it holds whenever v is an aggregate of several components satisfying all of these inequalities. When the component valuations are all-or-none and noncontradictory —i. e. satisfying vα+vα ≤ 1 for any α— these inequalities hold whenever these valuations are definitely consistent. Under this assumption
—that is quite reasonable in the jury scenario of § 1.1— the doctrine (53) is therefore unquestionable in the sense of § 4.3. In the general case, one has only v
∗ = v 00.

As a particular example, let us consider the following aggregate of consistent truth assignments:

$(v_{p},\,v_{q},\,v_{t};\,v_{\overline{{p}}},\,v_{\overline{{q}}},\,v_{\overline{{t}}})\;=\;.30\,\left(1,1,1;0,0,0\right)\,+\,.40\,\left(1,0,0;0,1,1\right)$  $+\;.25\,\left(0,1,0;1,0,1\right)\,+\,.05\,\left(0,0,0;1,1,1\right)$  $=\;(.70,\;.55,\;.30;\,.30,\;.45,\;.70)$.  
As one can see, these collective degrees of belief result in the inconsistent decision of accepting both p and q but rejecting t. In contrast, the corresponding upper revised valuation, namely

$$\mathbf{v}$$

(v
∗
p
, v∗
q
, v∗
t
; v
∗
p
, v∗
q
, v∗
t
) = (.70*, .55, .*55; .55, .70*, .70),* (58)
results in the consistent decision of accepting p and rejecting both q and t.

This decision holds up to a margin of .15, above which all three issues are left undecided. The classical example of § 1.1 results in all upper revised values being equal to exactly the same value, namely 23
, so that all three issues are then undecided even for a vanishing margin. All of this is in agreement with the concept of definite consistency defined in § 2.2.

5.2 Equivalence relation on a set A. Here the set of atomic propositions is Π+ = { exy | x, y ∈ *A, x* 6= y }, where exy stands for the proposition
'x is equivalent to y '. In order to follow the standard definition of an equivalence relation, one should include also the propositions exx , and constrain them to being true; however, this goes against our convention (D2) of avoiding unit clauses, which is why the definition of Π+ considers only pairs xy with x 6= y . The doctrine requires that for any pairwise different *x, y, z* ∈ A
one must have: exy ∧ eyz → exz (transitivity) and exy → eyx (symmetry).

This is equivalent to identifying exy with eyx and adopting the following clauses:
exy ∨ eyz ∨ exz, for any pairwise different *x, y, z* ∈ A. (59)
This doctrine has a definite Horn character and has the following autarkic sets: Π+ itself; Π− = { exy | x, y ∈ A, x 6= y }; Σa = { eax | x ∈ *A, x* 6= a }
for any a ∈ A.