There seems to be a distinction between studying logic mathematically and studying it philosophically and, in practice, it is reasonably clear which framework one is using when one studies logic.
I've used this distinction in the past to try to defend using very strong theories (classical FOL + ZFC) to analyze simple ones (that they arguably depend on) like propositional calculus by claiming that the apparent conceptual circularity is a philosophical concern and is not a mathematical concern. Now I'm wondering whether this position is wrong or naive.
I'm wondering whether this position (i.e. there is an important/meaningful distinction in practice between studying logic mathematically and studying logic philosophically) has a name and whether any philosophers have described, defended, or rebutted it specifically.
When you study logic mathematically, you assume ordinary mathematical reasoning in the background (which in practice usually means classical logic and ZFC or NBG or another mainstream set theory), and aren't particularly concerned about the meaning of the deductive system you're working with. You're just analyzing it as a mathematical object. Also, more to the point, when studying logic mathematically one is not bothered by conceptual circularity, such as using natural numbers at the meta-level even though logic is conceptually prior to natural numbers.
Studying logic philosophically would include questions that are not specifically about analyzing the behavior of a logical system, using a possibly-formalized background theory like ZFC or ordinary mathematics.
Here are some questions that I think illustrate the concept of investigating a logic philosophically. Many of them are extremely naive, but that's just because I made them up instead of grabbing them from somewhere.
- How well does some deductive system capture inferences that are valid in natural language?
- What are the ontological commitments of a deductive system (such as plural logic or second-order logic)?
- What's the right way to measure how "good" a logic system is?
- Is there one true logic?
- Is logical pluralism a coherent position?
- What, if anything, is the relationship between logic and truth?