1. INTRODUCTION

Natural languages are cultural instruments for the symbolic association of sounds and meanings for purposes of effective communication by the members of human societies. Like all other instrumental objects, therefore, it is the functions or purposive goals of languages that determine their essential properties and their range of possible variation and change. The science of linguistics, which takes natural language as its subject matter,¹ must consequently provide theories, metatheories, and metalanguages which are capable of expressing lawlike descriptive and explanatory generalizations not only about the forms and formal structures of the objects in its domain, but also about their functions and functional structures, and the systematic relationships that hold between these two types of structure. Like biology, economics, anthropology, and all other sciences that deal with the characteristics of physical or cultural instruments, linguistics can adequately account for its subject matter only by determining the precise nature of these operant relationships between form and function. One of the most important tasks for a theory of language, therefore, is the principled differentiation of those characteristics of languages that are naturally determined by their natural communicative function from those characteristics that are underdetermined by this function and thus free to vary arbitrarily within functionally established limits from language to language.

The hypothesis of Equational Grammar (Sanders 1971, 1972) constitutes a partial differentiation of this sort. This hypothesis asserts that all the variable, or functionally underdetermined, characteristics of natural languages are due exclusively to differences in the particular sets of naturally arbitrary symbolic equivalence relations that are linguistically institutionalized in one language community or another. Languages can differ, for example, as to whether ‘young male human’ and boy are symbolically equivalent or not, or whether word-final voiced and voiceless obstruents are equivalent or not. Languages cannot differ, according to this hypothesis, in any ways other than this—and in particular not in any ways that involve differences in the inferential or derivational functions and functional interactions of the symbolic equivalences of different languages. The equationality hypothesis thus claims that all non-universal rules of grammar are simple symmetric principles of equivalence or non-equivalence, and that all constraints on the proper use of such principles for purposes of inference, association of particular meanings with their expressions, proving theorems about words and sentences, or whatever must be fully determined by strictly universal principles of natural language function.

Evidence and arguments in support of this claim have been presented in various recent publications, including not only my own studies on equational grammar itself, but also my paper “On the notions ‘optional’ and ‘obligatory’ in linguistics” (1974a), and the large body of works on Universally Determined Rule Application, which deal particularly with the prediction of appropriate rule-ordering interactions in phonological derivations.² One purpose of the present paper will be to provide additional phonological support for the equationality hypothesis on the basis of evidence concerning the functions of phonological processes and the phonological equivalence rules that underlie them. For the most part, however, I will consider the equationality hypothesis itself here to be a sufficiently well-confirmed principle of natural-language grammar, and will be concerned solely with determining the precise nature of the relationships that hold between the forms of equational phonological rules and their directed uses in derivations. My primary goal will be the discovery and expression of the underlying principles of grammatical function that determine these particular derivational uses of grammatical rules from the structures and functions of the whole grammars that include them.

The hypothesis of equational grammar can be appropriately viewed both as a theoretical claim or empirical hypothesis about natural languages, and as a general metatheory, model, or metalanguage for all descriptions and analyses of such languages. We will actually maintain that the equationality principle is more highly valued than its contrary or contradictory hypotheses under both of these interpretations. The distinction, nevertheless, is an important one to bear in mind here, since failure to distinguish between theories and empirical hypotheses, on the one hand, and models or metalanguages, on the other, is one of the primary sources of counterproductive inquiry and argumentation in linguistics. Thus, for example, one cannot usefully try to compare the standard three-level phonological model of American structuralism with any of the various multiple-level models of standard generative phonology on any possible factual grounds, since these are not distinct theories generating different factual claims about natural languages, but rather different metalanguages for the description of such languages—and thus subject to evaluation only on the basis of such practical and essentially non-empirical criteria as clarity, economy, perspicuity, and appropriateness of expressive powers.

2. EQUATIONAL GRAMMAR AND ITS IMPLICATIONS

2.1  The basic principles and applications of Equational Grammar have been described and fairly thoroughly exemplified in my paper, “On the symmetry of grammatical constraints” (1971), and in the monograph titled Equational Grammar (1972). It was proposed and argued in these studies that the axioms and theorems of natural-language grammars are intrinsically equational in content and function. All linguistically significant facts about the words, sentences, and other linguistic objects constituting a particular language were shown to be expressible, as suggested in (1), by interpreted instances of the general theorem schema A = b, where A is a terminal semantic representation, a string of elements with distinct interpretations into non-null observation statements about the meanings of linguistic expressions, and b is a terminal phonetic representation, a string of elements with distinct interpretations into non-null observation statements about the sounds or pronunciations of linguistic expressions.

A valid proof for any theorem of the form (A = b) can be appropriately defined simply as a finite sequence of equations of the form (A = b, . . . , b = b) or (A = b,..., A = A), such that the equivalence of all equations in the sequence follows deductively from the axioms and rules of inference of some given theory. A proof that terminates in an equation between identical phonetically interpreted representations is called a phonetically-directed proof or phonetically-directed derivation. A proof terminating in an identity equation between semantically-interpreted representations is called a semantically-directed proof or semantically-directed derivation. It has been shown (Sanders 1972) that validity is decidable for any finite equation sequence and any finite theory; it has also been shown that if a theorem (A = b) has a valid proof at all, then it will have at least one valid proof that is phonetically-directed and at least one that is semantically-directed.

Moreover, for the effective proof of all true theorems of this sort, it was shown that the only language-specific linguistic principles, or rules of grammar, that are required are simple equations asserting or denying the symmetrical relations of symbolic equivalence between linguistic representations. All non-universal grammatical principles, in other words, are appropriately expressible as instances of the equational statement schemata in (2), and are appropriately interpretable into law-like linguistic generalizations in accordance with these schemata.

Equivalence statements, instances of the affirmative equation schema (2a), express ordinary affirmative rules of grammar. Non-equivalence statements, instances of the negative schema (2b), express well-formedness constraints, or prohibitive rules of grammar. Examples of the major types of equivalence and non-equivalence rules are given in (3) and (4).

(3) Equivalence Rules

(a) Redundancy rules; adjunction-deletion of constants

[NASAL] = [NASAL, VOICED]

[HUMAN] = [HUMAN, ANIMATE]

(b) Lexical and ordering rules; intermodal substitution

[YOUNG, MALE, HUMAN] = [[LAB, OBST, VCD] [VOC, BK] [VOC]]

[[ADJ], [NOUN]] = [ADJ] & [NOUN]

(c) Idempotency rules; identity adjunction-deletion

[NASAL] [CONS, X] = [NASAL, X] [CONS, X]

[X, NP, Y] = [NP, [X, NP, Y]]

(4) Non-Equivalence Rules

[SEG] [SEG] [SEG] ≠ [CONS] [CONS] [CONS]

[PRO, CLITIC] & [PRO, CLITIC] ≠ [1ST PERS] & [2ND PERS]

Language-specific equivalence rules will constitute at least part of the necessary axiomatic basis for the proof of grammatical theorems under all metatheoretical assumptions about linguistic description, both equational and non-equational alike. The need for axiomatic assertions of language-specific non-equivalence, on the other hand, which express global and prohibitive constraints only, has not been as definitively established. For present purposes, in any event, we can assume the more restricted version of equational grammar, without language-specific non-equivalence statements, and all subsequent discussions will be based on this more restricted version.

