I'm trying to understand this comment on this answer to this question I asked two days ago.
Specifically, I'm trying to understand the difference between the symbol $=$ and the symbol $\approx$ in a universal algebra setting.
The book I'm using Algebraizable Logics defines $\approx$ using a second-order formula $p \approx q \;\;\text{iff}\;\; \forall P \mathop. P(p) \leftrightarrow P(q)$, but I'm not sure whether $P$ is supposed to range over all definable predicates or what.
I'm coming at this specifically from the direction of trying to understand algebraic semantics for nonclassical propositional logics. I have a vague intuition that the elements in the algebras in my variety are supposed to be "similar" to truth values at one end of the spectrum and the full syntax of the propositional logic on the other, so we end up with some kind of lattice of structures with these two as "end points". I think.
My understanding of universal algebra is extremely limited and I mostly think of it as model theory without predicate symbols.
First-order logic with equality has a single distinguished logical predicate $=$. The very short introductory entries on MathWorld and Wikipedia do not mention multiple logical predicates.