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THE
LOGIC OF CAUSATION
©
Avi Sion, 2003. All rights reserved.
Phase Two: Microanalysis
Chapter
15 - Some More Three-Item Syllogisms
1.
Special Cases of Three-Item
Syllogism
2.
Dealing with Negatives.
In the previous chapter, we examined positive absolute three-item syllogism. In the present chapter, we shall look for other
syllogistic forms, involving relative or negative causative propositions. But
keep in mind that this endeavor does not exhaust the matter: we shall eventually
still be obliged to develop four-item syllogism.
1.
Special Cases of Three-item Syllogism.
We shall now consider special cases of three-item
syllogism, involving weak determinations
relative to the major and/or minor item. That is, three-item syllogism with
major premise of form Q(P)R or R(P)Q and/or minor premise of form P(R)Q or Q(R)P,
with eventual conclusions of absolute form PR or relative form P(Q)R.
Note well that such syllogisms still involve only
three items, even though one or more propositions in them are of relative
weak determination. Such cases may conceivably arise in practice, though
very rarely. For each of the three figures, we may conceive of seven such
subfigures, in addition to the already dealt with standard Aristotelian
arrangement. The eight subfigures are labeled ‘a’ and ‘j’ to ‘p’,
for convenience, in the table below.
Table 15.1.
Subfigures of syllogism with three items only.
|
|
With PR
conclusion
|
With P(Q)R
conclusion
|
|
Subfigures
|
a
|
j
|
k
|
l
|
m
|
n
|
o
|
p
|
|
Figure
1
|
QR
|
QR
|
Q(P)R
|
Q(P)R
|
QR
|
QR
|
Q(P)R
|
Q(P)R
|
|
|
PQ
|
P(R)Q
|
PQ
|
P(R)Q
|
PQ
|
P(R)Q
|
PQ
|
P(R)Q
|
|
|
PR
|
PR
|
PR
|
PR
|
P(Q)R
|
P(Q)R
|
P(Q)R
|
P(Q)R
|
|
Figure
2
|
RQ
|
RQ
|
R(P)Q
|
R(P)Q
|
RQ
|
RQ
|
R(P)Q
|
R(P)Q
|
|
|
PQ
|
P(R)Q
|
PQ
|
P(R)Q
|
PQ
|
P(R)Q
|
PQ
|
P(R)Q
|
|
|
PR
|
PR
|
PR
|
PR
|
P(Q)R
|
P(Q)R
|
P(Q)R
|
P(Q)R
|
|
Figure
3
|
QR
|
QR
|
Q(P)R
|
Q(P)R
|
QR
|
QR
|
Q(P)R
|
Q(P)R
|
|
|
QP
|
Q(R)P
|
QP
|
Q(R)P
|
QP
|
Q(R)P
|
QP
|
Q(R)P
|
|
|
PR
|
PR
|
PR
|
PR
|
P(Q)R
|
P(Q)R
|
P(Q)R
|
P(Q)R
|
Notice that in each figure there are two sets of four
subfigures, with conclusions of form PR or P(Q)R. Only subfigure ‘a’ has
been treated by macroanalysis, the other seven here were simply ignored in Phase
One (being very special cases, not likely to often arise). They become relevant
here only because they serve to clarify what we mean by three-item (as against
four-item) syllogism.
Moods with major premise of form Q(P)R or R(P)Q involve a
weak determination relative to the minor item P; and those with minor premise of
form P(R)Q or Q(R)P involve a weak determination relative to the major item R.
Their summary moduses are given in the following table. Note that the forms
P(Q)R and Q(P)R yield the same moduses.
Table 15.2.
Summary moduses for weak premises relative to the minor item P or to the
major item R.
|
|
|
|
|
pP
|
pP
|
pR
|
pR
|
qP
|
qP
|
qR
|
qR
|
|
Row
labl
|
P
|
Q
|
R
|
QR
|
RQ
|
PQ
|
QP
|
QR
|
RQ
|
PQ
|
QP
|
|
a
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
∙
|
∙
|
∙
|
∙
|
|
b
|
1
|
1
|
0
|
0
|
∙
|
∙
|
∙
|
∙
|
1
|
1
|
1
|
|
c
|
1
|
0
|
1
|
∙
|
0
|
0
|
∙
|
1
|
∙
|
∙
|
1
|
|
d
|
1
|
0
|
0
|
1
|
1
|
1
|
∙
|
∙
|
∙
|
∙
|
0
|
|
e
|
0
|
1
|
1
|
∙
|
∙
|
∙
|
0
|
1
|
1
|
1
|
∙
|
|
f
|
0
|
1
|
0
|
1
|
∙
|
∙
|
1
|
∙
|
0
|
0
|
∙
|
|
g
|
0
|
0
|
1
|
∙
|
1
|
1
|
1
|
0
|
∙
|
∙
|
∙
|
|
h
|
0
|
0
|
0
|
∙
|
∙
|
∙
|
∙
|
1
|
1
|
1
|
1
|
The above table may be proved by appropriate reshuffling of
columns and rows: in each case, we obtain the same summary modus of partial or
contingent causation. For instance, for pQRP, move column P to
the right of columns Q and R, then reorder the rows to ‘aebfcgdh’ (so that
the sequences of QRP are 111, 110, 101, 100, etc.) – the result is summary
modus 10.1.1.. as required to prove.