An equational grammar, then, is a finite set of symmetrical equivalence statements of the form A = B, (2a). Each such statement justifies by the principle of equal substitutability a pair of converse directed inferences, or transformations, or derivational processes, of the forms (5a) and (5b).

The basic hypothesis of Equational Grammar asserts that all empirically significant constraints on the derivational uses of such directed inferences are determined from their governing equivalence rules by some set of strictly universal principles of grammatical function.³ It asserts, in other words, that the grammars of natural languages can differ only in the particular sets of representational equivalence relations they postulate.

The metalanguage of Equational Grammar thus generates a much more restricted and more homogeneous set of possible grammars, or phonological components of grammars, than those generated by any of the various standard or non-standard metalanguages of generative phonology, including among others those particular versions proposed or exemplified in Chomsky (1967), Halle (1962), Chomsky and Halle (1968), Kiparsky (1971), Anderson (1974), and Vennemann (1971). All such non-equational metalanguages generate grammars which specify axiomatically not only a particular set of representational equivalence relations but also a particular set of inferential constraints on the directed use of these equivalences in the justification of derivational substitutions, on the optionality or obligatoriness of such directed substitutions, and on the relative order in which these substitutions can be appropriately made. These metalanguages, in other words, permit grammars to differ not only in their posited equivalence relations, or rules, but also in the extrinsic directionality, optionality, and ordering constraints that are imposed on their derivational functions. The set of distinct grammars so generated will be vastly larger and more diverse, therefore, than that generated by any otherwise comparable equational metalanguage, which can permit grammars to differ only in their representational equivalence rules, and not in any of their derivational uses or functions.⁴

2.2   The natural basis and primary implications of the equationality hypothesis can be seen quite easily. The equational nature of grammar can be shown to follow directly from the essentially equational characteristics of languages and of the linguistic objects that comprise them. Viewed extensionally, a language is just an infinite set of linguistic objects, and each such object—that is, each word, phrase, sentence, or discourse of a language—is just a conventional association, or symbolic equivalence pairing, between the meaning of that expression and its sound (or other publicly perceptible type of expression). Since sounds and meanings are totally different types of entities—sharing no physical, spatial, or temporal characteristics whatever—any use of a directed or non-symmetrical relation between the members of a sound-meaning pairing would be wholly redundant. Each linguistic expression, then, can be appropriately represented as a symmetrically related pairing, or equation, of the form (A = b), where the interpretation of A is a set of observation statements about the meaning of that expression, and the interpretation of b is a set of observation statements about its sound, or pronunciation.⁵

Grammars can be viewed as the intentions of languages, the systems or general principles that are necessary and sufficient to establish or determine the particular infinite sets of sound-meaning pairings that constitute their various culturally variable extensions. It is in their grammatical or intentional aspect, of course, that languages appear most obviously instrumental in character. Something can be considered to be a grammar, in fact, only if it has the function of finitely establishing, determining, or generating some unbounded set of sound-meaning pairings that could be efficiently used for purposes of communication by the members of some possible human society. This function is achieved, of course, for any given grammar and given language if and only if the grammar provides a sufficient non-universal basis for specifying or identifying exactly those pairings of sounds and meanings that are available for communicative use by the speakers of that language.

To specify a pairing between the interpretations of A and b in the sound-meaning pair (A, b) is simply to provide a proof of the theorem (A = b). To specify all such pairings for a given language, and thereby achieve the function of a grammar of that language, it is thus necessary merely to provide a finite basis for the proof of some particular infinite set of theorems of this type.

Every proof of a linguistic theorem has a well-defined function too—namely, the deductive derivation of a terminal representation in one interpretable alphabet—phonetic or semantic—from a terminal representation in the other interpretable alphabet. Phonetically-directed proofs or derivations thus have the inherent function of deriving fully-specified terminal phonetic representations from terminal semantic ones; and semantically-directed proofs or derivations have the converse function of deriving terminal semantic representations from terminal phonetic ones. The structure of any particular proof, then, will be partially determined or delimited in advance of any appeal to particular rules by its particular derivational function—just as the structure of any grammar will be partially determined or delimited by its essential function as a basis for determining the validity and truth of theorems about sound-meaning associations.

It is, therefore, not a matter of chance or arbitrary convention, for example, that the successive lines of a non-redundant proof of any grammatical theorem will each be progressively closer in their approximation to a well-formed terminal representation in one mode and progressively less like a terminal representation in the other mode. Thus the most general metaprinciple of grammatical function, which I have called Maximalization of Terminal Specificity (Sanders 1971:234-35), the basis for all other universal constraints on rule application, can be seen to follow directly from the natural function of grammars as instruments for the proof of sound-meaning equivalences. This principle of Terminal Maximalization asserts in essence simply that a grammatical rule (A = B) can be appropriately used to justify the derivational inference of representations of the form XBY from those of the form XAY if and only if XBY is a closer approximation than XAY to a well-formed terminal representation in the given terminal alphabet of the given derivation. It follows, therefore, from the natural function of grammatical derivations as instruments for the specification of sound-meaning pairings that the appropriate directed use of any particular equational rule in any particular derivation should always be fully predictable from the content of that rule, the mode of terminality of the derivation, and the rule-independent characteristics of all terminal representations in that mode. It is the power of the equationality hypothesis to motivate and direct the search for such universals of phonetic and semantic well-formedness that constitutes one of its most important metatheoretical and heuristic advantages over all non-equational models for linguistic description and explanation.

The specification of terminal well-formedness is partly determined by simple vocabulary and interpretability considerations alone. Such determination is effected in a particularly natural and straightforward way for those metalanguages that are consistent with the general hypothesis of Simplex-Feature representation (Sanders 1974). The Simplex-Feature hypothesis asserts in essence that all linguistic representations and rules, whether equational or not, must consist of finite strings of unanalyzable content or relational elements like NASAL, ANIMATE, X, &, and Ø, that every constant except Ø has a single, distinct, non-null interpretation into a distinct observation statement either about meaning or about articulation, and that every terminal, or empirically interpretable, representation of a linguistic object must consist wholly of constants that are interpretable into observation statements of the same type or mode, i.e., all semantic or all phonetic. It is clear, therefore, that for any model incorporating the Simplex-Feature hypothesis, terminal semantic representations and terminal phonetic representations will be completely distinct, the former consisting of strings of semantically interpretable elements, including the semantic relational element for simple grouping or association, the latter consisting of strings of phonetically interpretable elements, including the phonetic relational element for linear ordering. It follows, then, that the directed uses of any rule that is capable of justifying the substitution of elements of one interpretable type for elements of the other will be intrinsically determined for all derivations of either directionality.

