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THE LOGIC OF CAUSATION

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THE LOGIC OF CAUSATION

Phase Three: Software Assisted Analysis

Chapter 21 Causative Syllogisms.

1. Methodology.

2. 3-Item Syllogisms.

3. 4-Item Syllogisms.

4. About 5-item Syllogism.

Most of the limitations mentioned in the present chapter are overcome in the next chapter. Likewise, most the results obtained here are improved upon there. Nevertheless, I have not tried to rewrite the present chapter, considering that showing the process through which the study progressed is a good thing. This chapter has to be read to fully understand the next, because it breaks much new ground, uncovering issues and how to deal with them, and setting the stage for the finale in the next.

1. Methodology.

As we saw in phase II, as of chapter 14, validating or invalidating syllogistic arguments using the moduses identified through microanalysis is simple enough. In principle, any two or more premises might be put together, and their potential for ‘conclusion’ is the list of moduses they have in common. In practice, things are a bit more demanding.

For a start, the premises must have some item(s) in common; or else they will have no moduses in common. In the event that the argument is a sorites involving more than two premises, each premise must have at least one item in common with at least one other premise; and indeed the series must form a continuous chain reducible to two-premise syllogisms. Secondly, given that the two premises do have some item(s) in common, we need to check that they do indeed have moduses in common; if they have none, it means that the premises are incompatible and so cannot be put together to form an argument. Thirdly, if the premises do have some moduses in common (i.e. logically intersect), two things can happen.

It may be that these moduses are too ‘scattered’ – i.e. that some of them suggest a certain verbal conclusion and others of them suggest a contradictory verbal conclusion, the outcome being we have effectively no formal conclusion. This is not an inconsistency in the premises, but a sort of indecision in their joint implications. Alternatively, the moduses obtained from the conjunction of the premises all point to the same formal conclusion; in that case, we have a valid syllogism. Note well, to yield a valid conclusion, the common moduses of the premises must all be included in the list of moduses of the putative formal conclusion.

What I mean by a ‘formal conclusion’ is any propositional form. As we have seen, every propositional form ‘has’ a number (one or more) of alternative moduses. This means that each of ‘its’ moduses is enough to imply the proposition. It follows that if the premises jointly yield a certain set of moduses (one or more), and all of these moduses are included in the list of moduses for a given propositional form, that form is their conclusion. This is true, because each of their moduses is capable by itself of implying the form and they are all agreed in this implication. On the other hand, if any (one or more) of the moduses shared by the premises is not included in the list of moduses that imply the putative formal conclusion, it is not a valid conclusion.

The premises and conclusion of a syllogism may in principle be any relational statement; that is, each of them may be a conjunction of items (e.g. ‘P and R is possible’), or a conditional statement (e.g. ‘if P then R’), or a causative proposition (e.g. ‘P is a complete cause of R’). Likewise, the premises and conclusion may have either polarity, and they may have any combination of positive and negative items. In the present chapter, I am limiting our attentions to positive causative premises and (positive or negative) causative conclusions, to avoid an excess of information. I just want to demonstrate how syllogisms are validated or invalidated using a spreadsheet. But in principle, this limitation is artificial and we must study all conceivable combinations of premises and conclusions.

The main difficulty in researching syllogism through matricial analysis is in finding a conclusion that includes all the moduses the premises have in common. These moduses are easily ‘calculated’ using simple formulae; but to find the appropriate formal conclusion we must look at all available forms and see which one(s) include all the common moduses. This seems hard to do mechanically with a mere spreadsheet program; more sophisticated software or ad hoc programming seems required[1]. For this reason, I do not at this stage try to search for conclusions mechanically, but instead am content for now to verify the conclusions established manually in previous phases of the present work.

Given a putative conclusion, it is easy to test it in a spreadsheet program. We just write a formula for the conjunction of the two premises and the putative conclusion. If the result is a number of moduses (one or more), we know the putative conclusion is applicable. But we cannot yet be sure that it is a valid conclusion. We must still try to conjoin the same two premises with the negation of the putative conclusion. If the latter trial conjunction yields one or more moduses in common, our putative conclusion is invalid, for reasons already explained. If, however, the said trial conjunction yields zero moduses in common, then our putative conclusion is finally proven valid.

