According to the Property Theory of content, the contents of our beliefs, desires, etc. aren't propositions, but properties. The contents of de se, or irreducibly first-personal, beliefs are properties we self-ascribe; if I believe that I myself am the messy shopper, I self-ascribe the property of being the messy shopper. If I merely believe de dicto (or even de re) that Jason is the messy shopper, then what I believe is the property of being such that Jason is the messy shopper. If I'm suffering from selective amnesia and have forgotten who I am, I may believe the latter without believing the former.
I like the property theory. I'd like to believe it. I've even defended it against objections. But I'm worried that it's motivations are going to over-generate. The property theory can be thought of as taking thoughts best expressed with one indexical expression -- "I" -- and inserting a slot into the content of this thought for the bit associated with that expression. (E.g., "I am the messy shopper" expresses the content ____ is the messy shopper.) I'm worried the argument that gets us to add in this slot is going to drive us to add in other slots as well.
Consider what I take to be the strongest case for the property theory: Lewis's ("Attitudes De Dicto and De Se", 1979) case of the two gods. Zeus lives at the top of the mountain; Poseidon lives at the bottom of the deepest ocean. They both know all the true propositions. But neither knows who he is. Zeus knows that Zeus is at the top of the tallest mountain; but he doesn't know that he is at the top of the tallest mountain. Since he knows all the true propositions (Lewis argues), and since if he did know that he was at the top of the mountain he'd have a (new) true belief, whatever content Zeus fails to be belief-related to must not be a proposition. But properties: those could do the job. Zeus could believe all the propositions but not believe the property being on top of the mountain.
Properties aren't the only way to handle Lewis's two-gods case. We could instead have belief as a triadic relation between, roughly, a believer, a proposition believed, and a way of presenting that proposition to oneself. Then Zeus might believe the proposition that Zeus is on top of the mountain under one mode of presentation, but not under another, first-personal, mode. Why prefer the property theory to this one? Neil Feit (Beliefs About the Self, 2008) argues (inter alia) that the property theory is just more streamlined, more elegant, than the triadic theory. We'll come back to this in a mo.
Here's the case that's worrying me. We have one god, who is sitting in front of two ghostly spheres -- call them Bo and Luke. They're intrinsic duplicates and, gosh, wouldn't you know it, they're occupying the exact same region right now. But one of them is going to move here in a minute.
Beings like us will have a hard time ostending one of the co-located spheres. But that's no problem for a god! So this god ostends one of them and says, "I wonder if that one is going to be the one that moves in a minute."
It looks like we can repeat the Lewis-style worries here. Our curious god -- call her Daisy -- might well know that Bo is going to move in a minute, but not know whether she is ostending Bo or Luke. Indeed, it looks like she might know all propositions, but still not know whether that sphere is going to move in a minute. So -- by parity of reasoning -- if Lewis's gods case drives us to add a slot in for irreducible "I"-thoughts, shouldn't the Bo and Luke case drive us to add in a slot for irreducibly demonstrative thoughts? But I take it this would be a disaster (once we see the trick, it's a good bet this will get out of hand pretty soon), so we should resist drawing the property-theory lesson from Lewis's two gods case.
I expect the property theorist to respond: "If we're already property theorists, we can find a property that Daisy doesn't believe: the property of ostending Bo. Once she comes to know that property, since she also knows that Bo will move in a minute, she will be in a cognitive state that she is not in now --- and it's one that can serve the role of 'knowing that that sphere will move in a minute'".
But the simplest version of this won't work. Suppose Daisy wonders, "Will it be that sphere or that one which moves in a minute?", respectively ostending Bo and Luke in the process. Even if she knows that she has ostended Bo during her wondering, this won't improve her cognitive state (because she has also ostended Luke). So the property theorist will have to resort to a more linguistically fine-grained property for Daisy to believe, one along the lines of "the property of having first ostended Bo and then ostended Luke", or something like that.
I don't have any argument the property theorist can't make this move work. I rather suspect he can. What I want to point out now is that the property theorist is now relying on properties that seem to be close to the triadic theorist's modes of presenting a proposition. That is: there will need to be some sort of quasi-syntactical specification of the thought that Daisy is having, so that Daisy can learn how parts of this thought are related to the world (e.g., that this part is related to Bo, and that one is related to Luke). This isn't the same thing as the triadic theorist's view by a long shot; but it makes use of many of the same sorts of resources.
But once we're going down this line as property theorists to deal with Daisy's ignorance, what happens to the objection to the triadic theorist's treatment of Zeus's ignorance? The triadic theorist, in essence, says that Zeus doesn't know his mental tokens of "I" pick out Zeus; the property theorist (on the envisaged response) says that Daisy doesn't know that her (particular) mental tokens of "that" pick out Bo and Luke, respectively. If we're going down this line anyway, why not be triadic theorists from the get-go? Maybe the triadic theory is ugly, but if the property theory has to partake of this same ugliness, then there's no argument from ugliness in favor of properties over modes of presentation. And the property theory in fact looks worse, because the triadic theorist can treat what seem like similar phenomena -- indexical ignorance -- in a similar fashion, whereas the property theorist treats some cases of indexical ignorance very differently than others.