The vast majority of rules in any grammar are of precisely this sort. These are the lexical rules of the language, which specify the direct pairings between the distinctive phonological and non-phonological representations of individual morphemes, and its quasi-lexical rules of ordering, which specify direct equivalences between the groupings of individual constituent types into constructions and the relative orderings of those constituent types. It is logically impossible for a representation in one terminal alphabet to be derived from a representation in the other terminal alphabet unless all applicable lexical rules and ordering rules are used to justify substitutions of elements in the former alphabet for elements in the latter, since in no other way can elements not in the intended terminal alphabet be eliminated. The only possible way for lexical rules to be applied, in other words, is to justify the substitution of phonological representations for non-phonological ones in phonetically-directed derivations and the substitution of non-phonological representations for phonological ones in semantically-directed derivations.

The directed use of rules is intrinsically determined in this fashion not only for all ordinary context-free lexical rules like that indicated in (6a), but also for all context-sensitive rules of lexicalization, like those in (6b), and all of the various types of quasi-lexical rules of morphologically conditioned phonological alternation, like those illustrated in (6c).

(6) (a) [YOUNG, MALE, HUMAN] = [[LAB, VCD, OBST] [VOC, BACK] [VOC]]

(b)[OX] & [PLURAL] = [OX] & [en]
[HIT] & [PAST] = [HIT] & Ø

(c)guws & PLURAL = giys

liy [LAB, OBST, CNT, DIACRITIC] & [PLURAL] = liy [LAB, OBST, CNT, VCD] & [PLURAL]

bend & [PAST] = bent

[X [VOC] Y] & [PLURAL] = [X [VOC, FRONT] Y]

Thus all such rules have the inherent capacity to justify the elimination of phonetically uninterpretable elements (like PLURAL, PAST, and DIACRITIC) in favor of phonetically interpretable ones during the course of phonetically-directed derivations and the elimination of semantically uninterpretable elements (like LAB, VCD, and FRONT) in favor of semantically interpretable ones in semantically-directed derivations. Given the function of grammatical derivations in general, therefore, as expressed by the principle of Terminal Maximalization, these are the only possible directed uses of these rules.

It also follows from the Maximalization of Terminality principle that all phonetic or semantic redundancy rules, like those in (7), can be appropriately used for derivational purposes only to justify the addition of elements in the vocabulary of the final representation in a derivation and to justify the deletion of elements from the alphabet of its initial representation.

(7) [VOC, BACK] = [VOC, BACK, ROUND]

[HUMAN] = [HUMAN, ANIMATE]

For the vast majority of grammatical rules, therefore, the only facts about terminal well-formedness and relative degrees of terminality that are necessary for the complete prediction of their derivational uses are facts about which elements have phonetic interpretations and which have semantic interpretations—facts that are given by the rules of interpretation of whatever grammatical metalanguage is being employed. Nearly all of the really significant questions about rule function in grammatical derivations, however, revolve around non-elementary, or relational, characteristics of terminal representations. For example, in Sanders (1972), it was suggested that among non-phonological representations semantic terminality increases as the amount of internal grouping or bracketing of constituents decreases; and that for phonological representations, phonetic terminality increases, other things being equal, with increased element identity between adjacent segments. Other relational principles of relative terminality will be discussed in detail subsequently, with particular reference to Houlihan and Iverson’s (1977) segmental markedness constraint, and other related principles of rule function in phonology. It will been seen from these considerations that there are indeed certain highly significant relational characteristics of phonetic well-formedness which may make it possible in all cases to correctly predict the relative terminality relations of equivalent phonological representations and hence the appropriate derivational uses of all rules that express phonological equivalences.

3. PHONOLOGICAL PROCESSES AND THEIR FUNCTIONAL DETERMINATION

We will now turn to certain major representative types of phonological rules with the aim of determining the precise nature of the underlying principles of grammatical function that govern their directed uses in linguistic inference or derivation. We will thus be trying to find out exactly what relations hold between the forms and functions of phonological equivalence rules, and between the functions of particular derivational processes and the more general functions of the whole grammars that justify them.

The first subsection, which deals with apocope, will use this process as a basis for illustrating in some detail the general framework and pattern for the discovery and development of functional laws. The following subsections will then be chiefly concerned with certain other major types of phonological processes as a basis for testing and refining the particular functional laws that have been suggested. Throughout we will attempt to restrict our investigations to maximally clear cases of general phonological processes where both the facts and the governing equational rules can be maximally well established.

3.1 Apocope and Paragoge. Apocope is the loss of word-final vowels or consonants. Paragoge is the addition of sounds in final position. These two processes are derivational converses, therefore, constituting the two possible directed or inferential uses of equivalence rules of the type schematized in (8).

(8) W [SEGMENT, X] # = W Ø #

It follows from the Equationality Hypothesis, then, that for any given instance of this schema the specified equivalence statement can be used to justify apocope, or final deletion, only in derivations terminating in one mode of terminal representation—phonetic or semantic—and to justify paragoge, or final addition, only in derivations terminating in the other terminal mode. And it must be the case, moreover, that the empirically correct derivational uses can be made to follow in every instance from some set of true law-like generalizations about linguistic structure or function. I will attempt to show now how this might be achieved with respect to apocope and paragoge, on the basis of certain fundamental laws of grammatical function that may serve to determine the correct derivational uses of many other types of phonological rules as well.

There are a number of well-known patterns of morpheme alternation in various languages that can be derivationally accounted for most naturally by the assumption of a phonetically-directed process of apocope, or loss of segments in word- or possibly syllable-final positions. For example, in Samoan, where consonants can occur only in syllable-initial prevocalic positions (Pratt 1911:4), there are systematic patterns, as illustrated in (9), where consonants at the ends of morphemes alternate with nulls when the morphemes are word-final.

Since the meaning ‘passive’ and the form ia are perfectly covariant here, and since the active form of any verb is fully predictable from its corresponding passive form, but not vice versa, a natural and empirically defensible grammatical analysis of Samoan verb forms will be one which treats the final consonants of alternating verb stems as distinctive characteristics of their associated stem morphemes, and specifies are predictable phonetically-directed loss, or null phonetic manifestation, of these consonants by a general rule of null-equivalence for consonants in word-final positions.⁶

In other words, for Samoan it is reasonable to assume a general process of phonetically-directed apocope, which is expressible by directed rule as in (10).

This derivational rule is justified by the representational equivalence statement (11),

(11) W [CNS, X] # Y = W Ø # Y

from which it follows that Ø can be legitimately substituted for any subrepresentation of the form [CNS, X] in lines of the form W[CNS, X] # Y in otherwise well-constructed derivations, or proofs of linguistic theorems, for languages of which (11) is true. It also follows from this equation, of course, that the opposite substitution, as expressed by the directional rule (12), is a derivationally legitimate substitution too.

The facts of Samoan are consistent with both of these inverse substitution rules, (10) and (12), but only if (10) is considered to be the only appropriate use of rule (11) in phonetically-directed derivations and (12) is considered to be its only appropriate use in semantically-directed derivations. By the hypothesis of equational grammar, these constraints on the derivational use of (11) must follow, like all other constraints on the use of grammatical rules, from strictly universal principles of linguistic structure and function in conjunction with the particular set of strictly symmetrical assertions of equivalence or non-equivalence which constitute the only equationally permissible grammars of particular languages.

The required restriction here follows in the required fashion from the overriding functional principle of linguistics—Maximalization of Terminally—and the empirical generalization (13), that, other things being equal, words of the form [X] are phonetically more optimal, or communicatively more valuable as terminal phonetic structures, than words of the form [X & CONSONANT].

