 The proof of Theorem 4 is significantly more difficult than that of Theorem 2: Theorem 2 requires only basic probability theory, whereas Theorem 4 is proved using methods from linear programming (Adams and Levine 1975; Goldman and Tucker 1956). Theorem 4 subsumes Theorem 2 as a special case: if all premises are relevant (i.e. have degree of essentialness 1), then Theorem 4 yields the same upper bound as Theorem 2. Furthermore, Theorem 4 does not take into account irrelevant premises (i.e. premises with degree of essentialness 0) to compute this upper bound; hence if a premise is irrelevant for the validity of the argument, then its uncertainty will not carry over to the conclusion. Finally, note that since \(E(\gamma)\in [0,1]\) for all \(\gamma\in\Gamma\), it holds that 