THE LOGIC OF CAUSATION

Phase Three: Software Assisted Analysis

Chapter 23 Exploring Further Afield.

## 1. Possible Forms of Premises.

It is clear that the scanning technique we have developed in the preceding chapter is a thorough and foolproof method for finding the conclusion(s) from any set of premises with a common item, i.e. any syllogism. We have there demonstrated the method with reference to positive causative premises and limited our search for conclusions to the range of all possible causative and preventive forms with the major and minor items concerned. But obviously, with appropriate changes in the data and/or the formulae used, and the required patience and effort, we can use the same method to solve any syllogistic problem whatever with full confidence.

We can use as premises any forms that can be entirely expressed in terms of moduses: viz. those I call ‘pre-causative’ forms, i.e. conjunctive propositions with two or more items, like ‘P + Q is possible’ or ‘P + Q is impossible’, or their derived conditional propositions, like ‘if P, then Q’ or ‘if P, not-then Q’, which constitute the building-blocks of causative propositions[1], or we can use any sort of causative proposition. Needless to add, all that is said here concerns not only propositions with two terms (like P, Q), but equally those involving a complement (i.e. a third term, be it positive or negative).

The term ‘causative proposition’ can be (and has here been) used in both a narrow and a broad sense, note well. In a narrow sense, ‘causative propositions’ signifies those of the form ‘P is a (complete, necessary, partial and/or contingent) cause of Q’ or the negation of such form, i.e. whose stated items are intended as positive. But in a broad sense, ‘causative propositions’ refers to all forms involving a causative relation ‘is (or is not) a cause of’, whatever the intended polarities of the items involved, thus including ‘preventive propositions’ (with items P and notQ), and inverse causative propositions (with notP, notQ) and inverse preventive propositions (with notP, Q).

The form of causation is not essentially affected by the polarities of the items involved. Every term, whether conceived as positive or negative, is just ‘a term’ to the causative relation involved. The issue of polarity of items only acquires significance when we compare two or more causative propositions having the same terms but with different polarities. Thus, the issue of polarity of items is merely comparative. But it is not without importance.

Preventive propositions, I remind you, can be stated (for instance) as ‘P prevents Q’, or as ‘P causes notQ’; note the change of polarity of Q. As we have seen in matricial analysis, the moduses of ‘causative’ forms in the narrow sense and ‘preventive’ forms are not the same. This means that, although ‘preventive’ propositions are a species of ‘causative’ propositions in the broad sense, they stand in significant contrast to ‘causative’ propositions in the narrow sense. Therefore, it is often necessary to investigate those two sets of forms separately, if we want to be exhaustive in our treatment of logical processes.

On the other hand, matricial analysis has shown us that the moduses of ‘inverse causation’ are closely related to those of ‘causation’ (in the narrow sense), and likewise those of ‘inverse prevention’ are closely related to those of ‘prevention’. This means that these two forms of causative proposition (in the broad sense) rarely require separate treatment, and their processes can be inferred from those of causative (in the narrow sense) and preventive propositions without need for bulky matricial analysis.

Another possible application of scanning is to syllogisms with one or both premises of ‘vague’ form, or of ‘lone’ form, or otherwise out of the ordinary. Unusual conjunctions of positive and negative forms can also be studied in this way. I have in the past (in phase II) looked into such moods briefly, but now we can do the job with more certainty and thoroughness. I will not however do it, but leave it to others.

As well, any mixture of forms of pre-causative or causative propositions can in principle be used as premises and/or conclusions in causative syllogism. However, we tend in our natural discourse not to mix such large categories of form too much. We would rarely if ever mix pre-causative and causative premises, and would rarely seek conclusions of pre-causative form to premises of causative form or vice versa. On the other hand, mixtures of different sorts of causative, preventive, inverse causative and inverse preventive forms, do occasionally occur in discourse, and should be considered.

