A new puzzle about believed fallibility

I shall consider the phenomenon of believing ourselves to have at least one false belief: a phenomenon I call believed fallibility I shall first present a paradoxical argument which appears to show that believed fallibility is incoherent,, second, note that this argument assumes that we are committed to the conjunction of all our beliefs; third, sketch a more intuitive notion of commitment in which we are not committed to the conjunction of all our beliefs and argue that the original paradoxical argument is now defused; fourth, construct a new paradoxical argument showing that if we believe that we have at least one false belief we are committed to a contradiction, employing the preferable notion of commitment; and, fifth, suggest that perhaps we might avoid the new paradox by denying that closing our beliefs under conjunction is required by rationality.