You're going off a very strict "logically provable" definition which is generally incompatible. We use "logically provable" like this in cases for math equation

You're going off a very strict "logically provable" definition which is generally incompatible. We use "logically provable" like this in cases for math equations and algorithms (which has a lot of value!) but not for every day things. "My cat is sad" is not logically provable, but we feel comfortable that this idea is basically real enough. "vorpalhex has an internal monologue" is not logically provable, but we feel comfortable that it's probably true if I say it's true. To go back to your original ask, for something that can be shown via Bayesian thinking but not logical deductive reasoning: "Intelligent alien life has never visited Earth". This can't be logically proven because of the observer paradox - the strongest statement we could defend logically would be "There is no known evidence that intelligent alien life has ever visited Earth" (which is a weaker claim). If we allow Bayesian reasoning, then we can defend that the likelihood that intelligent alien life has visited Earth and never left behind evidence is much, much smaller than the likelihood that intelligent alien life has simply never visited us.