You are committing a fallacy fallacy fallacy. If an argument is fallacious, it definitely is wrong. The proposition supported by that argument might still be right however.
Classic fallacy fallacy. The fallacy fallacy is a fallacy that asserts that because an argument is fallacious, the conclusion of the argument is false.
As good as it is to identify these, it's worth pointing out that another fallacy is the assumption that the point a fallacious argument is trying to prove is wrong because it has a fallacy. It's possible to be right but still commit a fallacy in your argument.
If an argument is fallacious, it definitely is wrong.
F^4: Not necessarily. For example, some would argue that an argument is not necessarily "wrong" if it is logically sound and produces a correct conclusion, even though it is fallacious because unbeknownst to the participants there is both a false premise and countervailing unknown factor. See Gettier Problems: http://www.iep.utm.edu/gettier/.
Identifying errors in reasoning isn't the same as explaining why someone is wrong, though. Given the basic definition of an argument as a structure linking premises to a conclusion, a fallacy is simply a flaw in the structure. It doesn't invalidate the premises or the conclusion, only how they relate to one another. So while you're certainly free to attack the structure behind a conclusion, really you're better off attacking the conclusion itself with an argument of your own.
Not quite right: just because an argument is fallacious doesn't mean that the propositions on which it is based are false. Not a single proposition need be false, only the logic.
It would be the fallacy fallacy if I thought they were wrong, but I don't think they are. I think their reasoning behind their conclusion isn't completely sound, though.
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