Sterling: first, to come to the point raised at the end of your last post: Logic, formal logic, is concerned with inference primarily and only secondarily with truth insofar as truth is that which is, as it were, ‘carried’ by a valid argument.
Epistemology is concerned with knowledge and belief, the conditions for each, the relation between the two, and many other questions.
There is a field of study called the philosophy of logic which inhabits a no-man’s land between the two areas – for this, see, for example, Susan Haack’s “Philosophy of Logics”.
On your substantive point: one can believe things for a whole variety of reasons – because they are self-evident and just obviously so, or delivered to one by one’s tradition or culture, or something one read in a book or heard from a friend, and so on. There is no requirement that our beliefs be proved or provable before we are entitled to entertain them. Cardinal Newman, in his “A Grammar of Assent” challenged his readers (British, 19th century) by asking them how many of them believed Great Britain to be an island. Of course, the answer would have been, all of them. And then he asked how many of them had circumnavigated the island, or knew someone who had done so – and now, of course, the answer was, of course, practically zero. Yet, none of his readers doubted that GB was, in fact, an island. Newman wondered why this was so and pointed out that everything was for such a belief and nothing against it. All of English history, its trade, its relations with Europe, makes no sense unless GB is an island.
Most of our beliefs are held on little or no evidence at all – easy come, easy go. The flash point, as usual, is with religious beliefs. Recently, we have seen the emergence of a new and very hostile form of scientism, which is the 21st century version of the ultra-rationalist position Newman was reacting against. This holds that only that which is scientifically establishable may be reasonably believed. Of course, the very statement of this position is not itself scientifically establishable and so should, by virtue of that criterion, be abandoned! A corollary of this scientismic thesis is that only that can be believed which can be believed by all – but, once again, this position is incoherent.
Many religions have held that there is evidence for belief in God and in the holdings of particular faiths. Not everyone is persuaded of this but that in itself doesn’t establish the falsity of these claims.