Belief Merging and Judgment Aggregation in Fuzzy Setting

Social choice theory defines “preference aggregation” as forming collective preferences over a given set of alternatives. Likewise, “judgment aggregation” pertains to forming collective judgments on a given set of logically interrelated propositions. This paper extends beyond classical propositional logic into the realm of general multivalued logic, so that we can handle realistic collective decision problems (see Dietrich and List [1, 2], List [3], Beg and Butt [4], and Manzini and Mariotti [5]). List and Pettit [6, 7] were the first to give an axiomatic treatment to the problem associated with judgment aggregation. In their classic example, a set of propositions is expressed in propositional calculus as {p, q, p∩ q}. The set L = {(0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 1)} consists of all assignments of 0 or 1 to the propositions in {p, q, p ∩ q} that are logically consistent. A procedure for n judges to decide on the truthfulness of each proposition in {p, q, p∩q}amounts to an aggregator that maps Ln → L. The “Doctrinal Paradox” illustrates that proposition-wise majority rule leads to inconsistent collective decisions. This paradox has made the literature on “judgment aggregation” grow appreciably. Most of the discussions on this paradox have been in the domain of social choice theory, and a number of “(im)possibility theorems,” similar to those of Arrow [8] and Sen [9] have been proved. In fact, these theorems show that there cannot exist any judgment aggregation procedure that simultaneously satisfies certain minimal consistency requirements (see Dietrich [10]). List and Pettit [6] have shown that the majority rule is but one member of a class of aggregation procedures that fails to ensure consistency in the set of collective judgments. Van Hees [11] has further generalized the paradox by showing that there is even a larger class of aggregation procedures for which this is true. The aim of this paper is to resolve the paradox and also to illustrate optimal judgment aggregation. We abandon the assumption that individual and collective beliefs necessarily have a binary nature (true or false) and so our analysis is in a fuzzy logic framework. Pigozzi [12] discusses a possibility result in binary logic in which the paradox is avoided at the price of “indecision.” Distance-based aggregation procedures like that of Pigozzi [12] often result in dictatorship. Accordingly, aggregation procedures in fuzzy logic can help us make the collective judgment set more “democratic” in nature. In this paper we try to give the present literature in this area a more realistic touch by using fuzzy logic. The structure of the paper is as follows: Section 2 illustrates an example of the “Doctrinal Paradox” and its reformulation in a fuzzy setting. Section 3 illustrates how the paradox is resolved in a fuzzy framework to find optimal fuzzy aggregation functions. Section 4 further elaborates on the results in Section 3 to present a democratic fuzzy aggregation function. Section 5 presents the entire previous discussion in utility maximization framework.