(13) For any otherwise equivalent representations [W X # Y] and [W X [CNS] # Y] in any natural language, [W X # Y] is phonetically more optimal than [W X [CNS] # Y].

This generalization, like all other such assertions of differential communicative value, effectiveness, or efficiency for human systems of communication, is subject to both observational and experimental tests of various sorts, including tests of productive and receptive discrimination, measurements of physical time and energy expenditures in production and reception, word-confusion indices in sentence repetition and memory tasks and in spontaneous speech, errors by first- and second-language learners, etc. This type of evidence is necessarily indirect, of course, since the only real test of relative communicative value would require the controlled investigations of two otherwise identical human communities whose languages differ only in the particular characteristic whose relative value is at issue. Nevertheless, there are many relatively clear cases, like the present one, where indirect evidence, even of an informal and largely anecdotal sort, seems sufficient to support reasonably strong belief in the truth of the hypothesis in question.

The outlines of a general theory of grammatical function are first indicated in skeletal fashion by these clear cases of differential communicative value. The preliminary theory then serves as a basis for discovering and explicating other, initially unclear values for human communicational systems.

It is also possible to extend a theory of linguistic function by the discovery of new characteristics or kinds of terminal value if such values are found to be correlated with other more easily found or more easily observed characteristics of languages. One such correlation at least does appear to hold—the correlation between relative terminality and typological markedness.

Typological markedness relations are definable independently of any facts about the morphological alternations or phonological rules of any particular language and hence, a fortiori, of any empirical constraints on the ways such rules are appropriately used in the construction of particular derivations in any particular language. Thus A is typologically unmarked relative to B (and B is marked relative to A) if and only if every language that has B also has A and there is at least one language that has A but not B. There seems to be a regular correlation, however, between the relative typological markedness of two representations and the logically quite independent relation of relative terminality that holds between them in the derivations of particular sentences in particular languages.

Thus there is found to be a logically non-necessary correlation between derivational terminality and typological markedness, such that the phonetically more terminal (or more optimally phonetic) of two equivalent representations in a language will be the one that is typologically unmarked relative to the other. For example, in German voiceless word-final obstruents are phonetically more terminal in derivations than their voiced word-final counterparts. And voiceless word-final obstruents are unmarked relative to voiced word-final obstruents, since every language which has the latter (e.g., English, Dakota) also has the former while there are some languages (e.g., German, Thai) that have the former but not the latfer.

Examples like this are in fact consistent with the existence of a biconditional correlation between relative terminality and unmarkedness. This is expressed by the empirical law of Unmarked Terminality stated in (14).

(14)  Unmarked Terminality: For any language L and any [segmentally distinct, morphophonemically-related⁷] representations X and Y, such that X =Y in L, X is phonetically more terminal than Y if and only if X is phonetically unmarked (typologically) relative to Y—i.e., there exist languages with X and Y and with X but not Y, but no languages with Y but not X.

The evident correlation here between typological markedness and relative terminality in derivations appears to be essentially only a somewhat generalized version of the segmental markedness constraints on the functions of phonological rules proposed by Houlihan and Iverson in their paper “Phonological Markedness and Neutralization Rules” (1977). These constraints, which are put to a fairly extensive test both in their original paper and in the one presented at this conference, assert that “Phonologically-conditioned neutralization rules produce unmarked segments” (1977:14), and that “Phonologically-conditioned rules which produce exclusively marked segments are allophonic” (1977:15). Except for a few minor differences, mostly non-substantive, Houlihan and Iverson’s constraints seem essentially related as the segmental counterparts of the representational constraints on derivational function determined by Maximalization of Terminality and the Law of Unmarked Terminality.

The precise nature of this relationship will become more clear, hopefully, as we consider more of the complete range of types of phonological processes. Here it is sufficient to point out that functional constraints, like Houlihan and Iverson’s, or like those of Kisseberth (1970), Kiparsky (1973b), and others, stand merely as ad hoc excrescences in the context of Sound Pattern of English and all other such non-equational metalanguages for grammars, but play a fundamental and completely essential role with respect to equational metalanguages. Thus, in contrast to the standard models where the uses of phonological rules—the direction, obligatoriness, and relative order of the derivational substitutions justified by them—can be merely stipulated rule by rule and language by language without any need at all for universal principles of appropriate rule function, equational grammars require such functional principles to achieve even their simplest purpose of providing correct descriptions of the pronunciations of words and sentences in particular languages. The equationality hypothesis thus engenders an otherwise absent need for universal principles of rule function, and thereby provides a reason for their discovery and precise formulation, and hence for the explicit revelation of the highly significant linguistic generalizations that they embody.

In our discussion of apocope and paragoge thus far we have seen that there are some natural languages with morphemic alternations between consonants and null such that the substitution of null for a consonant is appropriate if and only if the derivation is phonetically terminated and the substitution of a consonant for null is appropriate if and only if the derivation is semantically terminated. Thus it follows from the equationality hypothesis that since there is at least one language that has phonetically-directed apocope of consonants and semantically-directed paragoge, it must be the case that all languages make the same uses of the rule in question, and hence that there can be no languages with phonetically-directed paragoge of consonants or semantically-directed apocope. This prediction seems correct, since in such languages every word would end in a consonant and there would be some morphemes that end in a vowel in non-final positions but end in a null-equivalent consonant finally. There appear to be no viable natural languages like this. In fact, when phonetically paragogic consonants have been claimed at all—as for Mongolian by Poppe (1970) or for certain Italian dialects by Rohlfs (1966)—there are no morpheme alternations or regular patterns of any sort involved, and no basis at all for the assumption of any general rule-governed phonological process of paragoge. The known facts about apocope and paragoge are thus consistent with the hypothesis of equational grammar and provide further substantiation for it.

It has been seen here that the specific universal principle which specifies the appropriate derivational uses of equations justifying the apocopation or counter-apocopation of consonants would follow directly by Maximalization of Terminality from the functional law of Unmarked Terminality, which entails that for any equivalent representations (here words of the form [#X#] and [#X [CNS]#]) the phonetically most terminal ([#X#]) is the one that is phonetically unmarked relative to the other. It is as if the goal or target of all phonetically directed derivations and rule applications were the achievement of terminal phonetic representations which are maximally rich in phonetic characteristics that are typologically unmarked. It has also been seen, moreover, that there is a general correlation between the relative unmarkedness of phonetic characteristics, as determined simply by their presence or absence in the languages of the world, and their relative communicative value or efficiency for purposes of effective expression of meanings in human systems of verbal communication.

There are thus three wholly distinct variables here, each logically independent of the others, and determinable from a logically independent body of factual observations:

(1) relative terminality—determined language by language on the basis of the facts about the particular morpheme alternations, phonetic inventories, and phonotactic generalizations of each language;

(2) relative unmarkedness—determined for the set of all known human languages on the basis of the occurrence and co-occurrence of particular phonetic characteristics in the members of this set;

(3) relative communicative value—determined for the set of all possible human systems of verbal communication on the basis of the physical, social, and psychological efficiency of particular characteristics for ease and accuracy of use, learning, and cultural transmission.

