Back to 4. Deductively Definable Inductive Logics in Preferred Partitions  What Logics of
Induction are There? John D. Norton 
Here's another illustration that shows converging changes in probabilities as well as
refinements that are of direct relevance to the content.
Consider some system in which we have a
probability distribution p(x) of finding
an outcome in the range of x from 0 to 1. It is plotted opposite. The particular system doesn't really matter. For concreteness, we could imagine that x is the time of year, with x=0 to x=1 spanning an entire year, and the outcome of interest is the time of the day of maximum precipitation. Since the curve represents a probability density, the probability is given by the area under the curve, which, in total, will be unity. 

A continuous probability distribution for x
lies outside the scope of the analysis here. We are considering
finite sets of propositions only. So we need to approximate this
distribution by finite sets of propositions. 

To improve the fit of our
logic to the distribution of interest, we form a disjunctive
refinement. We will posit that this new refinement is also
inductively adapted. As a result, its atoms will still be
equiprobable. The probability of each atom will be represented by a rectangular area in the figure. That means that range of x values associated with each atoms must vary if the probabilities of the atoms are to come closer to the curve. For example, the atom associated with the curve's peak must correspond to a narrower range of x values; and the atom associated with the trough must correspond to a wider range of x values. The refinement displayed opposite is b_{1} = x lies in (0, 0.2) b_{2} = x lies in (0.2, 0.3) b_{3} = x lies in (0.3, 0.5) b_{4} = x lies in (0.5, 0.75) b_{5} = x lies in (0.75, 1) What is important for our example is that each of these atoms are equiprobable, so that each has probability 1/5. As a result, in this new refinement, we have now altered the probability of a_{2} = (b_{4} or b_{5}) from its value of 1/2 to P(a_{2}) = P(b_{4}) + P(b_{5}) = 2/5 

By continuing this process of refinement, we can
come as close as we like to the target probability distribution. What this illustrates is how successive refinements can change the probabilities of the propositions; we have changed the probability of a_{2} in the last refinement. However as long as the refinements are crafted to approximate the original distribution more and more closely, we can be assured that the probability of a_{2} will stabilize asymptotically towards the probability assigned by the distribution for outcomes in the range x=0.5 to x=1. This stabilization is what asymptotic stability requires. Here is another partition that gives a picture of how close we can
come. This is the partition that arises if we want ten equiprobable
atoms. 
Back to 4. Deductively Definable Inductive Logics in Preferred Partitions  What Logics of
Induction are There? John D. Norton 