
## 3 Construction And Main Results

3.1 Construction. The upper revised valuation will be obtained by means of an iterative process whereby belief will be propagated along the implications contained in the doctrine. More specifically, we will consider the implications (3) and we will apply the following general principle:
(P) Consider an implication of the form p ←Vα∈Σ α with Σ ⊂ Π .

As soon as the right-hand side is satisfiable, this implication gives to p at least the same degree of belief as the weakest of the conjuncts α.

The hypothesis of satisfiability contained in this principle should be understood in relation to the doctrine under consideration: Among the truth assignments that make the doctrine true, at least one of them should make the conjunction Vα∈Σ α true. Unless otherwise stated, in the sequel 'satisfiability' should always be understood in this sense. In § 4.1 we will see that this amounts to requiring the conjunctive normal form that expresses the doctrine to be a prime conjunctive normal form. However, for the moment we need not be concerned with this question. In fact, principle (P)
will be used only to motivate certain definitions that can be applied to any conjunctive normal form.

In contrast, the present subsection already makes use of the assumptions
(D1, D2, D3) adopted at the end of § 2.1.

Let us look at the consequences of applying principle (P) to the implications (3). Starting with the degrees of belief given by v , we infer that every p ∈ Π should be believed at least in the degree v 0 p defined by

$$v_{p}^{\prime}\,=\,\operatorname*{max}_{\begin{array}{l}{C\in\mathcal{D}}\ \ \alpha\in C\\ {C\ni p}\ \ \alpha\neq p\end{array}}v_{\overline{{\alpha}}},$$
$$\left(7\right)$$
vα, (7)
where the max and min operators are ensured to deal with non-empty sets of values as a consequence of assumptions (D2) and (D3).

Lemma 3.1. *The transformation* v 7→ v 0 *has the following properties:*
(a) *It is continuous.*
(b) v ≤ w *implies* v 0 ≤ w 0.

(c) v ≤ v 0.

(d) *The image set of* v 0is contained in that of v .

Proof. Part (a): Since max and min are continuous. Part (b): Since max and min are monotone. Part (c): Because of the *tertium non datur* clauses