I'm curious about what math (mathematical logic, metamath) says about the way we reason. This is going to be a vague question because I have not yet explored mathematical logic myself. My question: can we somehow develop, or construct, all of the valid rules of inference
ways we reason about proofs (by contradiction, implication, and so forth) in some mathematical way? Can it be done from some simple principles?
Like, if you had some sort of model, and these ways of reasoning, as objects in that model, that could be derived. What would be the model?