From the above summary moduses, we can derive the
alternative moduses listed in the following table. The microanalyses of the
strong forms (m and n) are carried over from Table 14.3. Notice
the differences between the results below and those for relatives in Table 13.1.
Here mqrel = #s 42, 58,
170, 186, whereas there it = #s 42, 46, 170, 174; similarly, here nprel
= #s 149-150, 157-158, whereas there it = #s 149-150, 181-182. These differences
(two out of four moduses in each case) are simply due to the forms of strong
determination the weaks are combined with: there it was with PR, whereas here it
is with QR.
Table 15.3.
Enumeration of three-item alternative moduses weak positive premises
relative to the minor item P or to the major item R.
|
Determ.
|
Major
Q(P)R
|
Major
R(P)Q
|
Minor
P(R)Q
|
Minor
Q(R)P
|
|
m
|
10, 12, 25-28, 42, 44, 57-60, 130, 132, 138,
140, 145-148, 153-156, 162, 164, 170, 172, 177-180, 185-188
|
10, 14, 25-26, 29-30, 74, 78, 89-90, 93-94,
130, 134, 138, 142, 145-146, 149-150, 153-154, 157-158, 194, 198, 202,
206, 209-210, 213-214, 217-218, 221-222
|
66-68, 70-72, 74-76, 78-80, 130-132, 134-136,
138-140, 142-144, 194-196, 198-200, 202-204, 206-208
|
66-68, 82-84, 98-100, 114-116, 130-132,
146-148, 162-164, 178-180, 194-196, 210-212, 226-228, 242-244
|
|
n
|
10, 14, 25-26, 29-30, 74, 78, 89-90, 93-94,
130, 134, 138, 142, 145-146, 149-150, 153-154, 157-158, 194, 198, 202,
206, 209-210, 213-214, 217-218, 221-222
|
10, 12, 25-28, 42, 44, 57-60, 130, 132, 138,
140, 145-148, 153-156, 162, 164, 170, 172, 177-180, 185-188
|
66-68, 82-84, 98-100, 114-116, 130-132,
146-148, 162-164, 178-180, 194-196, 210-212, 226-228, 242-244
|
66-68, 70-72, 74-76, 78-80, 130-132, 134-136,
138-140, 142-144, 194-196, 198-200, 202-204, 206-208
|
|
prel
|
149-152,
157-160, 181-184, 189-192
|
147-148,
151-152, 155-156, 159-160, 211-212, 215-216, 219-220, 223-224
|
147-148,
151-152, 155-156, 159-160, 211-212, 215-216, 219-220, 223-224
|
135-136,
151-152, 167-168, 183-184, 199-200, 215-216, 231-232, 247-248
|
|
qrel
|
42,
46, 58, 62, 106, 110, 122, 126, 170, 174, 186, 190, 234, 238, 250, 254
|
74,
76, 90, 92, 106, 108, 122, 124, 202, 204, 218, 220, 234, 236, 250, 252
|
74,
76, 90, 92, 106, 108, 122, 124, 202, 204, 218, 220, 234, 236, 250, 252
|
98,
100, 102, 104, 106, 108, 110, 112, 226, 228, 230, 232, 234, 236, 238,
240
|
|
mqrel
|
42,
58, 170, 186
|
74,
90, 202, 218
|
74,
76, 202, 204
|
98,
100, 226, 228
|
|
nprel
|
149-150,
157-158
|
147-148,
155-156
|
147-148,
211-212
|
135-136,
199-200
|
|
prelqrel
|
190
|
220
|
220
|
232
|
|
wrel
|
42, 46, 58, 62, 106, 110, 122, 126, 149-152,
157-160, 170, 174, 181-184, 186, 189-192, 234, 238, 250, 254
|
74, 76, 90, 92, 106, 108, 122, 124, 147-148,
151-152, 155-156, 159-160, 202, 204, 211-212, 215-216, 218-220, 223-224,
234, 236, 250, 252
|
74, 76, 90, 92, 106, 108, 122, 124, 147-148,
151-152, 155-156, 159-160, 202, 204, 211-212, 215-216, 218-220, 223-224,
234, 236, 250, 252
|
98, 100, 102, 104, 106, 108, 110, 112,
135-136, 151-152, 167-168, 183-184, 199-200, 215-216, 226, 228, 230-232,
234, 236, 238, 240, 247-248
|
|
crel
|
10, 12, 14, 25-30, 42, 44, 46, 57-60, 62, 74,
78, 89-90, 93-94, 106, 110, 122, 126, 130, 132, 134, 138, 140, 142,
145-160, 162, 164, 170, 172, 174, 177-192, 194, 198, 202, 206, 209-210,
213-214, 217-218, 221-222, 234, 238, 250, 254
|
10, 12, 14, 25-30, 42, 44, 57-60, 74, 76, 78,
89-90, 92-94, 106, 108, 122, 124, 130, 132, 134, 138, 140, 142, 145-150,
151-160, 162, 164, 170, 172, 177-180, 185-188, 194, 198, 202, 204, 206,
209-224, 234, 236, 250, 252
|
66-68, 70-72, 74-76, 78-80, 82-84, 90, 92,
98-100, 106, 108, 114-116, 122, 124, 130-132, 134-136, 138-140, 142-144,
146-148, 151-152, 155-156, 159-160, 162-164, 178-180, 194-196, 198-200,
202-204, 206-208, 210-212, 215-216, 218-220, 223-224, 226-228, 234, 236,
242-244, 250, 252
|
66-68, 70-72, 74-76, 78-80, 82-84, 98-100,
102, 104, 106, 108, 110, 112, 114-116, 130-132, 134-136, 138-140,
142-144, 146-148, 151-152, 162-164, 167-168, 178-180, 183-184, 194-196,
198-200, 202-204, 206-208, 210-212, 215-216, 226-228, 230-232, 234, 236,
238, 240, 242-244, 247-248
|
Note that for prel,
qrel, prelqrel and wrel,
a major premise of form R(P)Q and a minor premise of form P(R)Q, of the same
weak determination, have the same alternative moduses. This is not surprising,
since these weak forms involve the same items as causes (P, R) and effect (Q). A
similar equation is not obtained where a strong determination is involved
because in such case the items involved are in fact only RQ and PQ (without
complements).
