Stratificational Grammar: A Definition and an Example 9783110811834, 9789027907127

Table of contents :
CONTENTS
ABBREVIATIONS AND SYMBOLS
0. INTRODUCTION
1. THE PRESENT STATE OF STRATIFICATIONAL THEORY
2. FORMAL DEFINITION OF THE THEORY
3. THE NUMERAL SYSTEM
4. SEMOLOGY
5. THE LEXEMIC STRATUM
6. THE MORPHEMIC STRATUM
BIBLIOGRAPHY
INDEX

Citation preview

STRATI FI CATION AL G R A M M A R : A DEFINITION AND AN EXAMPLE

JANUA LINGUARUM STUDIA MEMORIAE N I C O L A I V A N WIJK D E D I C A T A

edenda curat

C. H. V A N S C H O O N E V E L D INDIANA UNIVERSITY

SERIES MINOR 88

1970

MOUTON THE HAGUE • PARIS

STRATIFICATIONAL GRAMMAR A DEFINITION AND AN EXAMPLE

by

G E O F F R E Y SAMPSON

1970

MOUTON T H E H A G U E • PARIS

© Copyright 1970 in The Netherlands. Mouton & Co. N.V., Publishers, The Hague. No part of this book may be translated or reproduced in any form, by print, photoprint, microfilm or any other means, without written permission from the publishers.

LIBRARY O F CONGRESS CATALOG CARD NUMBER: 74-118282

Printed in The Netherlands by Mouton & Co., Printers,,The Hague.

CONTENTS

Abbreviations and symbols

6

0. Introduction

7

1. The present state of stratificational theory

16

2. Formal definition of the theory

21

3. The numeral system

35

4. Semology

48

5. The lexemic stratum

55

6. The morphemic stratum

59

Bibliography

64

Index

66

ABBREVIATIONS A N D SYMBOLS

CF DL DM DS s GAE HN / IC L L-... M M-... MN P P P-marker PSG QE RP S S-... s t TG UL UM US

•

#

()

context-free downward lexeme downward morpheme downward sememe see p. 23 General American English hypersemon identity element for concatenation immediate constituent lexeme label for lexotactic line morpheme label for morphotactic line morphon phoneme HN /power-of/ phrase-marker phrase-structure grammar Queen's English Received Pronunciation sememe label for semotactic line HN /sum-of / HN /times / transformational-generative upward lexeme upward morpheme upward sememe tactics-initial symbol DM /word-boundary/ HN /times/ DL /comma-intonation/

0.

INTRODUCTION

The present work offers an analysis of the English numeral system, as a brief example of a stratificational grammar. 1 In view of the fact that the transformational-generative (TG) theory of language developed by Noam Chomsky and his followers already exists in a high state of sophistication, it will be appropriate to begin by explaining why a linguist may be motivated to work in the framework of the relatively new stratificational theory. 0.1 As is widely known, transformational theory is centrally concerned with the problems of speech acquisition and linguistic universals; the main goal of the theory is to characterise as narrowly as possible the set of natural languages as a subset of the set of languages in the mathematical sense. To this end, grammars are written whose primary component (the 'syntactic' component) can be used to generate all and only the sentences of the language in question, by making random choices through a set of rewrite rules; the sentences thus randomly produced are then assigned semantic and phonetic INTERPRETATIONS via the semantic and phonological components respectively. A transformational grammar is not to be regarded as providing the semantic interpretation of a given phonetic matrix, or phonetic interpretation of a given semantic input, in whatever form this may be: Chomsky himself is quite 1 I am grateful to Professor Sydney Lamb, Professor Rulon Wells, Mr. Paul Black, Dr. Paul Kratochvil, and Professor John Lyons, who have between them made many helpful comments on various versions of this work. The responsibility for its shortcomings is of course my own. (This customary statement is particularly relevant to a work which consists largely of my own interpretations of another man's ideas.) This work was supported by the National Science Foundation of the USA.

8

INTRODUCTION

clear on this point. ('Both the phonological and semantic components are ... purely interpretive': Chomsky, 1965:16.) 0.2 The achievements of transformational theory during the last ten years have been remarkable, and I do not wish to belittle them. However, it is clear that the goals I have attributed to the transformational school, while interesting, are not the only possible goals the student of language may set himself ; I do not even think they represent a formulation of the (often unexpressed) goals of the majority of structural linguists before Chomsky. The reason for the existence of language, after all, is that it is a device that, for the speaker, converts 'meaning' or thought into speech-sound, or into other physical phenomena inherently unrelated to that thought (i.e. writing) ; and for the hearer or reader, performs the reverse conversion process. Stratificational theory aims to provide a fully formal account, in very general terms, of these two processes, called 'encoding' and 'decoding' respectively. A stratificational grammar is a device that accepts as input a set of semantic units, and associates with them the appropriate set of phonetic units as output, or accepts phonetic units as input and associates with them the corresponding set of semantic units as output. (It is of course required that a single model be equally capable of accounting for both the encoding and decoding functions.) In their excellent 1966 paper, Fodor & Garrett point out the shortcomings of the view that a model of speech production and recognition should consist of a transformational-generative grammar 'plus some further component or components at present unknown' (138); the only difference between their position and that of the present writer appears to be that Fodor and Garrett do require such a model to assign the identical structural descriptions to utterances as are already assigned them by a TG grammar (152). Stratificationalists do not make this assumption; it is hard to see why they should, as there is at present disagreement between transformational grammarians about structural descriptions. (For parallel criticism, see also Narasimhan, 1967a, especially §§3.3-5, & 1967b.)

INTRODUCTION

9

0.3 Abstracting from the above argument, one may draw a general distinction between 'generative' and 'communicational' descriptions of language. The basic fact, from which anyone wishing to describe language formally must set out, is that any natural language is an (infinite) set of potential utterances, and an utterance is a pair of (semantic) content and (phonetic or other) expression. Thus one may represent a language, say L, formally, as follows: L = { ( C 1 , E 1 ) , (C 2 , E 2 ), . . . } (Here, I beg the question of what form the representation of the content and the expression of utterances should take. I shall return to this question in §4.4 below; meanwhile, I will point out that this question is no more - and no less - of a problem for a communicational theory of language than for a generative theory.) 0.4 A GENERATIVE description of L will be a function that enumerates all and only the members of L; or, as a simpler task, all and only the members of the set L', where L' = {E 1 ( E 2 , ...}. 2 A COMMUNICATIONAL description of L, on the other hand, will be a function f such that, for any integer i, f(C,) = E, & f(E|) = C,. s The difference between enumerating the members of L and enumerating the members of L' is related to the distinction Chomsky (1965:60) draws between strong and weak generative capacity, respectively. He expresses the distinction as one between enumerating the set {Ei, E 2 , ...}, and assigning the members of this set their 'structural descriptions'; however, the only reason for assigning a structural description to an utterance is that the content of the utterance may be derived from it. The justification for establishing 'deep structures' for utterances, in the absence of a 'semantic component', is that the analyst can point to them and say: 'These, while not in themselves semantic interpretations, are structures containing all and only the information needed as input by a hypothetical semantic component'; and the analysis is convincing to the extent that readers agree with this statement. Similarly, many syntactic analyses of languages produce surface structures consisting of strings of morphemes or 'formatives', without giving the phonological component that would allow phonetic interpretations to be derived from these. These also are convincing to the extent that the reader accepts that such a component could be written to produce the desired phonetic output given the surface structures in question; the difference being only that the act of faith is far smaller in the latter case,

10

INTRODUCTION

(Note that I am here restricting my discussion to natural languages. Some 'languages' invented by logicians, which may be equipped with 'syntax' but no 'semantics' in the logical sense of those terms, as well as all languages in the mathematical sense, where a 'language' is defined as a subset of the free set of strings on a given 'alphabet' - see Chomsky & Miller (1963), especially p. 283, and Chomsky (1963) - will have only expression and no content, i.e. they may be fully represented as {E 1; E 2 , ...}. For such languages only a generative and not a communicational description is possible.) Now transformational-generative theory may be characterised as by far the most sophisticated theory of generative language-description; whereas stratificational theory is the most highly developed theory known to me of communicational descriptions of language. 0.5 Regarded in this light, the two theories, stratificational and transformational, are seen to be not so much incompatible with as irrelevant to one another. One may reasonably criticise either theory on the grounds that it does not fulfil its goals, or on the grounds that there is something inherently inappropriate about its goals.3 It is not reasonable to criticise either theory for failing

as one has a much better idea of what a 'phonetic interpretation' would be like than a 'semantic interpretation', either for any given utterance or in general. (This is also the reason why the third theoretically possible class of generative descriptions, those which would define for a language L the set L ' = { Q , C 2 , ...}, need not be taken into consideration.) The generation of deep and surface structures for utterances has no value in itself, because it is uncheckable; as Hockett (1967:80) correctly points out, 'the two interfaces [between 'sound' and the linguistic system, and between the latter and 'meaning'] are the only places where observations are possible. The machinery that links them can only be inferred'. 3 Stratificational theory has been criticised in the former way notably by Postal (1964), but not, as far as I know, in the latter. Postal's criticism has been influential, and therefore should perhaps not be passed over in silence, as there are several objections to be made to it. It is in the first place incorrect in many respects. Stratificational grammars, as defined in Lamb (1966b), are not equivalent to PSGs, because of their provision for anataxis and discontinuous ICs; and although I have not seen the earlier, 1962 version of Lamb's work,

INTRODUCTION

11

to fulfil the goals of the other. This is not to say that we should not hope that, at some time in the future, there will be a unified theory which will fulfil the goals of both stratificational and transformational theories; but until that time there seems no reason why the two theories should not co-exist peacefully. 0.6 Any description of language must assign structure to the collections of semantic and phonetic units which will occur in the semantic and phonetic representations of utterances; and, as Chomsky (1957) has demonstrated, a single utterance will in general have to be assigned more than one structure. The mechanism by

to which Postal refers, Lamb tells me that it was the same in this respect. Admittedly, the system I define in the present paper lacks some of the nonPSG mechanisms of Lamb (1966b) (see my §2.0 and fn. 6 on page 27 below); but the presence of upward-ands in tactics (see §2) in itself is sufficient for stratificational grammars as here defined not in general to be PSGs. Furthermore, even if it were true that the 'language' (in the mathematical sense) generated by a single one of Lamb's tactic patterns were a CF PSG, it would not be true, despite Postal (32), that the entire grammar would in general be a CF PSG, as the intersection of two CF PSGs is not necessarily a CF PSG (Chomsky, 1963:380); the language defined by a stratificational grammar is mathematically equivalent to the intersection of the set of 'languages' defined by the separate tactics (the 1962 version of stratificational grammars had three tactics each). Lastly, Postal is incorrect in asserting (31) that Lamb's permission of deletions interferes with the assignment of structure to utterances by a stratificational grammar. Lamb's theory, at least during the period I have been acquainted with it, has never advocated assigning P-markers to utterances by inspecting successive lines of their derivation, as has been normal in TG theory (cf. Postal, 1964:11-13); instead, in stratificational theory structure is assigned to utterances directly in the course of their derivation. I understand a similar proposal has recently been made for TG theory by McCawley. However, Postal's factual errors with respect to Lamb's theory are less important than the irrelevance of his criticism. He misses the point that the main purpose of a stratificational grammar is not to generate all and only the utterances of the language in question, but rather to provide the correct realisation for any content or expression structure which is appropriate for the language. One assumes that the brain does not normally present the human speech device with 'unencodable meaning' to be encoded; so the case with a stratificational grammar that the input is an anomalous content structure is, as it were, a 'don't-care state'. Conversely, if the input is an anomalous expression structure, we expect to derive from it not the empty content structure but an anomalous one: this is, after all, the analogue of the real-life situation.

12

INTRODUCTION

which this is achieved in a TG description is that each utterance 4 is assigned a single basic structure (the 'deep structure') by the 'base component', and this structure is simultaneously given as input to two sets of rules. One set of these, the 'projection rules' (cf. Katz & Postal, 1964:20 if.), output the semantic representation of the utterance; the other set, which consists of the 'transformational rules' (Chomsky & Miller, 1963:296 ff.) followed by the 'phonological rules' (Chomsky & Miller, 1963:313 ff.), output its phonetic representation. Thus only at one point is the class of structures occurring in the language defined (the base component defines the class of deep structures); the other structures are obtained via rules which alter structure (this is true obviously of the transformational rules; also however of the projection and phonological rules, although in the case of these latter the function of altering structure is subsidiary to the function of substituting units). 0.7 Such a mechanism cannot be used for a communicational description, because the structure-altering rules are intrinsically one-way rules. That is, although (for example) given a deep structure and a set of syntactic transformations, there is a simple algorithm for obtaining the appropriate surface structure, the converse does not hold: there is no algorithm for obtaining the deep structure, given the surface structure and these syntactic transformations. (One cannot, for instance, reverse the arrows on the transformations and then apply the same algorithm as is 4 Or to be precise, sentence. The fact that TG theory gives priority to the syntactic component leads to the sentence being considered as the primary linguistically significant stretch of speech, and to the relative neglect of relationships which cross sentence-boundaries. In stratificational theory, on the other hand, while the sentence is the stretch within which relationships hold at the lexemic level (the equivalent of TG surface structure), the corresponding stretch at the sememic level (roughly the level of Chomsky's deep structure or semantic interpretation, or somewhere between the two) is the whole discourse. (For an example of stratificational treatment of relationships which cross sentence boundaries, see Taber (1966).) This appears to represent a real advantage of stratificational over TG theory, though not, perhaps, one it would be difficult for TG theory to cope with.

