Only something that is meaningful may be false, but one may believe something that is clearly false to be true if and only if, at some level, he doesn't know what his belief means.
Analytic apriori principles are relevant here because they're... analytic. (I included the apriori part out of haste.) Analytic propositions show well the principle I'm trying to convey. It is clearly true, by virtue of the meaning of the terms alone, that all bachelors are unmarried. Thus, it would be possible to believe that some bachelor is married if and only if you didn't know what at least one of the terms in these propositions meant.
I don't think one need be completely ignorant of a proposition in order for it to be meaningless for him. Someone may be more or less conscious and consistent in his beliefs; e.g., Russell's paradox--I wouldn't say Frege was totally ignorant of what the propositions in his set theory meant, yet his system contained a fatal contradiction in its foundation, and so the whole system was meaningless by explosion.