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I don't see no reason why I shouldn't trust him.

Is the sentence above grammatical?

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I've derived that sentence from the other: My dad trusted him, and I don't see no reason why I shouldn't.

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"I don't see no reason" is an incorrect way of saying "I don't see any reason" or "I see no reason". This kind of misuse of the double negative is characteristic of uneducated English.

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Students: Are you brave enough to let our tutors analyse your pronunciation?

Thank you for the reply.

Try either

I see no reason why I shouldn't trust him.

or

I don't see any reason why I shouldn't trust him.

or

I don't see a reason why I shouldn't trust him.

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GPYThis kind of misuse of the double negative is characteristic of uneducated English.

It's also characteristic of Chaucerian, Shakespearean and certain dialects of English. It's not condoned in contemporary standard English. You can blame the 18th century grammarians.