INTRODUCTION

13

used in the original process.) The same restrictions are true of the projection and phonological rules. (Cf. in the latter case Lamb's objection to what he calls 'mutation rules' (Lamb, 1964), and the rejoinder by Chomsky & Halle (1965:111)); the reason why Lamb considers mutation rules appropriate for diachronic, but not for synchronic statements, is that language change is a one-way process, but that translation between meaning and sound is a two-way process.) Again, I should make it clear that I realise these restrictions are not arguments against TG theory, as the form of this follows logically, if one decides to describe language exclusively from the generative as opposed to the communicational point of view. It is the belief that linguists need to restrict themselves in this way that I, in common with Fodor & Garrett (1966), do not accept. 0.8 The mechanism stratificational theory adopts, in order to assign more than one structure to a given utterance, is that classes of possible structures are defined for more than one level in the semantics-to-phonetics dimension. In Lamb's terminology, a statement of the structures in which units of any given level may occur is called a 'tactics' or 'tactic pattern'; and unlike a TG grammar, which has a single base component, a stratificational grammar is allowed to have as many tactic patterns as necessary, each of which defines the class of structures appropriate to its own level. In particular, one tactics defines the class of structures appropriate for semantic representations of utterances, and one defines that appropriate for their phonetic representations. Any given utterance is separately assigned a structure at each level equipped with a tactic pattern; the definition of the theory in §2, below, explains how a structure at any given level is associated with a particular structure (or more than one structure, in the case of structural ambiguity or structural free variation, respectively) at the next level above or below. 0.9 The stratificational theory of language includes a small set of types of relation, and a simplicity criterion. A stratificational grammar of a particular language consists of a set of semantic

14

INTRODUCTION

units, a set of phonetic units, and a large unordered set of relations, of the types specified in the theory, whose terminals are either semantic or phonetic units, or terminals of others of the relations. The grammar must of course be faithful to the facts, in the sense that given semantic inputs should produce only appropriate phonetic outputs, and vice versa; and, of two grammars meeting this requirement, the simplicity criterion chooses between the two. (I fully agree with the comments Chomsky makes (1957: §6.1) about the weakness of the requirement to be placed on the relation between the general theory of language and specific grammars.) It is no doubt true that the characteristics of semantic and phonetic representations of utterances are universal, but it does not follow that we are forced to adopt a hypothesis which completely specifies these characteristics. Except insofar as empirical research (for instance, acoustic phonetics) shows that universals exist, the semantic and phonetic units, and the structures of sets of the respective units, are to be discovered independently for each individual language. I personally feel that by its insistence on universal Jakobsonian distinctive features (the correctness of which Chomsky was still maintaining in the summer of 1966 - although I gather now that when Halle & Chomsky (1968) appears, there may be some modification in this respect), and by its adoption of the principle of the preservation of meaning under transformation, transformational theory does unnecessary violence to the phonetic and semantic structures of language respectively.5 0.10 It is a claim of stratificational grammarians that, for any given language, a grammar of the type described in Lamb (1966b), including on the one hand a 'realisation portion', showing the 6

For criticism of the Jakobsonian distinctive features, see, now, Hockett (1967:fn. 3, p. 131). Like Hockett, I fail to see evidence not just for the correctness of those particular features, but in general for the correctness of the idea that phonetic (or for that matter, any other) features should be 'binary', in the sense that a given cell in a phonetic interpretation matrix may be filled in three ways (by + , —, or blank). Objections to the principle of the preservation of meaning under transformation, however, are met to some extent by Chomsky (1965: chapter 3, fn. 9, and cf. the discussion of 'topic' and 'comment' in chapter 2, fn. 32).

INTRODUCTION

15

various possibilities of diversification, neutralisation, portmanteau and composite realisation for different units, and on the other a certain number of 'tactic patterns', giving the set of permissible sequences of units at various levels, and thus determining the choices in the case of diversifications and neutralisations, will be both the simplest according to the simplicity criterion, and, in its internal form, will correspond well with our intuitions about language. (I accept Chomsky's defence of the necessity to admit nativespeaker's intuition as data in the study of language (cf. for instance Chomsky, 1965:18ff.), as long as this is not taken to justify any neglect of more objective research; but it is noticeable that one of the concepts which very many linguists appear to have found intuitively satisfying when applied to their own as well as to other languages, is that language contains a series of 'levels', with units at the different levels; and in particular that there exists such a level between that of morphophonemes and that of phonetic features. The refutation of the phonemic principle, discussed by Halle for Russian (1959:21-4), and further discussed by Chomsky (1964: chapter 4; and elsewhere), is cogent, given the assumptions of transformational theory; but I suggest that its cogency depends on a model of phonology which translates strings of lexical items into phonetic features, as a one-way process.)

1. T H E P R E S E N T STATE O F STRATI F I C ATI O N AL T H E O R Y

The idea that a language can be described as a set of relations occurring in the configurations called 'tactic patterns' and 'realisation portion' is basic to stratificational theory. The particular relations that one defines, and the simplicity criterion one uses, on the other hand, are matters to be settled empirically; we must experiment to see which set of relations (i.e. relation-types, not relation-tokens) account in the most general way for the facts of language, and which simplicity criterion, in combination with these relations, will give the lowest simplicity measure for the analysis which seems intuitively the most generalised. In fact, both the set of relations and the simplicity criterion have been refined since the appearance of Lamb (1966b). For the purposes of this work, I adopt the version of the theory current in the summer of 1967, when the bulk of the work was written; but of course further changes in the theory are to be expected as time passes. In §2, I shall give a complete formal definition of stratificational theory as exemplified in this work. In §1, I discuss, informally, the changes that have occurred in the theory since the appearance of Lamb (1966b); I shall assume a knowledge of that work. The sort of modifications discussed in this section have about the same status, in stratificational theory, as does, for instance, the adoption of the 'parenthesis notation' for disjunctive ordering of rules (now discussed in print by Chomsky, 1967:121), as opposed to the addition of some notation to mark disjunctive ordering between particular pairs of rules (the possibility implied by Chomsky, 1967:120), in TG theory: that is, each of the alternatives would allow the same set of facts to be accounted for, but one adopts the alternative which appears to permit the description to embody 'linguistically significant generalisations'.

THE PRESENT STATE OF STRATIFICATIONAL THEORY

17

AA V9 ^ ? M

TI

10

13

11

14

12

¥ ¥ y «( >K 15

16

17 Figure 1.

18

19

18

THE PRESENT STATE OF STRATIFICATIONAL THEORY

1.1 The most important change in the relations is concerned with the 'knot pattern', or connection between the realisation portion of a grammar and a given tactic pattern. It has for some time been apparent that the relation between the realisational role of an emic unit and its tactic role was something different from the relation expressed within a tactic pattern as an upward-and. We now use a special node to indicate all links between a tactic pattern and the realisational portion; and each tactic pattern, which is as a whole at a single level in the semantics-to-phonetics dimension, is thought of as extending at right-angles to this dimension. The knot-pattern node is symbolised as a diamond (see Fig. 1, 15-19), with four sides, each of which may contain up to one terminal: the upper left connects to higher in the realisation portion, the upper right to 'higher' (i.e. nearer the tactics-initial node) in the tactics, the lower left to 'lower' in the tactics, and the lower right to lower in the realisation portion. Not all possible combinations of terminals on a diamond will occur: obviously no diamond will appear with terminals only on two opposite sides, or it would be redundant and removed immediately by the simplicity criterion. Also, any diamond will have a connection higher in the tactics; for one not to do so would be equivalent to having a rule ' / - » x \ for some x, in the base component of a transformational grammar (/ being the identity element for concatenation). Apart from these logical constraints on the configurations of diamond nodes, no use has been found for the diamond with four terminals.1 1.2 The vast majority of diamonds will have connections upwards in the tactics and in both realisational directions. These represent units at some level which both form part of the realisation of some higher unit, and themselves have a realisation at a lower level. There will also be diamonds with realisational connections in only one direction. Those with a connection only downward are 'downward-determined units': units whose occurrence is predicted 1 As diamonds link tactics with realisation portion, it follows that there can be no question of diamonds with only one terminal.

THE PRESENT STATE OF STRATIFICATIONAL THEORY

19

by the relevant tactics, but which are not the realisation or part of the realisation of any unit, i.e. they have no 'meaning'. The classic example is the -i- of Hockett's Fox word poonimeewa 'he stops talking to him' (Hockett, 1947:332-35). The converse, 'upward-determined units', are units with no realisation, 'zerounits' such as the genitive plural ending of Czech zen (from zena, 'woman'). Finally, there are diamonds which have connections downward as well as upward in the tactics, and either upward or downward in the realisation: these are for units at a given level which are realised at the lower or higher level respectively not as units but as features of arrangement. An obvious example is the semon /interrogative/ in English, which in some circumstances is realised lexemically, not as any lexeme unit, but as a particular arrangement of the other units in the utterance. Also, both these latter types of diamond are used for different types of portmanteau realisation. The algebraic notation for diamond nodes is as follows: where a is the line connecting higher in the realisation, b the line higher in the tactics, c the line lower in the tactics, and d the line lower in the realisation, the formula is a* b / c* d. Whichever lines do not occur are simply left blank in the formula. 1.3 Another innovation (which happens to be of particular importance in the grammar of numerals) concerns optional elements. It often happens that of two lines below a downwardand, both are optional but at least one must occur; it does not usually happen that either both items, one, or no items may occur as expansions of a given tactic element. So instead of using unordered ors with zero to indicate optionality, we now use an arrowhead notation (see Fig. 1, 2-4 and 6-8); where a line on the plural side of an and has an arrowhead, the occurrence of the line in question does not depend on the occurrence of the other line(s) on the plural side of the node. Algebraically, an arrowhead is indicated by writing the symbol in question in angle brackets. Thus 'a / c' means 'a is realised as either b or b^c; 'a / ' means 'a is realised as either b or c or b^c; 'a/b c\ as before, means 'a is realised as 6"V. Where there are more than two lines on the

20

THE PRESENT STATE OF STRATIFICATIONAL THEORY

plural side of the node, if they all have arrowheads any combination of one or more may occur; where none have arrowheads, all must occur, as before. There are different possibilities for defining the situation where a plural side has more than two lines of which at least one has an arrowhead and at least one lacks one; it has not yet been determined whether the facts of language provide a good reason for adopting one of these possibilities, and in this work such a node-type will remain undefined. 1.4 The last innovation in nodes is that the initial line of a tactics, which fulfils a special function in the system, is written with a small square instead of an ordinary zero-element at the upper end. The small square node or TACTICS-INITIAL NODE occurs m no other position. I symbolise it algebraically as ' • ' . 1.5 There are also two new restrictions on the arrangement of nodes in the overall system. In the first place, no upward-ands are permitted in the realisation portion: the analysis of portmanteaus now involves diamonds of types 17 or 19 in Fig. 1. Secondly, no ordered ors are allowed in the alternation pattern. Except where the non-preferred element was zero, the work they did can be handled by ordered ors in the tactics. To cover the exception, it is now permitted to have an ordered or with zero as the non-preferred element immediately above the knot pattern. (Lamb suggests, in effect, that each line above a knot pattern should have such an or, and that this part of the linguistic structure should not contribute to the simplicity measure. But even if this proposal is adopted, it will still be convenient in presenting grammars to human readers as opposed to inputting them into computers - to distinguish those zero options which will at some time be taken, from the vast majority which would not, by writing in only the former.)

2.

