Definition talk:Equality

I didn't write $\forall P$ or $\forall S$ to keep the definition a first-order statement. If it's better to write it as a shorter, second order statement, please let me know and I will fix it. --GFauxPas 12:36, 3 November 2011 (CDT)
 * Don't need any of it because it's already in a) the page it redirects to, and b) in the Set Equality pages. So I rolled it back to what it was. --prime mover 13:21, 3 November 2011 (CDT)

Prime mover I was setting up this definition because the definition in this form allows me to prove the the reflexivity, transitivity, symmetry, and commutativity of equality, which I didn't see any proofs for. The definitions of equality already on PW aren't in the form of conditionals so how do I use them in logical proofs? --GFauxPas 13:33, 3 November 2011 (CDT)


 * You want the page Equals is Equivalence Relation. I'll put a link to it. The Axioms of Equality are something that arise from the basics of Predicate Calculus which I never got round to working my way completely through.
 * In any case, the page Definition:Equality is a redirect to Definition:Equals. The two things are the same. We don't need two different expositions for the same concept. That's what redirects are for. --prime mover 15:13, 3 November 2011 (CDT)

Thank you for the explanation prime.mover. In any event, what should I do about proving that equals is symmetric, transitive, and reflexive? Leave them to you? --GFauxPas 15:26, 3 November 2011 (CDT)


 * "... that equals is symmetric, transitive, and reflexive" is known as the Axioms of Equality. That page does not exist. If you want to write it, then feel free to go ahead. Although they are called "axioms", they follow (apparently) from the more basic axioms of propositional and predicate calculus. I believe that it may be necessary to put more groundwork in place (predcalc has hardly been touched upon yet) before we have a solidly rigorous basis yet. But feel free to have a go. --prime mover 15:42, 3 November 2011 (CDT)

Prime.mover, I'm using a logic book by Tarski. He says that if you allow Leibniz's law ($ x = y \iff \forall P : P(x) \iff P(y)$) as a definition, you can prove symmetric/transitive/reflexive using more basic axioms. But he says that Leibniz's law is a definition "only if the meaning of the symbol "$=$" seemed to us less evident than that of the expression [$P(x) \iff P(y)$]". Tarski says that Leibniz himself took the first approach, that the law is a definition of "$=$". Wikipedia has some interesting articles on other approaches, but they seem kind of meta. --GFauxPas 16:13, 3 November 2011 (CDT)


 * Interesting, I haven't thought about this properly before. Excuse me while I think out loud.


 * ... "if you allow Leibniz's law ($ x = y \iff \forall P : P(x) \iff P(y)$) as a definition". It's an approach. The philosophical question remains: if every single property of two items is the same for both items, does that mean the items are the same item? Interesting. I would say that they were the "same" item, because otherwise if they were not the same item, then object $a$ would have "the property of being object $a$" while object $b$ would not have "the property of being object $a$" unless object $b$ were object $a$, in which case objects $a$ and $b$ are the same object.


 * On the other hand, if you wish to define objects which are not the same objects as "equal" because they both have the same properties out of a predefined and separately understood collection of properties in a specifically defined context, then you can specify a particular, limited kind of equality which can only ever be something less than "true" equality. Examples come to mind: similarity: the set of angles of two triangles are equal. If you only allow yourself to consider the properties of a triangle to be the angles, then you can say that "similar triangles are equal". But you never do, so you can't.


 * On the other hand, again, if you consider two triangles to be equal if they have the angles and the sides matching between them, then it's less clear-cut. To my mind, they are different triangles which are not equal: they have equal area and they have equal sides and they may even have equal orientation but they are not "equal" as such unless they enclose the same specific piece of territory, i.e. they have the same position, in which case, yes, they are the equal triangles because they are the same triangle.


 * It might be worth adding Tarski's analysis into this page (I haven't read it myself, I may hunt it down), but I'd recommend against actually replacing what's there unless you really feel certain of your ground. And, good as Tarski might well be, that requires that you read a good deal more books on the subject than just Tarski, and make sure you understand all of them. And that's at the moment beyond what I feel confident to do myself, and I've read a fair way round this subject, enough to know that there is no dogmatic "this is the truth, all else is inaccurate or incomplete" because there are many approaches which may be used to come to the same basic truth, which is: Equals is Equivalence Relation. --prime mover 17:04, 3 November 2011 (CDT)

Very well put and your advice is noted, thank you. Tarski and Khan Academy address the distinction between two triangles being the same ($=$) and being the same shape/angle, but perhaps in a different position ($\cong$). --GFauxPas 17:36, 3 November 2011 (CDT)