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Craig is just purposely conflating the likelihood of a particular result and the likelihood of given the declaration of a result by the lottery officials, that result being true.

If you and I are flipping coins for a million dollars, it's going to take a lot of convincing evidence that I lost the coin flip before I pay up. You just cannot flip the coin in another room where I can't even see, and then expect me to pay up because, well, the probability of heads is 50% and I shouldn't be so surprised to learn that I lost.

Therefore, the actual likelihood of a particular set of lottery numbers is totally irrelevant in this discussion.

In any case, the only kind of "evidence" that we have been presented for miracles has always been of the form "person X says Y happened', which has been known as hearsay and dealt with without even bothering with probability theory.

I am at a loss about the true meaning of a "universally compelling argument", but from Eliezer's original post and from references to things such as modus ponens itself, I understood it to mean something that is able to overcome even seemingly axiomatic differences between two (otherwise rational) agents. In this scenario, an agent may accept modus ponens, but if they do, they're at least required to use it consistently. For instance, a mathematician of the constructivist persuasion denies the law of the excluded middle, but if he's using it in a proof, classical mathematicians have the right to call him out.

Similarly, YEC's are not inconsistent in their daily lives, nor do they have any undefeatable hypotheses about barbeques or music education: they're being inconsistent only on a select set of topics. At this point the brick wall we're hitting is not a fundamental difference in logic or priors; we're in the domain of human psychology.

Arguments that "actually convince (all) people" are very limited and context sensitive because we're not 100% rational.

Nitpicking: Modus ponens is not about "deriving". It's about B being true. (Your description matches the provability relation, the "|-" operator.) It's not clear how "fundamental" modus ponens it is. You can make up new logics without that connective and other exotic connectives (such as those in modal logics). Then, you'd ask yourself what to do with them... Speaking of relevance, even the standard connectives are not very useful by themselves. We get a lot of power from non-logical axioms, with a lot of handwaving about how "intuitively true" they are to us humans. Except the Axiom of Choice. And some others. It's possible that one day an alien race may find our axioms "just plain weird".

The never-learn-anything example that you quoted looks a bit uselessly true to me. The fact that once can have as prior knowledge the fact that the monkey generates perfect 1/4 randomness is utopia to begin with, so then complaining about not being able to discern anything more is like having solved the halting problem, you realize you don't learn anything more about computer programs by just running them.

I'm not well versed on YEC arguments, but I believe people's frustrations with them is not due to the lack of universally compelling arguments against them. Probably they're already guilty of plain old logical inconsistency (i.e. there's a valid chain of reasoning that shows that if they doubt the scientific estimates, then they should turn off their television right now or something similar), or they possess some kind of "undefeatable" hypothesis as prior knowledge that allows for everything to look billions of years old despite being very young. (If so, they should be very much bothered by having this type of utopic prior knowledge.)