“This sentence is false,” understood as talking about itself, is
often referred to as the Liar Paradox, but the name does not matter
so much as its rather curious logical properties: If
“this sentence is false” is true, then it’s false, but also, if
it’s false, then it’s true. But unless a sentence can be both
true and false, that’s surely unacceptable.
I shall claim
that neither Bill nor anyone else who has addressed the issue here
(or rather the theorists upon whom they rely, for nothing proposed
here is new to the literature on this subject) has successfully
resolved the paradox. More precisely, the paradox can be resolved in
one of these ways, but only at the price of falling into some related
paradox. Indeed, if I am correct, the paradox cannot be
resolved – cannot in a way that does not engender further
paradoxes, that is – because in it, we are struggling with the
boundaries of our conceptual scheme and attempting incoherently to
both remain within it and to stand outside it.
Two rather different lessons might be drawn. Some might conclude that our most basic notions of truth, meaningfulness and justification are themselves incoherent or in some other way radically mistaken. I, on the other hand, think that’s a decided over-reaction (if such basic notions are misguided, what is the standing of the reasoning that tells us they are?) I think the right lesson to draw is that such notions cannot – cannot in principle – be fully formalized. We do not have, and never will have, some formalizable guarantee against falling into the Liar Paradox (or one of its relatives, such as Russell’s Paradox). But that is no reason for despair. The fact that we can fall into paradoxes is no reason at all for casting doubt on the truth, meaningfulness, or possibility of justifying statements like “grass is green.” If we are trying to map the entire realm of statements that can be true (or false, etc.), then we will not be able to find any formal rules that insure that we will stay out of trouble. Perhaps, the best we can do is take a cue from the ancient map-makers who marked uncharted and vaguely indicated regions with the legend, “here be monsters.” But that doesn’t mean that there’s no such thing as staying out of trouble nor that there are no regions without monsters. In the end, this should be unsurprising: it’s a way of saying that there is no formalizable substitute for good judgment.
****
You apparently think we can avoid the paradox by claiming that “This sentence is false” is meaningless and therefore, neither true nor false. Avoiding the Liar Paradox is not so easy. You give two general reasons for saying that it’s meaningless. One is Ryle’s (though you don’t cite him). The other is verificationist.
You offer a version of Ryle’s criticism that an apparently self-referential sentence stands in need of what he called a “namely rider.” That is, before you know what the sentence means – and therefore before you can judge whether it’s true or false – you need to be able to fill in the ellipses in “This sentence, namely, …., is false.” For a sentence that’s not self-referential, this can readily be done, e.g., “This sentence, namely, ‘Bill has resolved the Liar Paradox,’ is false.” You can fill in the ellipses in the ‘namely’ clause by quoting the sentence. This strategy, however, cannot be applied to a genuinely self-referential sentence like “This sentence is false.” If you try, you get things like “This sentence, namely, ‘this sentence, namely, “this sentence …” ’ ” – and you never get around to saying that it’s false.
The Rylean criterion, though, is too stringent, because it would also rule out plainly true sentences like “this sentence contains five words” or “this sentence ends with a period.” Equally, it would rule out plainly false sentences like “this sentence contains six words” or “this sentence ends with a question-mark.”
About the verificationist point, you are less clear, but you object that we would have to know what would verify that the sentence is true (or not) in order to meaningfully claim that it is true (or not). However, verificationism, as a general criterion of meaningfulness, is deservedly in bad odor – and has been for forty years or more. There have been numerous attempts to make it work, and they have all failed. I assume you know this, since you have, in the past, favorably cited Brand Blanshard’s Reason and Analysis, which contains a devastating criticism of verificationism. Accordingly, it’s unclear what you’re doing in appealing to the supposed impossibility of verifying “this sentence is false” as an argument against its being meaningful. If some sentences can be meaningful without being verifiable, why not this one? Something more than an appeal to verification conditions is needed, and you haven’t provided it.
But, leaving aside your inconclusive arguments that “this sentence is false” is meaningless, let’s suppose that it is indeed meaningless. Does that really enable you to escape the Liar Paradox? At first sight, it does. However, there’s a related paradox – I think it’s been called the Strong Liar – that it does not escape. To wit:
“This sentence is false or meaningless.”
Plainly, this lands you back in exactly the same kind of problem. You can’t escape here by saying that the sentence is meaningless, because if it’s meaningless, then it’s true (and if true, then false or meaningless, and if false or meaningless, then not true).
In addition, the Strong Liar exemplifies a strategy that can be
reiterated to deal with any response offered. Suppose you say that
“this sentence is false or meaningless” is not true or false or
meaningless, but is just a string of words that doesn’t manage to
say anything. That succumbs to the Stronger Liar:
“This
sentence is false or meaningless or is just a string of words that
doesn’t manage to say anything.”
And so on.