The following tables list a selection of syllogisms one or
both of whose premises involve all three items. They ignore syllogisms with an
exclusively strong premise (such as moods 111-118, 121, 125, 126, etc.), not
because such syllogisms are impossible or uninteresting, but simply for
brevity’s sake (the reader is invited to look into such cases as an exercise).
The conclusions are implicit in the common moduses of the
premises. Once these common moduses, if any, are identified, the conclusion has
to be sought in Table 12.4, if of absolute form PR, or in Table 13.1, if of
relative form P(Q)R, these forms being those predicted in Table 15.1 above.
Forms without subscript are intended as absolute, those
with a subscript as relative to the mentioned complement (P, Q or R). A
conclusion of the form p or q or pq has absolute form PR; a
similar conclusion of relative form P(Q)R is not implied by it. A conclusion of
the form not-cQ means not-m
+ not-n + not-pQ
+ not-qQ, which does not
imply not-cabs. Lone
determinations are sometimes concluded, but these are of course relative and not
absolute. Note well that the subsidiary item S is never mentioned here, since we
have not performed four-item microanalysis yet.
Table 15.4.
Moduses of conclusions for selected relative weak positive minor premises
(subfigures j, n).
|
Mood
|
Major
|
Minor
|
Conclusion
|
Common
moduses
|
No.
|
|
1st
|
QR
|
PQ
|
PR
|
|
|
|
122
|
mq
|
mqR
|
inconsistent
|
None
|
0
|
|
123
|
mq
|
npR
|
not-cQ
|
147-148
|
2
|
|
124
|
mq
|
pRqR
|
inconsistent
|
None
|
0
|
|
127
|
mq
|
pR
|
not-cQ
|
147-148,
155-156
|
4
|
|
128
|
mq
|
qR
|
inconsistent
|
None
|
0
|
|
132
|
np
|
mqR
|
not-cQ
|
74,
202
|
2
|
|
133
|
np
|
npR
|
inconsistent
|
None
|
0
|
|
134
|
np
|
pRqR
|
inconsistent
|
None
|
0
|
|
137
|
np
|
pR
|
inconsistent
|
None
|
0
|
|
138
|
np
|
qR
|
not-cQ
|
74,
90, 202, 218
|
4
|
|
142
|
pq
|
mqR
|
not-cQ
|
76, 204
|
2
|
|
143
|
pq
|
npR
|
not-cQ
|
211-212
|
2
|
|
144
|
pq
|
pRqR
|
not-cQ
|
220
|
1
|
|
147
|
pq
|
pR
|
not-s
|
151-152,
159-160, 211-212, 215-216, 219-220, 223-224
|
12
|
|
148
|
pq
|
qR
|
not-s
|
76, 92, 106, 108, 122, 124, 204, 220, 234,
236, 250, 252
|
12
|
|
172
|
p
|
mqR
|
not-cQ
|
74,
76, 202, 204
|
4
|
|
173
|
p
|
npR
|
not-cQ
|
211-212
|
2
|
|
174
|
p
|
pRqR
|
not-cQ
|
220
|
1
|
|
177
|
p
|
pR
|
not-s
|
151-152, 159-160, 211-212, 215-216, 219-220,
223-224
|
12
|
|
178
|
p
|
qR
|
not-s
|
74,
76, 90, 92, 106, 108, 122, 124, 202, 204, 218, 220, 234, 236, 250, 252
|
16
|
|
182
|
q
|
mqR
|
not-cQ
|
76, 204
|
2
|
|
183
|
q
|
npR
|
not-cQ
|
147-148,
211-212
|
4
|
|
184
|
q
|
pRqR
|
not-cQ
|
220
|
1
|
|
187
|
q
|
pR
|
not-s
|
147-148,
151-152, 155-156, 159-160, 211-212, 215-216, 219-220, 223-224
|
16
|
|
188
|
q
|
qR
|
not-s
|
76, 92, 106, 108, 122, 124, 204, 220, 234,
236, 250, 252
|
12
|
|
2nd
|
RQ
|
PQ
|
PR
|
|
|
