The missing ingredient in a “small world” is roughly the continuity conditions that Jaynes calls “qualitative correspondence with common sense” in Chapter 2 of PT:TLoS. In terms of model theory, adding the coin means that the model now “has enough points”.
Here is one way to think about it: One of the consequences of Cox’s theorem is that
P(X) = 1 - P(~X)
Suppose you decided to graph P(X) against P(~X). But in a small world, there are only a finite number of events you can substitute-in for X. So your graph is just a finite set of colinear points—not a line. Many continuous functions can be made to fit those points. Add a coin to your world, and you can interpolate an event between any two events in your world. You get a dense infinity of points between 0 and 1. And that is all you need. Only a single unique function (y = 1 - x) can be fit to this data.
Interesting. Do they give a good intuition for why this change occurs?
The missing ingredient in a “small world” is roughly the continuity conditions that Jaynes calls “qualitative correspondence with common sense” in Chapter 2 of PT:TLoS. In terms of model theory, adding the coin means that the model now “has enough points”.
Here is one way to think about it: One of the consequences of Cox’s theorem is that
P(X) = 1 - P(~X)
Suppose you decided to graph P(X) against P(~X). But in a small world, there are only a finite number of events you can substitute-in for X. So your graph is just a finite set of colinear points—not a line. Many continuous functions can be made to fit those points. Add a coin to your world, and you can interpolate an event between any two events in your world. You get a dense infinity of points between 0 and 1. And that is all you need. Only a single unique function (y = 1 - x) can be fit to this data.
That was hand-waving, but I hope it helped.
It did help. I was expecting something like this. I still have to go look at the paper for some more clarification.