Drinker paradox

Apparent logical paradox
The drinker paradox (also known as the drinker's theorem, the drinker's principle, or the drinking principle) is a theorem of classical predicate logic that can be stated as "There is someone in the pub such that, if he or she is drinking, then everyone in the pub is drinking." It was popularised by the mathematical logician Raymond Smullyan, who called it the "drinking principle" in his 1978 book What Is the Name of this Book? The apparently paradoxical nature of the statement comes from the way it is usually stated in natural language. It seems counterintuitive both that there could be a person who is causing the others to drink, or that there could be a person such that all through the night that one person were always the last to drink. The first objection comes from confusing formal "if then" statements with causation (see Correlation does not imply causation or Relevance logic for logics that demand relevant relationships between premise and consequent, unlike classical logic assumed here). The formal statement of the theorem is timeless, eliminating the second objection because the person the statement holds true for at one instant is not necessarily the same person it holds true for at any other instant. The formal statement of the theorem is

where D is an arbitrary predicate and P is an arbitrary nonempty set.

The proof begins by recognizing it is true that either everyone in the pub is drinking, or at least one person in the pub is not drinking. Consequently, there are two cases to consider:

The paradox is ultimately based on the principle of formal logic that the statement is true whenever A is false, i. e., any statement follows from a false statement Since then it has made regular appearance as an example in publications about automated reasoning; it is sometimes used to contrast the expressiveness of proof assistants.