|
The Logician
© Avi Sion
All rights reserved
General Sitemap
Search
Contact
| |
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
2.
Dealing with Negatives.
Let us now sketch the application of our methods to the
solution of syllogisms involving a negative premise or two. We start as always
by identifying the alternative moduses of the various possible premises, then we
list their conjunctions in syllogisms of various figures, and with reference to
the common alternative moduses, if any, evaluate their possible conclusions.
We have learned in the preceding chapters and the
above section the lessons that: the number of common moduses in the premises may
be small, yet yield a conclusion; or it may be large, yet yield no conclusion.
The less number of alternative moduses there are in the premises, the less
likely are they to intersect, and be compatible and yield a conclusion. On the
other hand, the more number of alternative moduses in the premises, the more
often they will have some in common, but also the more likely will there be a
vague conclusion, or no conclusion at all by virtue of including moduses of both
causation and non-causation. These generalities are equally applicable to the
findings below.
In the following table, the alternative moduses of the
strong and absolute weak determinations are obtained by negation from Table
14.3. Those of the relative weaks are obtained by negation from Table 15.3
above. All the rest follows by intersection.
Table 15.8a.
Enumeration of three-item alternative moduses for negative premises, for
any figure of syllogism (generic forms).
|
Determination
|
Major QR
|
Major RQ
|
Minor PQ
|
Minor QP
|
|
not-m
|
2-9, 11, 13-24, 29-41, 43, 45-56, 61-129,
131, 133-137, 139, 141-144, 149-152, 157-161, 163, 165-169, 171,
173-176, 181-184, 189-256
|
2-9, 11-13, 15-24, 27-28, 31-73, 75-77,
79-88, 91-92, 95-129, 131-133, 135-137, 139-141, 143-144, 147-148,
151-152, 155-156, 159-193, 195-197, 199-201, 203-205, 207-208, 211-212,
215-216, 219-220, 223-256
|
2-65, 69, 73, 77, 81-129, 133, 137, 141,
145-193, 197, 201, 205, 209-256
|
2-65, 69-81, 85-97, 101-113, 117-129,
133-145, 149-161, 165-177, 181-193, 197-209, 213-225, 229-241, 245-256
|
|
not-n
|
2-9, 11-13, 15-24, 27-28, 31-73, 75-77,
79-88, 91-92, 95-129, 131-133, 135-137, 139-141, 143-144, 147-148,
151-152, 155-156, 159-193, 195-197, 199-201, 203-205, 207-208, 211-212,
215-216, 219-220, 223-256
|
2-9, 11, 13-24, 29-41, 43, 45-56, 61-129,
131, 133-137, 139, 141-144, 149-152, 157-161, 163, 165-169, 171,
173-176, 181-184, 189-256
|
2-65, 69-81, 85-97, 101-113, 117-129,
133-145, 149-161, 165-177, 181-193, 197-209, 213-225, 229-241, 245-256
|
2-65, 69, 73, 77, 81-129, 133, 137, 141,
145-193, 197, 201, 205, 209-256
|
|
not-pabs
|
2-13, 15, 17-28, 33-45, 47, 49-60, 65-73, 75,
77, 79, 81-88, 97-105, 107, 109, 111, 113-120, 129-133, 135, 137-141,
143, 145-148, 153-156, 161-165, 167, 169-173, 175, 177-180, 185-188,
