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John D. Norton

7. A Limit Theorem

There is a familiar limit theorem in Bayesian confirmation theory. It arises when we have an hypothesis H that deductively entails n items of evidence, E1, E2, ... ,En. If the prior probability P(H)>0, it is fairly easy to show that

P( En | E1 & E2 & ... & En-1 ) approaches 1 for large n.

This is a strong and even astonishing result. Pick some hypothesis that has infinitely many deductive consequences. While we may not know if the hypothesis is true, assume that we find it reasonable enough to give it some non-zero prior probability, no matter how small. That is enough to make us inductively audacious and even brazen about its consequences. While we learn the truth of some of the consequences, we will become successively more sure of the truth of the next as yet untested consequence. As this process continues we become more sure without limit; we need only to have finitely many successes before we become arbitrarily sure that the next untested consequence will prove true.

A fact about the deductive structure ...

This theorem and the generalization to be developed here express a fact about the deductive relations among H and E1, E2, ... , En. To see that fact, we need to define some auxiliary propositions:

F1 = E1
F2 = E1 & E2
F3 = E1 & E2 & E3
Fn = E1 & E2 & ... & En

Since H entails each of E1, E2, ... ,En, it follows that

H entails Fn entails Fn-1 entails ... entails F2 entails F1

This means that the number of atoms is non-decreasing as we proceed along the sequence

#H < #Fn < #Fn-1 < ... < #F2 < #F1

This is combined with a simple relation among the ratios of these atoms:

(#H/N) = (#H/#Fn) x (#Fn/#Fn-1) x (#Fn-1/#Fn-2) x ... x (#F3/#F2) x (#F2/#F1) x (#F1/N)

From this last relation, we arrive at the important fact about the deductive structure, if we consider what happens as the number n of propositions Fn grows. On the right hand side we are multiplying together a growing number of factors of the form (#Fi/#Fi-1). Each of these factors is less than or equal to unity. So their product will approach zero, in general, thereby forcing (#H/N) to approach zero as well.

If, however, we also require that (#H/N) approaches a stable value greater than zero, then we must have the one case in which the growing product on the right hand side does not approach zero. That case arises only if the ratio of atoms (#Fn/#Fn-1) approaches 1.

If (#H/N) approaches a stable value greater than zero, then #Fn approaches #Fn-1 for large n.

... re-expressed in the inductive logic

The basic fact is that the propositions Fn and Fn-1 approach one another in the natural measure of their atom counts. Any logic of induction that measures support through these these atoms counts must then reflect that approach in a corresponding approach of strengths of support. That will be the inductive limit theorem. This theorem appears in the inductive logics under consideration here. They are inductive logics that are asymptotically stable and deductively definable in preferred partitions. To find the theorem, let us consider the above fact in the context of the preferred partitions of this logic.

First we will assume that H has a non-zero generalized prior. That means that the limiting value of [ H | Ω ] as we proceed to larger partitions must differ from the minimal support [ contradiction | Ω ] that a contradiction is accorded by the background Ω. Closer examination of the continuity of the logic (given in the full paper) shows that this means that the ratio #H/N must approach a value greater than zero.

It now follows that #Fn approaches #Fn-1. If the inductive logic obeys the natural sense of continuity of the full paper, that means that

[ Fn | Fn-1 ] approaches [ Fn-1 | Fn-1 ] for large n.

However we expect that anything gets certain support from itself, so [ Fn-1 | Fn-1 ] equals "certainty," which is the maximum level of support. Combining, we have the most general form of the limit theorem:

If [ H | Ω ] approaches a stable value that exceeds the minimal value,
then [ Fn | Fn-1 ] approaches certainty for large n.

This form of the limit theorem does not quite match the original Bayesian version. That is because the Bayesian system conforms to a particular property of some inductive logics, "Narrowness." As defined earlier, that property tells us that [A&B|B] = [A|B]. Applying it to [ Fn | Fn-1 ] we find

[ Fn | Fn-1 ]
= [ E1 & E2 & ... &En-1 & En | E1 & E2 & ... & En-1 ]
= [ En | E1 & E2 & ... & En-1 ]

Thus the limit theorem now becomes

If [ H | Ω ] approaches a stable values that exceeds the minimal value in a narrow logic,
then [ En | E1 & E2 & ... & En-1 ] approaches certainty for large n.


It is quite informative to illustrate the limit process in a Venn diagram.

Below is the process showing the sequence of propositions E1, E2, ... ,En. The essential point about the sequence is that the individual terms of the sequence of conjunctions E1, E1 & E2, E1 & E2 & E3,... must always contain H. That means they must approach a limit proposition entailed by H, not shown in the figure below, but shown in the next figure at right.
That limiting proposition (which may never be realized in a finite algebra) is denoted by F below. It is more easily visualized as the limit of the sequence of conjunctions F1, F2, ... ,Fn.

The essential deductive fact expressed in the limit theorem is that the successive terms of the sequence of conjunctions approach one another arbitrarily closely in their atom counts as that limit is approached.

Since the degrees of support are defined in terms of atom counts, that approach must then also be reflected in an approach of inductive strengths.
Limit 1

limit 2

To get the full experience, you need to see the animation of the approach to this limit.


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John D. Norton