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| The '''drinker paradox''' (also known as '''drinker's principle''', '''drinkers' principle''' or (the) '''drinking principle''') is a [[theorem]] of [[classical logic|classical]] [[predicate logic]], usually stated in [[natural language]] as: ''There is someone in the pub such that, if he is drinking, everyone in the pub is drinking''. The actual theorem is
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| :<math>\exists x.\ [D(x) \rightarrow \forall y.\ D(y)]. \, </math>
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| where D is an arbitrary [[Predicate (mathematical logic)|predicate]]. The [[paradox]] 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?''<ref name="Smullyan">{{cite book
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| | title = What is the Name of this Book? The Riddle of Dracula and Other Logical Puzzles
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| | author= [[Raymond Smullyan]]
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| | publisher = [[Prentice Hall]]
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| | year = 1978
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| | isbn = 0-13-955088-7
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| | nopp = true
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| | pages = chapter 14. How to Prove Anything. (topic) 250. The Drinking Principle. pp. 209-211
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| }}</ref>
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| == Proofs of the paradox ==
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| The proof begins by recognizing it is true that either everyone in the pub is drinking, or at least one person in the pub isn't drinking. Consequently, there are two cases to consider:<ref name="Smullyan"/><ref name="Images of SMC Research 1996"/>
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| # Suppose everyone is drinking. For any particular person, it can't 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.<ref name="Smullyan"/><ref name="Images of SMC Research 1996"/>
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| # Suppose that at least one person is not drinking. For any particular nondrinking person, it still cannot be wrong to say that ''if that particular person is drinking, then everyone in the pub is drinking'' — because that person is, in fact, not drinking. In this case the condition is false, so the statement is [[vacuous truth|vacuously true]] due to the nature of [[Material_conditional#Definition|material implication]] in formal logic, which states that "If P, then Q" is always true if P (the condition or [[Antecedent (logic)|antecedent]]) is false.<ref name="Smullyan"/><ref name="Images of SMC Research 1996"/>
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| Either way, ''there is someone in the pub such that, if he is drinking, everyone in the pub is drinking''. A slightly more formal way of expressing the above is to say that if everybody drinks then anyone can be the [[witness (mathematics)|witness]] for the validity of the theorem. And if someone doesn't drink, then that particular non-drinking individual can be the witness to the theorem's validity.<ref name="Cameron1999">{{cite book|author=Peter J. Cameron|authorlink=Peter Cameron (mathematician)|title=Sets, Logic and Categories|url=http://books.google.com/books?id=sDfdbBQ75MQC&pg=PA91|year=1999|publisher=Springer|isbn=978-1-85233-056-9|page=91}}</ref>
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| The proof above is essentially [[First-order_logic#Semantics|model-theoretic]] (can be formalized as such). A purely syntactic proof is possible and can even be mechanized (in [[Otter (theorem prover)|Otter]] for example), but only for an [[equisatisfiable]] rather than an [[Logical equivalence|equivalent]] negation of the theorem.<ref name="Coq">Marc Bezem , Dimitri Hendriks (2008) [http://igitur-archive.library.uu.nl/lg/2008-0402-200713/preprint187.pdf Clausification in Coq]</ref> Namely, the negation of the theorem is
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| : <math>\neg [\exists x.\ [D(x) \rightarrow \forall y.\ D(y)]]\, </math>
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| which is equivalent with the [[prenex normal form]]
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| : <math>\forall x \exists y.\ [D(x) \wedge \neg D(y)]\, </math>
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| By [[Skolemization]] the above is equisatisfiable with
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| : <math>\forall x .\ [D(x) \wedge \neg D(f(x))]\, </math>
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| The [[Resolution_(logic)#Resolution_in_first_order_logic|resolution]] of the two [[Clause (logic)|clauses]] <math>D(x)</math> and <math>\neg D(f(x))</math> results in an empty set of clauses (i.e. a [[contradiction]]), thus proving the negation of the theorem is [[Satisfiability|unsatisfiable]]. The resolution is slightly non-straightforward because it involves a search based on [[Herbrand's theorem]] for [[Ground expression|ground instances]] that are [[Propositional calculus|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]]:<ref name="GrumbergNipkow2008"/>
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| : <math>\{ d, f(d), f(f(d)), f(f(f(d))), \ldots \}</math>
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| One can sketch the following [[natural deduction]]:<ref name="Coq"/>
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| <math>
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| \cfrac
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| {\cfrac
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| {\cfrac
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| {\forall x .\ [D(x) \wedge \neg D(f(x))]\, }
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| {D(d) \wedge \neg D(f(d))}
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| \forall_E
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| }
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| {\neg D(f(d))}
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| \wedge_E
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| \qquad
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| \cfrac
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| {\cfrac
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| {\forall x .\ [D(x) \wedge \neg D(f(x))]\, }
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| {D(f(d)) \wedge \neg D(f(f(d)))}
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| \forall_E
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| }
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| {D(f(d))}
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| \wedge_E
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| }
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| {\bot}\
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| \Rightarrow_E
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| </math>
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| Or spelled out:
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| # Instantiating ''x'' with ''d'' yields <math>[D(d) \wedge \neg D(f(d))]</math> which implies <math>\neg D(f(d))</math>
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| # ''x'' is then instantiated with ''f(d)'' yielding <math>[D(f(d)) \wedge \neg D(f(f(d)))]</math> which implies <math>D(f(d))</math>.
