Is there a word to define the verbal strategy that requests action of someone without using a direct command—e.g., Could you please open the door?
Steven Pinker calls it off-record indirect speech acts...
Note that Pinker's definition of indirect speech as quoted there isn't the same as the more well-known reported speech one. But if you Google "indirect speech act" it's obvious in that particular collocation Pinker's meaning is invariably what's meant.
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The term cajole is defined as
It is also defined as
Although the AHDEL doesn't license the word request in this usage
re·quest (r-kwst) tr.v. re·quest·ed, re·quest·ing, re·quests
... is as near as it gets ...
request [rɪˈkwɛst] vb (tr) to express a desire for, esp politely; ask for or demand [bolding mine]
Certainly, the expression
Could you please open the door?
contains two hedging devices, and
Could you please just open the door?