Anyone want to debate this formally with me?

[Chas]

Believing something for which there is no evidence, even indirect evidence, is irrational. Your turn.

Am I irrational if I believe my neighbor when he tells me he saw a mouse in his basement?

You have indirect evidence. You know that mice actually exist, that your neighbor has a basement, that your neighbor isn't a liar. It is rational to believe his claim.

If your neighbor tells you he saw a dragon in his basement, you would require evidence of the existence of dragons before considering his claim. Simply believing him would be irrational.