¹ P. Duhem: “…une deduction mathématique n'est pas utile au physicien tant qu'elle se borne à affirmer que telle proposition, rigoureusement vraie, a pour conséquence l'exactitude de telle autre proposition. Pour ětre utile au physicien, il lui faut encore prouver que la seconde proposition rest à peu près exacte lorsque la première est seulement à peu près vraie” (La Théorie Physique, p. 231).

² Plato: “Those who study geometry and calculation … use the visible squares and figures, and make their arguments about them, though they are not thinking about them, but about those things of which the visible are images. Their arguments concern the real square and a real diagonal, not the diagonal which they draw, and so with everything. The actual things which they model and draw … they now use as images in their turn, seeking to see those very realities which cannot be seen except by the understanding.” (Republic, 510—Lindsay's translation.)

³ “Toute loi physique est une loi approchée; par conséquent, pour le strict logician, elle ne peut ětre, ni vraie, ni fausse.” (Loc. cit. p. 280.)

⁴ “Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen sind sie nicht sicher, und insofern sie sicher sind, beziehen Sie sich nicht auf die Wirklichkeit.” (Geometrie und Erfahrung, p. 3.)

⁵ “All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life, but only to an imagined celestial existence.” (Vagueness, Australasian Journal of Philosophy, Vol. 1 (1923), p. 88.)

In this paper Russell contends that “all language is more or less vague.” Again the “laws of Excluded Middle is true when precise symbols are employed but it is not true when symbols are vague, as, in fact, all symbols are.” (Ibid. p. 85.)

⁶ N. R. Campbell, Measurement and Calculation, p. 131.

⁷ Baldwin's Dictionary of Philosophy and Psychology, II, 748.

⁸ In this paper reference will always be made to the vagueness of a word or symbol, but no important difference is involved in speaking of a proposition's vagueness. The proposition can be regarded as a complex symbol and its vagueness defined in terms of that of its constituents, or vice versa.

⁹ In the remainder of the passage Peirce explains that by an indeterminacy of habits he means the hypothetical variation by the speaker in the application of the proposition, “so that one day he would regard the proposition as excluding, another as admitting, those states of things.” But the knowledge of such variation could only be “deduced from a perfect knowledge of his state of mind; for it is precisely because these questions never did, or did not frequently, present themselves, that his habit remained indeterminate.”

¹⁰ H. G. Wells, First and Last Things, p. 16.

¹¹ E. Rignano, Psychology of Reasoning, p. 109.

¹² Cf. B. A. W. Russell “A vague word is not to be identified with a general word” (Analysis of Mind, p. 184). He adds, however, “that in practice the distinction is apt to be blurred” and blurs it himself in saying “a memory is vague when it is appropriate to many occurrences” (Loc. cit. p. 182). This confusion between generality and vagueness invalidates his neat definition “the fact that meaning is a one-many relation is the precise statement of the fact that all language is more or less vague.” (Vagueness, p. 89.)

The confusion may ultimately be traced to a certain uneasy nominalism in Russell's philosophy which tends to treat generality and vagueness indifferently as imperfections of symbolism in relation to the attempt to describe a universe composed exclusively of absolutely specific or atomic facts.

¹³ The variation of this amount with the choice of the observer, and with conditions affecting the same observer, strengthens the subsequent argument by introducing further indeterminacy into the operation of “drawing the line.”

¹⁴ This is, in part, a definition of the “normal” observer; we shall reject the testimony of an observer who claimed to have discovered the point at which the division was to be made. Cf. section 7 for a fuller discussion of this point.

¹⁵ See section 5, below.

¹⁶ Cf. the experiment described in Appendix I when the subject agrees beforehand to make a unique division, the “inducement” being desire to keep his word, or curiosity, or some other motive.

¹⁷ Stout, Manual of Psychology, p. 160. This manner of phrasing the situation suggest of course that the fault is in the language or in imperfect perception: there is an “exact point” where the transition occurs but we are unable to find it.

¹⁸ See Appendix II.

¹⁹ Russell assumes an infinite series of doubtful regions, each fringe having a fringe of higher order at its boundary, but does not pursue the consequences of this assumption.

²⁰ The hypothesis of an infinite series is considered later in this section.

²¹ Cf. M. Black, “The Claims of Intuitionism,” The Philosopher, July 1936.

²² This is Russell's assumption in the paper to which reference has already been made.

²³ The argument would need trivial adjustments if the field of reference, while having an infinite number of terms, did not constitute a continuum.

²⁴ Op. cit., p. 88.

²⁵ Op. cit., p. 88.

²⁶ Cf. section 9 below when Russell's argument on this point is further discussed.

²⁷ Cf. article on “Objective,” Baldwin's Dictionary of Phil. and Psych. II, 192.

²⁸ The qualification is necessary because some languages such as Chinese use differences of pitch as significant linguistic elements.

²⁹ Since O's state of mind may depend upon his environment, evidence about O's state of mind, may of course, indirectly yield evidence also about E.

³⁰ In practice this can never be completely successful.

