Editing Equality (mathematics) (section)

=== Axioms ===
[[File:Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg|thumb|[[Gottfried Wilhelm Leibniz|Gottfried Leibniz]], a major contributor to [[17th-century mathematics]] and [[philosophy of mathematics]], and whom the ''Substitution property of equality'' is named after.|upright=1.2]]
''[[Law of identity]]'': Stating that each thing is identical with itself, without restriction. That is, [[Universal quantification|for every]] <math>a,</math> <math>a = a.</math> It is the first of the traditional [[Law of thought#The three traditional laws|three laws of thought]].<ref>"Laws of thought".  ''[[The Cambridge Dictionary of Philosophy]]''.  [[Robert Audi]], Editor,  Cambridge University Press. p. 489.</ref> Stated symbolically as:

<math display="block">\forall a(a = a)</math>

''[[Substitution (logic)#Mathematics|Substitution property]]'': Sometimes referred to as ''[[Gottfried Wilhelm Leibniz|Leibniz's]] law'',<ref>{{Cite encyclopedia |title=Identity of indiscernibles |url=https://www.britannica.com/science/identity-of-indiscernibles |access-date=2025-01-12 |encyclopedia=Encyclopædia Britannica}}</ref> generally states that if two things are equal, then any property of one must be a property of the other. It can be stated formally as: for every {{mvar|a}} and {{mvar|b}}, and any [[Well-formed formula|formula]] <math>\phi(x),</math> (with a [[free variable]] {{mvar|x}}), if <math>a=b,</math> then <math>\phi(a)</math> [[Material conditional#Definitions|implies]] <math>\phi(b).</math> Stated symbolically as:

<math display="block">(a=b) \implies \bigl[ \phi(a) \Rightarrow \phi(b) \bigr]</math>

Function application is also sometimes included in the axioms of equality,<ref name="Grishin" /> but isn't necessary as it can be deduced from the other two axioms, and similarly for symmetry and transitivity. (See {{Section link||Derivations of basic properties}}) In [[first-order logic]], these are [[axiom schema]]s (usually, see below), each of which specify an infinite set of axioms.<ref name="Hodges1983">{{Cite book |last=Hodges |first=Wilfrid |date=1983 |editor-last=Gabbay |editor-first=D. |editor2-last=Guenthner |editor2-first=F. |title=Handbook of Philosophical Logic |publisher=Springer |location=Dordrecht |pages=68–72 |doi=10.1007/978-94-009-7066-3 |isbn=978-94-009-7068-7}}</ref> If a theory has a predicate that satisfies the Law of Identity and Substitution property, it is common to say that it "has equality", or is "a theory with equality".{{Sfn|Mendelson|1964|pp=82–83}}

The use of "equality" here somewhat of a [[misnomer]] in that any system with equality can be modeled by a theory without standard identity, and with [[indiscernibles]].<ref>{{Cite encyclopedia |last1=Deutsch |first1=Harry |title=Relative Identity |date=2024 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/identity-relative/ |access-date=2025-01-20 |edition=Fall 2024 |publisher=Metaphysics Research Lab, Stanford University |last2=Garbacz |first2=Pawel |editor2-last=Nodelman |editor2-first=Uri}}</ref><ref name="Hodges1983" /> Those two axioms are strong enough, however, to be isomorphic to a model with idenitity; that is, if a system has a predicate staisfying those axioms {{Em|without}} standard equality, there is a model of that system {{Em|with}} standard equality.<ref name="Hodges1983" /> This can be done by defining a new [[Domain of discourse|domain]] whose objects are the [[equivalence class]]es of the original "equality".{{Sfn|Kleene|1967|pp=158-161}} If the relation is interpreted as equality, then those properties are enough, since if <math>x</math> has all the same properties as <math>y,</math> and <math>x</math> has the property of being equal to <math>x,</math> then <math>y</math> has the property of being equal to <math>x.</math><ref name="Russell1910" /><ref>{{Cite book |last=Suppes |first=Patrick |url=https://web.mit.edu/gleitz/www/Introduction%20to%20Logic%20-%20P.%20Suppes%20(1957)%20WW.pdf#page=120 |title=Introduction to Logic |date=1957 |publisher=[[Van Nostrand Reinhold]] |location=New York |page=103 |lccn=57-8153}}</ref>

As axioms, one can [[Logical consequence|deduce]] from the first using [[universal instantiation]], and the from second, given <math>a = b</math> and <math>\phi (a),</math> by using [[modus ponens]] twice. Alternatively, each of these may be included in logic as [[Rule of inference|rules of inference]].<ref name="Hodges1983" /> The first called "equality introduction", and the second "equality elimination"<ref>{{Cite web |title=Introduction to Logic – Equality |url=http://logic.stanford.edu/intrologic/extras/equality.html |access-date=2025-03-01 |website=logic.stanford.edu}}</ref> (also called [[paramodulation]]), used by some [[Theoretical computer science|theoretical computer scientists]] like [[John Alan Robinson]] in their work on [[Resolution (logic)|resolution]] and [[automated theorem proving]].<ref>{{cite book |last1=Nieuwenhuis |first1=Robert |url={{GBurl|HxaWA4lep_kC |pg=PR9}} |title=Handbook of Automated Reasoning |last2=Rubio |first2=Alberto |date=2001 |publisher=Elsevier |isbn=978-0-08-053279-0 |editor1-last=Robinson |editor1-first=Alan J. A. |pages=371–444 |chapter=7. Paramodulation-Based Theorem Proving |editor2-last=Voronkov |editor2-first=Andrei |chapter-url=http://www.cmi.ac.in/~madhavan/courses/theorem-proving-2014/reading/Nieuwenhuis-Rubio.pdf}}</ref>