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THE
LOGIC OF CAUSATION
©
Avi Sion, 1999. All rights reserved.
Phase One: Macroanalysis
Chapter
8 - Matricial
Analyses.
1.
Matricial Analysis.
2.
Crucial Matricial Analyses in Figure 1.
3.
Crucial Matricial Analyses in Figure 2.
4.
Crucial Matricial Analyses in Figure 3.
3.
Crucial Matricial Analyses in Figure 2.
Evaluation of mood # 221.
(Similarly, mutadis mutandis, for mood
# 231.)
Major premise:
R is a complete and (complemented by S) a contingent cause of Q:
(i)
If R, then Q;
(ii)
if notR, not-then Q;
(iii)
where: R is possible.
(iv)
If (notR + notS), then notQ;
(v)
if (R + notS), not-then notQ;
(vi)
if (notR + S), not-then notQ;
(vii)
where: (notR + notS) is possible.
Minor premise:
P is a complete and necessary cause of Q:
(viii)
If P, then Q;
(ix)
if notP, not-then Q;
(x)
where: P is possible.
(xi)
If notP, then notQ;
(xii)
if P, not-then notQ;
(xiii)
where: notP is possible.
Putative conclusion
is P a complete or (complemented by S) a partial cause of R?
NO! P is not implied to be a complete
cause of R:
|
If P, then
R?
|
denied by
(vi) + (xi);
|
|
if notP,
not-then R
|
implied by
(iv) + (vii) + (viii);
|
|
where: P is
possible
|
same as (x).
|
Nor (complemented by S) a partial cause
of R:
|
If (P + S),
then R?
|
denied by
(vi) + (xi);
|
|
if (notP +
S), not-then R?
|
open;
|
|
if (P +
notS), not-then R?
|
denied by
(iv) + (viii);
|
|
where: (P +
S) is possible
|
implied by
(vi) + (xi).
|
|
Table
8.16. Evaluation
of mood 221.
|
|
P
|
Q
|
R
|
S
|
see
(i) + (iii), or (viii) + (x), or (xii)
|
|
P
|
Q
|
R
|
notS
|
implied
possible by (v) + (xi)
|
|
P
|
Q
|
notR
|
S
|
implied
possible by (vi) + (xi)
|
|
P
|
Q
|
notR
|
notS
|
implied
impossible by (iv)
|
|
P
|
notQ
|
R
|
S
|
implied
impossible by (i) or (viii)
|
|
P
|
notQ
|
R
|
notS
|
implied
impossible by (i) or (viii)
|
|
P
|
notQ
|
notR
|
S
|
implied
impossible by (viii)
|
|
P
|
notQ
|
notR
|
notS
|
implied
impossible by (viii)
|
|
notP
|
Q
|
R
|
S
|
implied
impossible by (xi)
|
|
notP
|
Q
|
R
|
notS
|
implied
impossible by (xi)
|
|
notP
|
Q
|
notR
|
S
|
implied
impossible by (xi)
|
|
notP
|
Q
|
notR
|
notS
|
implied
impossible by (iv) or (xi)
|
|
notP
|
notQ
|
R
|
S
|
implied
impossible by (i)
|
|
notP
|
notQ
|
R
|
notS
|
implied
impossible by (i)
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|
notP
|
notQ
|
notR
|
S
|
see
(ii), or (ix), or (xi) + (xiii)
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|
notP
|
notQ
|
notR
|
notS
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implied
possible by (iv) + (vii) + (viii)
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Evaluation of mood # 222.
(Similarly, mutadis mutandis, for mood
# 233.)
Major premise:
R is a complete and (complemented by P) a contingent cause of Q:
(i)
If R, then Q;
(ii)
if notR, not-then Q;
(iii)
where: R is possible.
(iv)
If (notR + notP), then notQ;
(v)
if (R + notP), not-then notQ;
(vi)
if (notR + P), not-then notQ;
(vii)
where: (notR + notP) is possible.
Minor premise:
P is a complete and (complemented by S) a contingent cause of Q:
(viii)
If P, then Q;
(ix)
if notP, not-then Q;
(x)
where: P is possible.
(xi)
If (notP + notS), then notQ;
(xii)
if (P + notS), not-then notQ;
(xiii)
if (notP + S), not-then notQ;
(xiv)
where: (notP + notS) is possible.
