John Smith, a former firefighter, said he believes that was not true.OK, so he believes it, and . . . d'oh! How about he thinks that was not true?
A lot of us, it seems, have had a lot of nonsense, sometimes contradictory, drummed into our heads about supposed distinctions between "thinking" and "believing" and "feeling." I'm not a big fan of "feeling," I admit, but I'll save the discussion of those nuances for another time.
My point here is that "believe" has an unfortunate transitive property that can result in the kind of rhetorical roller-coaster ride you see above, and that search-and-replace editors need to keep that property in mind before they go mindlessly replacing every "think" with a "believe."
The Smith example is especially interesting because it is immune to the fix that usually renders "believe" usable. "They believe Bush lied" does the Moebius/roller-coaster/insert-your-own-analogy thing, but "They believe that Bush lied" is just as good as "They think Bush lied."
With Firefighter Smith, however, such a fix would result in an unfortunate "that that." In that case especially, there's nothing wrong with thinking.