But I do NOT think that reasonable behavior is simply "following" the rules of logic (i.e., cookie-cutter thinking) as much as it is action that INCLUDES not violating the axioms of logic.
To be honest, I have not really ever learned much of what is the axioms of logic. If logic is a method of problem solving, it should relate primarily to the problem, not to principles of problem solving.
Then, solving a problem becomes a matter of understanding the problem; understanding it's different component may be a process of reductionism, but understanding why
it is a problem in the context of what is required in a solution, seems to me to be more holistically oriented.
I would not shy away from a solution that violates the axioms of logic if that solution worked, and I would consider any definition or axiom of logic that allows for such a thing an invalid definition or axiom, as they would serve as restrictions on one's capacity for problem solving. I'd say the only rules that matter are the restrictions provided by the subject that is the essence of the problem.
Historically, if we think about how axioms are postulated then later abandoned when progress shows them to be restricting to progress, I think axioms are just as likely to cause problems as to solve them. That doesn't mean they are useless, because they help identify problems. It's like a swimmer's diving board. It is useful to get him off to a swift start, but once he is in the water he can't really use it for anything.
I am trying to think of aspects of logic that relate to culture, but I am unsuccessful beyond thinking that the choosing of which subjects we apply logic to, how we define and outline problems, depends in part on cultural conditioning. Is the process of solving a problem once the problem is identified related to culture?