It seems obvious to me that after glancing at my watch I "know" what time it is. But this apparent "knowledge" can be explained away by infinitely many skeptical hypotheses. Perhaps I mistook the time. Maybe I was hallucinating and therefore some other incorrect time appeared on my watch. It is possible that I never glanced at my watch at all, and that my memory was fooled by some trick. How, then, do I "know" the time? Sure, I can go back and check. But there will always be infinitely many skeptical hypotheses that will prevent me from being "truly justified"—whatever that means.
So how do we justify our beliefs? It seems obvious that we do in fact "know" things, but how? The only escape I can see is pragmatism. I can continue to act in acceptance of a given hypothesis, and as long as it does not turn out false life goes on. But this isn't really a justification. What answer have philosophers put forth, here?