## 2. Dealing with Negative Items.

Let us now, therefore, consider to what extent the syllogisms, ‘causative’ in the narrow sense, dealt with in the preceding chapter, cover the ground of ‘causative syllogism’ in the broadest sense. The moods dealt with so far have been with positive causative premises with positive items, in the three main figures. That is, they had the forms (sometimes relative to a subsidiary term S, not here shown) listed below:

 Primary forms Figure 1 Figure 2 Figure 3 Major premise:: Q-R R-Q Q-R Minor premise: P-Q P-Q Q-P Conclusion(s): P-R + P-notR P-R + P-notR P-R + P-notR

Note the large form of the conclusion, which might include not only causative propositions but also preventive ones (our chosen range having been expanded thus). We do not need to mention inverse conclusions like ‘notP-notR’ or ‘notP-R’ because these are entirely inferable from the eventual conclusions of form ‘P-R’ or ‘P-notR’. For example, a conclusion m of form P-R is logically identical to a conclusion n of form notP-notR. We might thus view the inverse forms implicit in our eventual conclusions to represent subaltern moods of the same syllogism, and ignore them accordingly.

Let us now consider what happens when we change the polarity of the minor and/or major term in the premises, and so in the conclusion to those premises:

 Negate major term Figure 1 Figure 2 Figure 3 Major premise:: Q-notR notR-Q Q-notR Minor premise: P-Q P-Q Q-P Conclusion(s): P-notR + P-R P-notR + P-R P-notR + P-R

If we just change the polarity of the major item, we obtain, in the first and third figures, a preventive major premise, and in the second figure, an inverse preventive major premise, and in the conclusion, preventive and causative forms.

 Negate minor term Figure 1 Figure 2 Figure 3 Major premise:: Q-R R-Q Q-R Minor premise: notP-Q notP-Q Q-notP Conclusion(s): notP-R + notP-notR notP-R + notP-notR notP-R + notP-notR

If we just change the polarity of the minor item, we obtain, in the first and second figures, an inverse preventive minor premise, and in the third figure, a preventive minor premise, and in the conclusion, inverse preventive and inverse causative forms.

 Negate extremes Figure 1 Figure 2 Figure 3 Major premise:: Q-notR notR-Q Q-notR Minor premise: notP-Q notP-Q Q-notP Conclusion(s): notP-notR + notP-R notP-notR + notP-R notP-notR + notP-R

If we change the polarities of both the major and minor items, the premises are modified as above, become preventive or inverse preventive in form, and the conclusion becomes a conjunction of inverse causative and inverse preventive forms.

 Negate middle term Figure 1 Figure 2 Figure 3 Major premise:: notQ-R R-notQ notQ-R Minor premise: P-notQ P-notQ notQ-P Conclusion(s): P-R + P-notR P-R + P-notR P-R + P-notR

Similarly, if we negate the middle term in both the premises, no change occurs in the conclusion. And if we do this in addition to the above listed changes in major and/or minor premise, the conclusions found there remain the same. However, in all these cases, the categorization of the premises of course changes. Thus, where the polarity of the middle term is changed in both premises, but not that of the other terms, the premises all have preventive or inverse preventive forms (as shown immediately above). If now we change the polarity of the major and/or the minor term(s) as well, we will naturally again affect the forms of the premises in the various figures, some of them becoming causative or inverse causative. In this way, we gradually exhaust all possible forms of premises.

All the above concerns both 3-item and 4-item syllogism (and even 5-item syllogism). In 4-item (and likewise 5-item) syllogism, we must additionally take into consideration the complement(s) involved. When one of the premises is or involves a relative weak proposition with a complement S, we can similarly obtain an additional mood by replacing it with the complement notS. In such case, obviously, where the conclusion refers to S we will insert notS, and where it refers to notS we will insert S. Such substitutions of course further increase the number of valid moods at our disposal.