The advantage of doing this work with a spreadsheet is the speed of calculation and the increased certainty in the results obtained. Assuming no error is made in formula writing, once we have a formula in the first cell of a column, we just copy it all the way down the column and the work is done. Moreover, we can copy a given formula from column to column and make changes to it as appropriate. What manually takes days and weeks of painstaking work can now be done in a few minutes or hours, and the results are more credible. Of course, errors in formula writing are possible, but they can usually be readily spotted by comparing the number of moduses obtained in similar columns and checking whether they are symmetrical.

I should add that the results obtained by me mechanically in the present phase were all compared to results obtained manually in previous phases of the research, and I can report that they are consistent. This shows both that the earlier manual calculations were all accurate and that the present formula based calculations were all accurate. The three phases have, thus, I am happy to say, verified and confirmed each other’s results.

2. 3-Item Syllogisms.

Having already in phases I and II analyzed 3-item causative syllogism in considerable detail, it was easy to reproduce them in phase III and check the results. Regarding such syllogism, which is the main object of our research, there are, in each of the three figures (ignoring the fourth figure, as usual), 64 conceivable moods with positive causative premises involving positive items. All the pairs of premises listed are compatible, and so the remaining question concerning them is only whether they yield a formal conclusion or do not.

Note that in 3-item syllogism the premises and conclusions concerning weak causation (mq, np, pq, p, q) are all about absolute causation; relative causation can only be dealt with as of 4-item syllogism.

Having already found the applicable causative conclusions (mostly positive, though some negative) in previous phases of the research, our task here is just to verify them. This is done, firstly, by checking that the conjunction of the two given premises and the proposed conclusion yields one or more common moduses, and that these moduses are indeed all included under the putative conclusion. Secondly, the same is attempted with the negation of the proposed conclusion, and this should yield no common modus. If both these conditions are satisfied, the proposed conclusion is validated; otherwise, it is not.

The following four tables show the results obtained (mechanically) in phase III and their full consistency with results previously listed in Table 14.4 (obtained by manual method).

The first three of these tables list the valid moods for the three figures of the syllogism, and the fourth table shows the formulae used to produce the first three tables. The verification of these results was indeed done by me, by modifying the formulae in the three tables, so that the contradictory of the proposed conclusion (be it positive or negative) is tried instead; and I can report that in all cases, the result was zero common moduses. I have not bothered to produce additional pdf files showing these zero results, so as not to needlessly clutter the presentation of evidence; the reader can take my word for it or try doing the job independently.

It must be stressed that I have not here verified “nil” conclusions with equal meticulousness. Such non-conclusions from certain combinations of premises are, as already explained, due to the moduses found to be shared by the premises having scattered implications – some of them implying one formal conclusion and others implying a contradictory conclusion, so that no uniform conclusion from them is possible. All I have done here in such cases is list and count the moduses in common to the premises concerned. But I have not gone on to check that these moduses are indeed, as previously ascertained, too scattered for any finite conclusion[2]. I trust my previous manual check and see no point in repeating them.

The object of the present phase is to mechanize solutions to problems, remember. In the case of verification of “nil” conclusions, there is no doubt that such mechanization is technically feasible. This would proceed as follows. If the intersection of two premises result in a number of moduses (one or more), the program would have to check whether all these moduses fall within the modus list of any causative (or more broadly, propositional) form(s). If they do, we have a formal conclusion; if no such form is found, then we have no formal conclusion. This is not an easy task to perform with a spreadsheet program, since the program would have to automatically repeat the search and compare tasks for the full range of defined forms, before it could declare that the conclusion it found to be complete or that there was no conclusion.

For this reason, I limit the present stage of research to verification of previously manually validated conclusions. Moods previously found invalid are here accepted as such. And indeed no effort is made to expand the research and look for eventual conclusions of any form other than causative (positive or negative). It may be, for all I know, that where some or no causative conclusion is possible some other form of conclusion (whether pre-causative, or causative with some or all items of negative polarity) might be found valid; i.e. there might be a not purely causative form that includes all the moduses shared by the premises. This is certainly an important question, which ought to eventually be investigated in detail. But I have not attempted to do it here, because (to repeat) it does not seem mechanically feasible with my present resources[3].

