CHAPTER 9. SYLLOGISM: APPLICATIONS.
In this chapter we will list the valid moods of the syllogism, and make some generalizations and comments, so as to acquaint the reader with the central subject of our discussion. Thereafter, validation will be dealt with in a separate chapter. Please remember that we are dealing here specifically with one type of proposition, the actual, classificatory, categorical. Other types of proposition require eventual treatment, of course.
Our main concern here is classical logic in all its beauty, the showpiece of the science, which we owe to Aristotle and subsequent masters. There are related topics of lesser importance, these will be mentioned in the course of development.
Syllogism is inference from two propositions of a third whose truth follows from the given two. In categorical syllogism, we deduce a relation between two terms by virtue of their being each related to a third term. According to the direction of their relationship to the third term, the syllogism is said to form different figures, or “movements of thought” (Joseph). The polarities and quantities of the premises, because of their diverse ways of distributing their terms, generally affect the character and validity of the conclusion. These differences are used to distinguish moods of the syllogism in each figure, which may reflect a variety of approaches through which our minds analyze a subject to attain understanding of it.
In this section, we will list the principal valid moods of plural syllogism, that is, of syllogism both of whose premises are plural. They are the most important in this doctrine. Valid moods involving one or two singular premises will be listed in the next section. Derivatively valid syllogisms, of an artificial or subaltern nature, or involving atypical conclusions, will be discussed separately. Moods not included in these listings of valid moods are to be regarded as paralogisms, they are either nonsequiturs (‘it does not follow’ in Latin) or selfcontradictory.
a. First Figure.
Form:
Major premise 
MP 
Minor premise 
SM 
Conclusion 
SP. 
AAA 
AII 
All M are P 
All M are P 
All S are M 
Some S are M 
All S are P 
Some S are P 
EAE 
EIO 
No M is P 
No M is P 
All S are M 
Some S are M 
No S is P 
Some S are not P 
We may observe that the major premise is always universal, and the minor premise always affirmative, here. The principle of such reasoning, called the first canon of logic, could be expressed as ‘Whatever satisfies fully the condition of a rule, falls under the rule’. The condition here means ‘being M’, and the rule means ‘being P’ or ‘not being P’.
b. Second Figure.
Form:
Major premise 
PM 
Minor premise 
SM 
Conclusion 
SP. 
AEE 
AOO 
All P are M 
All P are M 
No S is M 
Some S are not M 
No S is P 
Some S are not P 
EAE 
EIO 
No P is M 
No P is M 
All S are M 
Some S are M 
No S is P 
Some S are not P 
We observe that the major premise is always universal, and the conclusion always negative. The second canon of logic, implicit in these moods, can be stated as ‘Whatever does not fall under a rule, does not satisfy any full condition to the rule’. The condition here meaning ‘being P’ and the rule ‘being, or notbeing, M’.
c. Third Figure.
Form:
Major premise 
MP 
Minor premise 
MS 
Conclusion 
SP. 
AII 
EIO 
All M are P 
No M is P 
Some M are S 
Some M are S 
Some S are P 
Some S are not P 
IAI 
OAO 
Some M are P 
Some M are not P 
All M are S 
All M are S 
Some S are P 
Some S are not P 
We observe that the minor premise is always affirmative, and the conclusion is always particular. Two more moods, AAI and EAO, are normally included by logicians with the above; but these are true only by virtue of the truth of AII and EIO, respectively, whose minor premises theirs imply; I have therefore chosen to exclude them. The principle here, our third canon, is expressed as ‘Rules following from the same condition are in that instance at least compatible’. The common condition being instances of subsumed M in both premises, and the rules being their relations to S and P.
d. The Fourth Figure.
Form:
Major premise 
PM 
Minor premise 
MS 
Conclusion 
SP. 
EIO 

