THE LOGIC OF CAUSATION
Phase Three: Software Assisted Analysis
Chapter 19 – Defining Causation.
2. The Puzzle of No Nonconnection.
1. Back to the Beginning.
In the present chapter, I propose to deal with some of the difficulties that have become apparent in the previous two chapters. Before doing so, however, it is perhaps wise to review our basic definitions of the four generic determinations of causation – complete, necessary, relative partial and relative contingent causation – and their two derivative concepts, viz. absolute partial and absolute contingent causation.
1.  complete causation by P of R  symbol: m  
a)  If P, then R  (P + notR) is impossible  
b)  if notP, notthen R  (notP + notR) is possible  <=> 2(c) 
c)  P is possible  (P + R) is possible  <=> 2(b) 
2.  necessary causation by P of R  symbol: n  
a)  If notP, then notR  (notP + R) is impossible  
b)  if P, notthen notR  (P + R) is possible  <=> 1(c) 
c)  notP is possible  (notP + notR) is possible  <=> 1(b) 
3.  partial causation by P (with Q) of R  symbol: p rel  => not1 
a)  If (P + Q), then R  (P + Q + notR) is impossible  <= 1(a) 
b)  if (notP + Q), notthen R  (notP + Q + notR) is possible  => 1(b) 
c)  if (P + notQ), notthen R  (P + notQ + notR) is possible  => not1(a) 
d)  (P + Q) is possible  (P + Q + R) is possible  => 1(c) 
4.  contingent causation by P (with Q) of R  symbol: q rel  => not2 
a)  If (notP + notQ), then notR  (notP + notQ + R) is impossible  <= 2(a) 
b)  if (P + notQ), notthen notR  (P + notQ + R) is possible  => 2(b) 
c)  if (notP + Q), notthen notR  (notP + Q + R) is possible  => not2(a) 
d)  (notP + notQ) is possible  (notP + notQ + notR) is possible  => 2(c) 
5.  partial causation by P of R (abs)  symbol: p abs  <= 3, => not1 
a)  If P, notthen R  (P + notR) is possible  <=> not1(a) 
b)  if notP, notthen R  (notP + notR) is possible  <=> 1(b), 6(c) 
c)  if P, notthen notR  (P + R) is possible  <=> 1(c), 6(b) 
6.  contingent causation by P of R (abs)  symbol: q abs  <= 4, => not2 
a)  If notP, notthen notR  (notP + R) is possible  <=> not2(a) 
b)  if P, notthen notR  (P + R) is possible  <=> 2(b), 5(c) 
c)  if notP, notthen R  (notP + notR) is possible  <=> 2(c), 5(b) 
Let us now explain and justify these definitions. To claim complete causation (m) implies we know (or think or believe) that one thing P is invariably accompanied by another thing R, i.e. that P without R is impossible (in the mode of modality concerned – be it logical, extensional natural, or whatever). However, that P implies R cannot by itself signify causation. We need to also know that notP does not imply R, i.e. that notP without R is not impossible, for if both P and notP implied R, then R would be independent of them. Thirdly, we need to know that P is possible, so as to ground the first implication in actuality; and given that P is possible and that P implies R, it follows that R is also possible, i.e. that the conjunction of P and R is possible.
I go into this in detail to make clear to readers that these definitions were not pulled out of the blue or arbitrarily imposed, but are the product of reasoning. Necessary causation (n) is very similar to complete causation, except that the polarities of all the items involved are inversed. It is a statement that without P, R cannot occur; i.e. that the conjunction of notP and R is impossible; in such cases P is called a sine qua non (without which not) of R. Here again we must on logical grounds add two more propositions to the definition to make it applicable correctly.
Note that complete and necessary causation share the last two of their defining clauses, but differ in their first clause. However, since these first clauses do not, according to the laws of thought, exclude each other, it follows that the generic determinations of complete and necessary causation can be combined into one specific determination mn. However, they do not formally have to be so combined; i.e. m may be true without n being true, and vice versa. This brings us to the concepts of partial and contingent causation.
The relationship of partial causation (p) is designed to resemble that of complete causation, except that the cause is not one thing P, but a conjunction of two things P and Q, the latter being called the complement of the former. The first clause in our definition is a claim that P and Q together bring about R. But for this to be true, we must also ascertain that Q without P and P without Q are not also always followed by R; otherwise one or both of them might be accidental to the occurrence of R (i.e. P or Q might alone cause R, or R might be independent of their conjunction). The second and third clauses in the definition guarantee the dependence of R on P and Q together. The fourth clause serves to ground the hypothetical relationship implied by the first; and together they tell us that the conjunction of the three items P, Q and R is possible.
