www.TheLogician.net © Avi Sion - all rights reserved
© Avi Sion
All rights reserved
In this closing chapter, my purpose is to break some
additional ground, discussing certain outstanding issues in causation without
attempting to exhaust them at this time.
Our attempt, in the preceding five chapters, to solve by microanalysis all the problems of causative logic with reference to only three items has evidently failed. Some problems have indeed been solved – in particular, the refutation of absolute lone determinations should be cited as an important breakthrough. However, many syllogistic problems have been left without solution – and worse still, we are faced with apparent contradictions between the results of macro- and microanalysis, at least superficially, though these fears may turn out to be unfounded after further scrutiny.
Evidently, we will not succeed in definitive solution of the technical issues relative to causation until we develop microanalysis with four items. This essential and daunting task remains to be done at the present time. Only through microanalysis including the subsidiary item (S), in addition to that involving the major, middle and minor items (P, Q, R), will be have full control of syllogistic argument involving relative weak determinations.
To develop such level of detail, i.e. for syllogisms with four items, we would have to build a much larger grand matrix – a table with 16 rows and 216 = 65,536 columns. This could be done easily enough through a computer spreadsheet. We would prepare the said number of cells, filling in 0s and 1s in a systematic pattern and labeling the rows a-p and the columns 1-65536.
But the difficulty comes after that, in view of the inhumanly large quantity of data involved. We would need to write a search and flag program identifying the alternative moduses for each determination (based on conditions and rules like those exemplified when developing three-item microanalysis). Then we would infer the moduses for derivatives, including absolute vs. relative weak determinations, joint determinations, negative determinations, and preventive vs. causative propositions. Finally, we would consider all conceivable combinations of premises and identify their common modus(es) if any (if they have no common moduses, they are inconsistent); and find out what conclusion(s) if any these results allow in each case. I am personally not a good enough computer programmer to even try that.
Instead, my intention is to try and develop a relational database, through some existing software. The ultimate goal pursued here is to develop a quick method for identifying applicable determinations and for computing fusions of determinations in a syllogistic context. If the software is made to classify the determination relating certain items, from an analysis of their observed relative presences and absences (alternative moduses), the job is as good as done. It should then be possible to input the syllogism under investigation (specifying its figure and mood) and have the software output all the conclusion(s) if any that can be validly drawn.
Such a program would have not only theoretical value, but also practical value. In this day and age, when computers are readily available, this would give us a universal tool for causative reasoning, inductive (formation of causative propositions from alternative moduses) and deductive (immediate or syllogistic inferences). I am currently developing this tool.
Note also that neither in Phase One nor so far in Phase Two of the present research have I made any systematic attempt to test for syllogistic conclusions other than the regular ones, involving the major and minor items (form PR) and where appropriate also a subsidiary item (form PSR). When I say that there is “no valid conclusion” to some syllogism, I mean that there is none of such form. And when I find and list a valid conclusion, it is only one of such form (or the inverse causative forms, notP.notR or notP.notS.notR).
But in many cases, there may be a valid conclusion or additional valid conclusions of irregular form, i.e. involving the negation of the subsidiary item (notS), or the negation of the minor (notP) or the major (notR); that is to say, of causative form P.notS.R or notP.S.R, or of preventive form P.notR or notP.R or P.S.notR or notP.S.R or P.notS.notR or notP.notS.R. Any program written to spew out conclusions from regular premises (PQRS) should also test for such irregular conclusions, i.e. should be made to tell us the full conclusion all forms considered.
The expression ‘law of causation’ can also be applied to each and every theorem we have proved concerning causation. All our reductions of causative propositions to simpler conjunctive or conditional propositions, or to specified alternative moduses, all the immediate or syllogistic inferences from causative propositions that we have established, constitute so many ‘laws’ about causation.
The ‘grand matrices’, of 15 possible moduses for any two items, or 255 possible moduses for three items, or 65,535 possible moduses for four items, and so forth, may be viewed as the nearest thing to a universal law of causation that we can formally guarantee:
Any two or more items must
be related by some modus(es) within these frameworks, although the modus(es) by
which they are related are not
necessarily those of causation (or prevention).
