I'm reading Kleene's introduction to logic and in the beginning he mentions something that I have thought about for a while. The question is how can we treat logic mathematically without using logic in the treatment? He mentions that in order to deal with this what we do is that we separate the logic we are studying from the logic we are using to study it (which is the object language and metalanguage, respectively). How does this answer the question? Aren't we still using logic to build logic?
And I have a feeling that the answer is to some extent that we use simpler logics to build more complex ones, but then don't we run into a paradox of what 's the simplest logic, for won't any system of logic no matter how simple be a metalanguage for a simpler language?