Contingent causation (q) is similarly constructed, but by analogy to necessary causation. The partial and contingent forms of causation are called weak determinations, in comparison to the complete and necessary forms (called strong determinations), because in the former (unlike in the latter) the cause P (or for that matter its complement Q) is not by itself strong enough to bring about the effect R. It is clear from the definitions of p and q that these relations are true relative to a specific complement Q. If we put notQ in place of Q, P and R remain cause and effect in a similar sense, but their exact relationship is considerably modified, note well. The complement Q (or alternatively notQ) signifies the conditions under which the (weak) causative relation between P and R comes into play.
Note too that p and q (relative to Q) do not share any defining clauses, unlike m and n. Since they refer to the possibility and impossibility of different sets of conjunction, there is no conflict between them, and they (as generic determinations) can logically be combined as (the specific determination) pq without infringing any law of thought.
Now compare the above listed definitions and implications of partial and complete causation. It is of course noteworthy that m and n involve only two items P and R, whereas p and q involve three items P, Q and R; but this does not prevent logical comparisons. We see that clause 1(a) formally implies clause 3(a), and clauses 3(b) and 3(d) respectively imply 1(b) and 1(d), but clause 3(c) negates clause 1(a). This means that p and m are on the whole contrary to each other, though they do share some elements of information. Similarly, q is incompatible with n, though they have some common aspects.
Now compare m and q. We see that 4(d) implies 1(b), and 4(b) implies 1(c), but no clause in q conflicts with 1(a), and none in m conflicts with 4(a) or for that matter 4(c). Similarly in the comparison between n and p, we find no notable opposition between them. This means that, formally speaking, nothing prevents the specific combinations of strong and weak forms mq and np from occurring (separately, of course).
Let us now turn our attention to the last two forms[1] – absolute partial causation (p abs) and absolute contingent causation (q abs), not to be confused with the preceding two forms of p and q relative to Q (or eventually to notQ), henceforth symbolized by p rel and q rel. The idea of absolute weak causation forms was generated by two related considerations. First, we wanted to express the weak determinations in terms of two items rather than three, for purposes of matricial analysis and direct comparisons to the strong determinations; and second, we wanted to express the weak determinations without regard to whether the complement is Q or notQ, or anything else for that matter.
Thus, the qualification of weak causations as ‘absolute’ here is only intended to mean that they are not relative, note well. It does not signify some stronger relationship, but on the contrary (as is soon apparent) a weaker relationship! Comparing the above definitions of p abs to p rel, we see that 5(a) is implied by 3(c), 5(b) is implied by 3(b), and 5(c) is implied by 3(d); but these implications are not mutual. Thus, p abs is a derivative of, i.e. a restatement of some but not all of the information in it. Notice especially the absence in p abs of any of the information contained in clause 3(a) of p rel, though this clause is the crucial part of it, the part most indicative of causation! All the same can be said of q abs and q rel, mutadis mutandis.
It is also noteworthy that if we change Q to notQ and vice versa in the clauses of the definition of p rel, the implied p abs is exactly the same. That is, p relative to Q and p relative to notQ yield the same subaltern p abs. This is of course to be expected, since neither Q nor notQ are mentioned in it. But additionally, p abs does not mention any other eventual third item – and so is identical for all eventual third items, X, Y, Z or whatever. Whence the characterization of it as ‘absolute’. Now, this should cause us alarm; how can we know something so general from so little information, we might well ask. But the truth is that in fact p abs tells us exactly nothing about Q or notQ or any other third item! All the same can be said of q abs and q rel, mutadis mutandis.
Now compare p abs with m. We see that 5(a) contradicts 1(a), though 5(b) and 5(c) are identical with 1(b) and 1(c) respectively; this tells us that, albeit their having some common ground, p abs and m are contrary to each other. Also compare p abs with n. We see that 5(b) is the same as 2(c) and 5(c) is the same as 2(b), while 5(a) and 2(a) do not affect each other; this means that p abs and n are compatible and can be conjoined. Similar results are obtained comparing q abs with n, and q abs and m. Thus, the compounds mq abs and np abs are logically conceivable.
As for the oppositions between p abs and q abs, 5(b) is identical with 6(c) and 5(c) is identical with 6(b), whereas 5(a) and 6(a) do not impinge on each other; thus the two forms are compatible, i.e. can be conjoined in a compound form pq abs. What does this compound form tell us? Simply, that each of the four conceivable combinations of P and R, viz. P+R, P+notR, notP+R, notP+notR, is possible.
