Outline of a semantic theory of Kernel sentences [Reprint 2018 ed.] 9783111729091, 9783110995503

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JANUA

LINGUARUM

STUDIA NICOLAI

VAN

MEMORIAE WIJK

edenda C.

H.

VAN Indiana

DEDICATA

curai

S C H O O N E V E L D University

iSeries Maior

71

OUTLINE OF A

SEMANTIC THEORY OF

KERNEL SENTENCES by

EMANUEL BUCHAREST

VASILIU UNIVERSITY

1972

MOUTON THE HAGUE • PARIS

This work is jointly published by Editura Academiei Republicii Socialiste Romania and Mouton & Co., N.V., Publishers

This is the revised version of "ELEMENTE EDITURA

DE

TEORIE

SEMANTICÄ

A L1MBILOR

ACADEMIEI REPUBLICII SOCIALISTE 3 bis, Gutenberg, Bucharest, 1970 All rights reserved

PRINTED IN ROMANIA

NATURALE" ROMÄNIA

PREFACE TO THE ENGLISH VERSION

Outline of a Semantic Theory of Kernel Sentences is the English version of the booh which appeared in Romanian under the title : Elemente de teorie semantica a limbilor naturale. The title of the English version, in contradistinction with that of the Romanian version, has the quality, I hope, of making clearer the topic of this work, so that no extra explanation is needed in the preface. I would like to point out that O u t l i n e . . . should be viewed more as an English version than as an English translation of the Romanian Elemente... The English version differs from the Romanian

one in several

respects:

1°. The Introductory chapter, together with the next two chapters (one devoted to structuralist theories of the linguistic sign and the other to a brief explanation of what is meant in transformational theory by semantics) are omitted in the English version. The presence of these chapters in the Romanian version was justified for pedagogic reasons : I intended to make it possible for students to find in that book besides my own point of view, a brief outline of the main linguistic ways of approaching semantics, in order to get an impression of the context in which the theory I was going to develop might be placed. 2°. Some of the expressions which in the Romanian version were treated as rules of inference of the system S, in the English version are viewed simply as theorems of the system 8 (that is rules RI 1, 3, 5, 7, 8, 9 are stated as theorems). 3°. In the English version, the Camapian notion of relative L-concept was introduced (see § 19); on the basis of this concept, many expressions which in the Romanian version were considered simply L-true or L-false because they were L-implied by, or contradicted some meaning postulates, are now characterized as L-true with respect to, or L-false with respect to some definite meaning postulates. This change was determined by the fact that I now consider the arguments given in E l e m e n t e . . . , p. 122 and offered in support of the idea that one could eliminate the relative L-concepts as being more intuitive than formal. 4°. Corollary 58—6 (p. 243 of the Romanian version) stating relation T is symmetric has been omitted.

8

OUTLINE OF A SEMANTIC THEORY OF KERNEL SENTENCES

5°. Proposition 39—3 from Outline... defining the notion of immediate constituent of 'O' has no correspondent in Elemente... Theorem 39—5 from Outline... is a revised version of 58—4 from Elemente... Proofs of 39—4, 5 from Outline... were given a form which is not identical to the form of the proofs given to 58—4, 5 in Elemente... 6°. It is hoped that at least a great part of the printer's errors occurring throughout the Romanian version have been eliminated in this book. Unfortunately I started to work out the English version only some months after Elemente... appeared. This means that I was unable to benefit from any review of the Romanian version. Many of Prof. Marcus' comments concerning Elemente... were extremely helpful to me in working out a version, which I hope has been improved. Some months after the Romanian version of this book appeared, Rudolf Carnap died. For anyone acquainted with his work, the deep, essential and decisive influence he had on my thought concerning semantics as well as the theory of language is, I think, obvious on every page of this book, and in many of the papers I have written in recent years. This acknowledgement is intended to be my modest homage to his memory. • I would like to express my gratitude to Mrs. Sorana Chivu, from the Publishing Souse of the Romanian Academy who was kind enough to read the entire English version and to suggest many improvements in my English style. E.V. Chicago,

March,

1971

PREFACE TO THE ROMANIAN VERSION

This book is the result of the contact I have had with some of Y. BarHillel's works ; this contact has enabled me to view Carnap's works from the vantage point of the linguist. At the same time, this book is the result of a long contact with Rudolf Carnap's works ; my technique of constructing logic languages and the meta-language describing them, as well as many of my views concerning the relations between logical languages and natural languages and even concerning the theory of language itself originate in Carnap's works. This book also makes connection with J. J. Katz's transformational semantic theory, although it is viewed from the standpoint of a semanticist with a Carnapian background. The phrase "Elements o f . . . " from the title of this book should be understood in two senses : a) that the aim of this work is not to develop a theory of a full natural language, but only of a part of such a language (the so-called kernel-sentences in Chomsky's sense); b) that this work is not aimed at developing all the theoretical aspects involved in the description of this sub-language. The title contains the phrase "of natural languages" only in order to make clear the fact that our attention is directed especially toward this class of languages. I am aware of the fact that a "semantic theory of natural languages" is but a particular instance of a "semantic theory of Language". I would like to acknowledge the important role of the many fruitful discussions I have had with my friend and colleague, Prof. Solomon Marcus, while I was working on this book; and I thank Mrs. Sanda Golopenfia-Eretescu (researcher in the Center for Phonetic and Dialectal Researches) for her consenting to read the manuscript of this work and for her valued comments. I ask these colleagues to find here the expression of my gratitude. E.V.

Bucharest, August, 1969

CONTENTS

Preface to the English version Preface to the Romanian version

7 9

INTRODUCTION

15

§ 1. Natural languages and formalized languages; structural analogies § 2. Methodological consequences § 3. The structure of the proposed semantic theory.

15 17 19

Chapter I CRITICAL APPROACH TO T H E TRANSFORMATIONAL THEORY

§ § § §

4. 5. 6. 7.

SEMANTIC

The meaning problem Object-language and meta-language The relations between semantics and syntax . Final remarks Chapter

8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

21 29 32 35

II

THE BASIC FORM OF A SEMANTIC SYSTEM S

§ § § § § § § § § §

21

The lexicon of the system S Formation rules Eules of designation Eules of truth Implication and equivalence L-Ooncepts Eules of inference Derivation and proof in 8 Primitive sentences in S Theorems concerning primitive sentences and transformation rules § 18. Some theorems in S § 19. Meaning postulates in S § 20. Extension and intension

37

37 38 41 42 43 44 48 49 49 51 53 54 57

12

OUTLINE OF A SEMANTIC THEORY OF K E R N E L

SENTENCES

Chapter III SYSTEM S IN RELATION TO K E R N E L SENTENCES

§ § § § § § § § § § § § § § §

21. 22. 22 a. 22 b. 22 c. 22 d. 22 e. 22 f. 22 g. 23. 23 a. 23 b. 23 c. 23 d. 24.

§ 25. § 25 a. § 25 b. § 25 c. § § § § § § § § §

26. 26 a. 26 b. 26 c. 26 d. 27. 28. 29. 30.

63

The structure of kernel sentences . . . Words and atomic sentences Indefinite pronouns Personal pronouns Demonstrative adjectivals . . . . . . . . The tense of the verbs Space determiners Time determiners Manner determiners Quantifications in L, The definite article Plural inflexions Noun determiners Final remarks Possibilities of representing in S quantifications from L, The quantifications in L, and their representation in S Quantity determiners Approximate quantification Other expressions containing numeral constituents The verbal moods Real/unreal Assertive/non-assertive Hortative (imperative), optative The presumptive Negative sentences Interrogative sentences Copula sentences Concluding remarks

63 64 65 68 75 77 79 84 89 89 90 91 92 93 94 97 97 100 101 104 105 107 109 110 Ill 114 116 117

Chapter IV • TRANSLATION FROM NATURAL LANGUAGES INTO S,

§ § § §

31. 32. 33. 34.

. . .

.

119

The grammar of kernel sentences . . . . Intensional identity and translation . . . Translation rules Final remarks

119 123 128 143

13

CONTENTS

Chapter V MEANING POSTULATES IN Si

§ § § §

145

35. 36. 37. 38.

Introductory considerations Semantic markers and meaning postulates. . Selection restrictions and meaning postulates. Other properties which may be expressed by meaning postulates § 39. Pinal remarks Chapter

40. 41. 42. 43.

Translatability from L, into S, Ambiguity and homonymy in Lj L-equivalence and synonymy in Lj Decidability in L, Chapter

CLOSING REMARKS

References Authors' index

160 168

VI

POSSIBILITIES OF CHARACTERIZING Li IN TERMS OF S, .

§ § § §

145 147 155

.

.

