8. Scale-Free Inductive Logics

We have required the inductive logics defined here to be asymptotically stable. That means that the inductive strengths must stabilize only as larger and larger inductively adapted partitions are defined. It is easy for us to define inductive logics that will stabilize this way merely by choosing a suitable function in (S).

For example,

[A|B] = #A&B/#B + 1/N

will eventually stabilize towards the same values as the probabilistic strengths of (P) in the corresponding inductively adapted partitions. However, since N is always finite, this new formula will always return strengths that differ by some small finite amount 1/N from the limiting, probabilistic values.

That further refinement will alter the values of the inductive strengths for some fixed A and B indicates that, by the logic's own standards, the present values of the these strengths are not the best values. They will change when the logic learns of more refinements.

Among asymptotic logics, there is a subclass that is free of this problem. These are the "scale-free" logics. They are characterized by their invariance under refinement to new, inductively adapted partitions. They have already converged to their final values.

For ease of analysis, it is convenient to develop the simplest case. These are the inductive logics that are invariant under uniform refinement. That is a refinement that replaces each atom by the same number of new atoms. So, if the refinement replaces one atom by a disjunction of ten new atoms, then that must be true of each of the remaining atoms in the original algebra.

We can define a scale-free logic by replacing the formula (S) by

(SF) [A|B] = g(#A&B/N, #A&~B/N, #~A&B/N )

for some suitable function g. That is, the strengths [A|B] may no longer depend directly on N. They are functions only of ratios like #A&B/N.

Inductive strengths defined by this formula will be scale-free. That is, they will be unchanged by a uniform refinement. Imagine, for example, that the uniform refinement replaces each atom by 8 atoms. Then each of the quantities mentioned in the formula (SF) will be altered as follows:

N to 8N#A&B to 8(#A&B)
#A&~B/N to 8(#A&~B/N)#~A&B to 8(#~A&B)

so that the ratios

#A&B/N #A&~B/N #~A&B/N

will be unchanged. Therefore [A|B] will have the same value before and after the refinement.

This shows that the formula (SF) is sufficient to generate a scale-free logic. It turns out that there is no other way. It is also necessary that each scale-free logic be associated with a function g in (SF).

Both of the logics that we have considered seriously here are scale-free. They are the probability logic for which

(P)[A|B] = #A&B/#B = (#A&B/N) / (#A&B/N + #~A&B/N)

and the specific conditioning logic for which

(SC)[A|B] = (#A&B/#B).(#A&B/#A)

which can also be expanded into terms that use solely #A&B/N, #A&~B/N, #~A&B/N

The term "scale-free" is widely used elsewhere. It designates systems that, in some suitable sense, look the same at all scales of magnification. The most familiar example is fractal curves, which always look the same no matter how much we magnify our point of view. This change of point of view in the case of inductive logics corresponds to moving to larger refinements. The logics are scale-free in the sense that the strengths that they assign are independent of the magnification provided by differing refinements.