A paraphrase that eliminates intentional objects only succeeds if the agent's mental state already has fully determinate propositional structure, which Chisholm's irreducibility thesis contests.

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Chisholm's irreducibility thesis(as used in philosophy of mind): Roderick Chisholm was a 20th-century philosopher who argued that certain mental facts (like intentionality—the 'about-ness' of thoughts) cannot be reduced to or fully explained by purely physical facts.
Determinate propositional structure(as used in logic and philosophy of language): A clear, fully spelled-out thought or claim that is precise and definite enough to be true or false—not vague or incomplete.
Intentional objects(Earliest Stoic philosophy of mind): Things that serve as the objects of thought and other mental states, which on the earliest Stoic view need not exist or obtain
Irreducibility(as used in metaphysics): The quality of something that cannot be broken down into simpler parts or fully explained by reducing it to something else.

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