In discussion with Jon Cohen in the past few weeks, I’ve realized a bit more explicitly the different ways logic seems to be treated in the three different disciplines that it naturally forms a part of. I intended to post this note as a comment on his recent post, but for some reason the combination of old operating system and browser on the computer I’m using here wouldn’t let me do it. So I’m making it my own post.
The thing that has struck me most in my experience with doing logic in both mathematics and philosophy departments is that in mathematics, “logic” is seen as just being classical propositional or first-order logic, while in philosophy a wide range of other logics are discussed. The most notable example is modal logic of various sorts, though intuitionist logic and various relevance logics and paraconsistent logics are also debated in some circles. But in talking to Jon I’ve realized that there are far more logics out there that very few philosophers are even aware of, like linear logics, non-commutative logics, and various logics removing structural rules like weakening, contraction, or exchange (which basically allow one to treat the premises as a set, rather than a multiset or sequence). In his sketch of history, it seems mathematicians are stuck in the 1930’s, and philosophers are stuck in the early 1980’s, in terms of what sorts of systems they admit as a logic. Of course, all three disciplines have developed large amounts of logical material relating to their chosen systems.
The reason for these divisions seems to be a disagreement as to what a logic is. Mathematicians just want to formalize mathematical reasoning in a sense, and so have fixed on classical logic, as it seems to best capture the work that mathematicians find acceptable and necessary. Philosophers on the other hand, have debates about whether classical logic, intuitionism, some sort of relevance, or some other logic is the “one true logic” (or one of the several true logics as advocated by Greg Restall and JC Beall). Although computer scientists study even more types of logic, they don’t seem to argue about which is an appropriate logic for doing their reasoning in – from what I understand, they do all their metareasoning in classical logic (or some approximation thereof). The various systems are studied to gain structural insights, and to model the capacities of various computational systems, but not to talk about truth per se.
Does this sound about right?