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THE LOGIC OF CAUSATION

© Avi Sion, 1999, 2000; 2003, 2005; 2008, 2010  All rights reserved.

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THE LOGIC OF CAUSATION

Phase Three: Software Assisted Analysis

Chapter 22 Scanning for Conclusions.

1.  Methodology.

2.  Forms Studied and their Oppositions.

3.  3-Item Syllogisms.

4.  4-Item Syllogisms.

In the present chapter, thank G-d, I finally find the way to achieve the major task of this whole study of causative logic, namely discovering and validating ALL causative (and preventive) positive (and negative) conclusions of 3- and 4-item positive causative syllogism!

1. Methodology.

After completing the preceding chapter, it occurred to me that with a bit (i.e. a lot!) more patience and effort on my part, I could validate or invalidate all possible conclusions to causative syllogism. I did not need to wait to develop an ad hoc computer program for this job, but could do it using spreadsheet software, provided I cut up the task into sufficiently small segments. That is, instead of awaiting a fully ‘mechanical’ solution to the problem, I could make-do with a semi-mechanical one. The present chapter is the result of this work – this crucial work, for after all it was one of the main goals of all my research into causative logic.

I am happy to say that I succeeded entirely. Eureka! I found all valid positive and negative causative and preventive conclusions to positive causative syllogism, and proved all remaining such propositions to be invalid conclusions. This for all three figures, for 3-item and for 4-items syllogism. Other work done in this context is presented further on.

The method I used is essentially no different from the ‘manual’ one I previously used, except that here I apply it more systematically. To determine the valid and invalid conclusions from a pair or set of premises (a ‘mood’ of the syllogism, in the language of logicians), we must ‘scan’ over a ‘range’ of possible conclusions, and see which fit and which do not. How is this done? Simply:

a) Find the moduses that the given conjunction of premises, AB, yields, using matricial analysis as demonstrated already. Suppose their number is x.

b) Find the moduses, if any, of the same conjunction of premises with a proposed conclusion, C, again by matricial analysis. Suppose that the conjunction ABC yields y moduses.

c) Now, if y=x, then C is indeed a valid conclusion of AB (since the conjunctions AB and ABC have exactly the same moduses); that is to say, C is logically implied by AB. On the other hand, if y=0 (zero), then C is an invalid conclusion for AB, i.e. not-C is a valid conclusion of AB (since the conjunction of AB with not-C is bound to have the same moduses, x in number, as AB alone); that is to say, C is logically denied by AB. If y neither =x nor =0, then AB neither implies nor denies C – i.e. neither C nor not-C is a valid or invalid conclusion of AB.

This, then, is what I mean by ‘scanning for conclusions’. Knowing the moduses corresponding to a given set of premises (here labeled AB), we mechanically test each form (such as C) within a chosen range of forms, to see whether when conjoined to the premises the resulting moduses are identical, entirely different, or in between. If the moduses are identical, we have a valid conclusion; if they are entirely different, then the contradictory of the form tested must be a valid conclusion; and if neither of these results is obtained, it means the moduses are too scattered for any conclusion of that form or its negation to be drawn.

One small improvement in this method is that there is no need to test both a form C and its negation not-C. If we start by finding the moduses for the conjunction of premises (AB) alone, then we need only test positive forms (C) and the status of their negations (not-C) follows as just explained. This saves us half the work. Moreover, we need only find the moduses of the premises AB once, and then compare them to the moduses with various conclusions (C1, C2, C3, etc.) conjoined to them. This simplifies formula writing in the spreadsheets.

The ‘range of conclusions’ investigated is here symbolized by C1, C2, C3, etc. In the preceding chapter, I had limited the research to causative conclusions of the form PR or PSR (i.e. the six forms m, n, p rel to S, q rel to S, p abs, q abs); this was the range there chosen, minimal in its ambition. In the present chapter, the range has been considerably expanded, since I have added causative conclusions with a negative complement, i.e. PnotSR forms (p rel to notS, q rel to notS), and a similar range of preventive conclusion, i.e. eight forms with items PnotR, PSnotR, and PnotSnotR.

