Law of identity
- This article uses forms of logical notation. For a concise description of the symbols used in this notation, see List of logic symbols.
In logic, the law of identity is the first of the three classical laws of thought. It states that: “each thing is the same with itself and different from another”: “A is A and not ~A”. By this it is meant that each thing (be it a universal or a particular) comprises it own unique set of characteristic qualities or features, which the ancient Greeks called its essence. Consequently, things that have the same essence are the same thing, while things that have different essences are different things. In its symbolic representation:(“A is A”), the first element of the proposition represents the subject (thing) and the second element, the predicate (its essence), with the copula “is” signifying the relation of “identity”. Further, since a definition is an expression of the essence of that thing with which the linguistic term is associated, it follows that it is through its definition that the identity of a thing is established. For example, in the definitive proposition:"A lawyer is a person qualified and authorized to practice law", the subject (lawyer) and the predicate (person qualified and authorized to practice law) are declared to be one and the same thing (identical). Consequently, the Law of Identity prohibits us from rightfully calling anything other than "a person qualified and authorized to practice law" a "lawyer".
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 difference 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 is difficult to form an interpretation at the desired level of specificity.
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: 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?
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." - (Metaphysics, Book IV, Part 4)
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.
Another possible allusion to the same principle may be found in the writings of Nicholas of Cusa (1431-1464) where he says:
... there cannot be several things exactly the same, for in that case there would not be several things, but the same thing itself. Therefore all things both agree with and differ from one another. 
Gottfried Wilhelm 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".
George Boole, in the introduction to his treatise An Investigation of 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. To a certain extent these elements are arbitrary. Their interpretation is purely conventional: we are permitted to employ them in whatever sense we please. 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.”
John 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" ...
See also 
- Thomas Aquinas
- René Descartes
- Keith Donnellan
- Thomas Hobbes
- David Kaplan
- Saul Kripke
- Gottfried Wilhelm Leibniz
- John Locke
- W.V. Quine
- Hilary Putnam
- Baruch Spinoza
- John Searle
- African Spir
- "Two things are called one, when the definition which states the essence of one is indivisible from another definition which shows us the other (though in itself every definition is divisible).” [Aristotle's Metaphysics, Book VI, Part 4 (c) - Translated by W. D. Ross]
- "Each thing itself, then, and its essence are one and the same in no merely accidental way, as is evident both from the preceding arguments and because to know each thing, at least, is just to know its essence, so that even by the exhibition of instances it becomes clear that both must be one.” [Aristotle's Metaphysics, Book VII, Part 6 - Translated by W. D. Ross]
- “For if a definition is an expression signifying the essence of the thing and the predicates contained therein ought also to be the only ones which are predicated of the thing in the category of essence; and genera and differentiae are so predicated in that category: it is obvious that if one were to get an admission that so and so are the only attributes predicated in that category, the expression containing so and so would of necessity be a definition; for it is impossible that anything else should be a definition, seeing that there is not anything else predicated of the thing in the category of essence.”[Aristotle's Topics, Book VII, Part 1 - Translated by W. A. Pickard-Cambridge]
- A definitive proposition is that wherein one term is the definition of the other, e.g., "Hope is the looking with pleasure into the future."
- Things are said to be named 'equivocally' when, though they have a common name, the definition corresponding with the name differs for each.
- De Venatione Sapientiae, 23.
- La philosophie éternelle ou traditionnelle, la métaphysique, la logique, la raison et l'intelligence
- Forschung nach der Gewissheit in der Erkenntniss der Wirklichkeit, Leipzig, J.G. Findel, 1869 and Denken und Wirklichkeit: Versuch einer Erneuerung der kritischen Philosophie, Leipzig, J. G. Findel, 1873.