It would thus be an empirical fact of the utmost significance if these three independent and independently determinable variables are actually correlated with each other in any way. And all indications thus far suggest that they are in fact correlated, and that they are correlated, moreover, in a quite direct and systematic way.

The evident triple correlation here is expressed in its strongest form by the Law of Unmarked Terminal Value, (15).

(15)Unmarked Terminal Value: For any language L and any (segmentally distinct, morphophonemically related) representations X and Y, such that X = Y in L, X is phonetically more terminal than Y if and only if X is phonetically unmarked relative to Y and if and only if X is a communicatively more valuable phonetic structure than Y.

This law appears to hold for a number of reasonably clear sets of facts about natural languages. Much of the remainder of this paper will involve further tests of (34) and its constituent sub-laws.

If these laws are true, it will be noted, then it obviously cannot be because typological characteristics cause one representation to be simpler or communicatively more valuable than another, or phonetically more terminal in particular derivations of particular languages. Nor could it be the case that such relative terminality characteristics cause one representation to be terminally more valuable than another or, alone, typologically unmarked relative to another. Instead, the direction of causation here would presumably have to be from relative communicative values to both terminality and markedness characteristics. The causal connection, moreover, would have to be by way of some extremely general cultural law of optimum availability of efficient means. I discussed the necessity for such a link generally with respect to linguistic universals in my paper on the typology of elliptical coordinations (Sanders 1976), where I showed that the correlation that exists between the relative ease of decoding of a type of elliptical coordination and its relative typological unmarkedness can be explained if and only if there is a law to the effect that nothing complex or difficult can be available to a language (or culture) unless all things that are simpler and easier are also available to it. Although we will have nothing more to say here concerning matters of analysis or explanation at this level of generality, it is appropriate, nevertheless, to keep these fundamental matters of cultural adaptation and instrumental efficiency in mind as a general background for all particular investigations of grammatical rule function that may be carried out.

3.2    Prothesis and Apheresis. The traditional term prothesis is generally used to refer to the phonetically directed addition of a specified vowel at the beginning of words that would otherwise begin with a sequence of consonants not normally permitted in word-initial positions. The standard example is of the type illustrated for Spanish in (16).

Thus, since there are words and potential words in Spanish that begin with the sequences esk, esp, est, etc., but no words or potential words beginning with sk, sp, st, etc., the initial e is wholly redundant in words like escala or estado and can be predictably added in all phonetically-directed derivations of Spanish sentences and predictably deleted in all semantically-directed ones.⁸ The grammar of Spanish, in other words, could appropriately be assumed to include redundancy-free lexical rules like those in (17) for all words of the type exemplified in (16), and a perfectly regular phonological equivalence rule of the form given in (18).

(17) scala = “ladder”

(18)#s [CNS] X= # [VOC] s [CNS] X

where italicized letters abbreviate distinctive phonological representations and (VOC, Ø) is the distinctive representation of Spanish e, the simplest or most neutral of its vowels.⁹

Given this analysis of Spanish, then, and the facts that make it defensible, it follows from the equationality hypothesis that there can be no language for which rule (18) could be used to justify counterprothesis, or apheresis, rather than prothesis in phonetically-directed derivations, or prothesis rather than apheresis in semantically-directed ones. In other words, given Spanish, the equationality hypothesis predicts that there can be no language like the *Counter-Spanish language illustrated in (19), a language having words beginning with clusters of s followed by a consonant, but no words in which such clusters are preceded by a word-initial neutral vowel.

(19) *Counter-Spanish

The correct use of rule (18) for languages like Spanish and the exclusion of such use as would yield non-natural languages like *Counter-Spanish would be effected simply by a functional constraint against phonetically-directed apheresis or semantically-directed prothesis—either constraint following from the other by the principle of opposite use for opposite directionality.

And the delimitation of the set of natural languages determined by these constraints, like all of the other language-delimiting implications of the equationality hypothesis, would appear to be consistent with the inclusion of all known natural languages while correctly excluding many otherwise expectable non-natural ones. Thus as in the case of purported phonetically directed paragoge, all purported instances of phonetically-directed apheresis seem to be only apparent counterexamples to the general laws of use for null equivalence rules rather than real ones. All of the cited examples I have seen, in fact, refer not to any even moderately general rules, processes, or relations between phonological structures, but rather to certain purely non-derivational relationships that hold between the sporadic alternate pronunciations of certain words in different styles or dialects. For example, in Webster’s Third New International Dictionary, apheresis is illustrated by “round for around, coon for raccoon, baby talk ‘top for stop.”¹⁰

It will be observed, moreover, that this constraint follows directly as we would expect from the overriding functional Law of Unmarked Terminality. Thus any language which has words of the form #sCX# (e.g., English) also has words of the form #VsCX# (or even #esCX#), and there are languages which have words of the latter type (e.g., Spanish) but no words of the former type. Structures of the form #VsCX# are thus phonetically unmarked relative to those of the form #sCX# and are phonetically more terminal, therefore, by the Law of Unmarked Terminality.

The facts about prothetic vowels in languages like Spanish are thus fully consistent with the functional laws of Unmarked Terminality and Terminal Optimality and the general principle of Equationality that underlies them.

It also follows from these laws, moreover, that phonetically directed prothesis should not be possible before prevocalic consonants in general, since this would require, contrary to Unmarked Terminality, that a typologically more marked structure of the form #VCVX# be phonetically more terminal in derivations than its relatively less marked equivalent of the form #CVX#.

It is thus predicted that there are no natural languages that have words beginning with predictable vowels but no words beginning with single consonants followed by vowels. This prediction certainly appears to be correct.

There can be prothetic consonants in languages as well as prothetic vowels, but apparently only under just those conditions that are demanded by the law of Unmarked Terminality—namely, where the prothetic consonant occurs immediately before an otherwise word-initial vowel, thereby satisfying the required correlation between the terminality and unmarkedness of #CVX# relative to #VX#—since all languages have words and syllables beginning with consonants, and some (e.g., Arabic, possibly Thai) have no words beginning with vowels.

It is also worth noting that wherever clear cases of consonantal prothesis can be found—as in Arabic, for example, or German or Thai—the prothetic consonant, as predicted in “The Simplex-Feature Hypothesis” (Sanders 1974b), seems always to be the simplest and most neutral of all consonants, glottal stop, which is represented in simplex-feature notation by the consonantally minimal representation [CONSONANTAL, Ø] or, perhaps, [OBSTRUENT, Ø].

In Egyptian Colloquial Arabic, for example, where syllables beginning with vowels are prohibited (Mitchell 1962), there are systematic morphological alternations between prevocalic syllable-initial glottal stop and non-syllable-initial null. Thus the morpheme meaning ‘you’ is ?inta in ?inta kitábt ‘you wrote’, where the otherwise morpheme-initial i would otherwise be in syllable-initial positions, but inta in šúɣlak inta ‘your work’, where the i would not otherwise be syllable-initial (suɣ-la-kin-ta). The required phonetically directed prothesis or epenthesis of glottal stop in Egyptian Arabic will be correctly determined by the law of Unmarked Terminality from the Egyptian Arabic phonological rule in (20), which correctly asserts that null and glottal stop can be symbolically equivalent in this language before otherwise syllable-initial vowels.