|
222
|
mq
|
mqR
|
not-cQ
|
74, 202
|
2
|
|
223
|
mq
|
npR
|
inconsistent
|
None
|
0
|
|
224
|
mq
|
pRqR
|
inconsistent
|
None
|
0
|
|
227
|
mq
|
pR
|
inconsistent
|
None
|
0
|
|
228
|
mq
|
qR
|
not-cQ
|
74, 90, 202, 218
|
4
|
|
232
|
np
|
mqR
|
inconsistent
|
None
|
0
|
|
233
|
np
|
npR
|
not-cQ
|
147-148
|
2
|
|
234
|
np
|
pRqR
|
inconsistent
|
None
|
2
|
|
237
|
np
|
pR
|
not-cQ
|
147-148,
155-156
|
4
|
|
238
|
np
|
qR
|
inconsistent
|
None
|
0
|
|
242
|
pq
|
mqR
|
not-cQ
|
76,
204
|
2
|
|
243
|
pq
|
npR
|
not-cQ
|
211-212
|
2
|
|
244
|
pq
|
pRqR
|
not-cQ
|
220
|
1
|
|
247
|
pq
|
pR
|
not-s
|
151-152, 159-160, 211-212, 215-216, 219-220,
223-224
|
12
|
|
248
|
pq
|
qR
|
not-s
|
76, 92, 106, 108, 122, 124, 204, 220, 234,
236, 250, 252
|
12
|
|
272
|
p
|
mqR
|
not-cQ
|
76, 204
|
2
|
|
273
|
p
|
npR
|
not-cQ
|
147-148,
211-212
|
4
|
|
274
|
p
|
pRqR
|
not-cQ
|
220
|
1
|
|
277
|
p
|
pR
|
not-s
|
147-148, 151-152, 155-156, 159-160, 211-212,
215-216, 219-220, 223-224
|
16
|
|
278
|
p
|
qR
|
not-s
|
76,
92, 106, 108, 122, 124, 204, 220, 234, 236, 250, 252
|
12
|
|
282
|
q
|
mqR
|
not-cQ
|
74, 76, 202, 204
|
4
|
|
283
|
q
|
npR
|
not-cQ
|
211-212
|
2
|
|
284
|
q
|
pRqR
|
not-cQ
|
220
|
1
|
|
287
|
q
|
pR
|
not-s
|
151-152,
159-160, 211-212, 215-216, 219-220, 223-224
|
12
|
|
288
|
q
|
qR
|
not-s
|
74, 76, 90, 92, 106, 108, 122, 124, 202, 204,
218, 220, 234, 236, 250, 252
|
16
|
|
3rd
|
QR
|
QP
|
PR
|
|
|
|
322
|
mq
|
mqR
|
inconsistent
|
None
|
0
|
|
323
|
mq
|
npR
|
inconsistent
|
None
|
0
|
|
324
|
mq
|
pRqR
|
inconsistent
|
None
|
0
|
|
327
|
mq
|
pR
|
inconsistent
|
None
|
0
|
|
328
|
mq
|
qR
|
inconsistent
|
None
|
0
|
|
332
|
np
|
mqR
|
inconsistent
|
None
|
0
|
|
333
|
np
|
npR
|
inconsistent
|
None
|
0
|
|
334
|
np
|
pRqR
|
inconsistent
|
None
|
0
|
|
337
|
np
|
pR
|
inconsistent
|
None
|
0
|
|
338
|
np
|
qR
|
inconsistent
|
None
|
0
|
|
342
|
pq
|
mqR
|
p
+ not-pQ
|
226,
228
|
2
|
|
343
|
pq
|
npR
|
q
+ not-qQ
|
136,
200
|
2
|
|
344
|
pq
|
pRqR
|
not-cQ
|
232
|
1
|
|
347
|
pq
|
pR
|
q + not-qQ
|
136,
151-152, 168, 183-184, 200, 215-216, 232, 247-248
|
12
|
|
348
|
pq
|
qR
|
p
+ not-pQ
|
106, 108, 110, 112, 226, 228, 230, 232, 234,
236, 238, 240
|
12
|
|
372
|
p
|
mqR
|
p
+ not-pQ
|
226,
228
|
2
|
|
373
|
p
|
npR
|
q
+ not-qQ
|
136,
200
|
2
|
|
374
|
p
|
pRqR
|
not-cQ
|
232
|
1
|
|
377
|
p
|
pR
|
q + not-qQ
|
136, 151-152, 168, 183-184, 200, 215-216,
232, 247-248
|
12
|
|
378
|
p
|
qR
|
p
+ not-pQ
|
106,
108, 110, 112, 226, 228, 230, 232, 234, 236, 238, 240
|
12
|
|
382
|
q
|
mqR
|
p
+ not-pQ
|
226,
228
|
2
|
|
383
|
q
|
npR
|
q
+ not-qQ
|
136,
200
|
2
|
|
384
|
q
|
pRqR
|
not-cQ
|
232
|
1
|
|
387
|
q
|
pR
|
q + not-qQ
|
136,
151-152, 168, 183-184, 200, 215-216, 232, 247-248
|
12
|
|
388
|
q
|
qR
|
p
+ not-pQ
|
106, 108, 110, 112, 226, 228, 230, 232, 234,
236, 238, 240
|
12
|
Table 15.5.