FORMAL D E F I N I T I O N O F T H E THEORY

Complaints have been heard that stratificational theory is not rigorously enough defined to qualify as a formal theory of language (cf. Postal (1964:31); and Chomsky has said, in a critique of Lamb (1966a), that the principle of simplicity, which Lamb adduces as determining a linguistic analysis, is so vague as to be worthless (Chomsky, class lecture at Linguistics Institute, UCLA, August 1966; and cf., now, Chomsky, 1967:fn. 10)1). It is certainly true that, in describing a theory, one is usually faced with a choice between an account that gives readers intuitive understanding of the system, and one which defines it rigorously; and it seems to be true that up to now stratificationalists have, very reasonably, tended to adopt the former approach. I wish, however, to forestall criticism of the present work on these grounds, by giving a formal definition of the system I am adopting. I shall define as much of 1

It is rather surprising that Chomsky chose to attack Lamb's article on these grounds, as it does offer a very simple, well-defined simplicity criterion, although not the one currently in use (see below). Possibly Chomsky meant to attack vagueness in the concept of generalisation, according to which the criterion was invented. Generalisation in a linguistic description is defined by Lamb as the elimination of repetition; but it is no doubt true that what constitutes repetition, or its elimination, is an intuitive judgement. However, it is precisely because an intuitive definition of generalisation cannot be used as an operational definition, that one replaces it with a formal definition such as the one given in §2.1; and no linguistic theory concerned with simplicity at all can do more than this, I think. Of course, one takes it for granted that in discussing generalisation, one means 'linguistically significant generalisation'; but I question Chomsky's suggestion (1965:43) that what constitutes this is always an empirical problem (though with respect to such an extreme example as the notation abbreviating his list (16), versus the hypothetical notation to abbreviate his list (17), empirical evidence is no doubt easily obtained).

22

FORMAL DEFINITION OF THE THEORY

Lamb's system as I need for my purpose. (In particular, neither unordered ands nor 'reduplication elements' (Lamb, 1966b: 26) will be treated. They are not needed in the analysis of English numerals; and the introduction of the reduplication elements would raise some controversial problems, whereas I wish to restrict myself to what is well established in the theory.) The system is first defined as an encoding device: that is, we define what it means for a phonetic structure to be a realisation of a given semantic structure; then the changes needed in the definitions for the system as a decoding device are indicated. 2.1 In order to define the term GRAMMAR, we must first define the relations ESSENTIAL CONNECTION and HIGHER THAN between lines. These relations are defined recursively. Two lines, a, b, are ESSENTIALLY CONNECTED if (i) there exist a node of one of the types 1-12 in Fig. 1, such that both a and b are connected to a terminal of the node; or if (ii) there exist a node of one of the types 15-19 in Fig. 1 (a 'diamond node'), such that a and b are connected to opposite terminals of the node; or if (iii) there exist a line c, such that a and c are essentially connected and b and c are essentially connected. A line a is HIGHER THAN a line b if (i) there exist a node n such that a connects to an upper terminal of n and b to a lower terminal of n; or if (ii) there exist a line c, such that a is higher than c and c is higher than b. We may now define the term GRAMMAR : a GRAMMAR, G, shall consist of an array of lines and nodes, such that each line connects to one upward terminal of a node and/or one downward terminal of a node; each node being of one of the types shown in Fig. 1. (In Fig. 1, node-types drawn with dots are understood to be allowed to have any number of terminals not less than two on their plural side.) Each line not connected to two terminals shall have a distinct symbol: lines connected to no downward terminal of a node are CONTENT UNITS, lines connected to no upward terminal of a node are EXPRESSION UNITS. 2 Further,

1

Here and elsewhere in the present paper, for 'content' and 'expression' might

FORMAL DEFINITION OF THE THEORY

23

it must be the case that the relation of essential connection partions the set of lines in G into n + 1 blocs B 0 , B l 5 ..., B n (where n is the number of nodes of type 13 in G) in the following way: (i) for all a e Bi (0 < i < n), and for all b 6 G, if a and b are essentially connected then b e Bj; (ii) B 0 contains all content and expression units, all lines connected to the upper left or lower right terminal of a diamond node, and no line connected to the upper right or lower left terminal of such a node; (iii) each of the blocs B 1 ; ..., B n contains one and only one line connected to the terminal of a type 13 node; (iv) the subset of the relation 'higher than' restricted to B 0 is anti-symmetric; and for any two lines a, b, such that a (b) connects to a terminal of a diamond node which also has a terminal connecting t o a line in B, (Bj) (i, j # 0), if i > j then a is higher than b.3 2.2 A unique simplicity value may be determined for G as follows: except for tactics-initial nodes and zero nodes (i.e. types 13 and 14 in Fig. 1), each node in G having t terminals shall be assigned the value 1 + (t — 3 ) . (1 — e), where e is a positive be substituted 'semantic(s)' and 'phonetic(s)' respectively. But as I do not wish to exclude the application of stratificational theory to written language, I prefer the more neutral Hjelmslevian terms. 3 The definitions of 'essential connection' and 'higher than' are used to ensure that tactic patterns are separated from the realisational portion and that the set of tactic patterns is well-ordered. (Note that because of the presence of so-called 'recursive loops' in tactic patterns, the relation 'higher than' is not in general anti-symmetric in the whole grammar.) When using the theory to describe natural languages, of course further constraints are imposed on the class of grammars; these constraints are discussed in Lamb (1966a, b) and in §1.5 above. For instance it would be impermissible, as the theory stands, to have a downward or immediately dominating a downward and in the realisation portion. Also, my definition permits a grammar to have any number, including zero, of tactic patterns; in fact a natural language will have at least three. But it is undesirable to write these constraints into the formal definition; rather, the latter is made as general as possible, and constraints are imposed as a consequence of empirical investigation into the structures necessary to describe natural languages.

24

FORMAL DEFINITION OF THE THEORY

quantity less than the reciprocal of the number of lines in G; the simplicity value of G is then the sum of the values of all nodes in G having values. The significance of '3' in this formula is that it is the smallest number of terminals any node except a diamond node may have, without being redundant. The criterion in effect prefers the grammar with the fewest nodes (not counting diamonds with only two terminals, which contribute a minimum of information and are assigned the value e) and 'extra lines'; where two grammars tie in this, it prefers the one with fewer nodes. (This criterion, which was invented by Peter Reich, can be shown in some specific cases to give better results than the line count of Lamb (1966a) or the external connection count of Lamb (1966b), in the sense that for some sets of data, the two latter counts will prefer an incompletely generalised description, whereas the present count will prefer the most generalised description. In the vast majority of cases, each of these counts will give the same result.) 2.3 One may now convert a grammar to an unordered set of rewrite rules4 and statements, as follows: first, to each line of the grammar not already having a symbol by §2.1 is assigned a distinct symbol. Each node is then translated according to Table 1, where a node of the form specified on the left is converted to a set of rewrite rules and/or statements of the form specified on the right. After this conversion has been completed, the resulting rewrite rules can be partitioned into a set corresponding to the realisation portion, and one set corresponding to each tactic pattern. (Any symbol for a line in the bloc B0 is a REALISATIONAL SYMBOL, and any symbol for a line in one of the blocs B t — Bn is a TACTIC 4

The fact that I define my system by means of rewrite rules should not be taken to imply that I am adopting a 'process' model of language. In a process model, the output symbols (e.g. the values of distinctive features) are themselves operated on by rewrite rules. In the system I define, any symbol which is rewritten cannot by definition be an output symbol. The distinction is not, in my opinion, a trivial one.

FORMAL DEFINITION OF THE THEORY

25

TABLE 1 N o d e type (see Fig. 1 on page 17)

Algebraic formula

Translation

1 2

a I b0 bi ... bn ( n > 1) a I 1)

3

a I b

a -*b0 bt ... bB. a (where x is any string of symbols b, b, ... bm, such that for any pair, bk, bu of adjacent symbols, 0 < k < 1 < n). a b c; a c.

4

a I c

a^>b\

5

b0 bi ... ba I a ( n > 1)

6

... I a ( n > 1)

bo bi ... bB -*a. x a (where x is as in the translation of type 2.) b -> c; ab^-c.

7

a / c

8

b/c

a ->• c; ab->

9

a/60,

10

a/6„ + 6n.,+...+6o(n>l)

11

Z>o, bu ..., ¿>„ / a ( n > 1)

12

¿>n + 6n-l + ••• + 6 o ( n > 1)

13 14 15

O/a a / 0 a *bI *c

16

a* bj *

17

a* bl

18

19

..., ¿ „ ( n > l )

c*

*a/*b

*aI b *c

be.

c.

a -^bi (for any i such that 0 < i < n). a -+bi (for any i such that 0 < i < n ; each rule a -+bi is assigned the preference value i). bi ->a, for any i such that 0 < i < n. bi -*a is assigned the preference value i. a is a tactics-initial symbol. a is deleted. a ->c; b is a terminal symbol; the mate of b is a. a is deleted; b is a terminal symbol; the mate of b is a. a is deleted; b ->c; the mate of b is a. a is a terminal symbol; the mate of a is 6; b may be inserted anywhere in the realisational string at any time (i.e. I -*b). a -+b\ the mate of c is b\ I-*c.

26

FORMAL DEFINITION OF THE THEORY

of the tactics corresponding to that bloc; any given rewrite rule will contain only either realisational symbols, or tactic symbols of a certain tactics.) Each of the content units will appear on the left-hand side, and each of the expression units on the right-hand side, of some rule in the realisational set; in each tactic set, one rule will have the initial symbol for that tactics on the left-hand side, and each of the terminal symbols will appear on the right-hand side of one of the rules. Some of the tactic symbols will be assigned a MATE among the realisational symbols; and some of the tactic symbols will be named TACTICS-INITIAL symbols. Some of the rules will be assigned a PREFERENCE VALUE. It should be stressed that there will be no extrinsic ordering in the rules; that is, there will be only the intrinsic ordering by which a rule with a string x on the left cannot be applied before some rule with x on the right has been applied, unless x is a tactics-initial symbol. (For extrinsic v. intrinsic ordering, cf. Chomsky, 1965: chapter 3, fn. 6.) SYMBOL

2.4 The set of realisational rules5 will be a device that accepts as input strings of content units (CONTENT STRINGS), and by successive applications of rewrite rules produces from them strings of expression units (EXPRESSION STRINGS). Let us call a sequence of strings such that the first is composed exclusively of content units, the last of expression units, and each string results from the preceding one by the application of one rewrite rule, a REALISATIONAL SEQUENCE, and each string of a realisational sequence a REALISATIONAL STRING.

2.5 A set of tactic rules will be a device that defines a set whose members are strings of terminal symbols of that tactics, together with their structural descriptions. Because of the presence of upward-ands in the tactic pattern, it will in general be possible to make choices, in applying the rewrite rules to the initial symbol, 5

'Realisational rule' here should not be identified with the 'realisation rule' of Lamb (1964). Realisation rules in the 1964 sense no longer form part of stratificational theory.

FORMAL DEFINITION OF THE THEORY

27

which lead to forced termination of the expansion before all the symbols in the string are terminal symbols; but we need consider only such expansions of an initial symbol as result in strings all of whose symbols are terminal: let us call such a string a TACTICSTERMINAL STRING. The structural description associated with a given tactics-terminal string will resemble the tree diagram of a phrase-structure grammar, except that it will in general contain upward as well as downward branching (because the tactics includes upward as well as downward ands). It will thus not in general be a TREE in the graph-theoretic sense, but will be a PLANAR GRAPH as no lines will cross6 (Ore, 1963:12). A tactics-terminal string with its structural description is called a TRACE. Each node of the trace, being one of the symbols assigned by §2.3 above, will correspond to a specific line of the tactic pattern; but some lines of the tactic pattern may be represented more than once in the trace, and some of course may not be represented at all (because a given rewrite rule may be applied more than once, or not at all). 2.6 We now define EQUIVALENCE between realisational strings and traces. A trace T with tactics-terminal string of the form t0 ti ... tm (m > 0), is EQUIVALENT to a realisational string, call it R, of the form r 0 r x ... r n (n > m), if the string T' is identical to R, where T' is obtained from T by

• This is the case as long as the upward-and, e.g. 'a blc\ is defined, as I define it above, to apply to a string of the form 'a x b' only if the substring x is null. This is a debatable point in current stratificational theory; the definition used earlier was equivalent to having 'a b/c' rewrite 'a x b' as 'x c' even if * is not null (cf. the discussion of 'interstratal anataxis' in Lamb (1966:23ff.)); Lamb now suggests that anataxis may be handled by unordered ands. The definition of (ordered) upward-and which I use is needed for the particular grammar presented in this article; under the alternative definition, the downward lexeme /and/ would never occur. (I am indebted to Ilah Fleming for pointing this out.) This does not mean that I regard the adoption of the definition I use here as necessary; but it means that the relative simplicity of my lexotactics, and the alternative lexotactics which would account for the same data under the alternative definition, is a piece of evidence to be taken into account when deciding which definition is the more appropriate.

