One of the traditional three laws of thought, the other two being the laws of contradiction and excluded middle.
‘Everything is what it is and not another thing’, or (where ‘P’ is any proposition) ‘If P then P’. The English empiricist George Edward Moore (1873-1958) took the first quotation above as the motto for his book Principia Ethica (1903), attributing it to Bishop Joseph Butler (1692-1752).
In comparison to the other two laws of thought, little attention has been paid to this one.
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.
It is used explicitly only once in Aristotle, in a proof in the Prior Analytics:
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
Aristotle believed the law of non contradiction to be the most fundamental law. Both Thomas Aquinas (Met. IV, lect. 6) and Duns Scotus (Quaest. sup. Met. IV, Q. 3) follow Aristotle in this respect. 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, 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”. Leibniz’s Law is a similar principle, that if two objects have all the same properties, they are in fact one and the same: Fx and Fy iff x = y.
John Locke (Essay Concerning Human Understanding IV. vii. iv. (“Of Maxims”) says:
[…] 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” […]
Hamilton was one of the last to dedicate much to the “three laws”
Afrikan Spir proclaims the law of identity as the fundamental law of knowledge, which is opposed to the changing appearance of the empirical reality.
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.
Objectivism, the philosophy founded by novelist Ayn Rand, is grounded in the law of identity, “A is A”.
In the Foundations of Arithmetic, Gottlob Frege associated the number one with the property of being self identical. Frege’s paper “On Sense and Reference” begins with a discussion on equality and meaning. Frege wondered how a true statement of the form “a = a”, a trivial instance of the law of identity, could be different from a true statement of the form “a = b”, a genuine extension of knowledge, if the meaning of a term was its referent.
Bertrand Russell in “On Denoting” has this similar puzzle: “If a is identical with b, whatever is true of the one is true of the other, and either may be substituted for the other without altering the truth or falsehood of that proposition. Now George IV wished to know whether Scott was the author of Waverley; and in fact Scott was the author of Waverley. Hence we may substitute “Scott” for “the author of Waverley” and thereby prove that George IV wished to know whether Scott was Scott. Yet an interest in the law of identity can hardly be attributed to the first gentleman of Europe.”
Martin Heidegger held a talk in 1957 entitled “Der Satz der Identität”, where he links the law of identity “A=A” to the Parmenides’ fragment “to gar auto estin noien te kai einai” (….for the same thing can be thought and can exist). Heidegger thus understands identity starting from the relationship of Thinking and Being, and from the belonging-together of Thinking and Being. Gilles Deleuze wrote that “Difference and Repetition” is prior to any concept of identity.
Rejection of the principle of identity
Schrödinger logics are logical systems in which the principle of identity is not true in general. The intuitive motivation for these logics is both Erwin Schrödinger’s thesis (which has been advanced by other authors) that identity lacks sense for elementary particles of modern physics, and the way which physicists deal with this concept; normally, they understand identity as meaning indistinguishability (agreement with respect to attributes).