171

171 177 188 197

VII 213

217 220

INTRODUCTION

§ 1. Natural languages and formalized languages; structural analogies. I t is well known that what we call today "symbolic logic" or "mathematical logic" or "logistic" is the result of a long mathematization process. At the same time, it is worth pointing out that the mathematization process of logic goes together with the idea that, in its symbolic, mathematized form, logic is but a l a n g u a g e . This idea occurs, for instance, in Boole, 1847 : " L a théorie de la logique est ainsi intimement liée à celle du langage. Une entreprise qui réussirait à exprimer des propositions logiques par des symboles, dont les lois des combinaisons seraient fondées sur les lois des opérations mentales qu'elles représentent, serait du même coup, un pas vers un langage philosophique" (ap. Cahiers pour Vanalyse, 10, 1969, La formalisation ; Travaux du Cercle d'épistémologie de l'Ecole normale supérieure publiés par la Société du Graphe, aux Editions du Seuil; Boole, L'analyse mathématique cle la logique, French translation by Y . Michaud, p. 28). Symbolic logic can be viewed as a s y n t a c t i c c a l c u l u s , that is a set of rules governing the use of a finite set of signs (the lexicon) ; by means of rules, one can construct sequences of signs called expressions in the given language and derive from a set of expressions in the given language another set in the same language. According to Carnap (1959, pp. 1—2), we speak about a logical syntax-, a logical syntax is a set of rules of the above mentioned kind : formation rules (i.e. by means of which expressions are constructed out of signs) and transformation rules (i.e. rules by means of which expressions are derived from other expressions). A symbolic logic system is called by Carnap (1958, p. 1) a language (not a theory) because it is " a system of signs and rules for their use" and it is not " a system of assertions about objects". When syntactic rules are supplemented by a set of semantic rules, that is rules assigning meanings to the well-formed expressions of the language, a "formalized language" in Church's sense (1965, p. 437) is obtained, that is a language having syntax and semantics, in contradistinction with a "logistic system", having syntax but no semantics (Church, 1965, p. 437). On the other hand, the development of the theory of natural languages progressively made obvious several deep structural analogies between

16

OUTLINE OF A SEMANTIC THEORY OF K E R N E L SENTENCES

natural languages and the artificial ones (especially the languages of the logic). First of all, the point that natural languages are but sign systems is a common idea of about all structural approaches in linguistics (notice, however, that "sign" means different things in different major structural theories, that is those rooted in de Saussure's Cours de linguistique générale, in the Bloomfieldian and neo-Bloomfieldian theories and in Hjelmslev's glossematic theory). No less important, from the point of view we are now interested in, was the endeavour of different structural schools to develop a full system of concepts related to the operation of reducing the — perhaps — infinite multitude of concrete entities varieties occurring in speech acts to a finite and relatively small set of invariants, or the infinite set of speech acts to a finite set of types, on the one hand, or, on the other hand, related to the operation of classifying these invariants and types arrived at by means of previous operations (as "instances" of such a conceptual device may be viewed structural theories like Saussure, 1922; Martinet, 1939; Troubetzkoy, 1957; Hjelmslev, 1963; Bloch& Trager, 1942 ; Trager & Smith, 1956 ; Harris, 1963). Many concepts of the structural linguistics were re-defined in terms of various mathematical analytical models (concerning the mathematical concept of "analytical model" or "analytical grammar" see Marcus, 1967, p. 1 ; an example for the formalization of some definite concepts belonging to structural linguistics may be found in Marcus, 1966, p. 57 — 71 ; various kinds of analytical models may be found in Dobruâin, 1957, 1961 ; Kulagina, 1958 ; Bevzin, 1962). Under such conditions, the structural analogies between (parts of) natural languages and different mathematical objects become obvious. The development of the generative theory made clear that natural languages can be described in terms of a finite set of rules according to which a finite set of signs are used in order to obtain strings of such signs, called sentences in the given language. Marcus (1967, p. 1) defines the concept of generative grammar as follows : "A generative grammar of O is a finite set of rules (called grammatical rules) specifying all strings (and only these strings) and assigning to each string a structural description that specifies the elements of which the string is constructed, their order, arrangement, interrelations, and whatever other grammatical information is needed to determine how the string is used and understood". This conception of grammar is fit not only for the description of natural languages, but for the description of any language, since in the most abstract way by language we denote any set of strings constructed out of signs (or symbols) in agreement with some definite rules given for the use of these signs (or symbols).

INTRODUCTION

17

It is obvious that natural languages together with different kinds of logic systems fall into the domain of what is now to be called "language", in agreement with the above mentioned conception. The deep structural analogies between natural languages (or "wordlanguages") and logical languages were often pointed out especially by logicians like Carnap (1959, pp. 2, 8), Church (1965, p. 444), and BarHillel (1967, p. 545). But these analogies do not belong only to the grammatical (or syntactical) level. Every logical syntactic system becomes a logical language, or a semantic system (or, using Church's terminology, a formalized language when some meanings are assigned by means of a set of explicit rules to the well-formed expressions specified by grammar. The meaning-assigning rules are the semantic rules of the language. In the same way, in the case of natural languages, a set of meaning-assigning rules may be attached to the grammar, so that the well-formed strings receive a semantic interpretation of the systems specified by means of syntactic grammatical rules). According to the above outlined considerations in the case of natural languages and of the languages of the logic, one can speak about deep structural analogies not only between grammars but also between the two categories of languages as a whole. It is to be noticed that structural analogies between natural languages and languages of the logic have nothing in common with the old conception concerning the parallelism between "language" and "mind" rightly criticized by many scholars (see, for instance, Saineanu, 1891; Serrus, 1941); the structural analogies I was talking about are analogies between languages, whereas the parallelisms between language and mind are analogies between languages and objects which are not languages, i.e. the mind. § 2. Methodological consequences. The semantic device allowing the interpretation of well-formed strings of natural languages is, in principle, of a different nature in comparison with the traditional approach to structural semantics, which is essentially related to the analytic-classificatory conception of grammar (see for instance such semantic approaches in Lyons, 1 9 6 3 ; Pottier, 1 9 6 4 ; Coseriu, 1964, pp. 150, 1 5 7 ; 1968, p. 9 ; Garvin & Brewer & Mathiot, 1967). For structural semantics the point to be reached is the reduction of the, perhaps, infinite multitude of meanings to some definite set of invariant meanings, the classification of these meanings and the correlations to be established between meaning(s) and the "signifiants". For a semantic device attached to a generative grammar all information which might be given in terms of a structural semantic theory is contained in a lexicon and it can be viewed as a kind of starting point in the construction of the rules of the system assigning meaning to the 2 -

C. 1402

18

OUTLrNE OF A SEMANTIC THEORY OF KERNEL

SENTENCES

well-formed expressions. Moreover, the specific form of the lexicon and even the kind of information given by the lexicon is determined by the semantic rules system. Otherwise phrased: the aim of a semantic device is neither a matter of establishing a reduction, or a classification procedure to be applied to meanings, nor a procedure mapping the set of meanings onto the set of "signifiants" ; the aim of a semantic theory is only to assign definite meanings to the well-formed expressions ( = strings) recursively specified by grammar. I could say that information like that given by structural semantics is only a component of the semantic device, but the semantic device, as a whole, is by no means required to be a procedure enabling us to "discover" information of a structural semantic type. Church (1965, p. 444) pointed out: "The difference of a formalized language from a natural language lies not in any matter of principle, but in the degree of completeness that has been attained in the laying down of explicit syntactical and semantical rules and the extent to which vagueness and uncertainties have been removed from them". In agreement with Bar-Hillel (1967, p. 545), Church's assumption seems to be too strong, but, again as Bar-Hillel pointed out, " a linguist could do worse than to make it perfectly clear to himself exactly where Church goes wrong". I think that Bar-Hillel's observation ought to be understood as follows : one cannot ask oneself where structural analogies between natural and formal languages stop unless the syntactic and semantic rules of natural languages were made explicit. If one of the possible forms of the transformational generative approach to natural languages is a proper way to construct a syntactic theory of natural languages, the same thing can not be said about the transformational semantic theory outlined by Katz (1965) and Katz & Fodor (1965); as may be seen in Chapter I of this book, there are many strong reasons to reject this attempt as grounds for a semantic theory of natural languages. On the other hand, the direct investigation of the semantic level of natural languages meets many difficulties due to the fact that the signs and the expressions belonging to natural languages are ambiguous. If such is the case, laying down explicit semantic rules (especially transformation or inference rules) becomes a very difficult (if not even impossible !) task. Notice that the elaboration of different kinds of languages of logic was, at least partially, determined by reasons of the same kind (the difficulties arising in the traditional "logic of terms" are discussed in Lewis & Langford, 1959, pp. 4 9 - 7 7 ) .