28

FORMAL DEFINITION OF THE THEORY

(i) rewriting every tactics-terminal symbol of T as its mate, or deleting it if it has no mate; and (ii) for every non-terminal symbol of T, say a, that has a mate, say b, adding to T a new line branching down from a, to the right7 of the line already dominated by a, terminating in b. 2.7 Where G is a grammar converted to rewrite rules and statements, we define a DERIVATION, D, in G, as including: (i) a realisational sequence R 0 , Ri, ..., R„; (ii) for each tactics-initial symbol in G, a trace generated from that symbol, such that T is equivalent to some R| (0 < i ^ n); (iii) a preference value V D which is the sum of the preference values of all the rewrite rxlles applied to form (i) and (ii); rewrite 7

This is one of the points over which there is likely to be some debate among stratificationalists: the line terminating in b could alternatively be defined as branching to the left, or the definition could allow b to appear in the terminal string at any point between these extremities. Where portmanteau realisation exists, of the two higher-level units so realised, one is usually subsidiary to the other: thus M /went/ for instance seems to be one of the realisations of L /g°/ rather than of L /preterite/ (assuming that went will be analysed as a portmanteau). My definition is based on the observation that, at least in English, of the two units realised by a portmanteau, the 'main' unit is usually the first, as determined by the evidence of parallel strings where portmanteau realisation does not occur, for instance liked. The definition does not prevent us recognising portmanteaus in which the 'main' realisate is the second, as there is another way of diagramming portmanteaus, valid no matter in which order the 'main' and 'subsidiary' realisates occur, involving an upward-and in the tactics. However, this latter type of diagram will have a higher simplicity count than the one used for portmanteaus in this work. Should it be true that, as a linguistic universal, the 'main' realisate of a portmanteau usually is the first, it would be a good feature of my definition that the other type of portmanteau, where it occurs, is costlier in terms of economy. (Postscript: At the time I wrote the present work, I had not realised that, as pointed out by Chomsky (1957:§5.3), the preterite affix, and indeed all the affixes in English which led me to the conclusion reported above, must be described as preceding rather than following the verbal base at a high - or in T G terms 'deep' - level. I would now wish to modify the system I define by stipulating that if b, the 'mate' in question, was assigned by a type 17 rather than type 19 node, the trace line dominating b should branch to the left rather than to the right. In connection with this revision it will turn out that the diamond with four connections is needed, for instance for the present participle in English morphotactics.)

FORMAL DEFINITION OF THE THEORY

29

rules not explicitly assigned a preference value have the value zero.8 2.8 We are now in a position to give a formal definition of the statement that an expression string is a REALISATION of a content string. We say: a string P is a REALISATION of a string Q in a grammar G if (i) there exist a derivation D in G, such that in the realisational sequence R 0 , R b ..., R„ of D, R 0 = Q and Rn = P(D is a DERIVATION FROM Q TO P ) ;

(ii) there does not exist a derivation D' in G with the realisational sequence R0', Ri, ..., Rp, such that R 0 ==Q, and such that the preference value VD' of D' is greater than the preference value VD of D. (It may happen that there exist two (or more) distinct derivations with realisational sequences R 0 , Ri, ..., R n , and R0', Ri, ..., Rp, such that R 0 = R0' and whose preference values are equal, and higher than that of any other derivation with a realisational sequence starting with that particular realisational string. In this case, both R n and Rp are realisations of R 0 = R0'. This corresponds to the situation in natural language called FREE VARIATION.)

2.9 This concludes the definition of a stratificational grammar as an encoding device. To turn it into a decoding device it is necessary, for every REALISATIONAL rewrite rule, to exchange the left- and right-hand sides. A rule of the form 'a is deleted' becomes 'a may be inserted anywhere in the realisational string at any time (i.e. I - t o ) ' and vice versa. In §§2.3, 2.4 above, for 'content' read 'expression', and vice versa, wherever they occur; also in the first sentence of the last paragraph. At the end of that paragraph, for 'free variation' read 'ambiguity'.9 8

Note that the preference value, which is a device used to formalise the 'ordered or' relation, has nothing to do with the simplicity value. The simplicity criterion chooses between grammars of a given language; the preference value chooses between derivations for a given input to a given grammar of a language. • Note, also, that if there exist a derivation from Q to P (and hence, obviously, also a derivation from P to Q), the fact that P is not a realisation of Q does not

30

FORMAL DEFINITION OF THE THEORY

2.10 So far I have given no algorithm for determining what string or strings, if any, are the realisation(s) of any given string (where the term 'realisation' is used interchangeably for the expression output corresponding to a content input, or the content output corresponding to an expression input). Such an algorithm would not be hard to write; it would involve, for any given string, applying the realisational rewrite rules to produce all the realisational sequences containing realisational strings equivalent to different sets of tactics-terminal strings; and, for each such set of tactics-terminal strings, applying the tactic rewrite rules, with arrows reversed, to eliminate all sets where at least one of the tacticsterminal strings could not be reduced to a tactics-initial symbol; and finally, out of the remaining set of derivations, eliminating those with a less than maximal preference value. As the grammar itself is finite, this algorithm could always be carried out in a finite number of steps; I do not propose to give it in full detail.10 2.11 It may be objected, however, that such a procedure is not realistic as a model of how speakers and hearers manipulate the language system, because for any given utterance it would involve far too many steps. Certainly stratificational grammarians, when examining the effect of their grammar on a given input, do not necessarily imply that Q is not a realisation of P. For instance, in the present grammar, there exists a derivation from DS/2 ordl/ to DM/two th/, and vice versa. This has a preference value lower than that of the derivation from DS/2 ordl/ to DM/second/, so DM/two th/ is not a realisation of DS/2 ordl/; nevertheless, DS /2 ordl/ is a realisation of DM/two th/. This corresponds to the fact that a speaker of English will utter second to denote the ordinal of two, but will understand a child or a foreigner who says twoth. It should be a test of a good stratificational grammar that incorrect utterances which can nevertheless be readily understood by a native speaker should have derivations to the desired content-string (whereas those that are not immediately understood would have no derivations, and those which are misunderstood would have higher-valued derivations to some other content-string). 10 Strictly speaking, to ensure the finiteness of the algorithm in question it is necessary to specify that the tactics contain only 'downward' rather than 'upward' recursion, that is, any recursive line must be attached to a lower terminal of an and (rather than or) node and must not have an arrowhead at that terminal. There appears to be no practical objection to such a constraint.

FORMAL DEFINITION OF THE THEORY

31

normally produce a list of possible outputs and then eliminate them one by one, rather they use intuition to avoid following 'blind alleys' very far beyond the point where they diverge from the correct path; and it seems probable that the brain has some mechanism permitting it to operate equally efficiently. But to make this objection would be to ignore the distinction Chomsky draws between COMPETENCE and PERFORMANCE.11 The system I have defined is concerned with competence, and I did not aim in the present paper to provide a mechanism for eliminating branches from the decisiontree of choices in applying the rewrite rules, before this tree is complete (which is what is required for a performance model). Certainly, though, if such a mechanism were not forthcoming, stratificational theory would not be very interesting. In fact it seems that, given some constraints on the type of grammar described in Lamb (1966b), such as removing from the realisational portion all upward-ands and ordering of non-zero lines in ors, it should be possible to produce such a mechanism; and work by the Yale Linguistic Automation Project on computer simulation of stratificational grammar as a performance model is now under way. 2.12 It may also be objected that, admitted that TG grammars are primarily intended as generative grammars, it will nevertheless

11

The terms 'competence' and 'performance' here refer to the distinction between what a speaker does and how he does it. As Fodor & Garrett (1966: 138-9), and Moravcsik (paper at LSA Summer Meeting, Ann Arbor 1967) have pointed out, the terms 'competence* and 'performance' cover several different distinctions. With respect to the distinction between the ideally grammatical utterances of a language and the imperfect utterances actually produced by speakers, both a transformational grammar, and a stratificational grammar with the algorithm discussed in this paragraph, deal with competence; although stratificational theory does appear to offer a more promising basis for attacking the problem of performance in this sense (cf. footnote 9; and cf. Ilah Fleming's paper on aphasia, also at the 1967 LSA Summer Meeting). For more on competence and performance from a stratificational point of view, cf. Peter Reich's and Sydney Lamb's papers on the subject, to be given at the LSA Annual Meeting, Chicago, December 1967.

32

FORMAL DEFINITION OF THE THEORY

be possible to produce an algorithm by which they can be used as communicational grammars. That is, given a phonetic interpretation, it might be possible to find an algorithm which would discover the deep structure or set of deep structures which would give rise (via the appropriate surface structure) to that phonetic interpretation, and hence discover the corresponding semantic interpretation(s); and vice versa given a semantic interpretation. Of course, in a sense a very simple algorithm to do this exists; one need only systematically produce a succession of different sentences (i.e. paired phonetic and semantic interpretations) from the grammar, and wait until one reaches a pair or more than one pair in which the appropriate half corresponds to one's 'given'. But, as the set of sentences defined by the grammar (as opposed to the grammar itself) is infinite, there is no limit to the number of steps that might be necessary to achieve the desired result; and if one wants the complete set of outputs corresponding to one's given input, and if one does not know how many members that set has (in general it may have any number from zero up), the process actually becomes infinite. Obviously, what one requires is an algorithm which would accomplish these tasks in a limited number of steps.

2.13 As I mentioned above, Fodor & Garrett (1966), and Narasimhan (1967a & b), have already pointed out the possibility or even probability that such an algorithm does not exist; on the face of it, there appears to be no reason why a model designed to fulfil one particular function, many features of which have been designed specifically to allow the model to fulfil that function and no other as economically as possible (e.g. the rejection of the phonemic level already referred to), should be capable also of fulfilling a quite separate function. Recent developments in TG theory suggest a further argument. In standard TG theory, as represented by Chomsky (1965), the base component stands at a point on the dimension between semantics and phonetics which, while nearer the former than the latter, is a certain distance from either extreme. (One can give a precise measurement of its location

FORMAL DEFINITION OF THE THEORY

33

on this dimension, to the extent that comparable economy measures exist for the different components within TG theory, by comparing the complexity of the semantic component on the one hand, with the total complexity of the syntactic transformational component together with the phonological component on the other.) Recently, however, the school of TG theorists including for instance Postal, Lakoff, and Ross, appear to have been revising TG theory in such a way that the base component is moving closer to the semantic end of the dimension; and in fact it seems to me likely that in the near future the base component will be generating semantic 'interpretations' directly, i.e. deep structures and semantic interpretations will coincide. Observe what will happen to the question of an algorithm by which TG grammars can be used as communicational grammars, should such a modification of TG theory be carried out. TG grammars, as generative grammars, will do half the work of communicational grammars: the transformational and phonetic components will carry out the conversion from semantic input to phonetic output with no further apparatus. However, the problem of getting from phonetic input to semantic output will be just as great as it is now; if not in fact greater, since the part of the task which currently involves getting from deep structures to semantic interpretations will no longer be done automatically. Even if an algorithm to solve this problem were found, the result would, it seems to me, be quite unsatisfactory as a communicational model, as it would be a model in which decoding was orders of magnitude more complicated a process than encoding; whereas in reality the two processes seem to be reasonably comparable. (I would not deny that there is some degree of difference in the complexity of the two processes; certainly, provided one is limited to a known vocabulary and set of constructions in either case, it is easier to speak a foreign language than to understand it spoken to one. But in stratificational theory the difference in complexity between encoding and decoding will be the ration of the number of diversifications to neutralisations (Lamb, 1966b: 17), representing choices to be decided in the two processes respectively, contained in the grammar. This will cause decoding to be more complex, because

34

FORMAL DEFINITION OF THE THEORY

of the large number of neutralisations represented by almost every morphon, but the differential will correspond more nearly with reality.) I should stress that the modification I discuss has not yet entered the canon of TG theory. 12

12

Such a proposal has now been made explicit by J. D. McCawley "The ròle of semantics in a grammar", in Emmon Bach & R.T. Harms 1968, eds.: Universali in linguistic theory (New York: Holt, Rinehart & Winston), pp. 124-69.

3.