The only alternative modus that is formally impossible is the one in each framework (labeled No. 1) consisting entirely of zeros: this the laws of thought interdict in advance for all items. Two (or more) items are always ‘tied together’ by one or more of moduses (each of which can be visually imagined as a sort of ticker-tape in which zeros and ones are punched), but we cannot predict how many and precisely which moduses are effective for that particular pair of items (or more).
A grand matrix represents all the ways that any two (or more) items might in principle, i.e. from an epistemological perspective, at first sight, be found to co-exist or not co-exist. But in practice, from an ontological point of view, after thorough research, not all these ways are applicable in every case: in each given case, only some alternative moduses are likely to be applicable.
As previously discussed, we can group the alternative moduses in various ways, according to what sort of relationship they signify between the items concerned. We can thus distinguish between ‘connective’ relationships (causation or prevention) and ‘non-connective’ relationships (one or more items incontingent), as shown in the table below for two items.
In one case, the last modus of any grand matrix (that involving only ‘1’ codes), i.e. modus #16 in a two-item framework (conventionally classified as absolute partial and contingent causation or prevention, i.e. the form pabsqabs), we cannot strictly say whether connection or nonconnection is ultimately involved (i.e. when more items are eventually taken into account, in a larger grand matrix). So this modus might be placed under either heading, or under neither of them.
Table 16.1. Possible relations between any two items P and R.
Some groups of alternative moduses signify incontingency (necessity or impossibility) of one (or more) of the items concerned, while the others signify contingency of the two (or more) items concerned. An incontingent item is independent of all others. Only where all items involved are contingent can causation or prevention (i.e. some connection) occur between them. Different combinations of moduses have been identified as different determinations of causation or prevention. These determinations have been classified in various hierarchies and polarities: strongs/weaks, absolutes/relatives, generics/joints/lones, positives/negatives, causative/preventive, each of which is signified by a certain group of moduses. But contingency of all items does not signify their connection.
Having thus put matricial analysis in perspective, it is easier for us to evaluate on purely formal grounds certain philosophical claims for or against causation that have arisen over the centuries. We shall here use the word ‘cause’ in the specific sense of causative connection, including in it both causation and prevention, but excluding other causal relations (such as volition). As we shall see, none of these claims can be formally established from our definitions of causation and all the properties of causation emerging from them.
1. Some philosophies have claimed that everything has a cause. This is commonly referred to as ‘the law of universal causation’ and is the position most widely adhered to. It is a claim that causation is to be found everywhere, that all things are ruled by it – i.e. that every thing is caused by some other things, themselves in turn caused by others, and so forth ad infinitum. There are different versions of this proposed law, “nothing is without cause”.
a) Oriental philosophies would opt for a radical interpretation, based on the belief that all things in the empirical world (dharmas) are impermanent, so that nothing exists that is independent. Clearly, this viewpoint eliminates a certain number of alternative moduses (those signifying incontingency) from consideration at the outset: but no formal grounds for such a narrowing of scope have been proposed. Indeed, if one reflects, the claim in question is self-contradictory, since it is itself put forward as a permanent fact, a necessity. So we can on formal grounds reject it.
b) Most Western advocates of universal causation would more moderately understand it as “nothing contingent is without cause”. They would allow that some things are necessary or impossible, but consider that those things which are possible but not necessary have to have a cause. It is important thing to realize that, contrary to what many of its advocates believe, this alleged law cannot be formally deduced from the definitions of causation. It is only conceivable on inductive grounds, by generalization from previously encountered cases.
Note that, unlike the radical version, the moderate version of universal causation is not inconsistent and does not prejudicially exclude any alternative moduses. What it does exclude at the outset, in advance of empirical research and without formal proof, is that some contingent item may exist that has no causative (or preventive) relation to at least one other item in the universe.