What all the above means is that the lists of causative syllogisms developed by scanning in the previous chapter can with relative ease be adapted to such changes of polarity in the items concerned. This method of substitution is a valid shortcut. There is no real need to deal with these moods separately, producing bulky spreadsheets all over again. Just change the polarities of the terms systematically as shown above, and thus re-define the forms of the premises and conclusions involved, and the job is equally well done. We can thus refer to such syllogisms as derivatives of purely causative syllogisms treated in the previous chapter.

I have not numbered the results of such multiplication of moods, but this is a job that eventual needs doing. We should anyway keep in mind the distinction between primary (independently validated) and secondary (derived) moods when we do count them.

It is probable that many, if not all, syllogism thus derived (by substitutions) can also be derived by direct reductions. I cannot predict with certainty offhand, in view of the fact that many moods have multiple (both positive and negative) conclusions or only negative conclusions. Perhaps some or all of the negative conclusions can be derived by indirect reduction (i.e. reduction ad absurdum). The matter would have to be investigated carefully in detail before we can say for sure. However, I am not about to do this job. I leave it too to others.

## 3. Preventive Syllogisms and their Derivatives.

But this substitutive method is not adequate for preventive syllogisms and their derivatives. The problem of negation of items is more complicated when the middle term is negated in only one of the premises. In such cases the conclusions are affected in ways not easily predicted by mere substitutions.

 Negate middle term Figure 1 Figure 2 Figure 3 Major premise:: Q-R R-Q Q-R Minor premise: P-notQ P-notQ notQ-P Conclusion(s): P-R + P-notR P-R + P-notR P-R + P-notR

 Negate middle term Figure 1 Figure 2 Figure 3 Major premise:: notQ-R R-notQ notQ-R Minor premise: P-Q P-Q Q-P Conclusion(s): P-R + P-notR P-R + P-notR P-R + P-notR

We can predict at the outset that it is irrelevant whether the middle term is negative in the major or minor premise. The forms involved will be named differently, according to where the middle term has been negated, but the syllogistic process and its conclusion will be the same. What is sure, however, is that the conclusions obtained will be different.[2]

I decided therefore to explore the issue further, and developed the following tables for 3-item preventive syllogism, with one causative premise (the major) and one preventive premise (the minor), which can all be viewed online at my website as usual. The first of these tables shows the moduses of the premises used; and the second table summarizes the next three, which detail the scanning work done in each figure. The important table is the second, viz. Table 23.2-0; please look at it carefully:

As can be seen, what I did here was simply change the data for the minor premise in the 3-item tables developed in the preceding chapter. The latter served as templates for the present investigation. No formulas were changed, note (which is why I have not produced a new ‘formulas used’ table for the tables posted here). Note too that the forms of conclusion scanned for (P-R + P-notR) are unchanged.

However, as the comparative Table 23.3 shows, the specific conclusions actually obtained are very different. What is interesting is that the conclusions here obtained are almost a mirror image of the conclusions previously obtained, except that the ‘spectrum’ has shifted from causative to preventive. That is, whereas with two causative premises the conclusions leaned on the side of causation, consisting roughly speaking of a mix of mainly positive causative conclusions and exclusively negative preventive conclusions – here, with a causative and a preventive premise, the conclusions lean on the side of prevention, consisting roughly speaking of a mix of mainly positive preventive conclusions and exclusively negative causative conclusions! We might express this shift by writing the conclusions in the form ‘P-notR + P-R’ (instead of ‘P-R + P-notR’) – i.e. with the preventive part first and the causative part second, though of course the order in which the parts are stated is formally irrelevant.

If you look again at the lists of conclusions and compare them, you can see that, apart from this ‘spectral’ shift, the conclusions are essentially the same except that the order they appear in the columns is not identical. But the differences are, as might be expected, orderly. In the first figure, the moods numbered 111-118, 141-148 reflect the moods with the same numbers (though with the spectral shift, of course – i.e. they are, as it were, laterally inversed ‘mirror images’); while the moods numbered 121-128, 131-138, 151-158, 161-168, 171-178, 181-188 respectively mirror the moods 131-138, 121-128, 161-168, 151-158, 181-188, 171-178. All statistics are consequently the same, mutadis mutandis.

