The Meta-Puzzle

post by DanielFilan · 2021-11-22T05:30:01.031Z · LW · GW · 26 comments

Contents

  The puzzle
  Solution to the puzzle
  The meta-puzzle
  Discussion of the meta-puzzle
  Homework exercise
  Acknowledgements
None
26 comments

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?

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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?

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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).

26 comments

Comments sorted by top scores.

comment by Dave Orr (dave-orr) · 2021-11-22T05:49:43.480Z · LW(p) · GW(p)

I found a different solution to the initial puzzle, which I won't spoil here, but post as a follow-on:

Same scenario, except after you hear the statement, you know the person is single -- but you don't know who they worship!

What was the statement?

Replies from: Vivek, Rochambeau, Richard_Kennaway, Yoav Ravid, joachim-bartosik, DanielFilan, Measure, AprilSR
comment by Vivek · 2021-11-22T06:23:17.625Z · LW(p) · GW(p)

My solution (rot13'd): "Vs lbh nfxrq zr vs V nz fvatyr, V jbhyq fnl lrf"

Replies from: dave-orr, Yoav Ravid, DanielFilan
comment by Dave Orr (dave-orr) · 2021-11-22T20:33:07.141Z · LW(p) · GW(p)

This is the Godel Escher Bach solution :)

comment by Yoav Ravid · 2021-11-22T10:12:53.454Z · LW(p) · GW(p)

I like it!

comment by DanielFilan · 2021-11-22T08:52:24.518Z · LW(p) · GW(p)

I didn't think of that one!

comment by Rochambeau · 2021-11-22T12:16:17.322Z · LW(p) · GW(p)

"I'm a rationalist."

JKJK

comment by Richard_Kennaway · 2021-11-22T08:47:22.475Z · LW(p) · GW(p)

"I am single if and only if I worship God."

If I worship God, then the sentence reduces to "I am single", and since I worship God, it must be true.

If I worship Satan, it reduces to "I am not single", and since I worship Satan, it must be false.

Replies from: DanielFilan, dave-orr
comment by DanielFilan · 2021-11-22T08:58:17.370Z · LW(p) · GW(p)

That was my solution.

comment by Dave Orr (dave-orr) · 2021-11-22T20:29:34.909Z · LW(p) · GW(p)

This was my solution! :)

comment by Yoav Ravid · 2021-11-22T10:11:30.704Z · LW(p) · GW(p)

Oh, I thought that was the version in the post. So I guess my solution to the post version [LW(p) · GW(p)] is also my solution here.

comment by Joachim Bartosik (joachim-bartosik) · 2021-11-22T09:03:32.776Z · LW(p) · GW(p)

"V jbefuvc fngna naq V'z zneevrq." ?

Replies from: Yoav Ravid
comment by Yoav Ravid · 2021-11-22T10:15:50.569Z · LW(p) · GW(p)

That sentence is impossible for anyone on that island to say - if they worship Satan they can't say it because it would be true, and if they worship God they can't say it because it would be false. On the other hand, both can (and would) say they worship God.

Replies from: Measure
comment by Measure · 2021-11-22T13:12:55.244Z · LW(p) · GW(p)

A single satanist could say it.

Replies from: Yoav Ravid
comment by Yoav Ravid · 2021-11-22T19:32:33.076Z · LW(p) · GW(p)

Oh, right, if you treat it as one statement then it's possible. But then it fails the modified question because you discover who they worship.

comment by DanielFilan · 2021-11-22T06:52:36.993Z · LW(p) · GW(p)

I actually thought about including that: similarly, in American logic, you can't take an arbitrary claim P and come up with a sentence S such that you can derive P from "S is true" and also from "S is false".

comment by Measure · 2021-11-22T13:04:03.004Z · LW(p) · GW(p)

"I worship God XOR I am married."

comment by AprilSR · 2021-11-22T05:58:55.740Z · LW(p) · GW(p)

rot13

Jnf lbhe nafjre fbzrguvat nybat gur yvarf bs "V jbefuvc Tbq be V nz zneevrq, ohg abg obgu"?

comment by gjm · 2021-11-22T18:59:16.012Z · LW(p) · GW(p)

Relevant post from the Sequences: The Parable of the Dagger [LW · GW]. (It's not making the exact same point as this one, but it's in the same territory.)

comment by Pattern · 2021-11-22T05:51:16.446Z · LW(p) · GW(p)

Depending on how the 'always lie' part is defined, a liar could say something 'impossible' like 'I am neither single nor married'.


and wherever you see “I worship God”, you can substitute “This sentence is true”—all the reasoning will go thru the same.

Not actually the case - both the honest and the liars will say 'I speak the truth'.

Replies from: DanielFilan
comment by DanielFilan · 2021-11-22T05:54:27.824Z · LW(p) · GW(p)

Depending on how the 'always lie' part is defined, a liar could say something 'impossible' like 'I am neither single nor married'.

I mean that everything they say is false.

both the honest and the liars will say 'I speak the truth'.

And they also both say "I worship God".

comment by Yoav Ravid · 2021-11-22T10:08:37.012Z · LW(p) · GW(p)

Rot13'd answer to the first puzzle: Vs V'z tbq-srnevat, gura V'z fvatyr. Vs V'z n fngnavfg gura V'z zneevrq.

I tried to find a sentence that only has one "statement" but couldn't, so this one has two. And I also assumed that the sentence isn't supposed to give you any information about whether the person is god-fearing or a satanist (meaning, both can say the sentence and you'd still know they're single).

comment by Randomized, Controlled (BossSleepy) · 2021-11-23T18:31:21.714Z · LW(p) · GW(p)

"All god fearing people are single and all satanists are married"?

Replies from: Yoav Ravid
comment by Yoav Ravid · 2021-11-23T19:05:37.361Z · LW(p) · GW(p)

If it's true and someone says that, then they're god-fearing, so they're single.

If it's false and someone says that, then they're satanists, but I'm not sure if it follows that they're single - cause if you treat it as one statement that only has to be false when taken together, then it can have parts that are true on their own, which means it might still be true that all satanists are married. But if that part is also false, then yeah, this seems to work.

comment by Ericf · 2021-11-22T14:05:55.750Z · LW(p) · GW(p)

In casual usage "or" is not well defined. Sometimes it means "exactly one" and sometimes it means "one or more"

Therefore, you cannot assume that the speaker is using the second definition when speaking especially in the middle of a tricky logic puzzle with a speaker who might be an always-liar.

Furthermore, speaking more naturally would be "X thing is true, or I am a liar" - which is a fairly common construction of an Oath even in America - just substitute "so help me god" for the second part, which is the same meaning.

Replies from: Pattern
comment by Pattern · 2021-11-22T15:56:16.576Z · LW(p) · GW(p)
Therefore, you cannot assume that

In order to make the problem simpler/solvable that assumption is often made. It is one of the cases.

comment by newcom · 2021-11-22T10:46:34.075Z · LW(p) · GW(p)

This comes really close to the 'This statement is false'-discussion in Gödel-Escher-Bach. On itself, this statement is circular (if it were true, it would become false and vice versa), but when combined with another true/false statement, the full statement is not always circular:

If in "This sentence is false, or I am single", the "I am single" part is false, then the truth-value of the full statement reduces to that of 'This sentence is false' and becomes circular. If 'I am single' is true, the whole statement is true independent of the first part, and so the statement is not circular (with 'this sentence is false' being true).

I guess the following happens in the Australia example: the Satan-worshipers must always say (full) statements which are false, so they can never set the truth-value of 'this sentence is false' to true. This way they are excluded from making "This sentence is false, or ..."-type statements.