(Is it Wednesday already? Oh well, better late than never....)
I've argued that, if (as I think) the truth predicate/operator is just a device used to assert things (just as the falsehood predicate/operator is just a device used to assert their negations), it can pretty clearly only be meaningfully applied when there is something there to be asserted--thus, it can only be meaningfully applied to claims about something other than truth. Thus, for example, in a Yablo-like series of sentences where each sentence ascribes truth to the next sentence in the series,
T1: T2 is true.
T2: T3 is true.
T3: T4 is true.
....and so on forever, all the sentences in the infinite series are literally as devoid of meaning as strings of nonsense syllables, or 'Colorless green ideas sleep furiously.' If, on the other hand, sentence T1000000 is "Snow is white," the rest of the sentences inherit their meanings (and, thus, truth-values) from that.
A semantic property pretty clearly *unlike* truth in this respect is meaningfulness itself. If the meaningfulness predicate only applied to meaningful sentences, it wouldn't fulfill its sole communicative function of separating out the meaningful sentences from the meaningless ones. This is important when we consider (18), which, by analogy to the Liar, we can call The Babbler:
(18) Sentence (18) is meaningless.
If (18) is true, it's both true and meaningless, therefore both meaningful and meaningless, and, of course, if it's meaningless, it's both true and meaningless, therefore both meaningful and meaningless. As such, on pain of contradiction, (18) had better just be false.
Fortunately, in light of the above, we have a good principled reason to think that this is indeed the case. If the function of the meaningfulness predicate is to separate out the meaningful from the meaningless sentences, it has to apply to all sentences. Therefore, it's meaningful to say of any sentence that it's meaningful or meaningless, regardless of the nature of the sentence we're talking about. As such, if all a sentence does is assert a view about the meaningfulness of some sentence, even itself, there's no reason for it not to be meaningful. Thus, (18) is false and (19):
(19) Sentence (19) is meaningful.
One important principle, underlying the whole business of revenge-paradoxology, is worth calling attention to here, since I've been implicitly using it a lot. Given these sorts of examples, or, better yet, cased like (20) and (21):
(20) This sentence has seven words in it.
(21) This sentence has twenty words in it.
....where it would be clearly absurd to assert about sentence (20), for example, that is seven words long, without granting that sentence (21) is true, we have what we can call the Meaningfulness of Self-Reference Principle: "If Sentence X has property Y, and Sentence X *states* that Sentence X has property Y, then Sentence X is true (and thus, of course, meaningful)."
With all that in mind, and the demonstrations in Parts II and III that it clearly is possible to engage in apparent reasoning about even the most clearly meaningless sentences--meaning that it's not a problem for meaninglessness solutions to the Liar that it's "clearly possible to reason about it, and we all know what does and doesn't follow from it"--let's turn to the apparently troubling revenge paradox for my view that I ended with last time:
(17) Sentence (17) is one that one would have to ultimately label as "false" if one treated it as being meaningful and went through the motions of "reasoning" about it without making the sort of mistake we would regard in normal contexts as a mistake in reasoning.
So, playing along with the game of treating it as meaningful for a moment, an obvious first question is this:
Does (17) take a stand on the question of its own meaningfulness? In other words, does it (a) say of itself that it's meaningful, (b) say of itself that it's meaningless, or (c) remain neutral on that topic?
The wording strongly suggests that (a) would be the wrong gloss--talk of treating it 'as meaningful' and 'going through the motions' strongly suggests that the point is to, at the very least, keep open the possibility that it's meaningless, if not to actively assert it. That said, if (a) is right--it's taking a stand on its own meaningfulness in the directing of asserting it--then to say that, if one went through the motions of reasoning about it, one would make something we would regard in other contexts--i.e. really reasoning about meaningful things--as a mistake, is to say that, if one reasoned about it and failed to come to the conclusion that it was false, one would be making a real, full-fledged mistake in reasoning--a factual mistake, landing us with the wrong answer. In other words, given (a), (17) is just a normal if un-usually phrased Liar sentence, the normal meaninglesness solution applies to it, and the Principle of the Meaningfulness of Self-Reference is not violated if we simultanneously say of it that, although meaningless, going through the motions seems to get us the result that it's false (and true), given that what it's saying is that this isn't a matter of going through the motions in an empty context, because it really is false. (It can be neutral about its own falsehood, given that it asserts its own meaningfulness and thus converts the neutral-sounding language about apparent mistakes into, in effect, the positive claim that one would be making a substantive mistake and getting the wrong result.)
If (b) is the case, then we have a disguised conjunction of two claims: (i) a claim about its alleged meaninglessness, and (ii) a claim about whether any possible analysis of it that (1) took it as meaningful and (2) failed to include any mistakes unrelated to the meaningfulness question would therefore (3) diagnose (17) as false. There's a lot to untangle here, but suffice to say that if it is meaningless, then the true 'first conjunct' asserting as much doesn't make the whole thing meaningful, for reasons examined when we looked at (2), above, and if it's not meaningless, the falsity of the first conjunct guarantees the falsity of the whole thing without fear of contradiction. Really, though, I think the most natural reading is (c), and that's where the real problem seems to be.
If (c) is, then, the case, as should be clear by now, (17) really amounts to a disguised disjunction between the claim that (i*) reasoning about (17) and failing to come to the conclusion that it's false would be a *factual* mistake, and (ii*) that 'reasoning' about (17) leads us to the apparent conclusion that it is false, but only because we're indulging a nonsensical category mistake. In other words, given (c), what we end up with is a disguised version of sentence (2), above:
(2) The sentence marked (2) is either false or meaningless.
...which we already dealt with in Part I. Since I enjoy the circularity of ending by directing back to the first post in the series, I think I'll just leave off there and throw open the floor to questions, comments and devastating objections.