Notes on Identity Gustav Bergmann Philosophy of Science, Vol. 11, No. 2. (Apr., 1944), pp. 123-124. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28194404%2911%3A2%3C123%3ANOI%3E2.0.CO%3B2-T Philosophy of Science is currently published by The University of Chicago Press.
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DISCUSSION
NOTES ON IDENTITY
Sir : I t is the purpose of this note to discuss two comments Church (J.of Symb. Logic, 1943, 8, p. 86) has made concerning my article "Notes on Identity," which appeared in an earlier issue of your journal (1943, 10, pp. 163-166). 1. Church remarks that I distinguish calculational from non-calculational or descriptive signs on the basis that any interpretation of a calculus rests upon the interpretation of its symbols of the latter kind. K O Mit~ is my belief, and has been so stated, that the (epistemological) signijicance of the distinction has something to do with the circumstance mentioned, namely, that the semantical interpretation of a formalized language hinges upon the interpretation of its descriptive signs and(or) expressions. The distinction itself, however, that is, the distinction between logical and descriptive signs is purely syntactical. This, by the 1%-ay, is one of the important points in Carnap7sLogical Syntax of Language, and has been restated in two of my earlier papers (this journal, 1942, 9, pp. 283293; 372-374). There are, a t least in all epistemologically interesting formal languages which have been proposed so far,-and I , for one, cannot imagine any other-three categories of signs, (a) logical constants, (b) variables, (c) descriptive constants (names and predicates). Categories (a) and (b) together constitute the class of the logical or non-descriptive symbols. The distinction this classification implies is not merely nominal and arbitrary but syntactically significant since there are clear-cut and very general differences in the way in which the t1vo classes of symbols, and in particular the descriptive constants and the variables "corresponding7' to them, are dealt with in the formation and transformation rules. To give an example with respect to the formation rules: In a syntactical formulation of the lower functional calculus with descriptive constants, '(a) f(a)' is a non-sentential design (syntactical nonsense), while '(x) f (x)' is a sentential design. In the realm of the transformation rules, the patterns usually referred to as rewriting and substitution rules yield further pertinent differences. I t might help to clarify what is meant by attributing epistemological significance to the syntactical distinction in question if it is pointed out that axiomatizations of set theory which start with one or several undefined predicates are, from this view point, entirely devoid of epistemological significance. As such, even the most ingenious constructions of this type are of mathematical interest only. Let it be said, finally, that I am aware of the neo-conventionalist argument that even so there is nothing a priori about the distinction between descriptive and logical symbols as there is, according to the same vie~vpoint,nothing a priori about the distinction between analytic and synthetic. One reason why this thesis seems idle to me is that in some sense it is certainly true while in some other sense it is as certainly false. I t is the analyst's task to distinguish and delimit these several aspects with ever increasing precision. 2. Church suspects that the attribution of the substitution property to what I have called semantical identity rests, strictly speaking, upon a confusion between
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DISCUSSION
use and mention. This is not the case. A semantical system formally considered does not mention anything a t all. What we loosely call the "names" of the expressions of the object language is, in a semantical system, merely a class of particulars. This class of particulars is, within pure semantics, subdivided and distinguished from other particulars, not by a non-formal reference to what it refers to, but by a set of definitions. Confusion between use and mention is thus precluded by the very nature of a semantical system. Concerning the conception of semantics which underlies these remarks, see also my forthcoming paper on "Pure Semantics, Sentences, and Propositions9'(Mind, 1944). I t is true, though, that the defined semantical relation which I have called semantical identity will not, strictly speaking, show unrestricted substitutivity, but only substitutivity with respect to the semantical predicates of the system such as truth and designation. Thus interpreted Church's criticism seems, a t least from the formalistic viewpoint, justified. GUSTAVBERGMANN. State University of Iowa