INTRODUCTION

19

Since — as we have just seen — direct investigation of the semantic level is difficultly accomplished, I am going to take another way, namely that suggested by Carnap (1959, p. 8 ; 1965, p. 432), i.e. the indirect semantic approach to natural language, by means of a semantic system (or "formalized language" in Church's terminology), S. § 3. The structure oi the proposed semantic theory. The "indirect investigation" from § 2 requires some explanations intended to make clearer its exact meaning. A first approximation in this sense would be that natural languages are compared with a semantic system S having the form of that described by Carnap (1958) as " t h e simple language A " or " t h e language B " or "the extended language C " or the system described by Lewis & Langford (1959, p. 122—198) or by Reichenbach (1966), etc. Semantic systems like those just mentioned are, of course, originally constructed independently of any natural language and are in fact independent of any such language. That is, whatever can be said in a natural language L cannot necessarily be said in a language S and, possibly, vice versa; for instance a sentence like go out cannot be expressed in terms of, let us say, Carnap's "Language B " . However, in principle a semantic system, S , , can be constructed satisfying the following condition : 3—1. For every sentence, from language L, (or from, the set L x , L 2 . . . . . . , L n ) there is at least one expression, G in S j , so that S were a translation of S into the system S r I f Sj satisfies condition 3—1, that is, if (£j is an expression in S, and if (Sj is a translation of a sentence Sj from L , , then everything which can be significantly said about ©j holds for Sj t o o ; in other words, if such is the case, every characterization of S j is at the same time a characterization of Sj. Obviously, this consequence of 3—1 holds only as far as a set of translation rules is established, putting in correspondence each sentence of the (set of) natural language(s) with at least one expression from Sj. These translation rules say that if sentence Sj is true in L t , the corresponding expression is true in S, and vice versa, or otherwise phrased, it never happens that Sj is true in L , and (Sj false in S[ or that Sj is false in L ( and @j true in S j . Since this agreement in truth value is established only by means of the rules of the meta-language in terms of which the semantic level of L, is described (translation rules belong to the meta-language) and does not require any observation of

20

OUTLINE OF A SEMANTIC THEORY OF K E R N E L

SENTENCES

facts, we may say — using Carnap's terminology — that translation rules express the logical equivalence (L-equivalence) between Sj and (£j. Further, since Sj and (Sj are L-equivalent, they have the same meaning. Let us now suppose in the meta-language an expression (1)

X -

- Y

which semantically characterizes expression Giij . Since equivalent to Sj, we may say that expression (2)

is logically

X - S, - Y

obtained from (1) by substituting Sj for , is logically equivalent to (1), that is says the same thing as (1), or has the same meaning as (1). The logical equivalence between (1) and (2) is an "explicatum" for what was meant by the phrase "every characterization of is at the same time a characterization of Sj". Up to this point, our considerations were concerned with natural languages viewed as the full set of sentences specified by a definite grammar. However, the aim of this book is not to develop a semantic theory of a natural language in its totality, I restrict myself to develop only a theory of what Chomsky (1963) called kernel sentences, that is sentences which do not contain any embedded sentence as their properconstituent. Therefore, the theory to be developed will be concerned only with a subclass of the class of sentences called "language". It will not be a theory of some definite language or of some class of languages, but a theory of any class of kernel sentences of any language which can be specified by means of a "rewriting rules grammar" which will be explicitly formulated. The structure of the theory may be outlined as follows : (i) description of a semantic system S ; (ii) non-formal comparison between kernel sentences from natural languages and the system S ; the aim of this comparison is to establish which is the form of a semantic system satisfying condition 3—1; the final result of this comparison is the construction of a new semantic system, Sj, satisfying condition 3—1 ; (iii) translation rules from any L¡ into Sj; (iv) semantic characterization of kernel sentences from L, in terms of the system S¡. The meta-language (ML) to be used is English supplemented with some special symbols, whose meaning will be explained at their first occurrence. Examples from natural languages are taken mostly from Romanian —which is the author's mother tongue—and also from some other European languages.

CHAPTER I

CRITICAL APPROACH TO THE TRANSFORMATIONAL SEMANTIC THEORY

SUMMARY.

In this chapter we critically examine the transformational point of view concerning : 1°. the concept of 'mianing'; 2°. the relationship between object-language and meta-Iangtiage; 3°. the relationship between semantics and syntax. A motivation is given for rejecting the transformational point of view as basis of a semantic theory of natural languages.

§ 4. The meaning problem. Within the transformational theory, the meaning is represented by means of a set of 'semantic markers' and a set of 'distinguishers' (see K a t z & Fodor, 1965, pp. 496, 497, 4 9 8 ; Katz & Postal, 1964, pp. 1 2 — 1 4 ) ; speaking of 'meaning' one has to distinguish between 1°, the meaning of specific formatives, and 2°, the meaning of the expressions ( = strings of formatives). I shall consider separately the two aspects of the meaning problem. 1°. a. The meaning of a formative is represented in the lexicon by means of a reading (see Katz & Postal, 1964, p. 13) constructed out of one or several semantic markers (optionally a reading might end with a distinguisher). If one formative is characterized by more than one reading, then it is called ambiguous. 'Semantic markers', in agreement with K a t z & Fodor, 1965, can be viewed as 'elementary concepts' and a reading is an ordered set of such 'elementary concepts'. The class of all semantic markers, by means of which the meaning of various formatives is specified, might be considered as a kind of lexicon of the meta-language. The meaning specification of a formative is then but a kind of translation from the object-language into the meta-language. One may easily find here some rough analogies between this meaningspecifying method and Carnap's 'rules of designation' (see Carnap, 1960, p. 4 ; 1965, p. 4 2 4 ; 1958, pp. 9 8 - 1 0 0 , and below § 10). However, there is an essential distinction between the transformational way of representing the meaning and the 'rules of designation', namely the fact that the 'rules of designation' are grounded on the concept of 'truth

22

OUTLINE OF A SEMANTIC THEORY OF KERNEL SENTENCES

value', whereas the meaning specification in terms of 'semantic markers' has nothing in common with such a concept. W e say that 'rules of designation' specify the meanings of various signs and expressions because such rules make explicit the conditions under which an expression constructed out of one or several signs may be said to be true, or, in other words, the conditions under which such an expression may be asserted (for more exact explanations concerning the relations among 'rules of designation', 'truth value' and 'meaning' see below

§§ 10, 11, 20). W h a t might be viewed within the transformational semantic theory as the analogous of 'rules of designation', i.e. the translation in terms of 'semantic markers', has no connection with any device specifying the conditions under which an expression constructed out of one or several formatives becomes true (or untrue). Indeed, simply knowing by means of translation the 'designata' of each formative of a sentence, one cannot say whether the sentence is true or false because one does not know the truth conditions for the expressions containing syntactic connectives (such as 'is' from 'A is B ' , 'and' from 'A and B ' , etc.). As a matter of fact, transformational semantic theory does not make any distinction between 'logical signs' (syntactic connectives belong to the class of 'logical signs') on the one hand and 'descriptive signs', on the other, so that such a treatment make us think either that syntactic connectives should be considered as something referring to some 'designata', or that syntactic connectives are in no relation with the so-called 'semantic interpretation'. Either way something goes obviously wrong : in the former case, because syntactic connectives have no 'designata', but can be defined only in terms of the 'truth value' of the descriptive signs they put into connection; in the latter case, because* as a matter of fact, syntactic connectives are in connection with semantic interpretation : in order to say whether an expression (constructed out of descriptive signs and syntactic connectives) may or may not be asserted, one is required to know not only the meaning of descriptive signs (that is to know whether the descriptive part of the expression is true or untrue) but also to 'infer' from the truth-value of the 'descriptive part' of the expression the truth value of the whole expression. This requirement implies the knowledge of the truth conditions established for syntactic connectives. W e are now in a position to ask the following question : could we interpret the leaving aside of the concept of truth value as a kind of 'significant generalization' about the structure of natural languages? Otherwise phrased : can the fact that the 'truth value' is disregarded within

