Using "I don't think" to express an opinion

Solution 1:

Maybe the answer is, they're two different sentences - but the second one is kind of truncated: "I don't think (so)". So, the second part is not attached to the first part, but reiterating the negative and confirming it.

Solution 2:

It's called negation, albeit a special form of it used more in informal communication.

It doesn't express disappointment (although your tone when you say it might).

This statement...

I think Hillary Clinton will not run for president.

...is the equivalent of this statement...

I don't think Hillary Clinton will run for president.

...and a less formal, more idiomatic way of saying that is...

Hillary Clinton will not run for president, I don't think.

Notice this formulation combines the other two. It's technically a double-negative. It's fine face-to-face, but don't use it in formal communication.

Read more about negation at the following website. There's a section on negating "think" at the bottom.

http://dictionary.cambridge.org/us/grammar/british-grammar/negation

Not … I don’t think

There are some cases where we can use reporting verbs such as imagine, suppose and think in end position, after the reported clause. In such cases, both clauses may have a negative verb:

He’s not a teacher, I don’t think. (or I don’t think he’s a teacher.)

Not: I don’t think he’s not a teacher.

I won’t be very late tonight, I shouldn’t imagine. (or I shouldn’t imagine I’ll be very late tonight.)

Not: I shouldn’t imagine I won’t be late.

Sometimes we use not in front position where a following reduced clause (a clause with something omitted but which is understood) also has a negative form:

A: Have you seen Leila?

B: Not today, I haven’t. (I haven’t seen Leila/her.)

A: Is Tony working at the university?

B: Not now, he isn’t. He used to.