Graphic representation and linguistic science

We have seen that it is entirely possible to speak of graphic representation as a distinct element in linguistic science; but what is important about the existence of three basic figures? What is their role in the development of linguistic science? To see where graphic representation fits into the structure of linguistic science, we must have recourse to the philosophy of science, the first of our two perspectives on graphic representation in linguistics.

Graphic representation and the philosophy of science

Any discussion of graphic representation in linguistic science has for its context the philosophy of science; for the nature, place, and function of graphic representation are determined by, and in turn determine, other components of linguistic science. Some sort of working definition, then, of such notions as ״theory," "law," and "model" is indispensable. Indeed, a working definition is the most that can be aspired to: the philosophy of science is a field (as Tennyson put it) whose margin fades forever and forever when we move. The definitions arrived at here, based on an eclectic, often expedient survey of the philosophy of science, are shaped to the discussion of graphic representation that follows.

There is controversy among philosophers of science even over what constitutes a scientific theory. Nagel (1961:117-152) discusses three different conceptions of scientific theory: "descriptive," "instrumentalist," and "realist." These are differences in orientation toward science itself. The instrumentalist view is particularly useful for graphic representation and for linguistic science in general. Toulmin (1953:108), for instance, sees scientific theory as a map in which

the diagrams present all that is contained in the set of observational statements, but do so in a logically novel manner: the aggregate of discrete observations is transformed into a simple and connected picture.

Such a view is an eminently sensible conception of the relationship between theory and reality. Although map-making demands a priorichoices--the cartographer's choice of method of projection, scale, and so forth--which render subjective a scientist's-eye view of reality, "the alternative . . . is not a truer map [but] no map at all" (Toulmin, 127). The instrumentalist position is captured in Hadamard's (1945 :xi-xii) characterization of scientific theory as invention rather than discovery.

We shall assume here, then, that a scientific theory is an artifice. Some further distinctions, largely irrelevant to a consideration of graphic representation, will not be taken up here. These are such questions as whether a theory constitutes description or explanation (Toulmin 1953:53); which of the various types of explanation--deductive, probabilistic, functional or teleological, genetic--a theory exhibits (Nagel, 20-26); whether explanation follows the "pattern model" or the "deductive model" (Kaplan 1964:332-338).

The distinction between theory and law (Nagel, 64-67) is, in linguistics, hard to discern. Does Grimm's Law, for instance, constitute a theory of sound change, as would seem to be the case when it is set beside the theory of Martinet (1955)? Or is it a law? If it is a law, what is its scope? If its scope is not restricted to the past, it is a poor law, for the phenomenon by no means invariably occurs (Nagel, 63; but cf. Campbell 1921:69). Moreover, linguistic science has not yet reached the point where ״universals" are verifiable, if indeed they are useful at all. If, on the other hand, the scope of Grimm's Law is restricted to the past--a reconstructed past, at that--it is not a law at all (Nagel, 63, but cf. Campbell, 69) . Yet it surely goes beyond description, though it falls short of explanation.

Less dispensable than a definition of theory or law is a definition of model. For one thing, as Chao (1962) has demonstrated, the term ״model" covers a whole flock of senses, which fly off in different directions when approached too closely; for another, graphic representation and models are (as we shall see) closely connected. The place of the model within scientific theory must be established first of all. Not unexpectedly, philosophers of science disagree: their conceptions range from an indispensable and even pretheoretic model to a peripheral, purely decorative one.⁸ Again the middle ground will serve us best; and again Toulmin's conception is useful. Beside his notion of the theory as a map we may set Toulmin's assertion (1953:35) that models supplement theory by providing a "clearly intelligible way of conceiving the physical systems we study." A model, then, is the bridge between "discrete observations" about the data and the "simple and connected picture" furnished by a theory.

The function of the model standing midway between data and theory is to mediate between the very concrete and the very abstract. An example is the wave theory of light. This theory provides a concrete, visualizable model--water waves--for an abstract mathematical formulation of the properties of light. It does so by means of the similarities between water and light; by means of the correspondences between a known thing and the unknown datum, the explicandum; by means, in short, of analogy. This notion of the relationship between model and data is formulated by Hesse (1966: 68-69) as "material analogy," a relationship in which the similarity between terms of the analogue and corresponding terms of the explicandum is independent of causal relations within each.⁹ For the wave theory of light, the likeness of the model to the explicandum can be elaborated in the following way (Hesse, 11) :

Each term of the analogue, water waves, serves to explain, translate, render visualizable, some aspect of the explicandum, light.

