In logic, the law of identity states that each thing is identical with itself. It is the first of the traditional three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, few systems of logic are built on just these laws.
History
Ancient philosophy
The earliest recorded use of the law appears 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: that they both are?
Theaetetus: Yes.
Socrates: Then do you think that each differs to the other, and the same as itself?
Theaetetus: Certainly.
Socrates: And that both are two and each of them one?
Theaetetus: Yes, that too.
It is used explicitly only once in Aristotle, in a proof in the Prior Analytics:
When A belongs to the whole of B and to C and is affirmed of nothing else, and B also belongs to all C, it is necessary that A and B should be convertible: for since A is said of B and C only, and B is affirmed both of itself and of C, it is clear that B will be said of everything of which A is said, except A itself.
— Aristotle, Prior Analytics, Book II, Part 22, 68a
Medieval philosophy
Aristotle believed the law of non-contradiction to be the most fundamental law. Both Thomas Aquinas (Met. IV, lect. 6) and Duns Scotus (Quaest. sup. Met. IV, Q. 3) follow Aristotle in this respect. 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 Suárez (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.
Modern philosophy
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."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 (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" [...]
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:
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.
Objectivism, the philosophy founded by novelist Ayn Rand, is grounded in three axioms, one of which is the law of identity, "A is A." In the Objectivism of Ayn Rand, the law of identity is used with the concept existence to deduce that that which exists is something. In Objectivist epistemology logic is based on the law of identity.
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.”
In his "Tractatus Logico-Philosophicus", Ludwig Wittgenstein writes that "roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing."
In the formal logic of analytical philosophy, 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.
Continental
Martin Heidegger gave a talk in 1957 entitled "Der Satz der Identität" (The Statement of Identity), where he linked the law of identity "A=A" to the Parmenides' fragment "to gar auto estin noien te kai einai" (for the same thing can be thought and can exist).[citation needed] Heidegger thus understands identity starting from the relationship of Thinking and Being, and from the belonging-together of Thinking and Being.
Gilles Deleuze wrote that "Difference and Repetition" is prior to any concept of identity.[citation needed]
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 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 restricts the law of identity in this way is Schrödinger logic.
See also
- Identity (philosophy)
- Tautology (logic)
- Rectification of names
References
- Wang, Hao (2016). From Mathematics to Philosophy (Routledge Revivals). Routledge. ISBN 978-1-134-88433-9 – via Google Books.
- Thomas, Ivo (1 April 1974). "On a Passage of Aristotle". Notre Dame J. Formal Logic. 15 (2): 347–348. doi:10.1305/ndjfl/1093891315 – via Project Euclid.
- De Venatione Sapientiae, 23.
- Curley, E. M. (October 1971). "Did Leibniz State "Leibniz's Law"?". The Philosophical Review. 8 (4): 497–501.
- 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.
- Rand, Ayn. For the New Intellectual. OCLC 969408226.
- "UNIFORM ABBREVIATIONS OF WORKS BY AYN RAND", Concepts and Their Role in Knowledge, University of Pittsburgh Press, pp. 269–270, doi:10.2307/j.ctt9qh7ww.18, retrieved 2021-09-01.
- Desilet, Gregory (2023). The Enigma of Meaning: Wittgenstein and Derrida, Language and Life. McFarland. p. 133.
External links
In logic the law of identity states that each thing is identical with itself It is the first of the traditional three laws of thought along with the law of noncontradiction and the law of excluded middle However few systems of logic are built on just these laws HistoryAncient philosophy The earliest recorded use of the law appears 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 that they both are Theaetetus Yes Socrates Then do you think that each differs to the other and the same as itself Theaetetus Certainly Socrates And that both are two and each of them one Theaetetus Yes that too It is used explicitly only once in Aristotle in a proof in the Prior Analytics When A belongs to the whole of B and to C and is affirmed of nothing else and B also belongs to all C it is necessary that A and B should be convertible for since A is said of B and C only and B is affirmed both of itself and of C it is clear that B will be said of everything of which A is said except A itself Aristotle Prior Analytics Book II Part 22 68a Medieval philosophy Aristotle believed the law of non contradiction to be the most fundamental law Both Thomas Aquinas Met IV lect 6 and Duns Scotus Quaest sup Met IV Q 3 follow Aristotle in this respect 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 Modern philosophy 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 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 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 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 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 Objectivism the philosophy founded by novelist Ayn Rand is grounded in three axioms one of which is the law of identity A is A In the Objectivism of Ayn Rand the law of identity is used with the concept existence to deduce that that which exists is something In Objectivist epistemology logic is based on the law of identity 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 In his Tractatus Logico Philosophicus Ludwig Wittgenstein writes that roughly speaking to say of two things that they are identical is nonsense and to say of one thing that it is identical with itself is to say nothing In the formal logic of analytical philosophy 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 Continental Martin Heidegger gave a talk in 1957 entitled Der Satz der Identitat The Statement of Identity where he linked the law of identity A A to the Parmenides fragment to gar auto estin noien te kai einai for the same thing can be thought and can exist citation needed Heidegger thus understands identity starting from the relationship of Thinking and Being and from the belonging together of Thinking and Being Gilles Deleuze wrote that Difference and Repetition is prior to any concept of identity citation needed Modern logicIn 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 x x x displaystyle forall x x x 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 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 restricts the law of identity in this way is Schrodinger logic See alsoIdentity philosophy Tautology logic Rectification of namesReferencesWang Hao 2016 From Mathematics to Philosophy Routledge Revivals Routledge ISBN 978 1 134 88433 9 via Google Books Thomas Ivo 1 April 1974 On a Passage of Aristotle Notre Dame J Formal Logic 15 2 347 348 doi 10 1305 ndjfl 1093891315 via Project Euclid De Venatione Sapientiae 23 Curley E M October 1971 Did Leibniz State Leibniz s Law The Philosophical Review 8 4 497 501 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 Rand Ayn For the New Intellectual OCLC 969408226 UNIFORM ABBREVIATIONS OF WORKS BY AYN RAND Concepts and Their Role in Knowledge University of Pittsburgh Press pp 269 270 doi 10 2307 j ctt9qh7ww 18 retrieved 2021 09 01 Desilet Gregory 2023 The Enigma of Meaning Wittgenstein and Derrida Language and Life McFarland p 133 External linksWikiquote has quotations related to Law of identity