As for premises about inverse causation, prevention and inverse prevention (separately or mixed together), they are also ignored here, so as to avoid a surfeit of information; the same can be said for pre-causative premises – i.e. possible conjunctions of items or conditional propositions. This does not mean that such moods are ultimately less interesting or important than purely causative moods; but only that there is no real need here to present all possible combinations of premises. There should be no difficulty for anyone to investigate such syllogisms, using the method of inference through moduses that we have here demonstrated with reference to purely causative premises. Indeed, it is possible to do the job merely by successive changes in the polarities of the terms in already established causative syllogisms.

To sum up, then, all I have done in the above mentioned tables is to verify previously identified formal conclusions from the main 3-item premises (positive causatives only). I have not tried to enlarge the research, but merely wished for now to demonstrate how syllogism can be validated using spreadsheet software. Of course, this was made easy thanks to the work already done (in the preceding chapters) in mechanically identifying the moduses corresponding to each and every form of proposition in the preceding chapters.

A statistical note: in each figure of 3-item causative syllogism, we have found 23 positive conclusions, 16 negative ones (not-m or not-n) and 25 nil conclusions. This being out of a total of 64 moods in each figure, the percentages were respectively: 36%, 25% and 39%. Although the total numbers of valid and invalid moods are the same in the three figures, the specific conclusions from superficially similar premises are of course not always the same.

3. 4-Item Syllogisms.

I have adopted the same minimalist approach for 4-item positive causative syllogism as I did for the 3-item arguments (in the preceding section). I tabulated the already known 4-item syllogisms, obtained in phase I through matricial analysis (i.e. through macroanalysis in this case – see detailed listings in chapter 6; or Tables 7.3, 7.4 and 7.5), and was content to here test mechanically whether the conclusions previously identified were reliable; they indeed all were. Actually, I went a bit further than this, and in certain cases sought out an additional conclusion, as will presently be explained. But some negative conclusions and all nil conclusions were simply passed over from 3-item syllogism to 4-item syllogism without attempt at improvement, as will be presently explained.

The work done is shown in the following five tables:

The first of these tables, Table 21.5, summarizes the premises and conclusions for the three figures. The next three tables (notice their lengths: remember, each has 65,536 rows!) show the lists of moduses obtained for each 4-item syllogism investigated, and the last table shows the formulae used to produce these three tables. Examining the results obtained here mechanically, we see that they perfectly match earlier results obtained manually (with a few exceptions explained below). The contradictory conclusion test was carried out throughout and further guaranteed the known conclusions (though here again, I ask you to take my word for it, because I do not want to publish too many tables, and especially not empty tables!).

Notice that both 3-item and 4-item syllogisms are listed in these tables, in order to compare results. Of course, the 3-item syllogisms involving absolute weak determinations in either or both premises yield absolute weak conclusions (if any), whereas the 4-item syllogisms here considered concern relative weak premises and conclusions (if any) – relative, that is, to a fourth item S – so they are not quite comparable. Nevertheless, when the 3-item (absolute) conclusions differ in form from the analogous 4-item (relative) ones, we would naturally want to double check the latter to be sure. This I did, and found the results obtained in the past essentially correct. Or more precisely put: none were incorrect, but some were incomplete.

In many cases, the absolute (i.e. irrespective of any complement) conclusion was found to have a relative (to complement S) analogue. For example, mood number 112, 1/mn/mq, yields the conclusion mqabs if its minor premise is absolute (3-item syllogism) and mqrel if its minor premise is relative (4-item syllogism). In some cases, this continuity does not hold. For example, mood number 125, 1/mq/m, yields the conclusion mqabs if its major premise is absolute (3-item syllogism) but only m if its major premise is relative (4-item syllogism). What do I mean by “only m”? I mean that, even though the 3-item conclusion mqabs remains valid, we cannot predict the conjunction of m with either qrel or not-qrel.

