Law of Identity

In logic, the law of identity is the first of the three classical laws of thought. It states that each thing is identical with itself. By this it is meant that each thing (be it a universal or a particular) is composed of its own unique set of characteristic qualities or features, which the ancient Greeks called its essence.

In its symbolic representation, "a = a" or "For all x: x = x".

In logical discourse, violations of the Law of Identity (LOI) 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 – even though the different meanings are conventionally prescribed to that term. In everyday language, violations of the LOI introduce ambiguity into the discourse, making it difficult to form an interpretation at the desired level of specificity. The LOI also allows for substitution.

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?
Theaetetus: Yes.
Socrates: Then do you think that each differs to the other, and is identical to itself?
Theaetetus: Certainly.
Socrates: And that both are two and each of them one?
Theaetetus: Yes, that too.

Aristotle takes recourse to the law of identity—though he does not identify it as such—in an attempt to negatively demonstrate the law of non-contradiction. However, in doing so, he shows that the law of non-contradiction is not the more fundamental of the two:

"First then this at least is obviously true, that the word 'be' or 'not be' has a definite meaning, so that not everything will be 'so and not so'. Again, if 'man' has one meaning, let this be 'two-footed animal'; by having one meaning I understand this:-if 'man' means 'X', then if A is a man 'X' will be what 'being a man' means for him. It makes no difference even if one were to say a word has several meanings, if only they are limited in number; for to each definition there might be assigned a different word. For instance, we might say that 'man' has not one meaning but several, one of which would have one definition, viz. 'two-footed animal', while there might be also several other definitions if only they were limited in number; for a peculiar name might be assigned to each of the definitions. If, however, they were not limited but one were to say that the word has an infinite number of meanings, obviously reasoning would be impossible; for not to have one meaning is to have no meaning, and if words have no meaning our reasoning with one another, and indeed with ourselves, has been annihilated; for it is impossible to think of anything if we do not think of one thing; but if this is possible, one name might be assigned to this thing."

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