Putative conclusion
is P (complemented by S) a cause of R?
NO! P is not implied to be a complete
cause of R:
|
If P, then
R?
|
open;
|
|
if notP,
not-then R
|
implied by
(i) + (xi) + (xiv);
|
|
where: P is
possible
|
same as (x).
|
Nor (complemented by S) a partial cause
of R:
|
If (P + S),
then R?
|
open;
|
|
if (notP +
S), not-then R?
|
open;
|
|
if (P +
notS), not-then R?
|
open;
|
|
where: (P +
S) is possible?
|
open.
|
Nor a necessary cause of R:
|
If notP,
then notR?
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denied by
(v) + (xi), or (iv) + (xiii);
|
|
if P,
not-then notR?
|
open;
|
|
where: notP
is possible
|
implied by
(v) or (vii) or (ix) or (xiii) or (xiv).
|
Nor (complemented by S) a contingent
cause of R:
|
If (notP +
notS), then notR
|
implied by
(i) + (xi);
|
|
if (P +
notS), not-then notR?
|
open;
|
|
if (notP +
S), not-then notR
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implied by
(v) + (xi), or (iv) + (xiii);
|
|
where: (notP
+ notS) is possible
|
same as
(xiv).
|
|
Table
8.17. Evaluation
of mood 222.
|
|
P
|
Q
|
R
|
S
|
see
(i) + (iii), or (viii) + (x)
|
|
P
|
Q
|
R
|
notS
|
see
(i) + (iii), or (viii) + (x), or (xii)
|
|
P
|
Q
|
notR
|
S
|
see
(vi), or (viii) + (x)
|
|
P
|
Q
|
notR
|
notS
|
see
(vi), or (viii) + (x), or (xii)
|
|
P
|
notQ
|
R
|
S
|
implied
impossible by (i) or (viii)
|
|
P
|
notQ
|
R
|
notS
|
implied
impossible by (i) or (viii)
|
|
P
|
notQ
|
notR
|
S
|
implied
impossible by (viii)
|
|
P
|
notQ
|
notR
|
notS
|
implied
impossible by (viii)
|
|
notP
|
Q
|
R
|
S
|
implied
possible by (v) + (xi), or (iv) + (xiii)
|
|
notP
|
Q
|
R
|
notS
|
implied
impossible by (xi)
|
|
notP
|
Q
|
notR
|
S
|
implied
impossible by (iv)
|
|
notP
|
Q
|
notR
|
notS
|
implied
impossible by (iv) or (xi)
|
|
notP
|
notQ
|
R
|
S
|
implied
impossible by (i)
|
|
notP
|
notQ
|
R
|
notS
|
implied
impossible by (i)
|
|
notP
|
notQ
|
notR
|
S
|
see
(ii), or (iv) + (vii), or (ix)
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|
notP
|
notQ
|
notR
|
notS
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implied
possible by (i) + (xi) + (xiv)
|
Evaluation of mood # 224.
(Similarly, mutadis mutandis, for mood
# 234.)
Major premise:
R is a complete and (complemented by P) a contingent cause of Q:
(i)
If R, then Q;
(ii)
if notR, not-then Q;
(iii)
where: R is possible.
(iv)
If (notR + notP), then notQ;
(v)
if (R + notP), not-then notQ;
(vi)
if (notR + P), not-then notQ;
(vii)
where: (notR + notP) is possible.
Minor premise:
P (complemented by S) is a partial and contingent cause of Q:
(viii)
If (P + S), then Q;
(ix)
if (notP + S), not-then Q;
(x)
if (P + notS), not-then Q;
(xi)
where: (P + S) is possible.
(xii)
If (notP + notS), then notQ;
(xiii)
if (P + notS), not-then notQ;
(xiv)
if (notP + S), not-then notQ;
(xv)
where: (notP + notS) is possible.
Putative conclusion
is P (complemented by S) a cause of R?
