Look at these sentences:

1.  The best outcome would be for the tyrant to die.
2.  The best outcome for the tyrant would be to die.

These sentences have not only a different meaning, but also different grammar.  In the second, ‘for’ is used as an ordinary preposition attached to ‘the tyrant’.  In the first, however, it is used (as a preposition? a conjunction?) to introduce a subject-infinitive construction, equivalent to ‘...that the tyrant dies’.  Such usage is common, but is it correct?  If so, how should it be analysed in grammatical terms?

1.  The best outcome would be for the tyrant to die.
2.  The best outcome for the tyrant would be to die.

The second sentence sentence is as you say: prepositional phrase “for the tyrant ” modifying “outcome.”
The first sentence, I believe, contains a prepositional phrase (with infinitive “to die” as a part of it) used as a nominal and subject complement of “outcome.”

For more information on prepositional phrases used as nominals, see http://www.yourdictionary.com/grammar-rules/Prepositional-Phrases.html

Thanks.  I have looked at the article you suggested. I query, however, whether ‘for the tyrant to die’ is really a prepositional phrase. Is ‘the tyrant to die’ the object of ‘for’? I have my doubts. Consider these sentences:

1. The most urgent task is for me.
2. The most urgent task is for the gas leak to be fixed.

Do ‘me’ and ‘the gas leak to be fixed’ perform the same function in the sentence? I don’t think so. ‘For’ means ‘down to’ or ‘applicable to’ in (1), but not in (2). It doesn’t seem like a preposition at all in (2) - more a kind of linking word attaching the subject and verb (‘the most urgent task is’) to the complement (‘the gas leak to be fixed’, in the sense of ‘the fixing of the gas leak’). Very puzzling!

Very puzzling indeed. Nonetheless, I’m content with the prepositional-phrase-as-a-nominal idea, both in the first set of sentences and in the second. The prepositional phrase in called a

nominal,

not a noun, because it functions in a noun-like way.

The most urgent task [subject] is for the gas leak to be fixed [subject complement].

I agree that ‘for the gas leak to be fixed’ is a nominal, but I question whether the ‘for’ is a true preposition. It seems more like a ‘dummy’ word, a mere place-holder. ‘A is for B’ doesn’t usually mean ‘A is B’, but it does here.

I agree with you on the odd use of “for,” but the task is “for the gas leak to be fixed” unless the sentence is rewritten (e.g., “that the gas leak be fixed” or something similar that stillexpress the thought).