Topics Language

Question Why is there anything weird about the sentence 'This very sentence is false'? If it is that the sentence seems to be true AND false, what makes it so different from certain ambiguous sentences which are true and false as well? If it is that the sentence seems to be neither true NOR false, what makes it so different from imperatives and questions which are neither true nor false as well? (The reformulation 'The proposition expressed by this very sentence is false' does not help, it seems, because it fails to express a proposition at all.)

Accepted:
September 28, 2006

Comments

Sentences of this kind are sometimes called `Liar setnences', and they give rise to the Liar paradox. What is "weird" about such a sentence is is that, if it is true, then it follows that it is false, and if it is false, then it follows that it is true. Or, at least, that's what intuition suggests. That seems to imply that, if the Liar sentence is either true or false, then it is both true and false. If so, then either one has to deny that the Liar is either true or false or accept that it is both true and false. (There are other options, in fact, but I won't discuss them here.)

I'm not sure what "ambiguous" sentences you have in mind. Perhaps you mean sentences like "John is bald", where one might have the intuition that he sort of is and sort of isn't. But does one have any inclination to say that he sort of is and isn't? I don't think so. So my own sense, for what it's worth, is that this intuition isn't very robust. And, in any event, we don't seem to have any inclination to say that, if he is bald, then he isn't, and if he isn't bald, then he is. So that's another difference.

As for the other case, imperatives and questions simply aren't the sorts of sentences that can be true or false, whereas the Liar clearly is. We can make this a bit more precise.

As Saul Kripke famously pointed out, there are plenty of examples of quite ordinary claims involving the notions of truth and falsity that are, as he put it, "risky": They are liable to be paradoxical if emprical circumstances are unfavorable. Here's a version of one of Kripke's examples. Suppose Bush says: (B) Everything Colin Powell says about Iraq is true. And Powell says: (P) Most of Bush's remarks about Iraq are false. That's the kind of thing they might well have said. Now it's obvious that these statements could easily have truth-values. Indeed, (B) is uncontroversially false: Even Powell now admits that much of what he told the United Nations was false. Whether (P ) is true is not so obvious. Maybe it is; maybe it isn't.

But suppose that everything Powell said about Iraq, other than (P), had been uncontroversially true. And suppose that exactly half of Bush's claims about Iraq, other than (B), were true. (We'll take "most" to mean: more than half.) Then if (P) is true, then, indeed, everything Powell said is true, so (B) is true; but (P) says that most of what Bush said is false, so (B) must be false if (P) is true. But if, on the other hand, (P) is false, then not everything Powell said is true, so (B) is false; but if (P) is false, then it must not be that most of Bush's remarks about Iraq are false, so (B) must be true. So if (P) is either true or false, (B) is both true and false. And one can establish, in the same way, that if (B) is either true or false, then (P) is both true and false. So if (P) is either true or false, (B) is both true and false and so either true or false, whence (P) is both true and false. And similarly, if (B) is either true or false, it is both true and false.

It's perhaps worth emphasizing that all of the foregoing can be rigorously proven. And, as Kripke also pointed out, the "self-referential character" of the example sentences is not what is at fault. For one thing, (B) and (P) are ordinarily unproblematic, and such pairs of claims get made all the time. For another, one can prove from very weak arithmetical assumptions that there are such self-referential sentences as (B) and (P), and the existence of such sentences plays a critical role in the proof of the Gödel incompleteness theorems.