The above six definitions for (i) causation by P or R can be modified to define (ii) prevention by P of R (by replacing R by notR, and notR by R, throughout them), (iii) inverse causation by P of R (by replacing P by notP, and notP by P, and R by notR, and notR by R, throughout them), and (iv) inverse prevention by P of R (by replacing P by notP, and notP by P, throughout them). Note in passing that pq abs has the same value in causation (and inverse causation) and in prevention (and inverse prevention), since it always just means that the four conjunctions of P, R, notP and notR, are all possible.
All this has been said before but is here repeated briefly to enable us to once and for all resolve a certain difficulty mentioned earlier. We shall see that the difficulty in question is upon closer inspection more apparent than real.
2. The Puzzle of No Nonconnection.
Looking at the interpretation table for the moduses in a 2item framework (Table 18.1, page 6), we see that only seven of the moduses refer to connection – and apparently not even one refers to ‘nonconnection albeit contingency’! Incontingency counts as nonconnection, of course; but what interests us here is to logically conceive nonconnection between two contingent items. Apparently, judging by the tabulated results, there is no such possibility! Note in passing that alternative words for connection and nonconnection are dependence and independence. Are we to think that all contingent items are mutually dependent in some way or other? Surely not! What does this mean, then? This result is indeed so surprising that I shall call it ‘the puzzle of no nonconnection’.
Considering that the logic of causation as here presented, i.e. through microanalytic tabulation, is entirely a formal product of the laws of thought, this is indeed mysterious. This result seems to fix in an apriori manner a detail about reality, by mere logical analysis, without need for empirical observation. Although some philosophers, indeed many of them across history, have adopted this position, it does not make sense. It would mean we cannot even imagine or theoretically conceive of nonconnection between contingent items, which certainly goes against our commonsense impression that we at least comprehend such nonconnection. All our concepts need contradictories to be intelligible. If we cannot even hypothetically formally define nonconnection between contingent items, the concept of connection itself becomes doubtful.
My discovery of this mystery is not new to phase III; I had already encountered it and made an effort to explain it in phase II (see Chapters 13.2 and 16.2). Here, I will succeed in going deeper into the question and remove all lingering doubt once and for all.
No doubt, seeing this puzzling result, believers in extreme determinism (which include many materialists and behaviorists still today) will rush to judgment and say: “See, we told you, since we cannot logically define indeterminism, it is not even open to debate – everything in the universe is determined, and there is no place in it for natural spontaneity or human freewill or any other indeterminism.” But if we consider the matter more closely (again, look at Table 18.1 page 6), we see that the seven cases with both items contingent refer to varying degrees of causation: 2 cases are mn (maximal determination), 4 cases are mq abs or np abs (medium determination), and 1 case is pq abs (minimal determination). Thus, only two relations are fully determining, whereas five others are partly undetermined, and we cannot draw an extreme determinist conclusion.
Another group likely to welcome this puzzling result are believers in the Buddhist viewpoint that everything is causatively related to everything else in an inextricable web of ‘interdependence’ (or ‘dependent origination’ or, in Sanskrit, Pratityasamutpada). They will say: “See! Since there is no such thing as nonconnection between some pairs of contingent items, any two contingent items taken at random may be considered, without any recourse to experience, as causatively related, at the very least through partial contingent causation (and similarly prevention), i.e. pq abs.” But such jubilation is premature and unjustified, as we shall now go on to show.
The simplest answer is that what we have called ‘partial contingent causation’ is not really causation! To see the truth of this, let us return to our initial definitions of p abs and q abs, in the previous section. What distinguishes these forms (numbered 5 and 6) from those preceding them (14) is that they lack an ifthen clause. They each specify the possibility of three combinations of P, R and their negations, but they distinctively do not specify the impossibility of any such combinations. Yet such ifthen or impossibility of conjunction constitutes the main clause of the definitions of strong causation and relative weak causation.
Thus, the absolute weak determinations are not forms of causation in the usual sense. This does not mean we ought to, or even can, just discard these two concepts. For it is clear that we formed them out of a real need. They do in fact play a role in causative relations – but their role is a supporting one. In combination with m or n, i.e. in mq abs or np abs, they are indicative of actual causation; but taken apart from the strong determinations, i.e. in the combination pq abs (i.e. p_{abs}q_{abs}), all they tell us is that the four basic conjunctions, viz. P+R, P+notR, notP+R, and notP+notR, are all possible, which is not a statement of actual causation but still leaves open the logical possibility of causation at a deeper level (as evident in Table 18.6).