CRITICAL APPROACH TO TRANSFORMATIONAL

THEORY

23

the transformational semantic theory be motivated by some adequacy reason? If the answer is 'yes', one could not see anywhere throughout transformational theory such a motivation; if the answer is 'no', then one might plainly consider that transformational semantic theory is illfounded. b. The 'meaning' conceived as plain 'translation' in terms of a lexicon of semantic markers and not as truth condition is not an adequate device for describing 'analyticity' in natural languages (see Yasiliu, 1968). According to Katz (1965) analytic sentences are characterized by a kind of 'semantic vacuity' of the subject-predicate relation, that is the predicate gives no more semantic information than the subject. Katz (1965, p. 531) gives the following definition of the concept 'analytic' with respect to copula sentences : "The copula sentence S is analytic on the reading r lt2 if, and only if every semantic element ex in p2 is also in p1 and for any complex semantic element { ^ U ^ L I • • • Ue„} in the path p2, there is a semantic element ei such that 1 j < n and ei is in the path j^." [Here p1 symbolizes the (derived) reading of the node NP, immediately dominated by S ; pz is the (derived) reading of the node V P ; by the 'amalgamation' of the readings plf p2 a new reading r1>2 is obtained, assigned to the node S ; thus r v z represents the meaning of the sentence.] I have already shown that the proposed definition is not fit for a hypothetical language where to each formative is assigned a reading ending with a 'distinguisher'. It happens to be so, because 'distinguishers' are what is 'ideosyncretic' in the meaning of different words; that is, if a distinguisher '[&,]' occurs at the end of a reading 'r,' of a formative 'fi', it does not occur associated with any other formative (not identical with 'f,') in the lexicon of the same language. Thus, if a copula sentence has the form

a

I

b

ih

24

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SENTENCES

(where 'a -> b [J^]' is the reading of the subject-NT, 'a b -> [S 2 ]' is the reading of the node VP, identical with t h a t of the node N P , excepting the distinguisher '[S 1 ]' non-identical with the distinguisher '[S 3 ]'), this sentence cannot be analytic, because there is an element of t h e derived reading of V P which is not 'contained', or which is not also an element of the reading associated to the subject-NP, and this element is exactly the distinguisher '[8 a ]'. Notice, from the very fact t h a t in our hypothetical language each reading ends with a distinguisher, one has to infer t h a t in a language satisfying this condition there is no analytic sentence (excepting, of course, sentences of the form a dog is a dog, a boy is a boy, etc., t h a t is sentences in which the nominal p a r t of t h e predicative is identical with the subject-noun). Moreover, since the representation of a semantic property either b y a 'semantic marker' or by a 'distinguisher' is a decision m a t t e r (there is no reason to qualify one or another representation as " t h e t r u e one"), one has to say t h a t even qualifying an arbitrary sentence, S,, as analytic or non-analytic becomes a decision matter, too. Accordingly, a sentence like (1) Bachelors are unmarried could be considered, in the same language, either as analytic or as synthetic, depending on whether the items bachelors and unmarried do or do not contain both (or only one of them) a distinguisher within their semantic representation. I t seems obvious t h a t such a conclusion can difficultly be accepted. As far as conditionals are concerned, one gets in trouble trying to characterize them from the analytic-synthetic standpoint, in agreement with t h e transformational theory. I n order to be characterized as analytic, the following conditions are laid down : (a) the derived reading r1 of the conditional clause, S1} is obtained by the amalgamation of the (derived) readings attached to t h e subject-NPj and predicate-VP a of this sentence, the derived reading r2 of the main sentence, S 2 , is obtained by the amalgamation of the (derived) readings belonging to subject-NP 2 and predicate-VP 2 of S 2 ; (b) all of the semantic elements belonging to the reading of N P 2 are elements of the reading of NPX and all the semantic elements belonging to the reading of VP 2 are elements of t h e reading of VP], (see Katz, 1965, p. 541). I t is easy to see t h a t we now have another t y p e of semantic vacuity. That means t h a t not only the subject-predicate syntactic relation may be

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25

semantically vacuous but some other ones, too. Under such conditions, we cannot decide for any sentence, Sj, whether it is analytic or not, unless the semantic theory is provided with a device recursively specifying all syntactic relations whose semantic vacuity is to be considered as a kind of 'mark' of the analyticity of a sentence. However, throughout the transformational semantic theory such a recursive specification cannot be found at all. One will also notice that Katz's definition of 'analytic conditional' does not cover conditionals of the form (2) [If ((If Si, then S 2 ) and S x )] then S 2 although any conditional of this form is obviously analytic. We must finally say that Katz's semantic theory does not establish any relation between an analytic sentence of the type (1) and an analytic sentence of the type (2) or of the type (3) The student is reading the book or the student is not reading the book and does not give any suggestion concerning the way of establishing such a relation. Obviously, sentences (1), (2) and (3) should be considered analytic as they are always true (and can never be false); but the concept of 'truth value' is, as already seen, completely extraneous to Katz's theory. c. I shall now consider the properties of two types of copula sentences. The first type is represented by (4) Bachelors are never-married men the second one, by (5) Books are physical objects (sentences (4), (5) are modified forms of some sentences characterized by Katz, 1965, as analytic). Sentences (6) I see a bachelor (7) I see a never-married man are synonymous, whereas sentences (8) I am reading a book (9) I am reading a physical object are not synonymous (moreover (9) might — perhaps — be qualified as 'anomalous', whereas (8) may not). It follows that although (4) and (5) are analytical and their analyticity is to be defined in the same way (both are copula sentences; in both the set of the semantic elements from the predicative are "contained" in the

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reading of the subject-NT), substituting in the same context the term from the right of the copula for the term from the left of the copula, we cannot always get synonymous constructions. The explanation of this fact is simple enough : in terms of a logical analysis : (4) is a logical equivalence, while (5) is an implication; we can then say that 'bachelors' and 'never-married men' are logically equivalent and that 'books' implies 'physical objects'. Since two expressions which are logically equivalent are synonymous, we must say that 'bachelors' and 'never-married men' are synonymous. Further, since by substituting synonymous elements within a given expression we obtain a new expression which is synonymous with the first one, we can say that (6) and (7) are synonymous. On the other hand, the two terms of an implication are not synonymous; accordingly, substituting non-synonymous terms within a given expression, we obtain a new expression, which is not synonymous with the first one; in our particular case, 'books' and 'physical objects' are not synonymous and, consequently, (8) and (9) are not synonymous, either. I think that transformational semantics cannot express the distinctions we have just made, for two major reasons : (a) the absence of any distinction between logical and descriptive signs (every copula is a logical sign); (b) the absence of any truth condition fixed for logical signs. It is easy to see that the transformational theory of analytic sentences "hides" some of the properties of the sentences characterized in terms of this theory. The fact of being analytic or not does not determine any kind of operations to be applied to the expressions of the object-language. The concept of analyticity seems to be a plain classificatory device, having no connection with the full semantic system. To conclude, I think that the discussion under 1° a—c gives enough reasons to reject the transformational theory of meaning and to disregard it as a possible basis for a semantic theory of natural languages. 2°. According to the transformational theory, the meaning of a sentence is a kind of 'sum' of the meanings belonging to each formative which is the ultimate constituent of this sentence. The 'sum' of the particular meanings is obtained applying what is called within the transformational theory 'projection rules' (see Katz & Fodor, 1965, pp. 503—516; Katz & Postal, 1964, pp. 30—70). I shall examine in more detail the functioning of the projection rules component.

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THEORY

(a) Let us consider the subsequent sentence together with its syntactic structure (represented by the tree) and with the reading associated to each formative :

According to the projection rules, the derived reading of the sentence ' A ' will be obtained as follows (within the constituent structure of ' A ' , ' D ' is a 'head', ' E ' is a 'modifier', ' F ' is a 'head', ' G ' is a 'modifier'; 'B' is a 'head', 'C' is a 'modifier') : (iii)

D +

E-^B->a->-b->c->d->e-^f

(iv)

F +

G->C->h->i-^j->a^k

(v)

B + C-^A->D + E->B->a-^b-^c->d->e->f->F

+

+ G C - > h i j ^ a^ k I t is to be noticed that, in (v), the fact th^t, for instance, the semantic element 'd' characterizes the formative *E' and does not characterize the formative 'D', is not expressed at all. On the other hand, the fact that 'h' belongs to the derived reading of 'C' and not to the derived reading of ' B ' is expressed solely by the insertion of a sequence of categorial symbols, i.e. 'F + G -> C', between 'f' (the last semantic marker of the reading of ' B ' ) and 'h' (the first semantic marker of the reading of 'C'). Herefrom one can infer that the presence of the categorial symbols within a derived reading of a sentence is a way for introducing structure in the derived reading; the complete absence of categorial symbols would prevent us from establishing a correlation between a definite piece of 'meaning' (represented by means of a semantic marker) and a definite syntactic function. A t the same time, it is also to be noticed that ( v ) gives by no means more information than diagram (ii), or than a labelled bracketing system like (ii')

[ [ ( A -> a

b

c) D

(d - » e

f ) E ] B _> [(h - » i - » j ) r

-> (a -> k ) u ] c ] A (which says the same thing as diagram (ii)).

