A General Approach to Aggregation Problems

We discuss a general approach to judgement aggregation based on lattice theory. Agents choose elements of a lattice, and an aggregation procedure yields a ‘social choice’ based on the individual choices. Settings traditionally studied in social choice theory can be thought of as implicational systems, and lattice theory provides an abstraction of such systems. In fact, traditionally studied settings correspond to certain atomistic lattices in our framework. Our aim is to systematically investigate how properties of a given lattice induce constraints on aggregation procedures that lead up to impossibility theorems. We allow for non-atomistic lattices and this raises some subtle issues. We will discuss how well our framework fits in with the traditional approaches to social choice theory, in particular with respect to generalizations of some of the well known axioms, and go on prove an impossibility result that highlights the role of certain lattice theoretical properties. These properties reflect some of the traditional axioms or other aspects of traditional systems.