193, 195, 197, 199, 201, 203, 205, 207, 225, 227, 229, 231, 233, 235,
237, 239
|
2-11, 13-15, 17-26, 29-30, 33-41, 43, 45, 47,
49-56, 65-75, 77-79, 81-90, 93-94, 97-105, 107, 109, 111, 113-120,
129-131, 133-135, 137-139, 141-143, 145-146, 149-150, 153-154, 157-158,
161, 163, 165, 167, 169, 171, 173, 175, 193-195, 197-199, 201-203,
205-207, 209-210, 213-214, 217-218, 221-222, 225, 227, 229, 231, 233,
235, 237, 239
|
2-81, 85, 89, 93, 97, 101, 105, 109, 113,
117, 121, 125, 129-145, 149, 153, 157, 161, 165, 169, 173, 177, 181,
185, 189, 193-209, 213, 217, 221, 225, 229, 233, 237, 241, 245, 249, 253
|
2-69, 73, 77, 81-85, 89, 93, 97-101, 105,
109, 113-117, 121, 125, 129-133, 137, 141, 145-149, 153, 157, 161-165,
169, 173, 177-181, 185, 189, 193-197, 201, 205, 209-213, 217, 221,
225-229, 233, 237, 241-245, 249, 253
|
|
not-qabs
|
2-11, 13-15, 17-26, 29-30, 33-41, 43, 45, 47,
49-56, 65-75, 77-79, 81-90, 93-94, 97-105, 107, 109, 111, 113-120,
129-131, 133-135, 137-139, 141-143, 145-146, 149-150, 153-154, 157-158,
161, 163, 165, 167, 169, 171, 173, 175, 193-195, 197-199, 201-203,
205-207, 209-210, 213-214, 217-218, 221-222, 225, 227, 229, 231, 233,
235, 237, 239
|
2-13, 15, 17-28, 33-45, 47, 49-60, 65-73, 75,
77, 79, 81-88, 97-105, 107, 109, 111, 113-120, 129-133, 135, 137-141,
143, 145-148, 153-156, 161-165, 167, 169-173, 175, 177-180, 185-188,
193, 195, 197, 199, 201, 203, 205, 207, 225, 227, 229, 231, 233, 235,
237, 239
|
2-69, 73, 77, 81-85, 89, 93, 97-101, 105,
109, 113-117, 121, 125, 129-133, 137, 141, 145-149, 153, 157, 161-165,
169, 173, 177-181, 185, 189, 193-197, 201, 205, 209-213, 217, 221,
225-229, 233, 237, 241-245, 249, 253
|
2-81, 85, 89, 93, 97, 101, 105, 109, 113,
117, 121, 125, 129-145, 149, 153, 157, 161, 165, 169, 173, 177, 181,
185, 189, 193-209, 213, 217, 221, 225, 229, 233, 237, 241, 245, 249, 253
|
|
not-prel
|
2-148, 153-156, 161-180, 185-188, 193-256
|
2-146, 149-150, 153-154, 157-158, 161-210,
213-214, 217-218, 221-222, 225-256
|
2-146, 149-150, 153-154, 157-158, 161-210,
213-214, 217-218, 221-222, 225-256
|
2-134, 137-150, 153-166, 169-182, 185-198,
201-214, 217-230, 233-246, 249-256
|
|
not-qrel
|
2-41, 43-45, 47-57, 59-61, 63-105, 107-109,
111-121, 123-125, 127-169, 171-173, 175-185, 187-189, 191-233, 235-237,
239-249, 251-253, 255-256
|
2-73, 75, 77-89, 91, 93-105, 107, 109-121,
123, 125-201, 203, 205-217, 219, 221-233, 235, 237-249, 251, 253-256
|
2-73, 75, 77-89, 91, 93-105, 107, 109-121,
123, 125-201, 203, 205-217, 219, 221-233, 235, 237-249, 251, 253-256
|
2-97, 99, 101, 103, 105, 107, 109, 111,
113-225, 227, 229, 231, 233, 235, 237, 239, 241-256
|
Readers are encouraged to continue this research as an
exercise, by filling in a table like that below for both absolute and relative
weak determinations. The negations of the joint determinations can be worked out
with reference to Table 14.3.
Table 15.8b.
Enumeration of three-item alternative moduses for negative premises, for
any figure of syllogism (specific forms).
