Theorem 4.16. *Assume that the original valuation satisfies*

vp = 1, vp = 0, for any p ∈ Σ, (49)
$\left(49\right)$ . 
where Σ *is an autarkic set. In this case, the upper revised valuation* v
∗ has the following properties: (a) For any p ∈ Σ *one has also* v
∗
p = 1 and v
∗
p = 0.

(b) For any q ∈ Π such that neither q nor q belongs to Σ, the upper revised value v
∗
q*coincides with the value which is obtained for a modified doctrine* that leaves out any clause that contains some element of Σ ∪ Σ.

Proof. Part (a) is a direct consequence of Proposition 4.14 and part (c) of Theorem 3.2. In order to prove part (b) it suffices to show that one can leave out the clauses that contain some element of Σ; in fact, by the definition of an autarkic set, those that contain an element of Σ are ensured to contain also some element of Σ. The statement of part (b) will be obtained by showing that it holds for all the iterates v
(n)
q with n ≥ 1. In this connection, we will make use of the fact that v
(n)
p = 0 for any p ∈ Σ and any n ≥ 0, which follows from part (a) because of the inequality v ≤ v
(n) ≤ v
∗. Let us assume that neither q nor q belongs to Σ. Recall that v
(n)
q is given by

$$v_{q}^{(n)}\,=\,\operatorname*{max}_{\begin{array}{l}{{C\in\mathcal{D}}}\quad\alpha\in C\\ {{C\ni q}}\quad\alpha\neq q\end{array}}v_{\overline{{{\alpha}}}}^{(n-1)}.$$
$$(50)$$
$$\left(51\right)$$

We claim that the value that results from this formula is not altered if the max operator of the right-hand side forgets about any clause C that contains some p in Σ. In fact, we know that v
(n−1)
p = 0 and p 6= q (because of the assumption that q 6∈ Σ). Therefore, we get minα∈*C, α*6=q vα
(n−1) = 0, which entails that claim.

4.6 Duality. The lower revised valuation ∗v announced in § 2.4 is obtained by means of a dual method —somehow it might be more appropriate to say 'codual'— which is based on the fact that the doctrine, i. e. (1) being true, provides the following implications originated at p (which should be compared to (3) ):

p →_ α∈C α6=p α, (51)
for any C ∈ D such that p ∈ C . Such implications can be dealt with by means of a principle dual to (P) that can be stated in the following way:
(Q) Consider an implication of the form p →Wα∈Σ α with Σ ⊂ Π .

As soon as the right-hand side is not always satisfied, this implication