I like dogs I know are smart
So, I was just talking with a student, and realized that I didn't have a good explanation for a very common case of a structure that I've explained at least a hundred times. I'm hoping that one of you excellent people can help me explain this case.
1. I like dogs *that* like to run.
2. I like dogs (that) my mom likes, too.
3. I like dogs (that) I know are smart.
In sentence 1, "that" is required, because the second "like" needs a subject. In sentence 2, the "that" is optional, because the second "like" already has a subject. In sentence 3, my instincts strongly tell me that the "that" is optional, and I can find dozens of similar examples from other natives. Obviously, "know" has a subject, but "are" does not.
I can't say:
WRONG: "I like dogs are smart"--I would need to say "I like dogs THAT are smart."
So, why is it acceptable to say "I like dogs I know are smart"? Doesn't that leave "are" without a subject?