Moduses of conclusions for selected relative weak positive major premises
(subfigures k, o).
|
Mood
|
Major
|
Minor
|
Conclusion
|
Common
moduses
|
No.
|
|
1st
|
QR
|
PQ
|
PR
|
|
|
|
122
|
mqP
|
mq
|
inconsistent
|
None
|
0
|
|
123
|
mqP
|
np
|
inconsistent
|
None
|
0
|
|
124
|
mqP
|
pq
|
qQ
|
170, 186
|
2
|
|
127
|
mqP
|
p
|
qQ
|
170,
186
|
2
|
|
128
|
mqP
|
q
|
qQ
|
170,
186
|
2
|
|
132
|
npP
|
mq
|
inconsistent
|
None
|
0
|
|
133
|
npP
|
np
|
inconsistent
|
None
|
0
|
|
134
|
npP
|
pq
|
pQ
|
150,
158
|
2
|
|
137
|
npP
|
p
|
pQ
|
150,
158
|
2
|
|
138
|
npP
|
q
|
pQ
|
150,
158
|
2
|
|
142
|
pPqP
|
mq
|
inconsistent
|
None
|
0
|
|
143
|
pPqP
|
np
|
inconsistent
|
None
|
0
|
|
144
|
pPqP
|
pq
|
pQqQ
|
190
|
1
|
|
147
|
pPqP
|
p
|
pQqQ
|
190
|
1
|
|
148
|
pPqP
|
q
|
pQqQ
|
190
|
1
|
|
172
|
pP
|
mq
|
inconsistent
|
None
|
0
|
|
173
|
pP
|
np
|
inconsistent
|
None
|
0
|
|
174
|
pP
|
pq
|
pQ
|
150-152,
158-160, 182-184, 190-192
|
12
|
|
177
|
pP
|
p
|
pQ
|
150-152,
158-160, 182-184, 190-192
|
12
|
|
178
|
pP
|
q
|
pQ
|
150-152,
158-160, 182-184, 190-192
|
12
|
|
182
|
qP
|
mq
|
inconsistent
|
None
|
0
|
|
183
|
qP
|
np
|
inconsistent
|
None
|
0
|
|
184
|
qP
|
pq
|
qQ
|
106,
110, 122, 126, 170, 174, 186, 190, 234, 238, 250, 254
|
12
|
|
187
|
qP
|
p
|
qQ
|
106,
110, 122, 126, 170, 174, 186, 190, 234, 238, 250, 254
|
12
|
|
188
|
qP
|
q
|
qQ
|
106,
110, 122, 126, 170, 174, 186, 190, 234, 238, 250, 254
|
12
|
|
2nd
|
RQ
|
PQ
|
PR
|
|
|
|
222
|
mqP
|
mq
|
not-cQ
|
74, 202
|
2
|
|
223
|
mqP
|
np
|
inconsistent
|
None
|
0
|
|
224
|
mqP
|
pq
|
not-cQ
|
90, 218
|
2
|
|
227
|
mqP
|
p
|
not-cQ
|
90, 218
|
2
|
|
228
|
mqP
|
q
|
not-cQ
|
74, 90, 202, 218
|
4
|
|
232
|
npP
|
mq
|
inconsistent
|
None
|
0
|
|
233
|
npP
|
np
|
not-cQ
|
147-148
|
2
|
|
234
|
npP
|
pq
|
not-cQ
|
155-156
|
2
|
|
237
|
npP
|
p
|
not-cQ
|
147-148,
155-156
|
4
|
|
238
|
npP
|
q
|
not-cQ
|
155-156
|
2
|
|
242
|
pPqP
|
mq
|
inconsistent
|
None
|
0
|
|
243
|
pPqP
|
np
|
inconsistent
|
None
|
0
|
|
244
|
pPqP
|
pq
|
pq
+ not-wQ
|
220
|
1
|
|
247
|
pPqP
|
p
|
pq
+ not-wQ
|
220
|
1
|
|
248
|
pPqP
|
q
|
pq
+ not-wQ
|
220
|
1
|
|
272
|
pP
|
mq
|
inconsistent
|
None
|
0
|
|
273
|
pP
|
np
|
