Drinker paradox - Alchetron, The Free Social Encyclopedia

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 which can be stated as "There is someone in the pub such that, if he 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 statement is not an actual paradox. Its apparently paradoxical nature 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 which 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 actual theorem is

∃ x ∈ P . [ D ( x ) → ∀ y ∈ P . D ( y ) ] .

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

Proofs

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:

Suppose everyone is drinking. For any particular person, it cannot be wrong to say that if that particular person is drinking, then everyone in the pub is drinking — because everyone is drinking. Because everyone is drinking, then that one person must drink because when that person drinks everybody drinks, everybody includes that person.
Otherwise at least one person is not drinking. For any nondrinking person, the statement if that particular person is drinking, then everyone in the pub is drinking is formally true: its antecedent ("that particular person is drinking") is false, therefore the statement is true due to the nature of material implication in formal logic, which states that "If P, then Q" is always true if P is false. (These kinds of statements are said to be vacuously true.)

A slightly more formal way of expressing the above is to say that, if everybody drinks, then anyone can be the witness for the validity of the theorem. And if someone does not drink, then that particular non-drinking individual can be the witness to the theorem's validity.

The proof above is essentially model-theoretic (can be formalized as such). A purely syntactic proof is possible and can even be mechanized (in Otter for example), but only for an equisatisfiable rather than an equivalent negation of the theorem. Namely, the negation of the theorem is

which is equivalent with the prenex normal form

By Skolemization the above is equisatisfiable with

The resolution of the two clauses D ( x ) and ¬ D ( f ( x ) ) results in an empty set of clauses (i.e. a contradiction), thus proving the negation of the theorem is unsatisfiable. The resolution is slightly non-straightforward because it involves a search based on Herbrand's theorem for ground instances that are propositionally unsatisfiable. The bound variable x is first instantiated with a constant d (making use of the assumption that the domain is non-empty), resulting in the Herbrand universe:

One can sketch the following natural deduction:

∀ x . [ D ( x ) ∧ ¬ D ( f ( x ) ) ] D ( d ) ∧ ¬ D ( f ( d ) ) ∀ E ¬ D ( f ( d ) ) ∧ E ∀ x . [ D ( x ) ∧ ¬ D ( f ( x ) ) ] D ( f ( d ) ) ∧ ¬ D ( f ( f ( d ) ) ) ∀ E D ( f ( d ) ) ∧ E ⊥ ⇒ E

Or spelled out:

Instantiating x with d yields [ D ( d ) ∧ ¬ D ( f ( d ) ) ] which implies ¬ D ( f ( d ) )
x is then instantiated with f(d) yielding [ D ( f ( d ) ) ∧ ¬ D ( f ( f ( d ) ) ) ] which implies D ( f ( d ) ) .

Observe that ¬ D ( f ( d ) ) and D ( f ( d ) ) unify syntactically in their predicate arguments. An (automated) search thus finishes in two steps:

D ( d ) ∧ ¬ D ( f ( d ) )
D ( d ) ∧ ¬ D ( f ( d ) ) ∧ D ( f ( d ) ) _ ∧ ¬ D ( f ( f ( d ) ) )

The proof by resolution given here uses the law of excluded middle, the axiom of choice, and non-emptiness of the domain as premises.

Explanation of paradoxicality

The paradox is ultimately based on the principle of formal logic that the statement A → 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 → B is true if B is true or if A is false (in classical logic, but not intuitionistic logic, this is also a necessary condition).

So as it was applied here, the statement "if he is drinking, everyone is drinking" was taken to be correct in one case, if everyone was drinking, and in the other case, if he was not drinking — even though his drinking may not have had anything to do with anyone else's drinking.

On the other hand, in natural language, typically "if ... then ..." is used as an indicative conditional.

History and variations

Smullyan in his 1978 book attributes the naming of "The Drinking Principle" to his graduate students. He also discusses variants (obtained by replacing D with other, more dramatic predicates):

"there is a woman on earth such that if she becomes sterile, the whole human race will die out." Smullyan writes that this formulation emerged from a conversation he had with philosopher John Bacon.

A "dual" version of the Principle: "there is at least one person such that if anybody drinks, then he does."

As "Smullyan's ‘Drinkers’ principle" or just "Drinkers' principle" it appears in H.P. Barendregt's "The quest for correctness" (1996), accompanied by some machine proofs. 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.