What is the difference complementing with prepositional phrase and with to-infinitive? [closed]

To-infinitive complements must be distinguished from PP complement with the preposition to. The following quotation illustrates the difference.

The global threat to our security was clear.
So was our duty to act to eliminate it.
(Angela Downing, English Grammar): PP: prepositional phrase

I don’t get what is the difference between complementing a noun with to-prepositional phrase and to-infinitive. Would you show me the difference?

Solution 1:

The noun threat may take either a prepositional phrase (PP) headed by to or a verb phrase (VP) headed by a marked infinitive as a complement—or both:

... the global threat [_PP to our security]
... Britain's threat [_PP to France] [_VP to stifle its trade]

The PP names the recipient of the threat: the person or entity against whom Britain directs the threat. The VP names the action which Britain says it will perform.

So may the noun duty:

... our duty [_VP to act]
... our duty [_PP to our society] [_VP to pay taxes]

The PP names the person to whom the duty is owed, or who imposes the obligation. The VP names the action which we are obliged to perform.

Note, however, that in your second example the second marked infinitive, to eliminate it, is an infinitive of purpose naming the purpose which our ‘acting’ is intended to achieve; it is not a complement of duty, but a constituent of the VP headed by the verb act:

... our duty [_VP [_VP to act] [_VP to eliminate it] ]

How you parse to eliminate it is ambiguous. Some might regard it as a complement of act, others as an adverbial adjunct modifying act.