The psychiatrist and philosopher Bill Fulford describes a patient who was the living embodiment of the logical paradox “this statement is false” during a discussion on the difficulties in assuming delusions are false beliefs, as described in the standard definition.
[There is an] even more fundamental sense in which delusions may not be false beliefs, namely that for some patients this would present us with a paradox.
I have reported one such case that occurred in Oxford… The patient, a 43-year-old man, was brought into the Accident and Emergency Department following an overdose. He had tried to kill himself because he was afraid he was going to be “locked up”. However, this fear was secondary to a paranoid system at the heart of which was the hypochondriacal delusion that he was “mentally ill”.
He was seen by the duty psychiatrist and by the consultant psychiatrist on call, neither of whom were in any doubt that he was deluded. Indeed, both were ready on the strength of their diagnosis to admit him as an involuntary patient.
Yet had their diagnosis depended on the falsity of the patient’s belief, as in the standard definition, they would have been presented with a paradox: if the patient’s belief that he was mentally ill was false, then (by the standard definition) he could have been deluded, but this would have made his belief true after all.
Equally, if his belief was true, then he was not deluded (by the standard definition), but this would have made his belief false after all. By the standard definition of delusion, then, his belief, is false, was true and, if true, was false.
From p211 of the book Philosophical Psychopathology (ISBN 9780262071598).
Link to Wikipedia article on the vagaries of delusion.