not-cQ
|
147-148,
211-212
|
4
|
|
274
|
pP
|
pq
|
not-s
+ not-qQ
|
151-152, 155-156, 159-160, 215-216, 219-220,
223-224
|
12
|
|
277
|
pP
|
p
|
not-s
+ not-qQ
|
147-148,
151-152, 155-156, 159-160, 211-212, 215-216, 219-220, 223-224
|
16
|
|
278
|
pP
|
q
|
not-s
+ not-qQ
|
151-152, 155-156, 159-160, 215-216, 219-220,
223-224
|
12
|
|
282
|
qP
|
mq
|
not-s
+ not-cQ
|
74,
76, 202, 204
|
4
|
|
283
|
qP
|
np
|
inconsistent
|
None
|
0
|
|
284
|
qP
|
pq
|
not-s
+ not-pQ
|
90, 92, 106, 108, 122, 124, 218, 220, 234,
236, 250, 252
|
12
|
|
287
|
qP
|
p
|
not-s
+ not-pQ
|
90,
92, 106, 108, 122, 124, 218, 220, 234, 236, 250, 252
|
12
|
|
288
|
qP
|
q
|
not-s
+ not-pQ
|
74,
76, 90, 92, 106, 108, 122, 124, 202, 204, 218, 220, 234, 236, 250, 252
|
16
|
|
3rd
|
QR
|
QP
|
PR
|
|
|
|
322
|
mqP
|
mq
|
inconsistent
|
None
|
0
|
|
323
|
mqP
|
np
|
inconsistent
|
None
|
0
|
|
324
|
mqP
|
pq
|
qQ
|
170, 186
|
2
|
|
327
|
mqP
|
p
|
qQ
|
170,
186
|
2
|
|
328
|
mqP
|
q
|
qQ
|
170,
186
|
2
|
|
332
|
npP
|
mq
|
inconsistent
|
None
|
0
|
|
333
|
npP
|
np
|
inconsistent
|
None
|
0
|
|
334
|
npP
|
pq
|
pQ
|
150, 158
|
2
|
|
337
|
npP
|
p
|
pQ
|
150,
158
|
2
|
|
338
|
npP
|
q
|
pQ
|
150,
158
|
2
|
|
342
|
pPqP
|
mq
|
inconsistent
|
None
|
0
|
|
343
|
pPqP
|
np
|
inconsistent
|
None
|
0
|
|
344
|
pPqP
|
pq
|
pQqQ
|
190
|
1
|
|
347
|
pPqP
|
p
|
pQqQ
|
190
|
1
|
|
348
|
pPqP
|
q
|
pQqQ
|
190
|
1
|
|
372
|
pP
|
mq
|
inconsistent
|
None
|
0
|
|
373
|
pP
|
np
|
inconsistent
|
None
|
0
|
|
374
|
pP
|
pq
|
pQ
|
150-152,
158-160, 182-184, 190-192
|
12
|
|
377
|
pP
|
p
|
pQ
|
150-152,
158-160, 182-184, 190-192
|
12
|
|
378
|
pP
|
q
|
pQ
|
150-152,
158-160, 182-184, 190-192
|
12
|
|
382
|
qP
|
mq
|
inconsistent
|
None
|
0
|
|
383
|
qP
|
np
|
inconsistent
|
None
|
0
|
|
384
|
qP
|
pq
|
qQ
|
106,
110, 122, 126, 170, 174, 186, 190, 234, 238, 250, 254
|
12
|
|
387
|
qP
|
p
|
qQ
|
106,
110, 122, 126, 170, 174, 186, 190, 234, 238, 250, 254
|
12
|
|
388
|
qP
|
q
|
qQ
|
106, 110, 122, 126, 170, 174, 186, 190, 234,
238, 250, 254
|
12
|
Table 15.6.
Moduses of conclusions for selected relative weak positive premises
(subfigures l, p).
|
Mood
|
Major
|
Minor
|
Conclusion
|
Common
moduses
|
No.