For this reason, i.e. with reference to the kind of conclusions obtained, I have called syllogism with two causative premises (and its derivatives, as above described) ‘causative syllogism’; and syllogism with one premise causative and the other preventive, ‘preventive syllogism’. Note well, these labels refer to the conclusions obtained rather than to the kinds of premises (for, as we have seen in the previous section, a syllogism may have some preventive or inverse-preventive premise(s) and yet be essentially causative). Note also the distinctive format of preventive syllogism – the middle term is antithetical, being positive in one premise and negative in the other.

Just as causative syllogism has a mass of derivatives, as detailed in the previous section, so with preventive syllogism. If we take the syllogisms listed in the above tables, and negate their minor and/or major term, and/or negate both occurrences of their middle term, and/or negate their subsidiary term(s), we get a mass of derivative syllogisms as before. This means that, combining the lists of positive causative and preventive syllogisms, including all such derivatives, we truly have an exhaustive overview of syllogisms to do with causation in the largest sense.

The above listed tables concern only 3-item syllogism, of course. Concerning 4-item preventive syllogism, we can already predict that the results will be very similar. Just as 3-item preventive syllogism yields a shift in conclusions from the causative to the preventive side, so with 4-item preventive syllogism we can expect such a shift. Similarly, the vertical order of the conclusions will be somewhat changed, though in an orderly manner as before. The 4-item results are sure to be analogous, because they have to be consistent with 3-item results. The following CONJECTURAL table shows the results one may expect offhand, based solely on analogy to the results given in Table 23.3:

Please note well that this table is conjectural, and not to be relied on! The only reason I have put it here is to give the reader an idea of the sorts of results to expect. But it has yet to be proved by the scanning method. The reason why we cannot be sure of this analogical argumentation is that there may be unexpected twists when the fourth item is involved – e.g. it may be that a surprising change of polarity occurs in such cases. The scanning work must eventually be done if we wish to be exhaustive in our research. But I leave it till later or for others.

As regards the number of valid moods, we can predict that the collections of causative and preventive syllogisms will be numerically the same. By the way, regarding syllogisms with a premise (or two) that compounds causative and preventive propositions: such compound premises need not be viewed as necessitating further research, but can be treated as ‘double syllogisms’. That is, the conclusion(s) of the compound premise(s) will be the sum of the conclusion(s) of the two (or more) causative and preventive syllogisms involved.

The same comment can be made here as in the previous section regarding the possibility of deriving syllogisms by direct or indirect reduction. It may even be that the above tabulated preventive syllogisms can be so derived from causative syllogisms; I have not tried. As far as I am concerned, matricial analysis, scanning and substitution are independent sources of validity: there is no formal need for the traditional methods of Aristotelian logic. However, such research would be interesting to pursue for its own sake, as it might well reveal the interdependence of all syllogisms in the various figures and with various polarities of terms. Intuitively, this seems obvious to me; but it has to be proved. We would then know what the shortest possible list of causative syllogisms includes.

## 4. Syllogisms with Negative Premise(s).

All the above concerns positive premises, note well. We have yet to consider what happens when one or both of the premises, be they causative and/or preventive, is/are negative. This is what we will briefly look into now. My intent here is only to sketch the way, without actually doing the whole job.

I have in the past, in phase II, briefly considered the issue. In Table 15.9, I list various negative propositions or compounds with negative elements that may appear as premises, without working out their moduses[3]. Then, in Table 15.10, I consider first figure syllogisms with both premises negative and elementary – negations of m, n, p abs or q abs and find (using the ‘manual’ method) no conclusions from them.

