choose vep so that vp < vep < v∗
p
. The hypothesis that v
∗
qincreases ensures that v
∗
q < ve
∗
q(this is true not only for vep in a neighbourhood at the right of vp , but also for any vep > vp , since part (b) of Theorem 3.2 ensures that ve
∗
qis a non-decreasing function of vp ). Let us consider now the valuation w that coincides with v for all literals except possibly p, for which we set wp = v
∗
p
. Since vep < v∗
p
, part (b) of Theorem 3.2 ensures that ve
∗
q ≤ w
∗
q
. We will arrive at contradiction by showing that w
∗
q = v
∗
q
. In fact, by combining this equality with some of the preceding inequalities, namely v
∗
q < ve
∗
q ≤ w
∗
q
,
one would arrive at the false conclusion that v
∗
q < v∗
q
.

The claim that w
∗
q = v
∗
q is again a consequence of part (b) of Theorem 3.2:
Since v ≤ w ≤ v
∗, that result ensures that v
∗ ≤ w
∗ ≤ v
∗∗ . On the other hand, the invariance of v
∗ by the transformation u 7→ u 0entails v
∗∗ = v
∗.

Theorem 3.14. Let us allow vp to grow while vα *is kept constant for* α 6= p.

In these circumstances, the acceptability of p*, that is the difference* v
∗
p − v
∗
p
,
either increases or stays constant.

Proof. By part (b) of Theorem 3.2, v
∗
p and v
∗
p have the property that each of them either increases or stays constant. When v
∗
p stays constant we are done. Now, when v
∗
pincreases, Lemma 3.13 ensures that v
∗
p = vp . Since ve
∗
p ≥ vep > vp , it follows that v
∗
pincreases too. However, according to Lemma 3.12 the only possible way for both v
∗
pand v
∗
pto increase is by having ve
∗
p = ve
∗
p = vep for any vep in an interval of the form vp ≤ vep ≤ vp + δ with δ > 0, which entails that v
∗
p −v
∗
p stays constant (and equal to zero).

$$\widetilde{v}_{p}\;>\;v_{p},\qquad\widetilde{v}_{q}\;=\;v_{q},\quad\forall q\in{\cal H}\setminus\{p\},$$

Corollary 3.15. Assume that the valuation v *is modified into a new one* ve such thatvep > vp, veq = vq, ∀q ∈ Π \ {p}, (21)
and let η be any number in the interval 0 ≤ η ≤ 1. If p is accepted [resp. not rejected ] in the decision of margin η *associated with* v
∗, then it is also accepted [resp. not rejected ] in the decision of margin η *associated with* ve
∗.

## 4 Technical Issues And Supplementary Topics

4.1 Which conjunctive normal form? In this subsection we look at the effect of the conjunctive normal form Φ that expresses the doctrine. In this connection, we will be led to make a particular choice, namely the so-called Blake canonical form.