NO! P is not implied to be a complete
cause of R:
|
If P, then
R?
|
denied by
(i) + (x);
|
|
if notP,
not-then R
|
implied by
(i)+(ix), or (i)+(xii)+(xv);
|
|
where: P is
possible
|
implied by
(vi) or (x) or (xi) or (xiii).
|
Nor (complemented by S) a partial cause
of R:
|
If (P + S),
then R?
|
open;
|
|
if (notP +
S), not-then R
|
implied by
(i) + (ix);
|
|
if (P +
notS), not-then R
|
implied by
(i) + (x);
|
|
where: (P +
S) is possible
|
same as
(xi).
|
Nor a necessary cause of R:
|
If notP,
then notR?
|
denied by
(iv) + (xiv), or (v) + (xii);
|
|
if P,
not-then notR?
|
open;
|
|
where: notP
is possible
|
implied by
(v) or (vii) or (ix) or (xiv) or (xv).
|
Nor (complemented by S) a contingent
cause of R:
|
If (notP +
notS), then notR
|
implied by
(i) + (xii);
|
|
if (P +
notS), not-then notR?
|
open;
|
|
if (notP +
S), not-then notR
|
implied by
(iv) + (xiv), or (v) + (xii);
|
|
where: (notP
+ notS) is possible
|
same as
(xv).
|
|
Table
8.18. Evaluation
of mood 224.
|
|
P
|
Q
|
R
|
S
|
see
(i) + (iii), or (viii) + (xi)
|
|
P
|
Q
|
R
|
notS
|
see
(i) + (iii), or (xiii)
|
|
P
|
Q
|
notR
|
S
|
see
(vi), or (viii) + (xi)
|
|
P
|
Q
|
notR
|
notS
|
see
(vi), or (xiii)
|
|
P
|
notQ
|
R
|
S
|
implied
impossible by (i) or (viii)
|
|
P
|
notQ
|
R
|
notS
|
implied
impossible by (i)
|
|
P
|
notQ
|
notR
|
S
|
implied
impossible by (viii)
|
|
P
|
notQ
|
notR
|
notS
|
implied
possible by (i) + (x)
|
|
notP
|
Q
|
R
|
S
|
implied
possible by (iv) + (xiv), or (v) + (xii)
|
|
notP
|
Q
|
R
|
notS
|
implied
impossible by (xii)
|
|
notP
|
Q
|
notR
|
S
|
implied
impossible by (iv)
|
|
notP
|
Q
|
notR
|
notS
|
implied
impossible by (iv) or (xii)
|
|
notP
|
notQ
|
R
|
S
|
implied
impossible by (i)
|
|
notP
|
notQ
|
R
|
notS
|
implied
impossible by (i)
|
|
notP
|
notQ
|
notR
|
S
|
implied
possible by (i) + (ix)
|
|
notP
|
notQ
|
notR
|
notS
|
implied
possible by (i) + (xii) + (xv)
|
Evaluation of mood # 241.
Major premise:
R (complemented by S) is a partial and contingent cause of Q:
(i)
If (R + S), then Q;
(ii)
if (notR + S), not-then Q;
(iii)
if (R + notS), not-then Q;
(iv)
where: (R + S) is possible.
(v)
If (notR + notS), then notQ;
(vi)
if (R + notS), not-then notQ;
(vii)
if (notR + S), not-then notQ;
(viii)
where: (notR + notS) is possible.
Minor premise:
P is a complete and necessary cause of Q:
(ix)
If P, then Q;
(x)
if notP, not-then Q;
(xi)
where: P is possible.
(xii)
If notP, then notQ;
(xiii)
if P, not-then notQ;
(xiv)
where: notP is possible.
Putative conclusion
is P (complemented by S) a cause of R?
NO! P is not implied to be a complete
cause of R:
|
If P, then
R?
|
denied by
(vii) + (xii);
|
|
if notP,
not-then R
|
implied by
(ii)+(ix), or (v)+(viii)+(ix);
|
|
where: P is
possible
|
same as
(xi).
|
Nor (complemented by S) a partial cause
of R:
|
If (P + S),
then R?
|
denied by
(vii) + (xii);
|
|
if (notP +
S), not-then R
|
implied by
(ii) + (ix);
|
|
if (P +
notS), not-then R?
|
denied by
(v) + (ix);
|
|
where: (P +
S) is possible
|
impl. by
(i)+(iv)+(xii), or (vii)+(xii).
|
Nor a necessary cause of R:
|
If notP,
then notR?
|
denied by
(iii) + (ix);
|
|
if P,
not-then notR
|
impl. by
(i)+(iv)+(xii), or (vi)+(xii);
|
|
where: notP
is possible
|
same as
(xiv).
|
Nor (complemented by S) a contingent
cause of R:
|
If (notP +
notS), then notR?
|
| | 
