    <Paragraph position="0"> To arrive at a formal theory of natural language semantics, we must start from the set of requirements enumerated earlier-particularly those concerning the relation of a message to its functional import. Intensions are the principal components of * messages, and they are classified according to their values along the descriptive, appraisive, and prescriptive dimensions.</Paragraph>
    <Paragraph position="1"> Prescriptive intensions have values which are actions of the communicating system, and therefore can be sensibly regarded only as pro~ for action. Appraisive intensions have values which are evaluations of one kind or another; the only sensible way to regard these, then, is as evaluation functions.</Paragraph>
    <Paragraph position="2"> Descriptive intensions have values which are objects, sets of objects, and relations among objects, where the objects may in turn be intensions. Here we adopt the operationist philosophy, in the formulation of Benjsmin \[5\], and assert that descriptive intensions are functional operations on a data space which yield elements of knowledge as their result.</Paragraph>
    <Paragraph position="3">  It is clear, then, that the most natural representation of intensions is as programs is some programming language. Since intensions are functions on models, the operations that constitute them will be performed on structures that represent models--and since intensions may be values of intensions, the structure of programs must be of the same form as the structures of models. Furthermore, the progran~ning system which interprets the language must interpret it n0ndeterministically, for natural language may al~rays specify alternative definitions of a concept, or alternative procedures for evaluation, or alternative ways to perform an action; with respect to the first, it is a fund~nental premise of modern operationism that one can arrive at the same item of knowledge by means of different operational procedures, and that, in fact, the utility of a concept is largely as an expression of the generalization that a class of different operational procedures produce identical results. 15 Nondeterministic operation and the existence of evaluation functions characterizes a class of artificial-intelligence programs that have been written to do game playing, theorem proving, and general problem solving 2 all of which are based on the paradigm method of goal-directed heuristic search. A programming system based on this method of program operation, similar to the one that Pople \[34\] has recently implemented, would thus be indicated as the basis for an operational formalization of natural language semantics.</Paragraph>
    <Paragraph position="4"> Let us now turn to a sketch of how the semantics of natural languages might be formalized within such a system.</Paragraph>
    <Paragraph position="5"> 15This view is expressed clearly in Bridgman \[8\].</Paragraph>
    <Paragraph position="6">  B. Models and Messages There are two basic issues to be decided in the formulation of any model: what information is to be contained in the model, and in what form that information is to be represented. In the formulation of a message language for conT.municating with the model, it must furthermore be decided what computational processes are to be performed in the model. A semantic theory will rise or fsll on the basis of the extent of information that can be represented in the model, the extent of information processing that can be formulated in the message language, and the ease with which translation algorithms can be formulated between the message language and the corresponding natural language subset.</Paragraph>
    <Paragraph position="7"> Because of its close similarity to both formal logic and the attribute-value list structures and relational associative structures that have been employed in many artificial-intelligence programs, as well as its demonstrated advantages for linguistic formulations, the Fillmore case structure appears to be the most useful starting point for representing both models and messages. Additional specifications must be added in to represent the logical features which are lacking in Fillmore's formulation: logical connectives, quantification and quantificational ordering~ other function-definition operators, the structure of the modality constituent, etc. The inventory of case relations must also be completely specified and, since case relations are all contextual, supplemented with a set of inherent relations that will enter into both extensional and intensional description--s~ne of which, like the &amp;quot;spatially contains' relation, will be converses of the case relations l themselves.</Paragraph>
    <Paragraph position="8">  The formal content of both intensional and extensional description is still largely an open question, to which the various attempts to formalize nattu~al language semantics can only suggest methods for solution. At the lowest level of semantic description, the actions that an operational semantic model will be able to perform will be computer actions and not human actions, and the evaluations that it will be able to make will a//nost certainly be pragmatic evaluations rather than aesthetic evaluations (this iS not to assert, however, that no way will ever be found to program a computer to simulate a human being's appreciation of poetry, art, or musi&amp;). It is the descriptive dimension that is most interesting, especially since both appraisive and prescriptive intersions -~xe instances of second-order (or higher) descriptive intersiors, with the consequence that the values and actions of humaa beings ma~, be described in an operational semantic framework, and perhaps as a consequence also modeled by analogy though not applied directly. On the descriptive dimension Benjamin \[5\] lists the following types of operations, which we shall characterize as intensions with corresponding extensions:  All these operations may be combined, of course, by function-definition operators in defining intensions. Models may be defined in this context in terms of situations, which are hierarchically structured configurations of events where the elements of each level of the hierarchy are connected by relations of co-occurrence and temporal succession, and different levels are connected by part-whole relations. With each node of a situational hierarchy will be associated that extensional information which applies (inherently or contextually) to all events below it. Some situations will be associated with goals of the communicating system, which are intensional descriptions for which the communicating system seeks to transform the situation in order to satisfy. These goals form the basis for the operation of programs in the system.</Paragraph>