Note that I did not look into what I call ‘pre-causative’ conclusions to causative syllogism, i.e. possible conclusions of conjunctive or conditional form, but only for causative conclusions (in the broad sense, meaning causative and/or preventive). This may be justified by saying that the central goal of causative reasoning is, after all, to look for causative conclusions – to look for other forms of conclusion constitutes at best an additional and lesser interest. It is quite feasible with additional work (using the same techniques), but I do not bother to do it here.

All the above mentioned work concerns positive causative syllogism, i.e. syllogism with both premises having positive causative form, note well. As will be seen, none of the results obtained here contradict past results; i.e. no errors were found here in past conclusions. All past conclusions are, happily, here confirmed – but more conclusions are now discovered, thus justifying the whole elaborate procedure. Throughout the work, I kept reminding myself that this would be a one-time effort on my part for all humanity and for all time. Once these syllogistic problems are solved, they are solved for all time; so the effort invested is worthwhile.

As we shall presently see, most of the additional conclusions found are negative; only a very few are positive[1]. The positive conclusions previously obtained (using a narrower range) remain the main conclusions, though all negative conclusions are also significant. What is important, anyway, is that at the end of this chapter we can confidently say that we have a definitive, complete list of positive and negative valid conclusions from all positive causative syllogisms – and we know for sure which conclusions are invalid and which moods yield no conclusions at all. This was, to repeat, one of the principal goals of our research.

This is true for all 3-item and 4-item moods in the three figures. Needless to say, 5-item moods are still not treated here, and there is definitely no possibility of dealing with them using spreadsheets in a personal computer – the number of moduses involved is just too great for these tools. I leave this also important task to future researchers!

2. Forms Studied and their Oppositions.

The premises and conclusions studied here are given in the following tables:

These two tables contain the basic data, which will be used in subsequent tables. They show, for each of the three figures, the moduses of the positive causative premises used and the moduses of the chosen range (causative and preventive) of putative conclusions from these premises. The data here displayed is simply selected, cut and pasted from past tables (namely, 18.4 and 18.9).

Moreover, we here examine the oppositions between the conclusions scanned. The reason for this is that, in order to reduce the conclusions obtained from a given set of premises to a minimum, we have to distinguish between primary conclusions and subaltern conclusions. The latter are formally implied by the former, so that their implication by the conjunction of premises is incidental. Thus, to better understand and rationalize the results of scanning work, we need to know the oppositions between all possible forms of conclusion – i.e. which of them imply which, and which of them are not-implied by the others.

This knowledge can be developed by a technique akin to scanning. Say we take two forms K and L, to know how they relate to each other, we must look into all their conjunctions, i.e. K and L (11), K and not-L (10), not-K and L (01), and not-K and not-L (00). If the forms concerned have no moduses in common, it means they are incompatible; inversely, they are compatible if they do have some moduses (no matter how many) in common. The results obtained by this technique are listed in the preceding two tables (22.1 and 22.2). The following two tables show the formulas actually used in them, for the record:

As it turns out, for the ranges of possible conclusions studied here, some KL conjunctions are impossible, but no remarkable cases were found where K implies L or vice-versa (other of course than obvious implications, like p rel implying p abs), and no pair of forms were found exhaustive (i.e. such that the conjunction of both their negations was impossible). This means we can often infer a form’s negation from another form’s affirmation – but we can never draw useful inferences from a form’s negation (other than obvious cases like not-p abs implying not-p rel).

This is evidenced in the following summary table, including possible positive causative and preventive conclusions from both 3-Item and 4-Item syllogisms:

The results in this table of practical interest to us are the following:

Ø Finding the causative (PR) conclusion m, then the causatives p rel to S, p rel to notS, p abs, as well as all preventive (PnotR) forms, viz. m, n, p or q rel to S, p or q rel to notS, p or q abs are all denied. The remaining causative (PR) forms, n, q rel to S, q rel to notS, q abs, are neither implied nor denied by m (PR).

Ø Similarly for n (PR), mutadis mutandis.

Ø Finding the causative (PR) conclusion p rel to S, then the causative p abs is implied, and the causatives m, p rel to notS are denied, as are the preventive (PnotR) forms m, p rel to S, q rel to notS. The remaining causative (PR) forms, n, q rel to S, q rel to notS, q abs, and likewise the remaining preventive (PnotR) forms, n, q rel to S, p rel to notS, p abs, q abs, are neither implied nor denied by p rel to S (PR).

Ø Similarly for q rel to S (PR), mutadis mutandis.

