Monday, September 18, 2017

 

Smullyan - Martian/Venusian Men & Women

In "Forever Undecided", Raymond Smullyan proposes a place occupied by Martian men & women, and Venusian women & men. The Martian men always tell the truth, the Martian women always lie, Venusian women always tell the truth, and Venusian men always lie. This is one of his puzzles:

The next [Martian/Venusian man/woman] who walked by made a statement from which George and Oona could deduce that [they] must be a Venusian female. What statement could it have been?

I was surprised that my answer was not the same as Smullyan's -- or rather, it didn't appear to be at first. Smullyan suggested the statement "I am either female or Venusian". The reasoning is that a Martian man could not make this statement because it would be false in that case and Martian men always tell the truth. A Martian woman couldn't make the statement, because they always lie, and the statement would be true for her since she is a female. Likewise, a Venusian man couldn't make the statement because it is true that he is Venusian, and he too must always lie. That leaves the Venusian woman as the only person who can make the statement (either-or can mean both).

My answer was "I am not a Martian Male", which is also correct. A Martian man can't say that because he would be lying. Martian women and Venusian men can't make the statement because it would be true for them and they have to lie, so again that leaves the Venusian woman as the only person who can make the statement.

Now, I said that my answer didn't appear to be the same as Smullyan's at first. I said that because I realized that they are the same by DeMorgan's laws. These laws basically show the relationship between AND, OR, and NOT. That is:
(NOT a) AND (NOT b) is the same as NOT (a OR b)
(NOT a) OR (NOT b) is the same as NOT (a AND b)

If I use M to represent Martian and m to represent male, then my statement, "I am not a Martian male" is simply: NOT (M AND m). If you think of Venusian as "not Martian", and female as "not male", then Smullyan's statement, "I am either Venusian or female", is (NOT M) OR (NOT m), which by DeMorgan's laws is the same as NOT (M and m).

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