Are these PPs or non-finite clauses – or something else entirely?

I'm wondering about the construction for [NP] to [VP], as illustrated in the following examples:

(1) I waited for you to come here

(2) He arranged for me to go there

(3) For him to do that took courage

(4) For you to apologise is not enough

My questions are:

• what is this construction from a formal point of view? I can't see how for can be anything other than a preposition, which would suggest that it is a PP – but prepositions can't take infinitive clauses as complements, can they?
• if it's not a PP, but rather a non-finite clause, then what function does for have in the clause, and what is for from a formal point of view?
• are we dealing with the same formal construction in all four examples?
• what are the functions of the relevant construction in (1) and (2)?

Solution 1:

CGEL would say that all four of your boldfaced phrases are to-infinitivals with an overt subject. For CGEL, the for is not a preposition there, but a 'subordinator'.

On p. 1177 of CGEL, we find the following:

[16]  i  I wanted to arrange for Kim to do it.

In [i] we have a chain of three verbs, with for Kim to do it complement of arrange and to arrange for Kim to do it complement of want. We apply the term 'catenative' both to the non-finite complement and to the verb in the matrix clause that licenses it, so that want and arrange here belong to the class of catenative verbs. The last verb in the chain, do, is not a catenative verb as it does not have a non-finite complement.

A bit later on, CGEL explains that

for Kim to do it

is a to-infinitival with an overt subject. According to CGEL (p. 1178),

To-infinitivals containing a subject are always introduced by the subordinator for:

[20]  i  [For them to withdraw now] would be a mistake.                                           [subject]
         ii  It's not necessary [for them to wait any longer].                        [extraposed subject]
        iii  The best plan would be [for them to go alone].                              [predicative comp]
        iv  I can think of no solution except [for them to sack him].        [comp of preposition]

Solution 2:

Yes, we have no prepositions today!

There aren’t any prepo­si­tional phrases here, be­cause al­though one typ­i­cally iden­ti­fies the words to and for as prepo­si­tions, they are here func­tion­ing not as prepo­si­tions but as com­ple­men­tiz­ers of in­fini­tive clauses.

Your sen­tences 1 and 2 do not con­tain prepo­si­tional phrases be­cause they are phrasal verbs (to wait for, to ar­range for) whose com­ple­ments are in­fini­tive clauses with overt subjects:

• (1) I [waited for]^{phrasal verb} [you to come here.]^{in­fini­tive clause}
• (2) He [ar­ranged for]^{phrasal verb} [me to go there.]^{in­fini­tive clause}

Those two uses of for are no more prepo­si­tional than the two uses of to are. You can test that those two uses of for are not prepo­si­tions by ex­chang­ing their in­fini­tive clauses marked above for sim­ple noun complements:

• (1a) I [waited for]^{phrasal verb} [day­light]^noun.
• (2b) He [ar­ranged for]^{phrasal verb} [travel]^noun.

When you do that, for day­light and for travel are not prepo­si­tional phrases be­cause they are not syn­tac­tic con­stituents.

Those two in­fini­tive clauses are not bare in­fini­tives be­cause they have a to. How­ever, some ma­trix verbs al­low for bare in­fini­tive com­ple­ments with­out that to marker:

• I made her come here.
• They saw me bake the cake.

Com­pare those with these, where you need the to:

• I forced her to come here.
• She wanted me to bake the cake.

Your sen­tences 3 and 4 do not con­tain prepo­si­tional phrases ei­ther. They in­stead use for in a dif­fer­ent way than 1 and 2 do, this time to mark the in­fini­tive clause as the sub­ject of the main verb:

• (3) [For him to do that]^{in­fini­tive clause} took courage.
• (4) [For you to apol­o­gise]^{in­fini­tive clause} is not enough.

The for there is not part of the main verb; it’s a manda­tory marker of the in­fini­tive clause be­ing used as the the sub­ject of a tensed verb. You can re­place both of those two sub­jects with the pro­noun this with no change in meaning:

• (3a) This took courage.
• (4a) This is not enough.