Ø Also note: although separately there are no implications between the causative and preventive forms of p abs and q abs, the compounds pq abs causative and pq abs preventive are identical.

Ø On the other hand, nothing can be inferred from denying m, n, p or q – except that denying p or q abs implies denial of, respectively, p or q rel to S or notS forms with the same items (i.e. PR or PnotR).

This knowledge, to repeat, helps us distinguish primary from subaltern conclusions, and so makes possible reducing statements of conclusion to as few words as possible.

3. 3-Item Syllogisms.

Scanning for conclusions was first applied to 3-item syllogism with positive causative premises (64 moods for each of the three figures, the moods being grouped in sets of eight, viz. #s 11-18, 22-28, etc. till 81-88). The collected results are given in the following summary table. This is the table requiring most of your attention:

The main conclusions implied by the premises concerned are shown in pink; those denied by them are shown in grey; conclusions subaltern to these main positive or negative conclusions are identified as implied or denied but not colored pink or grey. Cells which are not labeled ‘implied’ or ‘denied’ signify that the form concerned is not part of the overall conclusion of that mood (obviously, if all the cells opposite a given mood are empty, it means there’s no causative or preventive conclusion). That said for the first segment (3 pages) of the summary. In the second segment (next 3 pages), the results are repeated, all subaltern conclusions, if any, being eliminated, and color coding being removed; and the symbol for the valid positive or negative conclusions is written in place of the labels ‘implied’ and ‘denied’, respectively. The third segment of the table (last 3 pages) again repeats the results in the briefest possible way, i.e. stating only the essentials: these are collectively ‘the conclusion’ for the given mood (‘nil’ being used to signal no-conclusion).

The said summary table (22.6-0) is derived from the three tables listed next, where the actual scanning work is done.

These three tables were produced in succession, using the first as the template or model for the next two, changing only the premises as appropriate (see Table 22.1) and then recalculating the whole spreadsheet. As you can see, the moduses are found for each mood of the syllogism (e.g. the positive causative premises mn/mn), and then for those same premises combined with eight different test conclusions (the positive causative (i.e. PR) forms m, n, p abs, q abs, and the positive preventive (i.e. PnotR) forms m, n, p abs, q abs, as earlier explained). The calculations in the spreadsheets are done by means of formulas, as usual. A sample set of formulas (those used in Table 22.6-1, i.e. for Figure 1) is given in the next table:

Looking at the results obtained (return to Table 22.6-0), and comparing them to past results (which are listed in the last segment of Table 22.6-0 for this purpose), we see – as earlier mentioned – that the latter are essentially confirmed. That is, as regards causative conclusions; but now we have some additional negative preventive conclusions we were not aware of before.

The following statistics are now applicable to each of the three figures: 23 moods (36%) yield a positive conclusion (always causative, some elementary, some compound); 16 moods (25%) yield two negative (not-m and/or not-n) conclusions only, one causative and one preventive (previously, the latter were unknown); 16 moods (25%) yield only a negative (not-m or not-n) preventive conclusion (previously all supposed nil); and 9 moods (14%) are without any causative or preventive conclusion (previously, 25 were thought inconclusive). The net validity rate for 3-item syllogism is thus 86%.

Clearly, we now have a wider range of possible conclusions and we can say with full confidence not only which forms are affirmed, but also which are denied and which are neither affirmed nor denied. We thus obtain a truly definitive list of valid and invalid conclusions. It is as if we previously used a microscope to observe a certain phenomenon, and now we have found a more powerful microscope capable of giving us a more accurate image of the phenomenon.

4. 4-Item Syllogisms.

Scanning for conclusions was next applied to 4-item syllogism with positive causative premises (64 moods per figure, as before). However, here the job is exponentially bigger, since we are dealing with 65,536 moduses per column (instead of a mere 256). This makes all opening, saving and closing of spreadsheet files very slow; and it especially slows down calculations, forcing us to split files into more manageable portions and perform calculations in smaller segments (otherwise, if too much is asked of the program, execution might even be blocked).

