That is, in logic there are two basic ways of accepting a theory:

(1) it is consistent

(2) it is at least equi-consistent with some convenient theory, usually arithmetic

However, the human brain only worries about consistency as far as one has time to think (having evolved a logic that was more concerned with escaping predators than working out the next quantum gravity theory); a theory is accepted provisionally until something better comes along, and contradictions which may or may not appear with more thinking are not a cause of worry until it happens. So, if one wishes a logic to imitate human thinking, it would seem useful to have a formal logic which had some concept of "it's consistent up to a certain level of implication". However, the details might get sticky (just as finite models are often less convenient than infinite ones). So, does such a logic exist? If so, what is it called, and can anyone give a good link to explain the nitty-gritty of the logic? Thanks.