Perhaps the central problem to which the phenomenon of intentionality gives rise can be set forth in terms of an aporetic triad:
1. We sometimes think about the nonexistent.
2. Intentionality is a relation between thinker and object of thought.
3. Every relation R is such that, if R obtains,then all its relata exist.
The datanic first limb is nonnegotiable, a 'Moorean fact.' The other two limbs, being more theoretical, can be denied if one is willing to pay the price. But something has to give since they cannot all be true.
Brentano denied (2) with unpalatable consequences to be explored in a separate post. Why not accept (2), deny (3) and admit that there are abnormal relations, relations that connect existents with nonexistents?
Consider the round square, that well-worn example that goes back at least to Bernard Bolzano. Since there is no such thing, and cannot be, one will be tempted to say that the round square is an idea (presentation, Vorstellung) without an object. That is what Bolzano maintained using that very example of rundes Viereck. (Theory of Science, pp. 88-89) In section 5 of Zur Lehre vom Inhalt und Gegenstand der Vorstellungen (1894), Kasimir Twardowski criticizes Bolzano's position.
Twardowski distinguishes among the following: there is the expression 'the round square.' Then there is the mental act, the act of presentation (Vorstellungsact) that transpires in someone who uses the expression with understanding. Corresponding to the act is a content (Inhalt) which constitutes the meaning of the expression. But there is also a fourth item, that to which the expression refers, the round square itself, that which combines logically incompatible properties and whose existence one denies as soon as one advances from the presentation round square to a judgment about it. (Cf. the Brentanian theses that judgments are founded upon presentations, and that every judgment is existential, involving the acceptance or rejection of a presentation.)
This of course sticks in the craw. One hesitates to admit that there is something outside the mind to which 'round square' refers, something that has the property of nonexistence. It smacks of a contradiction. Clearly, 'There exists an x such that x does not exist' IS a contradiction, but this is not what a Meinongian will say.
Note that Twardowski has a couple of powerful reasons for not identifying the round square and its colleagues with mental contents. The first is that contents exist while nonexistent objects don't. So the round square cannot be identified with the content expressed by 'the round square.' The second reason is that we ascribe to the round square attributes that not only cannot be ascribed to the corresponding content, but are logically incompatible to boot. Thus no content is round and no content is square and of course no content is both round and square. Since contents exist, they cannot have contradictory properties.
These arguments, spelled out a bit perhaps, show that mental contents cannot go proxy for nonexistent items, whether merely possible like the celebrated golden mountain or impossible like the round square. One could extend the argument to cover abstract objects which are not mental contents or in any way mind-dependent. They too are unsuited to go proxy for nonexistents. For (1) abstracta exist while nonexistents do not, and (2) the properties of nonexistent concreta cannot be attributed to abstracta. Thus a flying horse is an animal, a golden mountain is a mountain, and a round square is round. But no abstract object is an animal or a mountain or round.
When I think about the round square or the golden mountain (in whatever psychological mode) the object of my thought is neither a mental content nor an abstract object. What is it then? Why, it is the round square or the golden mountain! As bizarre as this sounds, it makes a certain amount of sense. If I want to climb the golden mountain, I want to climb a physical prominence, not a mental content or an abstractum.
The position under examination, then, is not only that every mental act has a content, but that every mental act has an object as well. But not all of these objects exist. One obvious advantage of this approach is that it allows us to hold onto (2) of our opening triad in full generality: in every case, intentionality relates a thinker through a content to a transcendent object, and not to some surrogate object, either!
Why is this a good thing? Well, if intentionality is relational only in some cases, the veridical cases, then it cannot be essential to mental acts to be of an object: whether or not an act actually has an object will depend on contingent facts in the world beyond the mind. For Brentano, all mental acts are intentional by their very nature as mental. The Twardowski-Meinong approach upholds this.
But the price is very steep: one must accept that there are items that actually instantiate properties (not merely possibly instantiate them), and that these items nevertheless do not exist, or indeed, as on Meinong's actual view, have any mode of being at all. This is his famous doctrine of the Aussersein des reinen Gegenstandes, the 'extrabeing of the pure object.' Thus the golden mountain is actually golden and actually a mountain despite having no being whatsoever. It is a pure Sosein utterly devoid of Sein.
Some, like van Inwagen, think that Meinong's theory of objects is obviously self-contradictory. I don't believe this is right, for reasons detailed here. Even so, I find Meinong's theory incoherent. 'Some items have no being at all' is not a formal contradiction. Still, I cannot get a mental grip on the notion of an item that actually has properties, but is wholly beingless.
In addition, one must accept that there are genuine relations that connect existents to nonexistents.
The price is too steep to pay. The Twardowski-Meinong-Grossmann solution is just as problematic as the original problem.
REFERENCE: Reinhardt Grossmann, The Categorial Structure of the World, Indiana UP, 1983, p. 197 ff.