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I'm looking for a word to describe someone that is easily persuaded to believe things that they are aware is likely to be too good to be true. Something along the lines of an optimistic cynic, perhaps? More specifically, someone who expects the worse but wants to believe in things so badly that they allow their gullible side to override their common sense.
EDIT: To clarify, this differs from A word for a worldly wise person who pretends to be naïve? in that this person isn't necessarily worldly wise and is not pretending to be naive; their gullibility arises more from desperately hoping their belief will be justified. There is no intention to deceive others into thinking that they are more gullible than they actually are.
EDIT 2: Having looked at various answers linked in the comments, further clarification is needed; the person in question is not Pollyannaish in that they are aware that what they are believing in might not be logical, but because they are hoping against hope that this is true, the end result is a sense of gullibility.