|
|
1st
|
QR
|
PQ
|
PR
|
|
|
|
122
|
mqP
|
mqR
|
inconsistent
|
None
|
0
|
|
123
|
mqP
|
npR
|
inconsistent
|
None
|
0
|
|
124
|
mqP
|
pRqR
|
inconsistent
|
None
|
0
|
|
127
|
mqP
|
pR
|
inconsistent
|
None
|
0
|
|
128
|
mqP
|
qR
|
inconsistent
|
None
|
0
|
|
132
|
npP
|
mqR
|
inconsistent
|
None
|
0
|
|
133
|
npP
|
npR
|
inconsistent
|
None
|
0
|
|
134
|
npP
|
pRqR
|
inconsistent
|
None
|
0
|
|
137
|
npP
|
pR
|
inconsistent
|
None
|
0
|
|
138
|
npP
|
qR
|
inconsistent
|
None
|
0
|
|
142
|
pPqP
|
mqR
|
inconsistent
|
None
|
0
|
|
143
|
pPqP
|
npR
|
inconsistent
|
None
|
0
|
|
144
|
pPqP
|
pRqR
|
inconsistent
|
None
|
0
|
|
147
|
pPqP
|
pR
|
inconsistent
|
None
|
0
|
|
148
|
pPqP
|
qR
|
inconsistent
|
None
|
0
|
|
172
|
pP
|
mqR
|
inconsistent
|
None
|
0
|
|
173
|
pP
|
npR
|
inconsistent
|
None
|
0
|
|
174
|
pP
|
pRqR
|
inconsistent
|
None
|
0
|
|
177
|
pP
|
pR
|
p-aloneQ
|
151-152,
159-160
|
4
|
|
178
|
pP
|
qR
|
inconsistent
|
None
|
0
|
|
182
|
qP
|
mqR
|
inconsistent
|
None
|
0
|
|
183
|
qP
|
npR
|
inconsistent
|
None
|
0
|
|
184
|
qP
|
pRqR
|
inconsistent
|
None
|
0
|
|
187
|
qP
|
pR
|
inconsistent
|
None
|
0
|
|
188
|
qP
|
qR
|
q-aloneQ
|
106,
122, 234, 250
|
4
|
|
2nd
|
RQ
|
PQ
|
PR
|
|
|
|
222
|
mqP
|
mqR
|
not-cQ
|
74, 202
|
2
|
|
223
|
mqP
|
npR
|
inconsistent
|
None
|
0
|
|
224
|
mqP
|
pRqR
|
inconsistent
|
None
|
0
|
|
227
|
mqP
|
pR
|
inconsistent
|
None
|
0
|
|
228
|
mqP
|
qR
|
not-cQ
|
74, 90, 202, 218
|
4
|
|
232
|
npP
|
mqR
|
inconsistent
|
None
|
0
|
|
233
|
npP
|
npR
|
not-cQ
|
147-148
|
2
|
|
234
|
npP
|
pRqR
|
inconsistent
|
None
|
0
|
|
237
|
npP
|
pR
|
not-cQ
|
147-148,
155-156
|
4
|
|
238
|
npP
|
qR
|
inconsistent
|
None
|
0
|
|
242
|
pPqP
|
mqR
|
inconsistent
|
None
|
0
|
|
243
|
pPqP
|
npR
|
inconsistent
|
None
|
0
|
|
244
|
pPqP
|
pRqR
|
not-cQ
|
220
|
1
|
|
247
|
pPqP
|
pR
|
not-cQ
|
220
|
1
|
|
248
|
pPqP
|
qR
|
not-cQ
|
220
|
1
|
|
272
|
pP
|
mqR
|
inconsistent
|
None
|
0
|
|
273
|
pP
|
npR
|
not-cQ
|
147-148,
211-212
|
4
|
|
274
|
pP
|
pRqR
|
not-cQ
|
220
|
1
|
|
277
|
pP
|
pR
|
not-s
|
147-148, 151-152, 155-156, 159-160, 211-212,
215-216, 219-220, 223-224
|
16
|
|
278
|
pP
|
qR
|
not-cQ
|
220
|
1
|
|
282
|
qP
|
mqR
|
not-cQ
|
74, 76, 202, 204
|
4
|
|
283
|
qP
|
npR
|
inconsistent
|
None
|
0
|
|
284
|
qP
|
pRqR
|
not-cQ
|
220
|
1
|
|
287
|
qP
|
pR
|
not-cQ
|
220
|
1
|
|
288
|
qP
|
qR
|
not-s
|
74,
76, 90, 92, 106, 108, 122, 124, 202, 204, 218, 220, 234, 236, 250, 252
|
16
|
|
3rd
|
QR
|
QP
|
PR
|
|
|
|
322
|
mqP
|
mqR
|
inconsistent
|
None
|
0
|
|
323
|
mqP
|
npR
|
inconsistent
|
None
|
0
|
|
324
|
mqP
|
pRqR
|
inconsistent
|
None
|
0
|
|
327
|
mqP
|
pR
|
inconsistent
|
None
|
0
|
|
328
|
mqP
|
qR
|
inconsistent
|
None
|
0
|
|
332
|
npP
|
mqR
|
inconsistent
|
None
|
0
|
|
333
|
npP
|
npR
|
inconsistent
|
None
|
0
|
|
334
|
npP
|
pRqR
|
inconsistent
|
None
|
0
|
|
337
|
npP
|
pR
|
inconsistent
|
None
|
0
|
|
338
|
npP
|
qR
|
inconsistent
|
None
|
0
|
|
342
|
pPqP
|
mqR
|
inconsistent
|
None
|
0
|
|
343
|
pPqP
|
npR
|
inconsistent
|
None
|
0
|
|
344
|
pPqP
|
pRqR
|
inconsistent
|
None
|
0
|
|
347
|
pPqP
|
pR
|
inconsistent
|
None
|
0
|
|
348
|
pPqP
|
qR
|
inconsistent
|
None
|
0
|
|
372
|
pP
|
mqR
|
inconsistent
|
None
|
0
|
|
373
|
pP
|
npR
|
inconsistent
|
None
|
0
|
|
374
|
pP
|
pRqR
|
inconsistent
|
None
|
0
|
|
377
|
pP
|
pR
|
p-aloneQ
|
151-152,
183-184
|
4
|
|
378
|
pP
|
qR
|
inconsistent
|
None
|
0
|
|
382
|
qP
|
mqR
|
inconsistent
|
None
|
0
|
|
383
|
qP
|
npR
|
inconsistent
|
None
|
0
|
|
384
|
qP
|
pRqR
|
inconsistent
|
None
|
0
|
|
387
|
qP
|
pR
|
inconsistent
|
None
|
0
|
|
388
|
qP
|
qR
|
q-aloneQ
|
106, 110, 234, 238
|
4
|
We can similarly look into syllogisms involving vague
positive premises, absolute or relative weak. The following table gives some
examples.
