The logic we are mainly occupied with here is first-order logic (sometimes called first order predicate calculus (with functional terms and identity)). A case can be made that it is the main, or the one true, logic. Sowa writes:

Among all the varieties of logic, classical first-order logic has a priviledged status. It has enough expressive power to define all of mathematics, every digital computer that has ever been built, and the semantics of every version of logic including itself. Fuzzy logic, modal logic, neural networks, and even higher-order logic can be defined in [first-order logic]....

Besides expressive power, first-order logic has the best-defined, least problematic model theory and proof theory, and it can be defined in terms of a bare minimum of primitives....

... Quine demonstrated that first-order logic plus the membership operator ∈ provides enough power to define all of set theory and the foundations of mathematics ....

Since first-order logic has such great power, many philosophers and logicians such as Quine have argued strongly that classical [first-order logic] is in some sense the "one true logic" and that the other versions are redundant, unnecessary, or ill-conceived. [Sowa [2000] Knowledge Representation p.41]

Quite a lot of interesting and important work can be done by adding certain special symbols to first order logic and then augmenting the logic with some axioms which apply to the subject area in question. These specialized axiom sets are known as first order theories. There is benefit in having some exposure to these.

Maria Manzano [1996] Extensions of First Order Logic