Albert, Bernard, Cheryl and the theory of mind

I love a good maths/logic puzzle. I love it even more when it becomes a language puzzle as well, and it seems social media has picked up on one such today. Known as the Cheryl Birthday Problem, you get a different result depending on how you interpret a single word in the way the puzzle is framed. That word is the second occurrence of “know”, and the article linked describes the different interpretations as being either a statement of “deduction” or “fact”.

This can be framed as a deep philosophy puzzle: “What do we mean by knowing?” That question is one that cannot be answered easily even by the greatest minds. (Suppose Cheryl lied to one or both of them? Then neither truly know…)

It can be framed as a purely linguistic puzzle: “How does this setter use the word ‘know?'” That would lead to the intended correct answer, since in every other use of the word “know” in the puzzle, it is to be interpreted as “can (from this information) deduce”.

But it can also be treated as a psychology puzzle, to do with theory of mind and socialisation.

I read the puzzle and assumed a statement of fact, that Albert had from some source been informed that Bernard does not know, rather than that Albert had managed to deduce that Bernard doesn’t know. But why did I make that assumption?

I made this assumption because I generally do not assume it is possible to know another person’s state of mind unless they give some indication. But I also know that people generally do give indications of their state of mind. James Grime (the first Guardian link above) suggests that, “Maybe Cheryl told him”. But maybe Albert looks at Bernard’s face, screwed up in frustration and puzzlement, and says to himself, “There is someone who doesn’t know the answer!” Thus, I assume more communication between the participants than has been reported. (This example uses “know” in the colloquial sense of “is confident that”.) It is telling that, in Grime’s explanation of the second solution, “Albert taunts Bernard”.

The setters, in a “Voice of God” statement after the fact, added comment to rule this out:

They say Bernard did not reveal that he did not know the answer at the start of the conversation, so Albert cannot know this as fact.

(I like Grime’s response that, “I don’t think they have settled it, since this ignores the possibility that Albert knows by some other method, for example from Cheryl.” I also feel that it is unfair, because we know we haven’t been told everything – because we have to figure out a key part of the three-way conversation. So why shouldn’t we suspect that there is more to the conversation than reported?)

I said that I assume (a) that it is not possible without communication to know another’s mind, and (b) that people generally do communicate. But this is entirely a puzzle about knowing others’ minds: we must figure out what Albert has been told and what Bernard has been told. We are asked to play the role of an eavesdropping 4th person (Danielle, perhaps?) who somehow also has a list of the dates. If we were told from the start which month Albert has, and allowed to play a role in the story (or, vice versa, which number Bernard has) then we would have a different journey to the solution.

At the heart of this conundrum is a question of whether the puzzle is a question about people, or about information. Of course it is a question about information, but it is framed as occurring within the relationships between three people and so information about ways that people behave can be brought in by solvers who are prone to see the world in terms of people-problems. This is not the first time I have been caught out by maths puzzles that state a fact that I am supposed to demonstrate are deduced rather than revealed.

Which leads to the third person in the puzzle: why would a person choose such a roundabout way of revealing hir birthday? Cheryl’s behaviour is puzzling. Some people, it seems, have been ascribing motives (usually negative ones) to her behaviour:

But the same goes for Albert and Bernard. Why don’t they just pool their data? Instead, they taunt each other with their deducing skilz. It’s the worst of masculine competitiveness in action! They’re a pair of jerks.

I love mathematics, and logic puzzles. But here, the “correct” answer is of less interest to me than the thought process: what makes a person instinctively leap to the “information” or the “personal” interpretation of the puzzle? That is, a “deduction” or a “fact” meaning of “know”. I am sure the MBTI analysis would discuss this in terms of the T/F scale (Thinking vs Feeling). I’m not sure how the Five Factor Model would encode it, or if researchers would code it using the “Dark Triad” (Psychopathy, Machiavellianism, Narcissism, if I recall correctly) instead. I am fairly confident that economists assume people see the world as deductive, information-based problems whereas plenty of people, as this puzzle has demonstrated, are not like that.

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About ValeryNorth

I overthink everything.
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