One of the most interesting papers in the Alter and Walter collection is Martine Nida-Rümelin's "Grasping Phenomenal Properties", which gives a new argument against the materialist thesis that phenomenal properties are physical properties. Nida-Rümelin's argument uses the two-dimensional apparatus at various points in an auxiliary role, but she argues that her argument requires weaker and less controversial assumptions than my two-dimensional argument. Here I'll look into this a bit. (It might be worth looking at these two papers first, if you're not familiar with the issues.)
Nida-Rümelin's argument runs roughly as follows.
(1) A person who grasps a property via two distinct concepts is in a position to rationally judge that those concepts are necessarily coextensive.
(2) Phenomenal properties are grasped via phenomenal concepts.
(3) Any physical property can be grasped via a physical concept, by someone with relevant physical background knowledge.
(4) No amount of physical background knowledge puts one in a position to rationally judge that a phenomenal concept and a physical concept are necessarily coextensive.
(5) No phenomenal property is a physical property.
Here grasping a property requires having more than having a concept of the property: it requires being in a position to know its modal profile across possible worlds. For example, if one has the concept of water, one may or may not grasp the concept of water, depending on whether one knows that water is essentially H2O. Nida-Rümelin explicates the concept in two-dimensional terms: roughly, one grasps a concept when, in all scenarios compatible with one's background knowledge, the concept has the same secondary intension.
This argument bears a structural relationship to my two-dimensional argument, though it differs in important respects. Premise (2) plays roughly the role of the thesis that phenomenal concepts are semantically neutral (when a concept is semantically neutral, it has the same secondary intension relative to every scenario, so anyone who possesses it grasps it). Premise (3) serves to rule out hidden-physical-essence view such as Russellian monism. Premise (4) expresses a relevant epistemic gap between physical and phenomenal properties. The conclusion is weaker than the denial of materialism, but it is still a strong conclusion.
The key thesis of the argument is (1), which Nida-Rümelin calls the principle of cognitive transparency (CT). This plays roughly the role that the 2D conceivability-possibility thesis (CP) plays in my argument:
(CP) If S is ideally conceivable (i.e. ~S is not knowable a priori), then S is 1-possible (i.e. the primary intension of S is true at some centered metaphysically possible world).
Nida-Rümelin argues that (CT) is significantly weaker than (CP). This may be right, but there is a close relationship between them. In particular, (CT) is closely related to the (SN), a close cousin of (CP):
(SN) When S is semantically neutral, then if S is ideally conceivable, S is metaphysiclly possible.
(SN) follows immediately from (CP) as a special case. To see this, note that if S is semantically neutral, its primary and secondary intensions in effect coincide, so if S is 1-possible it is 2-possible, or metaphysically possible.
To see the connection between (CT) and (SN), we can apply (CT) to the special case of semantically neutral concepts (which we have seen are grasped by anyone who possesses them). It follows that when someone has two coextensive semantically neutral concepts, they are in a position to rationally judge that the concepts are coextensive. This argument does not require any a posteriori knowledge of the subject in question, so when transposed from concepts to language, it strongly suggests the following thesis:
(CT') When 'p=q' is a true property identity involving semantically neutral expressions, 'p=q' is a priori.
This is equivalent to the thesis that when (semantically neutral) 'p≠q' is ideally conceivable, 'p≠q' is false. Of course this entails (SN) in the special case of negations of property identities. So while (CT) and (SN) are not logically equivalent, it is fair to say that they are theses of similar strength.
In particular, those who antecedently deny (SN) are also likely to antecedently deny (CT). Furthermore, those who deny (CP) are also usually inclined to deny (SN). The most important potential counterexamples to (CP) also yield potential counterexamples to (SN), and are likely to yield potential counterexamples to (CT). This applies in particular to materialists who deny (CP). For example, Loar (1990/1997) appears to hold that there can be semantically neutral phenomenal and physical concepts that pick out the same property. If so, then he will deny (CT') and probably (CT).
Nida-Rümelin suggests at one point (section 13, or p. 327 in the book) that anyone who accepts the notion of grasping should accept (CT). This seems a bit too quick. Someone such as Loar could accept the notion of grasping while rejecting (CT), holding that the same property can be grasped under two different concepts. Under each concept, one will be in a position to identify the secondary intension of the concept, but this secondary intension will be described differently in each case -- physically described in one case, phenomenally described in another. Nida-Rümelin suggests that this sort of opacity cannot arise, as grasping a property involves "fully understanding what the property consists in". But I think Loar et al will deny that for every semantically neutral concept, one fully understands what its referent consists in. For example, if phenomenal properties are physical properties, as on their view, one doesn't fully understand these properties just by possessing the concept. Relatedly, they may hold that there are two notions in the vicinity of "grasping", on the weaker of which there can be violations of cognitive transparency, and on the stronger of which one doesn't grasp phenomenal properties with phenomenal concepts. Of course I think these claims are incorrect, and I agree with Nida-Rümelin that this sort of opacity cannot arise with semantically neutral concepts, but I think that this view is roughly equivalent in strength to (CP) and (SN).
Overall, I'd say that Nida-Rümelin's argument does involve slightly weaker premises than (CP), corresponding to the fact that (SN) is slightly weaker than (CP) (it makes no commitments about non-neutral expressions, and involves somewhat less apparatus). The premises are quite closely related, though. There's also a close relationship to George Bealer's "semantically stable" conceivability argument, which invokes a principle akin to (SN) but with Bealer's notion of semantic stabilility in place of semantic neutrality (elsewhere I've argued that semantic neutrality is really what's needed here). Both of these arguments start from the somewhat weaker premise than (CP), and as a trade-off, yield a somewhat weaker conclusion (the denial of phenomenal-physical property identity, rather than the denial of supervenience). So (CT) is closely bound up with these other principles. The antecedent plausibility of (CT) may provide further reason to accept these principles and to reject alternative views, though. In any case, Nida-Rümelin's rich paper will repay careful attention.