Models, however, not only serve but also influence theory. These two functions--the interpretive and the heuristic--are in some measure attributed to models by nearly all philosophers of science.¹⁰ Like the interpretive function, which we have been discussing, the heuristic function of models is rooted in material analogy The heuristic usefulness of the model, its usefulness in extending and developing a scientific theory, depends upon the incompleteness of the analogy; that is, the likeness of model to explicandum must constitute a material analogy only for some terms, leaving others still to be explored. Thus if we go on spinning out the correspondences between water waves and light, we eventually arrive at such likenesses as

These two correspondences are at the outset part of what Hesse calls the "neutral analogy''--elements of the model whose relevance for the explicandum is unknown. It is by virtue of the neutral analogy that a model furnishes suggestions for the development of a scientific theory. As it happens, neither ether density nor ether elasticity is a significant property of the medium in which light is propagated; these terms are accordingly relegated to the "negative analogy"--elements of the model irrelevant for the explicandum. Had they turned out to be significant, they would have been incorporated into the "positive analogy"--elements of the model relevant for the explicandum--along with frequency, amplitude, and so on. The heuristic function of models consists in the exploration of the neutral analogy, which develops and extends the scientific theory: it adds to or elaborates the positive analogy and at the same time circumscribes it, sets its boundaries, by establishing the negative analogy.

Form follows function. If the model is to perform an interpretive function with respect to the abstract formal theory--to constitute, in effect, a translation of the theory--model and theory must be isomorphic. Explicitly suggested by such theorists as Black (1962:238) and Kaplan (1964:263), this is implicit in any notion of model. If the model is to perform a heuristic function, it must be a partial isomorph for the explicandum, as well.

The model, then, is a Janus-like thing, whose place is midway between theory and explicandum; whose function is at once interpretive and heuristic; whose form is an isomorph in two directions. In a chronology of theory formation, the model may make its appearance before the formal theory has assumed its final shape--perhaps even prior to any formal theory. What then is the place of graphic representation in scientific theory? We have said that it is hard to apply the concept of scientific law to linguistic science. Most linguistic statements that are called laws (Grimm's Law, Grassmannfs Law, Verner's Law) will not, because they characterize events past or unique, meet the standards for scientific laws. Yet they are not simple description: they structure the data. Often this structure is expressed in a graphic configuration, like the figure for Grimm1s Law, the Kreislauf (not, in fact, Grimm's design but Schleicher's [1888: 97]). For linguistics, the closest thing to a scientific law is the statement of a discovered (better, an invented) regularity, often graphically expressed.

Similarly, the concept of model, even the relatively capacious one of the model as analogue, will not fit linguistic science without alteration Models presented as such are rare in linguistics--perhaps because it has had relatively little use, compared with physics or biology, for physical scale models, and less awareness therefore of the model as an element in scientific theory. Moreover, explicit models--the computer in descriptive linguistics (Yngve 1966) , human genealogy in historical linguistics--cannot be expected to serve as what Harré (1960:105) calls "candidates-for-reality." There is no hope of someday validating the model by discovering it to be an actual physical structure, like the double helix in biology. Implicit models, höwever, abound: when graphic representation accompanies a linguistic statement, an unexpressed "is like" couples picture and prose. For linguistics, the notion of model, like the notion of scientific law, is accessible by way of graphic representation. It is as a model that a figure can come to have a life of its own--can, in fact, like models in science generally, influence the theory out of which it grew. This is sometimes obscured by the prominence of a detailed, highly structured theory that is assumed to govern the model; but, as we shall see, linguistic theory is often demonstrably the product of its representation.

