Typically, mathematicians and logicians take the following attitude: When we set out the details of formal logic, we implicitly adopt in the background a *meta*-logic. The metalogic is the logic we use *in English*: it's way we reason about things as mathematicians, as humans, etc., and the hope, then, is that our *formal* theory of logic (set out in a *formal* language, not English) will *match up* with the metalogic we are using (*in English*). So yes, while it's true that we are using logic to set out the formal details of a *theory* of logic, still the hope is that the *theory* of logic accurately represents the way we are using logic in the background.

This, of course, requires admitting at the beginning what kind of logic you're employing as your metalogic. A clear manifestation of this comes when defining what the various formal connectives $\wedge, \vee, \neg$, etc. mean. Often, we'll give an inductive definition as follows:

$\mathcal{M} \vDash \varphi \vee \psi$ if and only if either $\mathcal{M} \vDash \varphi$ or $\mathcal{M} \vDash \psi$

This may look innocent enough, until we ask what the "**either...or...**" clause of the definition is. Is it a classical disjunction, or an iniutionisitic one, or neither? Surely, answers to such questions will affect our understanding of the given definition. Similarly, when we write:

$\mathcal{M} \vDash \neg\varphi$ if and only if it's not the case that $\mathcal{M} \vDash \varphi$

how are we to understand the English-negation "**it's not the case that...**"? If our metalogic is classical logic, then the negation is classical; but different choices of metalogic will reflect a difference in our understanding of these definitions.

There's a lot of philosophical literature revolving around this point. For instance, it really takes centerfield in certain nonclassical logics, e.g. *relevant logic*, where people working in this field typically wish for their theories of logic to be "taken seriously" as *genuine* logic (i.e. the logic we do and/or should use *in English*). There are theorems about the formal theory of relevant logic (for instance, the admissibility of distribution) for which there is a *classical* proof but not a *relevant* proof (or rather, there is a relevant proof of the *classical* admissibility, but no proof of the *relevant* admissibility). So which metalogic one adopts in describing a formal theory of *relevant* logic becomes crucial. In general, which metalogic one adopts, *as well as* which *formal logic* one wishes to study, can affect the results that are obtainable in one's pursuits (intuitionistic logicians have been particularly concerned with this matter as well).

Most of the time, for the mathematician, it suffices to just adopt a classical metalogic, since even then there's a lot that can be said about the formal theory of logic!