I filled my mandatory philosophy credit hours many years ago, so this might be a very silly question: what happens when a product of philosophy can be demonstrated to logically not follow? It was my impression that such a product would be discarded, making logic extremely important in philosophy - unlike, unfortunately, many other human endeavors.
I see it this way. Logic is a method. It is one of the ways _how_ philosophy does what it does. (You could argue it is the only way, and many would not disagree with you.) However, consider what other disciplines do not require logically coherent inferences. Most, no? Art is one of the few which does not. Therefore logic is not central to philosophy because it is central to practically all disciplines! What is central to philosophy is, as I said, _abstraction_. The willingness to go meta-, to stand back from what you're looking at and questions the conditions of its possibilities. Do you see where I'm coming from? To make a stronger point, and to not at the same time denigrate your position, philosophers go astray when they focus on logic. :)
Though people are used to ambiguity and love extracting meaning from arbitrary stories, so it's not all that obvious of an issue. At its core philosophy is not about extracting meaning through analysis it's about conveying what someone has already desided to be true.
I think the field strives for a little more than that, at the very least discussion is preferred to rote learning. Logic provides a common language for that discussion.
Personally, I think the means of abstraction and logical reasoning we take for granted today are the direct result of thousands of years of philosophy!
B) Randomly trying things for thousands of years is likely to produce some interesting results. That does not mean doing so is a good idea.