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If you had been my child, I would have demanded that such an irresponsible teacher be fired.

It is the subjunctive mood, I suppose. Why every verb in the sentence in the past, but be is in the present?

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    This is a "mandative subjunctive", discussed here: "It was recommended that" + verb in present tense. – StoneyB on hiatus Jun 28 '13 at 2:24
  • I'm closevoting as a Dup of that earlier question, but would just point out that be here isn't really "present tense" at all - it's just a "bare infinitive". In this exact context, a "tensed" version of the verb can't be used, but in similar constructions it's (informally, at least) possible. "If he hits my child, I will demand that such an aggressive teacher is fired". – FumbleFingers Reinstate Monica Jun 28 '13 at 16:00
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Subjunctive tenses are a little more abstract with respect to time than indicative ones are. This is a good overview of usage: http://en.wikipedia.org/wiki/English_subjunctive

With the verb to be, past tense implies a condition known not to be the case, whereas present tense doesn't imply that knowledge. For example:

If Abraham Lincoln were alive today, he probably wouldn't be able to run for president.

Be he alive or be he dead, I'll grind his bones to make my bread.

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