THE LOGIC OF CAUSATION
Phase Two: Microanalysis
Chapter 12 – Systematic Microanalysis
2. Moduses in a Two-Item Framework.
3. Catalogue of Moduses, for Three Items.
4.Enumeration of Moduses, for Three Items
4. Enumeration of Moduses, for Three Items.
The following table interprets the preceding, by enumeration of the alternative moduses of the main causative forms. It is based on the known characteristics of positive strong and weak generics, i.e. the moduses given in Tables 1-6 of the previous chapter. From this initial information, we can, using the processes of negation, intersection and merger, infer the alternative moduses of derivative forms, i.e. negatives, as well as joints and vaguer forms (s, w, c), and their negations.
We shall deal here only with the absolute weak determinations and their derivatives; relative weaks and their derivatives will be considered in the next chapter.
Table 12.4. Enumeration of three-item moduses for the generic determinations and their derivatives (form PR).
Determination | Modus numbers | Comment |
Strongs and their negations: | ||
m | 34, 36-40, 42, 44-48, 130, 132-136, 138, 140-144, 162, 164-168, 170, 172-176 | 36 alternatives, by macroanalysis. |
n | 34, 37-38, 50, 53-54, 98, 101-102, 114, 117-118, 130, 133-134, 146, 149-150, 162, 165-166, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 | 36 alternatives, by macroanalysis. |
not-m | 2-33, 35, 41, 43, 49-129, 131, 137, 139, 145-161, 163, 169, 171, 177-256 | All alternatives but those of m, i.e. 219 cases. |
not-n | 2-33, 35-36, 39-49, 51-52, 55-97, 99-100, 103-113, 115-116, 119-129, 131-132, 135-145, 147-148, 151-161, 163-164, 167-177, 179-180, 183-193, 195-196, 199-209, 211-212, 215-225, 227-228, 231-241, 243-244, 247-256 | All alternatives but those of n, i.e. 219 cases. |
Absolute weaks and their negations: | ||
p_{abs} | 50, 52-56, 58, 60-64, 98, 100-104, 106, 108-112, 114, 116-120, 122, 124-128, 146, 148-152, 154, 156-160, 178, 180-184, 186, 188-192, 194, 196-200, 202, 204-208, 210, 212-216, 218, 220-224, 226, 228-232, 234, 236-240, 242, 244-248, 250, 252-256 | 108 alternatives, by macroanalysis of p_{rel} then contraction and expansion. |
q_{abs} | 36, 39-40, 42, 44-48, 52, 55-56, 58, 60-64, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 132, 135-136, 138, 140-144, 148, 151-152, 154, 156-160, 164, 167-168, 170, 172-176, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 | 108 alternatives, by macroanalysis of q_{rel} then contraction and expansion. |
not-p_{abs} | 2-49, 51, 57, 59, 65-97, 99, 105, 107, 113, 115, 121, 123, 129-145, 147, 153, 155, 161-177, 179, 185, 187, 193, 195, 201, 203, 209, 211, 217, 219, 225, 227, 233, 235, 241, 243, 249, 251 | All alternatives but those of p_{abs}, i.e. 147 cases. |
not-q_{abs} | 2-35, 37-38, 41, 43, 49-51, 53-54, 57, 59, 65-99, 101-102, 105, 107, 113-115, 117-118, 121, 123, 129-131, 133-134, 137, 139, 145-147, 149-150, 153, 155, 161-163, 165-166, 169, 171, 177-179, 181-182, 185, 187, 193-195, 197-198, 201, 203, 209-211, 213-214, 217, 219, 225-227, 229-230, 233, 235, 241-243, 245-246, 249, 251 | All alternatives but those of q_{abs}, i.e. 147 cases. |
Joints (absolute) and their negations: | ||
mn | 34, 37-38, 130, 133-134, 162, 165-166 | Their 9 common alternatives, by intersection. |
mq_{abs} | 36, 39-40, 42, 44-48, 132, 135-136, 138, 140-144, 164, 167-168, 170, 172-176 | Their 27 common alternatives, by intersection. |
np_{abs} | 50, 53-54, 98, 101-102, 114, 117-118, 146, 149-150, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 | Their 27 common alternatives, by intersection. |