This concerns 4-item syllogism, remember. In some cases, we can go further than this and predict the conjunction of m with not-qrel – meaning that the possibility of mqrel is formally excluded even though the 3-item conclusion mqabs is still valid. An example of this is mood number 231, 2/np/mn. In some other cases, as we shall later see (in Table 21.10), the results are split up. Here, I am referring to moods that mix absolute and relative premises. In some of these cases, the results are the same when the absolute premise is the major and the relative premise is the minor, and vice versa. But in certain cases, the results differ – one way yielding an indefinite “only m” type conclusion, and the other way yielding a definite “m + not-qrel” type of conclusion; I have labeled such conclusions “at least m”. An example of this is mood number 232, 2/np/mq.

It should be noted that such details were not brought up in our phase I analysis of 4-item syllogism, for the simple reason that we were unable at that time to deal with negative causative propositions. For the same reason, many moods that seemed inconclusive in phase I (for 4-item syllogism) were found to yield a negative conclusion like not-m or not-n in phase II (for 3-item syllogism). Now, in phase III, we are able to mechanically generate such negative conclusions (for 4-item syllogism), as well as conclusions that negate relative weaks, i.e. which involve not-prel or not-qrel.

In case it is not clear to you, let me underline the following: the form qabs is compatible with both the forms qrel and not-qrel – note well, though qrel implies qabs and not-qrel does not imply qabs – the latter two forms are quite compatible. Similarly, prel and not-prel are compatible propositions. With reference to matricial analysis, remember, compatible propositions have one or more moduses in common. For this reason, a 3-item syllogism with a valid conclusion consisting of or involving an absolute weak causation (pabs and/or qabs) neither implies nor excludes the validity of a 4-item syllogism with a conclusion consisting of or involving a relative weak causation (prel and/or qrel).

Obviously, any absolute conclusion (whether positive or negative, strong or weak) found valid in 3-item syllogism remains valid in 4-item syllogism – since the relative premises of 4-item syllogism formally imply the absolute premises of 3-item syllogism. This is direct reduction. But (to repeat) it does not follow that the corresponding relative conclusions are valid (since an absolute proposition does not imply a relative one). It is also obvious that, although the 4-item syllogism can yield a more precise conclusion (i.e. one signified by fewer moduses) than the analogous 3-item one, it cannot in any case yield a contradictory or contrary conclusion.

Diagram 21.1. Nil Conclusions are Not Reducible.

It does not follow from the above, however, that any conclusion which is found to be invalid in 3-item syllogism is bound to be invalid in 4-item syllogism. We cannot use indirect reduction (i.e. reduction ad absurdum) in such cases, for if the latter’s conclusion is denied it does not follow that the former’s conclusion is also denied.

This is made clear by above diagram. Consider two premises A and B, and suppose their intersection (the area AB they share) signifies a number of moduses which are too scattered to form a conclusion (i.e. no form includes them all); call these moduses v, w, x, y, z. It is still conceivable that there are two more specific premises, say C and D, whose intersection (the area CD they share) yields a formal conclusion; for CD may signify only the moduses v, w (excluding x, y, z) and these may happen to be included in the same form, say E. Note well that this may happen, but does not necessarily happen; the conjunction CD may also be inconclusive. All that may equally be understood with reference to implications. If A, B are or involve absolute weaks, and C, D are or involve analogous relative weaks, then C implies A, and D implies B. Given that C + D have a certain relative (or even absolute) conclusion E, it does not follow that A + B have that same conclusion E or any subaltern conclusion to E. Thus, AB may well be invalid while CD is valid.

Notwithstanding all that, I have here assumed that when a mood of 3-item syllogism yields no conclusion, then the 4-item syllogism need not be investigated further. That is, I have arbitrarily (or rather, speculatively) declared, for the time being, at least, that the latter’s conclusion will likewise be nil. Similarly, by the way, when a 3-item syllogism yielded a not-m or not-n conclusion, I generally accepted the identical conclusion for the corresponding 4-item syllogism to be all that can be inferred, even while aware that in some cases some additional formal conclusion(s) might conceivably be found. Such assumptions may be taken as inductive probabilities, which have yet to be proven right or wrong deductively when appropriate mechanical means are devised[4].

