 If belief revision focuses on how one base changes following the addition of a new piece of information, belief fusion studies how to aggregate several different and potentially conflicting bases (like, for instance, different experts’ opinions, several databases, information coming from different sources etc.) to obtain a consistent base. Different approaches have been proposed in the literature. Here we briefly mention combination and arbitration before moving to the merging operators as defined by Konieczny and Pino Pérez, which have been applied to judgment aggregation. The first approach to the problem of dealing with aggregating different and possibly inconsistent databases (Baral et al. 1991; Baral et al. 1992) built on Ginsberg’s idea of considering maximally consistent subsets when facing an inconsistent theory (Ginsberg 1986), such as the one that may result from the union of the information coming from several self-consistent (but conflicting with one another) agents. Combination operators take the union of the knowledge bases (a finite set of logical formulas) and, if the union is logically inconsistent, select some maximal consistent subsets. The logical properties of such combination operators have been investigated in (Konieczny 2000) and compared to merging operators as defined in (Konieczny and Pino Pérez 1998, 1999). There are several differences between combining and merging knowledge bases. One difference is that the method by Baral et al. (1991, 1992) is syntax-dependent while merging operators obey the principle of irrelevance of syntax. According to the principle of irrelevance of syntax, an operation on two equivalent knowledge bases should return two equivalent knowledge bases. For instance, \(K_1 = \{a, b\}\) and \(K_{2}= \{a \wedge b\}\) have the same logical consequences. So, if \(K_3 = \{\neg b\}\), merging \(K_1\) with \(K_3\) or merging \(K_2\) with \(K_3\) will give two equivalent knowledge bases. On the other hand, the combination of \(K_1\) and \(K_3\) may not be logically equivalent to the combination of \(K_2\) and \(K_3\). Let \(E_1 = K_1 \bigsqcup K_3\) (\(\bigsqcup\) is the union on multi-set) and \(E_2 = K_2 \bigsqcup K_3\). The maximal consistent subsets of \(E_1\) are \(\{a, b\}\) and \(\{a, \neg b\}\), and those of \(E_2\) are \(\{a \wedge b\}\) and \(\{\neg b\}\). So each maximal consistent subset of \(E_1\) implies \(a\), but this is not the case for all maximal consistent subsets of \(E_2\) (example from Konieczny 2000). Another difference is that when combination operators are used, the information about the source of the knowledge bases is ignored. This means that, unlike merging, combination operators cannot take into account cardinality considerations. Suppose, for example, that four medical experts advise on the effectiveness of four vaccines for adults over 65 years old. Let the propositions \(a, b, c, d\) stand for “Vaccine A (respectively, B, C, D) is effective in over-65s”. If two experts agree that vaccines A and B are effective in over-65s, one expert esteems that vaccine D (but not vaccine A) is effective and, finally, the last expert agrees with the first two that vaccine A is effective and that, if B is effective in over-65s, so is vaccine C too, we can represent the four experts opinions as four knowledge bases: \(K_1 = K_{2}= \{a, b\}\), \(K_{3}= \{\neg a, d\}\) and \(K_4 = \{a, b\rightarrow c\}\). The union of these four bases is \(\{a,\neg a, b, b\rightarrow c, d\}\), which is clearly logically inconsistent. Considering maximal consistent subsets is one way to avoid inconsistency while retaining as much information as possible. In this example, the two maximal consistent subsets are: \(\{a, b,b\rightarrow c, d\}\) and \(\{\neg a, b, b\rightarrow c, d\}\). This means that we cannot decide whether to accept a or \(\neg a\). However, a majority of knowledge bases contained a, and only one base contained \(\neg a\). It seems intuitive that a should be in the resulting knowledge base as long as all knowledge bases are treated equally. If, for whatever reason, \(K_{3}\) is more trustworthy than the other knowledge bases, then we may prefer a combined base in which \(\neg a\) is accepted.