(20) $ [VOC] = $ [CNS][VOC] ($ V = $ ? V)

Additive use in phonetically-directed derivations and subtractive use in semantically-directed ones follows by the governing functional law from the fact that the properly including member of the equivalence pair represents a type of phonetic structure, consonant-initiated syllables, that is typologically unmarked relative to the properly included member, representing vowel-initiated syllable structures.

3.3   Epenthesis and Syncope. Epenthesis usually refers to the phonetically directed introduction of a neutral vowel into a sequence of consonants in such a way as to render a phonetically impermissible sequence of segments phonetically permissible. Thus, for example, in Egyptian Arabic, where sequences of three consonants are impermissible within phrases (Mitchell 1962:34), the neutral vowel i generally occurs as a conditioned alternant of null between the second and third members of an otherwise ill-formed sequence.¹¹ This is illustrated in (21).

The appropriate introduction of epenthetic vowels in phonetically-directed derivations of Arabic sentences and their appropriate elimination in semantically-directed ones can be fully justified by the equivalence rule (22).

(22) [CNS] [CNS] [CNS] = [CNS] [CNS] [VOC] [CNS] (CCC = CCiC)

Thus it correctly follows by the law of Unmarked Terminality that the longer, or including, member of the pair of structures equated here must be the phonetically more terminal one, since every language with phonetic structures of the form CCC (e.g., English) also has structures of the form CCVC, but some languages which have CCVC structures (e.g., Arabic, Yawelmani Yokuts) have no CCC structures.

The beneficial effects of phonetically-directed epenthesis like this in terms of achieving optimally well-formed phonetic representations are so obvious that this type of directed transformation or process has traditionally been employed as the classic example of phonological functionalism. Yet, in spite of this, no principled basis has been provided in the traditional theories, models, or metalanguages of phonology for explicating the functional naturalness of phonetically-but not semantically-directed epenthesis, or for predicting the specific functions of this type of rule by general principles which determine the functions of other types as well. The greater perspicuity and conceptual facilitation values of an equational metalanguage are thus seen to be particularly striking here. Moreover, given Arabic, the factual claim generated by this choice of metalanguage alone is correct, namely, that there is no natural language in which vowels alternate with null in such a way as to maximize the number of triconsonantal sequences occurring and minimize the number of CCVC sequences. The non-existence of such “Counter Arabic” languages alongside ordinary Arabic can be given no really natural or fully principled explanation in the context of non-equational models of linguistic description.

Syncope is the phonetically-directed loss of weak unstressed vowels in medial positions, typically following stressed syllables and when flanked by single consonants. A standard case is illustrated by the optional null-alternation of the post-tonic vowels in words like Minneap(o)lis and happ(e)nings as pronounced in many varieties of American English.

Though epenthesis can add vowels and syncope can delete them, these processes are clearly not converses of each other. This is because the governing conditions for the occurrence of epenthetic vowels are always fully statable in terms of their adjacent consonants and boundaries, while the governing conditions for vowel syncopation are statable only by making essential reference also to the vowels or vocalic prosodies of the adjacent syllables in the word or phrase.

All rules justifying vocalic epenthesis will thus be instances of the general rule schema (23), and all rules justifying syncope will be special cases of the entirely distinct general schema in (24).

(23) X C Ø C Y = X C V C Y   (where X and Y are free of vocalic reference)

(24) (V́) X Ø Y (V́) = (V́) X V Y (V́)

Since no equation can be an instance of both of these distinct rule schemata at once, there are true law-like generalizations about the appropriate derivational uses of each type—phonetically additive and semantically reductive for (23), semantically additive and phonetically reductive for (24)—which are fully consistent both with each other and with the general hypothesis of equational grammar itself.

A different and, to me, basically quite plausible general account of syncope is outlined by Semiloff-Zelasko in a paper called “Syncope and Pseudo-Syncope” (1973). She proposes, on various grounds that I will not attempt to describe or evaluate here, that all cases of “real” or “pure” syncope involve the deletion of “a weakly accented syllabic which has come to be alone in a syllable,” and suggests further that this process “does not depend on the nature of adjacent consonants, except incidentally, insofar as syllabication depends in turn on consonant-types” (1973:603). She also points out that intervocalic consonants generally if not always syllabify with the more stressed of the surrounding vowels or with the one that follows if both are equally stressed or destressed. This type of analysis is shown to be appropriate to several instances of syncope in different languages, the most interesting evidence being from French, where Semiloff-Zelasko argues that the alternative patterns of syncope in different pronunciations of an expression correspond exactly to alternative ways of segmenting the expression into sequences of well-formed French syllables, with different schwas constituting full syllables and hence being subject to syncope under different syllabifications. This analysis seems to accommodate the Minneap(o)lis type of syncope in English too, since the prescribed syllabification for such words will evidently always yield a segmentation (e.g., mi-ni-yæp-ə-ləs) such that the potentially syncopational (schwa) vowel is isolated between syllable boundaries. The typical “optionality” of syncopation is also explained here by the existence of alternative segmentations into sequences of well-formed syllables (e.g., hæp-ə-niŋz vs. hæ-pə-niŋz).

If this type of analysis is correct, then syncope is a syllable-structure-based process that would follow simply from the general rule (25).

(25) $ V $ = $ Ø $

The correct directed uses of this rule would then follow in a perfectly regular fashion by the general principle of Unmarked Terminality. Thus since a null flanked by boundaries obviously has the same phonetic interpretation as null—namely, nothing—and since all languages can have nothing in any position of any expression, while vowel-only syllables do not occur in some languages (e.g., Arabic), structures of the form $ Ø $ are typologically unmarked relative to any otherwise equivalent ones of the form $ V $ and are hence correctly predicted to be phonetically more terminal. It is thus correctly predicted that syncope is possible only in phonetically-directed derivations and counter-syncope only in semantically-directed ones.

We have seen now that for all of the processes involving segmental equivalences with null—prothesis, epenthesis, apocope, syncope—the distinct derivational functions of each process are correctly determined from the structural equivalence that justifies the process by a single overriding principle of grammatical function—the law of Unmarked Terminality. It also seems reasonably clear that all of these processes, and the equivalence rules that underlie them, are principles for accommodating the segmental structure of morphemes to the required syllabic structures of the words and phrases that include them. Even the Samoan rule for consonant apocope seems to be syllable-based in reality, as expressed in (26), and a precise counterpart of the vocalic syncope schema (25).

(26)$ C $ = $ Ø $

The basic functional consideration throughout may thus be the avoidance phonetically of the typologically marked “extra weak” syllable types V and C and the also marked “extra strong” types with margins consisting of more than one consonant. The simplest, most efficient, and least ambiguating remedies are clearly phonetically-directed deletion for the extra weak syllables and phonetically-directed neutral vowel insertion for the extra strong syllables. Structure and function, means and ends seem admirably balanced here to me, and admirably economical in their contribution to the overriding function of human communication.

