When asked whether he recognized equal rights of atheists, George H. W. Bush said1:
No, I don't know that atheists should be considered as citizens, nor should they be considered patriots. This is one nation under God.
I would understand if he had used another verb:
No, I don't think/believe that atheists should be considered...
I would also understand if he had chosen another conjunction:
No, I don't know if atheists should be considered...
Is the original quoted sentence grammatically correct? Is it common to use know in the sense of think? Or is this some dialect-specific meaning? Could you please give some more examples of this usage of know?