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 an 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)