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In this example:

Agamemnon comes home to be killed by his wife and her lover.

We are quite sure that he didn't plan on dying, so comes home to be killed is not the “infinitive of purpose”. What is it then? Is there a grammatical term for that?

Edited, another example:

She came home to pick her car.

Here, to pick is the “infinitive of purpose”. In contrast:

She came home to find the house has burned to the ground.

Here, though, to find is not the “infinitive of purpose”, so what kind of infinitive is it?

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He didn't plan on it, but someone might have done. e.g. Fate, or the Author. Just throwing it out there... – lessthanideal May 6 '13 at 0:26

I have seen this called an infinitive of result. That strikes me as a happy term: you could then think of the other uses as subsets:

  • The infinitive of purpose is an infinitive of sought (or desired) result.
  • Your house-burned-to-the-ground example is an infinitive of unexpected result.
  • Your Agamemnon example is an infinitive of unsought ( or ironic) result.
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I see a passive infinitive there, to be killed. As the action is not performed by Agamemnon himself, it doesn't necessarily have to be his choice.

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It could be improved with a comma:

Agamemnon comes home [after surviving the battle], [only] to be killed by his wife and her lover.

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