I think the title pretty much sums up what I'm looking for, but just to elaborate a bit, I'm looking for something that goes into the philosophical basis for logic. Something that investigates the philosophical justifications for the rules of logic that we commonly use, and explores perspectives that try to understand what logic actually is.
One mathematical book that explains the motivations behind the logical notation is Introduction to Logic by Suppes. This is for classical logic of course, and is also logically rigorous. It explains that we build logic as a precise language by describing it using our imprecise human language, with the very purpose of using logic instead of our human language to perform our reasoning, so that there is no avenue for imprecision or illogical reasoning as long as we have accepted the language of logic and the method of translating natural language sentences into logic.
Haacks Philosophies of Logics (or something similar, I don't remember the exact title) is a classic. Dummett's Logical Basis of Metaphysics, which Mauro Allegranza suggested, is one of the most exiting texts in the field, in my opinion. However, it is also difficult - very difficult in parts. So before reading that I'd suggest Dummett's philosophical basis of Intuitionism (also the final chapter of his elements of intuitionism) and justification of deduction. [Both of these papers are available in `Truth and Other enigmas].