Remark. When v is balanced, the condition vp >
1 2 > vp is equivalent to requiring simply that vp > vp . Generally speaking, however, the latter condition cannot be substituted for the former in the preceding result.

According to part (c) of Theorem 3.2, vp = 1 implies v
∗
p = 1. The following result shows that the converse implication holds when v is an aggregate of definitely consistent partial truth assignments. By an aggregate we mean a convex combination of several components v k, as in (6).

Proposition 3.10. Assume that v is an aggregate of definitely consistent partial truth assignments. In this case, having v
∗
p = 1 *implies* vp = 1.

Proof. We will proceed by induction on the iterates v
(n). More specifically, we aim at showing that

$${\mathbf{\nabla}}_{o}={\mathbf{\nabla}}{\mathbf{I}},$$

for any p ∈ Π , the equality v
(n)
p = 1 implies vp = 1, (19n)
where we emphasize that the statement includes the quantifier "for any p∈Π ".

For n = 0 this is trivially true since v
(0) = v . For n ≥ 1, (19n ) can be obtained from (19n−1 ) in the following way: Let us assume that v
(n)
p = 1.

If v
(n−1)
p = 1, the conclusion follows directly from (19n−1 ). Otherwise, the inequality 1 = v
(n)
p > v(n−1)
p entails the existence of a proper clause C , i. e. a clause different from the ones associated with the *tertium non datur* principle, such that v
(n−1)
α = 1 for any α ∈ C \ {p}. Now, (19n−1 ) ensures that vα = 1.

Notice that this implies v 0 p = 1, which entails vp = 1 once (191 ) is established; however, establishing (19n ) for any n ≥ 1 requires making use of the fact that v has the form (6), where the v k are definitely consistent partial truth assignments. Since ak ≥ 0 and Pk ak = 1, it is clear that the equality vα = 1 entails v k α = 1 whenever ak > 0, i. e. for any k that matters. Now, since partial truth assignments are balanced, it follows that v k α = 0. On the other hand, since the v k are definitely consistent with the doctrine, and the preceding conclusion is valid for any α ∈ C \ {p}, it follows that v k p = 1.

Finally, this certainly implies vp = 1.

An important consequence of the preceding result is the following property of respect for unanimity:
Theorem 3.11. Assume that v *is an aggregate of definitely consistent partial truth assignments. In this case, having* vp = 1 implies that p *is accepted* by the basic decision associated with the upper revised valuation v
∗.