Responses to the Liar Paradox usually fall into one of three categories: those resolutions that allow (under limited circumstances) for propositions to be true and false at once; those resolutions that say it is meaningless for some reason; and those resolutions that say it is false. There's no question that my sympathies lie with the last of these solutions; I think the paradox only arises when you make questionable assumptions about what can be inferred if the Liar is false. But there is a very interesting approach of the second kind that is worth considering, as well; I think it deserves more consideration than it usually seems to get, so here's a summary of it.
Consider the difference between the two following sentences:
There is a ghost in the parlor.
Everybody thinks there is a ghost in the parlor.
The first of these sentences expresses a proposition. The second, however, not only expresses a proposition, it has a reference to another proposition. Englebretsen, following Sommers, proposes that we mark this difference by what he calls 'propositional depth'. The first sentence doesn't express a comment on any propositions. It has a propositional depth of 0. The second, however, comments on the proposition of "There is a ghost in the parlor," in order to say that everyone thinks it. It has a propositional depth of 1.
It is possible to have a propositional depth as great as you please. For instance,
Everyone thinks that Tom thinks that Mary knows that everyone thinks the world will end
has a propositional depth of 4. However, the important question is this: Can a sentence express a proposition of indeterminate propositional depth. Englebretsen says not, and formulates the propositional depth requirement:
Every meaningful statement must be assumed to have a determinate propositional depth.
The reasoning is that if you take any sentence you please, of propositional depth n, any comment on that must have a propositional depth of n+1; any comment on that comment must have a propositional depth of n+2; and so forth.
Now, take a sentence like this:
L: [L] is false.
L doesn't have 0 depth, because it comments on itself. That would naturally lead one to think it has a depth of 1, but it doesn't have that, either, because by its self-reflexive nature for any depth n you assume, it comments on it. So it has no determinate propositional depth; and from this Sommers and Englebretsen condlue that it is expressively vacuous, makes no statement, expresses no proposition, has no truth value.
But wouldn't this eliminate all self-reflexivity? After all, there are fairly straightforward self-reflexive sentences which clearly have truth value and meaning:
This sentence is in French.
This sentence has five words.
Englebretsen's response to this is to deny that they embed propositions; sentences express propositions, and these are about sentences rather than the propositions they express. So they have a propositional depth of 0. There is no problem with self-referential sentences, only with self-referential (or mutually referential) propositions. Such propositions have no determinate propositional depth.
In any case, it's clear how one would treat the Liar sentence in this account.
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.
Englebretsen treats this as universal -- all self-referential propositions are vacuous. It isn't clear, though, that this is so. Consider this example:
D: This sentence expresses [D] in French.
This is clearly as false as the other. But it appears to have a propositional depth of 1. (It's equivalent to "The proposition of this sentence is expressed in French.") And even if it had not determinate propositional depth, it looks like it still must have a proposition, because if the sentence had no proposition, it would seem to be false, because it wouldn't be expressing its proposition in French; which means it expresses a proposition. So it seems that either we must admit that some self-referential propositions have a determinate propositional depth, or we must deny that lack of determinate propositional depth is sufficient for meaninglessness. I suspect Englebretsen would stick by his guns, and claim that D has no proposition and is meaningless, so it cannot be false and cannot express a proposition. But it's not very plausible for cases like this, unless we have a solid independent reason to accept the propositional depth requirement, which I don't think we do (although it is very plausible).
Another case that seems to be a problem for this view is the Cretan Liar. Suppose that the following is said by a Cretan, and that the Cretan is the only Cretan, and that it is the only thing ever said by a Cretan:
C: Everything a Cretan says is false.
C has no determinate propositional depth in these circumstances, because its proposition embeds itself. But suppose that a Cretan were to say two things, one of which is true, and one of which is verbally the same as C (we'll call it C2). C2 looks like it would be false (not everything a Cretan says is false), but it embeds itself, so it would seem that it has no determinate propositional depth. Now, suppose a non-Cretan were to say of this one Cretan (if he didn't say C):
X: Everything a Cretan says is false.
C and X look the same; but Englebretsen is committed to saying that they are not: C has no determinate propositional depth, while X has a propositional depth of 1; C expesses no proposition and is meaningless, while X expresses a proposition and is meaningful. However, suppose that the Cretan does say C and only C, and the non-Cretan says:
Y: Everything a Cretan says is false.
Now this proposition embeds a proposition [C] that has no determinate propositional depth, so it has no propositional depth, expresses no proposition, and is meaningless. But if C has no truth value, Y should be false. Further, we are in the awkward position of saying that whether a sentence makes any sense at all depends not only on the terms and syntax of the sentence but also on whether the sentence is said by a Cretan or a non-Cretan, and whether a Cretan says anything not false. In other words, 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. This is an awkward result, but one we seem committed to on this approach.
Nonetheless, as Englebretsen notes, the Propositional Depth response has a lot going for it as well. It can winnow out all the bad comments you please; it is not ad hoc, since it is part of a larger theory of truth and falsity; and it is clearly better than a great many of its competitors (e.g., unlike standared meta-langauge responses, it is not about sentences but about the propositions they express, and unlike meta-language responses it can apply directly and without adjustment to natural language).
