In 1992, the philosopher George Boolos gave what he called the “Hardest Logic Puzzle Ever”, which he attributed to Raymond Smullyan. In 2008, a clever paper by two graduate students, Brian Rabern and Landon Rabern, appeared in the philosophical journal “Analysis” which gave a simpler solution to the puzzle than Boolos gave—and furthermore claimed that a solution to a stronger puzzle was possible!
As its name implies, the “Hardest Logic Puzzle Ever” has a number of complicating factors which will be irrelevant for this discussion. Instead, consider the following much simpler puzzle which will do just as well.
You are on an island populated by knights and knaves. Knights always tell the truth; knaves always lie. You meet a inhabitant of the island who you know is holding up either 1, 2, or 3 fingers behind his back. You don’t know if this inhabitant is a knight or a knave. By asking two yes-or-no questions, determine how many fingers the inhabitant has behind his back.
There are a number of ways to solve this. One solution follows from the observation that, for any question Q, if you ask a inhabitant of the island
If I asked you question Q, would you say “yes”?
then you will get a truthful response to question Q. A knight will tell the truth about his truthful response to Q, whereas a knave would lie about his false response to Q.
So one solution would be to ask
If I asked you if you were holding 1 finger up, would you say “yes”?
If the inhabitant says “yes”, then you know he is holding 1 finger up. On the other hand, if the inhabitant says “no”, then you ask “If I asked you if you were holding 2 fingers up, would you say `yes’?” If the inhabitant says “yes”, then he is holding 2 fingers up, otherwise he is holding 3 fingers up.
This works, and it seems like you can’t do any better: There are three possibilities for how many fingers the inhabitant could be holding up, and since you can only ask yes-or-no questions, you couldn’t determine which of the three possibilities holds with only one question. But this is exactly what Rabern and Rabern claim you can do.
Consider what would happen if you asked a knight, “Will you answer `no’ to this question?”. If he is bound to answer the question, then he is in trouble, because no matter if he says “yes” or “no” he will have lied. Rabern and Rabern argue that in this sort of situation, the knight’s head would simply explode as there is nothing else he can do.
Assume that a inhabitant’s head explodes exactly when he cannot consistently answer “yes” or “no” to a question. Given this, we can solve the above problem by asking the single question. To make things simpler, suppose that we know that the inhabitant is a knight. Then we may ask:
Is it the case that you are holding up one finger or (you are holding up two fingers iff you will answer “no” this question)?
In this instance, if he answers “yes”, then he’s holding up one finger, if he answers “no”, he’s holding up three fingers, and if his head explodes, he was holding up two fingers.
I think Rabern and Rabern’s argument is really clever, but I don’t think it goes far enough. If we can observe inhabitants’ heads exploding and reason based on it, we should be able to ask inhabitants about it. Consider what would happen if we asked a knight:
Is it the case that you will answer “no” to this question and that your head will not blow up upon hearing this question?
If the knight’s head does not blow up upon hearing the question, then he can neither truthfully answer “yes” nor answer “no.” Therefore his head blows up. But if his head blows up only when he can’t answer a question, there’s a problem because given that his head blew up, he could have consistently answered “no” to the question.
So what happens? Well it must be the case that God’s (or whoever decides whether or not to blow up a head) head blows up because God won’t be able to decide whether or not the knight’s head should blow up. Similarly, if we suppose that SuperGod is in charge of deciding whether or not God’s head blows up, SuperGod’s head is in danger of blowing up due to a clever self-referential question, and so forth. So we may actually extract an unbounded amount of information from a single yes-or-no question by choosing the question carefully and then observing how much of the universe is destroyed by our asking it.
We can eliminate this silliness a bit by seeing how this applies to the usual Liar’s Paradox. The Liar’s Paradox is
This sentence is false.
(Or “This sentence is not true,” but the two will be equivalent for our purposes.) If it was true, then it would be false, and if it was false, then it would be true. Many people say that this sentence simply doesn’t have a well-defined truth value.
But consider this sentence, which I call the Second-Level Liar’s Paradox:
This sentence is false and it has a well-defined truth-value.
We can’t say that this sentence doesn’t have a well-defined truth-value, since if it doesn’t, then it is unproblematically false! Similarly, we can’t say that it has a well-defined truth value, since in that case it reduces to the Liar’s Paradox and can be neither true nor false. So, the Second-Level Liar’s Paradox has no well-defined (well-defined truth-value or not)-value.
Of course, we can iterate this. It seems that we should have at least the following truth values: true, false, paradoxical0, paradoxical1, paradoxical2, … and possibly more extending in to the ordinals depending on the expressive power of the language with respect to which sets of truth values it can refer to. (Here, a sentence is paradoxicali + 1 if it cannot consistently be paradoxicali.)
I’ve made a lot of assumptions here which I’m sure could be challenged about how we should reason in murky, paradoxical situations. One issue which I’m not quite sure how to resolve is the question of why, given that a sentence being false implies that it has a well-defined truth value, the Liar Paradox and Second Level Liar Paradox are not equivalent. But my guess would be that this can be made to give some sort of interesting paraconsistent logic, and probably some interesting puzzles as well.