THE LOGIC OF CAUSATION
Phase Three: Software Assisted Analysis
Chapter 18 – Moduses of the Forms.
1. 2-Item Framework Moduses.
The next set of tables was produced with several purposes in mind. As we have established in the past, all propositions relating to causation – and indeed all their underlying categorical (individual or conjunctive) and conditional bases and connections – can be expressed entirely with reference to the moduses in the relevant matricial framework, depending on the number of items involved.
Thus, the main task of causative logic is to systematically identify the moduses of all the forms of causation. This work has of course already been largely done in phase II, but here our task was to do it mechanically instead of manually and to develop it from the 2- and 3- item frameworks to the 4-item framework. I first produced the following two tables, which are on display at the The Logician website.
Table 18.1 – 2-item PR Moduses of Forms. (6 pages in pdf file).
Table 18.2 – 2-item PR Moduses of Forms – Formulae Used. (1 page in pdf file).
The first table comprises a mass of information previously scattered in several tables (notably, 12.1, 12.2, 13.3, 13.10, 13.11, 13.12, and 16.1). The second table is merely an auxiliary one, showing (for the record) the formulae used to generate the first table. See the notes at the bottom of each of these tables for further information. All the results obtained in Table 18.1 were compared to corresponding results obtained in phase II, and they were found in agreement – showing that no errors were made in either research.
Notice that Table 18.1, which totals 79 columns, can be divided into distinct segments: the first segment shows the 2-item matrix from which all subsequent values are calculated through transparent formulae. The next segment lists the moduses for each of the individual propositions involved (P is possible, P is impossible, etc.). The following segment similarly treats conjunctions of the two items (P+R is possible, P+R is impossible, etc.). Then comes the moduses of the generic forms of causation and their negations (m, n, p, q, etc.). This is followed by the moduses of the specific forms of causation and their negations (mn, mq, np, pq, etc.). The segment after that deals with absolute lone forms (they are confirmed to be non-existent) and vaguer forms of causation and their negations (s, w, c, etc.).
The next segments deal with prevention, inverse causation and inverse prevention, connection and their negations. Prevention by P of R, remember, means causation by P of notR. Inverse causation by P of R refers to causation by notP of notR; and inverse prevention by P of R refers to causation by notP of R. Although these are all derivable from causation, they needed to be shown here to obtain the very last segment of the table, which gives us a full verbal interpretation of each and every modus. The interpretations in this table will be discussed at length a bit further on (section 4).
A valuable insight I had while preparing Table 18.1, which I must mention here, is that when in it I equate, say, “if P, then R” to “P + notR is impossible”, I have in mind de dicta (logical) conditioning, which in accordance with common practice only requires the ‘connection’ to be specified. But it is clear that if we wish to deal with de re (extensional, natural or temporal) conditioning, we cannot make this equation, for here the ‘base’ that “P is possible” (whence, “R is possible” too) must be specified as well[1]. This is important only insofar as we are still dealing with conditional propositions as such; for as soon as we get into causative propositions, the ‘bases’ are always tacitly implied anyway, so the logic for de dicta and de re is the same.
2. 3-Item Framework Moduses.
Although Table 18.1 (and its auxiliary 18.2) contains little new information on the 2-item moduses of forms, its production was very useful to producing an equivalent table for the 3-item framework, because I could copy the formulae used in the former and paste them in the latter, and then expand them to apply to the enlarged framework. This saved a lot of time and trouble. Of course, the 3-item framework table involved many new forms, but even these could be derived from the preceding using appropriate equivalence tables. The result was a very large spreadsheet of 415 columns (including 10 for the matrix).
Rather than present all this data in one massive table, which readers would get lost in and so miss out on important information, I split it up for publication into smaller tables. These include Tables 18.3-18.6, which respectively deal with categorical and conditional forms, causation, prevention, and interpretation of the moduses. To which must be added two auxiliary tables, Tables 18.7-18.8, which spell out the formulae used and the equivalences exploited in producing the original big table. We thus have six tables[2] for the 3-item framework, which are as usual posted at the The Logician website for your scrutiny:
Table 18.3 – 3-item PQR Moduses of Forms – Categoricals and Conditionals. (12 pages in pdf file).
Table 18.4 – 3-item PQR Moduses of Forms – Causation. (18 pages in pdf file).
Table 18.5 – 3-item PQR Moduses of Forms – Prevention. (18 pages in pdf file).
Table 18.6 – 3-item PQR Moduses of Forms – Interpretation. (4 pages in pdf file).