No P is M 

Some M are S 

Some S are not P 

We note that the major premise is a negative universal, the minor is affirmative, and the conclusion a negative particular one. (The mood EAO might also have been included here, but its validity is only due to its minor premise implying that of EIO.) This figure is rather controversial. It formally has three more valid moods, AEE, IAI and AAI, but these are left out as too insignificant for such central exposure. This topic will be further discussed. No canon is normally formulated for this figure.
There are therefore a total of 4+4+4+1 = 13 moods of the plural syllogism which are valid, nonderivative, and significant.
If we consider the second and third figures, we see that transposition of the premises does not change the figure, although the conclusion if any will have transposed terms; the middle term remains common subject or predicate, as the case may be, of the premises. But in the first figure, if the major and minor premises are transposed, not only are the major and minor terms transposed in the conclusion, but a new figure emerges, the fourth. The reverse is also true, shifting from fourth to first. Yet, the order of appearances of the premises is essentially conventional, and should not matter.
It is doubtful whether anyone ever thinks in fourth figure terms, probably because of the double complication it involves. The minor term shifts from being a predicate in its premise to being a subject in the conclusion, and the major term switches from subject in its premise to predicate in the conclusion. While each of these changes does occur in the third and second figures respectively, in the fourth figure both of these mental acrobatics are required. We have difficulty in reasoning thus, whereas the process should be obvious enough for the mind to concentrate on content.
Some logicians have opted for ignoring the fourth figure altogether, on such grounds. Others have insisted on including it as a formal possibility, arguing that the science of logic should be exhaustive and systematic, and show us all the information we can draw from any given data.
My own position is a compromise one. The valid moods AEE, IAI, and AAI (which is implicit in IAI, incidentally), clearly do not present us with information not available in the first figure (after transposition of premises). Given the two premises, we are sure to process them mentally in the first figure, and then, if we need to, convert their conclusions as a separate act of thought. In the case of valid mood EIO (and likewise EAO, which is implicit in it), however, the conclusion ‘Some S are not P’ would not be inferable in the first figure, since Opropositions have no converse. It follows that it must be retained to achieve a complete analysis of possibilities, even if rarely used in practise.
This position can be further justified by observing the lack of uniformity in these five moods. They do not have clear common attributes like the valid moods of other figures; they rather seem to form three distinct groups when we consider their polarities and quantities. EIO (and EAO) make up one group; AEE, another; IAI (and AAI), yet another.
Under this heading we may firstly include the two third figure moods, AAI and EAO, and the fourth figure mood, EAO, which were mentioned earlier as mere derivatives. The reason why logicians have traditionally counted them among the principal moods, was that they inform us that in the cases concerned, only a particular conclusion is obtainable from universal premises; but I have chosen to stress rather their implicitness in the corresponding moods with a particular minor premise, so that from this perspective they give us no added information. They do not constitute an independent process, but are reducible to an eduction followed by a deduction, or vice versa. Note in passing that the insignificant mood AAI in the fourth figure is such a derivative of IAI, also insignificant.
We can also call subaltern, moods which simply contain the subaltern conclusion to any higher conclusion found valid. Thus, though valid, they are regarded as products of eduction after the main deduction. They are: in the first figure, AAI and EAO; in the second figure, AEO and EAO; the third figure has none; in the fourth figure, AEO.
Thus, there are altogether of 2+2+2+3 = 9 plural moods which, though valid, are subaltern, in the four figures.
These contain one or more singular propositions. The valid ones are as follows.
In the first figure, ARR and ERG; in the second figure, AGG and ERG. In these figures, we have singular conclusions, higher than in the corresponding valid particular moods (since singulars are not implied by particulars), and so novel syllogisms. They are worth listing.
First Figure:
ARR 
ERG 
All M are P 
No M is P 
This S is M 
This S is M 
This S is P 
This S is not P 
Second Figure:
AGG 
ERG 
All P are M 
No P is M 
This S is not M 
This S is M 
This S is not P 
This S is not P 
I would not regard the moods AAR and EAG in the first figure as valid, in spite of their apparent subalternation by ARR and ERG, respectively, because they introduce a ‘this’ in the conclusion which was not in the premises (so that there is an implicit third premise ‘this is S’). Likewise in the second figure for AEG and EAG, they are not true derivatives of AGG and ERG. This issue will be confronted more deeply later.
The subalterns of these valid moods, viz. in first figure, ARI and ERO, and in the second figure, AGO and ERO, are of course also valid, but not of interest.
In the third figure, the two moods RRI and GRI are worthy of attention. Each exceptionally draws a conclusion from two singular premises, without involvement of a universal premise; this is of course due to the position of the middle term as individual subject of both premises. This reflects the fact that one instance often suffices to make a particular point (and is sometimes enough to disprove a general postulate). Note that the conclusion is particular, and not singular, because the ‘this’ cannot be passed on from a subject to a predicate.
Third Figure.
RRI 
GRO 
This M is P 
This M is not P 
This M is S 
This M is S 
Some S are P 
Some S are not P 
Also valid in the third figure, are ARI, ERO, RAI, and GAO. But in these cases the conclusions from singular premise moods are no more powerful than those from their particular premise equivalents, so that we have mere subaltern forms.
In the fourth figure, ERO and RAI are valid, but as they offer no new conclusion, they may be ignored as subaltern. Because in this figure validation occurs through the first figure, after conversion of premises or conclusion, and a singular proposition converts only to a particular, there cannot be any special valid singular syllogisms.
The total number of valid singular moods, which are not subaltern, is thus 2+2+2+0 = 6. Additionally, we mentioned 2+2+4+2 = 10 subalterns.
Regarding syllogisms involving propositions which concern a majority or minority of a class, we get results similar to those obtained with singular moods.
Thus, in the first figure, there are four main valid moods, their form being: ‘If All M are (or arenot) P, and Most (or Few) S are M, then Most (or Few) S are (or arenot) P’. In the second figure, there are four main valid moods, too, with the form: ‘If All P are (or arenot) M, and Most (or Few) S arenot (or are) M, then Most (or Few) S are not P’.
In the third figure, we have only two main valid moods. They are especially noteworthy in that they manage without a universal premise. Their form is: ‘Most M are (or arenot) P, Most M are S, therefore Some S are (or arenot) P’. Note that the two premises are majoritive, and the conclusion is only particular. The validity of these is due to the assumption that ‘most’ includes more than half of the middle term class, so that there is overlap in some instances.
There are no nonsubaltern valid moods in the fourth figure. Subaltern versions of the above listed syllogisms, involving majoritive or minoritive premises, exist, but will not be listed here.
The following table lists the 19 moods of the syllogism in the four figures, which were found valid, nonsubaltern, and sufficiently significant. These may be called the primary valid moods, because of their relative independence and originality. Another 25 moods are valid, but are either subaltern to the primary syllogisms or insignificant fourth figure moods. These may be grouped together under the name of secondary valid moods.
Table 9.1 Valid Moods in Each Figure.
Figure 
First 
Second 
Third 
Fourth 
Primary 
AAA 
AEE 
AII 
EIO 
Moods 
EAE 
EAE 
EIO 


