Group Conflict as Social Contradiction - Conflict and Multimodal Communication: Social Research and Machine Intelligence - page 39

The consistency of the output of the aggregation is defined by the following
properties. An aggregation procedure F , defined on an agenda ˚ ,issaidtobe
collectively rational iff F is:

complete if F. J / is complete for every J 2 J.˚/ n ; -

consistent if F. J / is consistent for every J 2 J.˚/ n ; -

That is, collective rationality forces the outcome of the procedure to be rational in
the same sense of the individual rationality. Of course, the case of doctrinal paradox
violates collective rationality.
We now introduce a number of axioms that provide a mathematical counterpart
of our intuition on what a fair aggregation procedure is. The following are the most
important axioms for JA discussed in the literature (List and Pettit 2002 ; List and
Puppe 2009 ):

- Unanimity (U): If 2 J i for all i , then 2 F. J / .
- Anonymity (A): For any profile J and any permutation W N ! N we have
F.J 1 ;:::;J n / D F.J .1/ ;:::;J .n/ / .
- Neutrality (N): For any , in the agenda ˚ and profile J 2 J.˚/ n , if for all i
we have that 2 J i , 2 J i , then 2 F. J / , 2 F. J / .
- Independence (I): For any in the agenda ˚ and profiles J and J 0 in J.˚/ n ,if
2 J i , 2 J i for all i , then 2 F. J / , 2 F. J 0 / .
- Systematicity (S): For any , in the agenda ˚ and profiles J and J 0 in J.˚/ n ,
if 2 J i , 2 J i for all i , then 2 F. J / , 2 F. J 0 / .
Unanimity entails that if all individuals accept a given judgment, then so should
the collective. Anonymity states all individuals should be treated equally by the
aggregation procedure. Neutrality is a symmetry requirement for propositions that
entail that all the issues in the agenda have to be treated equally. Independence says
that if a proposition is accepted by the same subgroup under two distinct profiles,
then that proposition should be accepted either under both or under neither profile.
These axioms express our intuition concerning the fairness of the procedure, for
example, (A) forces the procedure not to discriminate between individuals. This
fairness condition may be used to model the arguments of an agent for accepting
to solve conflicts by means of such a procedure. Systematicity is simply the
conjunction of Independence and Neutrality and has been introduced separately as
it is the condition used to prove the impossibility theorem in judgment aggregation.
The impossibility theorem of List and Pettit ( 2002 ) is stated as follows.

Theorem 1 (List and Pettit 2002 ). There are agendas ˚ such that there is no
aggregation procedure F W J.˚/ n ! P .˚/ that satisfies (A), (S), and collective
rationality.

In particular, for any aggregation procedure that satisfies (A) and (S), there is
a profile of judgment sets that returns an inconsistent outcome. The majority rule,
that we have seen in the examples of Sect. 2.2 , satisfies (A) and (S); accordingly, the
discursive dilemma shows a case of inconsistent aggregation.