Moreover, although the grouping of moods into sets is maintained, in some cases (specifically, 5 out of 8 per figure) sets of 8 moods do not suffice, but must be expanded to sets of 13 moods. This occurs when moods have a weak causation (whether an element or part of a compound) in both premises; in such cases, as we saw earlier (see Table 21.10), a single mood becomes two (labeled ‘b’ and ‘c’, combining an absolute premise with a relative one, and a relative with an absolute, respectively). This is due, remember, to our dealing here with only four items, and not five (which my computer hardware and software cannot handle). In sum, here we are dealing with 89 moods per figure (instead of 64 before).

Furthermore, the workload is increased due to enlargement of the range of conclusions tested for each mood. For 3-item syllogism, we had 8 forms to test every time; here, in 4-item syllogism, we have 16 conclusions to consider, namely: the 8 causative (PR) forms m, n, p rel to S, q rel to S, p rel to notS, q rel to notS, p abs, q abs, and the similar 8 preventive (PnotR) forms. This ensures that every possible inference of causative and/or preventive form is found by us, and our results are truly exhaustive and definitive.

The collected results of such scanning work are given in the following summary table. This the table requiring most of your attention in the present context. Look at the premises and the conclusions carefully in each row, and absorb the meaning of it all.

We shall return to these results in a moment. For now note that this table summarizes the findings of the following 48 tables (16 per figure). Each table concerns a specific set of 8 or 13 moods. For each set, table A tests 8 causative conclusions and table B tests 8 preventive conclusions. Each pdf file produced is over 1000 pages long (making almost 24’000 pages per figure, compared to a mere 48 pages before); seeing this, you can understand why it was necessary to split the job up into smaller pieces, and why I long hesitated to do it. Nevertheless, though much time was spent doing it and much attention was required, the job was not so hard, because once the two templates (the first couple of tables) were produced, it was easy to reproduce them with appropriately modified data and recalculate them.

Figure 1 – 16 tables (8 for causative conclusions and 8 for preventive ones).

Figure 2 – 16 tables (8 for causative conclusions and 8 for preventive ones).

Figure 3 – 16 tables (8 for causative conclusions and 8 for preventive ones).

The sorts of formulas used in above tables can be seen in the following two tables. The first of these (drawn from Table 22.7-11A) concerns tables with sets of 8 moods (viz. 11-18, 51-58, 61-68), and the second (from Table 22.7-12A) sets of 13 moods (viz. 21-28, 31-38, 41-48, 71-78, 81-88). With minimal appropriate changes, the formulas in these samples are easily adapted to different contexts.

So much, thus far, for the technicalities. Let us now compare and analyze the results obtained. First let us compare the results shown for 4-item syllogism in these tables with those shown in the previous chapter. You might argue that such comparison is idle, or at best ‘academic’, since the rougher, older results are supplanted by present results, which are complete and definitive. But I think it is still worthwhile comparing them, insofar as it ensures I made no errors of inattention when writing the formulas that generated the latter results, and also because it helps justify the extra work I invested in producing them. Let us therefore look at Table 22.7-0; especially the last six pages, which show the new and old results opposite each other. We see that the conclusions newly obtained include all the conclusions previously obtained, so the results of both inquiries are consistent.

The large majority of additional conclusions obtained here are negative propositions, of forms not previously investigated: either the causatives p, q relative to notS, or any of the eight preventives. However, we do not find any new negations of m or n (causatives). But we do find a great many previously undiscovered negations of p and/or q relative to S (causatives). For instances, 222(c), 225, 233(c), 236, and so on. This is not surprising, in that no attempt was made to search for these before (due to the difficulties involved). We did however in four cases find such conclusions, viz. in moods 223(c), 232(c), 323(b), 332(b), which you may recall we looked into exploratively in Table 21.10.

There are no new positive causative conclusions of the forms m, n, p rel to S, q rel to S, p abs, q abs, and no positive preventive conclusions of any form. But there are some new positive causative conclusions of the form p and/or q rel to notS in the second and third figures; namely, for the 10 moods 221, 231, 241 271, 281, and 312, 313, 314, 317, 318. Concerning these conclusions, it should be noted that they all displace previously listed negative causative conclusions of the forms p and/or q relative to S. This is understandable, since p, q rel to notS deny p, q rel to S, respectively; i.e. the disappeared negations remain implicit as subaltern conclusions.

