www.TheLogician.net © Avi Sion - all rights reserved
© Avi Sion
All rights reserved
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
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
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 non-sequiturs
('it does not follow' in Latin) or self-contradictory.
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'.
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 not-being, M'.
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.
The Fourth Figure.
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 O-propositions
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
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,
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
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.
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.
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
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
Thus, in the first figure, there are four main valid moods, their form
being: 'If All M are (or are-not) P, and Most (or Few) S are M, then Most (or
Few) S are (or are-not) P'. In the second figure, there are four main valid
moods, too, with the form: 'If All P are (or are-not) M, and Most (or Few) S
are-not (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 are-not) P, Most M are S, therefore Some S are (or
are-not) 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
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.
Valid Moods in Each Figure.
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 thought-processes 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.
One premise is always affirmative. Two negative
premises are inconclusive.
If both premises are affirmative, so is the
If either premise is negative, so is the
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
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 half-dozen '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
Additional rules have been formulated, which focus on the distribution of terms.
These rules help explain the generalities encountered in the previous approach.
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, 'S-P'. 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