Underlying language to specify various types of logic

Question

There exist several different types of logic -- 1st order, 2nd and higher order with many different sets of inference rules possible.

What I'm having trouble understanding is what's the "underlying logic" we use to talk about the various different types of logic. For instance, if we were limited to thinking in terms of $i$th order logic, then we couldn't talk about $>i$th order logics but we can. Does this mean that the logic we think in terms of is more powerful than any $n$th order logic?

Sorry this is so vague but I'm having a hard time formulating my question exactly.

Edit: Turing machines cannot even decide all statements in first-order logic -- so does this imply some sort of relation between first-order logic and any other logic that we can think of (as humans are no more powerful than TMs)?