We claim that this doctrine is disjoint-resolvable in the sense defined in § 4.2, and that its Blake canonical form, with the *tertium non datur* clauses included, consists of all clauses of the form

$${\overline{{e}}}_{x_{0}x_{1}}\lor{\overline{{e}}}_{x_{1}x_{2}}\lor\cdots\lor{\overline{{e}}}_{x_{n-1}x_{n}}\lor e_{x_{0}x_{n}},$$
$$(60)$$
, (60)
with n ≥ 1 and all xi (0 ≤ i ≤ n) pairwise different (which restricts n to be less than or equal to the number of elements of A). In the following we will refer to such sequences x0x1 *. . . x*n as non-cyclic paths from x0 to xn , and a generic non-cyclic path will be denoted by means of the greek letter γ . In order to establish the preceding claim it suffices to check that: (i) starting from (59) one can arrive at (60) for any non-cyclic path by using only disjoint resolution; and (ii) any further resolution does not add any new clause (i. e. the would-be new clause is absorbed by some clause of the form (60) ). The last statement can also be strengthened to show that condition (a) of Thm. 4.8 is satisfied for any of the propositions exy and exy .

As a consequence, v
∗coincides with the result of applying once the transformation v 7→ v 0 determined by the Blake canonical form, which makes this doctrine unquestionable in the sense of § 4.3.

More particularly, the values of v
∗(exy) can be obtained by considering all possible non-cyclic paths γ from x to y and applying the formula

$$v^{*}(e_{x y})\quad=\quad\operatorname*{max}_{\gamma}\quad\operatorname*{min}_{0\leq i<n}\quad v(e_{x_{i}x_{i+1}}).$$
$$(61)$$

These values are easily seen to satisfy

$$v^{*}(e_{x z})\ \geq\ \operatorname*{min}\left(v^{*}(e_{x y}),v^{*}(e_{y z})\right)\qquad{\mathrm{for~any~}}x,y,z.$$

As one can easily check, this inequality is a necessary and sufficient condition for the following binary relations Eη (0 ≤ η ≤ 1) to be equivalence relations:

$$(62)$$
$$x y\in E_{\eta}\ \equiv\ v^{*}(e_{x y})\geq\eta.$$
∗(exy) ≥ η. (63)
These facts are closely related to the definite consistency of the unilateral decisions of margin η considered in § 4.4. In fact, one can easily check that:
Eη coincides with the set of pairs xy such that exy is accepted by an unilateral decision of margin smaller but near enough to η, and that this decision is definitely consistent if and only if Eη is an equivalence relation. Obviously, the equivalence relations Eη become progressively finer as η grows
(i. e. *η < ζ* implies Eζ ⊆ Eη in the sense of subsets of A × A). What we are obtaining is the so-called single-link method of cluster analysis [13 : 7.3]. This method is usually formulated in terms of the "dissimilarities" dxy = 1−v(exy)