THE NUMERAL

SYSTEM

Van Katwijk (1965:51) has pointed out that the system of numerals in a language is interesting to the linguist, because it forms a relatively small and therefore manageable subsystem within the entire structure of the language. It is of particular interest to the stratificational grammarian, because it is not only a small and selfcontained system, but also one where it is unusually easy to isolate and describe the relationships of the linguistic forms to their meanings. 1 I offer a stratificational analysis of the English 1 I am well aware that the peculiarities of this subpart of English, its selfcontainedness and its cut-and-dried nature, which make it relatively easy to describe, also make it quite untypical for language as a whole. For my present purposes, however, I do not think this matters. Questions about which components of the meaning of common nouns, for instance, are part of our knowledge of the language and which part of our knowledge of the world, are problems no less (and no more) for stratificational theory than for any other comprehensive account of language. I wish to avoid these problems, and also the problems raised in §4.4 below, as far as possible, because my main aim in writing this monograph is to make as explicit as possible and exemplify stratificational theory as it stands, rather than to develop it further, or to discuss semantics for its own sake. (Postscript: The above note is perhaps unduly optimistic about the difficulties involved in extending stratificational theory to account for all the facts of natural language. So far the theory has not attempted to deal with structuredependent re-ordering of elements in strings: i.e. the kind of re-ordering that shifts, say, a whole noun-phrase from one place in the string to another, rather than simply exchanging adjacent elements, one of which is fixed and the other drawn from a specifiable class, which can be done within the present framework. This type of phenomenon is handled within a generative theory by syntactic transformation rules, of the type which we cannot allow in our communicational model because they are one-way. But such phenomena of course are quite basic in natural language, and will be encountered as soon as we leave certain narrowly-restricted areas such as the numeral system. I am hopeful that stratificational theory will be capable of modification to meet this demand. What appears to be necessary is a device that will allow two-way structural

36

THE NUMERAL SYSTEM

numeral system, covering cardinals, ordinals, a n d fraction d e n o m i n a t o r s (i.e. reciprocals), a t all levels a b o v e t h e p h o n o l o g y .

(Clearly,

if t h e a n a l y s i s w e r e t o b e e x t e n d e d d o w n t o t h e p h o n o l o g i c a l level, it w o u l d b e i m p o s s i b l e t o restrict it a n y l o n g e r t o t h e n u m e r a l s ; it w o u l d b e n e c e s s a r y t o write a stratificational a n a l y s i s o f E n g l i s h p h o n o l o g y a s a w h o l e , a t a s k w h i c h at t h e t i m e o f w r i t i n g - S e p t . 1 9 6 7 - r e m a i n s t o b e d o n e . ) T h e d i a l e c t I d e s c r i b e is m y

own;

a s far a s I k n o w , t h i s is q u i t e s t a n d a r d f o r Q E 2 i n all r e s p e c t s r e l e v a n t t o this article ( w h e r e I k n o w G A E t o d e v i a t e , 1 shall p o i n t this o u t ) . 3 3.1

T h e a n a l y s i s is g i v e n , in g r a p h i c f o r m , in Figs. 2 a , 2 b ,

a n d 3-5, a n d in a l g e b r a i c f o r m i n T a b l e 2. ( F o r r e a s o n s o f s p a c e , n o t all t h e structure c a n b e s h o w n o n o n e d i a g r a m ; b u t e a c h d i a m o n d is s h o w n t w i c e , o n c e in t h e r e a l i s a t i o n d i a g r a m a n d o n c e in t h e a p p r o p r i a t e tactic d i a g r a m , a n d f r o m t h e l a b e l s it will b e clear modification of traces, so that a given utterance, rather than having a single structure at each tactic level, will instead have two, one valid with respect to upward realisation and one with respect to downward. One line of attack with which I have recently experimented starts by allowing certain specified lines in a tactic pattern to cross other lines in the formation of a trace (cf. footnote 6 of §2); and then permits the set of lines which may cross in an 'upward trace' to differ from the set which may cross in a 'downward trace'. In this way pairs of traces may be defined, which appear to be related in the way required. However, this is all extremely tentative at present.) 2 The term 'British English', frequently used by American linguists, appears to me to be a very poor one: the narrowest referent normally assigned to the adjective 'British' is the island of Great Britain, and within that area there are possibly as many widely differing dialects of English as in the rest of the Englishspeaking world; I doubt if they have any common factor distinguishing them from other dialects. I prefer to adopt the term 'the Queen's English' (QE), and define it, in accordance with popular usage, to cover, not just Her Majesty's personal idiolect, but rather that dialect of English, the phonology of which is commonly labelled 'Received Pronunciation' (RP). For RP, see Jones (1963 :xvxvi). I abbreviate what is frequently described as 'General American' (see Francis, 1958:128) as GAE. • There are some specialised usages, such as e.g. nineteen hundred instead of one thousand nine hundred in dates, and five-and-twenty instead of twenty-five (but not e.g. *four-and-twenty), in telling the time in some subdialects of Q E (including my own), which are not permitted by my analysis; but it could easily be modified to account for them.

THE NUMERAL SYSTEM

eight

Figure 2a.

THE NUMERAL SYSTEM

Figure 2b.

39

THE NUMERAL SYSTEM

S-I

S-4

S-5

S-6

S-7

Figure 3.

S-S

S-9

S-2

S-3

40

THE NUMERAL SYSTEM

Figure 4.

THE NUMERAL SYSTEM

Figure 5.

41

42

THE NUMERAL SYSTEM TABLE 2

REALISATION PORTION.

HN

Sememic alternation pattern. /ten/ / us /ten/,

us

/teen/,

us

/ten-z/

Sememic knot pattern. us

/ten/ * S-ten • S-teen /teen/ us /ten-z/ * S-ten-z HN /x/ * S-x values of x: 1, 2, 3, 4, 5, 6, 7, 8, 9, HN /x/ * S-x x: sum-of, times, power-of, (, ); * S-xU for each of the following values of

us

/ » DS/ten/ / » DS/teen / / * / * DS /x/for each of the following recip, ordl; I * for each of the following values of / S-xD * DS/x/ x: hundred, thousand, million.

Lexemic alternation pattern. DS

/1/ /2/ DS /3/ DS /4/ DS /ten/ DS

/ / / / /

(UL/one/ + 0), UL /a/, UL /lz/, UL/eleven/ /2/, UL/bi/, UL/twelve/, UL /half/ UL /3/, UL/tri/ UL /4/, UL/quarter/ UL /ten/, UL/ty/ UL

Lexemic knot pattern. UL

/lz/ * L-lz / * /x/ * L-x / • DL/x/ for each of the following values of x: one, a, eleven, 2, bi, twelve, half, 3, tri, 4, quarter, ten, ty; DS /x/ * L-x I * DL/x/ for each of the following values of x: 5, 6, 7, 8, 9, hundred, thousand, million; DS /x/ * L-xU / L-xD * for each of the following values of x: teen, recip, ordl; * L-x / * DL/x/ for each of the following values of x: th, and, (>), teen. UL

Morphemic alternation pattern. DL

/one/ /2/ DL /3/ DL /5/ DL

/ / / I

UM

/one/, ""/first/ /two/, UM/twen/, UM /three/, UM/thir/ UM /five/, UM/fif/ UM

UM

/second/

THE NUMERAL SYSTEM

43

TABLE 2, cont. Morphemic knot pattern. DL/4/

•

DL

*

M-four / * DM /four/ DM M-six /6/ / * DM/six/ DL * * /seven/ M-seven / DL * M-eight / * DM/eight/ /8/ DL * M-nine / * DM/nine/ /9/ DL * /million/ M-illion / * DM/illion/ DL * /th/ M-thU / * M-thD DM DL * M-x /x/ for /x/ / * values of x: a, eleven, half, twelve, bi, tri, quarter, ten, ty, teen, hundred, thousand, and, (0; UM /x/ * M-x / * DM/x/ for each of the following values of x: one, first, two, twen, second, three, thir, five, fif; * M-x / * DM/x/ for each of the following values of x: m, th, d, / ? /

SEMOTACTICS.

•

... S-Numeral2 ... (S-recip, S-ordl) S-sum-of S-( S-Numeral S-) S-6Dig S-times (((S-2, S-3) S-power-of) ) S-sum-of S-( ( ) S-) S-sum-of S-( ( (L-(-) L-NumPhr) L-2L (L-3R, L-Digit L-hundred L-2Dig) < (L-3L, L - 3 M ) > L-Digit L-hundred, L-2Dig (L-2R + L-and) (( ), L-teenU, L-ten)

L-3L + L-3M + L-3R L-2Dig

0

L-2R L-teenD L-Digit L-PlDig L-aL L-aR L-lzL L-lzR

(L-eleven, L-twelve) 4- L-PlDig L-teen L-aR, L - l z R , L-one, L-PlDig L-2, L-3, L-4, L-5, L-6, L-7, L-8, L-9 L-a L-lz

MORPHOTACTICS.

•

M-Word

I /

M-TerminalContour M-Proclitic M-StressedWord

/ / /

M-thD

/

M-RegOrd

/

M-RegTy

M-Word < > M-TerminalContour> ( M - # M-Word) .... M-('), ... ... M-RegOrd, M - t h U , M-three, M-five, M-half, M-quarter, ... (M-first, M-second, M-thir M-d) + (M-RegOrd, M-fif) M-th M-one, M-two, M-ten, M-eleven, M-twelve, M-hundred, M-thousand, M-twen M-ty, M-RegTy, (M-thir, M-fif) (M-ty, M-teen), (M-m, M-bi, M-tri) M-illion

(M-ty, M-teen)

how they match.) In the realisation diagram, Figs. 2a-2b, terminals on the right of diamonds connect higher in the tactics, those on the left connect lower. The algebraic notation is logically subsidiary to the graphic notation, for reasons discussed by Lamb (1966a:footnote 35): basically, while a unique graph is recoverable from any set of algebraic formulae, the reverse does not hold - and simplicity is defined on graphs. The algebraic notation is useful for practical reasons.

THE NUMERAL SYSTEM

45

3.2 The highest stratum I posit for English is the sememic stratum. The content units are the units at the top of the sememic alternation pattern, called 'hypersemons' (HN). (This name is used because the units at the top of an alternation pattern are named after the stratum above, and Lamb has suggested that there exists a 'hypersememic' stratum above the sememic for English; but it appears to me that such a stratum, being composed largely of the speaker's knowledge of the world, should be considered as outside linguistic structure. I therefore regard English as having five strata; from top to bottom, the sememic, lexemic, morphemic, phonemic, and hypophonemic strata.) 3.3 The units at the bottom of Figs. 2a and 2b, the 'downward morphemes' (DM), are the lowest units at the syntactic level;4 each one will be realised as a string of 'morphons' (the stratificational term corresponding to the traditional 'morphophoneme'). In a complete description of English, there would be two further strata, each with its tactics, below the morphon level; the expression units would be the 'hypophonemes' or phonetic features. For reasons explained above, I am not in a position at the present to give the morphonic realisations of the morphemes; but all that is necessary, for us to be able to discuss the semantic and syntactic levels, is that it be agreed that the phonetic alternations exhibited by any given morpheme could be economically handled by the lower strata (that is, the realisation of the units could be determined by their phonological environment), whereas the alternations between morphemes need non-phonological information for their determination. A discussion of some of the individual morphemes in this regard will be found below (§6.4ff.). Between the hypersemons and the downward morphemes, the three rows of diamonds are the sememic, lexemic, and morphemic knot patterns respectively. 4 The term 'level' is used in two distinct senses, which context should keep apart adequately; it is sometimes used to denote any particular point on the expression-content dimension, and sometimes (as here) is a cover term for pairs of related strata (i.e. 'syntactic', or, as Lamb prefers, 'grammatical' level = lexemic and morphemic strata, 'phonological level' = phonemic and hypophonemic strata).

46

THE NUMERAL SYSTEM

3.4 It is important to keep in mind that the only labels on the graphs which have any bearing on the validity of the analysis are the labels of the units at top and bottom. Labels occurring between the top and bottom do not affect the validity of the analysis. But, as the realisation portion has a great deal of internal patterning, it is convenient to think of horizontals crossing it at various levels, and to think of every realisation line which crosses a given horizontal as a unit of a given type; and it may be that these sets of units will correspond to our intuitions about language structure. Thus, we call a line connecting upwards in the realisation from a diamond of the Xemic knot-pattern an 'upward Xeme'; one connecting downwards a 'downward Xeme'; and in the case of diamonds with connections both up and down in the realisation (i.e. the majority of diamonds) we may call the set of two lines and the diamond an 'Xeme'. Lines immediately below the Xemic sign pattern (and above the alternation pattern of the stratum below) are called 'Xons'; however, no doubt because the numeral system is such a 'basic' part of language, it so happens that there are no instances in my analysis of composite realisation above the morphemic sign pattern; i.e. all higher sign patterns are trivial, and there are no Xons distinct from Xemes for X = 'sem-' or 'lex-'. (Another respect in which the structure given is untypical is that there are no instances of neutralisation, which in general may occur either at the bottom of an alternation pattern or below a knot pattern). 3.5 By convention, the label for any unit consists of a string between obliques, preceded by an abbreviation of the name of the type of unit. (Thus, for instance, a downward lexeme will have a label of the form ' DL /x/'.) These labels should not be confused with labels not between obliques, for tactic lines. The latter labels are assigned solely in order that the tactics may be rewritten algebraically, as in Table 2. The tactic labels and the labels for units other than those at the top and bottom of the system form a subset of the set of symbols assigned in §2.3 above: by using brackets, and by stating the occurrence of upward unordered ors only im-

THE NUMERAL SYSTEM

47

plicitly, one avoids the need for a symbol for every line in the algebraic statement of the structure. Some further conventions are adopted for labelling lines. Thus, labels for tactic lines will be preceded by an appropriate prefix ('S-', 'L-', or 'M-'); terminal lines of the tactics will have the same labels, except that these prefixes replace the obliques, as the downward eme they connect to (or upward eme in the case of an upward-determined unit); the labels for non-terminal lines of the tactics will each contain at least one capital letter; the lines above an ordered upward node will have labels ending in 'L', ('M'), and 'R' respectively; the tactic lines above and below a type 17 or 19 diamond will have labels ending in 'U' and 'D' respectively.