³¹ It needs, therefore, to be clearly distinguished from such features of symbolism as ambiguity. The latter is constituted by inability to decide between a finite member of alternative meanings having the same phonetic form (homonyms). The fact that ambiguity can be removed shows it to be an accidental feature of the symbolism. But any attempt to remove vagueness by a translation is defeated by the over-specification of meaning thus produced. Cf. an attempt to replace The hall was half full by The ratio of the number of persons in the hall to the number of seats was exactly half. The presence of one person too many would falsify the second, but not the first of the statements.

³² In ordinary language, vagueness is shown explicitly by the use of adverbs of degree- or number such as any, many, rather, almost, etc. These serve as a set of pseudo-quantifiers, generalisations as it were of the “respectable” quantifiers all and any, forming a slide ing scale which can be attached to any adjective. The method of the next section, which reduces to the conversion of propositional functions into propositional functions of an extra variable by the addition of a numerical parameter is thus the generalisation of a device already present in ordinary discourse.

³³ The “set of conventions” determining the vocabulary and syntax of such a language are the simplified expressions, in the imperative mood, of the empirically discoverable rules of usage. While the existence of such a language presupposes, by definition, some uniformity in the linguistic habits of its users, the empirical laws expressing the partial uniformity of such habits are complex, in process of variation, and heterogeneous in character. It is necessary to distinguish between rules of logic, grammar and good taste. The neglect of certain distinctions and discriminations habitually made by users of the language provides a simplified or “model” language bearing some, but not too much, resemblance to their actual habits. Then the first crude analysis can be corrected by a supplement which considers the facts neglected. Thus the definition proceeds by a series of successive approximations.

³⁴ Cp. Russell, Vagueness, “we are able to conceive precision; indeed if we could not do so we could not conceive of vagueness which is merely the contrary of precision” (p. 89).

³⁵ The final term in any case differs from its predecessors in some respects. The situation is indeed complicated in physics by the existence of a multiplicity of different methods for measuring length in accordance with the familiar tendency of a science to extend the meaning of a concept by the assimilation of new methods of measurement as they arc discovered. But consideration can be restricted to length measured by a ruler (the case of length ‘in general’ produces no difference in principle) and we can imagine this phrase substituted for length in the text.

³⁶ Cf. Appendix II for further details.

³⁷ On a frequency theory of probability the assertion of the value of C(L, x) for a given argument x could be interpreted as an assertion concerning the probability of L's application to x. This would involve interpreting every statement of the form Lx as statements of probability lacking a numerical parameter. Such a theory bears a formal resemblance to the theories of say Keynes or Reichenbach (Wahrscheinlickheitslehre). But the argument in the text is independent of any particular interpretation of probability.

³⁸ L. Bloomfield,Language, 1935, p. 51.

³⁹ Bloomfield, cf. cit., p. 44.

⁴⁰ Actually to do this would seem to involve examination of the limiting behavior of each sub-group. The practical difficulties this would involve and the subsequent modifications required in practice in determining the uses of a language in any specific case need not be considered in a discussion of the principles.

⁴¹ Thus if we adopt a recent definition of species in genetic terms as a “group of individuals fully fertile inter se, but barred from interbreeding with other similar groups by its physiological properties (producing either incompatibility of parents or sterility of the hybrids or both)” (T. Dobzhansky, ‘Critique of the Species Concept in Biology,’ Phil. of Sci., II, 353) we are compelled to admit the qualification that “neither the mechanisms providing incompatibility, nor those producing sterility, function on an all-or-none principle. For instance, sexual isolation may be incomplete, and individuals belonging to different groups may sometimes, though seldom, copulate. Similarly, some hybrids arc only semi-sterile or sterile in one sex only” (Ibid.). Nor is the situation fundamentally altered if a genetic definition of this sort is replaced by a taxonomic definition in terms of the possession by the members of the species of certain common characteristics.

⁴² In fact, of course, there are an indefinite number of criteria which might be applied; those described are merely the most ‘natural’ i.e. those which most people tend to use.

⁴³ Any tendency to make a division in the middle of a series could have been avoided by prolonging both ends of the series a considerable distance or by using a series pasted round a cylinder.

⁴⁴ The limiting process and the suppression of the argument are assumed to produce equivalent effects.

⁴⁵ If this principle of transformation is itself formalised (corresponding to the use of the law of excluded middle as a premiss as well as a logical principle) it will be necessary to introduce a further consistency variable. The generalisation of the assertion (X) (Px v ~ Px) will then be

Where f f is some specified function of c (nearly equal to c when C is near to 1, very small when c is large and very large when c is small. The exact form of f(c) would depend on the exact form of the consistency curves in the special case.)

⁴⁶ Since such diagrams habitually assume a two-dimensional field of application, the corresponding diagram of consistencies of application should strictly be three-dimensional and consist of a (polyhedral) surface obtained by joining the top of adjacent ordinates erected not upon an axis (OS in Fig. 2) but upon a plane of reference. The argument of the section can, however, be sufficiently illustrated by supposing that the field of application is a one-dimensional series as in Figs. 1-3 above.

⁴⁷ i(L, M) might be, for example, the ratio of the number of x's when M's consistency is greater than L's to the number when M's consistency is less than L's. The exact definition which is chosen is unimportant.

⁴⁸ This means roughly speaking that it tends to zero when the degree of vagueness of all the premises tend to zero.