Obviously, to be thorough in our research, we must systematically list all possible moods and then (mechanically) scan for the conclusions they imply in the full range of causative and preventive forms. In positive syllogism, we considered moods with both premises positive and among the eight forms mn, mq, np, pq, m, n, p, q (these forms being ordered from the strongest determination to the weakest, and numbered 1-8, even though the first four are compounds of the latter four). Our first job, therefore, would be to consider the negations of these eight compound and elementary forms, i.e. not-(mn), not-(mq), not-(np), not-(pq), not-m, not-n, not-p, not-q.

This I have done for 3-item causative syllogism, for all three figures. I started with the spreadsheets used to produce Tables 22.6-1, 22.6-2, 22.6-3. Using these as templates, I duplicated them 3 times each, and negated the premises successively in them: first the major premise, then the minor premise, then both the premises. The formulas were so easily adapted that I do not need to produce a listing of them for you. Lastly, I recalculated everything. The reliability of the results obtained is confirmed by their symmetry. The findings are summarized in the first table, and the detailed scanning work on which it is based is shown in the next nine tables. These tables can all as usual be viewed online at my website.

The summary in Table 23.5-0 should certainly be looked at. What we learn from it is that 24 moods out of 192 (12.5%) in each figure do yield a conclusion. It is always a single conclusion, and it is always negative and causative – i.e. neither positive nor preventive. 12 of these conclusions occur in syllogism with the major premise negative; and the other 12 of them occur in syllogism with the minor premise negative; no conclusions emerge when both premises are negative[4]. Of the said sets of 12 conclusions, 5 are not-m, 5 are not-n, and the other 2 are p abs and q abs. This is true in each figure, though the order of appearance varies.

This exploration in negative syllogism does not cover the whole field (see Table 15.9 again), but it is of course a good start. We can later push further afield. Note that not-(mn) means ‘not-m and/or not-n’, which allows for three consistent alternative outcomes ‘m + not-n’ or ‘not-m + n’ or ‘not-m + not-n’; and similarly for the other negated compounds not-(mq), not-(np), not-(pq). Thus, when we validate a conclusion for a negated compound, we also validate it for its three alternative outcomes; though this does not guarantee that the alternatives would not, when taken specifically, yield a more specific conclusion. Their syllogisms still need to be separately investigated.

Moreover, all this must be done for preventive as well as causative premises, as we did on the positive side. We can expect negative preventive syllogism to resemble negative causative syllogism, except that the negative conclusions obtained will be shifted to the preventive side and vertically displaced as before. Moreover, all this must be done for 4-item as well as 3-item syllogisms (and eventually also 5-item syllogisms). However, there is no cause for despair; the project is within the realm of the possible. It is big, but the scanning method is clearly known and perfect in its results. Furthermore, we (or at least I) have applicable templates – we can use the spreadsheets already developed for positive syllogisms, merely modifying them as appropriate for the various negative syllogisms.

Logicians are duty bound to solve all conceivable syllogisms. But I am not going to do all the work. I think I have done enough. I have laid the foundations of and greatly developed this important department of logic. It is a big field and there is room in it for other workers. I invite those interested to do their bit, and claim their share of the territory in history books on the subject. Look upon the task as an exercise. The best way to learn and fully understand is by doing – i.e. by taking up the challenge of actual research work.

## 5. Causal Logic Perspective.

Since I am here more or less putting an end to my share of the research, I would like to mention a further perspective. The logic of causation, or more briefly put ‘causative logic’, is only one department of the logic of causality, or ‘causal logic’. Two other departments of causal logic seem to me very important fields that logicians must develop further. They are the logic of volition, or ‘volitional logic’ and the logic of influence, or ‘influential logic’.

In my work Volition and Allied Causal Concepts (2004), I discuss the topic of volition philosophically and explain its relation to causation and influence respectively. I there debunk various widespread misunderstandings concerning these concepts and show how they can be credibly defined and correctly understood. I will not go into details here, wishing only to add a few general comments relevant to the present context.