2. Some philosophies have claimed that nothing has a cause. We may cite as proponents: in the East, the Indian philosopher Nagarjuna (2nd Cent. CE), and in the West, the Scottish philosopher David Hume (18th Cent.). This viewpoint is essentially a denial that there is any such relation as causation; it is a claim that the concept is meaningless, a human invention without corresponding reality, an error of reasoning. Here again, we could distinguish a radical version, which excludes incontingency in principle and so is internally inconsistent, and a moderate version, which reserves indeterminacy to contingents.
Either way, the negative thesis that ‘nothing is causatively related to anything else’ arbitrarily eliminates for any two (or more) items the vast majority of alternative moduses: all those signifying causation or prevention, and does so for all items past, present or future, everywhere in the universe. It gives no formal ground for such a sweeping measure, but bases it on denial of the possibility of conceptualization or generalization. This may be claimed as an empiricist posture, or may be coupled with skepticism about perceptual evidence. But, since any such claims themselves use concepts and appeal to generalities that could only be admitted by granting generalization, they are self-contradictory and therefore logically untenable.
The antagonists of causation attempt to mitigate this paradox by claiming that causative propositions are “conventional” (Nagarjuna) or “habitual” (Hume), ignoring that such explanations themselves rely on admission of the causative relation. Some instead argue that though causation may be theoretically meaningful, it is impossible to establish in practice. But as shown in the present work, a concept of causation can readily be constructed, using indubitable simpler concepts of presence, absence, conjunction and disjunction, possibility and impossibility. Moreover, the concept would have to be convincingly defined, before it could be declared empty! So it cannot be meaningless. As for the fear that causative relations have no actual instances or are in practice unknowable, we shall now explain it.
The deep reason for such antagonisms is the failure to understand causative propositions as simply records of conjunctions of presences and/or absences of two (or more) items. Such summaries are generalized from observation, subject to corrective particularization if new observation belies them. The antagonists have not emotionally reconciled themselves to the tentative, inductive nature of knowledge, and so set up and cling to badly defined and impossible deductive ideals of knowledge (without noticing that they themselves cannot possibly satisfy them).
Causative judgment is indeed based not only on empirical evidence, but also on ordering of information by the rational faculty, since it concerns not only presence but also absence, and all negation involves rational projection. This however only means that reason provides an ‘overlay’ (the grand matrix) through which to order (summarize and predict) events, but the evidence this overlay is laid over (i.e. the 1s and 0s exhibited by the items, the moduses applicable in their case) is empirical (ultimately, though we may thereafter get to know some by immediate or syllogistic inference from previous experiences).
3. Some philosophies have claimed that some things have a cause and some things do not have any cause. This position gained acceptance among physicists and consequent popularity in the 20th Century, after the advent of quantum mechanics (as interpreted by Niels Bohr) and Big-Bang astronomy (Stephen Hawking seems to advocate that the apparition of matter and its primeval explosion were simultaneous and causeless). This position, note well, is a compromise between the preceding two. It admits of non-causation in specific areas (the beginning of time) or at certain levels (the subatomic), together with causation in all other cases or situations. This position is formally neither provable nor disprovable: it is consistent and does no violence to matricial analysis.
It is neither more nor less conceivable than the (moderate) ‘universal causation’ thesis. They are simply – equally conceivable contrary hypotheses or predictions. Each of these two theses must be viewed as an epistemological postulate or an ontological generalization. ‘Universal causation’ is a generalization from cases where causation was apparently (after certain generalizations) found, to cases where it is not yet found; whereas ‘particular causation and particular noncausation’ emphasizes the cases where we have not yet found a credible cause, and suggests that we generalize this failure to ‘no cause will ever be found, because none exists’.
A grand matrix, remember, foresees every conceivable way two or more items might appear together or apart (refer Table 16.1 above). To establish that a given item has some cause, it suffices that we find one other item that has a relationship of connection with it (two-item moduses #s 7-8, 10, 12, 14-15, and some cases of #16). But to establish that a given item has no cause is much more difficult! It is not sufficient to show that one other item is not its cause (two-item moduses 2-6, 9, 11, 13, and some cases of 16): one has to show that this is true of all other items.
