## Notes to Belief Merging and Judgment Aggregation

1. The set *At* of
atomic propositions is assumed to be ordered, i.e., an order such that
*p* comes before *q* and *q* comes before
*r*. Thus \((1,0,0)\) represents an interpretation that assigns
true to *p* and false to *q* and *r*.

2. Technically, *d* is a
pseudo-distance as the triangular inequality \((\forall \omega,
\omega', \omega'' \in W\), \(d(\omega, \omega') \le d(\omega,
\omega'') + d(\omega'', \omega'))\) is not required to hold. Konieczny
and Pino Pérez (2002) showed that, if *d* satisfies the
triangular inequality, then \(D_{\Sigma}\) satisfies the
*iteration* axiom:

- (\(\IC_{\textit{It}}\))
- If \(\varphi \vdash \IC\) then \(\exists n \Delta_{\IC}^{n} (E,\varphi) \vdash\varphi\)

3. Similarly as the
*minimax* rules in decision theory, the *max* operators
aims at selecting the outcome that minimizes the worst disagreement.
Such operators do not satisfy all the postulates for merging, and
hence are called *quasi-merging* operators (Konieczny and Pino
Pérez 2002).

4. Belief revision and belief
merging have deep connections. In particular, merging operators can be
seen as a generalization of revision operators (Konieczny and Pino
Pérez 2002). More precisely, if \(\Delta\) is an *IC*
merging operator, then the operator \(*\) defined as \(\varphi * \IC=
\Delta_{\IC}(\varphi)\) is an AGM revision operator (Alchourrón
*et al.* 1985). The other direction (whether an AGM revision
operator defines an *IC* merging operator) has a more complex
answer, which can be summarized by saying that a revision operator can
define a merging *IC* operator if and only if the revision
operator is defined in terms of distances.

5. The majority preservation property ensures that, if the proposition-wise majority vote is a consistent judgment set, the judgment aggreegation procedures must select that same judgment set (Lang et al. 2017).