6.2 The fractional degrees of belief form a continuum of possibilities that stretches over the all-or-none framework of classical logic. This allows for p and p not being exactly the semantic negation of each other, but rather the opposite, or antithesis, of each other. One may object that this is not compatible with the excluded-middle principle p∨p. However, this principle somehow loses its character just as fractional valuations come in. In fact, its role in connection with the latter is only through the excluded-middle clauses that we systematically include in the Blake canonical form; and this has only the following two effects: (a) providing the trivial implications p → p and p → p, through which the upper revised degrees of belief become larger than or equal to the original ones; (b) forbidding any implication of the form p ∧ p ∧ χ → t, which would be a gratuitous source of belief. Effect (b)
occurs because clauses are restricted to be prime, which prevents them from containing p ∨ p.

When p is the opposite of p rather than its denial, then intermediate degrees of belief can somehow be identified with belief in intermediate possibilities of fact. For instance, if p means 'white' and p means 'black'
—and even more if p means 'rather white' and p means 'rather black'—
then (wp, wp) = (0.3, 0.7) can be interpreted as giving belief to the gray that combines white and black in the given proportions. When the valuation is not balanced, then we are adding a sort of intensity of belief that can go from a full lack of opinion to a self-contradictory one.

6.3 Let us look back on the special problem of collegial courts with which we started the article. As we did in § 1.1 and § 5.1, we take as archetype the doctrine t ↔ p ∧ q , where t means 'being guilty' of a certain offence.

The data are the fractions of the jury who adhere to each of the propositions in question and their respective negations. These numbers can certainly be viewed as degrees of collective belief. The problem is that the decision that is naturally associated to this valuation, namely accepting α, and rejecting α, whenever vα > vα , may be inconsistent with the doctrine. This may happen even when each member of the jury is expressing a consistent opinion.

Thus, in the particular case of the doctrine t ↔ p∧q we have seen examples of consistent individual judgments that result in having at the same time vp > vp and vq > vq (guilty by the standard premise-based criterion) but vt < vt(not guilty by the standard conclusion-based criterion).

In contrast, the revised valuation v
∗ has the property that the associated decision is always definitely consistent with the doctrine in the sense defined in § 2.2. In particular, for the doctrine t ↔ p ∧ q one is ensured to have v
∗
p > v∗
p and v
∗
q > v∗
q(guilty by the revised premise-based criterion) if and only if v
∗
t > v∗
t(guilty by the revised conclusion-based criterion).