Table 18.7 – 3-item PQR Moduses of Forms – Formulae Used. (8 pages in pdf file).
Table 18.8 – 3-item PQR Moduses of Forms – Equivalences. (1 page in pdf file).
Notice the subdivisions into segments within these tables. Some of the data in these tables has already been generated in phase II, notably in Tables 11.3, 11.4, 12.3, 12.4, 13.1, 13.3, 13.4, 13.7, 13.14, 13.15, 14.3, 14.5. See the notes at the bottom of the segments; notice that wherever information was already given in phase II, results were compared and found consistent. However, much data in these phase III tables is new, generated in pursuit of enlarged scope, more symmetry of treatment, and thorough interpretation of the moduses. The more mechanical nature of data generation in phase III enabled such increased ambition.
Thus, Table 18.3 (96 columns, plus the matrix) shows the 3-item moduses of all possible categorical propositions (P is possible, etc.), then of all 2-item conjunctions (P+R is possible, etc.), and then of all 3-item conjunctions (P+Q+R is possible, etc.) – many of which conjunctions of course signify conditional propositions[3]. Note that every combination and permutation of the three items P, Q, R are treated here, and in the subsequent tables.
Table 18.4 (130 columns, plus the matrix) shows the 3-item moduses of the generic and specific, absolute and relative, forms of causation, including lones and vaguer forms, and their negations (and, for the record, of inverse causation). Note the equal treatment here of forms relative to notQ; the motives of this and similar expansions of scope being, not mere curiosity, but (a) to make possible interpretation of all the moduses in Table 18.6 and (b) to enable us to draw as much conclusion as possible when we get to the syllogistic stage (whereas in phase II, we deliberately limited our possibilities of conclusion).
Table 18.5 (130 columns, plus the matrix) repeats the work of 18.4 with regard to prevention (i.e. causation by P of notR). This is done, again, both to facilitate interpretation of the moduses and to ensure maximization of conclusions at the later syllogistic stage.
Table 18.6 (43 columns) constitutes the crucial interpretation of the results obtained in the preceding two tables. It verbalizes and makes sense of all the information collected in them. The need to develop such a table for the 3-item framework propelled most of the work preceding it. Notice here the symmetry of the results for causation and prevention (as can be expected). Note the use of certain summaries of information on causation, lone causation, prevention and lone prevention. Note also the last columns, concerning modus 16. The interpretations in this table will be discussed at length a bit further on (section 4).
Table 18.7, to repeat, lists the formulae used in producing the original table comprising Tables 18.3-18.6. It did not seem necessary or useful to split this table in four. What is noteworthy here is that most formulae are written in terms of the initial matrix. Table 18.8 reveals how some of the formulae in Table 18.7 were derived from others, simply by reordering and/or changing the polarities of the terms involved. I include it here for the sake of transparency.
3. 4-Item Moduses of the Forms.
Clearly, the 2-item framework table is of value only to begin with, to teach us how to analyze the forms – but this information is not enough to produce all conceivable syllogisms. On the other hand, the 3-item framework does give rise to systematic syllogistic work, so that many forms have to be analyzed in their many guises, i.e. with respect to the various combinations and permutations of the three items P, Q, R, and their negations.
This is of course all the more true in the 4-item framework – but in the latter case we have to be more restrained, otherwise the tables would be far too large for comfort. With this reasoning in mind, I only analyzed in the 4-item framework a selection of forms, the minimum needed to answer some previously unanswered syllogistic questions. The resulting Table 18.9 and its auxiliary 18.10 can be viewed at the The Logician website, as usual:
Table 18.9 – 4-item PQR Moduses of Forms. (2408 pages in pdf file).
Table 18.10 – 4-item PQR Moduses of Forms – Formulae Used. (3 pages in pdf file).
Table 18.9 (173 columns, including 17 for the matrix) shows the 4-item moduses of selected generic and specific, absolute and relative, forms of causation, including some lones, and their negations. This is already a big mass of information to have to deal with. Table 18.10 lists the formulae used to produce Table 18.9.
Notice that the applicable moduses for any of the forms examined are in here signaled by a ‘1’ instead of by the modus number (as in similar tables for 2-items and for 3-items). The reasons for this are simply to avoid overly wide columns and to make the file as a whole more manageable in size. The 1s in the columns of the 4-item table are just meant to indicate that yes, the modus number opposite (to the left of) the cell concerned is a possible modus number for the form concerned. Accordingly, ‘0’ means the adjacent modus is not applicable.