4.

SEMOLOGY

The content structures permitted by the grammar are strings on the following alphabet of content units: the integers one to ten, the operators 'sum-of, 'times', 'power-of', 'ordinal', and 'reciprocal', and left and right brackets (in American terminology, left and right parentheses). 'Sum-of' is defined so that 'sum-of («j n2 ... " 0 ' = "1 + n2 + ... + «iJ 'power-of' so that 'a power-of V = b"; 'times' in the normal arithmetical way so that 'Wi times vi2 times ... times H/ = . n2 (in each case used, i > 1). As an example, letting 's' stand for 'sum-of', 'p' for 'power-of', '.' for 'times', a n d ' t ' for 'ten', the numeral expression written as '11,000,000,040, 001st' will be represented as a content string as H N /s(s(s(lt)). 2p(t . t . t . t . t . t)s(s(4 . t) . t . t . ts(l))) ordl /. (Where 'X' is the abbreviation for some unit-type, the string x/a1l " *la2l " ••• " x /aJ is abbreviated as ^¡ax a2 ••• a J.) 4.1 The words hundred, thousand, and million are treated as portmanteaus of HN /ten times ten/, HN /ten times ten times ten/, and so on. It may be objected that this is not necessary: that it is a fact of mathematics, but not of language, that 1000 = 1 0 x 1 0 x 1 0 . If I accepted this objection, and allowed hundred etc. to be content units, it would permit considerable simplification of the semotactics: however, I judged it wiser to put in too much structure than too little, as it is always easier to delete than to add. 4.2 The words billion and trillion are analysed hypersemonically as successive powers of million. This analysis would be impossible for GAE usage, where one billion is 109 rather than 10 12 . But for QE it will certainly not give incorrect results. However, it may

SEMOLOGY

49

seem inconsistent to analyse billion as 'second power of million', when hundred is analysed as 'ten times ten' rather than 'second power of ten' ; also, one may be unwilling to recognise the relatively subtle arithmetical concept of exponentiation as a basic unit of meaning. (In a typological description of some South American numeral systems, Salzmann (1950) finds no clearcut instances of exponentiation used as an operator; the instance he offers (82) as a possibility, Tacana tunka-tunka-tunka '1000' (tunka = '10'), is clearly a case of multiplication, not exponentiation.) I offer two reasons for my analysis. First, it is clear, from the existence of many words such as bipartite and tri-valent, that the pairs of units bi- and two, and tri- and three, will have to be in alternation at some high level of the structure, independently of the analysis of billion or trillion. Secondly, although a child will understand a word like hundred or thousand before he has a concept of exponentiation, a word like billion will not be learned until quite late, when the learner may very well understand this concept. (We may note that Salzmann discusses only the lower numbers; I challenge his claim (1950: footnote 10) that this will be sufficient to reveal the overall structure of the numeral system, as it appears to me that English provides a counter-example.) 1 4.3 In the structure as I give it, the highest possible numberword is trillion. This is because trillion is the highest numberword I myself would use without feeling that it was a nonceformation. However, it may be that one would wish to permit formation of further words such as quadrillion, quintillion, etc. as high as the speaker knows appropriate Classical numerical 1

A much better counterexample is provided by Chinese (Putônghuà), where the morpheme ling 'zero', the use of which is the most important distinguishing characteristic of the Chinese numeral system as against those of the European languages, is encountered as soon as (but not before) one reaches the threefigure level. I am not clear what factor Salzmann believes 'aboriginal' languages to share, that would distinguish them from languages such as English and Chinese with respect to their numeral systems, unless it be that aboriginal languages just do not have expressions for the higher numerals; but if this is the characteristic Salzmann has in mind, his claim is surely tautologous.

50

SEMOLOGY

prefixes; and it would not be hard to rewrite the structure to allow this. (Corstius's programme modelled on van Katwijk's (1965) grammar of Dutch numerals generates number-words as high as undeciljoen (Corstius, 1965:59). I am not clear whether this word is normal in Dutch, but its cognate undecillion certainly is not in my dialect of English.) 4.4 A pair of important and related objections may be (i) that left and right brackets are not in themselves units of meaning, and (ii) that the string is not an appropriate structure to represent the semantic content of utterances. The class of structures permitted by current stratificational theory, for either content or expression, is graph-theoretically the class of complete quasi-orders (Harary, Norman, & Cartwright, 1965:8) with the relation 'earlier than (in time) or simultaneous with'; that is, sequences of positions are permitted, each of which is filled by one or more units. For my analysis, simultaneity has not appeared useful, so I have not defined unordered ands; with the result that my definitions permit only strings of elements (obviously, the string is a special case of the complete quasi-order, where each position is filled by only one element). However, it seems questionable whether temporal succession should form part of the structure of content at all, except insofar as semotactic units such as Taber's 'text blocks' (Taber, 1966:91) will be related to each other by a relationship of succession. I would certainly agree with Lyons when he suggests (1966:227) that we should analyse the structure of content independently and find out empirically how this is related to surface structure, for which by its nature successivity is an important structural element; and I take it that the 'sememic networks' of some of Lamb's articles (e.g. 1964:105) imply a structure unrelated to time, in that the arrows represent not content units but relationships between content units, different from the relationship of concatenation (for instance the single-headed arrow represents a relationship which might be called 'modification'). So it would appear that some revision of the set of stratificational relations may be in order. For the time being, however, at least for this quite

SEMOLOGY

51

limited part of the total structure of English, it seems possible to get by using concatenation, making the mental reservation that this may not be interpreted temporally at the sememic level. 2 4.5 Even if non-temporal relations are introduced into the theory, it will still be necessary to include brackets among the content units. In the present grammar they are used to delimit the domains of the operators 'sum-of' and 'power-of. But if we introduce a relationship of 'modification', as above, it will be necessary to be able to state in the content-structure of, for instance, the little old man's hat, that the 'possessor' (or whatever term is used) of hat is not man but the little old man. However, rather than, as I have done in the present analysis, arbitrarily adding brackets to the structure wherever they happen to be necessary to avoid ambiguity (or, as in the example above, to account for ambiguity where it exists: clearly, providing it be possible to qualify a hat as 'little', the utterance could refer to a hat which is little, old, and a man's, so there should be two different content structures of which the phrase could be a realisation), I propose that we add to stratificational theory by saying that, in the topmost tactics of the 1 Since the bulk of this work was completed, I have become acquainted with two works having considerable bearing on the matters touched on in this paragraph: Chafe (1967), and Hockett (1967). The kind of relatively complex structuring of utterance-content which I suggest, but do not adopt, in this paragraph is admirably expressed by Chafe in terms of 'semological axes' (64ff.); in fact, the position adopted by his paper as a whole is very close to mine, except that he gives priority to encoding at the expense of decoding (cf. my §2.13 above). Hockett (§6.7) provides the first substantial discussion known to me of what structures will be necessary to represent content. (Note that Hockett's and Chafe's proposals are not identical: Hockett does not use 'axes', so that where Chafe represents, for instance, the relationship between verb and object by describing verb and object as an ordered pair linked by the axis 'complementation' (64-65), Hockett describes it as an ordered triple of verb, goal, object. I am not sure at present whether this is more than a notational difference.) One question I would raise with respect to Hockett's structures is whether bracketing is not necessary; cf. my next paragraph. Hockett also (§6.8) provides the first discussion of realisation between these relatively complex structures, and strings; this is obviously the major problem facing stratificational theory at present. Note that Hockett coins the term 'stepmatrix' (§3) for the structure which I describe as a 'complete quasi-order'.

52

SEMOLOGY

structure of a language, for every downward-and (which at present covers the relationships of concatenation and simultaneity, but may possibly be expanded to include other relationships) there will be added a left bracket on the left and a right bracket on the right, as upward-determined elements, and these will not add to the simplicity measure of the structure. (As the meaning of 'left' and 'right' depends at the moment on the linear concatenation relation, it might be better to think of the bracketing as a circle drawn round the elements below the and.) 4.6 This would have several advantages. For one thing, in the analysis of the numerals probably more bracketing should be introduced than I have actually given; disambiguation of the content structures at present involves giving the operators the order of precedence (exponentiation before multiplication before addition) which they are conventionally given in arithmetic, but there are perhaps few grounds for regarding this mathematical convention as inherent in language. So adoption of my proposal would greatly simplify this and no doubt any other semotactics. Secondly, if we establish a parallel principle for the lowest tactics (that its downward-ands have downward-determined bracketing), we may find that it solves some phonological problems. For instance, difficulties occur in connection with the definition of unordered ands. At the moment, 'a / b . c' (see e.g. Lamb 1966b) is defined as 'a is realised as b simultaneous with c'; if c happens to be realised as d~e, the overall result will be that a is realised as b and d simultaneously, followed by e. It would be more desirable from the point of view of natural language, however, for a to be co-terminous with the whole string d^e. For instance, Lamb has pointed out (1966a: 547) that voicing is a characteristic of consonant clusters, rather than of individual consonants (this applies to English as well as to Russian). If the hypophonotactics of English permits an optional devoicing element in unordered-and relationship with the line dominating consonant clusters, when the latter happens to be realised as a sequence of consonants rather than a single consonant, the devoicing element will be simultaneous with the first. We could

SEMOLOGY

53

interpret it as an instruction 'switch off voice', but then we will need an instruction for switching on voice after the cluster. This could be a downward-determined element associated with any vowel, but then it will be redundant in the many cases where it does not follow an unvoiced segment. On the other hand, under my proposal the consonant cluster would be a bracketed string; we could interpret the unordered and as the relationship of coterminousness, and no further specification would be needed. 4.7 This proposal involves introducing into the content and expression structures exactly that part of the information from the appropriate traces that Katz & Fodor (1963:197) and Halle & Chomsky (1968) use from the deep and surface structures in their semantic and phonetic interpretation rules respectively, namely the immediate constituency. (To be precise, both the cited works use labelled bracketings, i.e. they allow the possibility of using information about nodes, as well as immediate constituency; but in each case they actually make far less use of the labels than they do of the bracketing.) 4.8 The semotactics is designed to accept all strings of sememes which are realised as non-deviant utterances. (The set of upward sememes differs from that of hypersemons only in that HN /ten/ has three possible realisations, one of which is upward-determined.) There is one respect in which the semotactics cannot be said to accept only such strings. English numeral expressions, like numerals written in digital form, are composed of multiples of powers of ten, where the earliest power of ten is arbitrarily high, and each later power is lower than the preceding one. This is a fully general fact; however, there is no way of representing such a generalisation in a stratificational grammar. 4.9 There are independent reasons for having the semotactics generate the powers of ten from 100,000 down, and from 100,000 million down to million, etc., in the correct order (for instance, we wish to permit DS /hundred thousand/ but forbid e.g. * DS /thousand

54

SEMOLOGY

hundred/ as a realisation of HN/t. t . t . t . t/); but these do not force us to get the powers of million in the correct order. We are therefore faced with three options: (i) we could change or add to the definition of the relations, that is, we could alter the general linguistic theory. However, the changes that would be needed do not appear to have applications elsewhere, and it would be unreasonable to add to the general theory of language just to cope with this very small subpart of English. (ii) we could define a cut-off point; we could say that the system will permit no numeral expression higher than 10 24 -1, and we could write the semotactics to get the powers of ten in the correct order from that point down. But this seems intolerably ad hoc. (iii) we may do what I have chosen to do: namely we may say that this marks the border between linguistic and non-linguistic organisation of material. It has often been pointed out, not least cogently by Bolinger (1965:563-64) and by Ziff (1965), that utterances can be deviant in many ways, and that it is inconceivable that even a grammar with a semantic component could distinguish all deviant utterances from non-deviant ones. What I am claiming is that the deviance of (1) is a linguistic fact, as is that of (2), but that the deviance of (3) is a non-linguistic fact, as is that of (4). (1) (2) (3) (4)

*'three hundred, two thousand' *'the quickly bachelor' *'three million, two billion' *'the stainless-steel bachelor'

4.10 This analysis may seem the more reasonable when we remember that in many languages the order of the powers of ten does differ from the logical order at a low level; for instance in German units are given before multiples of ten; but I know of no language, and it seems intuitively unlikely that there would be a language, which would do something like giving numerals as trillions followed by billions but then numerals below one million before the millions.

