Biology Reference
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self-contained subsystem. Its output is, of course,
B
. Knowing
B
alone, however,
does not turn (
3.3
) into a self-contained subsystem. Substituting its value into (
3.3
)
leaves the variable
A
in place (despite the fact that
B
cannot have a well-defined
value unless
A
also has a well-defined value), and we have to substitute
A
directly
from (
3.1
). Thus,
A
directly causes
B
, and
A
and
B
directly cause
C
; so,
A
is both a
direct and an indirect cause of
C
. This, of course, is the causal structure of Fig.
3.1
.
Suppose, however, that we modify the system slightly by replacing Eq. (
3.3
)
with (
3.3
00
):
ð
3
:
3
00
Þ
C
¼ α
CB
B
:
Then,
A
directly causes
B
,and
B
directly causes
C
,but
A
only
indirectly
causes
C
:
Eq. (
3.1
) is nested in the complete system (
3.1
), (
3.2
), (
3.3
00
), but the self-contained
subsystem (
3.1
) and (
3.2
) intervenes between the self-contained subsystem (
3.1
)
and the self-contained complete system (
3.1
), (
3.2
), (
3.3
00
).
Simon's analysis assumes that the original way of writing the equations is
canonical. But he notices that the same functional relationships can be represented
by other sets of equations. So, for example, the self-contained subsystem (
3.1
) and
(
3.2
) could be replaced by
A
¼ β
A
þ β
AB
B
;
ð
3
:
4
Þ
B
¼ β
B
;
ð
3
:
5
Þ
which has the same numerical solution as (
3.1
) and (
3.2
) provided that
α
A
β
A
¼
Þ
;
ð
3
:
6
Þ
ð
1
α
BA
1
β
AB
¼
Þ
;
ð
3
:
7
Þ
ð
1
α
BA
and
β
B
¼ α
A
α
BA
:
ð
3
:
8
Þ
(Nothing depends on the fact that the
α
ij
. We could as
easily have started with Eqs. (
3.4
) and (
3.5
) and derived an analogous set of
restrictions defining the
β
ij
are defined in terms of the
β
ij
to guarantee identical solutions.)
The two sets of equations have the same solution, but under Simon's analysis
B
causes
A
in (
3.4
) and (
3.5
), whereas
A
causes
B
in (
3.1
) and (
3.2
). Indeed, since
every linear combination of Eqs. (
3.1
) and (
3.2
) is functionally equivalent, we can
easily write down systems that would be interpreted as having no causal
connections or as displaying mutual causation. This is the sense in which systems
α
ij
in terms of the
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