Abstract
Numerical identity is standardly considered to be a relation between things. This means that two things are identical if they are only one thing. It is not only Wittgenstein who finds this claim rather odd. Another possibility is to understand identity as a relation between names which denote the same thing; or as a relation between the senses of those names which are modes of presentation of the same thing. Or identity statements can be considered as expressions of the fact that there is exactly one thing that has certain characteristics. These conceptions are, in fact, closely interlinked. There are some rather straightforward proposals on how to give these approaches a formal pattern in relative harmony with classical logic. What is important to emphasize and keep in mind is the specific character of identity and essential dissimilarities between identity and relations of objects in logic.
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Notes
“Numerical identity (\(\ldots \)) can only hold between a thing and itself. (\(\ldots \)) Numerical identity can be characterised as the relation everything has to itself and to nothing else.” [18] However, this cannot be considered to be a definition of identity while the notion “itself“ already presupposes the notion of identity, cf.: “Aber natürlich kann man Identität nicht zirkelfrei durch ,Relation, in der jeder Gegenstand zu sich selbst und zu nichts anderem steht’ definieren; denn die Wendungen ,sich selbst’ und ,nichts anderes’ versteht nur, wer den Begriff der Identität bereits in seinem Repertoire hat.” ([15, p. 110]).
[3, 103a]. Aristotle here divides sameness into three kinds: “Sameness would be generally regarded as falling, roughly speaking, into three divisions. We generally apply the term numerically or specifically or generically (\(\ldots \))”.
[7, §8]. (The quotation is clearer in context: ”Equality of content differs from conditionality and negation by relating to names, not to contents. Elsewhere, signs are mere proxies for their content, and thus any phrase they occur in just expresses a relation between their various contents; but names at once appear in propria persona so soon as they are joined together by the symbol for equality of content; for this signifies the circumstance of two names’ having the same content. Thus, along with the introduction of a symbol for equality of content, all symbols are necessarily given a double meaning - the same symbols stand now for their own content, now for themselves.”)
“Daß a und b nur Zeichen oder Namen für ein und dasselbe Ding sind, wird durch das Zeichen a = b und ebenso durch b = a angedeutet.” [6, §1, p. 1].
[23, §24, p. 115, p. 116], emphasis mine. Cf. also his claim that an identity statement is true “because the two terms are names of the same objects” and that “for truth of a statement of identity it is necessary only that ’=’ appear between names of the same objects” [21, §35, pp. 208–209]. However, Quine says that “[it] is not to say that identity is a relation of expressions in language” and that “no linguistic investigation of the names in a statement of identity will suffice, ordinarily, to determine whether the identity holds or fails” (ibid. p. 209). It is important to keep in mind that what is taken into account are not linguistic expressions as such (e.g., that one expression is shorter than another one), but linguistic expressions as having meanings and denotations.
Cf. “Or, A and B are the same if they can be substituted for one another everywhere (excepting, however, those cases in which not the thing itself but the manner of conceiving the thing, which may be different, is under consideration; thus Peter and the Apostle who denied Christ are the same, and the one term may be substituted for the other, unless we are considering the matter in the way some people call ’reflexive’: for example, if I say ’Peter, insofar as he was the Apostle who denied Christ, sinned’, I cannot substitute ’Peter’ and say ’Peter, insofar as he was Peter, sinned’).” ([16, 552/130]). See also [1, pp. 94–100].
“The following analogy will perhaps clarify these relationships. Somebody observes the Moon through a telescope. I compare the Moon itself to the meaning; it is the object of the observation, mediated by the real image projected by the object glass in the interior of the telescope, and by the retinal image of the observer. The former I compare to the sense, the latter is like the idea or experience. The optical image in the telescope is indeed one-sided and dependent upon the standpoint of observation; but it is still objective, inasmuch as it can be used by several observers. At any rate it could be arranged for several to use it simultaneously.” ([9, pp. 160–161]).
“Per terminum non intelligo nomen sed conceptum seu id quod nomine significatur; possis et dicere notionem, ideam.” ([16, p. 288]).
“Terms are concepts or ideas which are substituted in propositions and are constituents of propositions.” ([14, p. 20]). See also ibid., pp. 24–25.
“Concept” in the Fregean sense: If there are general terms in an identity statement, the general terms should denote the same property or relation.
See [16, p. 831 or p. 846]. Leibniz even mentions that “eadem” is sometimes used for relating A and A, “coincidentia” for relating A and B (“Aliquando tamen A quidem et A vocantur idem; A vero et B si sint eadem vocantur coincidentia.” Ibid., p. 846).
“The relation of the meaning to the denotation involves certain rather curious difficulties, which seem in themselves sufficient to prove that the theory which leads to such difficulties must be wrong. (\(\ldots \)) Thus it would seem that ’C’ and C are different entities, such that ’C’ denotes C; but this cannot be an explanation, because the relation of ’C’ to C remains wholly mysterious; (\(\ldots \)) This is an inextricable tangle, and seems to prove that the whole distinction of meaning and denotation has been wrongly conceived. (\(\ldots \)) hence the meaning of ’the author of Waverley’ must be relevant as well as the denotation, if we adhere to the point of view to which this distinction belongs. Yet, as we have just seen, so long as we adhere to this point of view, we are compelled to hold that only the denotation can be relevant. Thus the point of view in question must be abandoned.” ([24, pp. 487–488]).
