
\section{Introduction}
 
Judgment aggregation \cite{KS86,KS93,List2007} studies the aggregation of individual judgments of small groups such as expert panels, legal courts, boards and councils. We talk about judgment aggregation whenever a group of individuals needs to make a collective decision on a finite set of issues, and these propositions are logically connected. The propositions are of two kinds: \textit{premises} and a \textit{conclusion}. The first serve as supporting reasons to derive a certain judgment on the conclusion. If, for example \cite{BovensRabinowicz}, your department has to hire a new lecturer and the decision rule is such that a candidate $X$ will be hired only if the candidate is good at teaching and good at research, we will say that ``hiring $X$" is the conclusion while ``good at teaching" and ``good at research" are the premises. 

How shall we derive a group decision given the individuals' opinions on premises and conclusion? It is assumed that each individual expresses yes/no opinions on the propositions while respecting the logical relations. If we now define the group opinion as the majority view on the issues, it turns out that the collectivity may have to endorse an inconsistent position. This means that your department may have to face a situation in which a majority does not deem $X$ a good candidate. However, it will not be possible to provide reasons for this as a majority of people agrees that $X$ is actually good at teaching and (another) majority deems $X$ to be good at research. An example of such situation is presented in Table \ref{tab:hire}.

\begin{table}[h]
\begin{centering}
\begin{tabular}{ |l|c|c|c|}\hline
                           & \small $a=$ $X$ is good at teaching & \small $b=$ $X$ is good at research   & \small $x=$ hire $X$ \\ \hline
\small prof. A    & \small \textit{yes}                   & \small \textit{no}                        &\small \textit{no} \\\hline
 \small prof. B    &\small \textit{yes}                   &\small  \textit{yes}                       &\small \textit{yes}\\ \hline
     \small prof. C &\small \textit{no}                  &\small  \textit{yes}                       &\small \textit{no} \\ \hline
 \small Majority  &\small \textit{yes}                 &\small  \textit{yes}                       &\small \textit{no}  \\ \hline

\end{tabular}

\end{centering}
\vspace{0.2cm}
\caption{\small Hiring committee example. The candidate $X$ is hired if and only if $X$ is good at teaching and $X$ is good at research.}
\label{tab:hire}
\vspace{-0.8cm}
\end{table}

The problem is avoided if we decide to let the majority vote on the premises to dictate the final decision on the hiring process, or if the agents express their judgments only on the conclusion. Unlike the aggregation procedure on the premises \cite{Mongin2006,DietrichMongin2007}, the aggregation on the conclusion has not been throughly investigated.

We claim that in many decision problems the conclusion is more relevant than the reasons for it. When deciding which candidate to hire in your department, you may be more concerned of which new colleague you will have in your department than of the reasons for choosing her. Considering only the individual judgments on the conclusions has also the advantage that it is a strategy-proof procedure. 
 The same does not hold when you aggregate on the premises.

The problem this paper addresses is how a group can make decisions on the conclusion while providing reasons in support of the collective conclusion. Our procedure prioritizes the individual judgments on the conclusion and outputs sets of premises that support the collective decision. 

The paper is structured as follows: in Section~2 we present the problem of judgment aggregation. Section 3 is devoted to our formal framework, and in Section 4 we prove some results about our procedure. Section 5 relates our approach with existing work and, finally, Section~6 concludes the paper and outlines directions for future work.



