preliminary unsorted thoughts on logic

Sam, I like this topic (a lot). However before I say something, I need to sort my ideas out. There is a whole discipline of psychology devoted to this question (psychology of reasoning); however, I believe that many times they miss the point of logic. Logic is the study of structure, and most of psychology does not look at structure. In other words, I think that psychology of reasoning does well at observing manifestations of reasoning, but they have not reached the core of reasoning. Because they have not reached the core of reasoning, there are three disputing schools of thought within psychology, and each can argue their view with significant empirical evidence, as well as discount the others. (I said that I would sort my thoughts out and obviously I became too anxious; I'm still going to sort my thoughts.)

Currently, I am at a lost. I want to say "yes," but I can't. Classic logic though powerful in forming mathematical foundations, appears limiting in the world that involves possibles and necessaries, deontics (should and oughts), and other modals. However, are those washed out, weak, or misinterpreted quantifiers? If logic forms a foundation for mathematics through quantificational logic and set-theory, and mathematics is the ground physics rests, then could it also be the correct grounding of human reasoning? I need to get to bed (i.e., sort out my thoughts before I say more).