I wish him to believe it.
What is the exact meaning of this sentence? Does it mean “I wish he believes it”?
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According to Cambridge, "wish" is synonymous to "want" when followed by a to-infinitive:
When we use wish followed by a verb in the to-infinitive form, wish means the same as want, but it is more formal.
So your sentence basically means "I want him to believe it".
"I wish he believes it" is not grammatical in Standard English and does not have the same meaning as the sentence above.
"Wish" suggests that the proposition expressed is impossible, counterfactual or does not actually happen in real life; "I wish he believed it" presupposes that he does not actually believe it.
This is what Huddleston & Pullum (2002) call the 'irrealis mood' and it has to be marked by the past tense form of verb. So it should be:
I wish he believed it.