If you’re interested in solutions to the Liar, I assume you know what it is, so I’ll be quick with the explanation. “This sentence is false” – is that sentence true or false? If it’s true, then it’s false, and if it’s false, then it’s true. Whoa!
The Liar is a serious problem for logicians and philosophers because it calls classical logic into question. The Liar does not go away unless we make some changes, either to the way that we refer to things, the way we understand truth, or to some laws of classical logic, such as the law of excluded middle or the law of non-contradiction.
There have been many proposed solutions to the problem, and I’ll be doing a survey of them. I’ll begin with the most tempting and popular solution: the argument that the Liar is (or should be considered) meaningless.
The solution comes in a few basic flavors, but they all hinge on the idea that the Liar is only a problem if it is in fact saying something. If it isn’t – in other words, if it doesn’t express a proposition (philosophy talk for “statement”) – then it’s just a bunch of words put together, like “mafpol govohav,” or “the dog.” In order for a sentence to be meaningful, it must have a subject – a thing it is about – and a predicate – something being said of that subject. The Liar, so the argument goes, has no subject.
At first glance, the sentence at least seems to be saying something. Its subject is the sentence itself – “This sentence is false” – and the predicate is “is false.” The sentence refers to its subject by using the pronoun “this” – an “indexical” in philosophical parlance (it just means semantic pointer). But there is usually nothing problematic about sentences that refer to their subjects via indexicals.
One way to argue for this solution is to say, simply, that cases of self-reference are semantically “empty.” If all a proposition refers to is itself, then it’s not really referring to anything. But why? If there’s a sentence, and it refers to itself, why is that necessarily problematic? When I refer to myself, that isn’t semantically “empty,” I am referring – to myself. So self-reference generally isn’t a problem. Why should the Liar’s attempt at self-reference fail?
One answer to this objection is that the sentence refers to itself prior to its own utterance. That is, when I say, “This sentence is false,” I say “this sentence” before I’ve finished saying the whole sentence, so I am referring to something that does not yet exist. But this is not a serious problem. Consider the sentence: “The word that comes 29 words after the end of this sentence will be the word ‘true.’” Though there was no fact of the matter when I first wrote that sentence, by the time I finish this sentence that sentence will turn out to be true. There is generally no problem talking about things that have yet to occur. If they do occur, the sentence refers, if they don’t, the sentence doesn’t.
The most plausible argument for the idea that the Liar fails to refer goes as follows: if there really is a subject, then we should be able to specify what the indexical points to (its “referent”). If what it points to is also a proposition that refers via an indexical, we ought to be able to follow that indexical to its referent as well. This process of indexical-specification should bottom out at something definite. If it doesn’t – if it leads us down an infinite regress or a semantic loop of indexical subjects – then the original sentence fails to refer. This is what happens with the Liar.
This argument, as stated, lacks justification. What does “this sentence is false” refer to? “This sentence is false.” What happens when you try to specify the indexical within that referent? Answer: who cares? All you need in order to qualify as a proposition is a subject and a predicate – you don’t need to inquire further into the nature of the subject.
The fact that a subject is not perfectly specified does not challenge the meaningfulness of a sentence. If it did, the problem of vagueness would ensure that most of our sentences be meaningless. Consider “The mayor weighs 75 kilos.” What does “the mayor” refer to? Is a drop of sweat on her body part of “the mayor”? What if that drop is in the process of evaporating? It is impossible to fully specify what exactly “the mayor” refers to. This does not make sentences about “the mayor” fail to refer. The requirement of complete subject specification is not only ad-hod, but also practically unworkable.
There is a more interesting justification for the specify-the-indexical argument, however. It’s possible to say that we don’t generally need to specify subjects, but we do need to in the special case of the Liar. This is because the subject is the (would-be) sentence, and whether Liar is a sentence at all is the question at hand. You might look at it like this: if the Liar refers to “This sentence is false,” then whether the Liar’s subject is a sentence is the whole issue that we’re examining.
The thing is, there’s no way to follow this line of reasoning without question-begging one way or the other. Let’s reimagine the Liar as follows: “there is some proposition such that this string of symbols expresses that proposition and that proposition is false.” Does a proposition exist such that that proposition is expressed by the string of symbols in the quote? It would seem to be that there is; but if there is, that assumes that the string of symbols expresses a proposition. Now, let’s say that there isn’t. Then, the string of symbols doesn’t say anything at all, so there’s nothing else to ask. So we can assume that the Liar does express a proposition, or that it doesn’t; nothing compels us one way or the other.
This leaves us with the task of stipulating whether the Liar’s subject expresses a proposition. Since it so obviously seems to, why not stipulate that it does? After all, when strings of symbols appear to have subjects and predicates, don’t we always assume that they express a proposition? Consider “The spoon is in my hand.” Of course, we could stipulate that this isn’t a sentence, in which case it would have no subject. But why would we do that? Of course it’s a sentence, it has a subject and a predicate. The same seems to be true of the Liar, regardless of what else we might say about its subject.
You might still think that all this odd loopiness is reason enough to stipulate that the Liar doesn’t express a proposition. There are, however, other very strong reasons not to consider the Liar meaningless. We’ll explore those in part 2.