Just as with 3-item syllogism, no attempt was made to extend the research and consider other premises and other conclusions than those already investigated, so here with 4-item syllogism further explorations are kept to a minimum. This restraint is largely due to the fact that my computer (hardware and software) capabilities have been stretched to their limit when dealing with four items, in addition to the more general need (to repeat) to find a mechanical way to systematically look for conclusions. In view of these limitations, I must relinquish at this time the ambition to be exhaustive and be satisfied with mere demonstration, i.e. with showing the way putative conclusions can be validated[5].

A notable change in the treatment of 4-item syllogism in the present phase III compared to the earlier phase I is the issue of moods with five items. In the earlier phase, when both premises involved weak causation, I assigned the fourth item (S) to one of them, and took one of the other three items as the complement for the other two. For example, see mood #124 in chapter 6.2, which has item P as both complement (of Q) in the major premise and minor term (complemented by S) in the minor premise. My intent there was obviously to construct a 5-item syllogism at all costs, even if it was rather illusory or at any rate a very rare or special occurrence. But now I think that this is a misleading approach, in that it diverts our attention from the more general problem, viz. that of dealing systematically with five distinct items (P, Q, R, S and T).

Remember that we can always in such cases draw the corresponding 3-item conclusion (if any) anyway, i.e. an absolute weak conclusion; the issue for us is whether we can do better than that. Lacking the means to investigate 5-item syllogism when both premises are partly or wholly weak, I have not here as in the past (as just explained) adopted the special case where the fifth item T is identical with one of the other four items. What I have done instead is to consider cases where either one of the premises is taken as relative (to the fourth item S) and the other is taken as absolute (i.e. without a fifth item needing to be specified). The latter artifice does yield some further conclusions.

I have again chosen to narrow the problem somewhat, by considering only some of the moods concerned. The moods susceptible to having mixtures of absolute and relative premises are those with some weak causation in both premises. Looking at Table 21.5, we see that there are 25 such moods in each figure. Looking at the results of 3-item syllogism, we see that, of these 25 moods per figure, 15 yield no conclusion at all, 8 yield only a negative conclusion and only 2 yield a positive absolute conclusion.

Let us suppose, as we did before, that the nil conclusions and negative conclusions found in 3-item syllogism are also applicable to 4-item syllogism. This assumption is admittedly presumptuous, but as already discussed it is inductively reasonable in the context of our current technical limitations. This leaves us with only (3 times 2 =) 6 moods to study more closely, whose numbers are 122, 133, 223, 232, 323, 332. These 6 become 12 moods, since each may have the major premise absolute and the minor one relative to S (this is called subfigure b), or the major premise relative to S and the minor one absolute (this is called subfigure c).

Our task is to see whether any of these 12 moods give us a relative conclusion, i.e. a more specific conclusion than the absolute conclusion we already know they give when both the premises are absolute (i.e. through 3-item syllogisms). This job is done in the following table:

As this table shows, 8 moods do not give a new conclusion (i.e. one relative to S, instead of absolute) – but, surprisingly, 4 moods do give us a new conclusion, though not the conclusions qrel (with m) or prel (with n), but their negations! The 4 moods are numbers 223c, 232c (in the second figure) and 323b, 332b (in the third figure). When we try a positive weak conclusion for them, we obtain zero moduses; whereas when we try for the contradictory conclusion, we obtain 16 moduses in each case. With regard to the 8 moods that do not yield conclusions (of which 4 are in the first figure), both the positive trial and the negative trial result in a number of moduses (namely, 4 and 12 respectively), so we do not know which way to lean, and so we must remain content with just the absolute given by 3-item syllogism.

To conclude, the following little table (not numbered) summarizes the validity rate found in the 4-item causative syllogisms we have studied in this section. Please refer to Table 21.5 again for an overview of the results.

