 First of all, note that this theorem subsumes Theorem 1 as a special case: if \(P(\gamma) = 1\) for all \(\gamma\in\Gamma\), then \(U(\gamma)=0\) for all \(\gamma\in\Gamma\), so \(U(\phi)\leq \sum U(\gamma) = 0\) and thus \(P(\phi) = 1\). Furthermore, note that the upper bound on the uncertainty of the conclusion depends on \(|\Gamma|\), i.e. on the number of premises. If a valid argument has a small number of premises, each of which only has a small uncertainty (i.e. a high certainty), then its conclusion will also have a reasonably small uncertainty (i.e. a reasonably high certainty). Conversely, if a valid argument has premises with small uncertainties, then its conclusion can only be highly uncertain if the argument has a large number of premises (a famous illustration of this converse principle is Kyburg’s (1965) lottery paradox, which is discussed in the entry on  epistemic paradoxes  of this encyclopedia). To put the matter more concretely, note that if a valid argument has three premises which each have uncertainty 1/11, then adding a premise which also has uncertainty 1/11 will not influence the argument’s validity, but it will raise the upper bound on the conclusion’s uncertainty from 3/11 to 4/11—thus allowing the conclusion to be more uncertain than was originally the case. Finally, the upper bound provided by Theorem 2 is optimal, in the sense that (under the right conditions) the uncertainty of the conclusion can coincide with its upper bound \(\sum U(\gamma)\):