The continuity of p → preference space is an additional property you must assert. Like the total ordering, you need to specify that this function is in fact of this form, since it isn’t determined by the premises
A monotonically increasing function from p → preference space will preserve the total order just fine and meet the end point criteria, even if discontinuous. And since those are the two properties you currently require you need to add another property to the theory if you want to eliminate discontinuity.
Intuitively, you probably want to do this anyway, since you haven’t said anything about how decisions are made except by listing basic decision pathologies and ruling them out. First was cycles (although a total order is arguably over-kill for this one). The second was incorporating probability (which necessitated an end-point condition—A >= pA + (1-p)B >= B). Now it’s time to add a new condition of continuity of the p mapping, based on the intuition that immeasurably small changes in probability should not cause measurable changes in decisions. But nothing you’ve laid out so far excludes such an agent.
(Good point. My error there. But do note, though, while b implies c, c definitely does not imply b.)
Edited to add: The intuition about measureable changes caused by immeasurable probability shifts removes all but point-wise discontinuitites. Those you can remove via adding something like c, i.e. probabilities are real numbers or the like.
The continuity of p → preference space is an additional property you must assert. Like the total ordering, you need to specify that this function is in fact of this form, since it isn’t determined by the premises
A monotonically increasing function from p → preference space will preserve the total order just fine and meet the end point criteria, even if discontinuous. And since those are the two properties you currently require you need to add another property to the theory if you want to eliminate discontinuity.
Intuitively, you probably want to do this anyway, since you haven’t said anything about how decisions are made except by listing basic decision pathologies and ruling them out. First was cycles (although a total order is arguably over-kill for this one). The second was incorporating probability (which necessitated an end-point condition—A >= pA + (1-p)B >= B). Now it’s time to add a new condition of continuity of the p mapping, based on the intuition that immeasurably small changes in probability should not cause measurable changes in decisions. But nothing you’ve laid out so far excludes such an agent.
(Good point. My error there. But do note, though, while b implies c, c definitely does not imply b.)
Edited to add: The intuition about measureable changes caused by immeasurable probability shifts removes all but point-wise discontinuitites. Those you can remove via adding something like c, i.e. probabilities are real numbers or the like.