A paradox is something that, as a result of an inherent contradiction, can't possibly be right. An example of a paradox would be the following two sentences:
The statement below this one is false.
The statement above this one is true.
These two sentences can't both be correct, as accepting the first statement will result in having to reject the second and thus reject the first one; rejecting the first statement will result in accepting the second one, which says the first one is true.
As a result of greater dependence on a 2000 year old book of fiction and refusal to accept facts and logic, paradoxes are far more widespread in the Conservative mind. Some examples of Conservative paradoxes include George W. Bush actually being elected or being pro-life yet pro-death penalty at the same time.