Selection of the culprit axiom.
The selection of the culprit axiom is straightforward. We order the axioms by preference, for example AD > AWP > AU > APDM. From ~a, we conclude that we have to drop one of the axioms. We drop the least preferred one. My discussion on Condorcet’s example should generate support for the rejection of APDM. Basically though, scientists can only advise on preferences, and the proper decision is up to the body politic.
Lemma A.II: If all agents have a > APDM then, with AWP, society has [AU, AWP, AD] > APDM. Note: here [x, y, z] means the unordered set.
Proof: obvious.
Discussion: When all people put AU, AWP and AD in any individual order, but all would have APDM below these, then society can reject APDM unanimously. In fact, the condition AU might as well be regarded as part of the definition of a SWF-GM, and similarly, AWP could as well be regarded as part of the definition of the notion of collective preference. So the real choice concerns AD and APDM, or between dictatorship or not.Here a selfish dictator and his associates would have ¬AD > APDM > AD. The Jorgenson quote suggests his preference for a benevolent and non-selfish dictatorship, but, also since such dictatorships tend to turn sour, my impression is that he would eventually be an associate of a real dictator. Most likely, he did not understand the situation when the quote was printed.
Note that ordering the axioms means that the deontic predicate O is not homogeneous. This means that deontic logic may be more related to preference theory than deontic theorists think.
Examples of consistent constitutions
Consistent constitutions violate one of the axioms of Arrow’s Theorem. Violating one of these axioms is to be considered useful for reasonableness and morality, rather than the reverse. (That is what we proved above.)
One general feature is a Status Quo that persists when there are ties.
One example already has been mentioned in the discussion of the Condorcet problem. With majority voting, a cycle means indifference, and there are various ways to solve ties. One possible solution is the persistence of the Status Quo.