Actually, the way you formulated it, completeness seems quite clear. If completeness is violated then there are A and B s.t.A<B and B<A which is an obvious money-pump. It is transitivity that is suspect: in order to make the Dutch book argument, you need to assume the agent would agree to switch between A and B s.t. neither A<B nor B<A. On the other hand, we could have used ≤ as the basic relation and defined A<B as ”A≤B and not B≤A.” In this version, transitivity is “clear” (assuming appropriate semantics) but completeness (i.e. the claim that for any A and B, either A≤B or B≤A) isn’t.
Btw, what would be an example of a relation that satisfies the other axioms but isn’t coherently extensible?
Actually, the way you formulated it, completeness seems quite clear. If completeness is violated then there are A and B s.t.A<B and B<A which is an obvious money-pump. It is transitivity that is suspect: in order to make the Dutch book argument, you need to assume the agent would agree to switch between A and B s.t. neither A<B nor B<A. On the other hand, we could have used ≤ as the basic relation and defined A<B as ”A≤B and not B≤A.” In this version, transitivity is “clear” (assuming appropriate semantics) but completeness (i.e. the claim that for any A and B, either A≤B or B≤A) isn’t.
Btw, what would be an example of a relation that satisfies the other axioms but isn’t coherently extensible?