# Smullyan's Drinking Principle

## 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.