It is important to understand that the 3-item framework is in principle sufficient to fulfill the task of causative logic. That is because two items suffice to define the genera of strong causation and three items suffice to define those of weak causation. From the start of our research, remember (see chapter 2.3), we conceived of partial or contingent causation as consisting of two causes, say P and Q, and one effect R. We arbitrarily viewed P as the main cause and Q as its complement, so as to conjoin the weak forms of causation with strong forms expressed in terms of P and R. Just as the effect R could be a mass of phenomena lumped together under this name, so could P and Q respectively be far from unitary. Thus, by definition, the complement Q was designed to accommodate any number of phenomena – of which Q would be the effective single resultant in the PQR causative formula.
In other words, when there are more than two partial and/or contingent causes, or more to the point when in addition to P we have several complements, Q1, Q2, etc. – we are called upon to first determine a resultant complement Q whose behavior within the causative proposition concerned would correspond to the behavior of the several narrower complements Q1, Q2, etc. By this artifice, we were able to reduce the problem of relative causation to only three items, P, Q, R.
The need for more items than these three arises only at the syllogistic stage of the study of causation, when we need to investigate how relative causation is transmitted from either premise to the conclusion (if any), and what perhaps happens (if anything) when both premises are relative causations each involving a different complement. Thus, conceivably, we might need matrices of four, and maybe even five, items to find all possible syllogistic conclusions.
This issue will be further discussed later on.
4. Interpretation of the Moduses.
We shall now interpret and discuss the individual moduses in the 2-item and 3-item frameworks. It is important to understand at the outset that each modus represents one complete situation – meaning that the two or three items whose relations are found to fit into the pattern symbolized by a certain modus may be said to be causatively or otherwise related as that modus signifies. For this reason, it is important to clearly identify the significance of each modus; such identification has enduring, universal value.
a. Regarding the interpretations of the 2-item moduses, please refer to Table 18.1 (page 6), an extract from which is printed here, below. This table is not new, since it corresponds to Table 16.1 presented in phase II.
This table teaches us that of the 16 moduses in a 2-item framework: one modus is logically impossible anyway (#1); eight moduses have one or both items incontingent (i.e. necessary or impossible) and so cannot signify any causative connection between them (since an item that is incontingent, in the mode of modality concerned, is independent of all else); three moduses signify a strong causation (mn, mq, np) and three more a strong prevention (ditto); and the last modus (#16) refers to both weak (pq) causation and prevention. The weak causations mentioned in this framework are of course all absolute, since we do not know the complement they concern, though it may be assumed that they do concern some complement(s).
For connection (i.e. causation or prevention) to occur and be claimed, the two items concerned have to both be contingent. This occurs only in seven of the moduses, namely numbers 7, 8, 10, 12, 14, 15, 16 (the remaining eight being either impossible or incontingent, as already pointed out). This result is very surprising, for it means that apart from incontingency, logic has found no place from non-connection! That is, this tabulation of possibilities being exhaustive, we are left with no way to rationally express a situation of non-connection between individually contingent items. This seems to imply that any two contingent items in the universe, taken at random, are somewhat connected together, by causation and/or by prevention, to whatever degree (i.e. as mn, mq, np or pq – the latter three referring partly or wholly to absolute weak determinations).
Though I am again alarmed upon encountering this result, it must be stressed that I had already noticed it and tried hard to explain it in phase II (see Chapters 13.2 and 16.2). I will try to propose new insights regarding it, further on, armed with a similar analysis for the 3-item framework.
Detail from Table 18.1 – Interpretation of the 2-Item Moduses[4].
ID | summary | |
1 | impossible | |
2 | incontingency | |
3 | incontingency | |
4 | incontingency | |
5 | incontingency | |
6 | incontingency | |
7 | only strong prevention | mn |
8 | joint s-w prevention | mq abs |
9 | incontingency | |
10 | only strong causation | mn |
11 | incontingency | |
12 | joint s-w causation | mq abs |
13 | incontingency | |
14 | joint s-w causation | np abs |
15 | joint s-w prevention | np abs |
16 | both causation and prevention | pq abs + pq abs |
For now, note one thing that I did not clearly realize before – it is that the last modus (#16) refers to both weak absolute causation and weak absolute prevention, and not as I previously wrote or implied to either the one or the other. This new observation is significant, in that it teaches us that causation and prevention at this low degree of determination are not mutually exclusive, but rather apparently occur in tandem (this is later confirmed in the 3-item framework).
