Notes on Identity Gustav Bergmann Philosophy of Science, Vol. 10, No. 3. (Jul., 1943), pp. 163-166. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28194307%2910%3A3%3C163%3ANOI%3E2.0.CO%3B2-N Philosophy of Science is currently published by The University of Chicago Press.
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NOTES ON IDENTITY GUSTAV BERGMANN
This is a brief sketch of the form mhich the analysis of the concept of identity would take under the impact of the most recent phase of Scientific Empiricism. The frame of reference is thus that of Carnap's Introduction to Semantics. &/lost characteristic of this approach is that it reaches its main clarifications by djstinguishing between the various connotations of the traditional terms along the three lines of syntax, semantics, and pragmatics. The notion of truth, for instance, must be split into syntactical truth or analyticality, semantical truth, and pragmatical truth. Only the last has something to do with belief and verification. I n a somewhat similar manner identity splits, upon closer examination, into three radically different meanings, but it will be seen that the third one has nothing to do with either pure or applied pragmatics. Some of the points I am trying to make have been dealt with, in an interesting manner, in a recent paper by &uine.l I. Syntactical Identity. I n syntax, the formal study of languages as uninterpreted calculi, 'identity' is the name of any calculational symbol mhich by virtue of the logic of the calculus is endowed with the four properties of symmetry, transitivity, rejlexivity, and what Quine calls substitutivity. The customary mark for such a constant2 are two horizontal strokes, as in 'a = b', where 'a' and 'b' function as neutral and typeless symbols. For, a calculus might either contain identities that connect only names and name variables, or it might contain identities between both names and predicates (and the respective variables). A calculus which does not contain any descriptive terms or, to say the same thing differently, is coextensive with its logic as for instance the Principia Mathematica, can, of course, also contain identities between those calculational symbols which me call variables. The significance of the expression 'calculational symbol' lies in the circumstance that any interpretation of a calculus rests upon the interpretation of its non-calculational or descriptive symbols (names and predicates). The finer determination of this dependence is one of the tasks of pure semantics as a branch of formal logic. At any rate the occurrence, within a calculus, of a descriptive relation possessing the four formal properties of identity must be carefully distinguished from the occurrence of an identity in the syntactical sense now under discussion. The opinion, shared by this writer, that no such descriptive relation mill ever appear in formalizations of the empirical language mhich we all speak about the world, underlies Wittgenstein's indiscriminate attack against the concept of identity. It is a corollary to this viewpoint that a calculus could contain several kinds of syntactical identities. As a matter of fact Language I1 of Carnap's Logical l"Notes on Existence and Identity." Journal of Philosophy, 40, 1943, 113-127. 2Calculational symbols are either constants, such as 'and', 'not', and ' =', or variables and operators. Leibnitz-Russell's identity of indiscernibles is a defined nondescriptive relation, not a calculational symbol.
163
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GUSTAV BERGMAXN
Syntax of Language sets out with two different identity constants, equivalence between statements and equality between names and predicates. I t is a property of this particular language not to become inconsistent and not even to undergo any significant changes if the two identities are later on not distinguished from each other. 2 . Semantical Identity. One may use or discover that other people use two different symbols for the same referent and express this in the following manner: (1)
(2)
Napoleon = Bonaparte, Man
=
featherless biped.
It is generally felt that one is here still dealing with a linguistic matter in contradistinction to factual or empirical affairs. However, reference to the referents of the symbols is in this case of the essence of the matter and our concern here is therefore with semantics rather than with syntax. The anticipation that such statements, though not syntactical, are yet formal or linguistic is borne out by the possibility of accounting for them within pure semantics. For pure semantics is a branch of formal logic which investigates, by studying certain relations between languages, but without a n y actual empirical interpretation, the properties of such terms as 'true', 'false', and 'designation'. Among these semantical terms is also the semantical identity informally expressed by (I) and ( 2 ) . Strictly one would have to write: (1')
'Napoleon' = 'Bonaparte'
where ' = ' is a semantical relation defined in terms of designation, which has the four formal properties of identity. Doubts concerning this analysis of ( 2 ) mill be discussed presently. 3. P-Identity. The first two meanings of the unanalyzed term 'identity' strictly correspond to the first two members of the trichotomy, syntax, semantics, and pragmatics, in the sense intended in Carnap's most recent work. P-identity, the third connotation, has something to do with meaning; it is therefore worth while to emphasize that this third connotation does not lie either in semantics or in pragmatic^.^ For its analysis one must turn to the descriptive part of the object calculus itself or, more specifically, to what Carnap calls its P-laws, i.e., to those of its generalized, descriptive, non-analytic statements which are either axioms or analytic consequences of such. This analysis of P-identity can be performed without any reference to interpretation and therefore, a fortiori, without any ontological import. It seems best to start by pointing out that the semantical interpretation of statements of the type ( 2 ) not only differs from the usual treatment but is also pointless in the sense that with this interpretation such statements are not likely ever to occur in any empirical language. Furthermore, such statements, no 3 T o pragmatics belong, in Quine's article, only some of the remarks about not purely designative occurrences. Concerning this point see also my discussion in Philosophy of Science, 9,1942,372-374, and the reference thcre t o Russell's recent treatment of 'A believes that p'.