|
Determination
|
Major QR
|
Major RQ
|
Minor PQ
|
Minor QP
|
|
not(mn)
|
|
|
|
|
|
not(mq)
|
|
|
|
|
|
not(np)
|
|
|
|
|
|
not(pq)
|
|
|
|
|
|
m
+ not-n
|
|
|
|
|
|
not-m
+ n
|
|
|
|
|
|
not-m
+ not-n
=
not(m or n)
|
|
|
|
|
|
m
+ not-p
|
|
|
|
|
|
not-m
+ p
|
|
|
|
|
|
not-m
+ not-p
=
not(m or p)
|
|
|
|
|
|
n
+ not-q
|
|
|
|
|
|
not-n
+ q
|
|
|
|
|
|
not-n
+ not-q
=
not(n or q)
|
|
|
|
|
|
m
+ not-q
|
|
|
|
|
|
not-m
+ q
|
|
|
|
|
|
not-m
+ not-q
=
not(m or q)
|
|
|
|
|
|
n
+ not-p
|
|
|
|
|
|
not-n
+ p
|
|
|
|
|
|
not-n
+ not-p
=
not(n or p)
|
|
|
|
|
|
p
+ not-q
|
|
|
|
|
|
not-p
+ q
|
|
|
|
|
|
not-p
+ not-q
=
not(p or q)
|
|
|
|
|
|
no-causation
|
|
|
|
|
Once the researcher has filled in the above table, he or
she may investigate all interesting combinations of premises, and see whether
they are compatible (i.e. have common alternative moduses), and if so what
conclusions, if any, may be drawn (i.e. whether the common moduses all fall
under some form of causation or non-causation). I will do the job in the
following table for some selected negative premises, and leave it to the reader
to finish the table with a more exhaustive treatment, involving mixtures of
positive and negative premises.
Table 15.9.
Moduses of conclusions for selected (generic, absolute) negative
premises, in Figure 1.
|
Fig.
|
Major
|
Minor
|
Conclusion
|
Common
moduses
|
No.
|
|
1st
|
QR
|
PQ
|
PR
|
|
|
|
|
not-m
|
not-m
|
no
conclusion
|
2-9, 11, 13-24, 29-41, 43, 45-56, 61-65, 69,
73, 77, 81-129, 133, 137, 141, 149-152, 157-161, 163, 165-169, 171,
173-176, 181-184, 189-193, 197, 201, 205, 209-256
|
187
|
|
|
not-n
|
not-n
|
no
conclusion
|
2-9, 11-13, 15-24, 27-28, 31-65, 69-73,
75-77, 79-81, 85-88, 91-92, 95-97, 101-113, 117-129, 133, 135-137,
139-141, 143-144, 151-152, 155-156, 159-161, 165-177, 181-193, 197,
199-201, 203-205, 207-208, 215-216, 219-220, 223-225, 229-241, 245-256
|
187
|
|
|
not-m
|
not-n
|
no
conclusion
|
2-9, 11, 13-24, 29-41, 43, 45-56, 61-65,
69-81, 85-97, 101-113, 117-129, 133-137, 139, 141-144, 149-152, 157-161,
165-169, 171, 173-176, 181-184, 189-193, 197-209, 213-225, 229-241,
245-256
|
193
|
|
|
not-n
|
not-m
|
no
conclusion
|
2-9, 11-13, 15-24, 27-28, 31-65, 69, 73, 77,
81-88, 91-92, 95-129, 133, 137, 141, 147-148, 151-152, 155-156, 159-193,
197, 201, 205, 211-212, 215-216, 219-220, 223-256
|
193
|
|
|
not-m
|
not-p
|
no
conclusion
|
2-9, 11, 13-24, 29-41, 43, 45-56, 61-81, 85,
89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 131, 133-137, 139,
141-144, 149, 157, 161, 165, 169, 173, 181, 189, 193-209, 213, 217, 221,
225, 229, 233, 237, 241, 245, 249, 253
|
127
|
|
|
not-n
|
not-q
|
no
conclusion
|
2-9, 11-13, 15-24, 27-28, 31-69, 73, 77,
81-85, 97-101, 105, 109, 113-117, 121, 125, 129, 131-133, 137, 141,
147-148, 161-165, 169, 173, 177-181, 185, 189, 193, 195-197, 201, 205,
211-212, 225-229, 233, 237, 241-245, 249, 253
|
127
|
|
|
not-m
|
not-q
|
no
conclusion
|
2-9, 11, 13-24, 29-41, 43, 45-56, 61-69, 73,
