The statement in this poll is not exactly equivalent to the Epimenides paradox, but it is messy just like that one. Assuming a "strong" definition of a liar (always lies)
I am Italian AND All Italians are liars --> I am a liar --> I am not Italian AND not all Italians are liars --> Since I am not Italian, now I may or may not have been a liar (my being a liar was based on my being Italian) AND depending on whether I lied, all Italians may OR may not be liars --> Complete lack of information
truth value: 0.5 (no information)
In Epimenides' paradox, the first question is not in the loop; it's a given. We
know that he is Cretan.
With a "strong" definition of a liar (always lies), we end up in an infinite loop in only one branch:
All Cretans are liars --> I am a liar --> Not all Cretans are liars --> I may or may not be a liar (i.e. I may or may not be telling the truth about Cretans)
Possibility 1: He is telling the truth about Cretans (i.e. the statement is TRUE) (go back to the beginning of the loop, which goes on forever);
Possibility 2: He is lying about Cretans (go back to "Not all Cretans are liars", and it is perfectly consistent for him to be a liar and not all Cretans to be liars) and there is no loop. The statement is FALSE
truth value: (0+1)/2 = 0.5
With a "weak" definition of a liar (lies often but not always), we end up with:
All Cretans are liars --> I am a liar --> Not all Cretans are liars (although I might be)
Possibility 1: He is telling the truth about Cretans just this time (but he tells lies often, just like all of them) --> His statement is TRUE (no loop)
Possibility 2: He is lying about Cretans (Not all Cretans are liars) --> His statement is FALSE (i.e. He is a liar, but not all Cretans are) (no loop)
truth value: (0+1)/2 = 0.5
Now a statement like "This statement is false" is a nice, little, clean paradox with a tight little loop. But the poll question and the Epimenides paradox are not as simple.
Edited by Twilight Bark, 06 December 2003 - 02:18 PM.