THE LOGIC OF CAUSATION
Phase Two: Microanalysis
Chapter 15 – Some More Three-Item Syllogisms
1.Special Cases of Three-Item Syllogism.
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 | RQ | PQ | PR | ||
3rd | QR | QP | PR | ||
We can similarly proceed for the same combinations of generic, absolute, negative premises in Figures 2 and 3. But there is no need to, for it is easy to predict that the conclusion will be “no conclusion (of the form PR)” in all similar cases. If you look at all the cells of Table 15.8a, you will observe that they have certain alternative moduses in common to all of them – for instances Nos. 2 and 237. Since modus 2 is an alternative of non-causation (not-cabs) and modus 237 is an alternative of causation (cabs), as shown in Table 12.4, no pair of the premises listed in the above table can yield any causative conclusion.
Rather, our next step would be to develop syllogisms with negative premises from Table 15.8b, as well as 15.8a, and then – and probably much more fruitfully – syllogisms with mixtures of positive and negative premises. This can and should be carried out till all possible combinations of all conceivable forms are carefully exhausted. I leave the job to other researchers (any reader willing to do it), so as to move on and deal with other issues in causative logic.
