First order logic: How could $=$ be both a logical symbol be both a predicate symbol and logical symbol? I'm studying first-order logic and I saw the textbook put equality symbol $=$ ads a logical symbol.
It's not a logical connective symbol, so $=xy$ is an atomic formula.
https://en.wikipedia.org/wiki/Atomic_formula
However, the textbook explained it in such a way:" For example, $=v_1v_2$ is an atomic formula, since $=$ is a two-place predicate symbol and...".
But $=$ was listed and was obviously a Logical symbol (  a "superset" of equality symbol), not a parameter( a "superset" of Predicate symbols ).
How could $=$ be both a logical symbol and predicate symbol?
 A: First, $=$ is a predicate symbol since it says something about two objects, namely that they are identical. Out of all predicate symbols, though, it is the only 'logical' one, so it is indeed not like any other predicate, nor is it like any other logical symbol.
OK, but why is it a logical symbol? One answer is that $=$ is a logical symbol in the sense that in logic it has a fixed meaning (identity!), just like the other logical symbols like $\land$, $\neg$, $\forall$ have a fixed meaning. Whereas the non-logical symbols, such as atomic propositions like $P$ and $Q$, or predicates like $P(x)$ or $R(x,y)$ do not have a fixed meaning: indeed, one can interpret those symbols in any way one wants, which is exactly what formal interpretations do.
... but that doesn't really answer the question ... we can still ask why $=$ is treated as a logical symbol ... why is its meaning fixed, unlike other predicate symbols?
Well, conceptually, identity is 'logical' in the sense that no matter what context or subject or domain one is talking about, identity is just that: identity. That is, in any domain, it will be true that every object is identical to itself, and that if $a=b$, and $a$ has some property $P$, then $b$ will have that property as well (of course! $a$ and $b$ are the very same object!). So, a statement like $a=a$ is logically true; it is true no matter how crazy of a world one imagines; no need to even know what $a$ stands for.
Contrast that with a statement like $1 < 2$: this is 'merely' true in the sense that we mathematically defined $1$, $2$, and $<$ in a certain way ... that is, it is a statement about mathematically defined objects, and its truth is therefore relative to a specific domain, and therefore not a pure logical truth.
A: This is a good question. Depending on who you ask, $=$ is either a special logical symbol, or a special predicate symbol. So it can be presented as either one.
Syntactically, it behaves like a two-place predicate. Two-place predicates, like $\equiv, <, \le$, etc. create a formula out of any two terms. $=$ is like this: for any terms $t_1$ and $t_2$, "$t_1 = t_2$" is a formula.
However, $=$ is a bit special in that it is not just any two-place predicate, but it also obeys special rules. The special rules boil down to two facts:


*

*For every term $t$, $t = t$.

*For every two terms $t,t'$, if $t = t'$ then $t$ may be replaced by $t'$ (and vice versa) in any term.
Because $=$ cannot be interpreted as any two-place predicate, and actually obeys special rules, it is therefore sometimes thought of as a logical symbol instead of as a two-place predicate. However, unlike other logical symbols, it operates on terms instead of formulas.
The bottom line is that the equality symbol is special and doesn't quite fit in with all the other symbols. It may be thought of as part of the logic, or as a special predicate symbol that satisfies additional logical rules.
