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That is really the crux for any attempt to provide a non-circular, non-metaphysical

justification of Ockham’s Razor. It is

to see that you do worse at finding the truth in

worlds if you assume that the truth is complex. But if you also do

in complex worlds by

inferring a complex theory, then the argument for simplicity requires a circular or question-

begging assumption that the truth is simple, which prompts metaphysical speculations. To break

out of the circle, it is necessary to show something apparently paradoxical―that presuming a

complex theory makes your truth-finding performance worse, even in worlds in which the

theory is true. But it really is no more paradoxical than driving north to get on to a

south-bound interstate.

To examine the same idea from a slightly different angle, consider the worst-case reversal

bounds for Ockham and for the violator. In general, over degree

, Ockham reverses opinion at

most

times, but the violator reverses her opinion at least

+ 1 times.

So Ockham

the

violator in terms of worst-case reversal bounds over answers to the question. Therefore, Ockham’s

razor is, in a sense, necessary for optimal worst-case reversal performance in the polynomial

degree problem. It is also sufficient: every convergent method for the polynomial degree problem

that never rules out the simplest possibility compatible with information is both reversal efficient,

in terms of worst-case bounds over answers. It is not hard to see that the result extends to any

problem in which the alternative theories are totally ordered by simplicity as T

1

, T

2

, T

3

, …. Call

back-and-forth results of that kind

.

The preceding argument can be generalized. First, the above Ockham efficiency theorem

holds only in the fairly simple setting of the polynomial degree problem, in which answers are

sequentially ordered by simplicity in an obvious way. One can extend the argument to a very

general class of theory choice problems. A theory choice problem involves three components: (1)

a set of possible worlds, or ways the world could be; (2) the possible information that would be

received, eventually, in each world; and (3) the question, or the range of possible theories among

which one is choosing. Simplicity reflects

of the traditional problem of induction that

one faces in such a problem. You face the problem of induction from one answer to another in a

theory choice problem if, regardless of how the first answer is true, you would never receive

information ruling out the second answer. The problem of induction from one answer to another

defines an order on answers, and you face an iterated problem of induction along paths in that

order

The polynomial degree problem has just one such path “degree 0,” “degree 1,” …, “degree

,” …, but that is a very special case. In general, simplicity orders, like freeway systems, can

. For example, suppose that the question is not merely to find the polynomial degree of

,

but the

of

, which is defined as the set of all

such that is non-zero in the normal form

polynomial expression of

.

Second, scientists don’t usually plump for a unique theory at every stage of inquiry. They

typically suspend judgment for a while by disjoining all answers compatible with experience until

a sufficiently long run of simple data

the simplest answer. So Ockham’s razor should be

stated generally enough to deal with

beliefs over branching simplicity orders. That

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