Again, the property of autarky is preserved by the operations of absorption and resolution (checking it is a little exercise). As a consequence, it passes on to the corresponding Blake canonical form. So, Σ ⊂ Π is autarkic for the latter if and only if it is autarkic for some logically equivalent conjunctive normal form. In such a situation, we can say simply that Σ is autarkic for the given doctrine.

For a definite Horn doctrine, Π+ is easily seen to be an autarkic set.

This is a particular case of the following more general fact: if u is a truth assignment consistent with D, then Σ = { α ∈ Π | up = 1 } is an autarkic set. Generally speaking, however, autarkic sets need not decide on every issue. Even so, their definition allows for the following generalization of Proposition 3.8:
Proposition 4.14. An autarkic set Σ *has the property that*

$\left(46\right)$ . 
$$\operatorname*{max}_{p\in\Sigma}\;v_{\overline{{p}}}^{*}\;\leq\;\operatorname*{max}_{q\in\Sigma}\;v_{\overline{{q}}}.$$
vq. (46)
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 (46): v 0 p ≤ maxq∈Σ vq for any p ∈ Σ. This follows immediately from
(7) because, by definition, Σ being autarkic requires every clause C that contains p with p ∈ Σ to contain also some q ∈ Σ, which ensures that minα∈C,α6=p vα ≤ maxq∈Σ vq .

Theorem 4.15. *Assume that the original valuation satisfies*

$$\left(47\right)$$
$$\operatorname*{min}_{p\in\Sigma}\;v_{p}\;-\;\operatorname*{max}_{p\in\Sigma}\;v_{\overline{{{p}}}}\;>\;\eta,$$
vp *> η,* (47)
for some autarkic set Σ *and some* η ∈ [0, 1). In this case, the decision of margin η *associated with the upper revised valuation* v
∗ accepts every proposition in Σ. Proof. By combining (46), (47) and part (c) of Theorem 3.2, it follows that

$$\operatorname*{min}_{p\in\Sigma}\;v_{p}^{*}\;-\;\operatorname*{max}_{p\in\Sigma}\;v_{\overline{{{p}}}}^{*}\;>\;\eta,$$
$$(48)$$
$\square$
p *> η,* (48)
$${\mathrm{which~implies~}}v_{p}^{*}-v_{\overline{{p}}}^{*}>\eta{\mathrm{~for~any~}}p\in\Sigma.$$
