Law of identity
- This article uses forms of logical notation. For a concise description of the symbols used in this notation, see List of logic symbols.
In logic, the law of identity states that each thing is identical with itself. By this it is meant that each thing (be it a universal or a particular) is composed of its own unique set of characteristic qualities or features, which the ancient Greeks called its essence. It is the first of the three classical laws of thought.
In its symbolic representation, "a=a", "Epp", or "For all x: x = x".
In logical discourse, violations of the law of identity result in the informal logical fallacy known as equivocation. That is to say, we cannot use the same term in the same discourse while having it signify different senses or meanings – even though the different meanings are conventionally prescribed to that term. Corollaries of the law of identity, "Epp", "p if and only if p", are the law of noncontradiction, "NEpNp", "not that p if and only if not p"; and the law of excluded middle, "JpNp", "p or not p, exclusively", in which the prefix "J" represents the exclusive or, the negation of the prefix "E", the logical biconditional.
In everyday language, violations of the law of identity introduce ambiguity into the discourse, making it difficult to form an interpretation at the desired level of specificity. The law of identity also allows for substitution.
The earliest recorded use of the law appears to occur in Plato's dialogue Theaetetus (185a), wherein Socrates attempts to establish that what we call "sounds" and "colours" are two different classes of thing:
Socrates: With regard to sound and colour, in the first place, do you think this about both: do they exist?
Socrates: Then do you think that each differs to the other, and is identical to itself?
Socrates: And that both are two and each of them one?
Theaetetus: Yes, that too.
Aristotle takes recourse to the law of identity—though he does not identify it as such—in an attempt to negatively demonstrate the law of non-contradiction. However, in doing so, he shows that the law of non-contradiction is not the more fundamental of the two:
First then this at least is obviously true, that the word 'be' or 'not be' has a definite meaning, so that not everything will be 'so and not so'. Again, if 'man' has one meaning, let this be 'two-footed animal'; by having one meaning I understand this:-if 'man' means 'X', then if A is a man 'X' will be what 'being a man' means for him. It makes no difference even if one were to say a word has several meanings, if only they are limited in number; for to each definition there might be assigned a different word. For instance, we might say that 'man' has not one meaning but several, one of which would have one definition, viz. 'two-footed animal', while there might be also several other definitions if only they were limited in number; for a peculiar name might be assigned to each of the definitions. If, however, they were not limited but one were to say that the word has an infinite number of meanings, obviously reasoning would be impossible; for not to have one meaning is to have no meaning, and if words have no meaning our reasoning with one another, and indeed with ourselves, has been annihilated; for it is impossible to think of anything if we do not think of one thing; but if this is possible, one name might be assigned to this thing.— Aristotle, Metaphysics, Book IV, Part 4, 1006a–b
When A belongs to the whole of B and to C and is affirmed of nothing else, and B also belongs to all C, it is necessary that A and B should be convertible: for since A is said of B and C only, and B is affirmed both of itself and of C, it is clear that B will be said of everything of which A is said, except A itself.— Aristotle, Prior Analytics, Book II, Part 22, 68a
Both Thomas Aquinas (Met. IV, lect. 6) and Duns Scotus (Quaest. sup. Met. IV, Q. 3) follow Aristotle. Antonius Andreas, the Spanish disciple of Scotus (d. 1320), argues that the first place should belong to the law "Every Being is a Being" (Omne Ens est Ens, Qq. in Met. IV, Q. 4), but the late scholastic writer Francisco Suárez (Disp. Met. III, § 3) disagreed, also preferring to follow Aristotle.
Another possible allusion to the same principle may be found in the writings of Nicholas of Cusa (1431-1464) where he says:
... there cannot be several things exactly the same, for in that case there would not be several things, but the same thing itself. Therefore all things both agree with and differ from one another.
Gottfried Wilhelm Leibniz claimed that the law of identity (sometimes called "Leibniz'Law"), which he expresses as "Everything is what it is", is the first primitive truth of reason which is affirmative, and the law of noncontradiction is the first negative truth (Nouv. Ess. IV, 2, § i), arguing that "the statement that a thing is what it is, is prior to the statement that it is not another thing" (Nouv. Ess. IV, 7, § 9). Wilhelm Wundt credits Gottfried Leibniz with the symbolic formulation, "A is A".
George Boole, in the introduction to his treatise The Laws of Thought made the following observation with respect to the nature of language and those principles that must inhere naturally within them, if they are to be intelligible:
There exist, indeed, certain general principles founded in the very nature of language, by which the use of symbols, which are but the elements of scientific language, is determined. To a certain extent these elements are arbitrary. Their interpretation is purely conventional: we are permitted to employ them in whatever sense we please. But this permission is limited by two indispensable conditions, first, that from the sense once conventionally established we never, in the same process of reasoning, depart; secondly, that the laws by which the process is conducted be founded exclusively upon the above fixed sense or meaning of the symbols employed.
[...] whenever the mind with attention considers any proposition, so as to perceive the two ideas signified by the terms, and affirmed or denied one of the other to be the same or different; it is presently and infallibly certain of the truth of such a proposition; and this equally whether these propositions be in terms standing for more general ideas, or such as are less so: e.g., whether the general idea of Being be affirmed of itself, as in this proposition, "whatsoever is, is"; or a more particular idea be affirmed of itself, as "a man is a man"; or, "whatsoever is white is white" [...]
- Thomas Aquinas
- René Descartes
- Keith Donnellan
- Steve Ditko
- Thomas Hobbes
- David Kaplan
- Saul Kripke
- Gottfried Wilhelm Leibniz
- John Locke
- Willard Van Orman Quine
- Hilary Putnam
- Ayn Rand
- Baruch Spinoza
- John Searle
- Afrikan Spir
- Things are said to be named 'equivocally' when, though they have a common name, the definition corresponding with the name differs for each.
- Jozef Maria Bochenski (1959), Precis of Mathematical Logic, rev., Albert Menne, ed. and trans., Otto Bird, New York: Gordon and Breach, Part II, "The Logic of Sentences", Sect. 3.32, p. 11, and Sect. 3.92, p. 14.
- Wang, Hao (2016). "From Mathematics to Philosophy (Routledge Revivals)". Routledge – via Google Books.
- Thomas, Ivo (1 April 1974). "On a passage of Aristotle". Notre Dame J. Formal Logic. 15 (2): 347–348. doi:10.1305/ndjfl/1093891315 – via Project Euclid.
- De Venatione Sapientiae, 23.
- Curley, E. M. (October 1971). "Did Leibniz State "Leibniz'Law"?". The Philosophical Review. 8 (4): 497–501.
- Forschung nach der Gewissheit in der Erkenntniss der Wirklichkeit, Leipzig, J.G. Findel, 1869 and Denken und Wirklichkeit: Versuch einer Erneuerung der kritischen Philosophie, Leipzig, J. G. Findel, 1873.