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Where and when can I use the word whether?

Which out of the two is the correct use of the word:

I don't know whether Peter will come to school.

or

I don't know that Peter will come to school.

Are either sentences correct above? Could you please explain it to me?

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    It should be come to school, not *come school. Come is not a transitive verb, and school is not a direct object. – John Lawler Dec 29 '16 at 22:31
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"Whether" is nearly always replaceable by "if", for embedded questions. So

I don't know whether Peter will come to school.

and

I don't know if Peter will come to school.

have exactly the same meaning: I don't know if it is true or false that Peter will come to school.

Your other sentence,

I don't know that Peter will come to school.

is grammatical, but in principle has a different meaning (though that meaning is unlikely with a first person subject).

If we switch to a third person subject:

Jane doesn't know if/whether Peter will come to school.

means that she doesn't know the truth of the question. But

Jane doesn't know that Peter will come to school.

implies that it is a fact (known to the speaker) that Peter will come to school, but that Jane doesn't know that fact.

Clearly, this distinction is unlikely for a first person subject, because the subject is the speaker. For this reason, I think, you do hear

I don't know that Peter will come to school.

with the same meaning as if/whether; but I would say this is very colloquial.

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Uh… ‘whether’ addresses both sides of the question because it necessarily carries the rider ‘… or not.’ Either answer should be clearly explicit and carry with it various considerations.

‘If’ addresses only the positive answer. It has no explicit interest whatever in the negative choice, which is addressed only implicitly and never considered further.

Whether anyone knows ‘that Peter will (anything)’ or ‘what Peter will do' are separate issues.

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