Back to 4. Deductively Definable Inductive Logics in Preferred Partitions What Logics of Induction are There?
John D. Norton

One More Illustration of Asymptotic Stability
Appendix to 4. Deductively Definable Inductive Logics in Preferred Partitions

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.
sine curve
sine curve

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 begin, let us consider just two:

a1 = x lies in (0, 0.5)
a2 = x lies in (0.5, 1)

These are two atoms of a partition that we will assume to be inductively adapted. As a result, they are equally probable outcomes. They both have probability 0.5.

P(a1) = P(a2) = 1/2

Areas on the figure opposite measure probability. The two rectangular areas corresponding to the atoms a1 and a2 are shown. This partition provides a rather coarse approximation to the distribution of interest.

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

b1 = x lies in (0, 0.2)
b2 = x lies in (0.2, 0.3)
b3 = x lies in (0.3, 0.5)
b4 = x lies in (0.5, 0.75)
b5 = 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 a2 = (b4 or b5) from its value of 1/2 to

P(a2) = P(b4) + P(b5) = 2/5
since curve
sine curve 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 a2 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 a2 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.

The atoms are

c1 = x lies in 0 to 0.09
c2 = x lies in 0.09 to 0.165
c3 = x lies in 0.165 to 0.235
c4 = x lies in 0.235 to 0.300
c5 = x lies in 0.300 to 0.370
c6 = x lies in 0.370 to 0.450
c7 = x lies in 0.450 to 0.545
c8 = x lies in 0.545 to 0.695
c9 = x lies in 0.695 to 0.880
c10 = x lies in 0.880 to 1

Back to 4. Deductively Definable Inductive Logics in Preferred Partitions What Logics of Induction are There?
John D. Norton