Fallacy fallacy

Argument from fallacy is the formal fallacy of analyzing an argument and inferring that, since it contains a fallacy, its conclusion must be false. It is also called argument to logic (argumentum ad logicam), the fallacy fallacy, the fallacist's fallacy, and the bad reasons fallacy.

Fallacious arguments can arrive at true conclusions, so this is an informal fallacy of relevance.

It has the general argument form:

Thus, it is a special case of denying the antecedent where the antecedent, rather than being a proposition that is false, is an entire argument that is fallacious. A fallacious argument, just as with a false antecedent, can still have a consequent that happens to be true. The fallacy is in concluding the consequent of a fallacious argument has to be false.

That the argument is fallacious only means that the argument cannot succeed in proving its consequent. But showing how one argument in a complex thesis is fallaciously reasoned does not necessarily invalidate the proof; the complete proof could still logically imply its conclusion if that conclusion is not dependent on the fallacy:

All great historical and philosophical arguments have probably been fallacious in some respect... If the argument is a single chain, and one link fails, the chain itself fails with it. But most historians' arguments are not single chains. They are rather like a kind of chain mail which can fail in some part and still retain its shape and function.

Both of Bill's rebuttals are arguments from fallacy. Ginger may or may not be a cat, and Tom may or may not be English. The fact that Tom's argument was fallacious is not, in itself, sufficient proof that his conclusion is false.

That one can invoke the argument from fallacy against a position does not prove one's own position either, as this would be an argument from fallacy itself, as is the case in Joe's argument.

Argumentum ad logicam can be used as an ad hominem appeal: by impugning the opponent's credibility or good faith, it can be used to sway the audience by undermining the speaker rather than by addressing the speaker's argument.

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Wikipedia