Are there any modern approaches to logic that are non-mathematical in nature or not grounded in math?
Readers coming at this material from the point of view of modern logic may be surprised to find very few of what are sometimes called “logical results” — that is, theorems about interesting general logical truths. In fact, you may think what we are doing isn’t really "logic" at all, but more philosophy of language, or even philosophy of mind or epistemology. Why then call it logic? The short answer to this of course is that they called it "logic," and they got there first! A less contentious response would be to point out how much the close connection between logic and the foundations of mathematics in the recent period has shaped our view of what logic is, to the point of making it hard sometimes for us to think of logic in any other terms.
I've seen other comments elsewhere about the success of the mathematical approach to studying logic and how it revolutionized the field, but this passage has a rather different tone. It suggests but does not directly claim that conflating an application of logic and logic itself obscures some important and interesting ideas.
I've really enjoyed learning bits and pieces of traditional syllogistic logic, particularly the pieces that differ radically from mathematical approaches as I understand them, such as not including syllogisms with weaker-than-possible conclusions from the set of valid syllogisms, and not having syllogisms with inconsistent premises.