    <Paragraph position="9"> The criteriality or noncriteriality of intensional attributes may be represented by associating with each attribution a quantifier; in an intensional definition these quantifiers represent levels of criteriality for attributions. In English and other natural languages there are five levels of criteriality for both positive and negative attributions that acquire lexical ek~pression~ these, as represented by generic determiners, adverbs of relative frequency and adjectives of possibility, are shown in the diagram below, along with their relations of implication and minLmal mutual contradiction, and their relation to the absolute  scale of probability (represented by the diagonal in the figure). almost all;  characterissome; many; most; .tically; all; sometimes; often; generally; almost always; possible likely probable certain certain</Paragraph>
    <Paragraph position="11"> no; few; most + not; not nearly not all; never; seldom; generally + all; not always; impossi- unlikely not; often + not; not certain ble improbable not nearly certain With a bit of intuition, patience, and attention to the requirements of commutativity and associativity, one may also construct a heuristic &amp;quot;multiplication table&amp;quot; that will define products of these levels of possibility, to handle conjunctive and disjunctive attributions.</Paragraph>
    <Paragraph position="12"> The most general possible explication of messages, and probably the one that will prove to be necessary for a semantic theory, is that they be simply any programs in the system. Other than performative operations and intensional evaluation operations, the set of operations that constitute messages will include finding an instance of an intension, creating a new instance of an intension, finding or creating an intension similar to a given intension, inserting or deleting quantified relations  between extensions and/or intensions, comparing extensions or intensions for equality, inclusion, or mutual exclusiveness, adding or deleting intensional definitions, modifying intensional definitions, rearranging and otherwise modifying situation structures, numerical computations, and the logical operations indicated by the function-definition operators. The specific form of the language in which all these operations may best be combined into programs is yet to be determined, but it may well turn out to be similar to Woods' procedural language, in which the basic statement form is a quantified &amp;quot;pattern-operation&amp;quot; rule. C. Decoding and Encoding The process of decoding natural language consists of three stages: syntax recognition, semantic translation, and application to the model representing the current discourse context. Syntax recognition includes recognition of both phrase structure and relations of gra~natical ~em~nt among the two or more constituents that are combined by a syntactic rule. Syntactic error correction might be handled by a method akin to Chomsky's \[l~\] notion of &amp;quot;degrees of gr~ticalness&amp;quot;: relaxation of first grammatical agreement and then syntactic categorization conditions could be allowed until a parsing leading to a semantically-acceptable decoding was obtained. Semantic translation of the cembination of constituents recognized into a functional form in the message language then proceeds by way of one or more interpretation functions associated with the rule of grammar. These interpretation functions will make tests for agreement among the inherent and contextual attributes of the intensions they combine; if any of these tests fails, metaphorical interpretation rules (which operate by analogizin~  and disanalogizing) might be invoked to attempt to resolve the conflict through appropriate construal of one or more of the constituents before the interpretation is rejected entirely. In the final step, application to the model representing the discourse context, antecedents of anaphoric and elliptical expressions are found through appropriate intensional evaluation, and further disambiguation may be achieved in case one or more of the alternative decodings contains presuppositions that are not satisfied in the model. Here additional rules of application may need to be introduced to add information to the model that is required by a presupposition but with respect to which the model is nonc cmnittal.</Paragraph>
    <Paragraph position="13"> The problem of encoding is that of transforming a message into a well-formed surface syntactic structure through lexical substitution and formation of syntactic constructions. Encoding may be formulated as a recursive top-down procedure, which operates from the outermost level of functional application in the message on inward to the point where each expression may be replaced by a lexical item, therea+-%er &amp;quot;unwinding&amp;quot; its way back outward, applying syntactic encodings followed optionally by syntactic transformations to each functional composition encountered on the way. There will, of course, be alternative paths that may be followed in the encoding of a message, because of the possibilities of alternative lexical substitutions, applications of alternative encoding rules, and optional application of transformations. Some of these paths may block because syntactic conditions on the application or output of the encoding rules are not satisfied, others because certain &amp;quot;performance-oriented&amp;quot; constraints, such as constraints on the level of certain types of embedding, are not met in the resulting surface structure.</Paragraph>