AII 
AOO 
IAI 


EIO 
EIO 
OAO 


ARR 
AGG 
RRI 


ERG 
ERG 
GRO 

Secondary 
AAI 
AEO 
AAI 
EAO 
Moods 
EAO 
EAO 
EAO 
ERO 

ARI 
AGO 
ARI 
AEE 

ERO 
ERO 
ERO 
AEO 



RAI 
AAI 



GAO 
IAI 




RAI 
The count of primary valid moods is thus (secondaries in brackets): 6 (+4) in figure one, 6 (+4) in figure two, 6 (+6) in figure three, 1 (+7) in figure four. Thus out of 864 imaginable moods, barely 2.2% are valid and significant. A further 2.9% are logically possible, but of comparatively little interest, for reasons already given. These calculations show the need for a science of Logic. If there is a 95% chance of our thoughtprocesses being in error, it is very wise to study the matter, and not leave it to instinct.
We may observe some characteristics the valid moods have in common, relating to polarity or quantity.
a. Polarity.
· One premise is always affirmative. Two negative premises are inconclusive.
· If both premises are affirmative, so is the conclusion.
· If either premise is negative, so is the conclusion.
b. Quantity.
· Only when both premises are universal, may the conclusion be so; though in some cases two universals only yield a particular.
· One premise is always universal. Two particular premises are inconclusive. (Exceptions occur in Figure Three, if both premises are singular or majoritive; the conclusion is in such cases particular.)
· If either premise is particular, so is the conclusion. (To note, additionally, a singular conclusion may sometimes be drawn from a singular premise, in Figures One and Two. Likewise for majoritives and minoritives.)
Comparing these, it is interesting to note how polarity relations are almost similar to quantity relations. Positive is a connection superior in force to negative, much like as universal is to stronger than particular.
These halfdozen ‘general rules of the syllogism’ (as they are called), together with the couple of specific rules mentioned above within each individual figure, are intended to be sufficient, if memorized, to allow us to reject moods which do not fit into any one of them. They apply to the main forms under discussion, though some exceptions occur in a wider context, as will be seen.
c. Distribution. Additional rules have been formulated, which focus on the distribution of terms. These rules help explain the generalities encountered in the previous approach. They are:
· The middle term must be distributive once at least. That is, there must be common instances between the members of the middle term class subsumed in the two premises; this explains the general need of a universal, as well as the mentioned exceptions.
· A minor or major term which was not distributive in its premise, cannot become distributive in the conclusion. That is, we cannot elicit more information concerning a class than was implicit in the given data.
Euler diagrams are very helpful in this context. Through drawing the extensions of the three classes, we can observe on paper the transition from minor to major via the middle.
If logic is viewed as having the task of drawing the most information from given data, then certain additional formal possibilities of deduction from some pairs of categorical premises should be mentioned. We have seen that normal syllogisms always yield a conclusion with the minor term as subject and the major as predicate, ‘SP’. We may ask if there are cases where such a typical conclusion may not be drawn, but the deduction of some other form of conclusion, at least, is still possible.
It is found that indeed this occurs in certain cases. The conclusion involved always has the form ‘Some nonS are nonP’, a particular proposition connecting as subject and predicate the negations of the minor and major terms, instead of the terms themselves. The list of such imperfect moods is as follows. In the first figure, EE, OE (and GE); in the second figure, AA, EE; in the third figure, EE, EO, OE (and EG, GE); in the fourth figure, EE.
These syllogisms are of course very artificial, and will not be discussed further.