p_{abs}q_{abs} | 52, 55-56, 58, 60-64, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 148, 151-152, 154, 156-160, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 | Their 81 common alternatives, by intersection. |
not(mn) | 2-33, 35-36, 39-129, 131-132, 135-161, 163-164, 167-256 | All alternatives but those of mn; i.e. 246 cases. |
not(mq_{abs}) | 2-35, 37-38, 41, 43, 49-131, 133-134, 137, 139, 145-163, 165-166, 169, 171, 177-256 | all alternatives but those of mq_{abs}; i.e. 228 cases. |
not(np_{abs}) | 2-49, 51-52, 55-97, 99-100, 103-113, 115-116, 119-145, 147-148, 151-177, 179-180, 183-193, 195-196, 199-209, 211-212, 215-225, 227-228, 231-241, 243-244, 247-256 | All alternatives but those of np_{abs}; i.e. 228 cases. |
not(p_{abs}q_{abs}) | 2-51, 53-54, 57, 59, 65-99, 101-102, 105, 107, 113-115, 117-118, 121, 123, 129-147, 149-150, 153, 155, 161-179, 181-182, 185, 187, 193-195, 197-198, 201, 203, 209-211, 213-214, 217, 219, 225-227, 229-230, 233, 235, 241-243, 245-246, 249, 251 | All alternatives but those of p_{abs}q_{abs}; i.e. 174 cases. |
Strong causation and its negation: | ||
s = m or n | 34, 36-40, 42, 44-48, 50, 53-54, 98, 101-102, 114, 117-118, 130, 132-136, 138, 140-144, 146, 149-150, 162, 164-168, 170, 172-176, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 | Their 63 separate and common alternatives (including overlap, i.e. mn), by merger. |
not-s = not-m + not-n | 2-33, 35, 41, 43, 49, 51-52, 55-97, 99-100, 103-113, 115-116, 119-129, 131, 137, 139, 145, 147-148, 151-161, 163, 169, 171, 177, 179-180, 183-193, 195-196, 199-209, 211-212, 215-225, 227-228, 231-241, 243-244, 247-256 | All alternatives but the preceding; i.e. 192 cases. |
Absolute weak causation and its negation: | ||
w_{abs} = p_{abs} or q_{abs} | 36, 39-40, 42, 44-48, 50, 52-56, 58, 60-64, 98, 100-104, 106, 108-112, 114, 116-120, 122, 124-128, 132, 135-136, 138, 140-144, 146, 148-152, 154, 156-160, 164, 167-168, 170, 172-176, 178, 180-184, 186, 188-192, 194, 196-200, 202, 204-208, 210, 212-216, 218, 220-224, 226, 228-232, 234, 236-240, 242, 244-248, 250, 252-256 | Their 135 separate and common alternatives (including overlap, i.e. p_{abs}q_{abs}), by merger. |
not- w_{abs} = not-p_{abs} + not-q_{abs} | 2-35, 37-38, 41, 43, 49, 51, 57, 59, 65-97, 99, 105, 107, 113, 115, 121, 123, 129-131, 133-134, 137, 139, 145, 147, 153, 155, 161-163, 165-166, 169, 171, 177, 179, 185, 187, 193, 195, 201, 203, 209, 211, 217, 219, 225, 227, 233, 235, 241, 243, 249, 251 | All alternatives but the preceding; i.e. 120 cases. |
Causation (absolute) and its negation: | ||
c_{abs} = m or n or p_{abs} or q_{abs} | 34, 36-40, 42, 44-48, 50, 52-56, 58, 60-64, 98, 100-104, 106, 108-112, 114, 116-120, 122, 124-128, 130, 132-136, 138, 140-144, 146, 148-152, 154, 156-160, 162, 164-168, 170, 172-176, 178, 180-184, 186, 188-192, 194, 196-200, 202, 204-208, 210, 212-216, 218, 220-224, 226, 228-232, 234, 236-240, 242, 244-248, 250, 252-256 | Their 144 separate and common alternatives (including overlap). |
not-c_{abs} = not-m + not-n + not-p_{abs} + not-q_{abs} | 2-33, 35, 41, 43, 49, 51, 57, 59, 65-97, 99, 105, 107, 113, 115, 121, 123, 129, 131, 137, 139, 145, 147, 153, 155, 161, 163, 169, 171, 177, 179, 185, 187, 193, 195, 201, 203, 209, 211, 217, 219, 225, 227, 233, 235, 241, 243, 249, 251 | All alternatives but the preceding; i.e. 111 cases. |
The results obtained in Table 12.4 can be made to conveniently stand out by color coding each form’s moduses in Table 12.3. This is left to the reader to do.