77, 81-85, 89, 93, 97-101, 105, 109, 113-117, 121, 125, 129, 131, 133,
137, 141, 149, 157, 161, 163, 165, 169, 173, 181, 189, 193-197, 201,
205, 209-213, 217, 221, 225-229, 233, 237, 241-245, 249, 253
|
121
|
|
|
not-n
|
not-p
|
no
conclusion
|
2-9, 11-13, 15-24, 27-28, 31-73, 75-77,
79-81, 85, 97, 101, 105, 109, 113, 117, 121, 125, 129, 131-133, 135-137,
139-141, 143-144, 161, 165, 169, 173, 177, 181, 185, 189, 193, 195-197,
199-201, 203-205, 207-208, 225, 229, 233, 237, 241, 245, 249, 253
|
121
|
|
|
not-p
|
not-m
|
no
conclusion
|
2-13, 15, 17-28, 33-45, 47, 49-60, 65, 69,
73, 77, 81-88, 97-105, 107, 109, 111, 113-120, 129, 133, 137, 141,
145-148, 153-156, 161-165, 167, 169-173, 175, 177-180, 185-188, 193,
197, 201, 205, 225, 227, 229, 231, 233, 235, 237, 239
|
127
|
|
|
not-q
|
not-n
|
no
conclusion
|
2-11, 13-15, 17-26, 29-30, 33-41, 43, 45, 47,
49-56, 65, 69-75, 77-79, 81, 85-90, 93-94, 97, 101-105, 107, 109, 111,
113, 117-120, 129, 133-135, 137-139, 141-143, 145, 149-150, 153-154,
157-158, 161, 165, 167, 169, 171, 173, 175, 193, 197-199, 201-203,
205-207, 209, 213-214, 217-218, 221-222, 225, 229, 231, 233, 235, 237,
239
|
127
|
|
|
not-p
|
not-n
|
no
conclusion
|
2-13, 15, 17-28, 33-45, 47, 49-60, 65, 69-73,
75, 77, 79, 81, 85-88, 97, 101-105, 107, 109, 111, 113, 117-120, 129,
133, 135, 137-141, 143, 145, 153-156, 161, 165, 167, 169-173, 175, 177,
185-188, 193, 197, 199, 201, 203, 205, 207, 225, 229, 231, 233, 235,
237, 239
|
121
|
|
|
not-q
|
not-m
|
no
conclusion
|
2-11, 13-15, 17-26, 29-30, 33-41, 43, 45, 47,
49-56, 65, 69, 73, 77, 81-90, 93-94, 97-105, 107, 109, 111, 113-120,
129, 133, 137, 141, 145-146, 149-150, 153-154, 157-158, 161, 163, 165,
167, 169, 171, 173, 175, 193, 197, 201, 205, 209-210, 213-214, 217-218,
221-222, 225, 227, 229, 231, 233, 235, 237, 239
|
121
|
|
|
not-p
|
not-p
|
no
conclusion
|
2-13, 15, 17-28, 33-45, 47, 49-60, 65-73, 75,
77, 79, 81, 85, 97, 101, 105, 109, 113, 117, 129-133, 135, 137-141, 143,
145, 153, 161, 165, 169, 173, 177, 185, 193, 195, 197, 199, 201, 203,
205, 207, 225, 229, 233, 237
|
103
|
|
|
not-q
|
not-q
|
no
conclusion
|
2-11, 13-15, 17-26, 29-30, 33-41, 43, 45, 47,
49-56, 65-69, 73, 77, 81-85, 89, 93, 97-101, 105, 109, 113-117, 129-131,
133, 137, 141, 145-146, 149, 153, 157, 161, 163, 165, 169, 173, 193-195,
197, 201, 205, 209-210, 213, 217, 221, 225, 227, 229, 233, 237
|
103
|
|
|
not-p
|
not-q
|
no
conclusion
|
2-13, 15, 17-28, 33-45, 47, 49-60, 65-69, 73,
77, 81-85, 97-101, 105, 109, 113-117, 129-133, 137, 141, 145-148, 153,
161-165, 169, 173, 177-180, 185, 193, 195, 197, 201, 205, 225, 227, 229,
233, 237
|
109
|
|
|
not-q
|
not-p
|
no
conclusion
|
2-11, 13-15, 17-26, 29-30, 33-41, 43, 45, 47,
49-56, 65-75, 77-79, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117,
129-131, 133-135, 137-139, 141-143, 145, 149, 153, 157, 161, 165, 169,
173, 193-195, 197-199, 201-203, 205-207, 209, 213, 217, 221, 225, 229,
233, 237
|
109
|
|
2nd
|
| |
