The Cretan Paradox is a form of the Liar Paradox. It is also called Epimenides’ Paradox.
Epimenides is a person who is also a Cretan (i.e. from Crete, in Greece). Epimenides informs you as you walk by on your way to Crete: “All Cretans are liars”. The interesting question to ask is this: is this sentence capable of being true?
Epimenides himself is a Cretan. If what he says is true, then he would have to be a liar in this instance. So, is the contradiction supposed to be that Epimenides is a liar and yet does not lie? Perhaps you think there is nothing puzzling here, since he did not say that Cretans always lie. Perhaps Epimenides lies enough to be a liar without telling a lie this time. Yet what if we amend what he says to be:
“All Cretans always lie.”
In this case, if he is telling the truth, then it seems that Epimenides would himself have to be lying, since this is what he says Cretans always do, and he is one. His lying implies that he speaks falsely. However, how could he be speaking falsely here, since we are supposing that he is telling the truth?
And so we have our contradiction: if we suppose him to be telling the truth, he is telling a falsehood. He cannot be speaking only the truth. We would get the same result with:
“All Cretans always tell falsehoods”
I think this is the clearest rendition of the paradox, so let’s stick with this utterance from here on out. Is it possible that this utterance is only true? That is, can it be true in a way that follows the principle of bivalence, in which sentences are either true or false (and not both)? But we just discovered it has to be both, no? Yet let me show you how the sentence cannot be applied to itself for being meaningless, even if it is meaningful when applied to the other sentences of all the Cretans. That it leads to a contradiction is only an illusion, one generated by a failure to track the properties, true and false. At least, I hope to clearly show this.
In order for “All Cretans always tell falsehoods” to be potentially true, we must suppose that all the other utterances of every Cretan must be false (including the others that Epimenides has made). If not, then the sentence is already false, since sometimes they tell the truth (or else say something not meaningful, e.g. “Is a horse”).
Given that all the other utterances of every Cretan must be false, this would leave just one remaining utterance that we need to examine: the utterance “All Cretans always tell falsehoods” itself.
What is truth, regarding sentences? A sentence is true if what it says is indeed the way things are, and it is false if what it says is not the way things are. This is typically done by specifying a property about some thing. No doubt some Cretan has said “The sky is green”. This sentence is false because the sky does not have the property green. I want to suggest a general logical form for such sentences as these that will make it easy to track its moving parts. As I said, the moving parts involve properties (such as green) and things (such as a sky). Let’s say that a sentence is true if it specifies a property P of a thing M that thing M, in fact, has, and false if it does not have it. Indeed, for a claim such as “The sky is green”, this is the only form it has. That is, “The sky is green” is true if and only if the specified M, the sky, has that specified P, the property green.
Now, our utterance is only slightly more complicated than “The sky is green”. It uses quantifiers such as ‘all’ and ‘always’. Basically, this means that the ‘thing’ that our utterance picks out is actually a set of things, and the property is supposed to apply not to the set itself, but rather to each member of that set. If that property applies to each member, then the sentence is true. If the property does not apply to each member, then the sentence is false. Which set does it pick out, specifically?–It’s the set of all the utterances made by Cretans (let’s call this set C). Our utterance says that each and every one of these utterances has the property, false.
We are up to the utterance “All Cretans tell falsehoods” itself, and we need to determine whether or not it can be false. This utterance can be false only if it specifies a property of a thing (or things) it does not have. That is, “All Cretans tell falsehoods” is false if and only if the property falsehood does not apply to all members of set C. It applies to all members of set C that are not “All Cretans tell falsehoods”. What about the liar utterance itself? Falsehood, however, amounts to specifying a property of a thing (or things) it does not have. The property, used on our liar utterance shows it to be saying that all members of set C are false. The members of C are false if they specify a property of a thing (or things) it doesn’t have. This is so for all members of C that are not “All Cretans tell falsehoods”. What about the liar utterance itself? Falsehood, however, amounts to specifying a property of a thing (or things) it does not have. The property, used on our liar utterance shows it to be saying that all members of set C are false. . . and on and on it will go.
The issue here is that falsehood fails to be finitely specified as a property of the liar utterance itself. The specification of the property is what goes on and on, and not some determination about whether or not it actually has a property that has already been specified. If a property cannot be specified, then it surely cannot be applied. Thus, no thing could be said to either have it or not have it.
We were searching into whether or not “All Cretans always tell falsehoods” is capable of being true. Part way through, we ended up searching whether or not it is capable of being false (by fallowing the meaning of truth and falsehood). What I found and tried to make clear is that the property of falsehood never gets specified for the utterance itself. Since the utterance includes itself in the set it speaks of (indeed it specifies the thing it is considering to have a property just fine), the utterance is meaningless. It is incapable of being true or false.
We are sometimes very charitable in our interpretation of the things people say to us. If Epimenides told me that all Cretans lie and I had reason to believe that they do, I would commend him for being the first to tell the truth. That is, I would interpret his utterance as being a sentence made about all the other Cretan’s utterances and not this one. Perhaps it is just strange when sentences include themselves as something they are about. Yet “All sentences specify properties of things” is a sentence I agree with, and it no doubt includes itself regarding what it is about. We need to use language as the only tool to speak about anything, and this would have to include speaking about language itself.