5. T H E L E X E M I C STRATUM

A string of downward sememes has as its realisation a string of upward lexemes. The upward sememes us /power-of, times, sum-of, (, ), ten-z/ are determined: they have no lexemic realisation, and their occurrence is in all cases fully determined by the semotactics. Several downward sememes have alternate realisations: for instance, Ds /ten/ is realised as UL/ten/ or UL/ty/ etc.; and DS/one/ will be realised as zero when used as a multiplier of ten. In all cases choices will be determined by the lexotactics, which accepts some strings of lexemes but not others. 5.1 In the lexotactics, the lexemes 'ordinal' and 'reciprocal' must still be kept apart, although most instances of either will eventually be realised phonetically as [0], because the reciprocal of two, half, differs from the ordinal, second (note however that e.g. twenty-second is both ordinal and reciprocal); and the reciprocal of four may optionally be quarter rather than fourth.1 Both these sets of numerals are distinguished from the cardinals with respect to the treatment of DS/1/ initial in the numeral expression. If a cardinal starts with DS/1/, it may optionally be realised as the indefinite article; one may say (5) or (6), but not (7) or (8). In the case of an ordinal or fraction denominator, the indefinite article 1 In connection with this point, it may be that I have not imposed enough structure on the system. It appears for my own idiolect that I would read 'J' as a quarter and as three quarters, but e.g. as seven fourths. Moreover it seems that three quarters has a single primary stress (on quart-), i.e. it is a single morphemic word, whereas seven quarters, when this is used (e.g. where quarter has the specialised meaning 'segment of an orange'), and seven fourths, certainly have two each.

56

THE LEXEMIC STRATUM

cannot occur, but DS /1/ may be realised as zero; thus one may say (9) or (10), but not (11). (5) (6) (7) (8) (9) (10) (11)

'one thousand one hundred' 'a thousand one hundred' *'one thousand a hundred' *'a thousand a hundred' 'the one hundredth day' 'the hundredth day' »'the a hundredth day'

These variations are handled by lexotactic lines L-aL and L-lzL, which may optionally occur at the beginning of a cardinal or an ordinal/reciprocal respectively. If they do occur, they will force L-Digit to be rewritten as L-aR or L-lzR respectively, and thus produce L / a / o r the upward-determined unit U L /lz/. Naturally these lines can only be chosen if there is a HN /1/ to be realised as UL /a/, VL/lz/, or UL /1/, and need not be chosen then. 5.2 The most interesting feature of the lexotactics is its handling of the downward-determined lexemes D L /and/ and what I call 'comma-intonation', DL /(')/- Comma-intonation is the fallingrising intonation pattern that occurs on the word thousand in (12), for example. It does not completely correlate with the occurrence of a comma in the written form of the numeral; for instance, (13) (i.e. 3,005,000,600) contains three commas in writing but no comma-intonation. Comma-intonation is the intonation pattern Jones (1956) calls 'normal fall-rise' (§463); for the intonation of thousand in (12), cf. that of paper in §465, second example, or that of finished in §469, second example. I distinguish the comma-intonation from the written comma by writing the former as '(>)'. (12) (13)

'one thousand (>) two hundred and three' 'three thousand and five million six hundred'

5.3 In the occurrence of and in numerals we find another difference between GAE and QE. In GAE, the occurrence of and in for instance two hundred and thirty four appears to be optional; for

THE LEXEMIC STRATUM

57

QE and is obligatory wherever it occurs in numerals. On investigation it turns out that the occurrence of comma-intonation and and in numerals is regular, and the two phenomena are linked. Their occurrence is determined in this way: wherever in a numerals expression there is the possibility of a numeral between one and ninety-nine, if one of these occurs it will be preceded by and, unless it is initial in the whole numeral expression or preceded by comma-intonation (it is necessary to specify the possibility of a number up to 99, because a single digit as a multiplier of hundred is never preceded by and); comma-intonation occurs after thousand or a power of million (i.e. million, billion, or trillion) except in two situations: (i) in final position in the numeral expression (ignoring the lexemes 'ordinal' and 'reciprocal'), or (ii) before either a multiple of hundred or a number below one hundred (but not both together), either in final position or followed by a power of million. For instance, there is no comma-intonation with the thousand of (14), because it precedes a multiple of hundred in final position; similarly in (15) thousand precedes a number below 100 in final position, and the and occurs because five is in a position where any numeral up to 99 could occur. On the other hand, in (16) comma-intonation does occur with thousand because it precedes not a multiple of hundred nor a numeral below 100 alone, but both together. (14) (15) (16)

'two thousand five hundred' 'two thousand and five' 'two thousand (>) five hundred and five'

5.4 With respect to the occurrence of and, the beginning of the numeral expression and the occurrence of comma-intonation are equivalent. So I allow the lexotactics to generate by means of a recursive loop any number of strings which I label 'numeral phrase', L-NumPhr, separated by comma-intonation. (My 'numeral phrase' is thus the equivalent, restricted to the domain of numerals, of Jones's 'sense-group' (1956:§469); it appears however to be a syntactic unit with no semantic relevance.) As the semotactics specifies the ordering of powers of ten, it is unnecessary and would

58

THE LEXEMIC STRATUM

be uneconomical for the lexotactics to do so also. Then line L-2L specifies that and cannot occur as the first element in a numeral phrase, but will otherwise always occur before the numerals below 100, when the choice is among any of them. This neatly accounts for the occurrence of and, and has the further merit of making comma-intonation a lexotactic boundary-marker, which it clearly is elsewhere.2 5.5 The numerals below 1000 are divided into two sets with respect to the occurrence of comma-intonation. Any of these numerals may occur initially in the numeral-phrase, but the round hundreds (e.g. two hundred) and the numerals below 100 (e.g. thirty-four) will intervene between thousand or a power of million and the end of the numeral phrase (as in (13) above, where the whole numeral expression is one numeral phrase). On the other hand, strings containing both a multiple of hundred and a numeral below 100 (e.g. two hundred and thirty-four), will occur only at the beginning of the numeral phrase. So the line for numerals below 1000 forks at a downward-or; the left-hand branch (L-3R) dominating the 34 and 200 types, the right-hand branch dominating the 234 types. The ordered upward-or below L-3L and L-3R ensures that when a 34 or 200 type string occurs, if there has immediately previously been a thousand or power of million for it to follow, it will follow by selecting L-3L; thus we get not (17) but (18). Only if no such word has occurred will L-3R be selected. (Line L-3M is a dummy line, whose purpose is to make the preference value of L-3L greater than that of L-2L. It might instead be desirable to establish a convention whereby ordered ors could be directly weighted.) (17) (18)

*'two thousand (>) one' 'two thousand and one'

1 Comma-intonation occurs in the environments I give when numerals are read slowly and deliberately. Probably many are omitted in faster speech; this however does not mean that we should alter the conditions in which the lexotactics generates DL/(')/> but rather that we should allow DL/(')/ to have zero realisation in certain circumstances in the phonology.

6. T H E M O R P H E M I C STRATUM

Below the lexemic knot pattern there is again an alternation pattern accounting for alternations such as that between twen, as in twenty, and two, also for the portmanteaus first and second. It should be stressed that there is no rule by which one can know at which level to account for any specific alternation in the grammar; the only criterion is overall economy. The analysis presented in this work in fact represents the latest in a number of stages of permuting the alternations between the strata to find the most economical solution. By offering this analysis, I am making the claim, not only that it represents the data correctly (in terms of effective information), but that the simplicity count (which happens to be 248-44e) could not be reduced without distorting the data (unless of course the theory were changed). Obviously, these are both empirical claims, which may be shown to be false at any time. 6.1 The morphotactics is abstracted and slightly modified from a relatively complete morphotactics of English at present in preparation at Yale University. (The fact that the tactic patterns given in this article are not complete for the language is marked by some dotted lines in each diagram, indicating that in reality more structure intervenes.) The morphotactics generates sequences of 'words' (M-Word), separated by the downward-determined element 'word-boundary' ( D M /#/). The word-boundary will determine the occurrence of various phonetic boundary phenomena, and it also marks the domain of stress. The word consists of a stress-bearing string (M-StressedWord) optionally preceded by some stressless units, of which the only two relevant to the numeral system are and and the indefinite article, and optionally followed

60

THE MORPHEMIC STRATUM

by one of the 'terminal contours', of which the only relevant one is comma-intonation. Of course, phonetically the intonation pattern is simultaneous with a word rather than following it. But it is most simply treated as a separate segment at the syntactic level, which in the phonology is made simultaneous with the word it had followed at the higher level. 6.2 This upper part of the morphotactics is general rather than being specific to the numeral system. Below the dotted lines is that portion of the morphotactics which forms number-words. Note that the m- of million is a downward-determined morpheme: the morphonic realisation of the lexeme L /million/ will start with the vowel, and will be preceded by m- only if none of the other prefixes, which are upward as well as downward morphemes, occur. This analysis has been chosen because, unlike the cases of bi- and tri-, there is no reason, apart from the word million, to regard m- as a realisation of 'one'; so it is more economical not to do so. It is not a case of 'concealed rewriting rules' that the lexeme L /million/ corresponds to a morpheme M /illion/, as there are only labels; the downward sememe could have been labelled DS /illion/, except that this would probably be confusing. 6.3 The only interesting problem in the morphotactics is whether two-digit numerals, e.g. twenty-four, should be analysed as one word or two. The option chosen was to analyse them as two words, as it seems that both parts of the expression have primary stress (although it may be objected that the stress on the first part is a kind of permanent contrastive stress, distinguishing twenty-four from thirty-four and so on); this analysis is of course more economical, as the statement that such pairs may occur in sequence is also made in the lexotactics. 6.4 Something should be said at this point about the morphonic realisation of the downward morphemes. In many cases these will be straightforward: for instance, DM /twen/ will undoubtedly be realised as a sequence of morphons M N /twen/, and these morphons

THE MORPHEMIC STRATUM

61

will be realised regularly as p / t w e n / respectively (although in other environments some of these morphons may have alternative realisations, for instance the phonemic stratum will probably be set up in such a way that MN /n/ is realised as p /q/ before velars, e.g. in concord). In some cases, however, 'special' morphons of more limited distribution will be posited to account for more unusual alternations. Thus DM /a/ will undoubtedly be realised as M N /a N/, with some M N /N/ that is realised as p/n/ before vowels, zero elsewhere ; also the initial morphon of M /ty/ and M /teen/ will no doubt be a special M N /T/ that is realised as zero after MN /t/, r/t/ elsewhere. (In QE, both fourteen and eighteen have a short [t] phonetically, i.e. a single stop phonemically, as opposed to some American dialects which have a geminated stop in the latter, and others which have a geminated stop in both words.) 6.5 On the other hand, elisions such as in fifth [fi0], twelfth [twel0], and sixth [sik9], and the devoicing of the MN /d/ in hundredth ['hAndratG], can no doubt be handled automatically, being determined by that part of the hypophonotactics which handles consonant clusters ending in [0] within a single syllable; that is, there will be no need for morphons of restricted distribution in these cases. 6.6 This does not exhaust the list of phonological alternations exhibited by the downward morphemes of this grammar; but it should be enough to suggest that this particular inventory of downward morphemes is a sound one. 6.7 Finally, it will be appropriate to illustrate the working of the grammar by referring to a sample numeral expression. For my example I shall take the numeral '11, 000, 000, 040, 001st', the content string corresponding to which is given in §4.0 above. The correct expression string corresponding to this written numeral is (19). It will be seen that there exists a derivation in the sense of §2.7 between the pair of strings: two intermediate stages in the realisation sequence will be (20) and (21), at the downward sememic

62

THE MORPHEMIC STRATUM

and downward lexemic levels respectively. The preference value of this derivation is six: one from the semotactic trace, four from the lexotactic, and one from the morphotactic. There is no alternative derivation from the expression string; however there are some alternative derivations from the content string. Nevertheless', the expression strings of these alternative derivations are not in free variation with (19), because these derivations all have lower preference values. For instance, there is a derivation terminating in (22); this has a preference value of five (the lexotactic trace of the derivation includes an L-2R instead of L-and, but also has to include an L-3R instead of L-3L). Similarly there is a derivation with the expression string (23), but the preference value of this derivation is also five, because of a difference in the morphotactic trace. Note that, in accordance with footnote 9 of §2, although (23) is not a realisation of the content string of §4.0, that content string is a realisation of (23); and indeed the native speaker of English would have little difficulty in understanding the incorrect utterance 'eleven billion, forty thousand and oneth'. (19) (20) (21) (22) (23)

DM

/eleven # bi illion (>) # four ty # thousand and first/ DS /1 teen 2 million 4 ten thousand 1 ordl/ DL /eleven bi million (>) 4 ty thousand and one th/ DM /eleven # bi illion (») # four ty # thousand # first/ DM /eleven # bi illion (>) # four ty # thousand # and one th/

6.8 As an example of free variation, the content string HN /s(s(s(l. t . t)))/ has derivations both to DM /a hundred/ and to DM / one # hundred/, in both cases with a preference value of zero, which is maximal for this content string. In the present grammar no cases of ambiguity are possible; however, if the grammar were generalised to allow complete fractions (with numerator and denominator), an expression string such as DM /twen ty # five # thousand th s/ would be required to have equally and maximally valued derivations to content strings representing both '20/5000' and '25/1000'.