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On the ground of our previous comments (under 2°) we may say that the output of the amalgamation process does not bring any new information in comparison with the information given by the input of the amalgamation process. Therefore, the operations called'projection rules' are plainly redundant. (b) One could perhaps make responsible for this redundancy only the form of the projection (or : amalgamation) rules. That is, these rules might be constructed in a manner in which categorial symbols would be left aside from the derived reading; in such a case, the information given by derived reading would be different from that given by diagram (ii). The result of applying a projection rule to A, would then become (ii")

A - ^ a - > b - > c - > d - > e - > f - > h - > i - > j - > a ->k

However, in such a case the information given by (ii") would be quite insufficient, because the simple order relation among semantic markers has no structural significance. Let us consider an example. The sentence (10) The cat eats the mouse may be represented as follows (10 a) S

d

the

mouse

8

[the letters 'a', 'b', 'c', 'd', 'g' stand for definite semantic markers]. According to the projection rules given by Katz & Fodor, 1965, we g e t : (vi) (vii) (viii) (ix)

Noun + Art -> NP a -> b -> c Noun + Art -» N P a -» g —> c Y + iJP YP PredP - > d - ^ b - > a - > g - > c NP + PredP - > S - ^ a - > b - > c ^ V + NP ->VP PredP - > d - > b - 3 - a - ^ g - > c

The derived reading in the form given in (ix) is obviously different from the derived reading of a sentence like

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29

THEORV

(11) The mouse eats the cat where we have the subsequent derived reading : (ix')

NP + PredP

->d->b->a->b^c

+

PredP

If for (10) a derived reading of the form (10')

S->a->b->c->d->b->a->-g->c

had been obtained, and for (11) a derived reading of the form (11')

S->a->g->c->d-»b->a->b->c

had been obtained, then we could say at most that between (10) and (11) there is a meaning difference, but we could not say which is this difference. The situation arrived at may be explained by the very fact that expressions like (10') or (11') do not display any formal indication permitting to decide whether the sequence of semantic markers 'a b - » c' is the reading of the subject or of the direct object; moreover, we cannot even decide whether this sequence is to be assigned to the subject alone or to the object and to nothing else; and further : on which ground may we take the sequence 'a b -> c' and ask ourselves "which is the node to be assigned t o ? " , and not the sequence 'b a g -> c' or 'c d b a' or anything else? I conclude my remarks under (a) and (b) by laying down the following dilemma : — either the derived reading is a set of symbols syntactically structured, and, in such a case the 'projection component' is redundant; — or the derived reading is a set of symbols syntactically unstructured, and, in such a case, the derived reading cannot be taken as a device describing the meaning of a sentence. I think this dilemma compels us to reject the transformational 'projection' (or 'amalgamation') component as a basis for a semantic theory of sentence meaning. I think, too, that Weinreich's well-founded criticism (1966) concerning the unstructured character of derived readings is somehow to be amended in agreement with the results of the analysis just made in the previous lines. § 5. Object-language and meta-language. I t is known that, when describing a language, a sharp distinction should be kept between the signs and the expressions belonging to the object-language and the signs and the expressions belonging to the meta-language. I t is also well known that when this sharp distinction is not observed, contradictory statements become possible.

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In my opinion, the above-mentioned distinction is not explicitly made within the transformational semantics. Let ns consider the following sentences : (12) Bachelors are unmarried (13) Stones are physical objects These sentences can be considered analytic in terms of Katz's theory (1965), for one of the readings of the formative bachelor is (i)

bachelor noun (Human) (Not young) (Never-married)

(see Katz, 1965, p. 523) and unmarried has, perhaps, the derived reading (ii)

unmarried A/(Married)

(for the symbolism lA/(Married)' see in Katz, 1965, p. 534, the definition of the 'antonymy operator'). At the same time, stones has, perhaps, the reading (iii)

stones noun (Physical object) I

and the noun-phrase physical objects has the derived reading (iv)

physical objects UP (Physical object)

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In the first case, on the ground of a never motivated identification of the semantic marker '(Never-married)' and the semantic marker 'A/(Married)' (but we leave aside this inaccuracy), one could say that all semantic markers of unmarried are included into the reading of bachelor. In the same way, in the second case, all semantic markers of physical objects are included into the reading of stones. It is to be noticed, however, that the above specified identifications are only the conditions under which Katz's interpretation could be accepted. But, as a matter of fact, it is doubtful whether these conditions could actually be satisfied. In order to agree with the analyticity of (12), we should agree that married is represented in the lexicon as married

(v)

I

(Married)

(this being the only possible motivation for representing unmarried by 'A/(Married)' which is difficult to accept, because the semantic marker '(Married)' could not be a real explanation for the meaning of married. If one could accept this kind of meaning explanations, one could also write down such explanations as house

horse

noun

noun

walk... verb

(House) (Horse) (Walk) which are obviously trivial. Let us now suppose that the meaning of married could be represented not only like in (v), but in terms of a reading containing also the markers a, b, c, etc., along with (Married); but in such a case, none of the markers a, b, c, etc. should be identical with a semantic marker of the reading attached to bachelor. If such is actually the case, then (12) could no longer be accepted as analytic. In a similar way, one could show that not necessarily all semantic markers of the derived reading of physical objects are also semantic markers of the reading of stones. As a matter of fact, it seems that (12), (13) are considered by Katz as analytic not because they actually satisfy the conditions above stated, but plainly because, on the ground of homonymy or of simple material similarity, he identifies the English words unmarried or physical objects with the semantic markers '(Never-Married)' and '(Physical Object)',

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which do not belong to English, but to the meta-language in terms of which he speaks about English. Using a more explicit phrase, we can say that Katz is mistaken when he is not making a sharp distinction between sentences like (14) The stone is a physical object belonging to English, and sentences like (14') lThe stone' is a '•(Physical object)'' belonging to the meta-language. Obviously (12) and (13) are analytical. Yet this- characterization seems impossible in terms of Katz's theory, because of the misuse of the symbols belonging to the object-language and those belonging to the meta-language. § 6. The relations between semantics and syntax. According to Katz & Fodor (1965) semantics could be defined by exclusion : all the properties of a sentence which do not fall into the range of a syntactic description and which, in a reasonable way, might be considered semantic are semantic properties (see Katz & Fodor, 1965, pp. 483, 484). Such a semantic theory is supposed to account for the semantic properties in the following way : (a) characterizing a sentence as semantically normal or as semantically anomalous (for example, the cat eats the mouse is normal, whereas the table eats the mouse is anomalous); (b) characterizing a sentence as being ambiguous or non-ambiguous and establishing its degree of ambiguity (for example, I see the bank is ambiguous whereas I see the dog is not ambiguous); (c) describing the synonymy relations among sentences (for example sentences two chairs are in the room and there are at least two things in the room and each is a chair are synonymous); (d) characterizing a sentence as being analytic, contradictory or synthetic. All these semantic properties are described by means of 'projection rules' applied to the readings of the formatives of a sentence, and the readings are, as already seen, sequences of semantic markers, distinguishes and selection restrictions. All these terms designate the sign categories used within the theory. However, from the fact that this sign system (representing a part of the semantic theory) is used in order to describe the semantic properties of the class of sentences specified by a grammar G, one can by no means infer that any use of this sign system, even outside the given semantic theory, necessarily describes semantic properties. Otherwise phrased : a semantic marker like '(Human)' is, by itself, no more than a descriptive sign of the meta-language; it is assigned to

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the class of signs belonging to 'semantics' because and only because we put it into a fixed correspondence with an object-language feature which is supposed to have something to do with what we usually call 'meaning'. But nothing can prevent us from deciding to put into correspondence the same sign with an object-language feature having nothing to do with 'meaning', for instance with a class of words. In the latter case '(Human)' is a sign for a class of words (like boy, girl, professor, teacher, etc.) from the object-language, no matter whether the words belonging to this class have or not a common semantic feature. We may consider that some signs of the meta-language belong to the category of semantic signs only because they are interpreted within a theory which for some reasons is considered semantic. In order to be more explicit, let us take an example. Let us suppose that in our semantic marker system we use the sign '(Count)' in order to make the distinction between the meaning of nouns like man, animal, tree, etc., (that is nouns having as designata objects which can be represented as space-time points) and the meaning of nouns like water, milk, wine, etc. (that is nouns having as designata objects which cannot be represented as space-time points); the meaning of the former class is to be characterized by the mark '(Count)', the meaning of the latter one by the mark '(Non-Count)'. Now, the difference between (15) John counts the trees and (16) John counts the water is to be described as follows : the verb to count requires that its direct object be characterized by the feature 'Count' (and never'(Non-Count)'); the selection restriction of the verb to count is not satisfied by the word water and, therefore, (16) is anomalous, whereas in (15) the selection restriction of to count is satisfied by the noun tree, and, therefore, (15) is normal. All these statements are concerned with meaning, because we consider '(Count)' as a sign whose designatum is a meaning feature, and because the selection restriction of to count is concerned with a semantic feature; furthermore the characterizations 'normal' and 'anomalous' are interpreted as referring to meaning. We say (15) is syntactically wellformed and semantically normal, whereas (16) is syntactically well-formed and semantically anomalous. However, the same signs may be used in order to describe purely syntactic properties. In Romanian, the word niste nearly covers the meaning of the English some; it has the following restriction concerning its 3 -