The influence of graphic representation on linguistic theory

A notion may be fleeting, doubtful, vague; once it is codified as a method of representation, a figure, it must be reckoned with in all its implacable detail. Graphic representation influences scientific theory in two ways: in its extension, and in its development.¹¹ The extension of a theory is its application to more kinds of data than that for which it originated; the development of a theory is its articulation in progressively greater detail, and its modification. The more negotiable a figure--the more readily the design applies to various kinds of data--the more it facilitates the extension of linguistic theory. The more suggestive a figure--the more avenues for further exploration the design unfolds--the more it contributes to the development of linguistic theory.

The extension of linguistic theory

The matrix illustrates extension. That the matrix is a negotiable figure is demonstrated by its transfer from traditional articulatory phonetics to distinctive feature analysis; from phonology to morphology to semantics. The theory of componential analysis is thereby extended to practically every aspect of language. For phonology, as we have seen, matrix arrangement and componential analysis are indispensable. For morphology, besides Jakobson1s matrix arrangement and distinctive feature analysis for case (Jakobson 1958, a much earlier version of which is Jakobson 1936), there are Sebeok's (1946) for case and Jakobson's (1957) and Joos's (1964) for conjugation. For semantics, besides the semantic theory of Katz and Fodor incorporated by transformational-generative grammar (Katz and Postal 1964), there are componential analyses in linguistics (Austerlitz 1959, Nida 1964) and in anthropology (Colby 1966, Kay 1969).

Pike (1962) attests the transfer of graphic representation per se and its effect on the extension of theory. He asks (221),

Can grammatical dimensions be charted like phonetic ones? . . . for grammar, what would be the analogy of

The matrix, then, is a negotiable figure indeed. The effect of its negotiability on the extension of the theory of componential analysis is plain.

The tree is also a negotiable figure; but its effect on the extension of linguistic theory is less happy. On the transfer of the figure itself, little need be said. We have already seen the negotiability of the tree in its migration from diachronic to synchronic linguistics; from structural linguistics to transformational-generative grammar; from syntax to phonology to semantics. Its itinerary is of interest for the extension of linguistic theory, and it is for this reason that we considered at length the provenience of tree diagrams. The negotiability of the tree is such, indeed, that the notion that "Chomsky got his tree from Schleicher" is not so absurd as Maher (1966:9) makes it sound.

How does the transfer of the figure effect the extension of linguistic theory? Not of a single theory; for the tree cannot--any more than the matrix, which comprises such diverse analytic techniques as taxonomie (articulatory) phonemics and generative phonology--be said to correspond to a single theory. With the spread of the tree there has, however , been the spread of a theoretical bias—what Holton (1964) calls a thema. The thema for the tree is its most general meaning: subordination, expressed, quite literally, by "placing under." Whatever its modification, the figure carries this theme, as the matrix carries the theme of componential analysis. (They are the two great themes of linguistic science, deriving ultimately from the same principle: divide and conquer.) But we have said that extension of theory--extension, that is, of the theme of subordination--is, in the case of the tree, unwarranted. If the measure of extension is the number of meanings conveyed by a figure, the measure of unwarranted- ness is the confusion that results. The chart at the end of the last chapter shows the extent of the synonymy and homonymy in which the meanings of the tree take part. Several diverse techniques of linguistic analysis employ the tree, often in more than one of its meanings. So wide an appli- cation of the figure has resulted in the accretion of conflicting meanings; they become impossible to dislodge, creating confusion, contradiction, ambiguity. To the extent that any given tree diagram provokes this confusion, the transfer of the figure has ended in unwarranted extension.

The development of linguistic theory

The negotiability of a method of representation influences the extension of a theory, widens it, as it were; the suggestiveness of a method of representation influences its development, deepens it. A heuristic value for the development of theory is taken for granted in discussions like that of Pike (1962:240) concerning the applica- bility of the matrix arrangement to grammar. It sounds as if the subject were some new piece of laboratory equipment :

a set of preliminary dimension charts for grammatical constructions helps the consultant to alert the beginner to gaps in his data by allowing him to see more clearly the over-all grammatical structural system.

