I will have him believe it
What does this mean? Does this mean “I will make him believe it”?
My friends are saying have + object form is used in the sense of wish. Is it correct?
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Your friends are wrong, but not because of the use of have + object specifically.
The distinction comes from the first verb in the sentence:
It's not only the use of have + object that determines the meaning, but its use in combination with the particular tense of the auxiliary verb being used.
Note that the second sentence can have more than one interpretation.
If you look at the Merriam-Webster definition of will (which includes its past tense) in its use as an auxiliary verb, you'll see that it's quite complex.
In the first sentence, it uses the "used to express futurity" sense. In the second sentence, it uses the "used to express desire, choice, willingness, consent, or in negative constructions refusal" sense.