Statistics for 4-item syllogism

Fig. 1

Fig. 2

Fig. 3

sum

percent

new valid positive conclusion

10

5

5

20

10

new valid negative conclusion

0

9

9

18

9

same valid positive conclusion

13

9

9

31

16

same valid negative conclusion

16

16

16

48

25

no formal conclusion (assumed)

25

25

25

75

39

total moods considered

64

64

64

192

100

This summary includes all moods in all three figures of 4-item syllogism. Out of the total of 192 moods, 31 (16%) yielded the same positive conclusion in 4-item syllogism as in 3-item syllogism, meaning a strong causation (with or without an absolute weak, as the case may be); and 48 (25%) yielded the same negative conclusion in 4-item syllogism as in 3-item syllogism, meaning a denied strong causation. More interesting were the 38 (17%) moods giving new conclusions, i.e. inferences peculiar to 4-item syllogism; these included 20 positive conclusions (10%) and 18 negative ones (9%)[6]. Finally, 75 moods (39%) gave no conclusions.

The important moods in the present context, I would say, are the 20 ‘new’ positive moods, and to a lesser degree the 18 ‘new’ negative moods, because they concern weak causation relative to S. These 38 moods show the need for 4-item microanalysis – i.e. the value of this work for causative logic. The remaining 79 ‘same’ valid moods and 75 invalid moods are not specific to 4-item syllogism, but rather validated or invalidated by 3-item syllogism.

The 20 new positive valid moods of 4-item syllogism are the following: in figure one: the 10 moods numbered 112, 113, 114, 117, 118, 121, 131, 141, 171, 181; in figure two: the 5 moods numbered 212, 213, 214, 217, 218; and in figure three: the 5 moods numbered 321, 331, 341, 371, 381. The 18 new negative valid moods are: in figure one: none; in figure two: the 9 moods numbered 221, 223c, 226, 231, 232c, 235, 241, 271, 281; and in figure three: the 9 moods numbered 312, 313, 314, 317, 318, 323b, 332b, 353, 362. These most significant valid moods have been highlighted in Table 21.5. Take a good look at them!

4. About 5-item Syllogism.

I would like to say a few words now concerning 5-item syllogism, i.e. moods whose premises are relative to two different complements, say S and T. To put things in a wider perspective, I would today modify Table 5.2 (see in Chapter 5.3) to look like this:

Table 5.2 (modified). Subfigures of each figure.

Subfigures

a

b

c

d

Figure 1

QR

QR

Q(S)R

Q(T)R

PQ

P(S)Q

PQ

P(S)Q

PR

P(S)R

P(S)R

P(ST)R

Figure 2

RQ

RQ

R(S)Q

R(T)Q

PQ

P(S)Q

PQ

P(S)Q

PR

P(S)R

P(S)R

P(ST)R

Figure 3

QR

QR

Q(S)R

Q(T)R

QP

Q(S)P

QP

Q(S)P

PR

P(S)R

P(S)R

P(ST)R

The main change here is the insertion of a fifth complement (T) instead of (P) in the major premise of subfigures (d); notice too the consequent change in the conclusions’ complements from (S) to (ST). Subfigures (d), thus, concern premises and conclusions all involving some relative weak determination (whether or not conjoined to a strong determination). Notice that the conclusion is in such cases presumably one with two complements (S and T) – though we can speculate that a conclusion with only one complement might in some cases be feasible.

Subfigures (a) are dealt with in 3-item syllogism. Note that they concern not only “both premises strong” as stated in the original definition, but are equally applicable to moods with some absolute weak premise(s) and/or conclusion[7]. Similarly, we should consider the “strong only” premise in subfigures (b) and (c) as possibly an absolute weak premise. In short, a premise or conclusion with only two terms need not be strong only, but may be a mixture of strong and absolute weak or entirely absolute weak. At the time I wrote phase I, the concept of absolute partial or contingent causation was perhaps not yet fully formed, if at all, in my mind.

We can state the following generalities concerning the relation between syllogisms with more or less items. If a mood with fewer (e.g. 3 or 4) items is valid, the corresponding mood with more (e.g. respectively 4 or 5) items may or may not be valid. Why? Referring again to Diagram 21.1, suppose the vaguer premises A and B constitute a valid mood, i.e. suppose all the moduses they have in common (the area of intersection AB in our diagram) are included in some causative form. There is no guarantee that the more precise premises C and D (which respectively imply, i.e. fall within, A and B) will likewise constitute a valid mood. Granting they have a common item, these premises will intersect (forming the area CD) and do so within the larger area AB (as on our diagram).

