The Meta-Puzzle

Nov 21, 2021

This post was inspired by the Raymond Smullyan book The Riddle of Scheherazade.

This post contains two puzzles: first, a puzzle to warm up, then a meta-puzzle about the first puzzle. I strongly encourage you to give your best shot at the first puzzle—it is achievable, and the rest of the post will spoil the solution. I also encourage you to try the meta-puzzle before reading my solution.

The puzzle

This puzzle is set in a far-off, mystical island called “Australia”. In Australia, everyone either worships God or Satan. The God-fearing only speak the truth, while the Satanists always lie and never speak the truth. Furthermore, in Australia, both Satanists and the God-fearing can either be married or single.

While in Australia, you meet a stranger. You don’t know who this stranger worships, or whether the stranger is married or not. The stranger says a sentence, and once you hear the sentence, you are convinced that the stranger is single, without needing to check for proof. What could the sentence possibly have been?

Solution to the puzzle

Take the sentence “I worship Satan, or I am single”—where “or” is used in the sense that both could be true. No Satanist could say such a sentence, because if a Satanist said it, it would be true. Therefore, the person who spoke it must have been God-fearing. Since the person is God-fearing, they do not worship Satan, so they must be single, since otherwise the sentence would be false.

The meta-puzzle

All of this is very similar to how normal logic works, but in some ways very different. First, note that in Australia, a sentence is true if and only if the person saying it worships God. So, wherever you see “I worship Satan”, you can substitute “This sentence is false”, and wherever you see “I worship God”, you can substitute “This sentence is true”—all the reasoning will go thru the same.

Now, let’s think about how logic works off the island. I, the author of this post, am currently in California, one of the united states of America—far away from Australia. Suppose I say to you “This sentence is false, or I am single”. You could reason thusly:

If the sentence is false, then the sentence is false or he is single—but then the sentence is true, because that’s what the sentence says. Therefore it’s true. If the sentence is true, then “this sentence is false, or Daniel is single” is true—but “this sentence is false” can’t be right, we just said it was true. Therefore, Daniel is single.

It seems like something must be going wrong: I could have replaced “I am single” with any other claim, and you would have concluded that that claim must be true. But, once we replace “this sentence is false” with “I worship Satan”, that’s the exact statement and reasoning we used in the solution to the earlier puzzle, and it seemed valid there. So what’s going on? What’s the relevant difference between the faulty reasoning in America, and the valid reasoning in Australia?

Discussion of the meta-puzzle

In the United States of America, not all sentences people say are true or false. For instance, if I say “Aaargh!”, that doesn’t really express a state of affairs that may or may not correspond to reality. A slightly less obvious example is the claim “This sentence is false” - I can say this, but it’s neither true nor false. Let’s say that the sentences that are either true or false ‘express propositions’, and that those sentences like “Aaargh!” or “This sentence is false” ‘fail to express propositions’.

Now, in the united states of America, the sentence “This sentence is false, or I am single” isn’t guaranteed to express a proposition: so we aren’t justified in making the inference that if it’s not false, then it’s true. However, in Australia, people only say sentences that express propositions—in particular, the God-fearing only say true sentences, and the Satanists only say false sentences. Therefore, nobody would ever utter “I worship Satan, or I am single” unless they worshipped God and were single. In Australia, that sentence corresponds to a specific state of affairs (at least, once you nail down the speaker), and the God-fearing would only say it if it were true, and the Satanists would only say it if it were false. That’s why the reasoning is justified in Australia, but not in California.

Homework exercise

Consider the sentence “This sentence fails to express a proposition, or it is false, or I am single”. Does that sentence fail to express a proposition? If it expresses one, then we’re back in trouble again, but if it doesn’t, then it seems like it’s true—can’t sentences only be true if they express propositions?

Acknowledgements

I recently read Mike Huemer’s book Knowledge, Reality, and Value, which IIRC emphasizes that not all sentences express propositions, and that can get you out of trouble (if it doesn’t emphasize that, then I’m pretty sure some of his other writing does).