Solutions to the Liar Paradox according to which paradoxical sentences are meaningless face all sorts of challenges. For one thing, the partisan of such a solution needs to have a plausible error theory to explain the widespread intuition that such sentences are meaningful. For another thing, they must find a way to defuse familiar "revenge" Liars, like $, below.
$ The sentence marked with a dollar sign is either false or meaningless.
These are major obstacles, and whether or not they can be plausibly overcome is a subject for another time. What I want to focus on is an objection which I find far less initially plausible, but which I hear a surprising amount of the time.
It goes, more or less, something like this:
"Even if Liar sentences are meaningless, they're still not true, right? Meaningless claims aren't true, so that solution doesn't even help with the Strengthened Liar. ('This sentence is not true.')"
Now, in whatever sense in which we are speaking sense when we say "meaningless sentences aren't true," surely it would be exactly equally correct to say that "meaningless sentences aren't false." Meaningfulness is surely a prerequisite for falsity, just as it's a prerequisite for truth.
Forget, for a moment, about the Liar and its kindred semantic paradoxes. Let's just think about a normal case of a sentence whose meaninglessness is much less controversial, like "Green ideas sleep furiously."
Now, given the two claims we just endorsed:
(1) Meaningless sentences aren't true.
(2) Meaningless sentences aren't false.
Given these two claims, Disjunctive Syllogism, Conjunction-Addition and the Principle of Bivalence (for every P, either P is true or P is false), we can easily derive a contradiction about a normal, non-paradoxical meaningless sentence like "Green ideas sleep furiously."*
Let's symbolize "Green ideas sleep furiously" as G. Given Bivalence, we've got our first premise:
1. Tr(G) v F(G)
Symbolizing (1), above, we've got our second premise:
From 1, 2 and Disjunctive Syllogism, we can conclude:
Symbolizing (2), above, we get:
And finally, of course, from 3, 4 and Conjunction-Addition, we conclude:
5. F(G) & ~F(G)
So, given Bivalence and a couple of basic logical rules, the claim that meaningless sentences aren't true or false entails contradictions. Perhaps the very notion of meaninglessness as a separate category from truth and falsity is inconsistent!
But wait. Even if we're willing to give up on the claim that any sentence anywhere is meaningless, what about questions. Surely questions exist. Can questions be true or false? How about bits of burning candle wax? Are they true? No? Are they false? Also no? Well, if G symbolized not a meaningless declarative sentence but a question or a bit of burning candle wax, we could use precisely the same five-step proof to derive an outright contradiction about the semantic status of the question or the bit of burning candle wax.
Clearly, something has gone horribly wrong in our reasoning.
Here's what it is:
When we say "meaningless statements aren't true," we might be making one of two claims:
1-Let M(P) mean "P is meaningful." For every P, if ~M(P), then ~Tr(P).
2. Meaningless sentences aren't the sort of thing to which truth talk meaningfully applies.
If you mean 1, you're confused. (It's significant that no dialetheist has ever used the proof above as an argument for the existence of true contradictions. And if that argument were available with them, why would they bother to swim in the murkier waters of semantic paradoxes?) When we try to symbolize a meaningless statement and perform logical operations on them, we're engaged in a nonsensical category mistake, of exactly the same sort that we'd be engaged in if we tried to symbolize and perform logical operations on a big of burning candle wax.
If you think meaningless sentences aren't true, and when you say that, you actually mean to assert of every meaningless sentence the negation of the claim that that sentence is true, you are necessarily saying something meaningless. After all, given the universal intersubstitutivity of P and Tr(P) for every P, if you say that "Green ideas sleep furiously" is not true, meaning ~Tr(P)--where P is "Green ideas sleep furiously"--then you are, in effect, asserting ~P. As the philosophers of the Vienna Circle were so fond of pointing out, the negation of nonsense is nonsense.
Unless you're willing to accept that green ideas fail to sleep furiously--and that there are true contradictions about the truth-value of every meaningless sentence--when you say that "meaningless statements aren't true", you'd better mean it in sense 2.
Now, like I said before, none of this helps the partisan of the meaninglessness view against revenge paradoxes crafted to fit the details of the view. (For example, given the discussion above, one might wonder about the following sentence, which we could call The Babbler: "This sentence is not the sort of thing to which truth talk meaningfully applies.") And that's fair enough.
Still, whether or not they are ultimately viable when we really look into the ins and outs of revenge paradoxes, intuitive difficulties and so on, meaninglessness solutions can't be batted away with the blunt instrument of pointing out that meaningless sentences aren't true.
*At least, that's one that most people take to be meaningless. (E.g. another commonly heard response to claims that Liar sentences are meaningless is "wait, you don't mean meaningless the same way that 'Green ideas sleep furiously' is meaningless, do you?") If, however, you hold semantic views on which 'green ideas sleep furiously' comes out as meaningless, please accept the following as a substitute:
Sentence S1: 'Green swimming red night fun fun fun!'**
**"But wait," I can hear some of you saying, "Sentence S1 isn't even well-formed!"
Well, I'd argue that any invocation of "well-formedness" as a consideration here misses several points at once. "Well-formed" means something fairly specific for symbolic formulas. It's not clear what it's significance is supposed to be when we start throwing it around with reference to natural language sentences. The closest natural language equivalent of the formation rules of formal systems would be the rules of grammar, and conformity to those is clearly neither necessary nor sufficient for meaningfulness. If someone accuses another person of having done something wrong, and the person being accused responds with Sentence S2:
Sentence S2: "Like hell I did!"
....everyone knows what is meant. If the accuser, trying to catch the accused person in an inconsistency, formalized Sentence S2 with a Greek letter, did the same with some of his other statements and and drew out some logical implications, no one would think the accuser was in the grips of any kind of deep conceptual confusion.
Now, someone trying to desperately hold on to some significant role for natural language "well-formedness" could try to say that the difference is that there are grammatically "well-formed" sentences that mean the same thing as Sentence S2, whereas no grammatically well-formed sentence means the same thing as Sentence S1, but, of course, by definition, no grammatically well-formed sentence *ever* means the same thing as any meaningless sentence, because meaningless sentences don't mean anything. That's what we mean when we call them "meaningless."