Q:

What article would be correct here? A belief that / The belief that

Can I change the definite article in the following sentence and remain grammatical?

Once you adopt the belief that there's nothing you can do to change
  something, you start to take a pernicious poison into your system.

I might mean that of all the beliefs someone might adopt you adopt one of them.
On this assumption, I might say ''There's a belief that there's nothing you can do etc.''
On the other hand, there might be only one belief that there's nothing you can do: the belief. If I can, what's the principle by which a native speaker chooses between the two? Thanks!

A:

Once you adopt the belief that there's nothing you can...

... is absolutely correct because you are specific about that belief. And, that is why it'd take the. 
Again, you are right! Now, you are introducing that belief and that's why you can put an indefinite article. It's the same as you say - there's a place called Times Square. There is the belief somehow does not look idiomatic, at least to me. 

