Challenge #41 Result
Dear Zeno's patrons: Here are the results, which, given Zeno's usual patron's replies, were not as robust....we only received 99 replies! Please keep this alive, and send me YOUR challenges if you wish!..Maybe this one was too hard!:) Ron Barnette
First the challenge, as written:
Our good man Charles has been concerned about his bar keepers drinking on the job at Zeno's. Seems that some have succumbed to the temptations of the fun evenings, and have interfered with the robust business side of Zeno's, which now serves the good Guinness brew (recall the previous challenge). As a result, Charles had an idea: he hired a young chap, Al, and instructed Al to pour a Guinness for all and only those who can't pour a Guinness for themselves. Charles reasoned that by following this proviso, Al could never touch the Guinness, and when Al was on duty there would never be a problem with drinking on the job. Clever idea, if sound. But is it? Is Charles correct?
Several intriguing responses were submitted regarding Charles' claim. Thanks are sent to all respondents!!!
*Tom wrote a formal solution which vindicates Charles' claim.
comments: Claim: Charles is right.
Pxy : x pours y's guinness.
a : Al
(all x)(Pax <--> ¬<>Pxx) |= ¬Paa
Assume there is a countermodel. In it we must have:
Paa <--> ¬<>Paa
which is a contradiction. Hence there are no countermodels and Charles is correct.
*Erik wrote a fun note, which I had to include!
comments: There is a problem with Charles's idea: Al is to pour a Guinness for all and only those who can't pour a Guinness for themselves. Al either (i) can or (ii) cannot pour a Guinness for himself. (i) If he can pour a Guinness for himself, then he cannot, by rule of the proviso, pour a Guinness for himself. But if he cannot, because of this proviso, then this constitutes a reason why he must pour himself a Guinness, because the proviso states firmly that he pour Guinness for *all* those who can't pour a Guinness for themselves. (ii) On the other hand, if he cannot pour a Guinness for himself originally, then he must pour himself one; but his inability to do so in the beginning in this case will stem from some physical inability to pour himself a Guinness (why cannot he pour himself a Guiness? Does he not have arms?), and so he probably will not be a good bar keep. The end result, then, of Charles's idea is that either Al will pour himself Guinness all night (along with !
the other patrons, at least) or he won't because he cannot pour anybody a Guinness. This doesn't sound like a good business decision on Charles's part.