Box diagrams

Besides tree diagrams and matrix diagrams, there is another sort of diagram in linguistics that is not so easily catalogued. Box diagrams are a more elusive quarry, for what we shall group under the heading of box diagrams are really three different figures. They are usually called cubes, Chinese box diagrams, and block diagrams; they are constructions making use of shape or area rather than (like tree and matrix diagrams) of line. Their provenience is less interesting than the provenience of tree and matrix diagrams; and it is various. We shall therefore leave it to be touched on in connection with each of the three figures.

Meaning

The meaning of box diagrams is the relation of parts of a system to each other and to the whole. They are structures built of rectangles (see following page). Each kind of box diagram assembles the parts--rectangles--into a whole in its own way. The relation of areas to each other in space expresses the relation of parts in a system. This most abstract meaning, the meaning of box diagrams as a graphic design, is converted in any given instance into one of three more specific meanings, depending on how the graphic elements of the design are construed.

Two of them we have met before: componential analysis and constituent analysis; the third meaning we shall call conflation.

Componential analysis in a cube

As for its provenience, Forchhammer employed the cube, in what was probably its first appearance in linguistics, as early as 1924 (Forchhammer 1924:42). Gleason's cube for the Turkish vowel system, however, is for our purposes a more representative box diagram for componential analysis:

(Gleason 1961:267)

Cube diagrams are constructed of rectangles in such a way as to convey a three-dimensional figure, and in this they are unlike any figure we have looked at so far. Their illusory depth--perspective--carries meaning. Gleasonfs diagram resolves into six squares (see following page). Illusory depth reassembles the six areas into a whole. We view the figure as six congruent shapes projected to three different planes. Vowel height is represented by one plane; rounding, by another; place of articulation, by a third. The opposite values of a component--[+High] versus [+High], for instance--are on opposite sides of the threedimensional figure. This establishes a set of axes:

The similarity of the six sides now reflects the uniformity of the stuff of the figure, phonetic

components; the difference in plane reflects the distinctness of phonetic components one from another. The opposition of two sides in a plane reflects the polarity of the two values of a component, plus and minus.

Plane opposes plane to express components: pole opposes pole to express values of components. This is perhaps more evident in Hockett's version of the diagram for Turkish vowels:

(Hockett 1958:96)

Here the set of points sharing one and only one component,

can be spanned, without travelling along the axes, by a single square. The set of points sharing no component,

can be spanned only by travelling along all three axes. The greatest distance between two points corresponds to the greatest difference between two units in the system.

Like the matrix, then, the cube expresses the pattern as well as the inventory of a sound system. The two figures are partly isomorphic. The cells of the matrix are the corners of the cube. The rows and columns of the matrix are the sides of the cube. The dimensions of the matrix are the dimensions of the cube--and therein lies the difference between the two figures. While the matrix represents a unit as the intersection of two dimensions, the cube represents it as the intersection of three: the matrix analyzes a unit as the product of two features; the cube, as the product of three. We may therefore expect the cube to replace the matrix when the componential analysis of a system requires three features rather than two.

This is not quite what happens in Jakobson's (1958) componential analysis of the Russian case system. The three features are: [±Directional] (napravlennost'); [±Quantitative] (objemnost') ; [±Peripheral] (periferijnost')· A grid for Jakobson's first version of a distinctive-feature analysis of the Russian case system, in which the number of cases is six, would look like this:

Two more cases--the "second genitive" (Gen₂) and the "second accusative" (Prep₂)--are then added. A grid for this second version, in which the number of cases has grown from six to eight, would look like this:

For eight cases, the two "holes" in the grid are filled in, so that all features are specified for all cases. With this change in inventory comes a change in pattern. For a six-case analysis, the pattern in which the cases stand can be represented by a matrix. The matrix underlying Jakobson1s analysis is:

But when the number of cases grows from six to eight, so that all three features are distinctive for every case, the matrix is stretched and pulled into a cube:

(Jakobson: 1963:149 [Roman letters replace Cyrillic])

There is no change in the number of features, only in the number of features that are distinctive in every case.

A three-feature analysis, then, is necessary but not sufficient for the matrix to give way to the cube. Is it that the three features must be distinctive for every unit in the system? Must there be no unused feature combinations (properly speaking, permutations)--no empty cells in either a matrix or a grid for the system? But this is not a sufficient condition either; for in cubes like this one,

(Austerlitz 1959:105)

there is clearly one combination of features, at any rate, that goes unused, and that is the one that would occupy the missing corner. In the cube, as in the matrix, empty corners are as meaningful as full ones, though there are limits to this. Jakobson's first version of the Russian case system might, it is true, have been shown in the fashion of Austerlitz's diagram:

But this looks odd: it has an uneasy disproportion.

Again like the matrix, the cube can be made to accommodate, within the limits set by the componential analysis it represents, more sounds than it necessarily shows. Unlike the matrix, however, the cube at first sight looks like a different figure when it is so expanded. Jespersen1s representation of the Danish vowel system, for example,

(Jespersen 1934:104)

appears to be a double cube. It is actually the expansion of a binary analysis to n-ary. "Stretching" the cube accommodates three degrees of vowel height instead of two. The cube is, in fact, infinitely elastic. Its expansion, mathematically formalizable, can accommodate a theoretically infinite number of feature values, and therefore of sounds (see figure from Kay and Romney, following page). But it cannot increase the number of features by a single one.

The limit is set by the design. Because it is a three-dimensional configuration, the figure itself sets the number of features at three and only three. Its expansion in no way changes the cube into another figure; the expansion of the

(Kay and Romney 1967:8)

number of possible feature-values in no way changes the three-feature analysis.

