Help with a riddle

Question

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?

Answer 1

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.

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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.