Vagueness and Logic Carl G. Hempel Philosophy of Science, Vol. 6, No. 2. (Apr., 1939), pp. 163-180. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28193904%296%3A2%3C163%3AVAL%3E2.0.CO%3B2-Q Philosophy of Science is currently published by The University of Chicago Press.
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Vagueness and Logic BY
CARL G. HEMPEL S IS rather generally admitted today, the terms of our language in scientific as well as in everyday use, are not completely precise, but exhibit a more or less high degree of vagueness. It is the purpose of this paper to examine the consequences of this circumstance for a series of questions which belong to the field of logic. First of all, the meaning and the logical status of the concept of vagueness will be analyzed; then we will try to find out whether logical terms are free from vagueness, and whether vagueness has an influence upon the validity of the customary principles of logic; finally, the possibilities of diminishing the vagueness of scientific concepts by suitable logical devices will be briefly dealt with. As starting point for the subsequent considerations we choose the clear and stimulating analysis of the concept of vagueness which has recently been carried out by Max Black ( ( I )),I and which has suggested the considerations of this paper. Distinguishing vaDueness from generality and ambiguity, 9
Mr. Black characterizes the vagueness of a symbol by "the existence of objects concerning which it is intrinsically impossible to say either that the symbol in question does, or does not, apply" (1 I ), p. 430). "Thus a word's vagueness is usually indicated . . . by some statement that situations are conceivable in which its application is 'doubtful' or 'ill-defined', in which 'nobody would 1 Numbers
in curved brackets refer to the bibliography a t the end of this paper.
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Vagueness and Logic
know how to use it'. . . " ( ( I ), p. 431). Therefore, Mr. Black states, "the vagueness of a word involves variations in its application by the users of the language in which it occurs" ( f I 1, p. 442) This observation leads him to a very interesting proposal for the numerical determination of the vagueness of a term by means of what he calls the consistency of application C(T, x) of a term T to some object x. Let, for example, the symbol be the word "plant" and the object one of those deep-sea organisms which belong to the border zone between plants and animals. If several observers are asked whether the term "plant" does or does not apply to that object, there will be a certain number m of affirmative answers and a certain number n of negative ones. The consistency of application of the term "plant" to that object is then defined by Mr. Black as the limit of the ratio
my when the group of observers n
is more and more extended, and the number of decisions made by its members is indefinitely increased.-If several objects are given to each of which a certain vague symbol T may be applied, then the objects may be put into a linear order in which the consistency of application of T steadily decreases; the series thus obtained will begin with those objects to which all or most observers apply the term T (i.e. for which C(T, x) has a large value), and it will end with those objects to which all or most observers apply the negation of T (for these objects, C(T, x) is small); the middle section will contain the >> >random variations>speakers>apply to the given load the term "60 kgs9'assert that that term does not applyanswers to the question as to whether the term "60 kg" applies to that load>some group of users of the symbol>inessentialabstraction > is a noun and "------ an adjective, designates to a certain degree the state of affairs that the designatum of the noun has the property designated by the adjective; and the degree is the product of the degrees to which the noun and the adjective designate their designata. Accordingly, the sentence "sol esti cal" would designate, to the degree 0.63, the state of affairs that the sun is hot. As a comparison with the preceding example shows, this sentence would not admit of a translation into English; in particular the statement we made about its designatum, does not furnish such a translation. Thus we see that if a theoretical linguistic system is built up by means of a gradable relation of designation, the language thus arrived a t is of a very strange kind; it is not an interpreted language in the usual sense. Therefore, there is no place for a gradable concept of designation and hence no place for a purely semantical concept of vagueness either, when a system of syntax and semantics is to be abstracted from a natural language, or when an artificial language is to be constructed which is to serve as an interpreted linguistic system for communicating empirical contents. Thus the question as to the influence of vagueness upon the validity of the principles of logic does not arise on the purely syntactical and semantical level of investigation, and no modification of the logical symbolism is necessary. 99
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8. There is, however, another respect in which the vagueness of the terms of a given language may suggest a modification of its logical structure. As has been pointed out in section 6, vagueness is, so to speak, an ineradicable feature of any interpreted language; it can never be completely suppressed. But vagueness (in the strictly semiotic sense explained in section I ) may assume higher or lower degrees, and there are certain ways of modifying a given language in such a way that its vagueness in practical use is decreased. As vagueness obviously is a serious obstacle in establishing hypotheses and theories which are intersubjective, i.e. which may be tested, with the same result, by different observers, it is particularly important to diminish as far as possible the vagueness of scientific terms. The method which is applied for this purpose, consists in that a vague term which has so far been used as an undefined concept or has been explained by means of other comparatively vague terms, is defined (or redefined) in terms of less vague concepts. There is one form of this procedure which deserves special mention here. I t consists in reintroducing such concepts which have originally been used as names of properties, in such a way that they become names of relations. Take, for example, the concept "hard" as applied to physical materials. In everyday language, this term is used as the name of a property which does not have a precise definition. I t will be said to apply to granite and not to cold cream at room temperature; but there will be many >>doubtful cases
