Can equality be defined in second-order logic as the intersection of all reflexive relations? Can equality be defined in second-order logic as the intersection of all reflexive relations? And if it can, can it also be proven that it is the same definition as the intersection of all equivalence relations? I would like to see a paper or text where this is done.
 A: To move this off the unanswered queue, yes, the equality relation on a set $X$ is the intersection of all reflexive relations on $X$. In fact, we can do one better: it's the smallest reflexive relation on $X$. This follows from the basic rules governing equality which are built directly into our logic.
Specifically, the relevant rules are:

*

*Reflexivity: $\forall x(x=x)$.


*Substitution: From a set of formulas $\{\varphi, t_1=s_1,...,t_n=s_n\}$ where the $t_i$s and $s_is$ are terms (possibly involving free variables) and no free variable in any of the $t_i$s or $s_i$s appears bound in $\varphi$, we may infer $\psi$ whenever $\psi$ is gotten from $\varphi$ by replacing some $t_1$-instances by $s_1$, ...., and replacing some $t_n$-instances by $s_n$.
Reflexivity trivially gives us half our goal. For the other half, suppose $E$ is reflexive. Let $x=y$; we want to show $xEy$. By reflexivity of $E$ we have $xEx$. Now apply substitution with $\varphi\equiv xEx$, $t_1\equiv x$, and $s_1\equiv y$, and $\psi$ gotten from $\varphi$ by replacing the second instance of $t_1$ by $s_1$. (I'm using "$\equiv$" here to denote equality in the "meta-framework.")
I don't know of a citation for this, though; given the immediateness of the argument, I suspect it's generally taken for granted. That said, there are lots of good texts which carefully discuss the logical treatment of equality in general, and if they don't go over this specific point explicitly they certainly develop the relevant ideas. I'll add a citation to such a text later on (I have to run at the moment).
