Usually when we think of a rational belief, we think of a belief that is held for good reasons. There are two ways a belief might be held for good reason, though. One way is that some evidence or line of sound reasoning resulted in the belief. That's true of a posteriori beliefs. Another way is that a person can, by reflecting on something, grasp the necessity of it. That's true of at least some a priori beliefs.
But what about other beliefs such as the belief in the reliability of our cognitive faculties, our sensory perceptions, and our memories? There's no necessity to these beliefs. Whether there is any line of reasoning that could lead to these beliefs is questionable. If there were a line of reasoning that lead to these beliefs, it's likely that the premises would be less certain than the belief itself, which means the belief is probably not held because of that line of reasoning.
Some beliefs appear to be hard wired. They're just built in. We automatically assume that past experience can tell us something about what to expect in the future. We automatically assume that we remember things because a past really happened. Some people become philosophers and begin to question these things, but even people who question them find it hard to do so. They have to continuously resist the temptation to attribute truth to these seemingly built in notions.
So what justifies them? Since they cannot be justified on the basis of sound reasons, nor on the basis of their necessity, it seems like the only way to justify them is on the basis of their causes. If they are caused by a reliable mechanism, then they can be trusted, but if they are caused by an unreliable mechanism, then they can't be trusted.
Suppose these beliefs are caused merely by the structure and activity of the brain. We would have to trust that the brain is built in such a way as to be reliable. We can trust a calculator to give us the right answers because calculators were designed by people with accuracy in mind. With brains, they had to have come about in one of two ways--either they were engineered or they arose through purely blind and natural processes.
Suppose the brain was engineered. Before we could trust the deliverances of our brains, we'd first have to presuppose the trustworthiness of the engineer. After all, it's possible somebody could stick a probe in your brain and cause you to believe all sorts of things that aren't true. Maybe we don't have the technology yet, but it's conceivable that the technology could exist. So if we held a belief that was caused by an engineer who designed our brains in such a way as to produce that belief or he somehow installed that belief into our brains, would that belief be rational?
I don't know, but I suspect you'd need the presupposition that the engineer was honest/reliable or that the engineer designed the brain in such a way as to produce true a priori beliefs. Otherwise, the belief might not be rational.
But then how would you justify the belief that the engineer was reliable and had truth in mind when designing the brain or installing the beliefs? It seems like the only way to justify that belief is purely on pragmatic grounds. We have no choice but to trust our cognitive faculties. We couldn't rationally doubt them unless we could trust them, so to doubt them is self-refuting. If I say, "My brain is an unreliable truth-generating machine," I would have no rational basis upon which to believe that statement is true since my brain is the only thing could possibly tell me whether or not it's true.
Maybe it's rational to trust our brains merely because it's irrational not to, and it's irrational not to because any claim that our brains are unreliable will necessarily be a self-refuting claim. Of course self-refuting claims aren't necessarily false. It depends. A claim can be self-refuting in one of two ways. It can be self-refuting in such a way that if it is true, then its false, in which case it's necessarily false, or it can be self-refuting in such a way that if it's true, then it can never be justified, in which case it isn't necessarily false. The self-refuting nature of claiming that the brain is an unreliable truth-producing machine is self-refuting in the second sense, but not the first. (Maybe we should call the second sense "self-undermining" instead of "self-refuting.") So while it may be irrational to doubt the reliability of your brain, that doesn't necessarily mean your brain is reliable. It just means it's more rational to affirm the reliability of your brain (since you can do so consistently) than it is to deny the reliability of your brain (since you cannot do so consistently).
If we must trust the reliability of our brains on pain of self-refutation, then it seems like we must also believe that the brain was produced in such a way as to guaranty its reliability. So if you believe the brain was engineered, then you must believe the engineer was honest and had truth in mind when he designed the brain. Otherwise, you're being irrational.
The alternative is to believe the brain evolved naturally in such a way as to be reliable. In this case, nobody meant for the brain to hold true beliefs. It's just that holding true beliefs was conducive to survival and reproduction. I don't want to explore this right now because it would take us into Alvin Plantinga's evolutionary argument against naturalism. I'll just say that if a sound argument can be made for why we should expect the brain to be reliable on the supposition that it evolved naturally, then that would give us a rational ground for our built in beliefs.
So basically, I think our beliefs can be rational on the basis of sound reasoning from true premises, intuitively grasping the necessity of them, or having them be built into our brains under the presupposition that the brain was produced in such a way as to be a reliable belief producing machine, whether that entails presupposing a brain engineer or a truth-favoring evolutionary process (or some combination of the two).
I have a lot more to say about this, but I guess this post is long enough already.