\section{Judgment aggregation}

In judgment aggregation agents are required to express judgments (in the form of yes/no or, equivalently, 1/0) over \textit{premises} and \textit{conclusion}. As in \cite{PST08b}, to represent the distinction between premise and conclusion in our language,  
we distinguish between premise variables $a,b,c,p,q \ldots$, and conclusion variable $x$.

In the hiring example, ``$X$ is good at teaching" is \textit{premise}~$a$, and ``$X$ is good at research" is \textit{premise} $b$. The decision rule can be formally expressed by the rule $(a\wedge b)\leftrightarrow x$, where $x$ is the conclusion about hiring $X$. Each member of the department expresses her judgment on the propositions $a$, $b$ and $x$ such that the rule $(a\wedge b)\leftrightarrow x$ is satisfied.

Suppose the three professors in the department make their judgments according to Table~\ref{tab:hire}. Each member expresses a consistent opinion, i.e. she says yes to~$x$ if and only if she says yes to both $a$ and $b$. However, propositionwise majority voting (consisting in the separate aggregation of the votes for each proposition $a$, $b$ and $x$ via majority rule) results in a majority for $a$ and $b$ and yet a majority for $\neg x$. This is an inconsistent collective result, in the sense that $\{a, b,\neg x,(a\wedge b)\leftrightarrow x\}$ is inconsistent in propositional logic. The paradox lies in the fact that majority voting can lead a group of rational agents to endorse an irrational collective judgment. The literature on judgment aggregation refers to such problems as the \textit{doctrinal paradox} (or \textit{discursive dilemma}).


The relevance of such aggregation problems applies to all situations in which individual binary evaluations need to be combined into a group decision. Furthermore, the problem of aggregating individual judgments is not restricted to majority voting, but it applies to all aggregation procedures satisfying some seemingly desirable conditions. For an overview, the reader is referred to \cite{ListPuppe2009}.


Two ways to avoid the inconsistency are the \textit{premise-based procedure} (PBP) and the \textit{conclusion-based procedure} (CBP) \cite{Pettit2001,Chap02}. According to the PBP, each agent votes on each premise. The conclusion is then inferred from the rule $(a \wedge b) \leftrightarrow x$ and from the judgment of the majority of the group on $a$ and $b$. If  the professors of the example followed the premise-based procedure, the lecturer would be hired.


Because in PBP the collective judgment on the conclusion is derived from the individual judgments on the premises, it can happen that PBP violates a unanimous vote on the conclusion.  In \cite{Nehring2005} Nehring presents a variation on the discursive dilemma, which he calls the Paretian dilemma. In his example, a three-judges court has

\begin{quote}
to decide whether a defendant has to pay damages to the plaintiff. Legal doctrine requires that damages are due if and only if the following three premises are established: 1) the defendant had a duty to take care, 2) the defendant behaved negligently, 3) his negligence caused damage to the plaintiff. (\cite{Nehring2005}, p.1)
\end{quote}

Suppose that the judges vote as in Table \ref{tab:paretian}.



\begin{table}[h]
\begin{centering}
\begin{tabular}{|c | c c c | c |}\hline
Agenda      &  $a$  & $b$  &  $c$  &$x=(a\wedge b \wedge c)$  \\ \hline\hline
   Judge A  & 1       &  1               & 0                 &  0\\ \hline
  Judge B    &  0              &  1              &  1              &  0\\ \hline
   Judge C    & 1             & 0            &   1                &  0\\ \hline
  Majority &  1              &  1            &   1                &  0 \\ \hline
\end{tabular}

\end{centering}
\vspace{0.2cm}
\caption{Paretian dilemma. Premises: $a$~=~duty, $b$ = negligence., $c$ = causation. Conclusion: $x=~(a\wedge b \wedge c)$ = damages.}
\label{tab:paretian}
\vspace{-0.8cm}
\end{table}



The Paretian dilemma is disturbing because, if the judges would follow PBP, they would condemn the defendant to pay damages contradicting the \textit{unanimous} belief of the court that the defendant is \textit{not} liable.

A CBP would not lead to such a unanimity violation. According to CBP, the judges decide privately on $a$ and $b$ and only express their opinions on $x$ publicly. The judgement of the group is then inferred from applying the majority rule to the agents' judgments on $x$. The defendant will be declared liable if and only if a majority of the judges actually believes that she is liable. In the example, contrary to PBP, the application of CBP would free the defendant. However, no reasons for the court decision could be supplied.

Unlike PBP \cite{Mongin2006,DietrichMongin2007}, CBP did not receive much attention in the literature. Here we aim at filling this gap. We propose a procedure that attempts to overcome the major limit of CBP, that is the lack of reasons supporting the decision.

