Some authors use those words interchangeably and write, for example, that “The debater persuaded listeners that the program was flawed.” But there’s a distinction: Persuade should be used when action is involved, and convince should be used when belief is involved. In our example, the audience came to believe that the program was flawed, thanks to the debater, so the sentence should be recast to “The debater convinced listeners that the program was flawed.” Persuade would be correct in a sentence such as “The debater persuaded listeners to vote to end the program,” which describes how the audience acted.
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