c . 1492

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distribution : it cannot occur before countable nouns in the singular but it can occur before countable nouns in the plural or before uncountable nouns ; then, phrases like (17) niste oameni "some men" (18) niste lapte "some milk" are well formed, while a phrase like (19) * niste orn "some man" is ill-formed. In order to prevent grammar from producing phrases like (19), we have to construct it in such a way as to have a context-sensitive rule allowing the rewriting of the symbol 'Det' as niste only in a context characterized either by the features '(Count)' and 'Plural' or by the feature '(Non-Count)'. Here, '(Count)' or '(Non-Count)' could simply be taken as names of two word classes, that is as having nothing in common with meaning. The feeling that in this case we are necessarily dealing with meaning features is given by the plain fact that we have got traditionally used to think primarily of the common 'meaning' of the class of words designated by terms like count or non-count, and not to think of the class itself. Changing the symbolism, that is using some "neutral" symbols (with respect to our habits in referring to signs), e.g. 'Noun b ' for "countable nouns" and 'Noun a ', for "uncountable nouns", we can restate the contextual restrictions as follows : the symbol 'Det' is to be rewritten as niste either in the context 'Nounft' or in the context 'Noun b ' and 'Plural'. Such a statement says obviously nothing about meaning but only about the syntactic contextual restrictions of the formative niste. To conclude our theoretical remarks we must say that some signs (of the meta-language) are semantic signs or refer to semantic features only as far as they are put into correspondence with semantic features by an explicit decision. In this point, I fully agree to the following statement from Katz & Fodor (1965, p. 497) : "Semantic markers are the elements in terms of which semantic relations are expressed in a theory". Everything seems to be clear enough, but, starting from Katz & Fodor's theory (1965), some scholars claim that syntax and semantics are inseparable, that there is a "deep interpénétration" between these two levels of the structure of language (see, for example Weinreich, 1966, proposing a new version of the transformational theory grounded on this supposed "deep interpénétration"). Chomsky himself saying "we call a feature 'semantic' if it is not mentioned in any syntactic rule"

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THEORY

35

immediately adds : "thus begging the question of whether semantics is involved in syntax" (see Chomsky, 1965, p. 142). I think that the idea of "interpénétration" has occurred as a consequence of the fact that transformational semantics does not define "semantic markers" only by saying they are signs of a meta-language in terms of which a semantic theory is expressed, but also by a kind of "inherence" of some semantic nature (see Vasiliu, 1969). For instance, one may read in Katz & Fodor, 1965, p. 496 : "The semantic markers and distinguishers are used as the means by which we can decompose the meaning of a lexical item (in one sense) into its atomic concepts". That means 'semantic markers' and 'distinguishers' are not conceived here as simple signs to which we assign by rules a fixed meaning, but as a kind of 'instrument' decomposing the meaning. Therefrom it is easy to infer that there is some natural and secret relation between such elements and meaning. Once this view accepted, one can say that whenever signs like '(Human)', '(Count)' occur, they necessarily refer to meaning, and we reject the possibility of simply referring to a word class. We may say that the transformational theory cannot express significantly the relations between semantics and syntax as it disregards the distinction to be made between two possible designata of the signs of the meta-language : class of words on the one hand, and common meaning feature of a word class, on the other hand. § 7. Final remarks. To conclude I wish to say that transformational semantics cannot be taken as a basis for a semantic theory of natural languages in the form given by Katz & Fodor (1965). This is the motivation of my attempt to construct a different theory, of a transformational type, too, but which is intended to be exempt from the shortcomings just discussed in the preceding paragraphs. However, it is worth stressing that despite the above criticism, or, maybe, just because of it, the transformational semantic theory provided many valuable suggestions concerning the basic lines of the theory which is going to be outlined below.

CHAPTER II

THE BASIC FORM OF A SEMANTIC SYSTEM S

SUMMARY.

The aim of this chapter is to describe the form ot "formalized language" (in Church's sense, 1965) or what Carnap (1960) calls a semantic system. This system is not constructed in such a way as to account for some properties of the natural language L 1( but it is described in its very general form, that is independently of the possibility to put it into correspondence with one or more natural languages. This is the meaning of 'basic' from the phrase 'basic form* at the head of the chapter. This general or "basic" form of the system will be symbolized by S, with no subscript, whereas Sj, Sj will be the symbols for the more specific semantic systems, constructed in order to account for the properties of natural languages. This specific form will be given later on (chapter IV), only after specifying those features of natural languages to be accounted for in terms of the semantic system. The topics of this chapter will therefore be : the lexicon of S ; the formation rules of S ; the rules of designation of S ; the rules of truth; the explanation of the L-concepts; the rules of inference of S ; the explanation of the concepts derivation and proof; the primitive sentences of S ; some theorems in S ; some theorems about S ; meaning postulates in S ; explanation of the terms extension, intension and extensionality.

§ 8. The lexicon of the system S. Among the signs of the system 8, we have to make a distinction between A. Descriptive signs and B. Logical signs. The descriptive signs belong to one of the subsequent classes (a) predicate constants; (b) individual constants. The individual constants are signs referring to individuals within the domain of natural languages. This domain, in the case of natural languages is infinite, including all objects a collectivity knows or thinks that it knows something about. The predicate constants refer to the properties or the relations characterizing the individuals of the domain.

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SENTENCES

The signs belonging to the above-mentioned classes and subclasses are the following : A. Descriptive signs (a) predicate constants : PR X , P R 2 , . . . , PR n (b) individual constants : a, b, c . . . ; or : ax, a 2 . . . , a j ; b1} b 2 . . . , b k ; B. Logical signs (a) variables (b) connectives (c) quantifiers. Each descriptive sign belonging to Aa, Ab has a fixed value in S, and, consequently, belongs to the class of constants. Along with the predicate constants (Aa) and individual constants (Ab), the lexicon of S contains a (finite) number of corresponding variables, viz.: Ba a : predicate variables: F, G, H . . . ; or : F x , F 2 . . . , F m ; G x , G2, . . . , Gn, etc. Ba p : individual variables: x, y, z . . . ; or : xx, x 2 , . . . x k ; y1; y» •••> yk> e t c Ba y : propositional variables: p, q, r , . . . ; or : p1? p 2 , . . . , pn ; q w q2, . . ., q m , etc. A variable can take as its proper value any of the constants belonging to a definite class; that is, the different values which an individual variable can take are represented by the different individual constants which can be substituted for i t ; the different values of predicate variable are the different predicate constants which can be substituted for i t ; the different values of a propositional variable are the different sentences (see below § 9) which can be substituted for it. The class of all constants which can be substituted for a variable represents the range of the variable. The connectives of the system S are the following: ' V ' = logical disjunction, ' •' = logical conjunction, = negation. (Several other connectives will be introduced later on, by means of definitions; see below § 12.) The quantifiers of the system S are '(x)' = the universal quantifier, '(3X)' = the existential quantifier. § 9. Formation rules (FR). The formation rules of 8 represent a part of the grammar of S, namely that part dealing] with the specification of the correct form of the expressions in 8.

T H E B A S I C FORM OF A SEMANTIC S Y S T E M

S

39

9—1. Definition. The range of the predicate. An arbitrary predicate 'PR,' with the range n ( n > l ) = Df 'PR,' followed by n signs from the class Ab ( = the class of individual constants) is an expression in S ; if 'PR,' is followed by 'n + m ( m > l ) ' signs from the same class, this sequence of signs is not an expression in S. F R 1. Every sequence constructed from an arbitrary predicate, 'PR,', of range n, followed by n signs from the class Ab is an atomic sentence in S. According to F R 1, if 'PR,' is of the range 2 then 'PR, ab' is an atomic sentence in S ; if 'PR,' is of the range 3, then 'PR, abc' is an atomic sentence in S ; if 'PR,' is of the range 1, then 'PR, b' is an atomic sentence in S, too. F R 2. Bach constituent of an atomic sentence, can be substituted by a variable from the corresponding class : a predicate constant by a predicate variable, an individual constant by an individual variable. In agreement with F R 2, if 'PRab' is an atomic sentence in S, then the following expressions are atomic sentences in S, too: 'Fab', 'PR,xb', 'PR,ax', 'PR.xy', ' F x y ' ; in the same way, if 'PR,a' is anatomic sentence, then 'PR,x', 'Fa', 'Fx' are atomic sentences, too. Let us now introduce in our meta-language the sign whose designatum is a well-formed expression of the system S. Let us suppose that 1) by means of which the string 'Kat 1 ~Kat 2 ~... ~Kat n ! is derived; (b) applying rules { r x , . . . , rm}, one derives the string 'Kat 1 ~Kat 2 ~ " . . . ~Kat n ' and only this string. Obviously, 'KatC' is again a variable and its proper values are the category symbols of the grammar G,. The difference between 'Kat' and

