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 "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.Things are said to be named 'equivocally' when, though they have a common name, the definition corresponding with the name differs for each. That is to say, we cannot use the same term in the same discourse while having it signify different senses or meanings without 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 (185a), wherein Socrates attempts to establish that what we call "sounds" and "colours" are two different classes of thing: Rejection of the principle of identity Schrödinger logics Schrödinger logics are logical systems in which the principle of identity is not true in general citations needed%DIFDELCMD < ]%%% . The intuitive motivation for these logics is both Erwin   Schrödinger 's thesis (which has been advanced by other authors) that identitylacks 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) .
In logic, the law of identity states that each thing is identical with itself. It is the first of the historical 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. History Ancient philosophy 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: Modern logic In first-order logic, identity (or equality) is represented as a two-place predicate, or relation, =. Identity is a relation on individuals. It is not a relation between propositions, and is not concerned with the meaning of propositions, nor with equivocation. The law of identity can be expressed as , where x is a variable ranging over the domain of all individuals. In logic, there are various different ways identity can be handled. In first-order logic with identity, identity is treated as a logical constant and its axioms are part of the logic itself. Under this convention, the law of identity is %DIFDELCMD < ]%%% a logical truth. In first-order logic without identity, identity is treated as an interpretable predicate and its axioms are supplied by the theory. This allows a broader equivalence relation to be used that may allow a = b to be satisfied by distinct individuals a and b. Under this convention, a model is said to be normal when no distinct individuals a and b satisfy a = b. One example of a logic that rejects or restricts the law of identity in this way is Schrödinger logic .
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historical
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It can also be written less formally as A is A. One statement of such a principle is "A rose is a rose is a rose."
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In logical discourse, violations of the law of identity result in the informal logical fallacy known as equivocation.Things are said to be named 'equivocally' when, though they have a common name, the definition corresponding with the name differs for each. That is to say, we cannot use the same term in the same discourse while having it signify different senses or meanings without 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.
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Rejection of the principle of identity
Modern logic
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Schrödinger logics
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Schrödinger logics are logical systems in which the principle
In first-order logic, identity (or equality) is represented as a two-place predicate, or relation, =. Identity is a relation on individuals. It is not a relation between propositions, and is not concerned with the meaning of propositions, nor with equivocation. The law of identity can be expressed as , where x is a variable ranging over the domain of all individuals. In logic, there are various different ways identity can be handled. In first-order logic with identity, identity is treated as a logical constant and its axioms are part of the logic itself. Under this convention, the law
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not true in general
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citations needed
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. The intuitive motivation for these logics is both Erwin
a logical truth.
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In first-order logic without identity, identity is treated as an interpretable predicate and its axioms are supplied by the theory. This allows a broader equivalence relation to be used that may allow a = b to be satisfied by distinct individuals a and b. Under this convention, a model is said to be normal when no distinct individuals a and b satisfy a = b.
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One example of a logic that rejects or restricts the law of identity in this way is
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's thesis (which has been advanced by other authors) that identitylacks 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)
logic