Hmm, so you answered "No" to 'If Boolbar had instead asked "if Tuj were to ask if it begins with P, would you say yes?" would you have said no?'. So you would not say "No" (ie. you would say "Yes"*) to the question "if Tuj were to ask if it begins with P, would you say yes?". Hence it does begin with "P".
* Or you might have answered "Maybe" which throws any attempt at logic out of the window.

[Boolbar] I always wondered if "Bool" meant a Boolean logical construct and "bar" meant the logical inverse of that construct, thus permitting the assumption about the local valid of "true" and "false" of the "Boolbar" referential. Is it true?
[Software] NO, unwanted dead or alive.
[Raak] It does not fly of its own accord.
[Bismarck] Dead or alive makes no difference, it's the name of the thing I'm after.