Your link confuses logic - which is about constructing syllogyms that preserve truth value - with
rhetoric, which is the use of language to construct argument.
Logic builds statements from basic axioms via transformations that preserve truth value. Math does the same thing. If math and logic do the same thing, then they're the same. There's just no difference between math and logic.
Logic isn't about language, it's about symbols. Rhetoric is about language, and is in fact philosophy.
On the other hand, you may be referring to informal logic, which we use to debate. I guess I'm referring to formal logic, which is mathematics.