The Drinker Paradox

Page last updated 13 Jun 2021

Raymond Smullyan

This paradox can be found in a puzzle book entitled “What Is the Name of this Book?”  (Footnote: Raymond Smullyan. What is the Name of this Book? Prentice-Hall. ISBN 0-13-955088-7, 1978 Smullyan, What is the Name of this Book? Details.) by Raymond Smullyan, who called it the ‘drinking principle’. This is perhaps the origin of this ‘paradox’. The ‘paradox’ is also referenced in some other books (Footnote: ▪ Sindy Dunbar, “Logic & Mathematical Paradoxes”, World Technologies, ISBN 13: 9788132343134, 2012 Dunbar, Logic & Mathematical Paradoxes: Details. ▪ Orna Grumberg, Tobias Nipkow, and Christian Pfaller, Formal Logical Methods for System Security and Correctness, IOS Press, ISBN 9781586038434, 2008 Formal Logical Methods: Details. ▪ Martín Escardó and Paulo Oliva, PDF Searchable Sets, Dubuc-Penon Compactness, Omniscience Principles, and the Drinker Paradox Computability in Europe Conference 2010.) and articles. (Footnote: ▪ Marc Bezem and Dimitri Hendriks PDF Clausification in Coq. ▪ Freek Wiedijk, Mizar Light for HOL Light Conference on Theorem Proving in Higher Order Logics, 2001. ▪ Henk Barendregt, PDF The Quest for Correctness in Images of SMC research, 1996. )

This paradox is quite different from most paradoxes, since of itself it is not a paradox at all, and it simply is a statement that is not correct. The claim that it is a paradox relies on the insistence that the statement is also a theorem of classical logic. Classical logic is a system that gives a general method of analyzing statements in a logical way. However, one of the crucial differences between statements of natural language and statements of classical logic is that classical logic assumes a fixed, non-changing world, where nothing changes over time. Obviously, in that respect, classical logic does not reflect real world situations that do involve changes over time.

The statement of the paradox is this:

S: “For any pub, there is always a customer in the pub such that, if he is drinking, every customer in the pub is drinking.”

Obviously, we know from real world experience that this is incorrect - and because of that, we would not, by any stretch of the imagination, call the statement S a paradox - we would simply say it is incorrect.

So where does the notion that the statement S is a paradox come from?

It comes from the naive notion that there is a theorem of classical logic that reflects precisely what the natural language statement S states. This theorem of classical logic can be written as:

T : ∃x, [D(x) ⇒ ∀y, D(y)]

which in English, essentially states:

“There is some x, where, if D applies to x then D applies to every y”.

In classical logic, this statement is precisely the same as:

T : ∃x, [¬D(x) ∨ ∀y, D(y)]

which in English, essentially states:

“There is some x, where, D does not apply to x or D applies to every y”.

Now, in classical logic, everything has a fixed truth value. So, either D applies to every y, or it does not. So, considering the two options:

1. Suppose that D applies to every y. In that case, the statement T is correct,
    or
2. Suppose that D does not apply to every y. In that case, there must be at least one x where D does not apply, so the statement T is again correct.

In either case, in classical logic, the statement T is correct.

But many people seem to believe that you can apply this theorem of classical logic to the real world situation of people drinking in a pub, where some people may not be drinking and then start drinking. They take the classical theorem of logic

∃x, [D(x) ⇒ ∀y, D(y)]

and say that it does apply to the real world, and they may write it such as this:

∃x, [Drinking(x) ⇒ ∀y, Drinking(y)]

where Drinking(x) means that the person x is drinking.

And yes, taken as a statement of classical logic, this can be proved in classical logic, where Drinking(x) is an abstract concept which has no connection to real world drinking. And in classical logic, every proposition has a fixed truth value, and every proposition must be either true or false, and always remains so. That presents no difficulties when dealing with situations where everything has a fixed truth value, for example, in the case of arithmetic, where 1+1 always equals 2. And in classical logic, for any given thing x, if D(x) or Drinking(x) applies, then it always applies and it never changes; and if the negation of D(x) or Drinking(x) applies, then that always applies and it never changes. But in the real world, which is the case with drinkers and pubs, the case of whether a person is, or is not drinking is subject to change over time and does not have a fixed truth value.

Some people have tried to get round this by claiming that the theorem of classical logic does apply in the real world for an “instant of time”. There is an article in Wikipedia on this drinker paradox, and after I pointed out in Inaccurate statements regarding theorems of classical predicate logic the inapplicability of the classical theorem of logic to the real world situation, the Wikipedia article now relies on this claim - that the classical theorem of logic applies to the real world at an instant of time - but it doesn’t even begin to explain what that might mean.

There are major difficulties with the notion of an actual real world “instant of time”. According to the theory of relativity, different observers see time in different time frames. And how might one define ‘drinking’ at an “instant of time”⁠? Is it when n molecules of drink have passed a certain molecule on someone’s lip by x angstroms? How many molecules? Which molecule on the lip do we choose? How are the exact positions of the molecules observed at the “instant of time”⁠? How does this apply to different orientations of the molecules?

But the main difficulty is that if a person is not drinking at some “instant of time”, and is drinking at some later instant, then there must be an interval of time between those two instants. But then that interval itself must be either an indeterminate state of neither drinking nor not-drinking, or else we can find within that interval further “instants of time” of not drinking followed by “instants of time” of drinking. We can continue this process further, and if there is always a time interval that is an indeterminate state of neither drinking nor not-drinking, then the theorem does not apply to the real world. On the other hand, if there is no such interval, then the state must change from not drinking to drinking in zero time - within one “instant of time” - so that the person is drinking and also not drinking in that same instant. This means that whichever way one looks at it, while the notion of an “instant of time” in the real world can be a useful hypothetical tool, there is no exact correspondence of that concept in the real world.

In summary, there is no logic in the assertion that this theorem of classical logic means that the natural language statement “For any pub, there is always a customer in the pub such that, if he is drinking, every customer in the pub is drinking” must be true.

The old adage that a little learning is a dangerous thing applies here. Beware of people who call themselves logicians, and who have learned some of the rules of classical logic, and who take it on themselves to tell the rest of the world that they know better than we do as to what we actually mean when we make statements about the real world.

Note: For an analysis of other paradoxes by Raymond Smullyan, please see Smullyan’s Paradoxes.

Other paradoxes