Remember, p abs is contrary to m, and q abs is contrary to n. When m is combined with q abs, we have the important information that, though there is causation, it is not of the powerful mn sort. Similarly with regard to np abs – the p abs part serves to deny the conjunction of m to the n part. It is significant to remember, too, that there are no absolute lone determinations, that is: absolute malone, i.e. m conjoined to neither n nor q abs, is logically impossible; similarly, absolute nalone is impossible, and so are the absolute weaks alone. Thus, p abs and q abs are formally needed for causative discourse in a 2item framework.
However, though these absolute weak determinations are implied by the corresponding relative determinations, they do not in turn imply them. They are mere subalterns, not equivalents. At best, p abs tells us that p rel might occur, and likewise q abs tells us that q rel might occur; the former certainly do not imply that the latter are bound to occur. And the issue here is not merely that we do not know whether Q or notQ is the applicable complement. As the definitions in the previous section make clear, p abs and q abs remain the same, even if we change the polarity of Q to notQ and of notQ to Q in p rel and q rel. But, moreover, as Table 18.6 makes clear, p abs and q abs can be true without implying either p rel or q rel in relation to Q or notQ !
The latter finding should by itself cause alarm: how could we, using a PQR matrix only, know about a weak causative relation between P and R through an intermediary other than Q or notQ ? Such a thing is unthinkable in deductive logic – there are no magical leaps, no windfall profits – we can only conclude things already given in the premises. But if we look more closely at instances of pq abs only, we see that they do not tell us anything about causation involving some unstipulated fourth item other than Q or notQ, because they do not imply that some causation between P and R (and/or their negations) is indeed operative. They merely specify the various possibilities of conjunction between these two items; this is valuable information, but it is not causation.
Thus, although p abs and q abs are relevant to causation in the compound propositions mq abs and np abs, they are not definitely indicative of causation as pq abs, in the 2item framework as modus #16 (see Table 18.1), or in the 3item framework as the 23 moduses #s 52, 56, 6061, 64, 103, 116, 120, 154, 180, 188, 196, 205208, 221222, 237, 239, 244, 253, 256 (see Table 18.6).
Note that the 2item modus #16 unfolds as 81 distinct moduses in the 3item framework. Among those 81, only the just mentioned 23 moduses (which include the last modus 256, note) are in turn empty of causative information. The remaining 58 moduses all involve some definite causation, whether through relative lone determinations (54 cases) or relative partial contingent causation or prevention (4 cases). For this reason, we can rightly say that the 2item modus #16 is ambiguous as to whether there is or not some causation or prevention deeper down in a 3item framework.
Similarly, each of the 23 said 3item moduses may or may not at a deeper level become a connection of some sort, ad infinitum. Thus, to call pq abs ‘causation’ (or ‘prevention’, as the case may be) is a misnomer – it is excessive, inaccurate, misleading to do so, because though this compound is sometimes expressive of causation – it is sometimes not so. Thus, the solution to our problem is that to regard pq abs as a form of connection is to misuse the term. We should therefore, strictly speaking, refer to the 2item modus #16 as possible connection and possible nonconnection (as I suggested in phase II); and likewise for each of the 23 above listed 3item moduses (as now proven in phase III).
We have thus clearly located where nonconnection between contingent items can be placed. Let me further explain this as follows, so it is fully understood. The essence of connection (causation or prevention) lies in the limitations of possibility to be found in nature or logic. When we say that an item, say P, ‘causes’ (or ‘prevents’) another item, say R, in some way, to some degree, we mean that in the presence or absence of P, the presence or absence of R is somewhat restricted. It is not the occurrence of the latter item or its negation that signify causation, but the fact that some other avenue of occurrence has been naturally (in some cases, volitionally) or logically blocked.
Thus, the ‘force’ of causality lies not so much in positive events as in the restrictions in the degrees of freedom offered to an item by the interference of another; i.e. in the negative boundaries the one sets on the other. In more formal terms, we can say: it is not so much the ‘1s’ (the bases) that matter as the ‘0s’ (the connections). Roughly stated, the more zeros, the stronger the causal relation; the less zeros, the weaker the causal relation. If no zeros are to be found at any depth, there is no causal relation. In cases involving strong causation, the restrictions are very evident, whereas in cases involving only weak causation, the restrictions are not always evident – and by extrapolation, we may at least conceive of cases without restriction.
We can also put it as follows, to show that it makes perfect sense. For two items to be connected in some way, there has to be some incompatibility between them and/or their negations, some conflict that forces one or the other of them to behave in an special manner. If the items and their negations are every which way compatible, then they do not impinge upon each other but coexist harmoniously. Thus, the pq abs compound, which signifies such thorough compatibility, is essentially indicative of nonconnection, though some connection at a deeper level is not excluded by it offhand.
