I think the grandparent silently assumes we've solved logical uncertainty, and can use this to decide an axiom does not produce a contradiction by observing the lack of explosion.

No. We don't decide an axiom does not produce a contradiction by observing the lack of explosion. We just raise the probability of noncontradiction by observing the lack of explosion.

EDIT: Axiom, not acronym. Holy God I need more sleep.

Though "routing through logic" seems like an incredibly bad way to say that.

-_-. The post itself was a sketch, not a finished, formalized publication. Please refrain from being anal about phrasings or imagining that I've done far more math than I really have. If I want to be precise, I'll just use symbols -- that's what they're for.