^* Paper is in the Department of Near Eastern Studies; Harary has a joint appointment in the Department of Mathematics and the Research Center for Group Dynamics. The analogy between phoneme sequences and ordered pairs was suggested by Paper; the mathematical concepts and numerical indices were suggested by Harary. The authors worked in close collaboration during the entire preparation of the article; but Paper did most of the writing, since the material is intended for an audience of linguists.

¹ M. Joos, Description of language design, Journal of the Acoustical Society of America 22.701–8 (1950).

² C. F. Hockett, Two models of grammatical analysis, Word 10.210–34 (1954), especially Section V. Mathematical Interlude, 225–7.

³ There are also several important articles by N. Chomsky dealing with the underlying mathematico-logical strategies of descriptive linguistics. These however were published in a form directed to logicians and mathematicians rather than to linguists. Chomsky's work is a significant contribution in bridging the gap between linguistics and mathematics. The authors wish to record their gratitude to him for his criticism of an early draft of this article.

See especially Chomsky's article Systems of syntactic analysis, Journal of symbolic logic 18.242–56 (1953).

Mention should also be made of J. Lotz, The semantic analysis of the nominal bases in Hungarian, Recherches structurales - Travaux du Cercle Linguistique de Copenhague 5.185–97 (1949).

⁴ Braces enclose members of a set.

⁵ Small epsilon means ‘is an element of’; thus epsilon between y and P means ‘y is an element of the set P’. A crossed epsilon means ‘is not an element of’. A vertical bar means ‘such that’.

⁶ For a diagrammatic representation of ‘union’, cf. Fig. la.

⁷ A convenient source for the mathematics employed here is R. L. Wilder, Introduction to the foundations of mathematics (New York, 1952) ; cf. especially 46–9 (for reflexive, symmetric, and transitive relations), and 57–62 (for set theory).

⁸ B. Bloch, Studies in colloquial Japanese IV: Phonemics, Lg. 26.86–125 (1950). Since we shall be referring to the Japanese data constantly, we have reproduced the two binary sequence matrices that appeared in Bloch's article (116, 123). All of the ensuing calculations based on Japanese data derive from the information provided by these two matrices (Tables 1 and 2).

⁹ In counting the number of phonemes that function as predecessors and as successors to each phoneme of Japanese, Bloch anticipated our ‘degree of alpha-completeness’ and ‘degree of beta-completeness’.

¹⁰ For the ‘alpha-field’ of x, read down each column in Tables 1 and 2. For the ‘beta-field’ of x, read across each row in Tables 1 and 2.

¹¹ For a diagrammatic representation of ‘intersection’, cf. Fig. lb.

¹² All tables are assembled at the end of the article.

¹³ For a diagrammatic representation of ‘symmetric difference’, see Fig. lc.

¹⁴ The double arrow means 'if and only if; the sign ⊆ means 'is contained in'.

¹⁵ B. Sigurd, Rank order of consonants established by distributional criteria, Studia linguistica 9.8–20 (1955). A specifically mathematical article written in conjunction with Sigurd's is fare for mathematicians: L. Gårding, Relations and order, Studia linguistica 9.21–34 (1955).

¹⁶ In fact, these theorems hold for any binary relation (set of ordered pairs) regardless of context. The relation may be linguistic, sociometric, numerical, abstract, etc.

¹⁷ Where nR equals the number of ordered pairs in R. (See formula 6.)

¹⁸ Note to Equation 11 : The degree of antisymmetry of a relation is equal to the number of antisymmetric ordered pairs in R divided by the total number of ordered pairs in R. The term is equal to the numerator of this fraction, namely the number of antisymmetric ordered pairs in R, since each antisymmetric ordered pair xy is counted twice (once for the predecessor phoneme and once for the successor phoneme).