Hi, I learnt at school 3 types of conditional sentences; one of them (the easiest) was like this:

" If clause" - past tense; "2nd clause" - would+infinitive

For example:

"If I studied more, I would learn quicker"

Well, I recently flied in an Iberia plane who lent us headphones for the flight in whose bag appeared this sentence:

"We would appreciate if you would return them to a member of the cabin staff once the flight is over"

According to the above rule, shouldn't Iberia state "We would appreciate if you returned them...."
Yes, Iberia should say it as you suggest to be perfectly correct. However, in this case, many people believe that "if you would return them ..." sounds more polite and elegant than the alternative.

Semantically, this is not a standard, run-of-the-mill "if-then" structure. It is a standard formula for politely giving an order. "if you would return ..." can be interpreted as "if you were willing to return ..." ("would" of volition).

I'm more used to hearing 'we would appreciate it if...'

Is that your experience too?

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Hmmmm. You make a valid point there, Mr. P. Now that I reread it, the "it" really is missing in the original. And yet, for some reason, I don't object to its absence here as much as I would have in "We would like if ...", which seems truly bizarre.

I suppose there is a gradation among verbs that take the "it if" construction -- everything from optional "it" through absolutely required "it".
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