Volition is a form of causality very different from causation. It cannot be reduced to causation. Volition cannot be fully represented by any particular modus or combination of moduses. However, volition is not entirely separate from or devoid of causation. That is to say, causation often (indeed, always) sets limits to the natural possibilities inherent in volition. Volition is never without limit (at least not that of humans or animals); it is always delimited, within a given ‘realm of the possible’. ‘Freedom of the will’ does not signify ‘unlimited freedom, period’ – but ‘full freedom within certain naturally (i.e. causatively) set limits’. This must always be kept in mind, if we do not want to fall victim to fallacious skeptical arguments.

Volition goes beyond causation, into specific areas that causation does not entirely rule. But the relationship between the agent of volition, i.e. the conscious being (to whatever degree) doing the willing, and the immediate causal product of volition, that i.e. which the agent actually wills, is not devoid of causation. Specifically, we can say that without this agent – or, in some cases, another like him – the thing willed would not have occurred. Thus, we can say that the agent is causatively related to the thing willed at least in this respect – i.e. in being a necessary cause (or one possible instance of a kind of thing that is a necessary cause) of the thing emerging from the act of will.

‘If not for this agent willing it, this willed thing would not have occurred’. Therefore, the agent is not only the willer of that thing, but a causative of it – specifically a necessary cause of it. However, his relation to the thing in such cases is not like that of a lifeless ‘complete cause’. We cannot express the relation between the agent and his will’s immediate products with reference to the moduses of complete causation. If we seek a modus to formally apply to causation it can only be (as discussed in the present volume, chapter 19) the ‘last modus’ – i.e. modus 16 in a 2-item framework, or more deeply modus 256 in a 3-item framework, and so forth. But this modus is too vague; it is not exclusive to volition, so it does not tell us anything about it.

Note that I have simplified the matter a bit, when I suggested that the agent (or some such agent) is necessary to the immediate effect of the will. In some cases, it is true, the same effect might equally well have been produced through entirely causative (i.e. non-volitional) means. In such cases, the agent is may be said to be only one of the many possible ‘sine qua nons’ to the kind of effect concerned. Nevertheless, with reference to that given instance of the effect concerned, the agent is definitely the one and only necessary cause. If someone or something else had caused a similar effect, it would have caused a similar but distinct instance of the effect. The instantiation is entirely identified to the actual cause.

Thus, though volition may involve some causation in that the agent is a necessary cause of what he or she wills, its relation cannot be confused with complete causation. Volition is not complete causation – it replaces the natural function of complete causation with a quite distinct form of causality. That is why we must name it distinctively and study it separately. Through an act of volition, the agent does the same job as a complete causative of ‘bringing’ into actuality something that was previously inactual and that would have otherwise probably remained inactual; but that does not qualify it as a complete causative. That is to say, we cannot say of it ‘if the agent, then necessarily the effect’ – because the agent may choose not to produce the effect, unlike a lifeless causative. Where volition is applicable, we can only say ‘if the agent, possibly the effect’.

Let me now remind you of the basic insights regarding ‘influence’ in my theory of volition. Influence is a causal relation – it is the relation between various objects (natural or endowed with volition) and what the agent it (i.e. the influence) impinges on wills. Influence is the intermediary between the domain of causation and that of volition. However, it radically differs from causation in this: something is an influence on volition only through the intermediary of consciousness. It must be cognized somewhat to qualify as an influence.

A causative may well affect a volition, by virtue of setting limits to its operation. Every volition is affected by some causatives, in this respect. But influence is something else entirely. An object, whether mental or physical, can only influence us if we are (to whatever degree or extent or depth) aware of it. Once we are aware of it, it becomes one of the factors affecting the act of will proper, i.e. the effort of the agent concerned. How does an influence affect a volition? It naturally makes it easier (positive influence) or more difficult (negative influence) for the agent to will that will. The agent must put less or more effort, respectively, to achieve the same result, i.e. the volition concerned. This is clearly very different from mere causation; it is a much weaker (and more personal) causal relation.

