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Whenever three terms are so related to one another that the last
is contained in the middle as in a whole； and the middle is either
contained in； or excluded from； the first as in or from a whole； the
extremes must be related by a perfect syllogism。 I call that term
middle which is itself contained in another and contains another in
itself： in position also this comes in the middle。 By extremes I
mean both that term which is itself contained in another and that in
which another is contained。 If A is predicated of all B； and B of
all C； A must be predicated of all C： we have already explained what
we mean by 'predicated of all'。 Similarly also； if A is predicated
of no B； and B of all C； it is necessary that no C will be A。
But if the first term belongs to all the middle； but the middle to
none of the last term； there will be no syllogism in respect of the
extremes； for nothing necessary follows from the terms being so
related； for it is possible that the first should belong either to all
or to none of the last； so that neither a particular nor a universal
conclusion is necessary。 But if there is no necessary consequence；
there cannot be a syllogism by means of these premisses。 As an example
of a universal affirmative relation between the extremes we may take
the terms animal； man； horse； of a universal negative relation； the
terms animal； man； stone。 Nor again can syllogism be formed when
neither the first term belongs to any of the middle； nor the middle to
any of the last。 As an example of a positive relation between the
extremes take the terms science； line； medicine： of a negative
relation science； line； unit。
If then the terms are universally related； it is clear in this
figure when a syllogism will be possible and when not； and that if a
syllogism is possible the terms must be related as described； and if
they are so related there will be a syllogism。
But if one term is related universally； the other in part only； to
its subject； there must be a perfect syllogism whenever universality
is posited with reference to the major term either affirmatively or
negatively； and particularity with reference to the minor term
affirmatively： but whenever the universality is posited in relation to
the minor term； or the terms are related in any other way； a syllogism
is impossible。 I call that term the major in which the middle is
contained and that term the minor which comes under the middle。 Let
all B be A and some C be B。 Then if 'predicated of all' means what was
said above； it is necessary that some C is A。 And if no B is A but
some C is B； it is necessary that some C is not A。 The meaning of
'predicated of none' has also been defined。 So there will be a perfect
syllogism。 This holds good also if the premiss BC should be
indefinite； provided that it is affirmative： for we shall have the
same syllogism whether the premiss is indefinite or particular。
But if the universality is posited with respect to the minor term
either affirmatively or negatively； a syllogism will not be
possible； whether the major premiss is positive or negative；
indefinite or particular： e。g。 if some B is or is not A； and all C
is B。 As an example of a positive relation between the extremes take
the terms good； state； wisdom： of a negative relation； good； state；
ignorance。 Again if no C is B； but some B is or is not A or not
every B is A； there cannot be a syllogism。 Take the terms white；
horse； swan： white； horse； raven。 The same terms may be taken also
if the premiss BA is indefinite。
Nor when the major premiss is universal； whether affirmative or
negative； and the minor premiss is negative and particular； can
there be a syllogism； whether the minor premiss be indefinite or
particular： e。g。 if all B is A and some C is not B； or if not all C is
B。 For the major term may be predicable both of all and of none of the
minor； to some of which the middle term cannot be attributed。
Suppose the terms are animal； man； white： next take some of the
white things of which man is not predicated…swan and snow： animal is
predicated of all of the one； but of none of the other。 Consequently
there cannot be a syllogism。 Again let no B be A； but let some C not
be B。 Take the terms inanimate； man； white： then take some white
things of which man is not predicated…swan and snow： the term
inanimate is predicated of all of the one； of none of the other。
Further since it is indefinite to say some C is not B； and it is
true that some C is not B； whether no C is B； or not all C is B； and
since if terms are assumed such that no C is B； no syllogism follows
（this has already been stated） it is clear that this arrangement of
terms will not afford a syllogism： otherwise one would have been
possible with a universal negative minor premiss。 A similar proof
may also be given if the universal premiss is negative。
Nor can there in any way be a syllogism if both the relations of
subject and predicate are particular； either positively or negatively；
or the one negative and the other affirmative； or one indefinite and
the other definite； or both indefinite。 Terms common to all the
above are animal； white； horse： animal； white； stone。
It is clear then from what has been said that if there is a
syllogism in this figure with a particular conclusion； the terms
must be related as we have stated： if they are related otherwise； no
syllogism is possible anyhow。 It is evident also that all the
syllogisms in this figure are perfect （for they are all completed by
means of the premisses originally taken） and that all conclusions
are proved by this figure； viz。 universal and particular；
affirmative and negative。 Such a figure I call the first。
5
Whenever the same thing belongs to all of one subject； and to none
of another； or to all of each subject or to none of either； I call
such a figure the second； by middle term in it I mean that which is
predicated of both subjects； by extremes the terms of which this is
said； by major extreme that which lies near the middle； by minor
that which is further away from the middle。 The middle term stands
outside the extremes； and is first in position。 A syllogism cannot
be perfect anyhow in this figure； but it may be valid whether the
terms are related universally or not。
If then the terms are related universally a syllogism will be
possible； whenever the middle belongs to all of one subject and to
none of another （it does not matter which has the negative
relation）； but in no other way。 Let M be predicated of no N； but of
all O。 Since； then； the negative relation is convertible； N will
belong to no M： but M was assumed to belong to all O： consequently N
will belong to no O。 This has already been proved。 Again if M
belongs to all N； but to no O； then N will belong to no O。 For if M
belongs to no O； O belongs to no M： but M （as was said） belongs to all
N： O then will belong to no N： for the first figure has again been
formed。 But since the negative relation is convertible； N will
belong to no O。 Thus it will be the same syllogism that proves both
conclusions。
It is possible to prove these results also by reductio ad
impossibile。
It is clear then that a syllogism is formed when the terms are so
related； but not a perfect syllogism； for necessity is not perfectly
established merely from the original premisses； others also are
needed。
But if M is predicated of every N and O； there cannot be a
syllogism。 Terms to illustrate a positive relation between the
extremes are substance； animal； man； a negative relation； substance；
animal； number…substance being the middle term。
Nor is a syllogism possible when M is predicated neither of any N
nor of any O。 Terms to illustrate a positive relation are line；
animal； man： a negative relation； line； animal； stone。
It is clear then that if a syllogism is formed when the terms are
universally related； the terms must be related as we stated at the
outset： for if they are otherwise related no necessary consequence
follows。
If the middle term is related universally to one of the extremes；
a particular negative syllogism must result whenever the middle term
is related universally to the major whether positively or
negatively； and particularly to the minor and in a manner opposite
to that of the universal statement： by 'an opposite manner' I mean； if
the universal