LINGUIST List 30.2146

Wed May 22 2019

Sum: Matrices in Linguistics

Editor for this issue: Everett Green <>

Date: 18-May-2019
From: David Denison <>
Subject: Matrices in Linguistics
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My question (LINGUIST List 30.1801, Sat Apr 27 2019) was prompted by Randolph Quirk's (1965) use of matrices of pluses, minuses and question marks to describe and perhaps explain similarities and differences of morphosyntax between linguistic items. I'm planning some work on the ideas there. Meanwhile, in case of interest to LINGUIST readers, here's my current understanding of what came before it. Thanks to Peter Matthews (private communication before I posted to LINGUIST), Magdalena Zoeppritz, Bruno Maroneze, Herb Stahlke, Mareike Keller and Dick Hudson for their helpful responses; further information or correction to this summary would be welcome.

Tabular arrays are a convenient, economical descriptive device. According to Dick Hudson, 'I don't think you get tables, at least not in printed grammars, till the nineteenth century because printing technology couldn't cope with straight vertical lines'. Often a linguistic table is populated with a direct representation of the item or property concerned, but space will be saved if item and property define the rows and columns and the table cells are filled only with '+' and '-' symbols (or equivalently, 'x' vs. blank) to indicate whether or not a given item has some property, possibly with the option of '?', etc. Such tables may be called 'matrices', but most linguistic tables don't count as matrices in the mathematical sense of a rectangular array of numbers or algebraic symbols potentially subject to addition, subtraction, multiplication and other transformations. Genuinely algebraic matrices appeared in the work of Pike in the 1950s, as also now in the recent 'matrix syntax' of Juan Uriagereka and colleagues (Orús et al. 2019; reference due to Mareike Keller).

Distinctive features in phonology go back to Trubetzkoy (1939), according to Magdalena Zoeppritz, and visual inspection of the reprint shows arrays of phonetic symbols (e.g. 1939: 153) where the rows and columns represent shared features; I don't see pluses and minuses. I haven't yet had a chance to check the first edition of Jakobson, Fant & Halle (1952), but the 3rd edition presents such distinctive feature matrices (1965, 6th printing: 43ff.). Jakobson & Halle (1956) has distinctive fatures, but no matrices either by name or in tabular form (to judge from a reprint of the 2nd, rev. edn.). The term 'distinctive feature matrix' occurs in Halle (1959: 34, etc.), with pluses, minuses and zeros, and recurs in work of the mid-sixties by Halle and by Chomsky. One column of (almost always) binary features represents a segment. That also appears to be the usage in Chomsky & Halle (1968). Multi-column 'matrices' are used for two distinct purposes: (i) to represent a lexical item, where the relationship between columns is the linear sequence of segments, and (ii) for a tabular presentation of (some of) the phonemes found in a particular language variety, displaying relationships of similarity and contrast.

In semantics, componential analysis is a structural approach modelled 'on the phonological methods of the Prague School' (Wikipedia), of whom Trubetzkoy and Jakobson, among others, were members, and it can use the same kind of display. Among the few tables in Lyons (1963: 112), one (1963: 180) has '1' and '0' to indicate presence or absence; in principle that would allow for mathematical operations on it qua matrix. Goodenough (1956) uses formulas for set intersection and union but not, apparently, +/- tables. Pace Bruno Maroneze, +/- tables are not found in Pottier (1964) either, though that work does have tabular representations of semantic and other properties (e.g. 1964: 115) among a variety of other kinds of table and figure. Lamb (1964: 74) has one +/- table.

By 1965, then, such tables or matrices are definitely 'in the air' in several branches of linguistics, but it's morphosyntax that Quirk was doing, and the most pertinent forerunner of all is Kenneth Pike. Herb Stahlke pointed me to Pike (1959), which employs 'multiplication matrices', with combinatorial properties that are truly mathematical. Even closer (and actually cited by Quirk 1965: 209 n.9) is Pike (1962), which has matrices of syntactic properties that -- as a presentational device, at least -- may well have stimulated Quirk's discussion in the same journal a few years later. In the same footmote, Quirk also cites Pike's follower Longacre (1964). Longacre's has x-blank tables (1964: 60, 64, 120), which he calls 'matrices'. This, then, seems to be the context in which Quirk was writing.


Brend, Ruth M. (ed.). 1972. Kenneth L. Pike, Selected writings: To commemorate the 60th birthday of Kenneth Lee Pike. (Janua Linguarum, series maior, 55.) The Hague and Paris: Mouton.
Chomsky, Noam & Morris Halle. 1968. The sound pattern of English. New York: Harper and Row.
Goodenough, Ward H. 1956. Componential analysis and the study of meaning. Language 32, 195-216.
Halle, Morris. 1959. The sound pattern of Russian: A linguistic and acoustical investigation.
Jakobson, Roman, Gunnar M. Fant & Morris Halle. 1952. Preliminaries to speech analysis: The distinctive features and their correlates. (Massachusetts Institute of Technology, Technical Report No. 13). Repr. 3rd edn., 1965.
Jakobson, Roman & Morris Halle. 1956. Fundamentals of language. 2nd, rev. edn, 1971. Berlin and New York: Mouton de Gruyter.
Lamb, Sydney M. 1964. The sememic approach to structural semantics. American Anthropologist 66.3 (special issue on Transcultural Studies in Cognition), 57-78.
Longacre, Robert E. 1964. Grammar discovery procedures: A field manual. (Janua Linguarum, series minor, 33.) The Hague: Mouton.
Lyons, John. 1963. Structural semantics: An analysis of part of the vocabulary of Plato. (Publications of the Philological Society, 20.) Oxford: Basil Blackwell for the Society.
Orús, Román, Roger Martin & Juan Uriagereka. 2019. Mathematical foundations of matrix syntax. arXiv 1710.00372v2. (
Pike, Kenneth L. 1959. Language as particle, wave, and field. Texas Quarterly 2.2, 37-54. Repr. Brend (1972: 129ff.).
Pike, Kenneth L. 1962. Dimensions of grammatical constructions. Language 38.3.1, 221-44. Repr. Brend (1972: 160ff.).
Pottier, Bernard. 1964 Vers une sémantique moderne. Travaux de Linguistique et de Littérature publiés par le Centre de Philologie et de Littératures Romanes de l'Université de Strasbourg 2, 107-137.
Quirk, Randolph. 1965. Descriptive statement and serial relationship. Language 41.2, 205-17.
Trubetzkoy, N. S. 1939. Grundzüge der Phonologie. (Travaux du Cercle Linguisticque de Prague, 7.) Prague. (

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