Constituent analysis in box diagrams

The meaning of constituent analysis is expressed by two different box diagrams: Chinese box diagrams and block diagrams. The Chinese box, named by Francis (1958:293), looks like this:

(Francis 1958:312)

Of its provenience, Francis says (personal communication) that it derives from the representation for immediate constituent analysis used by Fries (1952),

Chinese boxes are constructed of rectangles in such a way as to show the relationship of inclusion by graphic inclusion. The inclusion of one element in another--a constituent--and the inclusion by one element of another--a construction--are shown as the inclusion of areas in space. One rectangle is seen as containing or contained by another, allowing

The second sort of box diagram for constituent analysis is almost, but not quite, a Chinese box.

(Hockett 1958:158)

This is what we have called, following Nida (1964), a block diagram. That it, too, is constructed of rectangles is more easily seen from Hockettf s "empty box" for the same sentence:

(Hockett 1958:159 [suprasegmentals omitted])

A construction-- new hat, for instance-- is represented by means of areas related in space. This relatedness of areas in space is accomplished, not by the illusion of depth, as in the cube, but by exploiting the linearity of the two-dimensional page. We begin with six elements, she, bough-, t, a, new, and hat. These are then progressively amalgamated by welding two areas into a single area. So new and hat become new hat, which is thereupon joined by a; and these two are amalgamated into a new hat; and so on.

The serial nature of the block diagram recalls the tree.

And it is true that the two are isomorphic. Nodes correspond to rectangles; branches, to the size and position of rectangles:

Moreover, the serial nature of block diagrams is such that they, like tree diagrams, can be made to convey explicitly the notions of substitutability:

(Gleason 1961:130)

and constituent function:

(Nida 1964:38)

None of this, however, alters the fact that block diagrams are graphically box diagrams, and not trees at all. They are close kin to Chinese boxes. The representation from which the Chinese box derives,

can with just as little tinkering serve to derive the block diagram:

This time it is not a matter of filling in the tops but rather of making boundaries common boundaries, so that abutting rectangles share a boundary. The two sorts of box diagram used for constituent analysis, then, are similar figures with the same origin. Block diagrams for the meaning of conflation have a different origin.

Conflation

The block diagram for the meaning of constituent analysis, as we have seen, can be matched with a tree. There is, as well, a block diagram for the meaning of conflation, which can be matched with a matrix.

(Jakobson 1958:136-140 [Roman letters replace Cyrillic])

These figures display patterns of case conflation against an "understood״ matrix.⁷ This is the matrix proposed earlier,

This matrix, as we have said, expresses a componential analysis of the case system of Russian, with three distinctive features, [±Directional], [±Quantitative], [±Peripheral]. Comparison with an understood matrix is what gives conflation diagrams their meaning. The position and shape of the rectangles (of course, not all the blocks in such diagrams are, properly speaking, rectangles) signals the flowing together of certain formerly distinct entities. To show conflation requires the graphic elements not only of the block diagram, but also of the matrix. Rectangles are plotted on the matrix as the aggregate of certain cells (the area encompassing the Prepositional Locative ending is not the same for Jakobson's Figure 1 as for Figure 2, showing that it is conflated with different case endings in the two systems). Empty space signifies absence just as in the matrix (the Accusative ending does not occur for the paradigm represented by Figure 3). The boundary of the conflation diagram is not significant, but boundaries between cells are: removal of the boundary between abutting cells indicates irreversible conflation (of Accusative with Nominative, in Figure 2); a permeable boundary--shown by dotted lines--indicates a case subsumed now under one, now under another ending (the Accusative in Figures 3 and 4).

The block diagram for conflation, then, is composed of figure (the conflated system) and ground (the implied matrix for a full system). Unlike the block diagram for immediate constituent analysis--or the matrix itself, for that matter--the block diagram for conflation is not static. The contrast between the conflated system and the full system gives conflation diagrams an ineluctable element of process, of change. Pike's diagram for suspicious pairs in phonological analysis perhaps shows this more clearly (see following page). Sounds enclosed in a ring are suspicious--that is, they are likely to be allophones of a single phoneme; if they are, they are conflated. Pike's diagram is a sort of blueprint of all possible conflated systems that jnight occur, given this particular full underlying matrix.

(Pike 1947:70)

The full system expressed by the implicit matrix that underlies a conflation diagram need not, strictly speaking, be an earlier stage of the conflated system. It can be simply the fully-specified system of contrasts implicit in the feature analysis itself:

Here egoconflates the two theoretically possible feature combinations [1₁ S₁ g₃] and [1₁ S₂ g₃]; uncle, aunt, nephew, and niece each conflate two possible combinations; cousin conflates ten. The underlying full matrix, here as in Jakobson's diagrams, is easily inferred from the conflated matrix because it is implicit in the feature analysis.

In the taxonomy of diagrams in linguistics proposed here, the correspondences between form and meaning are seen to be far more complicated than might have been supposed:

The relationship between form--figures--and meaning--analytic techniques--is, as we have said, far from isomorphic. All of the figures are synonyms for some meaning: the tree, the matrix, the grid, and the cube, for componential analysis; the tree, the Chinese box, and the block diagram, for constituent analysis. Conversely, two of the figures are homonyms: the tree, for genesis, taxonomy, componential analysis, and constituent analysis; the block diagram, for constituent analysis and conflation. Synonymy and homonymy tangle the correspondences between form and meaning. This, then, is the situation in linguistics at present. What is its significance for linguistic science?