THE LOGIC OF CAUSATION
Phase One: Macroanalysis
Chapter 7 – Reduction of Positive Moods.
The method of reduction was first theoretically identified by Aristotle, though of course it had been practically used by human beings long before. Reduction, in its broadest sense, consists in showing that an argument is valid or invalid because another argument is valid or invalid. Thus, reduction is not a primary process of evaluation but a method for transmitting validity or invalidity, and therefore presupposes that we have some other means for establishing certain fundamental validities or invalidities.
The ‘other means’, in our case, is matricial analysis; we shall use this method for a number of validations and invalidations, but before we do so we want to find out the minimum number of moods of causative syllogism which have to be so treated. For as already said, matricial analysis is a cumbersome, though essential and certain, method; and we wish to facilitate our task. Furthermore, while this method treats each mood as ‘an island onto itself’, reduction reveals the precise interrelations between moods, which we ought to be aware of.
Reduction is a shortcut. In the field of causative syllogism, reduction has many guises. The broad Aristotelian distinction between direct reduction and indirect (or ad absurdum) reduction is of course applicable here; but we may find fit to subdivide the concept of direct reduction.
Within the domain of positive moods of any figure, the validity of conclusions is transmitted by direct reduction and the invalidity of conclusions is transmitted by indirect reduction, within the same figure. The main implication which concerns us here is subalternation by joint determinations of generic determinations (thus, mn implies m and n, mq implies m and q, np implies p and n, and pq implies p and q). Since subalternation is oneway implication, different implications are used for validations and invalidations.
But there is also reduction from one figure to another, for which the eductive process of conversion is appropriate. Some second figure moods may be directly reduced to first figure moods, by conversion of the major premise; and some third figure moods may be directly reduced to first figure moods, by conversion of the minor premise. Since conversion works both ways, such reductions serve for both validation and invalidation.
We can thus distinguish between two sorts of direct reduction of positive moods, with reference to the precise sort of implication appealed to, i.e. subalternation (within the same figure) or conversion (across figures).[1]
It should be noted that the validity of any conclusion implies the invalidity of conflicting putative conclusions (thus if m is true, p cannot be, and vice versa; and if n is true, q cannot be, and vice versa); though note well that the invalidity of a putative conclusion does not imply the validity of its opposite, i.e. both m and p or both n and q may be invalid). We can on this basis save ourselves some work; this also might be viewed as a sort of indirect reduction.
Before going further, let us point out that some moods are composed of incompatible premises. Such moods may be declared invalid without further ado. This occurs specifically in subfigure (d) of each figure, where the premises have two items in common (namely, P and Q). Here, if the minor premise has a strong component, it may conflict with the weak component(s) of the major premise.
We shall now identify the implications between moods within any of the figures, due to inclusion within compound forms (joint determinations) of their constituent forms (generic determinations). The following table lists all implications between premises (note well) of moods; it is based on information given in Table 5.6, developed in the chapter on causative syllogism, listing the 64 moods conceivable in each figure.
Table 7.1.Implications between premises of moods, in all figures.  
The mood numbers on the left imply the adjacent mood numbers on the right.  
Along rows of table listing all moods:  
11  51  12  52  13  53  14  54  15  55  16  56  17  57  18  58 
11  61  12  62  13  63  14  64  15  65  16  66  17  67  18  68 
21  51  22  52  23  53  24  54  25  55  26  56  27  57  28  58 
21  81  22  82  23  83  24  84  25  85  26  86  27  87  28  88 
31  61  32  62  33  63  34  64  35  65  36  66  37  67  38  68 
31  71  32  72  33  73  34  74  35  75  36  76  37  77  38  78 
41  71  42  72  43  73  44  74  45  75  46  76  47  77  48  78 
41  81  42  82  43  83  44  84  45  85  46  86  47  87  48  88 
Down columns of table listing all moods:  
11  15  21  25  31  35  41  45  51  55  61  65  71  75  81  85 
11  16  21  26  31  36  41  46  51  56  61  66  71  76  81  86 
12  15  22  25  32  35  42  45  52  55  62  65  72  75  82  85 
12  18  22  28  32  38  42  48  52  58  62  68  72  78  82  88 
13  16  23  26  33  36  43  46  53  56  63  66  73  76  83  86 
13  17  23  27  33  37  43  47  53  57  63  67  73  77  83  87 
14  17  24  27  34  37  44  47  54  57  64  67  74  77  84  87 
14  18  24  28  34  38  44  48  54  58  64  68  74  78  84  88 
Note well that each implication may in turn imply others, i.e. one must follow up implications of implications. For instance, 11 implies 15 and 16, and 51 and 61; in turn, 15 implies 55 and 65, and 16 implies 56 and 66; also, 51 implies 55 and 56, and 61 implies 65 and 66. Similarly for other premises, as shown in the above table.
Also note, some of the implications shown in the above table may be useless in practice for a given figure: this occurs when a mood referred to has inconsistent premises.
The following are the principles for inference of validity or invalidity. Note well the condition that the validating or invalidating mood be internally consistent; as we explained, it can happen, in a given figure, that they are not so.
1. If the premises of one of the above moods, say Y, are consistent and imply those of another, say X, then any validated conclusion of X, say c1, is also a valid conclusion of Y. (But an invalidated conclusion of X, say c2, cannot be inferred to be an invalid conclusion of Y.)
Proof: Since Y implies X and X implies c1, it follows that Y implies c1.
(But that X does not imply c2, does not mean that Y does not imply c2.)
2. If the premises of one of the above moods, say Z, are consistent and imply those of another, say Y, then any invalidated conclusion of Z, say c2, is also an invalid conclusion of Y. (But a validated conclusion of Z, say c1, cannot be inferred to be a valid conclusion of Y.)
Proof: Since Z implies Y and Z does not imply c2, it follows that Y does not imply c2;
for given that Z implies Y, if Y implied c2, Z would imply c2.
(But that Z implies c1, does not mean that Y implies c1.)
One should be careful not to confuse the premises of a mood with a mood as a whole. Referring to the above rules, in case (1), while Y implies X, the validity of X+c1 implies the validity of Y+c1 (this is a direct reduction). In case (2), while Z implies Y, the invalidity of Z+c2 implies the invalidity of Y+c2 (this is an indirect reduction).
Generally, then, to establish a mood Y+c1+notc2 by reduction, we must look for two moods X and Z, such that (1) Y implies X, which concludes c1, and (2) Y is implied by Z, which fails to conclude c2. The following diagram illustrates these principles:
Diagram 1. Pathways of Reduction, for Validation (right) and Invalidation (left).
3. The above applies to reductions within a given figure, by subalternation. In the special case of direct reduction across figures, by conversion of the major premise (to derive figure 2) or the minor premise (to derive figure 3), the implications between the premises concerned are twoway; it follows in such case that both validity and invalidity are transmitted by the same mood of figure 1.
The following table, based on the preceding one, shows more explicitly the possible sources of validity or invalidity by reduction, for each mood within any figure. It should be noted that we cannot (as far as I can see) predict from it, at the outset for all figures, which moods will require matricial analysis; such knowledge has to be acquired in each figure by judicious trial and error.
Table 7.2.For each mood (central col.), those which imply it and those it implies.  
Moods implying central mood (if any of them is invalid, the central mood is also invalid)  Mood  Moods implied by central mood (if any of them is valid, the central mood is also valid)  







