There are several cases, and which ones can happen depends on exactly how the fairy made the spells work.
Case 1: Pinocchio is telling the truth (he knows that his nose will grow). Then his nose will grow. There's no stipulation that Pinocchio's nose grows only when he tells a lie, so this is quite possible. If that stipulation is in place, then Pinocchio's nose will not grow, contradiction.
Case 2: Pinocchio doesn't know whether or not his nose will grow, but this doesn't count as a lie. Then his nose will not grow (or it might grow for no reason at all; see Case 1).
Case 3: Pinocchio doesn't know whether or not his nose will grow, and this counts as a lie. Then his nose will grow.
Case 4: Pinocchio knows that his nose will not grow. Then his nose will grow, contradiction.
What we have proven is that Pinocchio can never know for sure that his nose will not grow when he makes the statement "My nose will grow." Under the additional condition that Pinocchio's nose grows only when he tells a lie, then it is also the case that Pinocchio can never know for sure that his nose will grow when he makes that statement. In the latter case, the magic that causes Pinocchio's nose to grow injects enough uncertainty into the system that we can never say he really "knows" what's going to happen when he makes certain statements.
There are several cases, and which ones can happen depends on exactly how the fairy made the spells work.
ReplyDeleteCase 1: Pinocchio is telling the truth (he knows that his nose will grow). Then his nose will grow. There's no stipulation that Pinocchio's nose grows only when he tells a lie, so this is quite possible. If that stipulation is in place, then Pinocchio's nose will not grow, contradiction.
Case 2: Pinocchio doesn't know whether or not his nose will grow, but this doesn't count as a lie. Then his nose will not grow (or it might grow for no reason at all; see Case 1).
Case 3: Pinocchio doesn't know whether or not his nose will grow, and this counts as a lie. Then his nose will grow.
Case 4: Pinocchio knows that his nose will not grow. Then his nose will grow, contradiction.
What we have proven is that Pinocchio can never know for sure that his nose will not grow when he makes the statement "My nose will grow." Under the additional condition that Pinocchio's nose grows only when he tells a lie, then it is also the case that Pinocchio can never know for sure that his nose will grow when he makes that statement. In the latter case, the magic that causes Pinocchio's nose to grow injects enough uncertainty into the system that we can never say he really "knows" what's going to happen when he makes certain statements.
http://forums.xkcd.com/viewtopic.php?f=3&t=35947
ReplyDeleteGoogle haha :) I saw it as well
In which case you should have included it as part of your post. That way you have a cartoon, a question, an analysis and thus a discussion.
ReplyDeleteThat would make a much better post than simply posting a cartoon and a question.