Even if they are (which is apparently disputed), pointing out that someone is using a fallacy does not make them automatically wrong. Next time, you should try saying something more substantive than "Fallacy!"
Correctly identifying fallacies is a useful skill. Doing it incorrectly and thinking you are correct is a detriment to yourself and conversation. (Kinda like how certain people say "fake news" to refer to things they don't agree with, even if it is factual).
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.
>just because an argument's a logical fallacy, that doesn't make it incorrect. //
An argument that uses fallacious reasoning is an incorrect argument, but pointing out a fallacy doesn't negate a conclusion (that would be the fallacy fallacy).
So, we can't tell if the conclusion is wrong when someone uses a fallacy.
Pointing out fallacies doesn't work because doing so itself uses the fallacy of "I have proven you wrong, therefore I am right" when the most likely truth is both are horrendously off-base.
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