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I Think I understand why the passive infinitive of " amo " is not " esse amatus" : "being loved" is not perfect ( without any play on words).

So we need something else than " amatus" . And we have " amatum/a/um esse ".

I'd have 3 questions to ask about this somewhat curious " amatum" :

Is it correct to translate ( literally) " amatum " as " having been loved" ? In such a way that the whole perfect passive infinitive would mean literally " being having been loved"?

Is there any link between " amatum " and the supine of " amo" , namely " amatum"?

And finally, are there cases where we need a plural for " amatum" ? And if so, what is/ are its plural form(s) ?

Thanks in advance!

Note . I am using Deleani & Vanmander Initiation à la langue latine et à son système. The book does not give much details regarding this kind of infinitive :" We do not give the translation of these infinitives, since they, so to say, never correspond to our [french] infinitive perfect passive ( ex: " ayant été aimé") " ( p. 53) .

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I don't think it's 100% right to translate infinitives as participles, as you're doing. Esse is less "being" and more "to be," and the difference isn't trivial. Esse, for example, cannot be used attributively, whereas participles can. It's the difference between étant and être.

For your specific questions, yes, in English the perfect passive participle is often taught to be translated as having been X'ed, though this is just to differentiate it from the simple past, as it can also simply be translated with the English past passive participle. Amatus, -a, um is simply "loved," though in the sense of "the loved child fared better than the hated child" and not "they loved one and hated the other."

For the infinitive, it's often best translated as to have been X'ed. So, amatum esse is "to have been loved," such as in the misquote, "It is better to have been loved..."

Concerning supines, that question has been answered already here (with additional links for your perusal):

Origin of supine form?

The perfect passive participle, being participles and therefore adjectives, decline according to gender, number, and case. The plural simply follows the 1-2 adjective endings:

puer amatus, puella amata, donum amatum

pueri amati, puellae amatae, dona amata

Perfect passive infinitives will decline according to what's being modified. This is especially important in verbs of indirect speech. "He says that she has been loved" will be in Latin: Dicit eam esse amatam. If the indirect verb were active, we'd get instead: Dicit eam amavisse, "He says that she has loved."

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Your question gives me the impression that you might be confused about the following points:

The perfect passive infinitive of "amo" is formed with the infinitive esse and a form of the perfect passive particicle amatus/amata/amatum. Any perfect passive participle inflects like an adjective of the first and second declension, with a masculine nominative singular ending in -us, a masculine accusative singular ending in -um, a feminine nominative singular ending in -a, a feminine accusative singular ending in -am, and so on.

No Latin word has a paradigm of the form "amatum/a/um".

The supine is not involved in the formation of the perfect passive infinitive.

Wiktionary lists the perfect passive infinitive of amō as "amātum esse"; I assume this is meant to be a masculine accusative singular participle (although the neuter nominative or accusative singular would have the same form). The plural equivalent would therefore be "amātōs esse". The case of nouns, adjectives, and participles used with esse depends on the surrounding context in the sentence: here are some previous questions about that: Verb + esse + predicate nominative, Is the complement of esse in nominative or accusative when esse is a subject?

I'm not sure what you mean by "'being loved' is not perfect". If you mean that the English constructions "being loved" or "to be loved" usually only have a passive and not a perfect meaning (the perfect being expressed in English by "having been loved" or "to have been loved"), remember that Latin forms the present passive infinitive of amo as amari. Latin, unlike English, often uses non-periphrastic inflection to indicate passive forms. So you shouldn't try to translate Latin passive constructions into English word-for-word, or vice versa.

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