As for the reason why Horatio is unable to believe in ghosts, that's a bit overstated. He doesn't say he does not believe. He says, "O day and night, but this is wondrous strange!"

That means he's sort of stuck at the stage when he says, "Wow, what was that!" He might disbelieve, though, and if he does, Hamlet's comments that follow tell us he does so because of his "philosophy": he's been trained in rational/logical thought, and this would lead him to reject the idea of ghosts.