Table 15.7.
Moduses of conclusions for selected combinations of relative weak and
absolute vague positive premises (various subfigures).
|
Fig.
|
Major
|
Minor
|
Conclusion
|
Common
moduses
|
No.
|
|
1st
|
QR
|
PQ
|
PR
|
|
|
|
|
s
|
wR
|
not-s
|
74, 90, 147-148, 155-156, 202, 218
|
8
|
|
|
w
|
wR
|
not-s
|
74, 76, 90, 92, 106, 108, 122, 124, 147-148,
151-152, 155-156, 159-160, 202, 204, 211-212, 215-216, 218-220, 223-224,
234, 236, 250, 252
|
31
|
|
|
c
|
wR
|
not-s
|
74, 76, 90, 92, 106, 108, 122, 124, 147-148,
151-152, 155-156, 159-160, 202, 204, 211-212, 215-216, 218-220, 223-224,
234, 236, 250, 252
|
31
|
|
|
wP
|
s
|
inconsistent
|
None
|
0
|
|
|
wP
|
w
|
wQ
|
106, 110, 122, 126, 150-152, 158-160, 170,
174, 182-184, 186, 190-192, 234, 238, 250, 254
|
23
|
|
|
wP
|
c
|
wQ
|
106, 110, 122, 126, 150-152, 158-160, 170,
174, 182-184, 186, 190-192, 234, 238, 250, 254
|
23
|
|
|
wP
|
wR
|
wQ
|
106, 122, 151-152, 159-160, 234, 250
|
8
|
|
2nd
|
RQ
|
PQ
|
PR
|
|
|
|
|
s
|
wR
|
not-s
|
74, 90, 147-148, 155-156, 202, 218
|
8
|
|
|
w
|
wR
|
not-s
|
74, 76, 90, 92, 106, 108, 122, 124, 147-148,
151-152, 155-156, 159-160, 202, 204, 211-212, 215-216, 218-220, 223-224,
234, 236, 250, 252
|
31
|
|
|
c
|
wR
|
not-s
|
74, 76, 90, 92, 106, 108, 122, 124, 147-148,
151-152, 155-156, 159-160, 202, 204, 211-212, 215-216, 218-220, 223-224,
234, 236, 250, 252
|
31
|
|
|
wP
|
s
|
not-s
|
74, 76, 147-148, 202, 204, 211-212
|
8
|
|
|
wP
|
w
|
not-s
|
74, 76, 90, 92, 106, 108, 122, 124, 147-148,
151-152, 155-156, 159-160, 202, 204, 211-212, 215-216, 218-220, 223-224,
234, 236, 250, 252
|
31
|
|
|
wP
|
c
|
not-s
|
74, 76, 90, 92, 106, 108, 122, 124, 147-148,
151-152, 155-156, 159-160, 202, 204, 211-212, 215-216, 218-220, 223-224,
234, 236, 250, 252
|
31
|
|
|
wP
|
wR
|
not-s
|
74,
76, 90, 92, 106, 108, 122, 124, 147-148, 151-152, 155-156, 159-160, 202,
204, 211-212, 215-216, 218-220, 223-224, 234, 236, 250, 252
|
31
|
|
3rd
|
QR
|
QP
|
PR
|
|
|
|
|
s
|
wR
|
inconsistent
|
None
|
0
|
|
|
w
|
wR
|
w
|
106, 108, 110, 112, 136, 151-152, 168,
183-184, 200, 215-216, 226, 228, 230, 232, 234, 236, 238, 240, 247-248
|
23
|
|
|
c
|
wR
|
w
|
106, 108, 110, 112, 136, 151-152, 168,
183-184, 200, 215-216, 226, 228, 230, 232, 234, 236, 238, 240, 247-248
|
23
|
|
|
wP
|
s
|
inconsistent
|
None
|
0
|
|
|
wP
|
w
|
wQ
|
106, 110, 122, 126, 150-152, 158-160, 170,
174, 182-184, 186, 190-192, 234, 238, 250, 254
|
23
|
|
|
wP
|
c
|
wQ
|
106, 110, 122, 126, 150-152, 158-160, 170,
174, 182-184, 186, 190-192, 234, 238, 250, 254
|
23
|
|
|
wP
|
wR
|
wQ
|
106, 110, 151-152, 183-184, 234, 238
|
8
|
Note that some combinations are inconsistent. Some moods
yield conclusion w in relative (which
implies absolute) form, and some only in absolute form. In other cases, though
the number of common moduses may vary considerably, there is no conclusion other
than the form not-s, which (remember)
means ‘not strong causation’, implying both not-m
+ not-n; it can also be read as the disjunction wabs
or not-cabs.
Next Section
(continuation of same chapter)
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