Page 152 - MODES of EXPLANATION

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suggests some alternative formulations of Ockham’s razor. Ockham’s
greedy
razor requires only
that one’s disjunctive belief state includes
some
simplest theory compatible with current
information. Thus, a greedy search for such an answer in the space of possible answers suffices.
Ockham’s
horizontal
razor requires, more ambitiously, that one’s disjunctive belief state include
every
simplest theory compatible with current information. Ockham’s horizontal razor entails the
greedy version, but still allows for simplicity gaps, where a
gap
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
vertical
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
chains
in the simplicity order and the horizontal razor pertains to
anti-
chains
―hence the names.
A more refined system of Ockham efficiency theorems emerges in the generalized setting.
Ockham’s
horizontal
razor is mandated by
reversal
efficiency, but not by cycle efficiency. For
suppose that the simplicity order has just two paths,
A
<
B
<
D
and
A
<
C
<
D
(it looks like a
diamond with
A
at the bottom and
D
at the top). Suppose that current information rules out
A
, but
none of the remaining possibilities. Finally, suppose that you violate the horizontal version of
Ockham’s razor by inferring
B
rather than
B
or
C
. Nature can now force you to plump for
C
and
then for
D
, so you produce the sequence (
B
,
C
,
D
) instead of the Ockham sequence (
B
or
C
,
C
,
D
),
which has one fewer reversal, because
B
or
C
is not contradicted by
C
. 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 (
B
,
C
,
D
) does not constitute a
cycle.
In contrast, Ockham’s
vertical
razor is mandated by
cycle
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
f
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
disjunctive
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
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