Every meaningful statement must be assumed to have a determinate propositional depth...
L: [L] is false.
This has no determinate propositional depth. If we assume that L has a propositional depth of n, we find that, since L embeds itself, it must have a propositional depth of n+1.
I like this sort of account. (Alex recently suggested that one might arbitrarily choose whether or not to believe the proposition P: I believe [P]. My immediate response was to doubt that there really is any proposition here. The 'depth' account can explain why.)
A standard objection to this sort of view is that there would seem to be some true self-referential propositions. In an old guest post, Rad Geek suggested:
EM: [EM] is true or [EM] is false.
Now, it's not entirely clear to me that the above constitutes a wholly meaningful claim. (What is this '[EM]' it speaks of? I get stuck in an infinite loop if I try to fill it out.) But perhaps we can apply a lesson from Yablo and say that it is partly true. (I doubt his truthmaker account can actually accommodate this, but never mind that for now.)
Subtract out the self-reference, and what remains ("__ is true or __ is false") is true about logical forms, i.e. insofar as it claims that the law of excluded middle holds. The particular application is meaningless, but we can abstract away from that part of what's said.
Another puzzle case:
M: [M] is true and grass is green.
We certainly don't want to say that this is wholly meaningless. It's partly true: grass is green. Again, it seems that the thing to do is simply to subtract away the meaningless self-referential component.
P.S. Towards the end of his post, Brandon worries that the propositional depth solution commits us to the view that "whether the sentence has the same meaning, or any meaning at all, depends on purely contingent facts about the world that we may not be aware of." We should embrace this result, though, as Michael Sprague once pointed out to me:
It's worth noting that all liar sentences are dependent on context. For example, an instance of the liar sentence next to an arrow pointing to another sentence (like, say, "All ravens are orange") may be true. Context determines the sentence to which "this sentence" refers; the truth or falsity of that sentence then determines the truth of the liar.