To obtain paradoxes, it is you that would need access to more proofs than I do.
From an evidence-based point of view, as a contrapositive of the usual argument against P=0 and P=1, we can say that if it's possible to convince me that a statement might be false, it must be that I already assign it probability strictly <1.
As you may have guessed, I also don't agree with the point of view that I can be convinced of the truth of any statement, even given arbitrarily bizarre circumstances. I believe that one needs rules by which to reason. Obviously these can be changed, but you need meta-rules to describe how you change rules, and possibly meta-meta-rules as well, but there must be something basic to use.
So I assign P=1 to things that are fundamental to the way I think about the world. In my case, this includes the way I think about probability.
I was very interested in the discussions and opinions that grew out of the last time this was played, but find digging through 800+ comments for a new game to start on the same thread annoying. I also don't want this game ruined by a potential sock puppet (whom ever it may be). So here's a non-sockpuppetiered Irrationality Game, if there's still interest. If there isn't, downvote to oblivion!
The original rules:
Enjoy!