Once this crucial new insight is grasped, it is easy to see why some modus(es) in any framework (such as the last modus in the 2item framework or the stated 23 in the 3item one) are the reasonable place where nonconnection (in whatever sense) between contingent items may be found. Partial contingent causation or prevention are indeed possibly housed in such modus(es); but we must admit that diverse forms of nonconnection are possibly housed there too. Their correct interpretation is thus ambiguous, and it is an error to interpret them only one way – as only connection, or for that matter as only nonconnection.
Furthermore, we should point out that the 2item modus #16 and the analogous 23 moduses in the 3item framework signify both pq abs of causation and pq abs of prevention, and not merely one or the other. This fact should not be swept aside as insignificant, although of course it does not go against the laws of thought. It is, however, unthinkable that something might be both a partial contingent causative and a partial contingent preventative of something else, relative to the same complement or even contradictory complements. This we know by looking at Tables 18.5 and 18.5 (pages 78), which teach us that the four forms pq rel to Q, pq rel to notQ, for causation, and pq rel to Q, pq rel to notQ, for prevention, have each only one modus, namely respectively moduses 190, 232, 127 and 220, and no modus in common. Causation and prevention are thus essentially antithetical, not only in their stronger forms but even in their weakest form.
This shows us that, even if pq abs of causation and pq abs of prevention are superficially compatible (indeed, they are identical, having in common the 2item modus #16 and all their 81 moduses in the 3item framework), such compatibility must not be interpreted as meaning that they can ever be realized together relative to any specific complement(s) Q and/or notQ. Such realization (i.e. going from absolute to relative) is logically impossible, so that the apparent compatibility between causation and prevention is purely illusory. Thus, the conceptual joining of p abs and q abs is, from the causative point of view, an abstraction without concrete referents. The generic forms are valid abstractions, because they can be validly joined to n and m, respectively, in the specific causative forms np abs and mq abs; but they do not produce a common causative form pq abs. The latter is meaningful (as a statement of possibility of conjunction every which way), but not as causation or prevention, and least of all as both causation and prevention.
It should also be stressed that when we here refer to the possible nonconnection between two specified items P and R, we are in no way making a general claim about the nonconnection of each of these items to some other unspecified items. The contingent item P may be unconnected to the contingent item R, but still be connected to one or many other contingent items X, Y, Z. Nonconnection does not imply universal nonconnection: it is here clearly intended as a characterization of the relation between a specified pair of contingent items.
Thus, this finding about the logically possible existence of nonconnection must not be taken as an apriori statement that ‘some contingent things are not connected to any others’, or more extremely that ‘nothing is connected to anything else’. These would be generalizations beyond what we have sought to establish here – which is only that, taking any two contingent items at random, there is no logical necessity that they be connected in a real sense (i.e. one stronger than the misnamed pq abs). The said moderate and extreme generalizations do however remain open to debate.
The extreme proposition ‘nothing is connected to anything else’ has been put forward in philosophy by Nagarjuna, David Hume, and others. I firmly reject it on the formal ground that they do not explain how all the other logical possibilities – i.e. those of connection between contingents – have been excluded from consideration by them; such skepticism is manifestly arbitrary.
The moderate position ‘some contingent things are not connected to any others’ is certainly not deductively proven here, either, but it remains quite conceivable, since we have identified the moduses within which such disconnection might occur and we do not claim an exclusive universal application. It formally opens the door to claims of occasional natural spontaneity (as in Niels Bohr’s interpretation of the uncertainty principle), and to claims of circumscribed human freewill and similar powers of volitions (which most people adhere to).
The antithesis to this would be the claim that ‘every contingent thing is connected to some other(s)’. Many philosophers throughout history have advocated this determinist thesis, calling it ‘the law of causation’ – but it is important to realize that, from a formal point of view, it is just a hypothesis. Moreover, what does ‘connected’ mean here – i.e. what degree of connection is intended? The extreme version of this thesis would affirm that ‘for any given contingent item R, there must be some item P that is a complete and necessary cause of it’.
A more moderate version might be postulated, however, that affirms such strong connection in most cases, but allows for exceptions, whereat natural spontaneity and/or volition may come into play next to determinism. I personally believe such combination of theses is the most credible alternative, being closest to commonsense belief. Our causative logic is thus, in any event, quite capable of assimilating all philosophical discourse concerning causation, note well.
