How does studying logic improve our reasoning ability?

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The Greek Philosophers did a good job of classifying and clarifying the way we use our brains. Logic is, etymologically, connected with how we use words (logos), and fallacies (falsehoods) can be uttered by language; rhetoric deals with the connotations of words and utterances; ethos deals with "right and wrong" behavior (in the public arena). Studying, really studying, these elements is invaluable, perhaps even essential, to our ability to "reason."

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