Sebeok (1946:18) makes a similar claim for its application to case. The matrix arrangement, he says, "will make comparison of the Finnish with other systems visually feasible and will also reveal certain structural characteristics of Finnish itself." Hockett's apologia (1958 :109) gives this use of graphic representation a name, the "principle of neatness of pattern." "If we are confronted with two or more ways of identifying allophones as phonemes, both or all of which equally well meet all other criteria," he says, "we should choose that alternative which yields the most symmetrical portrayal of the system." And Trubetzkoy himself claimed for the matrix a more than decorative import (1939:65; quoted in Cairns 1972:123):

die Ordnung, die durch Aufteilung der Phoneme in parallele Reihen erreicht wird, existiert nicht nur auf dem Papier und is nicht bloss eine graphische Angelegenheit. Sie entspricht vielmehr einer phonologischen Realität.

Graphic representation--at least, the matrix diagram--is seen not as a mere embellishment but as a tool.

There are many more instances in which such a view is implicit. Jakobson's diagrams for case conflation, which we looked at earlier, in fact parlay the matrix into a theory of case confiation. The "conflated field structures" of Pike and Erickson (1964) are the mirror image of Jakobson's procedure: the whole search for pattern in the data consists simply in inspection and rearrangement. The use of such structures asserts that sheer scrutiny of the data can render it intelligible, if matrix arrangement is the lens through which it is viewed.

The suggestiveness of graphic representation has played a role in historical linguistics from its beginnings. Considerations of pattern and symmetry are more pressing when it is a matter of reconstruction, because there is not the counterbalance of an actually existing system in all its unruliness. Saussure1s famous monograph of 1879 apparently rests upon such a use of matrix arrangement.

Graphic representation can also work the direct development of linguistic theory, without going by way of the data. We have mentioned the transfer of the matrix from traditional articulatory phonetics to distinctive feature theory. In both, it has proved to be productive for the development of theory. In both, the development is from a synchronic theory into a diachronic theory. The traditional articulatory matrix grounds Austin's (1957) theory of sound change, which says that sounds shift by travelling along the horizontal or vertical, never the diagonal. Thus a sound x cannot become y in fewer than two moves. We can express״ this as

The matrix arrangement underlying Martinet's (1955) theory of sound change is that of distinctive feature analysis. But the theory has the same components as Austin's: movement and direction; and phonological space is the backdrop against which the diachronic drama is played out. Martinet gives no figure, but one from King (1967:3) can serve:

Both Austin's and Martinet's conceptions of sound change grow out of the matrix figure. The components of the diachronic theory, movement and direction, can be traced to the figure as convincingly as to the synchronic theory.

It is illuminating to set beside the theories of Austin and Martinet a third, much earlier conception of sound change as movement against a matrix. This is Anton Pfalz's (1918) notion of Reihenschritte . Vowels that are the same in vowel height undergo the same change: if [i] becomes [ei], [u] becomes [ou]. In other words, sounds in the same row in a matrix arrangement behave in the same way. Pfalz lists the sound changes governed by this principle in what is itself a matrix arrangement :

(Pfalz 1918:28)

Here the matrix figure clearly directs the choreography of vowel change.

The process by which a figure effects the development of linguistic theory is the process by which a model effects the development of any scientific theory. It is the exploration of the neutral analogy. In all three theories of sound change, the matrix figure has furnished the neutral analogy for the development of a diachronic theory out of a synchronic one; the route from one to the other is by way of the neutral analogy--by investigating aspects of the model, the matrix figure, whose relevance for the explicandum, sound change, is not known. Out of them, the two dimensions, horizontal and vertical, have been assigned to the positive analogy; while the figure's static quality has been assigned to the negative analogy, to be replaced by the notion of sound change as sound movement. For Martinet's theory, exploration of the neutral analogy yields in addition the notions case vide, marge de sécurité , champ de dispersion, chaine de propulsion, chaíne de traction. A case vide is of course an empty cell; marge de sécurité is suggested by the boundaries between cells; champ de dispersion is the flat ground or "field" furnished by the matrix; and a chain-like progress of sounds across the matrix is the only conceivable sort. The notion of chaîne includes only the columns of the matrix figure, however; rows are relegated to the negative analogy, inasmuch as position on the horizontal dimension has little significance for sound change (Martinet, personal communication).