Obviously, for those of us who make no claim to omniscience, this is an impossible task. We can only – either appeal to a law of universal causation or accept the possibility that some things are causeless. In any case, generalization is doubly involved: first, in the inductive proof that any modus is applicable to the set of items observed; second, in the inductive passage from those items to items not observed. Denying both these principles is not a viable third alternative, as already explained.
To repeat, neither of the coherent doctrines can be proved or disproved deductively; they are neither self-evident nor self-contradictory. They may only conceivably be established inductively, through generalization from respectively “we have found causes for everything encountered so far” (which is far from the case) or “there are things for which no causes have so far been found” (which is true, but since “there are things for which causes have eventually been found” is also true, we are inhibited from quick generalization). There is a standoff.
Since no formal ground for either position is evident, the science of Logic must make formal allowance for both positions. Its task is to provide the formal means for open-minded debate of this topic (as of all others): it cannot prejudicially exclude the one or the other from language and block discourse in advance.
It is important to be clearly aware where in a grand matrix causelessness or spontaneity is allowed for. Refer to the last three rows of Table 16.1 above, where one or both items are contingent, yet the moduses of causation or prevention do not apply to them. If item R is the contingency under scrutiny, our table implies that R might be without cause if its relation to P falls under modus 4 or 13 or under modus 16. For an item to certainly be causeless, it would have to have one of these relations not only to P, but also to all other items in the universe – P1, P2, P3, etc.
Absolute noncausation of R can be expressed in the form “nothing causes R”, which collects together innumerable statements of relative noncausation, of the form “P does not cause R; P1 does not cause R; P2 does not cause R;...etc.”, where P, P1, P2,... are all existents other than R.
Now, to formally deny that there exists anything such that nothing causes it, one would have to find an inconsistency in the said mass of statements, or in their summing up in one sentence. No such inconsistency arises. Therefore, we have to admit absolute non-causation as a formal possibility, i.e. as at least conceivable. It may still be factually false, i.e. there may indeed be no such animal. The issue must therefore remain open; that is, formal logic may and must proceed without resolving it. Epistemology and ontology may still, nevertheless, postulate the one or other position with reference to wider considerations.
With regard to the question: what would the relationship be between the causation apparent at the level of our ordinary sensory experiences and the spontaneity assumed by physicists to be operative at a deeper, subatomic level (known indirectly, by postulates and experimental observations) – the answer is simple enough. It is the relationship implied by a dilemmatic argument like “whether X or Y occurs, nevertheless Z is bound to occur” – that is: whether X or Y blossoms spontaneously at the subatomic level, they both have the same effect Z, or equally fail to affect Z, at the commonplace level. Here, X and Y may refer to events underlying Z, or to magnitudes or degrees of certain events (namely, velocity and position), whose average result is equally Z or of which Z remains independent. Thus, the deeper level may be open to spontaneity while the more superficial level remains governed by causation, without any incoherence being implied.
The universal causation doctrine predicts that every existent has at least some causative relation(s) to some other existents. This is usually understood in a moderate sense as only some other things cause each thing, but Buddhism understands it more extremely as all other things cause each thing. This ‘universal universal causation’ is referred to as the interdependence (or codependence) of all things.
We normally suppose that only the past and present can cause the present or future; and indeed this principle should primarily be read that way. But some might go further and claim that time is transcended by causation, and that literally everything causes everything; I am not sure Buddhism goes to that extreme. Note also that, in truth, Buddhism intends its interdependence principle restrictively, as applicable only to dharmas, i.e. the transient phenomena constituting the world of appearances; in the higher or deeper realm of the quiescent and undifferentiated “original ground” there is no causation.
