The earliest recorded use of the law appears 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: that they both are?
Wilhelm Wundt credits Gottfried Leibniz with the symbolic formulation, "A is A.
"[4] 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.
("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" [...]Afrikan Spir proclaims the law of identity as the fundamental law of knowledge, which is opposed to the changing appearance of the empirical reality.
[5] 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.
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 three axioms, one of which is the law of identity, "A is A."
[7] 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.
Martin Heidegger gave a talk in 1957 entitled "Der Satz der Identität" (The Statement of Identity), where he linked 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).
Gilles Deleuze wrote that "Difference and Repetition" is prior to any concept of identity.
[citation needed] In first-order logic, identity (or equality) is represented as a two-place predicate, or relation, =.