 11  15  16  51  55  56  61  65  66 







 12  15  18  52  55  58  62  65  68 







 13  16  17  53  56  57  63  66  67 







 14  17  18  54  57  58  64  67  68 





 11  12  15 

 55 

 65 







 11  13  16 

 56 

 66 







 13  14  17 

 57 

 67 







 12  14  18 

 58 

 68 









 21  25  26  51  55  56  81  85  86 







 22  25  28  52  55  58  82  85  88 







 23  26  27  53  56  57  83  86  87 







 24  27  28  54  57  58  84  87  88 





 21  22  25 

 55 

 85 







 21  23  26 

 56 

 86 







 23  24  27 

 57 

 87 







 22  24  28 

 58 

 88 









 31  35  36  61  65  66  71  75  76 







 32  35  38  62  65  68  72  75  78 







 33  36  37  63  66  67  73  76  77 







 34  37  38  64  67  68  74  77  78 





 31  32  35 

 65 

 75 







 31  33  36 

 66 

 76 







 33  34  37 

 67 

 77 







 32  34  38 

 68 

 78 









 41  45  46  71  75  76  81  85  86 







 42  45  48  72  75  78  82  85  88 







 43  46  47  73  76  77  83  86  87 







 44  47  48  74  77  78  84  87  88 





 41  42  45 

 75 

 85 







 41  43  46 

 76 

 86 







 43  44  47 

 77 

 87 







 42  44  48 

 78 

 88 



 11 

 21 

 51  55  56 







 12 

 22 

 52  55  58 







 13 

 23 

 53  56  57 







 14 

 24 

 54  57  58 






11  12  15  21  22  25  51  52  55 








11  13  16  21  23  26  51  53  56 








13  14  17  23  24  27  53  54  57 








12  14  18  22  24  28  52  54  58 









 11 

 31 

 61  65  66 







 12 

 32 

 62  65  68 







 13 

 33 

 63  66  67 







 14 

 34 

 64  67  68 






11  12  15  31  32  35  61  62  65 








11  13  16  31  33  36  61  63  66 








13  14  17  33  34  37  63  64  67 








12  14  18  32  34  38  62  64  68 









 31 

 41 

 71  75  76 







 32 

 42 

 72  75  78 







 33 

 43 

 73  76  77 







 34 

 44 

 74  77  78 






31  32  35  41  42  45  71  72  75 








31  33  36  41  43  46  71  73  76 








33  34  37  43  44  47  73  74  77 








32  34  38  42  44  48  72  74  78 









 21 

 41 

 81  85  86 







 22 

 42 

 82  85  88 







 23 

 43 

 83  86  87 







 24 

 44 

 84  87  88 






21  22  25  41  42  45  81  82  85 








21  23  26  41  43  46  81  83  86 








23  24  27  43  44  47  83  84  87 








22  24  28  42  44  48  82  84  88 








Remember that breaks will occur in such implications, if any mood is invalid due to inconsistency between premises.
The following tables summarize the results obtained by such reductions, for each of the figures. Conclusions not validated or invalidated by such means must be evaluated through matricial analysis (which is done in the next chapter). The tables below may be read as follows:
yes = element of conclusion (m, n, p or q) are implied by the given premises.
no = element of conclusion (m, n, p or q) are not implied (which does not mean denied) by the given premises.
by = by any sort of reduction to (number of mood used) or MA (matricial analysis).
Elements of conclusions for which matricial analysis is required are shaded.
since = for given premises, if an element of conclusion is valid (yes), then its contrary element is invalid (no).
** = incompatible premises.
nil = no valid conclusion.
Next section (continuation of same chapter)
[1] Negative moods might be evaluated by indirect reductions to the positive moods, across figures. Imperfect moods might be evaluated by means of direct reductions consisting of eductive processes which result in negations of the complement. We’ll check into that later.