Law of identity


In logic, the law of identity states that each thing is identical with itself. It is the first of the three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, no system of logic is built on just these laws, and none of these laws provide inference rules, such as modus ponens or DeMorgan's Laws.
In its formal representation, the law of identity is written "a = a" or "For all x: x = x", where a or x refer to a term rather than a proposition, and thus the law of identity is not used in propositional logic. It is that which is expressed by the equals sign "=", the notion of identity or equality. It can also be written less formally as A is A. One statement of such a principle is "Rose is a rose is a rose is a rose."
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 and introducing ambiguity into the discourse – even though the different meanings are conventionally prescribed to that term. The law of identity also allows for substitution, and is a tautology.

History

Ancient philosophy

The earliest recorded use of the law appears to occur in Plato's dialogue Theaetetus, wherein Socrates attempts to establish that what we call "sounds" and "colours" are two different classes of thing:
It is used explicitly only once in Aristotle, in a proof in the Prior Analytics:

Medieval philosophy

Aristotle believed the law of non contradiction to be the most fundamental law. Both Thomas Aquinas and Duns Scotus follow Aristotle in this respect. Antonius Andreas, the Spanish disciple of Scotus, argues that the first place should belong to the law "Every Being is a Being", but the late scholastic writer Francisco Suárez disagreed, also preferring to follow Aristotle.
Another possible allusion to the same principle may be found in the writings of Nicholas of Cusa where he says:

Modern philosophy

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, arguing that "the statement that a thing is what it is, is prior to the statement that it is not another thing". 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 says:
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:
Objectivism, the philosophy founded by novelist Ayn Rand, claims to be grounded in the law of identity, "A is A".

Contemporary philosophy

Analytic

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.”

Continental

wrote that "Difference and Repetition" is prior to any concept of identity.
= Rejection of the principle of identity =

Schrödinger Logics

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 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.
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