In practice, to say that a belief refutes its own justification is usually not to say it's inconsistent, but rather, the alternative was.
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For example, pyrrhonian skepticism: if you buy an argument for it, then you don't buy any arguments for anything. It's a perfectly consistent position once you're there. The position you left, by buying the argument, must have been inconsistent tho.
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So "x refutes its own justification" doesn't seem like an argument against x, rather, an argument against the position which the justification assumed - which will usually be an anti-x position, if the pyrrhonian example is typical. So, favourable sign for x.
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