Moreover, it is well to remember in this context, before moving on, that causation here includes both causation by P of R and inverse causation by P of R, i.e. causation by notP of notR, since these have the same moduses, though mq becomes np and vice versa. Similarly, prevention, here includes both prevention by P of R, i.e. causation by P of notR, and inverse prevention by P of R, i.e. causation by notP of R, since these have the same moduses, though mq becomes np and vice versa. That is to say, the above table of 16 moduses covers every logical possibility.
b. Let us now look at the similar interpretations of the 3-item moduses. These may be examined in detail in Table 18.6. This is new material that has not been previously researched. I will begin by listing some statistics drawn from it:
First, apart from the formally impossible modus (#1), there are 48 moduses signifying non-connection due to the incontingency of P and/or R. In some of these cases Q is also incontingent; but if both P and R are contingent, the incontingency of Q does not impede a connection, however tenuous, between P and R. This leaves us with 207 moduses signifying a connection of some sort, whether causation only, prevention only or a mix of both, as the following extract shows:
Detail from Table 18.6 – Interpretation of the 3-Item Moduses.
causation and/or prevention | stats |
causation only | 63 |
prevention only | 63 |
both causation and prevention | 81 |
neither causation nor prevention | 49 |
total | 256 |
Next, we see that, for each of causation or prevention (their behavior must be similar, since they are mirror images of each other), there are ten logically possible causative formulas – mn, mq abs only, etc., and each of these has a certain frequency of occurrence in the moduses, as shown on the next table detail. Note that the joint strong-weak relations may be relative to Q or to notQ, and also that they are rather rare, compared to the absolutes. The most frequent relation is pq abs only (79 moduses). The total number of such general relations is 144, note. This is all true, to repeat, for causation and again for prevention.
Detail from Table 18.6 – Interpretation of the 3-Item Moduses.
for each of causation or prevention | stats |
mn | 9 |
mq abs only | 19 |
np abs only | 19 |
pq abs only | 79 |
mq rel to Q | 4 |
np rel to Q | 4 |
pq rel to Q | 1 |
mq rel to notQ | 4 |
np rel to notQ | 4 |
pq rel to notQ | 1 |
total | 144 |
The next detail table shows us the variety and frequency of lone determinations in causation and prevention respectively (again the patterns are as can be expected repetitive).
Detail from Table 18.6 – Interpretation of the 3-Item Moduses.
lones in each of causation or prevention | stats |
m-alone rel to Q and notQ | 19 |
n-alone rel to Q and notQ | 19 |
m-alone rel to Q only | 4 |
n-alone rel to Q only | 4 |
m-alone rel to notQ only | 4 |
n-alone rel to notQ only | 4 |
p-alone rel to Q and q-alone rel notQ | 4 |
q-alone rel to Q and p-alone rel notQ | 4 |
p-alone rel to Q | 7 |
q-alone rel to Q | 7 |
p-alone rel to notQ | 7 |
q-alone rel to notQ | 7 |
total | 90 |
Note well that there are no moduses signifying absolute lone determination, as already established in Chapter 12.2. All the lones that do arise (90 in all) are relative. They may arise relative to Q (30 cases) or to notQ (30 cases) or even to both (38 cases). The latter should not surprise – it is logically consistent, and indeed most common. Here as well, all this is true for causation and again for prevention.
The next statistical table shows the conjunctions possible between the joint determinations and the lone determinations, for each of causation and prevention. This teaches us that – except for 54 cases, viz. mn (9 cases), some cases (43) of pq absolute and all cases (2) of pq relative, all joint determinations occur in tandem with some lone determination(s), and conversely no lone determination occurs without a joint determination underlying it.
Detail from Table 18.6 – Interpretation of the 3-Item Moduses.
formulae found in each of causation or prevention | stats | |
mn | 9 | |
mq abs only | m-alone rel to Q and notQ | 19 |
mq rel to Q | m-alone rel to notQ only | 4 |
mq rel to notQ | m-alone rel to Q only | 4 |
np abs only | n-alone rel to Q and notQ | 19 |
np rel to Q | n-alone rel to notQ only | 4 |
np rel to notQ | n-alone rel to Q only | 4 |
pq abs only | p-alone rel to Q and q-alone rel to notQ | 4 |
pq abs only | q-alone rel to Q and p-alone rel to notQ | 4 |
pq abs only | p-alone rel to Q | 7 |
pq abs only | p-alone rel to notQ | 7 |
pq abs only | q-alone rel to Q | 7 |
pq abs only | q-alone rel to notQ | 7 |
pq abs only | no lone | 43 |
pq rel to Q | 1 | |
pq rel to notQ | 1 | |
total | 144 |
Moreover, as the table above reveals, these conjunctions follow certain patterns. For instances, mq abs only (i.e. without q rel to Q or to notQ) is always paired off with m-alone relative to Q and notQ; if mq rel to Q occurs, it is always accompanied by m-alone rel to notQ only; if mq rel to notQ occurs, it is always accompanied by m-alone rel to Q only; and similarly for np abs, and in other cases.