Be it said in passing, this version of “karmic law” must be distinguished from the more narrow statement, which most of us agree with, that actions have consequences. The latter does not imply the former! More deeply, I think what the Buddhists really meant by their law of karma was that each human (or other living) being is somewhat locked within recurring behavior patterns, very difficult (or impossible) to get out of. This is another issue, concerning not causation but volition.
That is the sense of “the wheel”: our cultural and personal habits as well as our physical limitations, keep influencing our behavior and are reinforced by repetition. Much meditation and long-term corrective action are required to change them; they cannot be overcome by immediate measures, by a sheer act of will. We are thus burdened by a “baggage” of karma, which we carry out through our lives with usually little change; it may be lightened with sustained effort, but is more likely to be made heavier as time passes.
If we logically examine the claim that “everything causes everything”, we see that if everything is causatively connected to everything else, then nothing is without such connection to any other thing, let alone without causative connection to anything whatsoever. That is, this doctrine is effectively a denial that relative as well as absolute noncausation ever occurs, which no one in Western culture would admit. To evaluate it objectively, let us look back on the findings in the present volume.
First, in defense of the idea of interdependence, it should be recalled that when we discussed the significance of the “last modus” in any grand matrix (modus #16 for two items, or #256 for three, etc.), which declares any combination of the items concerned or their negations as possible (code 1 in every cell of the modus), we saw that there was an uncertainty as to whether this indicated causation (or more broadly, connection) or its absence. If the last modus is could be shown on formal grounds to indicate causation in all cases, then all contingents in the universe would have to be considered as causatively related to all others (i.e. any two contingents taken at random could be affirmed as causatively related, specifically in the way of the partial contingent determination, pq).
However, since such formal demonstration is lacking, and the idea is anyway disagreeable to common sense (at least that of non-Buddhists), we estimated that the science of Logic had to keep an open mind and grant the possibility of the alternative interpretation, namely that two items may or may not be causatively related to each other (i.e. relative noncausation is possible), and moreover that spontaneity (i.e. absolute noncausation) is at least conceivable in some cases. However, in this context, the Buddhist thesis of interdependence, remains a legitimate formal postulate. But note well, only a possible alternative hypothesis; and not a very probable one for most observers (those of us who believe in freewill, for example; as well as physicists who reify the Heisenberg Uncertainty Principle).
An important formal criticism we can level against the notion of interdependence is to ask what manner or degree of causation is meant by it. The term ‘causes’ in ‘everything causes everything’ is used very vaguely. Is only causation intended, to the exclusion of volition? And if causation is intended, surely this is meant broadly to include prevention? And are the different determinations of causation admitted, i.e. strong (complete and/or necessary) as well as weak (partial and/or contingent)? The definition of causation traditionally attributed to the Buddha is:
When this is, that is; this arising, that arises. When this is not, that is not; this ceasing, that ceases.
This definition would suggest that only complete necessary causation is intended. But other discussions within Buddhism suggest that this definition is only intended as a paradigm, as the most obvious case, and partial and contingent causation is also in practice admitted, as use of the plural in the expression “causes and conditions” testifies. We may regard prevention as formally subsumed by all these concepts, by negation of an item. Some discourses also seem to accept volition, but this need not concern us here. Focusing, then, on causation in a broad sense, we may make the following criticism.
If everything is causatively related to everything else, then the only conceivable kind of causation would be weak (both partial and contingent). For strong causation (complete and/or necessary) surely implies a certain exclusiveness of relationship between the items. If all items are involved to some degree in the existence of a given item, then none of those causes can be claimed to predominate. So finally, it seems to me, this Buddhist doctrine of multilateral causation requires all bilateral causative relations to be weak, and ultimately abandons strong determinations (including mixtures), and all the more so the strongest determination (which it originally rightly claimed as the definition of causation).
One way to show that the interdependence theory implies specifically a ‘universal weak link’ is as follows. If we claim interdependence to apply indiscriminately to all ‘things’, i.e. not only to experiential things (dharmas), but also to abstract things, we fall into formal difficulties as soon as we suppose some causative relations to be strong. For then such abstract relations (i.e. causations) also count as ‘things’, and are therefore subject to interdependence. We might thus ask how a cause can be complete or necessary when that relationship is itself dependent on some yet other cause: we are forced to contradict our premise and conclude that the cause is not as complete or necessary as it seemed.
