There are "correct" answers, those that can be formally proven in some axiomatic system like mathematics, and there are "best" answers, which can be proven optimal in non-axiomatic systems but which generally cannot be proven "correct". A feature of "best" answers in mathematics is that there is no requirement that they be the same as "correct" answers for a given question, and in fact this feature accounts for most failure modes of rationality and inductive reasoning. Nonetheless, lacking access to a formally provable "correct" answer, the "best" answer has the highest probability of being the correct answer of any assertion that could be made short of formally proving the correct answer. And while we can prove there are a lot of questions to which we can never prove a "correct" answer, we can always generate a "best" one -- a "soft" proof of unprovable assertions. "rational" is usually defined as selecting the "best" answer (which has a formal selection process) in non-axiomatic reasoning systems.
Every bit of reasoning we do about the real world (not the fake world of mathematics) is non-axiomatic ("inductive") since axiomatic reasoning requires a certain amount of omniscience to work, though we fake a bit of first-order logic using it. For that reason, we cannot prove the correctness of any assertion about the universe we live in. Non-axiomatic reasoning systems have two very useful characteristics that make them superior to axiomatic reasoning systems in the real world. First, they are intrinsically adept at reasoning under uncertain, incomplete, or incorrect information and can rapidly adapt to new information -- because there are no axioms, breaking a single assumption does not collapse the whole system. Second, it is computationally much cheaper to compute the "best" answer than it is to prove the "correct" answer in the general case. This is the old 80/20 rule affect; the "best" answer works out just fine the majority of the time while being vastly cheaper. The cost of occasionally being grossly incorrect is worth the cost savings of not proving correctness.
There has been a lot interesting research lately on the concept of pervasively non-axiomatic computation -- computing with "best" answers instead of axiomatic evaluation. They have a number of provable properties, not the least of which is robustness, adaptability, and non-brittleness, which make them very interesting but wildly counter-intuitive from the perspective of classical computer science. Perhaps more interesting are the proofs that all intelligent systems must have this form.
In the form of links to course material, or Scientific American level expositions?
The material isn't too hard, but I don't have time to learn an entire new nomenclature and sets of obscure typographical symbols...
See my earlier posts about having a family, a life, etc. :-)
Cheers!