The problem brought up in the OP is essentially the problem of hard solipsism. It's an unfortunate reality that the laws of logic are not self-confirming - we cannot take them as given until we assume them as given. And at a certain level, we do in fact have to have faith. Even if we can put two stones with two stones and end up with four stones, how can we tell that the stones even exist? How do we know we're not brains in a vat?
The logical absolutes must be assumed, as they are the basis of any understanding of the world. However, as always, Ockham's Razor is a very useful tool here. We need to assume something. Assuming that the logical absolutes are true gives us a very basic, very useful framework to work off of. Assuming that god exists does not, as even once you assume that, you have to proceed to assume the logical absolutes, or assume quite a lot about god's character.
At the end of the day, though, I'm interested in what works, and going this far into the abstract seems rather pointless to me. You can assume that the logical absolutes aren't true, that reality is an illusion, and that science doesn't work. And then you'll almost certainly get killed the next time you try to cross the street.