3.4  Final Simplification. It is a well-known fact about human languages that the ends of words, phrases, and syllables are phonologically less complex and less highly differentiated than their beginnings. Thus there are many examples of contrasts being maintained initially in a language but not finally—for example, voice contrasts for obstruents in German, Russian, Thai, Dakota, or Vietnamese; or aspiration, affrication, and continuancy contrasts in Thai and Vietnamese. The reverse, moreover, seems never to obtain; there are no known languages, for example, in which there is a contrast between voiced and voiceless stops in word-final position but no contrast word-initially; the same is true for aspiration, continuancy, affrication, glottalization, and all other characteristics of manner and, evidently, position of articulation as well. It is part of the nature of human languages, in other words, that oppositions can be maintained initially and neutralized finally, but not the reverse.

It is the case, moreover, that in such instances of final neutralization what occurs in final position generally is the simplest member of the neutralization set—the member pronounced with the fewest and most easily executed independent articulatory gestures.¹² Thus when the voice opposition is neutralized finally, it is the voiceless obstruents that always occur in final positions, not the voiced ones. Similarly, it is the non-continuants that occur under final neutralization rather than the continuants, the non-affricates rather than the affricates, the non-aspirated stops rather than the aspirated ones, etc. A particularly striking illustration of this general pattern of final simplicity is provided by Thai, where in word-initial (and syllable-initial) positions there are oppositions based on continuency (s vs. t, t^h, d, č, č^h), affrication (č, č^h vs. t, t^h), voicing (d vs. t), and aspiration (t^h vs. t), and where all six of these initially distinctive obstruents are represented finally only by t, clearly the simplest of the six, being non-continuant, non-affricated, non-voiced, non-aspirated—and for good measure, and with equally good reason, non-released as well. The fact of final simplicity must thus be accommodated in any adequate description or characterization of natural language.

In the Simplex-Feature Hypothesis (Sanders 1974b), I suggested that all these facts about final simplicity are determined as consequences of a single functional principle governing the derivational uses of all rules about word-marginal null equivalences. This principle asserts simply that “in phonetically directed derivations, elements that are equivalent to null can be added but not deleted in word-initial positions, and can be deleted but not added in word-final positions” (Sanders 1974b: 151).

The full set of typological generalizations determined by the marginality principle are indicated by the schematic abbreviation in (27), where p and b represent the simpler and more complex members, respectively, of any pair of obstruent sounds differing only in the absence or presence of voicing, aspiration, glottalization, affrication, etc.

The typological laws generated by Unmarked Terminality sort out the nine language types as indicated in (28).

Unmarked Terminality thus agrees with the Marginality principle in predicting the non-existence of languages with final neutralization to the more complex member of a neutralization set rather than the simpler member—as in Type (g)—or with the more complex allophones of a phoneme in final positions rather than initially—as in Type (h)—or with complex sounds without their simpler counterparts at all—as in Type (i). But Unmarked Terminality would preclude the phonetically directed addition of voicing, etc., in initial positions as well as final ones, thereby excluding any languages of Types (c) and (d) also, both of which are allowed by the Marginality principle.

Concerning type (d), it seems to be the case that there are some languages of this type, but that their existence is due to allophonic rather than morphophonemic processes and thus not really contrary to Unmarked Terminality at all. But the same situation obtains also in fact for the much more interesting language types (g) and (h), which have been indicated in (27) and (28) as excluded by both Marginality and Unmarked Terminality. One possible example of a type (h) language is the variety of Central American or Caribbean Spanish which has the pattern of nasal distribution illustrated in (29), with labials and Unguals in contrast and the two linguals in complementary distribution. (There is also a phonemically distinct palatal lingual nasal, which is irrelevant, though, to the present issue.)

But the only rules in question here are clearly redundancy rules, whose intrinsic function is always to justify the addition of the specified redundant elements in derivations terminating in the alphabet to which those elements belong, and to justify the deletion of the redundant elements in derivations of the opposite directionality. Principles like both Marginality and Unmarked Terminality could thus be viewed as simply irrelevant to the application of such intrinsically directed rules. Under this intepretation, languages of type (h) would be possible under both principles, but only if, as in the Spanish case, the complex segment that is derived word-finally stands in an allophonic relation to its simpler counterpart rather than a morphophonemic one.

This interpretation would also allow for a possible subclass of the otherwise excluded type (g), a subclass including at least Dakota, a language that has voiced rather than voiceless stops word-finally, but always in an allophonic relationship to other non-morpheme-initial stops. (For details, see Boas and Deloria 1941; and Steyaert 1976, 1977.) In other words, exactly as in the case of Houlihan and Iverson’s (1977) segmental markedness constraint, the phonetically directed conversion of a simpler or less marked structure into a more complex or more marked one is precluded only in those cases where the conversion would have the effect of neutralizing an otherwise operant contrast. Where no neutralization, or morphophonemic effect, could result, rule use so as to increase complexity or markedness is permitted—and in fact required.

It remains to be seen, of course, whether there are other languages of types (g) and (h), and if so whether they in fact all belong to the same analytic subtypes as Spanish and Dakota. Extensive systematic investigations of language typology are needed here now, as they are indeed in all other areas of linguistics as well. But if such investigations should provide continued support for Unmarked Terminality, the Marginality constraint, or any combination of these and the various other functional laws that have been discussed here, it would still be necessary, of course, to specify how these various principles are causally or instrumentally related to each other and to the overriding functional imperatives of viability and selective adaptation. The outlines of a partial synthesis of this sort have been suggested earlier with optimal simplification of articulatory and perceptual distinctions as the functional basis from which both Unmarked Terminality and Terminal Optimality follow. The constraints on final complexity and final contrasts should certainly also be derivable from the same functional basis, the operant cultural metalaw for all these derivations being the principle of optimal availability of efficient means, or again “harder only if easier,” which seems to govern the non-linguistic instruments of human societies as well as the linguistic ones.

A complete and fully explicit expression of such a synthesis must obviously await the results of further typological research and extensive testing of various particular hypotheses about grammatical rule functions. I would like to conclude the present paper, nevertheless, by providing a rough sketch at least or tentative working model of this synthesis in the form of what appears thus far to be the most general of all possibly true principles of phonological rule function in natural language grammar. This principle is stated, as the law of Terminal Simplification, in (30).

(30)   Terminal Simplification: For any natural language L, and any pair of phonological representations WAY and WBY that are equivalent in L, WAY is phonetically more terminal than WBY if and only if either

(1) A is a redundantly specified expansion of B (i.e., A = XCZ, B = XZ, and there are no lexical [underlying] representations for L of the form XCZ) or

(2)A is simpler than B (i.e., either (a) A = XZ and B = XCZ, as in apocope, syncope, final simplification; or (b) A is also a part of W and/or Y but not B, as in agreement or assimilation, or (c) A = [C, D] and B = [C, X] [D, Z], as in contraction).

It follows from this law that phonological rules can be used to justify the phonetically-directed addition of features or (epenthetic and prothetic) segments only if the added elements are wholly redundant, or else mere copies of certain elements in their environment. Under all other circumstances, rules can be used in phonetically-directed derivations only for the purpose of justifying deletion of features, segments (as in apocope or syncope), or boundaries between segments (as in contraction). The overriding function of phonological rules in general is thus suggested here to be the function of associating optimally distinct and redundancy-free morphemic representations with fully-specified pronunciations that are both optimally rich in redundancies and as easy to pronounce and understand as possible.

