Can you solve the Pinocchio "green hats" logic puzzle?

Sure, but the logic puzzle is also making the implicit statement that “At least one of (A), (B), (C), (D), or (E) can be concluded”, and only one of “At least one of (A), (B), (C), (D), (E) can be concluded” and “Pinocchio is making a statement that he has at least two hats and each one is green” can be true. I would argue “At least one of (A), (B), (C), (D), or (E) can be concluded” is the stronger implicit statement. Word problems are horrible for this sort of thing.

All that we can conclude is that Pinocchio either has no hats at all or that he has n hat(s), where the number of hats that are green is lesser than or equal to n - 1.

Yeah, I’m sorry… but my none mathmatical take would be “C”. If Pinocchio always lies then you would have to assert the opposite to what he has said. In that one sentence… “All of my hats are green”… he’s making two assertions:

  1. He has hats. (“My hats”)
  2. The hats are green.

If the opposite of point one is he has no hats, then point two is moot.

Sorry folks, but Pinocchio has no hats.

Also in the exclusive footage I have acquired below, you’ll see Pinocchio talking about his nonexistent hats while his nose grows.

Italian Puppet GIF by Madman Films

IMO, if he has no hats… then saying that these hats he doesn’t have are green just makes it even more of a lie. They’re imaginary hats at that point and any reference to them would be imaginary as well… or a lie.

So, if he had two blue hats, he would not be lying if he said “All my hats are green”?

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There is no reason for this assumption though. If he says something that is not true, then he is lying even if the opposite is also not the truth.

The real question is: is he making one statement or two? I believe that he is making only one statement with two (purported) facts. If either fact is not true, then the statement itself is a lie.

If a blue duck says, “I am a brown dog,” that is just as much a lie as saying, “I am a blue dog.”


“IMO” is the whole basis of this controversy. The simple resolution is this: mathematicians collectively use language to have extremely specific and formal meaning. Non-mathematicians don’t. You’re interpreting the statement informally, and that’s fine. When non-mathematicians converse, they are frequently very loose about meaning, and much is implied but not said. And the non-mathematical world has (mostly) done fine with this for thousands of years.

To a mathematician (or a logician), however, a statement like this has a very precise meaning. It boils down to the fact that every statement you make about an element of the empty set is true (this is what “vacuously true” means BTW). In particular, for example, every element of the empty set is a bicycle. And every element of the empty set is a hat, and in fact a green hat. So the set of Pinocchio’s hats is empty, and he says every element of that set is green, then he is not lying. Since Pinocchio always lies, this is a contradiction, from which we conclude that the set of Pinocchio’s hats is not empty.


Answer (A) is provably correct.

Answer (A) doesn’t say everything that can be concluded, but it is nonetheless true.

In logic, “A and B” implies “A”. Logically, we can conclude that Pinocchio must have at least one hat that is not green. And that implies that Pinocchio has at least one hat.

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That argument assumes that one of the options (A through E) must be true, and there is nothing in the framing of the question that justifies that assumption. The question is: “What can we conclude,” and we are given five options, but we are not told that we have to choose one of them. Indeed, to conclude it would require us to eliminate all other possibilities, and there are possibilities outside of the options given.

He could have half of a hat that is blue, but that is not included among the options. He could have a green baseball glove that he calls a hat because he is lying about what is and is not even a hat.

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Can someone please explain to a non-mathmatician why the answer is not E?

To me, that is the simplest answer, which is perhaps also why I am not a mathmatician.

@SheiffFatman @Jesse13927 @mdavis I’m just glad that certain death isn’t on the line for correctly deciding Pinocchio’s headwear dilemma.


He could have one blue hat and one green hat and the statement would still be a lie, even though he does have a green hat.


Oh, duh, of course.
Many thanks to you.

I feel like vacuous truth is an attempt to rescue formal logic from Gödel instead of just admitting that there are problems with truth values other than true/false. But then again, my glrennorts are frimvle today, so you might consider that when deciding how much you should trust my assertions.

Yeah I guess what I should have said explicitly is that the puzzle’s implicit statement (that one of A, B, C, etc. is true) is not true. It’s a bad puzzle.

I assume that’s the answer they’re looking for, yes. But this assumes the word “lie” means “to make a statement that is false according to the rules of first-order propositional logic,” but that is not specified in the problem and is definitely not the only or obvious choice of definition for that word.

If Pinocchio is the one choosing to lie, then his is the definition that counts, and he’s not a logician. If the lying is some sort of externally enforced compulsion, then the relevant definition is that of whatever entity or force compels him.

We have extensive philosophical and literary traditions exploring this kind of thing, from Gettier problems (would it be a lie to say you don’t know something, if you have a justified true belief that is only true for unknown reasons unrelated to your justification?) to the Aes Sedai in the Wheel of Time being known as prolific liars despite a magical oath to speak no word that is not true.

@Neil_Austin Vacuous truth in formal logic is much older than Godel and doesn’t help there, at all. It is a way to force all well-formed statements to fit into a model with only two possible truth values, though. I’m curious about this, though. I often see people argue that vacuously true statements should be regarded as false, and sometimes as neither, which I agree with in everyday life generally. But are there formal systems of logic that represent this without giving up all the results that come from use of vacuously true statements in math?

Some models of logic use more than two truth values,

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It’s a question from a Mathematical Olympiad of public schools. I’m pretty sure first-order logic is assumed here.

However, you’re correct that there are other valid logical systems in which different interpretations are possible. These are not idle curiosities, either; they’re active areas of research in foundations; moreover, there’s no obvious “natural” choice of which of them is “correct” - they’re all just different sets of rules for playing a game. Mathematicians have used (without always realizing it) first-order logic for centuries, and it’s permitted the proofs of some amazing and beautiful things. But some are uncomfortable with the extremely abstract nature of those conclusions (for example, they allow you to assert the existence of things with certain properties even though there’s no way to give a specific example of something with those properties). Constructive mathematics is growing in popularity as an alternative to this.


I read into this the opposite premise; to me, that’s one whole statement, that is not separable, or rather doesn’t need to be separated, to determine an answer. IMO, the number of hats he has or whether he has any at all is a MacGuffin, which is why I think the given possible answers are nonsensical. He could have any number of hats including zero, but the lie hinges on if all of them are green, not the number represented by “all”.

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