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（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 statement is negative； the particular is affirmative： if
the universal is affirmative； the particular is negative。 For if M
belongs to no N； but to some O； it is necessary that N does not belong
to some O。 For since the negative statement is convertible； N will
belong to no M： but M was admitted to belong to some O： therefore N
will not belong to some O： for the result is reached by means of the
first figure。 Again if M belongs to all N； but not to some O； it is
necessary that N does not belong to some O： for if N belongs to all O；
and M is predicated also of all N； M must belong to all O： but we
assumed that M does not belong to some O。 And if M belongs to all N
but not to all O； we shall conclude that N does not belong to all O：
the proof is the same as the above。 But if M is predicated of all O；
but not of all N； there will be no syllogism。 Take the terms animal；
substance； raven； animal； white； raven。 Nor will there be a conclusion
when M is predicated of no O； but of some N。 Terms to illustrate a
positive relation between the extremes are animal； substance； unit：
a negative relation； animal； substance； science。
If then the universal statement is opposed to the particular； we
have stated when a syllogism will be possible and when not： but if the
premisses are similar in form； I mean both negative or both
affirmative； a syllogism will not be possible anyhow。 First let them
be negative； and let the major premiss be universal； e。g。 let M belong
to no N； and not to some O。 It is possible then for N to belong either
to all O or to no O。 Terms to illustrate the negative relation are
black； snow； animal。 But it is not possible to find terms of which the
extremes are related positively and universally； if M belongs to
some O； and does not belong to some O。 For if N belonged to all O； but
M to no N； then M would belong to no O： but we assumed that it belongs
to some O。 In this way then it is not admissible to take terms： our
point must be proved from the indefinite nature of the particular
statement。 For since it is true that M does not belong to some O； even
if it belongs to no O； and since if it belongs to no O a syllogism
is （as we have seen） not possible； clearly it will not be possible now
either。
Again let the premisses be affirmative； and let the major premiss as
before be universal； e。g。 let M belong to all N and to some O。 It is
possible then for N to belong to all O or to no O。 Terms to illustrate
the negative relation are white； swan； stone。 But it is not possible
to take terms to illustrate the universal affirmative relation； for
the reason already stated： the point must be proved from the
indefinite nature of the particular statement。 But if the minor
premiss is universal； and M belongs to