3. The Definition of Causation.
In the preceding section, we saw that moduses that mean no more than “pq abs” (i.e. p_{abs}q_{abs}) cannot rightly be counted as signifying a causative connection, but at best only a possible connection, which is also a possible nonconnection. We saw the truth of this with reference to the 2item modus #16, which was found to give rise to 81 moduses in the 3item framework, of which 58 moduses were indicative of some causation or prevention (as well as pq abs), whereas 23 moduses signified no more than pq abs.
However, here we must admit that such ambiguity cannot be tolerated. If we want to produce a clear definition of causation, which is one of the goals of our study, we must make up our minds and declare moduses that mean “pq abs only” to signify either a connection or a nonconnection. So far in our tables, we have opted for the designation of the 2item modus #16 and its equivalents 3item moduses to signify connection. But in view of our analysis in the preceding section, we must now reverse this policy if we wish to produce an accurate definition. This is reasonable, since two items related only by way of pq abs cannot be guaranteed to be causatively related, and so may be counted as not so related (unless or until more specific conditions are specified that imply them to be causatively related).
On this basis, the tables concerning the broad concepts of causation, prevention and connection, and their respective negations, must be rewritten with all cases of pq abs only moved over from the positive to the negative side, whether manually or by modifying the calculation formulae as appropriate. Thus, for instance, the 2item modus #16 must be moved from the columns of causation, prevention and connection to those of noncausation, nonprevention and nonconnection. Similarly for 23 moduses in the 3item framework. We shall tag these new columns as concerning ‘strict’ causation, prevention and connection and their negations – so that the corresponding old columns can be left unchanged, except that we understand that they concern causation etc. in a ‘loose’ sense.
The outcome of this revision are the following two tables, derived from earlier ones as just explained, which are posted at the website as usual:
Table 19.1 – 2item PR Moduses of Forms – Strict Moduses. (1 page in pdf file).
Table 19.2 – 3item PQR Moduses of Forms – Strict Moduses. (5 pages in pdf file).
Having done this, we can now proceed with constructing definitions of the concepts of causation, prevention and connection in their strict sense (i.e. with ‘pq abs only’ not counted as causation, etc.). The following extract from Table 19.1 suffices for this purpose:
Details from Table 19.1 – Causation, prevention and connection.
relation  summary  moduses  notable features  
strict causation  1001  1011  1101  outers both 1, inners one or both zero 
strict non causation  all other moduses, except #1 
 
strict prevention  0110  0111  0110  inners both 1, outers one or both zero 
strict nonprevention  all other moduses, except #1 
 
strict connection  strict causation or strict prevention  features of both  
strict nonconnection  all other moduses, except #1 

We see here that, strictly speaking, causation is applicable to three moduses (Nos. 10, 12, 14, to be specific), whose common features are that their summary moduses start with a 1 (for P+R) and end with a 1 (for notP+notR), and have one or two 0s in the middle (for P+notR or notP+R). Similarly, strict prevention concerns three moduses (Nos. 7, 8, 15), featuring two 1s on the inside and one or two 0s on the outside. Connection accordingly covers these six moduses, and is thus definable by the sum of their features. The negations of these relations refer to all remaining moduses, except #1 (consisting of four 0s, which is universally impossible). Modus #16 (consisting of four 1s) always falls in the negative relation (strictly speaking) – its lack of any 0 puts in doubt any causative relation in it.
We may express these results concerning strict causation in words as follows: causation is the relation between two items, if and only if they are found to have the following set of features: (a) the first cannot occur without the second and/or the second cannot occur without the first, and in any case (b) the first and second can occur together and their negations can occur together. If these conditions are satisfied, this first item is called cause and the second is called effect. The relation of prevention refers to causation of negation; and the relation of connection refers to either causation or prevention. The negations of all these relations can accordingly be defined. Note well that if the two items and their negations are compatible together every which way, they cannot strictly be said to be causatively related in any way; for such relation to be recognized, some incompatibility between the items and/or their negations must be established.
Of course, the here stated definition of causation (and thence those of prevention and connection) could be argued to be rather rough, being based on Table 19.1 only, that is to say on the configuration of ‘absolute’ causation between two items, comprising strong causation (mn) and its combinations with absolute weak causation (mq abs and np abs). It ignores causation relative to a third item, which is more complex and difficult to define. The simplest way to do it would be to say: ‘relative’ causation requires a more complicated and subtle definition, and rather than try and formulate one I refer you to Table 19.2. Alternatively, we could try and construct a verbal definition with reference to the original forms listed in section 1 of the present chapter.
