Let me attack yesterday's puzzle from a different angle. The puzzle in one sentence: we think about things that do not exist; but how is this possible given that they do not exist?
Here is the problem set forth as an aporetic hexad:
1. When I think about Frodo, as I am doing right now, I am thinking about, precisely, Frodo: not about some semantic or epistemic intermediary or surrogate or representative. I am thinking about a concrete, albeit nonexistent, item. I am not thinking about an idea in my mind, or a mental image, or any mental content; nor am I thinking about an abstract entity of any kind such as a property; nor am I thinking of a word or a phrase or anything linguistic.
2. Thinking about (thinking of) is a relation the relata of which are a subject who thinks and an object thought of. Thinking is triadic: ego-cogito-cogitatum.
3. Every relation is such that if it obtains, then all its relata exist/are.
4. There are no different modes of existence/being. This is the ontological counterpart of the semantic thesis of the univocity of 'exists' and 'is' and cognates.
5. To exist is to exist extramentally and extralinguistically, where the minds in question are finite.
6. Frodo, a purely fictional item, does not exist.
The limbs of the hexad are individually plausible but jointly inconsistent. To solve the problem we must reject one of the limbs. But which one? (6) is a datum, and (5) is an unproblematic definition. So the the candidates for rejection are (1)-(4). I'll take these in reverse order.
Deny (4): There are two modes of being, esse reale and esse intentionale. When we say, with truth, that Frodo does not exist, we mean that he lacks esse reale. But we can still think about him in a manner to satisfy (1)-(3) since he has merely intentional being.
Deny (3): Twardowski-Meinong-Grossmann Solution. There are items that have no being at all, and there are genuine relations that connect existents such as minds to beingless items in the realm of Aussersein.
Deny (2): Thinking-of is not relational, whether or not the obtaining of a relation requires that all its relata exist. This can be developed in different ways. Adverbial theories, Brentano's theory, Butchvarov's theory.
Deny (1): One way to deny (1) is via abstract artifactualism. A number of philosophers, including van Inwagen, have been putting forth some version of this view. The idea is that purely fictional items such as Frodo are created by the authors of works of fiction in which they figure. They are a peculiar species of abstract object since they come into being, unlike 'standard' abstract objects. They exist, but they are abstract. Meinong, by contrast, held that they are concrete but do not exist or have any being at all. Here is a paper that defends artifactualism against some objections by Sainsbury.
Now, gentlemen, pick your poison! Which limb will you deny? I claim, though this is but a promissory note, that no theory works and that the problem, though genuine, is insoluble.