3.3 Respect for consistent majority decisions and respect for unanimity. A very natural property to ask for is the following: If the majority criterion applied to the original valuation v decides on each issue and is consistent with the doctrine, then this decision should hold. This property of respect for consistent majority decisions will be obtained as a consequence of the following fact:

Proposition 3.8. For any valuation v , and any consistent truth assignment u*, one has*
$$\operatorname*{max}_{u_{p}=1}\,v_{\overline{{{p}}}}^{*}\,\leq\,\operatorname*{max}_{u_{q}=1}\,v_{\overline{{{q}}}}.$$
vq. (15)
Proof. Since v
∗is obtained by iterating the transformation v 7→ v 0, it suffices to show that this transformation has the following property analogous to (15): v 0 p ≤ maxuq=1 vq whenever up = 1. This follows immediately from (7) because the consistency of u requires every clause C that contains p with up = 1 to contain also some q with uq = 1, which ensures that minα∈C,α6=p vα ≤ maxuq=1 vq .

The property of respect for consistent majority decisions corresponds to the case θ ≥
1 2 ≥ θ − η of the following more general result:
Theorem 3.9. *Assume that there exist* θ ∈ (0, 1) and η ∈ [0, 1) such that every p ∈ Π satisfies either vp > θ ≥ θ − η > vp *or, contrarily,*
vp > θ ≥ θ − η > vp . If the decision associated with v (which contains no undecidedness and has a margin η) is consistent with the doctrine, then it agrees with the decision of margin η associated with the upper revised valuation v
∗.

Proof. Let u be the truth assignment that corresponds to the decision associated with v , that is:

$$u_{p}=\begin{cases}1,&\mathrm{if}\;\;v_{p}>\theta\geq\theta-\eta>v_{\overline{{{p}}}};\\ 0,&\mathrm{if}\;\;v_{\overline{{{p}}}}>\theta\geq\theta-\eta>v_{p}.\end{cases}$$
$$(16)$$

From this definition it follows that

$\left(17\right)$. 
$$\operatorname*{lim}_{u_{p}=1}\;\;v_{p}\;-\;\operatorname*{max}_{u_{p}=1}\;\;v_{\overline{{{p}}}}\;>\;\eta.$$
min up=1 vp − max up=1 vp *> η.* (17)
By combining this inequality with (15), which holds because of the consistency of u, and part (c) of Theorem 3.2, it follows that

$$\operatorname*{min}_{u_{p}=1}\;v_{p}^{*}\;-\;\operatorname*{max}_{u_{p}=1}\;v_{\overline{{{p}}}}^{*}\;>\;\eta,$$
$\left(18\right)^3$
$=\;1$ . 
p *> η,* (18)
which implies v
∗
p − v
∗
p > η whenever up = 1.