Beliefs and Public Office

In light of the upcoming VP debate tonight, here's one I've been thinking about.

The LA Times reports on a secondhand claim that Sarah Palin believes "that men and dinosaurs coexisted on an Earth created 6,000 years ago -- about 65 million years after scientists say most dinosaurs became extinct."

Let's make no claim about the veracity of the report. Maybe the music teacher misremembers, maybe Governor Palin has changed her mind...don't know. But if one were to hypothetically assign that belief to someone, would that disqualify the candidate from being qualified to hold the office? Would not believing something for which there is a great body of scientific evidence make one unfit for high office? If so, on what grounds?

What other sentences x might there be such that if candidate P believes (or fails to believe) x, P should not be in high office?