NOTES ON IDENTITY
165
matter how interpreted, do not occur in the empirical language if a t least one of the two descriptive predicates which are asserted to be identical is undefined. If, however, both predicates are defined, as 'man' and 'featherless biped' would have to be in a so-called sense-data language, then two cases must be distinguished. ITirst, the statement might turn out to be an analytic equivalence after all its defined terms have been eliminated. Second, if the statement is factual, then we are, as in the case of (2), faced with a P-law of the form which Russell and Whitehead call formal equivalence (coextensionality). I n both cases the unanalyzed "identity" sign is thus recognized as a calculational symbol. If the language is extensional then formal equivalence has the four formal properties of identity, otherwise only the first three. I n the latter case one would probably not speak of an identity at all. TWOfurther alternatives may be distinguished: either the P-law which the "identity" in question turns out to express is one of the axioms of the language or it is a deductive consequence of these axions. The historical progress of science is accompanied by changes in the empirical language which make more and more P-identities derivable in this sense. It was in order to emphasize that P-identity has no symbol of its own and has nothing to do with "ontological identity" that I have first dealt with P-identity between predicates, as instanced by the customary nonsemantical interpretation of (2). I n the case of names, statements of the type: Evening Star = Morning Star, (3 on the empirical or P-character of which Quine rightly insists, seem a t first sight to defy reduction to formal equivalence. Some people would indeed hold that these statements express some kind of supposedly ontological identity and therefore constitute a class of their own. In contradistinction to such views it must be emphasized that the reduction of this class to those previously considered is, within the framework of contemporary empiricism, a reasonable expectation. To make it plausible one first has to observe that in empirical languages P-identities occur only between defined names. Between undefined names they are as pointless as between undefined predicates. When syntactical identities do connect two different undefined terms of the empirical language it is always possible to interpret these statements, without constraint, as informal expressions of semantical identities of the type exemplified by (It), instead of incorporating them among the axioms. It is, however, essential that we can express in the object language, by use of a syntactical identity, that undefined names are P-identical with descriptions, as in: (4)
Napoleon
=
husband of Josephine.
In agreement with common usage 'Napoleon' is here regarded as an undefined name and can be so regarded since Napoleon actually existed. The empirical or P-character of (4) is o b v i o ~ s . ~The point is that expressions such as 'Pegasus'
* If the calculus does not contain a syntactical id en tit,^, the Leibnitz-Russell relation can be used to express (4). This is one of the main features of the theory of identity and description as developed in the Prlncipia Mathernatica.
166
GUSTAV B E R G M A ~ ~ N
and 'Evening Star' are neither names nor descriptions, but conjunctions of predications and P-laws. "Identities" such as (3) are then, typically, formal equivalences between statements which one obtains by applying these predications to different sets of variables. These formal equivalences are in turn consequences of the P-laws which are part of the "names" in question. Which descriptions are singled out for this specious kind of proper names is, formally considered, a matter of choice. In point of fact it seems safe to say that we are inclined to assign names when the referent thus distinguished exhibits, by virtue of accepted ?-laws, those general features of individualization, spatial localization, and temporal persistence which are the comprehensible connotations of ontological substantiality. The inclusion of Quine's example, 'Pegasus', makes it clear that in spite of the spatio-temporal element in this formulation one is here not dealing with actual existence. This is but an extension of the insight that the phrase 'the present emperor of the United States' is not necessarily syntactical nonsense. This section could be sunlmarized in the following manner: Properly speaking P-identity is not a third category coordinated to the syntactical and semantical concepts. This agrees with the earlier observation that there is no descriptive identity in the empirical language. I n the eighth chapter of his recent Inqz~iry into Meaning and Truth, Russell tries to orient himself with respect to certain epistemological problems by appealing to what he considers the analytical nature of the identity of indiscernibles. The introduction of the threefold distinction here indicated would necessitate a thorough reorganization of Russell's otherwise interesting argument. 4. Two meanings of meaning. Since meaning has once been mentioned in this context it might not be inappropriate to append an observation which is but loosely connected with the preceding remarks. It concerns two particular ones among the many meanings of meaning, confusion between which sometimes bedevils the methodology of science. I also shift for the moment to the terminology now favored by scientists. I n this usage one would say that the operational definition, and therefore the meaning, of the Evening Star is different from that of the Morning Star. This is the perfectly good meaning of meaning which is synonymous with operational definition and physicalistic reduction. I n this sense the intelligence quotient means a testing procedure, and mass the quotient of weight and gravitational acceleration. But sometimes scientists express themselves as if they thought one knew the meaning of mass only after he had realized at least some of its properties, for instance that it is constant in Newtonian mechanics. In the same sense the I&is held to be meaningful only insofar as it correlates with other achievement measures, or still more frequently, insofar as it agrees with certain prescientific notions of intelligence. Arguments which center around such apparently opposed viewpoints are plainly verbal and due to the failure to distinguish between meaning as synonymous with the operational definition of a concept on the one hand, and the class of the P-laws n ~vhichthis concept occurs on the other.
State University of Iowa.
