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Law of identity
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== History == ===Ancient philosophy=== The earliest recorded use of the law appears in [[Plato]]'s dialogue ''[[Theaetetus (dialogue)|Theaetetus]]'' (185a), wherein [[Socrates]] attempts to establish that what we call "sounds" and "colours" are two different classes of thing: {{quote|Socrates: With regard to sound and colour, in the first place, do you think this about both: that they both are? <br>Theaetetus: Yes.<br>Socrates: Then do you think that each differs to the other, and ''the same as itself''?<br>Theaetetus: Certainly.<br>Socrates: And that both are two and each of them one?<br>Theaetetus: Yes, that too.}} It is used explicitly only once in Aristotle, in a proof in the ''[[Prior Analytics]]'':<ref>{{cite book|url=https://books.google.com/books?id=h6FTDAAAQBAJ&pg=PA137|title=From Mathematics to Philosophy (Routledge Revivals)|first=Hao|last=Wang|date=2016|publisher=Routledge|isbn=978-1-134-88433-9 |via=Google Books}}</ref><ref>{{cite journal |last=Thomas |first=Ivo |date=1 April 1974 |title=On a Passage of Aristotle |url=http://projecteuclid.org/euclid.ndjfl/1093891315 |journal=Notre Dame J. Formal Logic |volume=15 |issue=2 |pages=347–348 |doi=10.1305/ndjfl/1093891315 |via=Project Euclid |doi-access=free}}</ref> {{quote|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|source = ''[[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: {{quote|...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.<ref>De Venatione Sapientiae, 23.</ref>|}} ===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."<ref>{{cite journal |last1=Curley |first1=E. M. |date=October 1971 |title=Did Leibniz State "Leibniz's Law"? |journal=The Philosophical Review |volume=8 |issue=4 |pages=497–501}}</ref> [[Identity of indiscernibles|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 if x = y. [[John Locke]] (''[[Essay Concerning Human Understanding]]'' IV. vii. iv. ("Of Maxims") says: {{quote|[...] 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.<ref>''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.</ref> [[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: {{quote|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.<ref>{{Cite book |last=Rand |first=Ayn |url=http://worldcat.org/oclc/969408226 |title=For the New Intellectual |oclc=969408226}}</ref> In Objectivist epistemology logic is based on the law of identity.<ref>{{Citation |title=UNIFORM ABBREVIATIONS OF WORKS BY AYN RAND |url=http://dx.doi.org/10.2307/j.ctt9qh7ww.18 |work=Concepts and Their Role in Knowledge |pages=269–270 |publisher=University of Pittsburgh Press |doi=10.2307/j.ctt9qh7ww.18 |access-date=2021-09-01|url-access=subscription }}.</ref> ===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 (philosophy of language)|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 [[Walter Scott|Scott]] was the author of [[Waverley (novel)|''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."<ref>{{cite book |last1=Desilet |first1=Gregory |title=The Enigma of Meaning: Wittgenstein and Derrida, Language and Life |date=2023 |publisher=McFarland |page=133}}</ref> In the [[logical form|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 [[Singular term|term]] rather than a [[proposition]], and thus the law of identity is not used in [[Propositional calculus|propositional logic]]. It is that which is expressed by the equals sign "=", the notion of [[Identity (philosophy)|identity]] or [[Equality (mathematics)|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).{{cn|date=October 2023}} 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.{{cn|date=October 2023}}
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