But I do not see the value of such a wordy and intricate definition in practice. Definitions should effectively lead us to the intended object, and not mystify us by their complexity. I think the rough definition proposed here suffices for most purposes; and when we do need to get very precise, we can just point to the original forms or the said table, without attempting a formal summary. One more thing needs doing, however – we need to explain the application of the proposed definition of causation (and its derivatives) in terms of generic ‘possibility’ and ‘impossibility’ to the different modes of causation, and say more about the way knowledge of causation is acquired in them.
With regard to the logical or ‘de dicta’ mode of causation, the modal specifications of ‘possibility’ and ‘impossibility’ refer of course to some or no ‘contexts of knowledge’. In this domain, our inductive practice is to assume modus #16 to be true, until and unless we manage to demonstrate another relation to be true. The truth of this principle can be seen in the theory of ‘opposition’, where we assume two propositional forms to be fully compatible (i.e. neutral to each other) if we do not manage to specifically prove them (if only by some logical insight) to be contradictory or contrary or subcontrary or implicant or subalternative.
Turning our attention now to the ‘de re’ modes, we can say: in extensional causation, ‘possibly’ means in some cases and ‘impossibly’ means in no cases; in natural, temporal and spatial causation, these modalities refer respectively to some or no circumstances, some or no times, and some or no places. In these modes, our inductive practice is the exact opposite of that for the logical mode. That is to say, here we assume the items concerned to be incompatible if we do not succeed in directly or indirectly finding empirical grounds to consider them as compatible. For example, we do not affirm that ‘some X are Y’ if we have not directly observed any such cases, or at least (more indirectly) empirically confirmed a theory that implies this proposition.
Thus, modus #16 is not taken for granted as easily for the de re modes as it is for the logical mode. In the logical mode, it is used as the default option when no other option is established. Whereas, in the de re modes we are not allowed to make such assumptions offhand, but rather remain in a state of ignorance until some good reason to accept modus #16, or any other modus, whether of causation or of noncausation, is found. In this sense, the logical mode is more ‘a priori’ and the de re modes are more ‘a posteriori’. But as regards their formalities they differ little.
I think we need not belabor this topic further, except to point out, once again, how much more accurate our definitions are from those implied by David Hume and from other past attempts.
4. Oppositions and Other Inferences.
Once we have analyzed each and every possible form of causation and its sources and derivatives in matricial analysis, it is very easy to compare forms and determine their oppositions, eductions, syllogisms and any other sorts of inference.
We can formulate general rules of opposition, from which the oppositions between any pair of forms can be determined, as follows[2].
· Implicance: two forms all of whose alternative moduses are identical may be said to imply each other; i.e. they are implicants. For example, m in causation and n in inverse causation are equivalent, having the exact same moduses (2item moduses #s 10, 12), no more and no less. It follows necessarily, note, that their negations are also implicants. For example, notm in causation and notn in inverse causation are equivalent (2item moduses #s 29, 11, 1316).
· Subalternation: if one form has more moduses than another, and its list of moduses includes all the moduses of that other and none of the moduses of its negation, the second form may be said to imply but not be implied by the first; i.e. they are subalternatives: respectively, subalternant and subaltern. Note well that it is the (narrower ranging, more precise) form with less moduses that implies the (broader ranging, vaguer) form with more moduses, and not vice versa. For example, “P is a complete cause of R” (2item moduses #s 10, 12) subalternates “if P, notthen notR” (moduses 916). It follows necessarily, note, that their negations are also subalternatives, though in the opposite direction. For example, “if P, then notR” (moduses 28) subalternates “P is a not complete cause of R” (moduses 29, 11, 1316).
· Contradiction: if two forms do not share any modus and if their moduses together make up the total number of moduses in the framework concerned (minus the universally impossible first modus), they may each be said to imply the other’s negation (i.e. to be incompatible) and their negations each to imply the other’s affirmation (i.e. to be exhaustive); that is, they are contradictories. For example, m has 2item moduses #s 10, 12 and notm has moduses 29, 11, 1316; therefore, m and notm are contradictory.
· Contrariety: if two forms do not have any modus in common, and if their moduses together do not add up to the total number of moduses in the framework concerned (minus the universally impossible first modus), their affirmations may each be said to imply the other’s negation, though their negations do not each imply the other’s affirmation; that is, they are incompatible but not exhaustive, i.e. contraries. For example, m (2item moduses 10, 12) and p abs (moduses 14, 16) are contrary forms. Note that if two forms are contrary, their negations are necessarily subcontrary.
