Self-reference is a slippery topic. Two recent Chris Maslanka puzzles in the Saturday Guardian bear this out.
Last weekend, he asked,
“Byker” Gove guesses every answer in the new I level multiple choice. What are his chances of getting this question right?
a) 1/2 b) 1/4 c) 1/3 d) 1/4
He gave the answer “no chance at all”, with which presumably few of his readers would disagree. But perhaps at least one saw the problem. Yesterday he gave a variant:
A) 1/2 B) 1/4 C) 1/4 D) 0?
Before giving the answer, let’s have a couple of classics. First, the Liar paradox: “This statement is false”. We easily see that we cannot assign a truth value to this: if it is true, then it is false, but if it is false, then it is true. (Epimenides the Cretan famously said “All Cretans are liars” – his statement is reported in the Bible.)
But what about the variant which says “This statement is true”? In a way this is even more worrying. If it is true, then it is true, and you may be tempted to stop there and say “OK, it is true”; but if it is false, then it is false. So either truth value is consistent.
(If you came to Hay-on-Wye, you might have heard me mention this in a debate about self-reference there.)
The statement refers only to itself, and we could fix the problem by simply declaring that self-referential statements are banned. But what about the classic puzzle, where I give you a card, one side of which says “The statement on the other side of this card is true”, and the other side says “The statement on the other side is false”. Again you can’t consistently assign truth values. But neither proposition refers to itself, and the self-reference only comes by taking them in conjunction. If the two sides of the card had said, “The statement on the other side of this card is true” and “Epimenides formulated the Liar Paradox”, there doesn’t seem to be a problem.
Back to Chris Maslanka. His solution to the second puzzle read:
An impossible set of questions: this is a distributed version of the Bertrand Russell autological/heterological paradox. Thanks to reader Charli for sending in this variation of last week’s [puzzle].
The Russell paradox referred to is a reformulation of the liar paradox. Let us call an adjective “autological” if it describes itself (e.g. “short”, “English”), and “heterological” if it does not (e.g. “long”, “French”). Now ask: is “heterological” autological or heterological? Either answer leads to a contradiction, just as in the liar paradox. The variant would be to ask: is “autological” autological or heterological? This time either answer appears to be consistent.
That said, let us have a third version of the “Byker” Gove puzzle:
As before “Byker” Gove answers at random. What are his chances of getting the right answer to this question:
α) 1/2 β) 1/2 γ) 1/4 δ) 0?
Now if the answer is 1/2, then it is 1/2; if it is 1/4, then it is 1/4. Which is it?
This might make you worry about the proposed answer to the first Maslanka puzzle. (Indeed, I hope it does!)