We need not repeat here what was said before, with reference to the similar table for a two-item framework (Table 12.2); the same comments apply, because the relationships there established are true irrespective of framework. We will, however, highlight something which was less visible before, namely the consistency between various results.
There are never overlaps between contradictory propositions, and their alternatives sum up to 255; also, each generic sums up to two joints (since absolute lones do not exist). For instance, m comprises 36 alternative moduses, the 9 of mn plus the 27 of mq_{abs}; while not-m has the 219 remaining alternatives. Similarly, with regard to n. Likewise, p_{abs} comprises 108 alternatives, the 27 of np_{abs} plus the 81 of p_{abs}q_{abs}; while not-p_{abs} has the 147 remaining alternatives. Similarly, with regard to q_{abs}.
Moreover, the number of moduses corresponding to the vaguer forms are predictable. Thus, s (= m or n) comprises the 36 moduses of m plus the 36 of n, less the 9 of mn[4], a total of 63 alternatives; and its negation has 255 – 63 = 192 alternatives. We can similarly predict the moduses of w_{abs} (= p_{abs} or q_{abs}) to be 108 + 108 – 81 = 135; and a residue of 120 alternatives for its negation. For c (= s or w_{abs}) we have 63 + 135 – 2*27 = 144 (the 54 subtracted being those of mq_{abs} and np_{abs} – i.e. of sw_{abs}); for its negation, 111.
Thus, incidentally, causation in all its forms covers more than half the matrix, but still leaves a large space to non-causation.
Now let us compare the results in Tables 2 and 4. They are essentially the same tables, except that each modus of the first is, as it were, further subdivided into a number of moduses in the second. However, the subdivision is evidently not proportional, say in the ratio 16:256; you cannot just say that to each two-item modus there corresponds 16 three-item ones. The following table makes this disproportionality clear:
Table 12.5. Numbers of Moduses for Positive Forms, in Different Frameworks.
Framework | m,n | p_{abs},q_{abs} | mn | mq_{abs},np_{abs} | p_{abs}q_{abs} | s | w_{abs} | c |
Two-Item | 2 | 2 | 1 | 1 | 1 | 3 | 3 | 4 |
Three-Item | 36 | 108 | 9 | 27 | 81 | 63 | 135 | 144 |
The explanation is easy. Expansion of a two-item alternative modus into a number of three-item moduses depends on how many zero or one codes it involves. For, as we saw in the previous chapter (with the proviso of appropriate locations), each ‘0’ in a two-item framework has a single expression (‘0 0’) in the three-item framework; whereas each ‘1’ in the former has three expressions in the latter (‘0 1’, ‘1 0’ or ‘1 1’ – i.e. any but ‘0 0’).
Thus, if a two-item modus involves four ‘zeros’ and no ‘one’, its three-item equivalent will consist of 1*1*1*1 = 1 (equally impossible) modus; if the former involves three zeros and a one, the latter will consist of 1*1*1*3 = 3 moduses; if the former involves two zeros and two ones, the latter will consist of 1*1*3*3 = 9 moduses; if the former involves one zero and three ones, the latter will consist of 1*3*3*3 = 27 moduses; and if the former involves no zero and four ones, the latter will consist of 3*3*3*3 = 81 moduses.
Whence, the strongs m, n, which each involves two two-item moduses, one with two zeros (No. 10) and one with a single zero (no. 12 or 14), will have 9 + 27 = 36 three-item moduses; whereas, the weaks p_{abs}, q_{abs}, which each involves two two-item moduses, one with a single zero (no. 12 or 14) and one with no zero (No. 16) and will have 27 + 81 = 108 three-item moduses.
The numbers of three-item moduses for the conjunctions and disjunctions of these forms follow. The joint mn (two-item modus No. 10) will have 9 of them; mq_{abs} (modus No. 12) and np_{abs} (modus No. 14) will each have 27; and p_{abs}q_{abs} (modus 16) will have 81. The vague form s (moduses 10, 12, 14) will have 9 + 2*27 = 63; w_{abs} (moduses 12, 14, 16) will have 2*27 + 81 = 135; and c (moduses 10, 12, 14, 16) will have 9 + 2*27 + 81 = 144.
