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I should clarify. I'm not exactly worried that people will mix up the colloquial meaning of logic with the mathematical meaning of mathematical logic. I just want people to taboo "logic" because I think it is frequently used to label a particular style of bad argument in order to mask certain kinds of weaknesses that such arguments have. Studying mathematical logic is one way to recognize that there's something off about how people colloquially use the word "logic," but I suppose it's not the only way.
Would the quote sound as bad to you if “logic” was replaced with “reasoning”?
As per Postel's law, if a word has both a colloquial meaning and a technical meaning, the latter is not what I want, and there's a decent synonym for the former, I personally use the synonym instead (e.g. “usefulness” instead of “utility”, “substantial” or “sizeable” instead of “significant”, etc.), but as per Postel's law I don't demand that other people do the same, especially if the colloquial meaning is way more widespread overall.