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| Observe that <math>\neg D(f(d))</math> and <math>D(f(d))</math> [[Unification (computer science)|unify]] syntactically in their predicate arguments. An (automated) search thus finishes in two steps:<ref name="GrumbergNipkow2008">{{cite book|editor=Orna Grumberg, Tobias Nipkow, Christian Pfaller|title=Formal Logical Methods for System Security and Correctness|url=http://books.google.com/books?id=QTc3WtqXXwQC&pg=PA123|year=2008|publisher=IOS Press|isbn=978-1-58603-843-4|pages=123–124|author=J. Harrison|chapter=Automated and Interactive Theorem Proving}}</ref>
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| # <math>D(d) \wedge \neg D(f(d))</math>
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| # <math>D(d) \wedge \underline{\neg D(f(d)) \wedge D(f(d))} \wedge \neg D(f(f(d)))</math>
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| The proof by resolution given here uses the [[law of excluded middle]], the [[axiom of choice]], and [[Empty domain|non-emptiness of the domain]] as premises.<ref name="Coq"/>
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| == Discussion ==
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| {{Original research|section|date=February 2011}}
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| This proof illustrates several properties of classical predicate logic that do not always agree with ordinary language.
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| === Excluded middle ===
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| The above proof begins by saying that either everyone is drinking, or someone is not drinking. This uses the validity of [[excluded middle]] for the statement <math>S =</math> "everyone is drinking", which is always available in classical logic. If the logic does not admit arbitrary excluded middle—for example if the logic is [[intuitionistic logic|intuitionistic]]—then the truth of <math>S \or \neg S</math> must first be established, i.e., <math>S</math> must be shown to be [[Decidability (logic)|decidable]].<ref>{{cite book |chapter=Choice in Dynamic Linking |author1=Martin Abadi |author2=Georges Gonthier |author3=Benjamin Werner |title=Foundations of Software Science and Computation Structures |editor=Igor Walukiewicz |page=24 |year=1998 |publisher=Springer |isbn=3-540-21298-1}}</ref>
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| === Material versus indicative conditional ===
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| Most important to the paradox is that the conditional in classical (and intuitionistic) logic is the [[material conditional]]. It has the property that <math>A \rightarrow B</math> is true if ''B'' is true or if ''A'' is false (in classical logic, but not intuitionistic logic, this is also a necessary condition).
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| 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.
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| In natural language, on the other hand, typically "if...then..." is used as an [[indicative conditional]].
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| === Non-empty domain ===
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| It is not necessary to assume there was anyone in the pub. The assumption that the [[empty domain|domain is non-empty]] is built into the inference rules of classical predicate logic.<ref>{{cite paper |url=http://www.cs.bham.ac.uk/~mhe/papers/dp.pdf |title=Searchable Sets, Dubuc-Penon Compactness, Omniscience Principles, and the Drinker Paradox |author1=Martín Escardó |author2= Paulo Oliva |publisher=Computability in Europe 2010 |page=2}}</ref> We can deduce <math>D(x)</math> from <math>\forall x D(x)</math>, but of course if the domain were empty (in this case, if there were nobody in the pub), the proposition <math>D(x)</math> is not well-formed for any [[closed expression]] <math>x</math>.
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| Nevertheless, if we allow empty domains we still have something like the drinker paradox in the form of the theorem:
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| :<math>(\exists x.\ [x=x]) \rightarrow \exists x.\ [D(x) \rightarrow \forall y.\ D(y)]</math>
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| Or in words:
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| :''If there is anyone in the pub at all, then there is someone such that, if he is drinking, then everyone in the pub is drinking''.
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| === Temporal aspects ===
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| Although not discussed in formal terms by Smullyan, he hints that the verb "drinks" is also ambiguous by citing a postcard written to him by two of his students, which contains the following dialogue (emphasis in original):<ref name="Smullyan"/>
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| {{quote|
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| ''Logician'' / I know a fellow who is such that whenever he drinks, everyone does.<br/>
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| ''Student'' / I just don't understand. Do you mean, everyone on earth?<br/>
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| ''Logician'' / Yes, naturally.<br/>
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| ''Student'' / That sounds crazy! You mean as soon as he drinks, at ''just'' that moment, ''everyone'' does?<br/>
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| ''Logician'' / Of course.<br/>
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| ''Student'' / But that implies that at some time, ''everyone'' was drinking at ''once''. Surely that never happened!}}
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| == History and variations ==
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| Smullyan in his 1978 book attributes the naming of "The Drinking Principle" to his graduate students.<ref name="Smullyan"/> He also discusses variants (obtained by substituting D with other, more dramatic predicates):
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| * "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.<ref name="Smullyan"/>
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| * A "dual" version of the Principle: "there is at least one person such that if anybody drinks, then he does."<ref name="Smullyan"/>
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| 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.<ref name="Images of SMC Research 1996">{{cite book|title=Images of SMC Research 1996|year=1996|publisher=Stichting Mathematisch Centrum|isbn=978-90-6196-462-9|url=http://oai.cwi.nl/oai/asset/13544/13544A.pdf|contribution=The quest for correctness|author=H.P. Barendregt|pages=54–55}}</ref> 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]].<ref name="Coq"/><ref name="GrumbergNipkow2008"/><ref>
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| Freek Wiedijk. 2001. [http://www.cs.ru.nl/~freek/mizar/miz.pdf Mizar Light for HOL Light]. In Proceedings of the 14th International Conference on Theorem Proving in Higher Order Logics (TPHOLs '01), Richard J. Boulton and Paul B. Jackson (Eds.). Springer-Verlag, London, UK, 378-394.</ref>
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| == See also ==
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| * [[List of paradoxes]]
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| * [[Reification (linguistics)]]
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| * [[Temporal logic]]
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| == References ==
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| {{Reflist}}
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| [[Category:Predicate logic]]
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| [[Category:Paradoxes]]
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