We can proceed in a like manner to predict expansions of negative forms. Furthermore, given the two-item modus(es) of a form, we can predict not only how many moduses it will have in a three-item framework, but precisely which moduses it will have. Thus, a table of equivalencies between the two frameworks can be constructed without difficulty. In short, we have here a functioning calculus.
The precise three-item modus(es) corresponding to each two-item modus are given in the following table:
Table 12.6. Correspondences between two- and three item frameworks.
Two-item modus | No. of zeros in it | Corresponding three-item modus numbers | No. of moduses |
1 | 4 | 1 | 1 |
2 | 3 | 2, 5, 6 | 3 |
3 | 3 | 3, 9, 11 | 3 |
4 | 2 | 4, 7, 8, 10, 12-16 | 9 |
5 | 3 | 17, 65, 81 | 3 |
6 | 2 | 18, 21-22, 66, 69-70, 82, 85-86 | 9 |
7 | 2 | 19, 25, 27, 67, 73, 75, 83, 89, 91 | 9 |
8 | 1 | 20, 23-24, 26, 28-32, 68, 71-72, 74, 76-80, 84, 87-88, 90, 92-96 | 27 |
9 | 3 | 33, 129, 161 | 3 |
10 | 2 | 34, 37-38, 130, 133, 134, 162, 165-166 | 9 |
11 | 2 | 35, 41, 43, 131, 137, 139, 163, 169, 171 | 9 |
12 | 1 | 36, 39-40, 42, 44-48, 132, 135-136, 138, 140-144, 164, 167-168, 170, 172-176 | 27 |
13 | 2 | 49, 97, 113, 145, 177, 193, 209, 225, 241 | 9 |
14 | 1 | 50, 53-54, 98, 101-102, 114, 117-118, 146, 149-150, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 | 27 |
15 | 1 | 51, 57, 59, 99, 105, 107, 115, 121, 123, 147, 153, 155, 179, 185, 187, 195, 201, 203, 211, 217, 219, 227, 233, 235, 243, 249, 251 | 27 |
16 | 0 | 52, 55-56, 58, 60-64, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 148, 151-152, 154, 156-160, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 | 81 |
16 | Total number of moduses | 256 |
Needless to say, each modus will occur only once in the above table, making a total of 16 or 256 moduses, according to the framework. Clearly, if we had developed this table earlier, we could have derived Table 12.4 from Table 12.2.[5]
Obviously, we can follow the same procedures to expand three-item alternative moduses (of which there are 256) into four-item alternative moduses (of which there are 65,536 – as seen earlier).
The number and configuration of the latter will emerge from the each of the former, in accordance with the number of zero and one codes it contains and the way they are arrayed within it (i.e. the incidence, prevalence and locations of zero and one codes in it). A table of correspondences can thus be constructed, which details the results obtained in each case.
We have above identified the main lines of what might be called the two-three (2/3) table of correspondences, emerging from the operation of expansion of ‘0’ into ‘0 0’ and ‘1’ into ‘0 1’, ‘1 0’, ‘1 1’ (all pairs but ‘0 0’). We could thereafter, step by step, build similar tables of correspondence of size 3/4 or 4/5… and so forth on to infinity, if need arise to resolve eventual issues.
For instance, from a three-item matrix (which has 8 rows) to a four-item matrix, each combination of zeros and ones will result in a product of eight factors of 1 (for ‘0’ codes) or 3 (for ‘1’ codes) – e.g., a modus with 1 zero and 7 ones will become 1*3*3*3*3*3*3*3 = 2187 moduses, in various possible permutations. These are long-winded techniques, which may or may not be needed.
[4] So as to avoid double accounting of mn, which is implicit in both m and n.
[5] I would like to slip in an unrelated comment here, regarding summary moduses, for the record. We could predict all conceivable summary moduses, within a two-item or three-item framework. Such lists would include all alternative moduses, since summary moduses may also be free of dots, and thus constitute enlarged grand matrices with additional columns and numbers. I do not do this in view of the inadequacies, which we encountered in the previous chapter, of the notation system adopted in this work for summary moduses.