I suppose the proposed state of affairs (universal interdependence) is formally conceivable, although I do not see on what grounds we could possibly allow such rejection in one fell swoop of a large number of moduses (i.e. all alternative moduses concerning the strong determinations). Unless a reasonable formal or empirical ground is provided, there is no justification in such a radical measure: it would constitute prejudice. The Buddhist claim is of course based on a meditative experience; but since this is esoteric, not readily available to all observers at will, we must remain critical and view it as speculative. We cannot categorically eliminate it on firm rational grounds, but we cannot just take it on faith.
It should be realized that causation is a conceptual object, not a percept. Before we can discern a causative relation between two or more percepts (and all the more so between concepts) we have to distinguish the percepts from each other (and conceptualize them by comparison and contrast of many percepts, in the case of concepts). Also, causation refers to negation, which is a product of rational as well as empirical factors. Thus, if we approach the issue of causation with respect to the phenomenological order of things, we must recognize that it is a rather high-level abstract, although of basic importance in the organization of knowledge. It is not something we just directly see or otherwise sense. For this reason, we may remain skeptical that there is some flash of insight that would instantly reveal the causal relations of all things in the universe.
Thus, while the interdependence doctrine apparently does not give rise to formal inconsistency, we have good reason to doubt it with reference to normal human knowledge development. Causation is ordinarily known only gradually, through painstaking observation and analysis of particular data, always subject to review and revision as new data makes its appearance and possible contradictions are encountered. Our minds are not omniscient or rigidly deductive, but cumulative and flexibly inductive: we proceed by trial and error, constantly adjusting our positions to match up with new input and logical insight. Therefore, we cannot rely on sweeping statements, like that about interdependence, without being very careful.
Of course, some philosophers would argue back that causation as such is a man-made illusion, since pure experience only reveals undifferentiated presence. Differentiation into ‘distinct’ percepts, and finding that some sought things are ‘absent’, and conceptualization on the basis of ‘similarities and differences’, are all acts of reason. Indeed, if all perceived appearances are regarded as mere wave motions in a single, otherwise uniform substrate of existence (the ‘original ground’ of Buddhists or the Unified Field of physicists), then the boundaries we think we perceive or conceive for individuated things are in fact mere fictions, and all things (including even our fantasies about causation) are ultimately One in a very real sense.
So let us keep an open mind either way, and cheerfully move on. I just want to add one more small set of reflections, which the Buddhist idea of interdependence generated in me. This idea is often justified with reference to causal chains. I tried therefore to imagine the world as a large body of water, like Lake Geneva say. According to this theory, supposedly, a disturbance anywhere in the lake eventually ripples through the whole lake, to an ever-diminishing degree but never dampening to zero. I then translated this image into the language of causal chains, for purposes of formal evaluation.
Looking at the results of macroanalysis, one would immediately answer that the Buddhist expectation is wrong. As we have seen, a cause of a cause of something is not necessarily itself a cause of that thing; and even if it is a cause, it may be so to a lesser degree. Many first figure syllogisms yield no causative conclusion, although their premises are compatible. Some do yield a conclusion, but that conclusion is often weaker in determination than the premises. Thus, we have formal reasons to doubt the idea of interdependence, if it is taken to imply that ‘a cause of cause of something is itself in turn a cause of that thing’.
All the same, I thought, thinking of the movement of disturbances in the lake, there is some truth in the contention. I then thought that maybe we should conceive of ‘orders of causation’ – and postulate that even “if A causes B and B causes C, but nevertheless A does not syllogistically cause C” is true in a given case in terms of first-order causation, it can still be said that A causes C in second-order causation. And we could perhaps continue, and declare that if the latter (meaning, causes a cause of) is not applicable in a given case, we could appeal to a third order of causation, etc. We might thus, in an attempt to give credence to all theories, explain the Buddhist notion as involving a diluted sense of ‘causation’.