Thus, of the 144 cases of causation, 54 are without adjacent lones. Note also that there are a total of 36 cases of pq abs only (i.e. not implied by pq rel) conjoined with lones, as against 43 not so conjoined. All the same can be said for prevention, of course. Now consider the following table:
Detail from Table 18.6 – Interpretation of the 3-Item Moduses.
all lones, with sw rel, sw abs or pq abs | stats |
all lones, in causation and/or prevention | 162 |
lones without sw rel, sw abs or pq abs | 0 |
lones with sw rel to Q or notQ | 32 |
sw rel to Q or notQ, without lone | 0 |
lones with sw abs | 76 |
sw abs, without lone | 0 |
lones with pq abs | 54 |
-of which, in both causation and prevention | 18 |
lones with pq rel to Q or notQ | 0 |
pq abs or pq rel, without lone | 27 |
This table, counting both causation and prevention, shows us again that no lone occurs without an associated joint strong-weak (sw) connection, whether relative or absolute, or without at least a pq absolute connection (162 = 32 + 76 + 54). Note that the joint determinations sw rel and sw abs never occur without conjunction of one or more lones. On the other hand, lones are never conjoined with pq relative to Q or notQ. Note also that the 54 cases of lones with pq abs in this table coincide with the 36 cases for causation and 36 more for prevention in the preceding table; this just tells us that there is overlap between 18 such cases of causation and prevention.
The last row of the above table tells us of pq abs (23 cases) or pq rel (4 cases) that are without lone. These 27 cases reappear in the next table, together with 54 cases of absolute causation and/or prevention associated with one to four lone determinations (which are always associated with pq abs):
Detail from Table 18.6 – Interpretation of the 3-Item Moduses.
analysis in the 3fw of modus 16 of the 2fw | stats |
lone causation and/or prevention with 1-4 lones (with pq abs) | 54 |
pq relative causation or prevention (the other = pq abs) | 4 |
only one of causation or prevention with pq abs | 0 |
absolute causation and prevention without lone | 23 |
total | 81 |
Now, it is interesting to note that the 81 moduses of the 3-item framework corresponding to item #16 of the 2-item framework (identified in Table 17.4) coincide with the 81 moduses where causation and prevention overlap in Table 18.6. Examining the latter table, we see that these 81 moduses include the 4 moduses #s 190, 232, 127, 220, in which one side has “pq rel to Q” or “pq rel to notQ” and the other side has “pq abs”; and the 77 moduses with “pq abs” on both sides, of which 54 also involve lone determinations, on one side and/or the other, while 23 moduses involve no lone determination.
While the other 54 + 4 cases signify a relation to the third item Q and/or its negation notQ, the 23 cases make no mention of Q or notQ. Note, too, that the 3-item modus #256 is among those 23. The said 23 moduses are of especial interest, because they will help us solve the earlier described problem of apparently having no modus with which to account for non-connection between contingent items.
In any case, it is now evident (looking at table 18.6) that this problem is not limited to the 2-item framework, but recurs in the 3-item framework. Here too, we see that none of the 256 moduses refer to non-connection between contingent items. They all refer to either incontingencies or to causative and/or preventive connections. We shall have to deal with this issue in more detail in the next chapter.
[1] I do not think I realized that during phase II – see for instance Table 13.12. Needless to say, this insight applies not only to the 2-item framework but equally to the 3-item framework and on.
[2] More tables are introduced in later chapters.
[3] Or more precisely, the ‘connection’ of such propositions, without their ‘bases’, as mentioned in the previous section. This is consistent with the usual formulation of logical conditioning; but for de re forms of conditioning, we would have to include consideration of the underlying possibilities before identifying the conjunctives with conditionals.
[4] Note my use here and elsewhere of ‘mq abs’ instead of mq_{abs}, ‘np abs’ instead of np_{abs}., ‘pq abs’ instead of p_{abs}q_{abs} – no new meaning is intended in such cases; I just find it more convenient. Similarly of course with regard to ‘p rel’ and ‘q rel’, and their compounds, later on.