There’s a general problem for those who account for the Liar’s Paradox of “This sentence is false” with the conclusion that “This sentence is false” is neither true nor false (or that it is meaningless, using slightly different terms). The problem is that there’s a Liar-like sentence readily available in terms of this result:
“This sentence is neither true nor false”.
Since the sentence refers to itself (is self-referential), if it is to possibly mean anything, it should be rendered as equivalent with the following claim: “The sentence, ‘This sentence is neither true nor false’ is neither true nor false”.
Consider for a moment that such a sentence is neither true nor false; in that case, this is exactly what the claim says it is; thus, it is true that it is neither true nor false; however, in that case, it would be true, and not neither true nor false; thus, it would false that it’s neither true nor false; on the other hand, if it’s false that “This sentence is neither true nor false”, then this would contradict an adequate account of truth (such as that of correspondence, from my previous post on the Liar’s Paradox) or else plausibly contradict any other account of “This sentence is false” that leads to the conclusion that it’s neither true nor false. Thus, such an account would nonetheless lead to the conclusion that it’s true.
In sum, if “This sentence is neither true nor false” is true, it is false (T→ F), and, if it’s false then, provided an account of the original Liar’s Paradox “This sentence is false” that concludes that it is neither true nor false, it’s nonetheless true (F→ T). That is, (T← → F).
We may wish to revise our account of truth such that, if “This sentence is neither true nor false” is false, then it is not, nevertheless from our account, true. However, such a revision of the account would need to avoid validating the original contradiction we wished to avoid of “This sentence is false”. That doesn’t seem likely!
From “This sentence is neither true nor false” we are led to (T← → F). Notice how the contradiction of the claim’s being both true and false inevitably follows.
To see this, first notice that for “This sentence is neither true nor false” we must either assume the initial truth that it is neither true nor false, or else assume that it is either true or false, each in turn. If the former, then the contradiction stands as long as the account of “This sentence is false” as itself being neither true nor false stands. If the latter, then the contrary of each in turn would follow and lead to a contradiction, unless again we amend our account of truth and falsity to allow that “This sentence is false” is a claim that is either true or false. If we do amend our account, however, (it seems almost certain anyway–this would be difficult for me to prove) the original contradiction of the Liar’s Paradox would stand. We’re damned if we do, damned if we don’t.
Other possible variants:
“This sentence says nothing.”
“This sentence is meaningless (Truth-functionally speaking).”
“This sentence fails to be either true or false (Under an assumed account of truth).”
Thus, if you think it’s true that “This sentence says nothing” says nothing for the same reason that “This sentence is false” says nothing, then a contradiction would seem readily derivable. And same for the other variants.
Under my account of truth (see my original post on the Liar’s Paradox for more on that), “This sentence is false” fails to be a claim (even about itself) for failing to specify any property or scope 𝟇 of itself. It fails to be something that could be false.
The new sentence that we now must unravel does not purport to involve a claim to be false, but is rather a claim to be not true, as well as not false.
“This sentence is neither true nor false.”
How can I analyze this? Under my account of propositions under The Principle of Bivalence, there is an ambiguity when we say that something is not true. To illustrate, a chair is not true and “The sky is cheese” is not true. However, the untruth of the former does not imply that it is false (chairs aren’t false), while the untruth of latter does imply that it is false. In short, one sort of thing (a proposition) follows The Principle of Bivalence while anything else does not. Indeed, following the principle of bivalence is the essential signifier that the thing in question is a proposition! As far as I’m concerned, the two are synonymous. This has favorable consequences for how we are to analyze “This sentence is neither true nor false”.
The sentence claims to be not true and it claims to be not false. In what sense of truth and falsity?–is it in the sense equivalent with a claim to be a non-propostion (i.e. to be something that fails to follow the principle of bivalence), or in the sense that it follows the principle of bivalence?
If it’s the latter, then we can take care of it just as we took care of the original Liar, “This sentence is false”. We would only add that this sentence is also true. More precisely, we’d negate the supposed proposition that it is false and the supposed proposition that it’s true. Since we covered this sense in The Liar’s Paradox, I’ll just restate the conclusion that using the sense of truth and falsity that follows the principle of bivalence for ‘not true’ and ‘not false’, “This sentence is neither true nor false” fails to be a proposition. That is, it fails to be a claim, despite initially appearing to be one. Thus, if the sense is one that follows the principle of bivalence, it fails to be a claim.
On the other hand, if the sense of ‘not true’ and ‘not false’ is in the sense of not following the principle of bivalence, then this is only because it fails to be a proposition (recall that the two are synonymous). Thus, if it’s a claim about not being a proposition, then this is to be analyzed:
“This sentence specifies no property 𝟇 (of itself or at all).”
Is this sentence specifying some property? Indeed it is. Let’s make this explicit:
“This sentence specifies the property 𝟇1: that it specifies no property 𝟇 (of itself or at all).”
We can obviously truthfully say about many things that they specify no 𝟇 at all. For example, it’s true that chairs aren’t propositions, because it’s true that chairs specify no 𝟇 at all. Now, the sentence above is specifying that it specifies nothing, which is self-defeating. That is, the specification of not specifying a 𝟇 is made, and it’s made about itself, which makes the specification. I’m inclined to believe that the proper evaluation of this sentence is that it specifies a non-contradictory 𝟇 and that it doesn’t have it. Thus, it’s false. Even if we think that the property specified is a contradictory one, this would simply lead to the conclusion that the sentence necessarily fails to have the property that it specifies.
The lesson is that if we assume a non-propositional meaning of ‘not true’ and ‘not false’ and have the proposition specify that about itself, as in “This sentence is neither true nor false”, then it becomes a meaningful proposition, one that is false. How?– ‘not true’ and ‘not false’ in the meaningless sense are indeed meaningfully specifiable. By now it shouldn’t be mysterious why this is–they are simply properties that things have or fail to have. If a proposition specifies this property upon itself the result is simply that it won’t have it, for its specifying a meaningful property. If it appears paradoxical that “This sentence is neither true nor false” is false, then I diagnose that this is merely due to a tendency to conflate the two senses of being ‘not true’. Such a sentence ascribes the meaningless sort of non-truth and non-falsity upon itself, and it doesn’t have them for ascribing anything at all, and thus it is false in the meaningful sense.
Keep track of which sense you’re using, and there’s no paradox left!