Smullyan's Drinking Principle

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Theorem

Suppose that at least one person is in the pub.


Then there is a person $x$ in the pub with the property that if $x$ is drinking, then everyone in the pub is drinking.


Proof

There are two cases:

$(1): \quad$ Everyone in the pub is drinking.
$(2): \quad$ Someone in the pub is not drinking.

Suppose first that everyone in the pub is drinking.

Then $x$ can be chosen to be any person in the pub.


Suppose instead that someone in the pub is not drinking.

Then $x$ can be chosen to be any person in the pub who is not drinking.

$\blacksquare$


Source of Name

This entry was named for Raymond Merrill Smullyan.