Delusions are a symptom of psychiatric disorders such as dementia and schizophrenia, and they also characterize delusional disorders.
As in some cases of erotomania, the content of a delusion can be mundane, and does not even need to be false: one can have the delusion that one's spouse is unfaithful or that one's neighbor is a terrorist, and these may turn out to be true beliefs. As in the Cotard delusion, the content of a delusion can be bizarre.
Here are some other examples: mirrored self misidentification is the delusion that the person in the mirror is not one's reflection but a stranger, and the Capgras delusion is the delusion that the spouse or a relative has been replaced by an impostor.
All types of delusions are rigid to some extent, that is, they are not easily given up because they tend to resist counter evidence.
Delusions may be inconsistent with a person's beliefs and behavior, are typically unresponsive to both counter evidence and counterargument, and are often defended by weak evidence or argument. The empirical literature suggests that the reasoning performance of people with delusions reflects data-gathering and attribution biases.
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The most prominent delusion I can think of right now is the "birther" movement, followed closely by the "death panelists". Can you think of other examples?