THE MORPHEMIC STRATUM

63

6.9 It is to be hoped that the foregoing will suffice to make clear what are the assumptions of stratificational linguists, to establish that the theory merits serious consideration as a formal account of language, and to stimulate further discussion.

BIBLIOGRAPHY

Bolinger, Dwight (1965), "The atomisation of meaning", Lg. 41:555-73. Chafe, W. L. (1967), "Language as symbolisation", Lg. 43:57-91. Chomsky, Noam (1957), Syntactic structures (= Janua linguarum, no. 4) ('s-Gravenhage: Mouton). Chomsky, Noam (1963), "Formal properties of grammars" ( = chapter 12 of Luce, Bush, & Galanter (1963)). Chomsky, Noam (1964), Current issues in linguistic theory (= Janua linguarum, no. 38) ('s-Gravenhage: Mouton). Chomsky, Noam (1965), Aspects of the theory of syntax (Cambridge, Ma.: the MIT Press). Chomsky, Noam (1967), "Some general properties of phonological rules", Lg. 43:102-28. Chomsky, N. & Halle, M. (1965), "Some controversial questions in phonological theory", JL 1.97-138. Chomsky, N. & Miller, G. A. (1963), "Introduction to the formal analysis of natural languages", ( = chapter 11 of Luce, Bush, & Galanter (1963)). Corstius, H. B. (1965), "Automatic translation of numbers into Dutch", Foundations of Language 1:59-62. Fodor, J. & Garrett, M. (1966), "Some reflections on competence and performance" in J. Lyons & R. J. Wales, eds.: Psycholinguistic papers, 13379 (Edinburgh: University Press). Francis, W. N. (1958), The structure of American English (New York: Ronald Press). Halle, Morris (1965), The sound pattern of Russian ('s-Gravenhage: Mouton). Halle, M. & Chomsky, N. (1968), The sound pattern of English (New York: Harper & Row). (N.B.: most of my knowledge of the contents of this work is derived from Chomsky's course on English phonology at the Linguistic Institute, UCLA 1966.) Harary, F., Norman, R. Z., & Cartwright, D. (1965), Structural models: an introduction to the theory of directed graphs (New York: Wiley). Hockett, C. F. (1947), "Problems of morphemic analysis", Lg. 23:321-43. Hockett, C. F. (1967), Language, mathematics, and linguistics (= Janua linguarum, no. 60) ('s-Gravenhage: Mouton). Jones, Daniel (1956), The pronunciation of English, 4th ed. (Cambridge: CUP). Jones, Daniel (1963), English pronouncing dictionary, 12th ed. (London: Dent). van Katwijk, A. (1965), "A grammar of Dutch number names", Foundations of Language 1:51-8.

BIBLIOGRAPHY

65

Katz, J. J. & Fodor, J. A. (1963), "The structure of a semantic theory", Lg. 39:170-210. Katz, J. J. & Postal, P. M. (1964), An integrated theory of linguistic descriptions (= Research monograph no. 26) (Cambridge, Ma.: the MIT Press). Lamb, S. M. (1964), On alternation, transformation, realisation, and stratification (= Monograph series on languages and linguistics, no. 17). Report of the 15th Annual RTM on Linguistic and Language Studies, 105-22 (Washington: Georgetown University Press). Lamb, S. M. (1966a), "Prolegomena to a theory of phonology", Lg. 42:53673. Lamb, S. M. (1966b), Outline of stratificational grammar (Washington: Georgetown University Press). Luce, R. D., Bush, R. R., & Galanter, E. (1963), Handbook of mathematical psychology, volume 2. (New York & London : Wiley). Lyons, John (1966), "Towards a 'notional' theory of the 'parts of speech'", JL 2.209-36. Narasimhan, R. (1967a), On the non-relevance of transformational linguistic theory to psycholinguistics. (= Technical report no. 22). Computer group research and development; Tata Institute of Fundamental Research, Colaba, Bombay-5 (Mimeographed). Narasimhan, R. (1967b), An appendix to technical report no. 22. (= Technical report no. 22-a). Computer group research and development ; Tata Institute of Fundamental Research, Colaba, Bombay-5 (Mimeographed). Ore, 0ystein (1963), Graphs and their uses (New York: Random House). Postal, Paul (1964), Constituent structure: a study of contemporary models of syntactic description. (= UAL, 30.1 part 3) (Bloomington, In.: Indiana University). Salzmann, Zdenëk (1950), "A method for analysing numerical systems, Word 6:78-83. Taber, C. R. (1966), The structure of Sango narrative. (= Hartford studies in linguistics, no. 17) (Hartford, Ct: Hartford Seminary Foundation). Ziff, Paul (1965), "About what an adequate grammar couldn't do", Foundations of Language 1:5-13.

INDEX

A page-reference in italics refers to the point where a technical term is defined. acoustic phonetics, 14 addition, 52 alternation, 59 alternation pattern, 20, 45, 46, 59 ambiguity, 13, 29, 62 anataxis, lOn, 27n 'and', 27n, 56-7, 59 and node, 30n aphasia, 31n binary feature, 14n Bolinger, D., 54 Cartwright, D., 50 Chafe, W. L., 51n Chinese, 49n Chomsky, A. N., 7-16, 21, 28n, 31-2, 53 comma-intonation, 56-8, 60 comment, 14n communicational description, 9, 10, 12-3 communicational grammar, 32, 33 competence, 31 complementation, 51n complete quasi-order, 50, 5In composite realisation, 15, 46 consonant, 52 consonant cluster, 52-3, 61 content string, 26, 29, 61-2 content structure, 53 content unit, 22, 23, 26, 45, 48 context-free PSG, l l n Corstius, H. B., 50 Czech, 19

decoding, 8, 22, 29, 33, 51n deletion, l l n derivation, 28, 29, 30n, 61-2 determined element, 55 deviance, 54 discontinuous IC, lOn disjunctive ordering, 16 diversification, 15, 33 downward-and, 19, 23n, 27, 52 downward-determined element, 18, 52-3, 56, 60 downward lexeme, 46 downward lexemic level, 62 downward morpheme, 45, 60-1 downward-or, 23n, 58 downward sememe, 55, 60 downward sememic level, 61-2 Dutch, 50 effective information, 59 encoding, 8, 22, 29, 33, 51n e, 23, 24 equivalence, 27 essential connection, 22, 23 exponentiation, 49, 52 expression string, 26, 29, 61-2 expression structure, 53 expression unit, 22, 23, 26 external connection, 24 extrinsic ordering, 26 Fleming, I., 27n, 31n Fodor, J. A., 8, 13, 31n, 32, 53 Fox, 19 Francis, W. N., 36n

INDEX

free variation, 13, 29, 62 Garrett, M., 8, 13, 31n, 32 General American English, 36, 48, 56 generative description, 9, 10, 13 German, 54 grammar, 22 grammatical level, 45n graph theory, 27, 50 Halle, M„ 13-5, 53 Harary, F., 50 higher-than, 22, 23 Hjelmslev, L., 23n Hockett, C. F., lOn, 14n, 19, 51n hypersememic stratum, 45 hypersemon, 45, 48, 53 hypophoneme, 45 hypophonemic stratum, 45, 45n hypophonotactics, 52, 61 identity element, 18 immediate constituency, 53 initial symbol, 26 interstratal anataxis, 27n intrinsic ordering, 26 Jakobson, R., 14 Jones, D., 36n, 56-7 van Katwijk, A., 35, 50 Katz, J. J., 12, 53 knot pattern, 18, 20, 46 label, 46 Lakoff, G., 33 Lamb, S. M„ lOn-lln, 13-4, 16, 20-2, 23n, 24, 27n, 31, 31n, 33, 44, 45n, 50, 52 level, 45n lexeme, 19, 60 lexemic knot pattern, 45, 59 lexemic stratum, 12n, 45, 45n, 55-8 lexotactics, 27n, 55, 60 lexotactic trace, 62 Lyons, J., 50 mate, 26, 28

67

McCawley, J. D., l l n , 34n Miller, G. A., 10, 12 modification, 50-1 Moravcsik, J. M. E., 31n morpheme, 9n, 45 morphemic knot pattern, 45 morphemic sign pattern, 46 morphemic stratum, 45, 45n, 59-62 morphon, 33, 45, 60-1 morphophoneme, 15, 45 morphotactics, 28n, 59-60 morphotactic trace, 62 multiplication, 49, 52 mutation rule, 13 Narasimhan, R., 8, 32 neutralisation, 15, 33, 46 Norman, R. Z., 50 numeral phrase, 57-8 optional element, 19, 52, 56 ordered or, 20, 29n, 58 ordered upward-and, 27n ordered upward-or, 58 'ordinal', 48 Ore, 0., 27 or-node, 30n, 31 performance, 31 phoneme, 15 phonemic level, 32 phonemic stratum, 45, 45n, 61 phonological alternation, 61 phonological level, 45n phonology, 9n, 58n, 60 phrase-marker, l l n phrase-structure grammar, 1 On-lln, 27 planar graph, 27 portmanteau realisation, 15, 19, 20, 28n, 48, 59 Postal, P. M., lOn-lln, 12, 21, 33 'power-of', 48, 51 preference value, 26, 28-30, 62 Queen's English, 36, 36n, 48, 56-7, 61 realisation, 29,30

68

INDEX

realisation portion, 14,18,20,23n, 24, 46 realisational sequence, 26, 28-30, 61 realisational string, 26, 27 realisational symbol, 24 Received Pronunciation, 36n 'reciprocal', 48 recursion, 23n, 30n reduplication element, 22 Reich, P. A., 24, 31n Ross, J. R., 33 Russian, 15, 52 Salzmann, Z., 49 semantic level, 45, 45n sememe, 53 sememic alternation pattern, 45 sememic knot pattern, 45 sememic level, 51 sememic network, 50 sememic stratum, 12n, 45, 48-54 semological axis, 51n semon, 19 semotactics, 48, 52-5, 57 semotactic trace, 62 sentence, 12n sign pattern, 46 simplicity, 21, 44 simplicity count, see simplicity value simplicity criterion, 13, 15-6, 20, 52 simplicity value, 23, 28n, 29n, 59 stepmatrix, 51 n stress, 59-60 string, 10, 48, 50, 51n strong generative capacity, 9n 'sum-of', 48, 51 syntactic level, 45, 45n Taber, C. R., 12n, 50 Tacana, 49

tactic pattern, lln, 13, 15, 18-20, 23n, 24, 26, 27, 44, 46 tactic symbol, 24-5 tactics, see tactic pattern tactics-initial node, 20, 23 tactics-initial symbol, 26, 28, 30 tactics-terminal string, 27, 30 tactics-terminal symbol, 28 terminal contour, 60 terminal symbol, 26 text block, 50 'times', 48 topic, 14n trace, 27, 28, 36n, 53 transformational-generative grammar, 7-8, 10-5, 28n, 31-4 tree, 27 universal, 7, 14 unordered and, 22, 27n, 50, 52-3 unordered or, 19 unordered upward-or, 46 upward lexeme, 55 upward morpheme, 60 upward sememe, 53, 55 upward-and, l l n , 18, 20, 26-7, 28n, 31 upward-determined element, 19, 52-3, 56 voice, 52-3 vowel, 53 weak generative capacity, 9n word, 59-60 word-boundary, 59 zero element, 19, 20, 23, 31, 55 zero realisation, 58n Ziff, P., 54