The theories of Austin, Martinet, and Pfalz all illustrate the development of a synchronic theory into a diachronic one. Another direction suggested by the matrix analogue for the development of linguistic theory is illustrated by the work of Pike. Here the matrix suggests a finished theory that outgrows the matrix figure. Thus the following figure,

(pike 1962:234)

Like the cube, Pike's tetrahedron replaces the matrix so as to display a system with a greater number of features. Each of the four clause types, intransitive, transitive, semitransitive, and equational, is specified with respect to three features, T, O, and P. Each of the four faces of the pyramid represents a clause type: the clause type intransitive, for instance, is represented as

Thus each face of the figure constitutes the intersection of three features; its faces are to the tetrahedron what cells are to the matrix.

Alternatively, the finished theory suggested by the matrix, though demanding a new sort of representation, may not be representable by a figure. Pike's "matrix multiplication" (1962:230) is an instance. In matrix multiplication, a matrix is modified to display units as the intersection of more than two features. The features that lie along the two dimensions of the matrix are multiplied by a constant consisting of a"kernel matrix." The whole is then the product of two matrices. It is a sort of kangaroo figure: we could see it as

That it is a theoretical concept rather than a new figure is indicated by Pike's formulating it not graphically but mathematically. A multiplied matrix,

(Pike 1962:227)

for instance, is expressed not as a figure but as (228)

(1,2,3) · (a,b,c) = M_k
(Ι,ΙΙ) · (A,B) · M_k = M_z

Matrix multiplication remedies the characteristic inability of the matrix to handle a great number of distinctions. A matrix represents a unit as the intersection of two distinctive features; a cube or tetrahedron represents a unit as the intersection of three distinctive features; a multiplied matrix represents a unit as the intersection of a theoretically unlimited number of distinctive features. The development of theory that produces the tetrahedron and the multiplied matrix, then, results from exploration of the neutral analogy furnished by the matrix figure. In this case, the aspect of the model whose relevance for the datum is under scrutiny is in the end relegated to the negative analogy. This is the fixing of the number of distinctive features at two. It is built into the matrix figure; relegating it to the negative analogy allows the matrix to grow into the tetrahedron or to multiply itself. More than the tetrahedron, the multiplied matrix follows the logic of the matrix design, for it is a matrix arrangement of matrices.

An attempt to overcome the same limitation--the fixing of the number of distinctions that can simultaneously be made, at two--for the version of the matrix figure that we have called the grid, is to be found in the abstract lexical representation of Chomsky and Halle (1968). In effect a modification of the figure, it combines the transformational grid, "in which the columns stand for the successive units and the rows are labeled by the names of the individual phonetic features," with a set of phonological rules that alter that grid, for each lexical item, "by deleting or adding columns (units), by changing the specifications assigned to particular rows (features) in particular columns, or by interchanging the positions of columns" (296) . The net result is a matrix in motion, in which columns come and go and change places, and cells contain changeable information. The phonological rules in the theory proposed by Chomsky and Halle have the effect of making a moving picture out of a series of ״stills--intermediate phonological matrices--of the matrix. Such a modification of the figure with an eye to getting beyond the limitations of its two-dimensionality thus has as well the virtue of overcoming the static quality of the matrix figure in general, relegated to the negative analogy by Austin, Martinet, and Pfalz for diachronic theory. The development of the neutral analogy presented by the matrix figure here takes a different direction from that proposed by Pike: in one case, the fixed two-dimensionality of the model is supplemented by the notion of movement; in the other, by the addition of a third dimension. Nevertheless, the matrix figure has in both cases proved suggestive for the development of linguistic theory; it has in both cases served the heuristic function of models.

It is for the development of transformational-generative theory that the tree has been most suggestive. Its suggestiveness, unlike that of the matrix, has two sources. One is of course the figure itself; the other is the metaphor attaching to the figure. Transformational theory shows first of all the fruits of a literal interpretation of the tree figure. "Extraposition" of elements, for instance, alludes to what goes on in the diagram;¹² so also do the various -fronting, -raising, -spreading, -insertion, -attachment, -placement, and -movement transformations--¹³all named from the model-on-paper, from the operations being performed on the diagram. The problem of where to attach constituents in reordering transformations is a problem of diagramming, as are other alterations in the tree that are necessitated by the application of a transformation, over and above the modification of the tree directly wrought by that transformation--for example, nodereduction (related to tree-pruning), and node- relabelling. Various constraints on the application of transformations, as well, are formulated in terms of the model-on-paper. Postal (1971), for instance, discusses constraints on the movement of NPs as what he terms "cross-over phenomena"--named from what goes on in the figure: the reference is to the movement of NP nodes across each other along the horizontal dimension of the tree. Other constraints exploit the vertical dimension of the tree; "stacking," for instance, refers to the piling up of like constituents (nodes) by self-embedding (Stockwell, Schachter, and Partee 1973:442), as in the configuration for relative clauses:

(Stockwell, Schachter, and Partee 1973:442)

Another example is Rosenbaum1s "principle of minimum distance" (Stockwell, Schachter, and Partee, 532-536), itself the outgrowth of the "erasure principle," the reference of which to the model-on-paper is patent; the "principle of minimum distance" exploits the vertical dimension of the diagram by counting the number of branches in the path between two nodes.

Even the assumption that constituents are ordered in deep structure may well be a response to the exigencies of graphic representation: in a two-dimensional medium, linearity is inherent. Constrained to linear order, we choose left-to- right because this is more natural in our culture, although, as Weyl (1952:24-25) points out, ',those laws of which we can boast a reasonably certain knowledge are invariant with respect to . . . the interchange of left and right." Such terms as "right-" and "left-branching" constructions also reflect this constraint. (That it is a very real constraint, attaching solely to the two-dimensionality of graphic representation, we shall see in the next chapter.) Finally, the common misconstruction of the term "generative" as "productive" is probably a product of the figure, which in transformational-generative theory is always upside-down and is naturally read from top to bottom--from S down through intermediate constructions to a particular sentence, rather than from the given sentence up through its analysis to its characterization as an S.¹⁴

With the tree as with the matrix, the response to the fixed two-dimensionality of the figure has been a development of transformational theory in the direction of movement. As phonological rules have the effect of making a moving picture out of a series of "stills" of the matrix, so transformations have the effect of making a moving picture out of a series of "stills"--intermediate phrase-markers--of the tree. Again we have the notion of rules as an attempt to overcome the limitations of a two-dimensional model. The intermediate phrase-markers which the addition of rules to the grammar makes possible are related to each other in sequence, as phases, so that the model then requires that mutually exclusive representations of the explicandum-- for this is what different stages in the derivation of a given sentence in fact are--be held simultaneously in the mind. (The dichotomy between deep and surface structure itself, fundamental to transformational-generative theory, makes the same demand: that two different characterizations of the datum be considered simultaneously.) Augmenting the model by the addition of rules, then, can be seen as an attempt to get over into a third dimension; and the elaboration of the concept of rule by the notion of the transformational cycle appears in this context as a further implementation of the concept of movement, imposing, as it were, movement upon movement, so that the rules that create a moving picture of the tree are themselves in motion.

Beyond such modifications of theory as these, suggested by the exploration of the neutral analogy presented by the figure itself, the model-on-paper, are other modifications that proceed from the metaphor attached to the figure. Such components of transformational-generative syntactic theory as "embedding," "pruning," "chopping" and "grafting," preservation of the "root," substitution of "vines" for trees-- all are culled from the metaphor of the tree.

"Embedding" of sentences is postulated to account for the phenomenon of recursion. Its metaphor is the grafting of a shoot--really a tree in microcosm, after all. Like Pike's multiplied matrix, it is a sort of kangaroo diagram :

It is replication according to the logic of the figure. We might call it a multiplied tree: interpolation at a node is to the tree what interpolation at a cell is to the matrix.

"Pruning" (Ross 1969) is justifiable on graphic grounds, it is true. This configuration,

(Ross 1969:297)

in which a node S dominates a non-sentence, is less elegant than this one:

(Ross 1969:298)

But if the concept of pruning is based on graphic considerations, its name comes from the metaphor.¹⁵

So also "chopping" and "grafting" (Ross 1967) . Chopping is removing a construction or part of a construction, which is then grafted onto the same tree somewhere else.

To the "root" of the tree corresponds the initial S of a phrase structure tree diagram (Emonds 1970) --its unique beginner. Here the analogy is clearly with the metaphor rather than the figure, inasmuch as the tree in transformational-generative grammar is upside-down, and can hardly be considered to have its root at the top.