· Subcontrariety: if two forms do have some modus(es) in common, and if their moduses together do add up to the total number of moduses in the framework concerned (minus the universally impossible first modus), their negations may each be said to imply the other’s affirmation, though their affirmations do not each imply the other’s negation; that is, they are exhaustive but not incompatible, i.e. subcontraries. For example, notm (2item moduses 29, 11, 1316) and notp abs (moduses 213, 15) are subcontrary forms. Note that if two forms are subcontrary, their negations are necessarily contrary.
· Unconnectedness: if two forms have some modus(es) in common, and their negations have some modus(es) in common, and the affirmation of each of them has some modus(es) in common with the negation of the other, these forms may be said to be unconnected with each other, for this simply means that the four stated combinations are possible, i.e. that each form and its negation is compatible with the other form and its negation. For example, “if P, then R” (2item moduses #s 24, 912) and “if P, notthen R” (moduses 58, 1316) are both unconnected to both “if notP, then notR” (moduses 2, 56, 910, 1314) and “if notP, notthen notR” (moduses 34, 78, 1112, 1516).
Remember, this last category of opposition, viz. unconnectedness, also called ‘neutrality’, means that the forms concerned do not imply each other, and their negations do not imply each other, and their affirmations do not imply their negations, and their negations do not imply their affirmations; i.e. the two forms are compatible in every which way and exhibit no incompatibility in any way – that is why they are said to be unconnected or neutral. This covers all leftover cases, i.e. it applies when neither implicance, nor subalternation either way, nor contradiction, nor contrariety, nor subcontrariety relate the two forms under scrutiny.
Let me remark here: the word ‘opposition’ was initially intended (in everyday parlance) to mean ‘conflict’ – i.e. it referred to contradiction or contrariety. The sense was then slightly enlarged by logicians so as to include subcontrariety (which refers to contrariety of negations). Then, it was further enlarged to enable the inclusion of implicance and subalternation; this changed the meaning of ‘opposition’ to ‘faceoff’. Finally, the theory of opposition naturally called for a further concept, one denying all the preceding forms of opposition – i.e. a concept of ‘unconnectedness’ or neutrality (see my Future Logic, chapter 6.1). This relation too, though negative, can and must be regarded a form of ‘opposition’ in an enlarged sense (i.e. faceoff).[3]
Note that the above definition of unconnectedness in terms of moduses justifies my thesis earlier in the present chapter that there has to be room in causation theory for nonconnection, since it demonstrates that there is one more relation of ‘opposition’ than the six traditionally listed. For opposition theory (and more broadly, inference theory) is nothing other than causation theory in the realm of logical modality[4]; it concerns causes in the special sense of ‘reasons’. What is true for this de dicta mode of modality is equally true for the de re modes, since there is no formal difference between them in the present context.
Eduction is immediate inference from one (or more) forms with identical terms. When one form implies another, the latter can be educed from the former. When one form is incompatible with another, the negation of either can be educed from the affirmation of the other. When two forms are exhaustive, the affirmation of either can be educed from the negation of the other. From these principles we can likewise, with reference to moduses, determine all possible eductions.
We can similarly work out all syllogism (i.e. mediate inference, through a middle term) with reference to moduses, as already explained in chapter 14.1 and demonstrated thereafter. If the premises have no moduses in common, or if the premises do have some moduses in common but these moduses imply contradictory conclusions (i.e. some imply one conclusion and others the negation of it), they are incompatible and therefore cannot make up a syllogism. But otherwise, the conclusion is generally the common ground of the premises, i.e. the moduses they have in common.
Thus, matricial analysis – more precisely, microanalysis – provides us with a practical way to correctly interpret all conceivable situations in causative logic.
[1] The definitions of complete and necessary causation are first given in chapter 2.1. Those of relative partial and relative contingent causation are introduced in chapter 2.3. The definitions here put forward of absolute partial and absolute contingent causation are not found till chapter 13.4, although the concepts are developed much earlier, as of chapter 11.3.
[2] See chapter 13.3 for applications of this technique in phase II.
[3] I must in passing deride the couple of people who have written scholarlylooking articles where they seem to deny my concept of ‘unconnectedness’ to be a logically possible relation between propositional forms and a needed category of ‘opposition’! This is not an issue open to choice, but (to repeat) a natural demand to exhaust the logical alternatives. Such people allow themselves to be misled by mere words, thinking that opposition must needs signify conflict since that is the popular sense of the term. Or they are pettily annoyed that this additional category does not fit into their pretty ‘squares of opposition’. This is the kind of silliness that focus on trivia produces.
[4] Implicance and subalternation each way are logical causation; and contradiction, contrariety and subcontrariety are logical prevention. In each case, the determinations are respectively mn, mq and np.