Every valuation w gives rise to a (partial) decision in the following way, that depends on a parameter η in the interval 0 ≤ η ≤ 1: For any p ∈ Π ,
p is accepted and p is rejected whenever wp − wp *> η,* (4)
p and p are left undecided whenever |wp − wp | ≤ η. (5)
We will refer to it as the decision of margin η associated with w, and we will identify it with the corresponding partial truth assignment, which will be denoted as µη(w). In the case η = 0 we will call it the basic decision associated with w, and the corresponding partial truth assignment will be denoted as µ(w). In tune with these definitions, the difference wp − wp will be called the acceptability of p according to w. If the valuation w is balanced, then the basic decision criterion is equivalent to the majority rule of §1.2, namely accepting p and rejecting p whenever wp >
1 2
.

2.3 Valuations are often an aggregate of several components (members of a jury, decision criteria, etc.), i. e. an average of the form

$$v_{p}\,=\,\sum_{k}a_{k}\,v_{p}^{k},$$
$$(6)$$
k
, (6)
where v k are the component valuations, and ak are the corresponding relative frequencies or weights, satisfying ak ≥ 0 for any k and Pk ak = 1. If the component valuations have an all-or-none character, then vp is the fraction of components where p is considered valid.

The doctrinal paradox points out the possibility that the average valuation be not consistent with the doctrine even when all component valuations are consistent with it.

2.4 Our main aim can be stated in the following way: *Given a valuation* v ,
build a revised one that achieves consistency with the doctrine while staying as near as possible to v .

In the method presented below, the consistency of the revised valuation will hold in a general sense that entails the definite consistency of the associated decisions in the sense of the definition given in § 2.2.

Generally speaking, we will obtain two revised valuations v
∗ and ∗v satisfying the inequalities ∗v ≤ v ≤ v
∗. We will call them respectively the upper and lower revised valuations. For a balanced original valuation, the upper and lower revised valuations give rise to exactly the same decision.

However, in the unbalanced case they can lead to different decisions.

For the moment, we will be concerned only with the upper revised valuation v
∗, the lower one being introduced at the end by duality.