4. SUMMARY AND CONCLUSION

We have investigated certain fundamental issues here concerning the structure and function of natural language grammars, with particular reference to their various types of phonological rules and the various directed uses, or derivational processes, that are appropriately determined or justified by them. We have carried out these investigations in the context of the general metatheory of Equational Grammar, a model for linguistic description and analysis which limits the content of all grammars to lawlike rules about representational equivalences, and thus requires that all constraints on the derivational use of such rules be determined by strictly universal natural principles of grammatical function. As an empirical hypothesis, Equational Grammar thus claims that human languages can vary only in formal characteristics and not in functional ones, and hence predicts that there can be no pairs of languages that have the same equivalence rules but different constraints on their directed application or applicational interaction in derivations. All available evidence indicates that this claim and prediction is correct, and hence that the equationality hypothesis expresses a true synthetic generalization about natural languages that is denied by the traditional contrary or contradictory hypotheses of directed grammar. Moreover, simply as a metalanguage, or determinant of linguistic metalanguages, Equational Grammar has also been found to be consistently superior to its available non-equational alternatives, on grounds of clarity, conciseness, appropriateness of expressive power, and facilitation of ascent to higher levels of analysis and explanation. In fact, by elevating all grammatical rules to the status of lawlike generalizations—which cannot naturally be done for the rules of non-equational grammars—the equationality hypothesis makes it possible for all linguistic terms, concepts, and statements to be subject to the kind of rigorous interpretation and verification standards that have standardly been required in all other empirical sciences.

Our primary purpose here, however, has not been to establish either the truth or the utility of the equationality hypothesis itself. Our chief intention, rather, has been to investigate the specific derivational functions of certain major types of phonological rules, and to try to discover how the functions of these particular rules are governed and determined by the functions of the whole grammars that include them. We considered in particular the types of rules that justify apocope, syncope, prothesis, and epenthesis, and their semantically-directed converses. The empirically appropriate functions of such rules were found to be in general accordance not only with the general hypothesis of equational grammar itself and the general functional principle of Maximalization of Terminality, but also with the more specific functional laws of Unmarked Terminality and Terminal Optimality, as well as Houlihan and Iverson’s Segmental Markedness constraint on phonological rule applications. On further investigation, though, with respect to facts about word-and syllable-final simplification processes and the converse uses of phonological redundancy rules, certain questions of adequacy were seen to arise with respect to each of these hypotheses about phonological functions, as well as the previously proposed marginality constraint on the adjunction and deletion of elements in word-initial and word-final positions. A tentative restatement or reanalysis or partial synthesis of the governing principles of phonological rule function was then suggested, based primarily on the proposed law of Terminal Simplification, from which the laws of Unmarked Terminality and Terminal Optimality may be derivable as theorems.

A large number of very interesting and very challenging questions remains to be investigated, of course, concerning the individuals and combined functions of phonological rules and their relations to the structures and functions of the whole grammars that include them. There is every reason to believe, though, that these questions will be just as amenable to productive inquiry and analysis as those which have been raised thus far, and that the most appropriate context for such inquiry and analysis, as for the study of grammatical structure and function in general, will continue to be the metatheoretical framework and climate established by the general hypothesis of equational grammar.

NOTES

I am grateful to Kathleen Houlihan, Gregory Iverson, and Linda Schwartz for helpful comments and discussion during the preparation of this paper.

1. This remains quite true, I believe, in spite of the many recent prescriptions to the contrary, chiefly by Chomsky and his close associates, who have often tried to redefine linguistics as the study of human linguistic behavior rather than the study of human language.

2. See, for example, Hastings (1974), Iverson (1974), Koutsoudas, Sanders, and Noll (1974), and the papers by Iverson, Koutsoudas, Norman, Ringen, and Sanders in Koutsoudas (ed.) (1976).

3. The type of directed-rule relation that Vennemann (1972a) calls “inversion” is not a converse relation (i.e., the relation holding between members of rule pairs of the form A B and B A), and the existence of such inversion relations is in fact entirely consistent with the equationality hypothesis. Thus Vennemann (1972a:212) defines rule inversion as the relation which holds between pairs of rules of the form AB/D and BA/D̄, where the context D̄ is the complement of context D. But it is clear that these two directed processes follow from two distinctively different equational rules, namely, (A, D = B, D) and (A, D̄ = B, D̄), and thus do not constitute converse uses of any single rule of grammar. Facts about rule inversion are thus irrelevant to any issues of present concern. The same is true, needless to say, for proposals, like that of Leben and Robinson (1977), which postulate semantically-directed uses of axiomatically directed rules rather than phonetically directed ones.

4. Thus for any given set of n equivalence pairs there are 2ⁿ x 2ⁿ x 2! possible non-equational grammars, but only 2ⁿ possible equational grammars with both equivalence and non-equivalence rules, and only one possible equational grammar with equivalence rules alone. This at least shows quite clearly and dramatically where the burden of proof lies—with the richer and more complex metalanguage of non-equational grammar, and not with the much more restricted and economical metalanguage of equational grammar.

5. The upper case letters here stand for terminal semantic representations or the meanings that constitute their interpretations; the lower case italicized letters stand for terminal phonetic representations or the pronunciations that constitute their interpretations. These notational distinctions have no systematic significance whatever and are made use of here simply for purposes of expository convenience.

6. Kiparsky (1971), following Hale (1973), attempts to argue against this type of analysis in a quite parallel situation from Maori. The evidence he presents, however, seems to me to be clearly insufficient to justify the intended conclusion.

7. This law is not intended to apply to representations that differ only allophonically, or just in the presence or absence of elements that are completely redundant, or predictably present simply from the presence of other characteristics of their including structures.

8. The pattern of predictability here may even govern a few morpheme alternations. Harris (1969:141) cites checoslovaco: eslovaco in support of his own proposal of phonetically-directed e-prothesis for Spanish.

9. A more general version of (18), which would also incorporate the process of epenthesis before prefinal s, . . . . would be simply ($ s $ = $ [VOC] s $). See Saltarelli (1970) for outlines of a related type of epenthesis analysis for Spanish, and Harris (1970) for criticisms of Saltarelli’s proposals.

10. Either apheresis refers to non-systematic derivationally unrelated variants like these, or it is explicitly defined more as a contraction of adjacent vowels than as an inverse of prothesis; for example, Marouzeau defines it in his Lexique de la terminologie linguistique as “suppression . . . d’ un phonème ou groupe de phonèmes à l’initiale du mot, par exemple d’une voyelle après voyelle finalle du mot précédent,” giving as an example English I’m for I am.

11. In a small number of situations an evidently epenthetic vowel has the quality a or u instead of i, perhaps as a result of assimilation to an adjacent vowel (or perhaps copying by a combined epenthesis copying process of the sort seen in Mohawk; for discussion see Postal 1968 and Sanders 1974b).

12. This should be compared with the related but nevertheless distinct characterization of the relation between neutralization and simplicity given by Trubetzkoy (1969).