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Do inconsistent premises in an argument hurt a person's case while they try to defend something they believe in because their statements contradict each other?

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    Yes. If people believe contradictory things at least one of them must be wrong, and that does not bode well for their system of beliefs as a whole. – Conifold Jan 20 at 22:17
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If the premises are supposed to be taken for granted, and they are contradictory, then anything at all can be logically shown to follow from them (https://en.wikipedia.org/wiki/Principle_of_explosion). So this would clearly make the argument suspect.

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