If a philosophical framework causes you to accept a basilisk, I view that as grounds for rejecting the framework, not for accepting the basilisk.
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To state it yet another way: to me, the basilisk has the same status as an ontological argument for God. Even if I can’t find the flaw in the argument, I’m confident in rejecting it anyway.
Despite the other things I’ve said here, that is my attitude as well. But I recognise that when I take that attitude, I am not solving the problem, only ignoring it. It may be perfectly sensible to ignore a problem, even a serious one (comparative advantage etc.). But dissolving a paradox is not achieved merely by clinging to one of the conflicting thoughts and ignoring the others. (Bullet-swallowing seems to consist of seizing onto the most implausible one.) Eliminating the paradox requires showing where and how the thoughts went wrong.
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Despite the other things I’ve said here, that is my attitude as well. But I recognise that when I take that attitude, I am not solving the problem, only ignoring it. It may be perfectly sensible to ignore a problem, even a serious one (comparative advantage etc.). But dissolving a paradox is not achieved merely by clinging to one of the conflicting thoughts and ignoring the others. (Bullet-swallowing seems to consist of seizing onto the most implausible one.) Eliminating the paradox requires showing where and how the thoughts went wrong.