Another comparison worth making is between our final lists of conclusions for 3-item and 4-item syllogisms; these are posted together for comparison in the last six pages of Table 22.7-0. All conclusions obtained are unsurprisingly consistent. And, needless to say, conclusions are much richer in detail in 4-item syllogism than in 3-item syllogism. But more specifically, one question asked in the previous chapter can now be answered: are there any ‘nil’ conclusions in 3-item syllogism that become positive or negative conclusions in the corresponding moods of 4-item syllogism? This question concerns 9 moods in each figure, viz. 44, 47, 48, 74, 77, 78, 84, 87, 88. The answer is no – all these moods remain without valid positive or negative conclusion in 4-item syllogism, too. It is still of course conceivable that some or all of these moods yield some conclusion in 5-item syllogism, but at least we are now sure they do not yield any in 3- or 4-item syllogism.

Another question we can ask here is: are there any 4-item moods whose conclusions are not more informative than the corresponding 3-item moods? There are indeed such cases. Mostly, these occur because the 4-item mood is in fact a 3-item mood, so the conclusion has predictably the strong forms m, n, or their negations, alone or in combinations (note the preventive as well as the negative sides of each conclusion). In some cases, when a mood is split in two (abs/rel and rel/abs), we may find one of the 4-item moods has identical conclusion(s) to the 3-item mood, but the other is has additional negative conclusions. But on the whole we can say that 4-item syllogism generally adds some new conclusion, either raising a p or q conclusion from absolute to relative, or adding some negative conclusion in the causative and/or preventive column.

Let us now finish our analysis of Table 22.7-0 with some statistics for 4-item syllogism. We shall compute these with reference to 89 moods per figure (i.e. counting moods that are here split in two as abs/rel and rel/abs as two). There are 18 nil conclusions (20% of the 89 moods) in each figure – i.e. moods with no conclusions, whether positive or negative, causative or preventive[2]. So the number of moods with some sort of conclusion is 71, and the overall validity rate for 4-item syllogism is 80%.

Of these 71 valid moods, 65 involve some positive and/or negative causative proposition(s) and 46 involve some negative preventive proposition(s). Of the 71 valid moods, 25 are causative only (+ve and/or –ve), 6 are preventive only (all –ve), and 40 are mixtures of causative (+ve and/or –ve) and preventive (all –ve). Note that all positive conclusions are causative and none are preventive. Of the 65 causatives, 17 are positive only, 8 are positive and negative mixtures, and 40 are negative only.

As can be seen in the following little table (unnumbered), there are 25 moods (28%) per figure with some sort of positive conclusion. Note that in the first figure, relative weak conclusions are all 10 relative to S, whereas in the second and third figures 5 are relative to S and 5 are relative to notS. There are 8 absolute weak conclusions in each figure (4 in mq abs and 4 in np abs – none standing alone).

+ve conclusions

fig. 1

fig. 2

fig. 3

mn

1

1

1

mq rel to S

2

1

1

mq rel to notS

0

1

1

np rel to S

2

1

1

np rel to notS

0

1

1

pq rel to S

2

1

1

pq rel to notS

0

1

1

mq abs

4

4

4

np abs

4

4

4

pq abs

0

0

0

m

3

3

3

n

3

3

3

p rel to S

2

1

1

p rel to notS

0

1

1

q rel to S

2

1

1

q rel to notS

0

1

1

p abs

0

0

0

q abs

0

0

0

Total

25

25

25

percent out of 89 moods

28%

28%

28%

Note lastly that some moods have a conclusion composed of as many as 7 negative elements (maximum 3 on the causative side, and 4 on the preventive side). Some moods, of course, have only one negative element or even none. Although negative conclusions are not as interesting as positive ones, they are still logically significant, telling us not to expect the opposite positive proposition to arise. They are also useful for consistency checking, since, having reached a negative conclusion, if we discover that we elsewhere, earlier or later, uphold the opposite positive proposition, we know we have made a mistake in our reasoning or observation somewhere. So it is important to keep negative conclusions in mind. They can also, by the way, become premises in subsequent syllogisms, as we shall point out in the next chapter.



[1] These are not earth-shattering news, but they are interesting in that they explain (i.e. imply) previously known negative conclusions (obtained in the narrower range of consideration).

[2] Though some or all of these moods might for all we know imply a mere conjunctive or conditional fragment of a conclusions. The matter is not here investigated, though it could be done with a certain amount of work, because our interest is centered in causative and preventive propositions (rather than their building blocks).

2016-06-13T11:04:14+00:00