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!)

The Jug of PunchBein’ on the twenty-third of June,

As I sat weaving all at my loom,

Bein’ on the twenty-third of June,

As I sat weaving all at my loom,

I heard a thrush, singing on yon bush,

And the song she sang was

The Jug of Punch.What more pleasure can a boy desire,

Than sitting down beside the fire?

What more pleasure can a boy desire,

Than sitting down beside the fire?

And in his hand a jug of punch,

And on his knee a tidy wench.

When I am dead and left in my mould,

At my head and feet place a flowing bowl,

When I am dead and left in my mould,

At my head and feet place a flowing bowl,

And every young man that passes by,

He can have a drink and remember I.

Juvenal asked who will watch the harem watch. Does the NSA scan its own metadata? And if it is turtles all the way down?

I said it was a slippery topic!

More metadata: who scores the scorer? If the machine is stateless there is no stop state and no contradiction. As long you keep flipping the card over you’ll be fine.

All Liar, No ParadoxAccording to my understanding of it, the so-called Liar Paradox is just the most simple-minded of fallacies, involving nothing more mysterious than the acceptance of a false assumption, from which anyone can prove anything at all.

Let us contemplate one of the shapes in which the re*putative Liar Paradox is commonly cast:

Somebody writes down:

1. Statement 1 is false.

Then you are led to reason: If Statement 1 is false then by the principle that permits the substitution of equals in a true statement to obtain yet another true statement, you can derive the result:

“Statement 1 is false” is false. Ergo, Statement 1 is true, and so on, and so on, ad nauseam infinitum.

Where did you go wrong? Where were you misled?

As it happens, graphical reasoning does help to clear this up — at least, it did for me — if only because the process of translating the purported reasoning into another form of representation gave me a crucial clue as to where the wool was being pulled.

Just here, to wit, where it is writ:

1. Statement 1 is false.

What is this really saying? Well, it’s the same as writing:

Statement 1. Statement 1 is false.

And what the heck does this dot.comment say? It is inducing you to accept this identity:

“Statement 1” = “Statement 1 is false”.

That appears to be a purely syntactic indexing,the sort of thing you are led to believe that you can do arbitrarily, with logical impunity. But you cannot, for syntactic identity implies logical equivalence, and that is liable to find itself constrained by iron bands of logical law.

And you cannot, not with logical impunity, assume the result of this transmutation, which would be as much as to say this:

“Statement 1” = “Negation of Statement 1”

And this my friends, call it “Statement 0”, is purely and simply a false statement, with no hint of paradox about it.

Statement 0 was slipped into your drink before you were even starting to think. A bit before you were led to substitute you should have examined more carefully the site proposed for the substitution!

For the principle that you rushed to use does not permit you to substitute unequals into a statement that is false to begin with, not just in the first place, but even before, in the zeroth place of argument, as it were, and still expect to come up with a truth.

Now let that be the end of that.

Then what do you make of “1. Statement 1 is true”, where we cannot decide if it is true or false?

Also, I stick to the word “paradox”, meaning something which has to be looked at a little more closely…

For the moment, I am viewing these questions merely as matters of classical propositional logic, even just Boolean formulas.

If we have a Boolean formula like then we do not know whether and are true or false, but we do know that the formula as a whole is true, because we adopted axioms beforehand to make it so.

In that perspective, a form like “1. Statement 1 is true” is just a way of expressing the formula “Statement 1 = (Statement 1 = true)”, which has the form which is true on the adopted axioms.

The question is “Byker” Gove is re-taking a multiple choice test, answering each question randomly. What are his chances of getting the right answer to this question:

A) 1/2 B) 1/4 C) 1/4 D) 0?

Since the question mark comes after the multiple choices, the multiple choices are within the question. The first sentence tells us that it is a multiple choice test which implies that one of the answers A,B,C, or D is correct.

If the correct answer is 1/2 then there is a 1 in 4 chance of being right

if the correct answer is 1/4 there is a 1 in 2 chance of being right

If the correct answer is 0 there is a 1 in 4 chance of being right

There is a 1 in 3 chance of each answer being right, so the overall chance of Gove getting the right answer at random is

1/3×1/4 + 1/3×1/2 + 1/3×1/4 = 4/12 = 1/3

I think you are over-interpreting the question mark. I followed Maslanka who used the two different positions for it in the two puzzles; I am fairly sure he intended them to mean the same thing.

I think that “Is it a bird? Is it a plane? Is it Superman?” can be tranlated either as “Is it (a) a bird, (b) a plane, (c) Superman?” or as “Which is it? (a) a bird, (b) a plane, (c) Superman.

But there is a point here. A “Sudoku” which doesn’t have a unique solution is not technically a Sudoku. Similarly, a multiple choice question which has no correct answer should be disqualified (this is the case with Maslanka’s second puzzle), and arguably one with more than one solution should be as well (as with my variant), though I am not so sure about this.

In any case, I think there is an issue with self-reference in these puzzles, and simply disqualifying them on a technicality should not be allowed to hide this.