TRANSLATION FROM NATURAL LANGUAGES INTO S

129

'KatC' is concerned, on the one hand, with the range of the predicates ('Kat' and 'KatC' are to be viewed as predicates): 'Kat' is a one-place predicate, whereas 'KatC' is a many-places predicate; on the other hand, 'Kat' is a preterminal symbol, whereas 'KatC' is a non-terminal symbol which is not at the same time pre-terminal; moreover, if 'Kat(X)' means " ' X ' is directly dominated by 'Kat'", 'KatCtKat^ K a t 2 , . . . KatJ' does not necessarily mean 'Kat x , K a t 2 , . . . , Kat n ' is directly dominated by 'KatC' (that is, the string might be indirectly dominated by (Kat'). According to 33—3 we may write down expressions like: (i a)

GN (QU, Det, N)

,i b)

GN (QU, Pers)

The fact that 'f,' and a, are in relation % will be symbolized as follows : (ii)

% [a,, Kat (/)]

We shall now explain some more symbols introduced in ML : is a predicate constant; the specification of its range and level — is not necessary because they result from the structure of the argument expression following the predicate. — a is any sign (constant or variable) which is the argument of a predicate. The superscript refers to the level of the sign; for example a0 refers to an individual sign; furthermore the predicate of an expression like (a0)' is a predicate of level one, the predicate of an expression like (a1) is a predicate of level two ; in a more general form, the predicate of an expression like (an) is a predicate of level n + 1 . — The subscripts of expressions like and a refer to the place the sign occupies within a sequence. — qn refers to any quantifier; the superscript refers to the level the variable belongs to. I shall establish the following rules for translation from L, into S,. The term "translation" is used here only as "interpretation" of the relation defined under 32—4. Trl : (a)

% [$r(a?, . . . aS), V (X)] if and only if : Y (X) j

(b) n is identical with the number of "cases" with respect to which the verb is sub-categorized. 9-C. 1482

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In agreement with TrRl, if a verb like spune "to say" is subcategorized with respect to 4 cases (subject, direct object, indirect object in the dative, indirect prepositional object), that is subcategorized with respect to 4V~C1~C2~CS~C4' and if relation ' 2 ( S P , spune)' holds, then spune from a sentence like (129) studentul spune ceva profesorului despre Ion "the student tell the teacher something about John" should be translated into S, by (129') SP (x, y, z, w) TrR2 :

(a0), Indef (X)] if and only i f : Indef (X)].

On the basis of TrR2, cineva "somebody" should be translated into S, by 'PEx' if we have £(PE, cineva) TrR3 : £[ 1 and where for any pair of rules, r i ; r,-, we may have r, = r,, if 1 < i < j < n. Definition 40—1 says that the level of a constituent is determined by the number of rules which should be applied for generating this symbol ; not all the rules from the set r 0 , r 1 ; . . . r n are required to be distinct from all the others. We might say t h a t it is more relevant that a rule is applied than that it has some specific form with respect to all of the others. From 40—1 the subsequent propositions follow : 40—2. Proposition.

E v e r y initial symbol is a constituent of level zero.

40—3. Proposition. If O is a constituent of level n then ' is an immediate constituent of O if and only if :

(a) O' is a constituent of O ; (b) is a constituent of level n + 1 .

POSSIBILITIES

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Let us now assume the same symbol may occur in the same derivation as constituent of level n and as constituent of level n — 1 ; we have then Of and O?- 1 . 40—4. Lemma. If Of is an immediate constituent of Oi - 1 , then G, contains at least one recursive rule r, by means of which Of is derived from Of - 1 . We shall now try to establish the conditions under which we may consider that any sentence generated by G, is translatable into Sj. Let ax, a 2 , . . . , an be the total number of the signs contained by the lexicon of the system Sj; let f 2 , . . . , f m ' be the total number of the formatives contained by the terminal vocabulary of L„ V t i . We shall now consider that Si is constructed in such away as to satisfy the following condition: (i) For every formative ft (1 < i < m) from Vt, there is at least one sign a, (1 < i < n) for which the following holds : % {% f,). Obviously, (i) is fulfilled if and only if the following additional condition is fulfilled too : (ii) The number of entries of must be at least equal with the number of entries of V t , , that is the following should hold : n ; > m. Let us assume Ox, 0 2 , . . . , On are (terminal or non-terminal) symbols of G,; OJ is a constituent 0 3 of level k ; let be an expression in S,. We now assume that the set of translation rules (TrR) from Lj into S, satisfies the following condition: (iii) If @j is an expression for which we have

and if Of is an immediate constituent of 0 £ - 1 , then expression @m for which the following holds :

is a part of an

We shall now use the symbol 'fni referring to a formative which is a constituent of level n of a sentence S, in L,. 40—5. Theorem. A sentence generated by the grammar Gi is translatable into Sj if and only if there is at least one expression @ fulfilling the following conditions with respect to sentence S :

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(a) for every formative 'if' of the sentence S, there is an expression (Sf, so that £ (ffif, ff) holds; (b) a. if (1 < j < n ) has 'ff' as its constituent there is an expression '(£{' for which we have and (3.

£ (®i, oj) contains @,n as its proper part; is a part of an expression @J_1 for which we have %m-\

of- 1 )

Y- if 0} is an immediate constituent of S, there is an expression (S}, for which we have £ and is a part of an expression 1 a. if ®a (k) is provable in Si, then 'E' may be either L-true, or L-false, or L-indeterminate in L,;

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b. if ®o (k) is refutable in S„ then 'E' is L-false in L,; c. if (k) is undecidable in S„ then: a. if (k) is F-true, then 'E' may be either F-true or F-false in L,; p. if (k) is F-false, then 'E' is also F-false in L. Theorem 43—4 and its corollary 43—5 are an essential formal characterization of the semantics of natural languages; they should be meant as explicata for what we usually call the "ambiguity" of natural languages. On the other hand, 43—4,5 give evidence for the adequacy of the theory developed throughout the previous chapters. This theory accounts for an essential property of natural languages (namely that of being ambiguous) and enables us to establish explicata for the less exact previous characterizations. At the same time, from 43—4,5 follows that simply putting into correspondence the expressions from L, with their corresponding ambiguity range enables us only to a small extent to establish the "truth conditions" for the expressions in L,; applying such a procedure we can only say that if the ambiguity range of an expression in Lj is false, then the expression in L, is also false. Therefore, we are going to disregard the correspondences between the expressions in L, and their full ambiguity range, and we shall take another way. Since the ambiguity range of any expression, 'E', in Lj is finite and since the expressions in S, belonging to the ambiguity range of 'E' may be recursively specified by means of TrR's, we shall proceed by referring successively any expression 'E' to each of the expressions belonging to its ambiguity range (in agreement with theorem 41—12). We are then going to consider that the translation into S i "disambiguates" a sentence 'E' in the same way in which the translation of a word like Er. bois by Germ. Holz or Walci makes the sense of 'E' unambiguous. In such a way, establishing the truth-conditions of an ambiguous expression 'E' will become closely similar to establishing the truth-conditions for unambiguous expressions (see 43—4.A, 43—5.A). In order to make possible this approach, we have to consider that among the truth-conditions must be included the disambiguation itself, expressed by specifying a definite translation among all the possible translations from the ambiguity range. We can lay down a new theorem, analogous to 43—5. A. Let V ... V be the ambiguity range of an arbitrary expression 'E'. 43—6. Theorem. If belongs to the ambiguity range of 'E' and if S (®„ B)

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is true, then the following hold : a. if (£] is provable in Si5 then 'E' is L-true in Li; b. if is refutable in S„ then 'E' is L-false in L,; c. if 'E' is L-true or L-false, then 'E' is L-determinate in L,; d. if is undeeidable in S„ then 'E' is L-indeterminate (or factual) in L,; e. if 'E' is L-indeterminate in L,, then 'E' is F-true if is derivable in S, and 'E' is F-false if ~ (S, is derivable in Sj. The proof of 43—6 is based on the fact % expresses the L-equivdlenee between and 'E' (see 40 —7a). From 43—6 it follows that to the extent that we can establish some decidability theorems with respect to the expressions (5 in Sj, these theorems express the truth-conditions for those expressions in L, which are translatable by (5. In order to take the next step in examining the decidability problems raised by the expressions in Lj, it is worth mentioning that the expressions @ in S, which correspond, by translation, to expressions 'E' in L, (generated by G,) are either conjunctions of atomic sentences, or implications (and conjunctions) of atomic sentences. To the full expression may be prefixed (by translation rules) non-extensional signs like ' : ' , 'HO :', ' O P : ' , 'OR :', ' t :' (that is signs belonging to the class see TrR25, § 32) or a modal (and also non-extensional) sign like ''. Since the truth-value of an expression belonging to the class 9Jlo does not depend on the truth-value of the immediate following expression and since Sj has no truth-rules for the symbols belonging to this category (the semantic interpretation of expressions of the form 9Jlol) . . . [¥t J

)D P-

1

belongs

(»!,•• •)]•

W e shall further assume that for we have the meaning postulate (ix) expressing a "selection restriction" with respect to the predicates from the class SU'-1: (ix) (bl) . . . ...)=)(t) x ,...)]. Let us suppose (vii) is an existential expression. In conformity with theorem

POSSIBILITIES OF CHARACTERIZING L ,

IN TERMS OF S,

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T14—1 Ld. (3) from Carnap, 1958, p. 55, we should obtain from (viii) (x) OM . . . W (»„. . • )D (3b!) . . . W - 1 to, . . . ). From (vii), we obtain by 17—2 and E I 1 (xi)

( 3 ^ ) . . . ^ (&!,•••)•

From (x) and (xi), we obtain by E I 4 : (Xii)

(3t.x) . . . ^

to,..