However, it does not follow from the fact that the more numerous moduses of AB do yield a vague formal conclusion that the fewer moduses of CD must yield an analogous, more precise formal conclusion. The moduses of CD may be too scattered to be all included in the list of moduses of a specific conclusion. In such case, the mood CD will still of course yield the same vague conclusion as AB – but it will not be able to produce a more precise one. For example, as already seen, mood number 125, 1/mq/m, yields the conclusion mqabs if its major premise is absolute (3-item syllogism), yet it does not yield the conclusion mqrel if its major premise is relative (4-item syllogism).

So direct reduction of the latter to the former is impossible. In other words, we cannot readily derive syllogisms with more items from syllogisms with less items. We must develop ad hoc means for each increase in the number of items. And these means increase exponentially our need for computer space and power, in hardware and software. Nevertheless, 5-item causative syllogism research is technically within the realm of the possible. The necessary computing resources are sure to be found in many existing research centers.[8]

The problem of 5-item syllogism can only be adequately solved when we are able to develop 5-item matrices. To do so, we would need software or a program, and hardware capabilities, able to deal with tables with over 4 billion rows and dozens of columns. My computing resources are barely able to handle 4-item syllogism (which have 65,536 rows). To generate the latter, I had to delete all data (i.e. columns) not directly needed to produce the tabulated conclusions, and I had to use a separate spreadsheet (i.e. a separate file) for each figure; otherwise, the software got stuck. It follows that matricial analysis with 5-items is not feasible for me at this time.

This limitation does not worry me greatly, because I intuitively doubt 5-item syllogism will yield any (or many) new conclusions; I may be wrong, of course. By ‘new conclusions’ I mean conclusions that are specific to 5-item syllogism. As can be seen in the tables of the preceding section, most (though not all) of the conclusions obtained in 4-item syllogisms were in fact conclusions of the corresponding 3-item syllogisms. We can expect 5-item syllogism to yield similarly rarified results (which still leaves us with some hope of new conclusions).

If we examine Table 21.5 again, we see that in each figure there are 25 moods capable of giving rise to 5-item syllogism (being in subfigure d). We know regarding the 3-item syllogisms corresponding to these 25 moods in each figure that 2 (8%) yield a positive absolute conclusion, 8 (32%) yield a negative conclusion, and the remaining 15 (60%) yield no conclusion at all. Further investigation of the 6 moods (in the 3 figures) yielding positive absolute conclusions, through 4-item syllogisms with mixed absolute-relative or relative-absolute premises (in which event they become 12 moods), shows that their conclusions can be differentiated, with 8 of them yielding a bare m or n conclusion, while 4 of them yield a more precise m+qrel or n+prel conclusion.

Admittedly, as already discussed at length, many of the moods without conclusion in 3-item syllogism may upon further scrutiny turn out to have a conclusion in 4-item or even in 5-item syllogism. Similarly, some negative or even positive conclusions might at a more specific level yield additional information. But the trend as we increase the number of items seems to be one of diminishing returns, so I believe that the chances of getting some new inference in 5-item syllogism, though not zero, are very slim indeed. This, I think, is as far as we can go for now trying to predict 5-item syllogism.



[1] Actually, I do finally manage to do the work of scanning for conclusions by means of (many and bulky) spreadsheets – in the next chapter.

[2] To repeat, this work is done in the next chapter.

[3] Again note: I belie this assumption in the next chapter.

[4] All this is indeed fully dealt with in the next chapter.

[5] Here again, I must stress that all these issues are successfully resolved in the next chapter.

[6] Concerning the latter conclusions, note that for 4 of them (2 in each of figures 2 and 3) I only here intend as new their negative element (negation of relative weak causation). In fact, they include positive elements (strong causation) already found in 3-item syllogism, which are here glossed over; if we wanted to count them, we would have to add them to the earlier category of ‘same valid positives’ and there would be some overlap. See Table 21.10.

[7] Note that when I do not mention the subfigure, it is probably subfigure (a).

[8] Note that I do not go further than this in the next chapter. 5-item syllogism remains an open issue in the present volume.

2016-06-13T11:02:56+00:00