¹ This division is the natural one for a linguist; it is necessary also for psychology; see A. P. Weiss, A theoretical basis of human behavior, second edition, Columbus, 1929, p. 307.

² It is necessary that we understand that writing is not “language,” but a device for recording language utterances. The utterances that are made in any one language consist structurally of a finite number of recurrent units, and this on two levels. In the first place, the utterances in a language consist of various combinations of smallest units (words) that can be spoken alone. Of these there are in any one speech-community some tens of thousands. Hence a language can be replaced by some tens of thousands of unit signals (say, visual marks), each of which replaces the utterance of one word. This is the principle of Chinese writing. In the second place, the words in any one language consist of various combinations of a few dozen typical sounds (phonemes). Hence a language can be replaced by a few dozen unit signals (say, visual marks), each of which replaces the utterance of a phoneme. This is the principle of alphabetic writing. All writing is a relatively recent invention, whose use until yesterday, was confined to a few favored persons. A system of writing opens the possibility of graphic notations that cannot be successfully paralleled in actual speech. This is true because visual symbols possess characteristics that are foreign to the sound-waves of speech: chiefly, they provide an enduring instead of an immediately vanishing stimulus, and offer possibilities of arrangement (tabulation) that cannot be matched in the succession of acoustic stimuli. Graphic notations that cannot be matched in actual speech have arisen in the case of classical Chinese (which is unintelligible in modern pronunciation, but is read and written by Chinese scholars; cf. B. Karlgren, Sound and symbol in Chinese, London, 1923) and in the case which here interests us, of mathematical and allied notations.

³ A less elementary point of the same sort appears in the “heterological” contradiction: An adjective which describes itself is autological (e.g., short is autological, since the adjective short is actually a short word). An adjective which is not autological is heterological (e.g., long is not a long word). Is the adjective heterological heterological? If it is heterological, it describes itself and is therefore autological. If it is autological, it does not describe itself and therefore is heterological.—The fallacy is due to misuse of linguistic terms: the phrase “an adjective which describes itself” makes no sense in any usable terminology of linguistics; the example of short illustrates a situation which could be described only in a different discourse. E.g.: We may set up, without very rigid boundaries, as to meaning, various classes of adjectives. An adjective which describes a phonetic feature of words is morphonymic (e.g., short, long, monosyllabic). A morphonymic adjective which describes a phonetic feature of itself is autological. A morphonymic adjective which is not autological is heterological. The adjectives autological and heterological designate meanings of adjectives and not phonetic features; hence they are not morphonymic.—Contrast the following sensible discourse: A hakab is a word that ends in a bilabial stop (p, b). A word that is not a hakab is a cowp. The words hakab and cowp are hakabs.

⁴ The parallel phrase without not of the last example is There are berries over there. The plus of the word any in the example illustrates the special twists, often quite bizarre, which we find in different languages; in setting up rules for scientific speech, we exclude these features or try to make them innocuous. A striking example is the troublesome adjectival no of English (No cat has nine tails, etc.).

⁵ Inhibition is here a physiological term (Pavlov): all English-speaking persons have been trained so that hearing the word not inhibits them in certain definable ways.

⁶ This recognition appears in an admirable way, both explicitly and in the structure of the discourse, in O. Veblen and J. W. Young, Projective geometry, Boston, 1918.

⁷ On demonstration, see Weiss, op. cit., 21.

⁸ This is not strictly true; the numeral thirteen, for instance, has another phase of meaning in our superstitious response; however, these non-serial phases of meaning are easily eliminated for scientific use.

⁹ On vectors, see E. Study, Einleitung in die Theorie der Invarianten, Brunswick, 1923.

¹⁰ In its everyday meaning, the term continuous is not particularly descriptive of convention (2); it was chosen, of course, on account of the simple geometrical bearing of this convention.

Translated into everyday language, Dedekind's postulate amounts roughly to this: We shall use in numerical discourse any word or phrase which we can define in such a way that its before-and-after relation to any rational number whatever is unmistakably determined.—It is true that at the bottom of our discourse there lie undefined words which may arise to trouble us. However, our terminology of numbers is sound enough to prevent this in all but the most unusual situations, and even in these unusual situations a realistic diagnosis will show also the cure.

¹¹ In linguistics the term statement applies only to sentences of a certain type (John runs). When the phrase John runs appears as part of a sentence (If John runs, he will fall), then this phrase is not a statement. In logic, however, the term statement is used also of expressions which are part of larger expressions; the term is so used in III, above.

¹² For sentence-types, see L. Bloomfield, Language, New York, 1933, p. 170.

¹³ Ibid., 158.

¹⁴ Ibid., 194.