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We define individual judgment sets as follows.
Definition 2.2.
A judgment set J on an agenda ˚ is a subset of the agenda J ˚ .
We call a judgment set J complete if A 2 J or :A 2 J , for all formulas A in the
agenda ˚ , and consistent if there exists an assignment that makes all formulas in J
true, namely we assume the notion of consistency that is familiar from propositional
logic.
These constraints model a notion of rationality of individuals, i.e. individuals
express judgment sets that are rational in the sense that they respect the rules of
(classical) logic. 2
Denote with J.˚/ the set of all complete consistent subsets of the agenda ˚ , namely
J.˚/ denotes the set of all possible rational judgment sets on the agenda ˚ .Given
aset N Df1;:::;ng of individuals , denote with J D .J 1 ;:::;J n / a profile of
judgment sets, one for each individual. A profile is intuitively a list of all the
judgments of the agents involved in the collective decision at issue. For example, the
profile involved in the paradoxical example of the previous section is the following:
.fA;A ! B;Bg; fA; :.A ! B/;:Bg; f:A;A ! B;:Bg/ .
We can now introduce the concept of aggregation procedure that is, mathemat-
ically, a function. The domain of the aggregation procedure is given by J.˚/ n ,
namely, the set of all possible profiles of individual judgments.
Definition 2.3. An aggregation procedure for agenda ˚ and a set of n individuals
is a function F W J.˚/ n ! P .˚/ .
An aggregation procedure maps any profile of individual judgment sets to a
single collective judgment set (an element of the powerset of ˚ ). Given the
definition of the domain of the aggregation procedure, the framework presupposes
individual rationality : all individual judgment sets are complete and consistent.
Note that we did not yet put any constraint on the collective judgment set, i.e. the
result of aggregation, so that at this point the procedure may return an inconsistent
set of judgments. This is motivated by our intention to study both consistent and
inconsistent collective outcomes. For example, in the doctrinal paradox of the
previous section, the majority rule maps the profile of individual judgments into
an inconsistent set:
.fA;A ! B;Bg; fA; :.A ! B/;:Bg; f:A;A ! B;:Bg/ 7!fA;A ! B;:Bg
7!
fA;A ! B;:Bg
2 Of course this may be a descriptively inadequate assumption. However, on the one hand, these
requirements are to be understood in a normative way, e.g. we exclude that a representative would
vote for a proposal A and a proposal :A at the same time. Moreover, the agenda may contain
very simple logical propositions: as we shall see, it is sufficient to assume very minimal reasoning
capacity to get the paradoxical outcomes.
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