Drinker paradox

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: == Explanation of paradoxicality ==

The paradox is ultimately based on the principle of formal logic that the statement A \rightarrow B is true whenever A is false, i.e., any statement follows from a false statement (ex falso quodlibet). What is important to the paradox is that the conditional in classical (and intuitionistic) logic is the material conditional. It has the property that A \rightarrow B is true whenever B is true or A is false. In classical logic (but not intuitionistic logic), this is also a necessary condition: if A \rightarrow B is true, then B is true or A is false. So as it was applied here, the statement "if they are drinking, everyone is drinking" was taken to be correct in one case, if everyone was drinking, and in the other case, if they were not drinking—even though their drinking may not have had anything to do with anyone else's drinking. == History and variations ==

Smullyan in his 1978 book attributes the naming of "The Drinking Principle" to his graduate students. 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. Non-empty domain In the setting with empty domains allowed, the drinker paradox must be formulated as follows: A set P satisfies :\exists x\in P.\ [D(x) \rightarrow \forall y\in P.\ D(y)] \, if and only if it is non-empty. Or in words: :If and only if there is someone in the pub, there is someone in the pub such that, if they are drinking, then everyone in the pub is drinking. == See also ==