The question of what constitutes or defines knowledge is another big problem in philosophy, and on its own forms what is called epistemology. I leave aside (for now) the central epistemic questions in order to discuss certain common perspectives; I will return to the broader questions in a few days.
Rationalism is, loosely defined, the view that knowledge and truth come from reason, logic, etc. By taking axiom #8 we are inherently taking what is at least a partly rational view, although there is somewhat more to it than that. However, on its own this axiom gives us a whole slew of tools in the category of what I shall call “logical systems”. These include propositional and predicate logic, algebra, set theory, probability and all the other systems that are strictly abstract (despite possible practical applications).
The critical point is that these systems are only tools. While they do produce internal “truths” (such as the statement that two plus two is four), all of these internal truths are at root truths by definition. Even the weirder, hard-to-prove theorems do eventually fall out of starting definitions; that’s how we define a proof. More interestingly, these tools give us methods of “lossless” operation on existing truths. The logical form of modus ponens takes two truths and produces a third truth of the same strength; no certainty is lost in the deduction.
These tools are extremely powerful, but like all tools they are useless on their own; they require raw materials to operate on. There are perhaps certain interesting facts that can be derived from our core set of eight axioms, but I would be very surprised if you could get very far (I have not even bothered to try). Instead, we need some other system to generate facts for us to operate on. Next time I will look at where we can get raw materials to feed our rational tools.