Nancy Lebovitz: If I say I feel something, I'm talking about an emotion.

That prohibits you from saying "I feel that X". No emotion is spoken of in saying "I feel that the Riemann hypothesis is true", or "I feel that a sequel to The Hobbit should never be made", or "I feel that there is no God but Jaynes and Eliezer (may he live forever) is His prophet", or in any other sentence of that form. "I feel" and "that X" cannot be put together and make a sensible sentence.

If someone finds themselves about to say "I feel that X", they should try saying "I believe that X" instead, and notice how it feels to say that. It will feel different. The difference is fear.