    <Paragraph position="14">  The rules for lexical substitution can probably be formulated along the lines proposed by Gruber, those for encoding into nominal structures along lines suggested by Celce and Schwarcz \[ll, 12\], and those for encoding into clauses and sentences along the lines suggested by Fillmore.</Paragraph>
    <Paragraph position="15"> Both decoding and encoding may be formulated most neatly as non-deterministic procedures employing heuristic search and evaluation. The rules employed in both, furthermore, are of the pattern-operation type, the syntactic structures they operate on are of the form of situation structures, and the semantic structures they operate on are, of course, components of models and messages.</Paragraph>
    <Paragraph position="16"> Therefore both procedures and rules for decoding and encoding should be formulatable, and perhaps formulatable most elegantly, in the message language, since it is a general-purpose programming language containing all these features. Such a formulation would have the further advantages of parsimony with respect to computer implementation of the semantic theory and easy modifiability through the ability to use natural-language statements to effect changes in these procedures and rules.</Paragraph>
    <Paragraph position="17"> D. Implications The approach described above, though it has not yet been implemented, can be regarded as a sincere attempt to meet the requirements set forth for a formal theory of natural language semantics in the first part of this paper--an achievement that no other approach advanced to date can claim, despite the many valuable ideas these approaches have produced. Evaluation of this attempt as a semantic theory must, of course, await the satisfaction of the final requirement: that the proposed system,  when progrmmned, engage successfully in purposive co~unlcation with speakers of a natural language. Simply as an approach that holds the promise of adequacy as a semantic theory, h0wever , it can provide a unifying direction for research in a number of areas, including linguistics, lexicology, logic, theory of computing, and artificial intelligence. The unification of such a diversity of directions of exploration, along with the rigorous test that the approach implies for an operationist theory of knowledge and meaning, should render the approach an interesting and fruitful one for philosophical study and exploration. Adopting an approach such as this, or any approach satisfying the requirements for a semantic theory, as ~ metatheoretical basis would also help greatly to resolve the confusion that exists today in linguistic theory. This state of disarray is a result of the fact that, with a very few exceptions, linguists have basically ignored the fundamental fact of language as being a tool for communicating sc~ethin~ to somebody. They have almost without exception ignored the interface between language and the speaker's or hearer's model of the universe of discourse. Operating in this sort of vacuum, linguists are under too few empirical constraints to determine any theory of grammar, let alone one that is meaningful. Only a semantic metatheory that takes the con~nunicational significance of language explicitly into account can provide a satisfactorily sound basis for a theory of gra~nar.</Paragraph>
    <Paragraph position="18"> The exploration of the approach offered here would bear very much upon the interests of cognitive psychologists, too, in that it offers a unified framework for a theory of language and cognitive processing. The heuristic-search-and-evaluation mode  of operation is the paradigm that has emerged from an extensive amount of empirical research on human thinking and problem solving; its successful extension to explicating the understanding and production of language would lend support to the view that the mechanisms employed in language processing are the same as those employed in human thinking in general. The specific forms of model and message structures would, conversely, provide a basis for a formal theory of cognition that would receive support from the linguistic side as well.</Paragraph>
    <Paragraph position="19"> Finally, the formulation of the approach as a general-purpose programming system implies that it would be usable, in principle at least, for any application of computers to linguistic and semantic information processing, including all the ones mentioned earlier in the paper. Availability of suitable computer hardvare and operating systems would, of course, be essential to any application of the approach on a realistic scale. A more demanding requirement, however, is that of encoding the definitions of the thousands of different words that make up any natural language into an appropriately structured lexicon. A standard dictionary is one possible source of this information, but it will obviously not contain enough to define every word of a language operationally. The work of Olney etal \[BBS should be very helpful, however, in determining what can be gleaned from a standard dictionary and whether this information can be appropriately supplemented to yield an adequate operational lexicon, or whether a major new lexicographic effort, more rigorous in its requirements than any that have gone before, will have to be undertaken. Nhatever the case, the operational lexicon, once created, would be usable for all varieties of applications--and its construction, as well as the programming of applications, would be made 'easier by the capability implied by this approach to program the system in a  natural language once it had been supplied with sufficient information to define the semantics of a suitable &amp;quot;base&amp;quot; subset of the language.</Paragraph>
    <Paragraph position="20"> This approach, of course, is only one of many that might be taken to formalizing the semantics of natural languages. Like all other approaches that have been attempted or proposed so far, it will surely reveal its limitations somewhere along the way. But at a time when linguistics, semantics, and computational linguistics are all anxiously searching for a paradigm to follow, this may well be a fruitful one to try.</Paragraph>
  </Section>
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