The other important point to keep in mind concerning influence is that it is never 100% determining. This is a requirement of consistency. If the will is free, it cannot be extremely influenced – some leeway must remain for it. If the will is truly determined by some factor external to the agent of will, then what occurs is not influenced volition but mere causation. The postulate of freedom of the will is that though volition may indeed be influenced it is never overpowered by influences, but retains some degree of liberty to go ahead in one direction or the other or to abstain from such courses of action. Generally, the reason why skeptics concerning freedom of the will err is that they fail to make this distinction between causation and influence.

All right. Why have I reviewed these basics concerning volition and influence in the present context? Simply because I want to briefly discuss the confluence of causative logic and the logics of volition and influence, and thus set the stage for causal logic in its widest sense. In Volition and Allied Causal Concepts, I lightly touched upon the formal logic relating to the concepts there studied; but there is still of course much, much work to do in that field. In the present volume, we have gone into causative logic in great detail, though there is still room for further research.

What I want to point out here is the need for a ‘formal logic’ style study of the matter. That is, one capable of drawing exact conclusions from syllogisms involving a mixture of causative premises with volitional and/or influential ones. What conclusions can be drawn from different combinations of these forms (and other related factors)? This is a big field that yet needs intensive study.

To give a formal example: if agent A willed W (under influences X, Y, Z) – and W is a complete cause (or partial cause with certain complements) of some phenomenon P – can A be said to have willed P? Offhand, I would say: no. Only if P was intended (i.e. thought about as a goal of W) would I say it was ‘willed’. If P was thought of as a very likely or inevitable effect of W, P may only be said to have been incidental to A’s action; and if it was neither intended nor thought of as possible or probable effect, then P was accidental to A’s action. Note that the thought of P is more influential in intention than it is in incident – and plays no role in accident.

To give a material example: when I drive my car to the supermarket, it is my intention to get there and shop there. I know this will cause some pollution, but that is not my intent; indeed, I wish I didn’t, but not enough to stop me driving; so this is incidental to my driving. I could of course knock my car into something on the way; but this is not very likely and I do my best to avoid it; so such occurrence would be an accident. (Incidents and accidents can also of course be positive events, note.)

Broadly speaking, then, given that A wills W and W causes P; what ought we to conclude regarding the causal relation of A and P? What we need are precise, complex, formal, accurate, consistent answers to all such questions. Only then would we be able to claim to have really mastered the whole field of causal logic. I leave it at that for now. Thank you for your attention.

[1] Please note that though I do not here develop the logic of pre-causative forms, I do not mean to imply that it is unimportant. The reason that I gloss over it is simply that this logic is not the topic of the current work. Moreover, the logic of conjunction and conditioning has been dealt with adequately in my Future Logic, and is in any case pretty well known and understood by logicians in general nowadays. Nevertheless, we could redo the whole of this pre-causative logic using the methods developed in the present volume for causative logic. The techniques of matricial analysis and scanning are very innovative, and would constitute a welcome renewal for pre-causative logic, which might well lead to previously unknown results. However, being human, I cannot do all this work, and I leave it to others.

[2] In truth, at first I was not sure of that: I assumed offhand that most premises so generated would be incompatible; but it turned out they were all compatible and most yielded conclusions. Thereafter, it seemed obvious that this should be so!

[3] I have in phase III worked out the 3-item moduses for the possible premises listed in this table, namely in Table 18.4.

[4] Note that the investigation in Table 15.10 taught us the same lesson for two negative elementary premises, not-m, not-n, not-p, not-q; but here the scope is larger, since we also prove the inconclusiveness of two negative premises, when one or both of them are negative compound premises, not-(mn), not-(mq), not-(np), not-(pq).