.).

From (ix), we obtain according to Oarnap, 1958, p. 55 (the same theorem as before) : (xiii) (3bJ . . . to,.. .) D ~ (3t)j). . . W - 1 to,. . . ) From (vii), we obtain by 17—2 and E I 1 : ^ckto,---)-

(xiv)

From (xiv), by a theorem of existential inference (see Carnap, 1958, p. 55, T 1 4 - 1 L, b(l)) we obtain : (xv)

O^t'to,

...).

From (xv) and (xiii) we obtain, in conformity with E I 4 : (xvi)

- ( 3 M ••• ^ - > 1 , . . . ) .

One should notice that from (vii), in agreement with postulates (viii), (ix), the rules of inference and theorems of S,, both (xi) and its negation (xvi) are provable. In agreement with 17—5 (which states that every sentence from which both i , - - - ) ] then, every expression „...)D

(»!,...)]

Let us now try to see what follows for (xxix) from 1°—3°. If, together with (xxviii), is asserted also : (xxxii)

(

(&„...)

it can be proved that (xxviii) and (xxxii) are incompatible with respect to (xxix)—(xxxi), or otherwise phrased, (xxviii) and (xxxii) are mutually exclusive, in agreement with (xxxiii)

(

» ^ ( ^ . . . j D - «

)¥t1(tou...)D¥rl(l)1)...))

If (xxxii) is not asserted, it can be proved that (XXXiv)

~(t>!)-

• • ( « > ! , . . . ) )

is true. However, since according to the previously established translation rules, expressions like (xxxii) —(xxxiv) have no direct correspondent in L„ we shall consider here that the possibility of proving (xxxiii) and/or

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(xxxiv) is i r r e l e v a n t for L, and, consequently, no decidability theorem will be established with respect to those expressions in L, which should be translated in S, by expressions of the form (xxviii). Let us now consider an expression of the form (

(XXXV)

)...iPt1(t)1,...)D(

) ...

(»!,...)

and a meaning postulate of the form (xxxvi)

to)

...

to,

to,

•••)]•

It is easy to show that each of the following forms (xxxv) may have, viz.: (3b,) . . . [ ^ ( « „ . . . J D I M ^ , . . ) ] (xxxvii)

to)...

(»!,...)=>

to)...

¥r„to,...)

to) . . . $ t x t o , . . . ) D (and . . . $ r a t o , . . . ) (3th) . . . to,...)D (3»i) . . . $ r , t o , . . . ) is derivable from (xxxvi) and, consistently, provable in Sj with respect to (xxxvii). Moreover, the negation of each of the expressions under (xxxvii) is refutable in S„ with respect to (xxxvi). One should notice that, in case one of the expressions under (xxxvii) is quantified either by '(31 b)', or by '(3s 0)', or by '(3" t>)', that expression becomes undecidable. This is because the occurrence of such a quantifier within that expression transforms it into a conjunction. For example, an expression like (xxxviii)

(31 x) Fx D Gx

is equivalent with (xxxix)

(3x) F x • (y) (Fy = Ixy) z> Gx

in agreement with the definition given to i (3 1 x)' (22b—7). But it is easy to see (xxxix) cannot be derived from a postulate of the form (xl)

(x)(Fx D Gx). Let us finally consider an expression of the form

(xli)

(

)...$tto...).

Any expression of the form (xlii)

(

) . ' . . OPt to...) D $ t t o . . . ) )

is provable in Si, in agreement with 18—9T.

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Since from (xlii) can be derived (t>i)...

(toi,... )=> (&i)...

(t»i) . . . ?t(t)x,

(xliii)

(3t»x) ...

(3»0 . . . [?Pr (th, (30!)

Oh,...)

(»1,...)]

(»!,...) 3 ( 3 ^ ) . . . ?Pt (to:,...),

all these expressions are provable in Sj. In conformity with what we previously discussed underjb, we are now in a position to establish the following theorems concerning the expressions which are translations from L, and contain implications. Let 2ik be any expression of the form (xliv)

( );...

)

or of the form (xlv)

( ) ...

and which is a part of an expression @k for which the following holds: (xlvi)

% (@k, E).

43-10. Theorem. 'E' is: (a) L-false in L„ if 2tk is refutable in S, (b) L-indeterminate in L t , if 2Ik is undecidable in S, and if S k is

not refutable on the basis of 43—8,9. For proving this theorem we should refer to the fact that, if 3tk is a part of an expression @k and if (£k is a translation, it follows (in agreement with the TrB's) that those parts of @k which precede or follow 2lk are connected with 2Ik by means of the conjunction sign, 1 • ' ; we have then (xlvii)

@i*...

• • • ,(Sn>

where . . . are atomic sentences or complex sentences. Further, according to ET 3 (§11), we may decide about the truth-value of (xlii): if is provable, then 2tk is refutable. But since 2ik is refutable, (that is L-false), (xlvii) is also refutable (that is L-false) because of ET3. On the other hand, if 9lk is undecidable, the whole expression (xlvii) must also be undecidable, if it is allowed to be so by 43—8,9 (which specifies only the conditions when a conjunction is refutable', if it is not refutable it is

undecidable).

Let us consider the following expressions: (xlviii)

(tot) • - - [ ^ ( ^ - i D l ' t V - ) ]

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and an expression 2ik with one of the following forms : (öj . . . [ ^ ( ^ . . . ( D l J ' f e , . . . ) ] (xlix)

(3M ... H i t f o , . . . ) Z>

J

( 0 l ) . . . [«T (t l f . . . ) ] D (t) 1 )...[ip'(b j ? ...)l (»,) . . . [¥r(»i, . . . ) ] = > (3»i) ••• [ ^ ( t ' i , •••)]•

43—11. Theorem. Expression 2lk i s : (a) provable in S, with respect to (xlviii), if (xlviii) is a moaning postulate in S i ; (b) undecidable in S,, if (xlviii) is not a meaning postulate in S r The immediate consequences of 43—11 are expressed in terms of the following corollary : Let @ t be an expression for which the following relation holds : (1)

2

E).

Let 9lk be a part of (£k. 43—12. Corollary. ' E ' i s L-indeterininate if and only if @k is not refutable on the basis of 42—8,9. I t is easy to show that, if Stk were provable with respect to (xlviii) (43 —11(a)), Qrk would still be undecidable (see above the comments under 43—10); on the other hand, if 2ik were undecidable in S, (43—11 (b)), would also be undecidable; of course, both statements hold only to the extent that @k is not refutable on the basis of 43—8,9 (see the comments under 43—10). I t can be shown that an expression like (li) (öi)... is but an instance of 43—10. According to the theorems under b, an expression like

(182) Orice pisicd este o masa "any cat is a table"

is L-indetermined in L„ in spite of the fact we have in S, meaning postulates like (Iii)

(x) ( P i x D ANIMx)

(liii)

(x) (MAx Z> OBx)

(liv)

(x) (ANIx Z> ~ OBx)

and for pisicä (lv)

" c a t " and masä "table" we have

% (PI, pisicä) % (MA, masä).

14 - c. 1482

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It is because expression (182)' SIM xag '(x) PIx Z> (3x) MAx is undecidable. On the other hand, we have (lvi)

% (ANIM, animal)

and if (lvii)

(x) (PIx D ANIMx)

is a meaning postulate, then (183) Orice pisica este mi animal "any cat is an animal" is L-true with respect to (lviii) because of 43—11. In the same way, sentence (184) Orice pisica este o pisica "any cat is a cat" is also L-tme, becauseof 43 — 11 and because the translation (184')

(x) P I x D (3x) PIx

is a particular case of 43 — 11. c. According to 26—6,8, where the truth conditions for an expression of the form O© were established, we can write down the following theorems. Let be an expression of the form (lviii) for which we have (lix)

o

©„), £ (®„ E).

4 3 - 1 3 . Theorem. E, is : (a) provable in S, with respect to 1° a and 2° from 43—3, if