Basic HTML Version

suggests some alternative formulations of Ockham’s razor. Ockham’s

razor requires only

that one’s disjunctive belief state includes

simplest theory compatible with current

information. Thus, a greedy search for such an answer in the space of possible answers suffices.

Ockham’s

razor requires, more ambitiously, that one’s disjunctive belief state include

simplest theory compatible with current information. Ockham’s horizontal razor entails the

greedy version, but still allows for simplicity gaps, where a

in a disjunctive belief state is an

answer compatible with current information that is excluded from the belief state even though it is

simpler than some answer included in the belief state. Ockham’s

razor is just Ockham’s

greedy razor, along with the requirement that one’s belief state is gap free (i.e., closed downward

in the simplicity order restricted to answers compatible with current information). Ockham’s

horizontal and vertical razors both entail Ockham’s greedy razor, but neither entails the other: the

vertical razor pertains to

in the simplicity order and the horizontal razor pertains to

―hence the names.

A more refined system of Ockham efficiency theorems emerges in the generalized setting.

Ockham’s

razor is mandated by

efficiency, but not by cycle efficiency. For

suppose that the simplicity order has just two paths,

<

<

and

<

<

(it looks like a

diamond with

at the bottom and

at the top). Suppose that current information rules out

, but

none of the remaining possibilities. Finally, suppose that you violate the horizontal version of

Ockham’s razor by inferring

rather than

or

. Nature can now force you to plump for

and

then for

, so you produce the sequence (

,

,

) instead of the Ockham sequence (

or

,

,

),

which has one fewer reversal, because

or

is not contradicted by

. Horizontal Ockham

methods never commit more reversals in a given node of the simplicity order than the height of

the node in the order, whereas violators commit at least one more reversal. Horizontal Ockham’s

razor is not enforced by cycle efficiency, since the extra reversal (

,

,

) does not constitute a

cycle.

In contrast, Ockham’s

razor is mandated by

efficiency, but not by reversal

efficiency, so the ancient authors of the Katha Upanishad were right to mention both! For suppose

that there is a simplicity gap in one’s belief state―e.g., “degree 2 or degree 4.” Then, since “degree

2” is compatible with current information, nature has available the strategy to present information

from some polynomial

of degree 3 until the convergent scientist plumps for “degree 3,” on pain

of never converging to the truth. Then nature is free to force the scientist to plump for “degree 4.”

The pattern (2 or 4, 3, 4) is a

cycle, in the sense that 2 or 4 is accepted, reversed to 3,

and then entailed by 4. Methods that satisfy the stronger, vertical version of Ockham’s razor never

perform disjunctive cycles. On the other hand, minimization of reversals does not enforce

Ockham’s vertical razor. Ockham methods can be forced along the path (2, 3, 4), but the gappy

method’s worst-case performance is (2 or 4, 2, 3, 4), which still has only 2 reversals (since 4 does

not contradict 2 or 4).

Thus, we have the following, interesting extension of the Ockham efficiency theorem to

the general setting of disjunctive answers and branching simplicity: assuming that the methods

6