[24, pp. 479–493, p. 487]. Cf. also “The difficulty in speaking of the meaning of a denoting complex may be stated thus: The moment we put the complex in a proposition, the proposition is about the denotation; and if we make a proposition in which the subject is ’the meaning of C,’ then the subject is the meaning (if any) of the denotation, which was not intended.” Ibid., pp. 486–487.
“[I]t is also the case that in relation to inference, and where the laws of logic are concerned, (...) concepts differ only in so far as their extensions are different. (\(\ldots \)) They [intensionalist logicians] forget that the laws of logic are first and foremost laws in the realm of meanings and only relate indirectly to sense. If it is a question of the truth of something—and truth is the goal of logic—we also have to inquire after meanings”. ([8, p. 118, p. 122]).
“No one outside a logic-book ever wishes to say ‘x is x’” [24, p. 492].
The presupposition of uniqueness of what falls under a definite description is hidden, according to Russell, in using the definite article “the”. It seems to be the point of Russell’s analysis of statements containing definite descriptions: what is stated by them is the existence of a unique object (see [24, p. 492]); Russell does not agree with Frege’s approach that it is necessary to artificially postulate a denotation in cases when the definite description does not denote at all (ibid., p. 484).
Russell presupposes that sentences containing definite descriptions are simply false if there is nothing denoted by the definite description. This approach is sometimes criticized because sentences with non-denoting definite descriptions seem strange rather than false. Is the sentence “The king of France is the man who proved Goldbach’s conjecture” simply false? Or is it merely not “used to express a true or a false proposition” [25, p. 326]? And what about “fictional existence” - is the sentence “Clark Kent is Superman” false because Superman does not really exist? However, accepting this kind of criticism would lead to some shifts in the understanding of the notion of proposition and existence rather than to a change of the notion of identity.
Identity symbol is usually interpreted as standard identity: “Most logicians include ’=’ as a part of the vocabulary of every language for predicate logic, and interpret it always to mean standard identity.” ([13, p. 71]). The reason is that axioms for identity can ensure the “defined” relation to be only an equivalence relation, not to be necessary standard identity (ibid., pp. 68–70, see also, e.g., [20, pp. 621–633]). Without a metatheoretical assumption of the uniqueness of an assigned object as a condition of identity, it is possible to reach only “relative identity” in Geach’s terminology (cf. [10]). On the problem of various notions of identity see also, for example, [4].
It is possible to find an interesting exposition of this issue in [26, pp. 1–10].
It can happen that the first sense and the second sense present the same thing—but even in this case we usually consider the thing both as a “subject” and as an “object” of the relation, and we have to ask whether there is or there is not the relation of the thing as subject to itself as object: “Does Romeo love himself?” In the case of identity it is senseless to ask in this way (“Is Romeo identical with himself?”).
The authors understand quantification, in accordance with Frege, as substitutional quantification, the truth of quantified sentences is therefore determined by the truth of particular substitutional variants of corresponding open formulas.
See [12, p. 230]. Hintikka offers three interesting examples from common language where this kind of “exclusive” quantification is used: “(1a) Any two points of a straight line completely determine that line; (2a) He is John’s brother if he has the same parents as John; (3a) Mazzini did more for the emancipation of his country than any living man of his time” (p. 225).
Instead of looking for the appropriate definite description, it is possible to simply replace a proper name (e.g., “Pegasus”) by a corresponding singular concept (“is-Pegasus” or “pegasizes”). See [22, p. 27].
Cf. “My central point is that a relation-ascription view of identity statements leads to confusion because there is a fundamental disanalogy between identity statements and other two-termed statements, which modern philosophy—in calling them all relational, and bringing the same logical and semantic ideas to bear on them—has obscured.“ ([11, p. 270]).
Cf., for example, “Frege originally conceived of the quantifier \(\forall \) as a monadic predicate that is true of a first-level concept F under which only objects fall if, and only if, all objects fall under F; likewise, Frege assimilated \(\exists \) to a monadic predicate that is true of a first-level concept if, and only if, at least one object falls under F.” [27].
Even individual variables can be considered as “temporary names” due to the procedure of valuation. The Tarski-style valuation function makes any variable a temporary name of the assigned object, i.e. “x” is a name of the object v(x) in the given valuation v.
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Acknowledgements
This work was supported by the Czech Science Foundation under Grant GA CR GA20-18675S. I would like to thank my colleague Jaroslav Peregrin for his very helpful comments on a previous draft of this paper. I thank the anonymous reviewers for their thorough reading of the manuscript, contributive comments and careful revision.
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Vlasáková, M. What is Identical?. Log. Univers. 15, 153–170 (2021). https://doi.org/10.1007/s11787-021-00272-7
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DOI: https://doi.org/10.1007/s11787-021-00272-7