~q, and we may take q = Op.
When the axioms would be morally desirable, then the derived contradiction would be morally desirable - but nobody can be asked to do the impossible. Hence the axioms are not morally desirable. This is a seemingly simple reasoning scheme, but destructive to the accepted view.
Rejection of the Arrow Reasonableness Claim (ARC)
Theorem A.2: For a reasonable society, the ARC is invalid.
Proof: Given AF, infeasible choices are not considered. Since ~a, apparently a is not feasible, and the Arrow constitution is not reasonable. So it is invalid that the axioms would be reasonable. Q.E.D.
Discussion: As we stated above, we have enlarged the commodity domain with constitutions, and hence the axiom of feasibility becomes a bit stronger. The extension itself is rather weak, since we only extend on consistency (and not empirical validity). But the conclusion is strong. No reasonable society in its right mind would want to accept Arrow’s axioms as its constitution. Supposedly at a chaotic Boston Tea Party a constitution c = a might be tried, but pretty soon rational people would see that they should make another constitution, for otherwise the situation will remain chaotic, and the Tea Party will not go down into history as a notable event.
Note that Arrow adopts feasibility, but also wants to impose infeasible conditions.
When Arrow’s axioms would be reasonable, then they would have to be consistent as well. However, they are inconsistent. Thus they are not reasonable. This seems a rather simple scheme of reasoning, but it destroys the impact of the Theorem.
For the axioms, there is the subtle difference between ‘reasonable’ and ‘seemingly reasonable when considered by itself’. The following is a good analogy. For a bicycle we want round wheels for when it rides. For a bicycle we also want square wheels, so that it does not fall when it stands still. But there are no round squares ! Ergo, conditions that seem reasonable by themselves, create something impossible and decidedly unreasonable when combined. To conclude ‘there is no good bike’ would however be absurd. Admittedly, it is a good teaching method to first convince students that something would be reasonable, and then have them derive a contradiction. As with the buying of a bad second-hand car, the students learn to be careful, and they learn a respect for science and the value of modesty. This teaching method however overshoots when people remain believers of the reasonableness of the assumptions - as apparently happened with the assumptions of Arrow’s Theorem. A paradox is only a seeming contradiction. Thus there must exist a system that we are willing to accept as the optimal one.
Many mathematicians have been sensitive to the distinction between ‘reasonable’ and ‘seemingly reasonable when considered by itself’, but the literature also abounds with instances where this distinction is not applied with sufficient care. Part of the accepted view thus is a case of bad communication of the incrowd with the larger public. (Given above quotes, the incrowd however might be small. Quis custodet custodes ?)