"He paid for the damage to the window."
Translation:Li pagis por la damaĝo al la fenestro.
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'por' means «purpose, destination, an object of exchange» and is the best translation for 'for', because the payment est destinated for the damage. 'Kontraŭ' could be understood in the context, but is not logic enough. But one could use 'damaĝo kontraŭ la fenestro'
Why is "damagxo" accepted here as an alternative? When I've tried using the x system elsewhere, it hasn't been accepted.
Normally x works always. I suppose there is a special program which transforms the x and it does not work properly in some versions of some browsers.
Which prepositions can be used here for the window? I used 'je' thinking that 'al' implied actual movement.
A man made an action towards (al) the window.
I see the damage of (de) the window (damaĝo de la fenestro, fenestrodamaĝo).
There is a damago on (sur) the window (damaĝo sur la fenestro).
"damage" is mostly an abstract uncountable noun in English, and damage will likely result from somebody moving something to hit something but the 'damage' itself doesn't move or move anything.
The action is from him to the windows, the result is a damage and a fine.
He paid for the damage (he has made) to the window."
Li pagis por la damaĝo (farita) al la fenestro.
Li pagis por la damaĝo (kiun li faris) al la fenestro.
The sentence is not about an action to the window, it's about damage with this preposition used to describe the location of damage. Things can become damaged without movement. I don't think you can do damage in a direction, except in RPG games where they use the word in a countable sense.
So I think the choice of 'to' in English for this situation is reasonably arbitrary and isn't about movement. Prepositions don't always map neatly from one language to another (e.g. por / pro / kontraŭ can be used in ways that 'for' would be used in English), so maybe something else is better here?
Oddly enough, in the two other sentences you gave above, I would always use 'to' and not 'of' or 'on' in English. 'damage of the window' would imply the damage that the window did to something else, and 'damage on the window' - damage is abstract; you can't pick it up or scrape it off, and doesn't sit literally atop or inside the window (as 'on' implies): it's inherent to the window.
Yes, « the choice of 'to' in English for this situation is reasonably arbitrary ». This exists in all languages. Sometimes there is no good preposition, so one use a preposition which seems good. If you use « al / de / por / pro / kontraŭ / je » between damaĝo and fenestro, I always understand.
The 2 other sentences I gave are OK en Eo, but KO en English.