No, "I saw him do it" is a perfectly valid example of an independent clause. It contains one subject (I) and one predicate (saw him do it). If I'm not mistaken (help me out here Faust) "him do it" functions as the direct object of the verb "saw".
For a sentence to be independent it doesn't necessarily have to make a lot of sense to the reader. An example of a dependent clause that can't be used in conjunction with a semicolon would be something like "When I saw him do it", which can't really be parsed in any meaningful way without an additional independent clause, e.g. "When I saw him do it, I remembered the trick."
You're actually right there dada. As long as it's grammatically correct as an independent sentence then it's absolutely fine. Again we have a situation where JOHNNY FOREIGNER knows the rules our language much better than 99% of its native speakers.
...Although, counting on Faust to help you out of a grammar situation is probably a bit like asking Tony Soprano for advice on being a good father! He thinks he's the shit, but it turns out he's just a massive douchebag!
Still, as long as the government keeps cutting those pay cheques!!!!!
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English buffs, help!