Logic doesn't dictate what truth is. It's only a tool for testing the validity of arguments.
"If someone commits murder and does not ask forgiveness, they will go to hell" is a logically valid statement. However it'snot sound because the rules for going to hell varies between denominations, some religions don't even have Hell and Hell can't be proven to exist anyways. We can't confuse something being logical with it being truth. That isn't to say that things that are true aren't logical or things that don't make sense are true. Logic isn't enough on its own.
Induction is tested differently from deduction. "I haven't found evidence for God, so he must not exist." I don't think that's a reasonable argument. But this is the problem of induction. But it's good because it relies on evidence before you make conclusions. When new evidence appears, it is taken into account and worldviews are adjusted accordingly. It's a lot better than "I believe it to be true no matter what because my faith is strong."
God is not falsifiable. Can't prove he exists, can't prove he doesn't. In that sense, it's not worth anything. What's the difference between something you can't prove and something that doesn't exist? Other than the way I would be treated, there's no distinguishable consequence between saying there are undetectable miniature pink elephants following me around and no elephants at all. In both cases, neither are detectable or have any effect on the real world. And given all we know about the animal kingdom, biology and physics, leaning toward them not existing would be wiser.
Brendan said that God is testable though. I agree. I don't think the Christian God or any religion on Earth is possible. However I can't-not be open to the possibility of a grand creator, or giant finger that pushed over the first domino some time ago. Not only because it's not falsifiable but because a being like that would be indistinguishable from nature. Super-advanced aliens playing around with a big-bang machine? I'm open for evidence, though I lean toward simpler explanations. Abrahamic deity who specifically created humans and watches over us to make sure we don't act against neolithic morals? Don't see it happening.
"Astronomy offers an aesthetic indulgence not duplicated in any other field. This is not an academic or hypothetical attraction and should require no apologies, for the beauty to be found in the skies has been universally appreciated for unrecorded centuries."