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Question: What is a simple yet precise definition for "=" as a binary relation?

My try: I find two definitions for "equality relation" which seems to be contradictory.

The first one I learnt in school is that equality is what is called an equivalence relation, that is, it satisfies three axioms: Reflexivity, Symmetry, Transitivity.

The second definition contains an axiom of extensionality.

The third definition I heard contains this additional axiom: $x=y$ implies $P(x)=P(y)$

Thanks to the comments, I also learnt a definition using first-order logics:

These equality axioms are:

  1. Reflexivity.
  2. Substitution for functions.
  3. Substitution for formulas.

This is close to the things I am looking for. However, this definition looks weird as the second axiom is a special case of the third. Axiomatic system usually don't use redundant axioms.

dodo
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    The first two aren't definitions of equality: there are equivalence relations other than $=$ (consider "has the same parity as"), and the axiom of extensionality is a characterization of equality rather than a definition of it (okay, I suppose one could view $\in$ as more primitive than $=$ and then take extensionality as a definition of $=$ in terms of $\in$, but I personally don't buy that - if only because non-extensional set theories are also interesting). – Noah Schweber Feb 12 '24 at 02:36
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    Recently there was a similar query, my anser there applies here also – m-stgt Feb 12 '24 at 03:35
  • You are right that equality is an equivalence relation, but it is not the only one of course. – Alex Feb 12 '24 at 03:46
  • in my experience, in logic, $=$ is often used to denote the identity relation, that is, $a=b$ iff $a$ and $b$ are the very same object. In turn, this implies every predicate possessed by $a$ is also possessed by $b$ and vice versa. The identity relation is an example (or instance) of an equivalence relation, but there are equivalence relations that are not identity relations. – RyRy the Fly Guy Feb 12 '24 at 06:06
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    equality is a basic concept of human language and thought. In mathematics, it can be formalized using predicate logic. – Mauro ALLEGRANZA Feb 12 '24 at 06:28
  • For equality in set theory see here and here – Mauro ALLEGRANZA Feb 12 '24 at 06:30
  • See also Defining equality – Mauro ALLEGRANZA Feb 12 '24 at 06:32
  • @NoahSchweber I agree with your comments so what is a definition for equality? – dodo Feb 12 '24 at 09:32
  • @MauroALLEGRANZA Thanks for you comments. I only need a simple definition for "=". – dodo Feb 12 '24 at 10:02
  • @dodo -- IMO there can't be a single definition of the $=$ sign, rather a list of usage. (It's like the word "mean" where my dictionary shows 9 different explications.) As with the equality sign it's similar with the term equality -- its meaning evolves from the context. – m-stgt Feb 12 '24 at 10:09
  • @MauroALLEGRANZA Your first link is similar to what I am trying to look for. Though I have one question: in your link, there are three axioms, and the second axiom is a special case of the third axiom. This is a bit weird, as axiomatic system usually don't use redundant axioms. – dodo Feb 12 '24 at 10:10

1 Answers1

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Re your addendum to the question.

Yes, the two basic axioms for equality are reflexivity: $x=x$, and substitution for formulas: $s = t \to (\varphi [s/z] \to \varphi [t/z])$.

Note that, using Universal instantiation, we have that reflexivity holds for terms whatever: $t=t$.

Now, if $f$ is a function symbol (assuming for simplicity that it is a unary function symbol) from reflexivity we have $f(x)=f(x)$.

Thus, having that $f(x)=f(x)$ is $(z=f(x))[f(x)/z]$ and $f(y)=f(x)$ is $(z=f(x))[f(y)/z]$ we can use susbstitution to get:

$x=y \to f(x)=f(y)$.

The substitution property is crucial for equality: if the binary relation has reflexivity, symmetry and transitivity but lacks the substitution property its is an equivalence relation.

Mauro ALLEGRANZA
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  • Many thanks! So there are two equivalent definitions: The first one is reflexivity+substitution for formulas. The other one is reflexivity+symmetry+transitivity+substitution for functions? – dodo Feb 13 '24 at 04:37
  • How is substitution for function relates to substitution for predicates ($x=y\implies P(x)=P(y)$)? – dodo Feb 13 '24 at 04:38
  • @dodo - subst for predicate is a case of subst for formulas: $x=y \to (P(x) \to P(y))$. If $P$ is a predicate symbol, $P(x)=P(y)$ is wrong. – Mauro ALLEGRANZA Feb 13 '24 at 06:43
  • Oh, I didn't know this before. Good point! – dodo Feb 13 '24 at 18:22