I'm having a discussion about the following riddle with a friend.

There is a boy behind each girl, and a girl behind each boy. What is the smallest number of children required to do this?

I say the answer is 0, they say 2 (if they stand back to back). I think it's 0 because the statement is then automatically true and it was never stated that there was at least one child. Am I correct and is there an easy way to explain this?

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    $\begingroup$ Sure, $0$ is logically correct (though I wouldn't be surprised if your friend promptly said, "Ok, what's the smallest number greater than $0$ which works?"). To explain it, I'd just remark that the only way "There is a boy behind each girl" could be False is if there were some girl with no boy behind her. If there are no girls at all, then there certainly isn't one without a male shadow. $\endgroup$ – lulu Dec 5 '16 at 16:52

You are correct. A statement like "Every X is Y" is called "vacuously true" if there is no $X$.

Essentially, you are saying that you have a set $S$:

For all $X$ in $S$, if $X$ is a boy then there exists a $Y$ in $S$ such that $Y$ is a girl and $Y$ is behind $X$.

and likewise the other way.

But if there are no boys in $S$, this statement is "vacuously true."

So the statement, "Every alien from Mars in the room has a peacock" is likely true. Unless you are near Roswell, New Mexico.

However, human language is funny. The above is true in logic, but human language often teems with hidden assumptions, and one can argue that there is a hidden assumption that there is at least one child. This is even true, sometimes, in mathematics.

For example, in classical first order logic, we do not allow for empty domains.

  • $\begingroup$ Your reasoning is correct, but only by pre-supposing that the question by the OP falls in the category of mathematical logic. However, the OP describes it as a "riddle" and has tagged it as a "puzzle". $\endgroup$ – M. Wind Dec 5 '16 at 17:10
  • $\begingroup$ Thanks for the response, I feel like it is reasonable to apply the mathematical logic in this case and try and avoid too many assumptions. I am generally annoyed by these kinds of riddles since they are often poorly worded but this one caught my interest. Thanks again! $\endgroup$ – Tim Trewartha Dec 5 '16 at 17:32

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