Infinite liars

Regrettably, this chapter consists entirely of lies—every sentence in it, including this one, is false.

When my daughter Hypatia was very young, she liked to greet visitors to our home at the door with a friendly challenge:

Answer yes or no: will you answer no?

The guest would return her cheerful smile, yet stand flustered in the doorway with ponderous uncertainty. They were stymied by her question, for if they should answer yes, they would be saying that they should instead have said no, and if they should answer no, then it would mean that they should have said yes. My daughter had trapped them in an impossible logical labyrinth.

Epimenides paradox

A much earlier logical labyrinth occurred on the island of Crete in the 6th century BC, when Epimenides announced

All Cretans are liars.

Being a Cretan himself, you see, Epimenides was in particular calling himself a liar. Does this mean that the statement itself is a lie?

Well, for someone to be a liar, must everything they say be a lie? I don't think so. If someone tells lies only some of the time, then I would call them a liar, even if at other times they tell the truth. (I don't feel similarly about describing someone as honest or as a truth-teller—they must be reliably honest for this designation.) So it seems that Epimenides could consistently be telling the truth with this particular statement, even if, as seems dubious, every Cretan including Epimenides himself has told a lie at some point.

Some accounts of Epimenides have him saying:

Cretans are always liars.

Does this change things? I don't think so, not much. After all, a musician is still a musician when the song has ended; an artist remains an artist when the brush is put down. And so also with liars. If all Cretans have told a lie, then they will always be liars from now on, even if they sometimes assert truths.

Consider, in contrast, the more extreme statement:

All Cretans always lie.

This is a totally different situation. Epimenides cannot truthfully make this statement, for if he is truthful here, then in virtue of the meaning, since he is Cretan, his statement must also be a lie. But can the statement be a lie? Well, it certainly seems so, and in fact I find it very likely that the statement is not true, since I believe that most Cretans are good honest people, who seldom lie. This statement could merely be one regrettable case of a Cretan untruth, with the truth being that only some Cretans lie and even then only sometimes.

The liar paradox

The liar paradox refers to the troubling conundrum we face when trying to assign a coherent truth value for the following sentence, known as the liar sentence or sometimes personified simply as the liar.

This sentence is false.

The liar thus asserts its own falsity.

The central problem is that if the sentence were true, then precisely because of what it asserts, it would also have to be false, contrary to the assumption; but if it were false, then again precisely because of what it asserts, it would have to be true. It seems that the sentence, therefore, can be neither true nor false, posing a fundamental challenge for a complete and successful theory of truth.

The no-truth-value solution strategy

Perhaps one might hope to resolve the liar paradox by saying simply that the liar sentence has no truth value. It expresses a proposition that is neither true nor false, a gap in the truth values. Can one dissolve the liar paradox this way?

The liar's revenge

Some philosophers rebut that response by introducing the following sentence, known as the liar's revenge:

Either this sentence is false or it has no truth value.

If the liar's revenge were true, then in light of what it asserts, it would either be false or have no truth value, but both of those contradict the assumption that it is true. If it were false, in contrast, then in light of the first clause it would have to be true, which is again a contradiction. Finally, if the liar's revenge had no truth value, then precisely because of what it asserts in the second clause, it would be true, which would contradict the assumption that it had no truth value. So it seems wrong to say of the liar's revenge that it is true, wrong to say that it is false, and also wrong to say that it has no truth value. Crazy!

The liar's revenge thus seems to undermine the no-truth-value strategy of answering the original liar paradox. If that strategy were robust, then we should expect it also to work for the liar's revenge, which doesn't seem to be the case.

False versus not true

Some philosophers prefer to state the liar sentence as the assertion:

This sentence is not true.

That is, they prefer to take the liar to assert its own untruth as opposed to its own falsity. They do so in order to avoid a possible objection that could be made by someone who finds a distinction between “not true” and “false.”

A straightforward approach to truth and falsity might take false just to mean not true, especially when applied to sentences with a definite meaning, and with this account the need for a modified statement of the paradox may seem less urgent. But meanwhile, if there will be question of meaningfulness, then perhaps it will be better to use “not true” in place of “false.” For example, would you say that the following sentence is false?

'Twas brillig, and the slithy toves

Did gyre and gimble in the wabe.

Or how about this one—would you say it is a false sentence?

Shillyhumph ghghx88shhs.

It would seem reasonable to say instead that these assertions are nonsense, without any meaning—they are neither true nor false, because they do not express any proposition. But in this case, the sentences would reveal a distinction between falsity and untruth, for we have said that these sentences are not false and yet they are not true. The sentences thus show that being false is not the same as being untrue.

But notice how the not-true formulation of the liar paradox thus seems to take on board much of the liar's revenge reasoning. Namely, if the sentence were true, then it should not be true; and if it were not true, then it should be true. This latter reasoning seems to work whether or not the sentence has a truth value, since if the sentence had no truth value, then in particular it wouldn't be true, and so what it says would be the case. So it would be true after all. For this reason, the not-true formulation of the liar sentence can be seen already as an efficient form of the liar's revenge.

Obvious truth

Meanwhile, if you should reflect on the matter, the following sentence

This sentence is not obviously true.

does seem indeed to be true. If it were obviously true, then in particular, it would also be true, which in virtue of what it says would mean that it is not obviously true, a contradiction. So it isn't obviously true, which is just what it says. So it is true, but not obviously true.

Consider in contrast the sentence:

This sentence is obviously not true.

It cannot be true, since then it wouldn't be. That is obvious. So it is obviously not true. But that is just what it asserts!

I find it interesting to consider the obvious operator and how it interacts with logic. For example, it seems clear that “obviously not φ” implies “not obviously φ,” but not always conversely, since perhaps the question of φ or not φ is confusing—not obvious either way—in which case we would have both “not obviously φ” and “not obviously not φ.” It seems reasonable to claim that obviously (A and B) is equivalent to (obviously A) and (obviously B), but the similar claim for disjunction “or” seems wrong. That is, obviously (A or B) can be true even when neither A nor B is obvious. What is the logic of obviousness?

The both-true-and-false solution strategy

Some philosophers embrace a solution strategy for the liar paradox by declaring that it is both true and false. In paraconsistent logic, for example, there can be true contradictions, statements that are both true and false. These would be truth-value gluts, as opposed to gaps, as in the no-truth-value solution strategy we considered earlier. In the paraconsistent approach to a theory of truth, one develops not only the theory of truth, but also a simultaneous theory of falsity, with the idea in mind that these designations might overlap. And indeed they do overlap in the liar sentence, which is declared both true and false. Does this resolve the paradox? To my way of thinking, no, it does not bring us insight, but can be seen rather merely as pointing at the paradox, underlining it and taking it fully on board by incorporating it into the very foundations of the logic, but not resolving it in any way. Paraconsistent logic itself, after all, is as paradoxical as the liar paradox—it does not seem to help us find our way out of paradox.

Avoiding self-reference

People often observe that the liar sentences exhibit a self-referential nature, with each sentence making an assertion about itself. And perhaps it is natural to become suspicious of self-reference when it seems to be featured prominently in such paradoxes. Is self-reference itself the source of paradoxicality?

The twin liars

Some philosophers attempt to eliminate the self-reference from the liar paradox by various means. Consider, for example, the following pair of sentences, which I dub the twin liars.