Pro­nouns can stand in for noun phrases. They can­not stand in for prepo­si­tional phrases, and in­deed a prepo­si­tional phrase “can­not” be the sub­ject of a clause. They can only ful­fill mod­i­fier roles in a sen­tence, not sub­stan­tive ones.

Be­cause we could make that swap here, those are not prepo­si­tional phrases. Another in­di­ca­tion that this for is not a prepo­si­tion is that you can no more ex­change it here for any other prepo­si­tion than you can ex­change to for one.

Lastly, if you re­ar­range the first two sen­tences with the phrasal verbs so that their in­fini­tive clause becomes the main verb’s subject, now you are forced to in­sert an “ex­tra” for to keep the sen­tence grammatical:

• (1c) For you to come here is what I waited for.
• (2c) For me to go there is what he ar­ranged for.

Now the first and last words are both for. You need both. These don’t work:

• (1d) *You to come here is what I waited for.
• (2d) *Me to go there is what he ar­ranged for.

And these are com­pletely un­gram­mat­i­cal be­cause they are no longer the phrasal verbs from the original:

• (1e) For you to come here is what I *waited.
• (2e) For me to go there is what he *ar­ranged.

See also this re­lated an­swer by John Lawler.

Solution 3:

This issue is covered in some detail in Radford's Analysing English Sentences (p86-88). I am offering a synopsis here to complement the accepted answer by @linguisticturn.

The text extracts included below are from the Complementisers section of Radford's chapter Words, which sets out to "explore how we determine what grammatical category a given word belongs to in a given use".

Radford states that a complementiser is used to introduce a complement clause. He gives three examples:

I think [that you may be right]

I wonder [if you can help me]

I am anxious [for you to receive the best treatment possible]

He notes the differences between complementisers that/if and for:

...for is an inherently infinitival complementiser, and so can be used to introduce a clause containing infinitival to, but not a finite clause containing a tensed auxiliary like (past tense) should:

*I am anxious [for you should receive the best possible treatment]

After some discussion of the grammatical functions that complementisers serve, Radford asks whether:

we really need to assign words such as for/that/if to a new category of _C/complementiser, or whether we couldn't simply treat (e.g.) for as a preposition... .

The answer is 'No', because there are significant differences between complentisers and other apparently similar words. For example, one difference between the complementiser
for and the preposition for is that the preposition for has substantive lexical semantic content and so (in some but not all of its uses) can be intensified by straight/right, whereas the complementiser for is a functor and can never be intensified: cf.

He headed straight/right for the pub [for = preposition]

*He was anxious straight/right for nobody to leave [_for = complementiser]

Moreover, the preposition for and the complementiser for also differ in their syntactic behaviour. For example, a clause introduced by complementiser for can be the subject of an expression like would cause chaos whereas a phrase introduced by a preposition cannot: cf.

For him to resign would cause chaos [=for clause]

*For him would cause chaos [=for phrase]

What makes it even more implausible to analyse infinitival for as a preposition is the fact that (bold-printed) prepositions in English aren't generally followed by a [bracketed] infinitive complement, as we see from the ungrammaticality of:

*She was surprised at [there to be nobody to meet her]

*I am not sure about about [you to be there]

*I have decided against [us to go there]

There then follows a lengthy section discussing yet another difference between the preposition for and the complementiser for, namely that preposition for can be pied-piped whereas complementiser for cannot.

Radford concludes:

So, there is considerable evidence in favour of drawing a categorical distinction between the preposition for and the complementiser for: they are different lexical items belonging to different categories.

With reference to the OP's sentences, we can say that Radford agrees with the CGEL that for introduces a to-infinitival complement clause and is clearly not a preposition. Conversely, Radford calls for here a complementiser, whereas for the CGEL it is a subordinator.