This idea seemed plausible for a while, until I got into microanalysis. In the latter approach, conclusions are given in terms of alternative moduses. There is no room for a fanciful, more abstract, additional order of causation: the result would be identical, still the same number (one or more) of legitimate alternative moduses. No useful purpose would be served in inventing new (narrower or broader) sets of alternative moduses, and giving such groups new names. We could only at best regard all moduses in a grand matrix (other than the first, composed of all zeros) as indicative of some ‘causation’ (in a maximal sense), and so say that any alternative modus found at the conclusion of a syllogistic intersection is ‘residual causation’.
But having reached this bottom line, we see how trite the suggestion is.
Before closing the present chapter, I would like to add some brief comments on some features of causation that should be further highlighted.
a) Parallel Causation. This concept was presented in some detail in our initial discussion of the generic and specific determinations, and thereafter no longer mentioned. I here just wish to remind the reader of the possibility that different causes, which are not necessarily causatively related to each other, may nevertheless have a causative relation to the same effect. That is, two things, say A and B may separately (strongly or weakly) cause some third thing C, and yet A does not cause B and B does not cause A. As the proverb says, many roads lead to Rome. If this is forgotten, one may easily get confused and think of ‘pluralities of causes’ as only possible within a single weak causation or in a chain of (weak or strong) causations.
This feature of causation is implicit in the microanalytic approach, insofar the possibility of several grand matrices having common items is not formally excluded.
b) Degrees of Causation. We have developed the concept of weak causation without distinction between the different possible degrees of such weak causation. That is, we have to also ask: what is more effective, what plays a larger part in producing the effect, the item (or collection of items) called partial and/or contingent cause or the item (or collection of items) called the ‘complement’? We did set up a gross hierarchy between the joint determinations, mn being the strongest, mq and np being middling, and pq being the weakest. But we also mentioned that in weak causation, the participant items may have unequal shares in the causation.
This feature of causation has not been made apparent in matricial analysis so far, and therefore needs to be accounted for in some way. I would suggest offhand that the way to include it may be to consider the degrees of probability underlying each possibility mentioned in the alternative moduses concerned. Thus, instead of a code ‘1’ in each cell of an alternative modus, we might have some as worth 20%, others 40%, etc., with all non-zeros adding up to 100% probability. For example, if P and Q are complementary partial causes of R, P without Q may be more likely to be followed by R than Q without P.
In some cases, the issue may be dealt with by considering concomitant variations (see below). In any case, this topic requires further attention.
c) Reciprocity and Direction.
A cause and effect may (in some cases, not all) be interchangeable. For example, if we refer to the ‘ideal gas equation’ PV/T = constant, and consider a gas at constant temperature, we know that if the pressure is varied (increased or decreased), then the volume varies accordingly (decreased or increased). It is also true that if the volume is varied, the pressure is proportionately affected. This is mutual causation. Some things in a cause-effect relation do not have similar reciprocity. For example, no matter what we do, entropy further increases: our relation to entropy is one-way.
It should be stressed that even if we acknowledge that the direction of causation may only go in the direction of time, cause and effect are often simultaneous events (this is especially common in the extensional mode of causation, but also occurs in the natural mode). Cases of mutual causation, as well as cases of non-reciprocity, may occur either way, i.e. with cause temporally before effect or with both at the same time.
The essence of causation is certain possibilities and impossibilities of conjunctions – it does not concern questions of reciprocity or direction. These issues are left implicit in matricial analysis, acknowledged as formally possible by virtue of being ignored.
d) Concomitant Variation. We analyzed J. S. Mill’s fifth method, that of Concomitant Variations, in some detail in Appendix 1. Although I mentioned this there, I want to here stress that this method concerns not only strong causation, but also weak causation. The above mentioned ‘ideal gas equation’ is an excellent example. In strong causation, the concomitant variation between cause and effect is one-to-one, although not necessarily proportional. In cases of weak causation, where two or more causes together produce the effect, the part played by each factor is clarified by (if possible) holding any other factor in check (i.e. constant) while varying the one examined. This is of course not always possible.
