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PECOS-9 comments on Useful Questions Repository - Less Wrong
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Not-A for publicly declaring that one hates homosexual behavior isn't "publicly declaring that one loves homosexual behavior". It's just "not publicly declaring that one hates homosexual behavior". Your A-or-not-A has to cover all the possibilities, including remaining silently at home, awkwardly evading questions about homosexuality, making positive statements about heterosexuality but none directly about homosexuality, etc.
Just to finish your thought: Because of this, it's possible that "A" and "the opposite of A" actually can both raise your estimate of p(B), even though "A" and "not A" can't, as BlueSun stated.