restricts the degree of belief of p to be less than or equal to the maximum degree of belief of the disjuncts α.

This leads to a downward revision of the degrees of belief according to the transformation v → 0v defined by the formula

$$\begin{array}{r l}{{}^{\prime}v_{p}\,=\,\operatorname*{min}_{\begin{array}{c}{{C\in\mathcal{D}}}\end{array}}\operatorname*{max}_{\alpha\in C}v_{\alpha},}\\ {{}}&{{}}\\ {{}}&{{C\ni\overline{{{p}}}}}\end{array}\alpha\neq\overline{{{p}}}$$
$$(52)$$

whose iteration leads to the lower revised valuation ∗v .

Equivalently, ∗v is given by the formula ∗vp = 1 − vˆ
∗
p
, where ˆv is related to v by the formula ˆvp = 1 − vp (ˆv believes p to the extent that v does not believe p). This characterization allows to obtain the results for ∗v from those for v
∗.

When v is a balanced valuation, then ∗vp = 1 − v
∗
p
, so that ∗vp −∗vp =
v
∗
p − v
∗
p
. Therefore, the upper and lower revised valuations lead then exactly to the same decisions.

## 5 Application To Specific Domains

In this section we apply the preceding ideas and results to three specific domains. Unfortunately, there is no space in this article for developing other applications.

5.1 One proposition being equivalent to the conjunction of two other ones. This is the problem with which we started the article, i. e. Π+ =
{*p, q, t*} with the doctrine t ↔ (p ∧ q). By rewriting the connective ↔ in terms of ∧, ∨, ¬, one easily arrives at the corresponding Blake canonical form, which has a definite Horn character:

$$(\overline{{{p}}}\vee\overline{{{q}}}\lor t)\,\land\,(p\vee\overline{{{t}}})\,\land\,(q\vee\overline{{{t}}})$$
$$\left(53\right)$$

$$\begin{array}{l}{(54)}\\ {(55)}\\ {(56)}\end{array}$$
(p ∨ q ∨ t) ∧ (p ∨ t) ∧ (q ∨ t) (53)
The one-step revision transformation, which uses the clauses above and also the *tertium non datur* ones, is given by

v 0 p = max (vp, vt), v0p = max (vp, min(vq, vt)), (54) v 0 q = max (vq, vt), v0q = max (vq, min(vp, vt)), (55) v 0 t = max (vt, min(vp, vq)), v0t = max (vt, vp, vq). (56)
One easily sees that the equality v
∗ = v 0 holds whenever v satisfies the following inequalities: vt ≤ vp , vt ≤ vq , vt ≥ vp , vt ≥ vq . More particularly,