Judgment aggregation: a general theory of collective decisions

aggregation theory. Most theorems of the literature on judgment aggregation can be stated in an abstract and logic-free model, which is indeed done by di¤erent authors. There are many variants of the abstract model; they involve no underlying logic L but in one form or another an agenda. Let me here present one variant, Dokow and Holzman’s binary evaluations model (other variants being Wilson’s 1976 model and Nehring and Puppe’s 2002 property space model). Let K be a non-empty set of issues, let an evaluation be a function v : K ! f0; 1g (assigning a position, 0 or 1, to each issue), and let E be a non-empty set of ‘admissible’ evaluations. One may then look for aggregation functions f : En ! E, mapping pro…les (v1; :::; vn) of admissible individual evaluations to admissible collective evaluations (where n is the number of individuals). To see the connection to judgment aggregation, let K be the set of non-negated propositions in the agenda X (so that K consists of exactly one member of each proposition-negation pair p;:p in X), identify every evaluation v with the opinionated judgment set A X that contains each p with v(p) = 1 and each :p with v(p) = 0, and call an evaluation v admissible if the corresponding opinionated judgment set is consistent (hence fully rational). Under these identi…cations, abstract aggregation functions f : En ! E correspond uniquely to judgment aggregation functions that have universal domain and generate fully rational collective judgment sets. Adaptations of this abstract model can also cope with rationality violations (on the individual or collective level), in particular with ‘abstentions’(corresponding to incompleteness of judgment sets). It is worth noting that a loss of information is involved in moving from the judgment aggregation model to an abstract model. It becomes impossible to (i) refer to propositions outside the agenda, and (ii) refer to a proposition’s syntactic form, e.g. distinguish atomic from compound propositions. (The loss of the syntax actually also happens when moving to a semantic model.) This informational loss detaches the model to some extent from philosophical questions that one may raise about judgments and their aggregation; and one cannot anymore state conditions or theorems that draw on additional information (such as Mongin’s (forthcoming) independence condition restricted to atomic propositions). However, most conditions and theorems proved up to now do not draw on additional information, and from this perspective one might regard the informational slimness of abstract models as an appealing feature. General attitude aggregation. We are back now to the logic-based framework, with a language L and an agenda X L of propositions under consideration. The standard model allows exactly two attitudes on each proposition in X: acceptance or rejection. But belief need not be a binary on-o¤ a¤air. This calls for a generalisation. Following Dietrich and List (forthcoming), consider an arbitrary non-empty set V of possible attitudes: in the binary case V = f0; 1g, in the case of probabilistic attitudes V = [0; 1], in the case of Spohnian ranks V = f0; 1; :::g [ f1g, and so on. Let F be a set of functions f : L ! V, the valuation functions; in the binary case these are the truth functions, in the probabilistic case the probability functions (or perhaps the Dempster-Schaefer belief functions or the capacities),5 in the case of Spohnian Probability functions (that is, …ninitely addive ones) require a Bollean algebra as their domain. To ensure L forms an (abstract) Boolean algebra (modulo logical equivalence), I here make a small additional richness assumption on the language: conjunctions can be formed (i.e. p; q 2 L implies