Should old tweets be fair game for firing?

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Last week, Amy Powell was fired as the president of Paramount TV. She is alleged to have said some comments in work meeting that included racially charged language, and then lied about making those statements when later confronted by her boss. Powell has denied making any such statements. Then, in a completely separate incident, 'Guardians of the Galaxy' director James Gunn was fired by Disney after a right-wing blogger resurfaced some decades-old tweets. Many of Gunn's friends and cast members have come to his defense. The whole incident raises a couple of questions--how did Disney not know about these tweets to begin with, and why did they hire Roseanne, when the company clearly knew about many of her past problematic tweets? Now, some people, including Dan Harmon are deleting their Twitter accounts altogether.