Solution to the Liar's Paradox

By David R. Tribble, circa 1982

There is a solution to the Epimenides Paradox (a.k.a. the Liar's Paradox), wherein it is no longer a paradox.

The paradox goes like this:

1. Epimenides is a Cretan.

2. Epimenides states, "All Cretans are liars."

On the face of it, this appears to be a paradox. Epimenides, being a Cretan, must either be a liar or a truth-teller. Thus his statement must be either true or false. But if it's true, then he (being a Cretan) must be a liar, so the statement can't be true. On the other hand, his statement is false, then he can't be a liar, so the statement must be true. This is a paradox.

Or so it would seem. Actually, the trouble lies in the interpretation of the statement "The statement 'All Cretans are liars' is false".

The solution goes like this:

p1. Epimenides is a Cretan.

p2. Epimenides is either a liar or a truth-teller.

p3. His statement is either true or false.

Assume that there is more than one Cretan:

p4. There is more than one Cretan.

Also assume that Epimenides is indeed a liar:

p5. Epimenides is a liar.

p6. Thus Epimenides's statement is false.

p7. Thus "All Cretans are liars" is false.

p8. Thus not all Cretans are liars.

p9. Thus some (one or more but not all) Cretans are not liars.

p10. Thus at least one (but not all) of them is a liar.

p11. Thus Epimenides, a Cretan, could be a liar.

We assumed that Epimenides was a Cretan (p1) and a liar (p5). Therefore, there is no paradox.

Another way of looking at it is to realize that, unless there is only one member of a set, then the negation of "all members of the set", i.e., "not all members of the set", is not "no members" but "some members".

If Epimenides is the only Cretan (so the set of Cretans has only one member), then there would be a paradox, since "not all Cretans" would mean "no Cretans".

Update, Sep 2004

After years of reading books about math, logic, and paradoxes, I found the same solution by Raymond Smullyan, a well-known logician and puzzle-meister, in his book What is the Name of This Book? (1978).

For many of the logic puzzles in his book, he makes use of a fictional island populated by knights and knaves, where knights always tell the truth and knaves always lie.

Excerpt from Smullyan's book, Part Four (Logic is a Many-Splendored Thing), Chapter 15 (From Paradox to Truth):

253. The Liar Paradox
The so-called "Liar Paradox," or "Epimenides Paradox," is really the cornerstone of a whole family of paradoxes of the type known as "liar paradoxes". (Boy, that sounded pretty circular, didn't it?) Well, the original form of the paradox was about a certain Cretan named Epimenides, who said, "All Cretans are liars."
In this form, we really do not get a paradox at all -- no more than we get a paradox from the assertion that an inhabitant of an island of knights [truth-tellers] and knaves [liars] makes the statement, "All people on this island are knaves." What properly follows is: (1) the speaker is a knave [liar]; (2) there is at least one knight [truth-teller] on the island. Similarly, with the above version of the Epimenides paradox, all that follows is that Epimenides is a liar and that at least one Cretan is truthful. This is no paradox.
Now, if Epimenides were the only Cretan, then we would indeed have a paradox, just as we would have if a sole inhabitant of an island of knights and knaves said that all inhabitants of the island were knaves (which would be tantamount to saying that he is a knave, which is impossible).

He goes on to say:

A better version of the paradox is that of a person saying, "I am now lying." Is he lying or isn't he?

While Smullyan doesn't explicitly mention set theory by name, my argument and conclusion are equivalent to his. This discovery brings me the satisfaction of knowing that this solution is correct, and that I discovered it independently while I was a young college student.

The author can be reached by email at david@tribble.com or at the web page here.