Bill Vallicella comments: "One cannot eat without eating something, and indeed something that exists. And one cannot desire without desiring something -- but in this case the thing desired needn't exist." But then we have the problem of 'something desired which doesn't exist', which seems contradictory. Is the problem about ontology at all?
(1) Jake is searching for a gold mine near Cripple Crow Creek.
(2) There are no gold mines near Cripple Crow Creek.
There is not even even a whiff of contradiction or paradox here, and we are not tempted to posit 'non-existent objects' or suchlike. Of course, if we try to analyse them in terms of predicate calculus, we do get a contradiction:
(1a) For some x, [x is a gold mine near Cripple Crow Creek, and Jake is searching for x].
(2a) Not for some x, [x is a gold mine near Cripple Crow Creek].
The first sentence logically implies 'for some x, x is a gold mine near Cripple Crow Creek', which directly contradicts the second. But that suggests a problem with the analysis of (1) into an inappropriate formalisation such as (1a), rather than any question of 'ontology'. As soon as we even ask the question about the 'ontological status' of the sought-for gold mine, we are already on the metaphysical sandbank. E.g. we might try resolve the contradiction between (1a) and (2a) by replacing (2a) with
(2b) Not for some x, [x is an existent gold mine near Cripple Crow Creek].
This resolves the contradiction. But by now we are well into the Meinongian jungle. This is entirely a problem of language and logic, as I see it. What is the deep structure or logical analysis of sentence (1) which makes it transparently clear that it is not inconsistent with (2)? Note that we do get inconsistency if we turn the grammatically active sentence (1) into a passive without getting 'existential implication'.
(1b) Some gold mine near Cripple Crow Creek is sought for by Jake.
In its most natural reading, this contradicts (2). So, what is the true semantic structure of sentence (1)? That is the real question, and it has nothing to do with metaphysics. As a suggestion, consider
(3) Jake says that there is a gold mine near Cripple Crow Creek.
which we can analyse into
(3a) Jake says that for some x, x is a gold mine and x is near Cripple Crow Creek.
This, unlike (1a) above, is not inconsistent with (2a) above, since it does not imply that for some x, etc. Could there not be some analogous analysis of (1) into
(1c) Jake is searching-that-there-is a gold mine near Cripple Crow Creek ?
Could it be that 'searching for' has an embedded that-clause which invalidates the inference to 'for some x there is ...', but which is not visible at the surface level of the sentence? That seems a much cleaner way to resolving the difficulty than all this intentional objects nonsense. For nonsense it is.