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In logic, the law of identity is the first of the three classic laws of thought. It states that an object is the same as itself: A → A (if you have A, then you have A). While this can also be listed as A ≡ A (A is equivalent to A), this is redundant. Any reflexive relation upholds the law of identity. When discussing equality, the fact that "A is A" is a tautology.
Socrates: How about sounds and colours: in the first place you would admit that they both exist?
Socrates: And that either of them is different from the other, and the same with itself?
Socrates: And that both are two and each of them one?
Parmenides the Eleatic (circa BCE. 490) formulated the principle Being is (eon emmenai) as the foundation of his philosophy.
Aristotle identifies the law in Book VII of the Metaphysics:
Now "why a thing is itself" is a meaningless inquiry (for—to give meaning to the question 'why'—the fact or the existence of the thing must already be evident—e.g., that the moon is eclipsed—but the fact that a thing is itself is the single reason and the single cause to be given in answer to all such questions as why the man is man, or the musician musical, unless one were to answer, 'because each thing is inseparable from itself, and its being one just meant this.' This, however, is common to all things and is a short and easy way with the question.)
Aristotle highlights "the fact that a thing is itself" because the objective of his inquiry at that point in the Metaphysics concerns "substance" and to provide answers to the question "what kind of thing substance should be said to be", given that "substance is a principle and a cause" of being. He further argues that while it is true that the question, "why a thing is itself" is meaningless, "the fact that a thing is itself" has meaning because we can then restate the why question to inquire "why something is predictable of something" given that each something is itself unique. For Aristotle, "substance is actuality" and it is the actual "thing that is itself" something that proceeds another such something in time.
Aristotle makes the claim that, "the fact that a thing is itself" allows for "a fixed constant nature of sensible things", and thus when confronted with the pronouncement that "this is bread" one can proceed to eat with confidence and not demand that for each and every such pronouncement evidence be provided to demonstrate that "this is not bread". Thus "the fact that a thing is itself" perhaps finds its greatest utility to man by providing "a fixed constant nature of sensible things" in an ever changing universe of being---Metaphysics, Book VII, Part 17
Thus in Aristotle we see the first logical presentation of the law of identity, "the fact that a thing is itself", to help answer the question "what kind of thing should be said to be". However, Aristotle never claimed that [A = A, 1 = 1, or A ≡ A], none of which correspond symbolically to "the fact that a thing is itself," for the simple reason that Aristotle never explicitly made the claim "thing is thing."
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 Suarez (Disp. Met. III., § 3) disagreed, also preferring to follow Aristotle.
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."
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" ...
J.S. Mill formulates the law as: "Whatever is true in one form of words, is true in every other form of words, which conveys the same meaning" (Exam. of Hamilton, p. 409). Shakespeare has Juliet Capulet state the same idea as "A rose by any other name would smell as sweet", in the 1597 play Romeo and Juliet.