When it is possible, the standard technique is to tabulate or graphically represent the results of experiment and then try and express them in a mathematical formula, like PV/T = constant, which summarizes a mass of if-then statements as already explained. Epistemologically, this constitutes generalization from observation. When such simple approach is not possible, because we cannot directly control the situation (for instance, in some sociological or medical researches), we resort to adductive methodology. We posit certain postulates, construct a formula out of them, and then test that formula with reference to empirical data.
It should be seen that concomitant variation deals essentially in concepts, rather than percepts. A percept is only what it is: if change occurs, another percept has replaced it. A concept, on the other hand, is an abstraction, which may well have different particular values in different cases or situations. Our formulae are algebra, not arithmetic.
We shall have to analyze concomitant variation further with
reference to matricial analysis. Can the latter method be enlarged or clarified
to include consideration of the former within it?
The Logic of Causation is a research and book project that I started several years ago, and which will no doubt take a few more years to complete. It is itself just a stage within my larger Causal Logic research and book project.
I published, on a small scale, an “unedited and unfinished draft” of The Logic of Causation back in 1999 (Phase One). The present “revised and expanded edition”, published on a small scale in 2003, corrects some errors found in the 1999 version relating to the issue of lone determinations, and adds new developments of 2000 (Phase Two), as well as some recently written material such as chapters 10 and 16.
The reason why I “pre-publish” like this is that I am periodically forced to leave this research work to earn my living by other means. I do not know when I will get another chance to continue it, and wish to share with other people the results already obtained, if only through my Internet site, www.TheLogician.net. Furthermore, knowing that life is unpredictable and often short, I want to make sure the work already done is not lost to humanity, if my days happen to come to an end prematurely. I pray, however, that G-d allows I finish this work (and more still) long before!
Phase Two is in truth far from over at this time. We have here introduced the basic principles and formulas of microanalysis, but only listed most of the significant three-item syllogisms. But a very important development still in process is four-item syllogism. For this, because of the enormous matrices involved, I have to work with complex relational databases. Only after this work is completed can we compare Phases One and Two, and make sure that all previous work is consistent and error-free.
After all these technicalities are finished, and the facts of the case are settled, I will be able to devote my full attention to remaining philosophical issues relating to causation. Thereafter, I shall turn to volition and other issues in causality.
Anières (GE), Switzerland, 2003.
 I say ‘may’ – I mean ‘hopefully will’. If it turns out that there are indeed contradictions, then they are due to human errors (errors of inattention on my part). I say this with certainty, because I do not believe that any of the postulates adopted in developing macro- and microanalysis are faulty; and they cannot be in mutual contradiction, being essentially the same in both instances.
 Other logicians might prefer to resolve the issues of causative logic through symbolic formulas. Although such an effort would not be without value, I personally eschew such means, which to my mind are obscure, lacking in transparency. Rather than express things in such esoteric language, I have preferred to make the message explicit, so that everyone can see for him or her self with a minimum of mental effort just what is meant and why it is true.
 See Chapter 13.2.
 Note well that this is a relatively late realization of mine, in Chapter 13.2: that the last modus is not necessarily always to be interpreted as signifying causation; it is only indicative of possible causation. Consequently, my classification of the 2-item modus #16, or the 3-item modus #256, etc., under the heading of causation was not accurate and could be misleading. It should more precisely be classified as ‘indefinite’.
 This refers to absolute noncausation, which is not to be confused with relative noncausation: it means no cause whatsoever, and not merely not this specified cause.
 See for instance Thich Naht Hanh, The Heart of Understanding (Berkeley, Ca.: Parallax, 1988).
 I always feel a certain affection for that example, which I learned in my teens. It shows how education has an impact on us.