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ERRATA: Paper [EndrissHBCOMSOC2016]
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U. Endriss. Judgment Aggregation. In F. Brandt, V. Conitzer, U. Endriss,
J. Lang, and A.D. Procaccia (editors), Handbook of Computational Social
Choice, Cambridge University Press, 2016.
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Lemma 1 states several correspondences between structural properties of
the space of winning coalitions on the one hand and axioms satisfied by
an independent aggregator on the other. Unfortunately, the structural
properties corresponding to complement-freeness and completeness are
stated incorrectly. The correct statements are as follows:
(v) f is complement-free if and only if C \not\in W_phi or complement(C)
\not\in W_{~phi} for all coalitions C and formulas phi.
(vi) f is complete if and only if C \in W_phi or complement(C) \in
W_{~phi} for all coalitions C and formulas phi.
Thus, in both cases the final W_phi should be replaced by W_{~phi}.
A simple counterexample for the false claims made in the book is the
aggregator (defined for an odd number of agents) that accepts a positive
formula whenever it is accepted by an odd number of agents and that
accepts a negative formula whenever it is accepted by an even number of
agents. The space of winning coalitions corresponding to this aggregator
satisfies the properties stated in the book, yet this aggregator is
neither complement-free nor complete.
The claims made in the book are correct once we also assume neutrality
of the aggregator, as we then get W_phi = W_{~phi}. Fortunately, all
later uses of parts (v) and (vi) of Lemma 1 are in contexts where
neutrality is assumed as well, so the correctness of later results is
not affected by this mistake.
Thanks to Adriaan de Vries for noticing the mistake.
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