Sadly, Bill finally had enough of my comments on intentionality. I'm not sure exactly what underlies it, but I suspect it is my persistence on the 'Brentano equivalence'. There is more here and very much more here. Briefly, a Brentano equivalence holds when we can convert existential sentences of the form 'an A-B exists' and categorical sentences of the form 'Some A is B'. There is much evidence that such an equivalence holds in ordinary subject-predicate sentences. Thus 'some buttercups are yellow' is convertible with 'yellow buttercups exist' and 'no buttercups are blue' with 'blue buttercups do not exist'. And as I commented here, there seems little difference between 'Tom's worship has an object' and 'there is an object of Tom's worship' or 'there exists an object of Tom's worship'.
Hence my objection to Bill's claim that 'We sometimes think about the nonexistent'. 'The nonexistent' is an abstract noun phrases built out of an adjective, like 'the unemployed', 'the French', 'the damned', which appear to refer to a whole class of things (unemployed people, French people, damned souls). If the same is true of 'the nonexistent', then it refers to a whole class of things - things which do not exist - and we can divide all things in reality into those which do exist (mountains, horses, yellow buttercups) and those which do not (golden mountains, unicorns, blue buttercups, four-leaf clovers and so on). You can see the idea in the picture on the left (from the Logic Now and Then symposium in 2008 - I don't remember who the speaker was).
Clearly, no one who divides reality up in this way can consistently hold the Brentano equivalence, according to which any thing is also an existing thing. If 'the nonexistent' refers at all, it refers to nonexistent things. But a nonexistent thing is a thing which does not exist, and so is a thing such that there is no such thing. But there is no such thing. So 'the nonexistent' does not refer. Accordingly, we cannot think of the nonexistent, if the Brentano equivalence is valid.
Clearly there is a problem, as true sentences like 'the Greeks worshipped Zeus' or 'Tom is thinking of Pegasus' suggest. But I will talk about this in the next post.