A contrast to these notions, which merely add to the theory as it stands, is the replacement of trees by "vines" (Morin and O'Malley 1969). The vine derives from the tree; but, unlike the tree, it may--though it need not necessarily--be multirooted. The following are all examples of vines:

(Morin and O' Malley 1969:182)

A tree with more than one beginner, the vine is offered as a better fit between the representation and the thing represented. The sentence I accuse you of mailing the letter (183), for example, requires two disjoined trees,

(Morin and O'Malley 1969:183)

but a single vine,

(Morin and O'Malley 1969:183)

It is at once a more compact structure and one better suited to the representation of the datum: for a single sentence, a single diagram. (In this respect, the vine is to the tree what the tetrahedron is to the matrix, each being a more capacious model, accommodating a more complex datum in a single diagram.) The figure has suggested its own modification in the course of exploring the neutral analogy. Like the limiting of features in the matrix, the limiting of beginners in the tree is relegated to the negative analogy; and with the removal of this limitation, the tree, like the matrix, gives way to a different figure.

Nevertheless, vines, like roots, chopping and grafting, and pruning, are named from the metaphor of the tree. And who knows what further elaboration of linguistic theory lies in that metaphor? We shall look more closely, therefore, at the process by which the metaphor is taken literally: the exploration of the neutral analogy. Graphic representation functions for a linguistic theory as a model; in the case of the tree, the model is attended by a metaphor. To the extent that the metaphor replaces the figure as bearer of the neutral analogy, it is a model once removed. The encroaching realism that results from giving prominence to the metaphor nearly equals that of the diagram on the following page for the languages of the world. This tree recalls Schleicher's first version, which he later discarded, the sioh verästelnde Baum. The model that furnishes the

(Von Ostermann and Giegengack 1936)

neutral analogy is no longer a figure, no longer schematic; it is a real tree. Replacing a figure with its metaphor may not in itself lead to unwarranted development of theory. But if the neutral analogy as it unfolds is then assigned all to the positive analogy, none to the negative, it is not surprising that aspects of the resulting theory have little to do with the explicandum, English sentences.

Even without a metaphor, carrying off all of the neutral analogy into the positive analogy can result in unwarranted development. We see this in such aspects of the comparative method as the abruptness, completeness, and simultaneity of separation posited for related languages (Bloomfield 1933:318); these notions seem to proceed from the figure itself, the graphic design. Thus all Schleicher's care to avoid a too-suggestive figure has availed little. The substitution of a more schematic tree perhaps stayed the metaphor from the course it took in synchronic linguistics; but, if so, removing the metaphor as the object of literal interpretation did no more than make room for the figure itself.

Graphic representation cannot lead to warranted development, then, unless the notions it suggests are checked against the explicandum. Checking them against the figure is begging the question. But even if this is done, as with the notion of vines, the fact that the name of a concept derives from the metaphor is unfortunate. Once settled on such a name, the temptation is to develop the concept itself on grounds, not of grammar, but of botany. Or take the notion of pruning: it is justified metaphorically; it is justified graphically; but unless it is justified linguistically, the bridge between analogue and explicandum has not been built. This is what it means to explore the neutral analogy offered by a model. A good figure has a wild civility; it cannot otherwise be suggestive; the problem is to keep its suggestiveness within bounds. Thus, for the tree, Darwin at the outset (1859:116-130) carefully charts the elements both of metaphor and of design. Each graphic element--node, branch, and so on--is accorded a meaning, to which limits are set. For scientific theory requires a model with room to grow in, but not without all semblance of fit.

Because a figure is assumed to have been tailored to its theory, the dangers of model-making are greater if the analogue is graphic representation. Martinet warns (personal communication) that a diagram is likely to "harden״ the pattern represented, though actually parts of it are "softer" than others; that the compulsion is strong to let representation replace reality. The result is a random harvest of notions like vines and roots, a perhaps harmlessly bizarre nomenclature that nevertheless interposes itself between theory and explicandum.¹⁶ Because a figure can be tailored to its theory, what we have said about graphic representation has consequences for its design. We have considered linguistic theory as a product of graphic representation; now we shall consider graphic representation as a product of linguistic theory.