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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 no O； and not to some N； it
is possible for N to belong either to all O or to no O。 Terms for
the positive relation are white； animal； raven： for the negative
relation； white； stone； raven。 If the premisses are affirmative； terms
for the negative relation are white； animal； snow； for the positive
relation； white； animal； swan。 Evidently then； whenever the
premisses are similar in form； and one is universal； the other
particular； a syllogism can； not be formed anyhow。 Nor is one possible
if the middle term belongs to some of each of the extremes； or does
not belong to some of either； or belongs to some of the one； not to
some of the other； or belongs to neither universally； or is related to
them indefinitely。 Common terms for all the above are white； animal；
man： white； animal； inanimate。
It is clear then from what has been said that if the terms are related
to one another in the way stated； a syllogism results of necessity；
and if there is a syllogism； the terms must be so related。 But it is
evident also that all the syllogisms in this figure are imperfect： for
all are made perfect by certain supplementary statements； which either
are contained in the terms of necessity or are assumed as
hypotheses； i。e。 when we prove per impossibile。 And it is evident that
an affirmative conclusion is not attained by means of this figure； but
all are negative； whether universal or particular。
6
But if one term belongs to all； and another to none； of a third；
or if both belong to all； or to none； of it； I call such a figure
the third； by middle term in it I mean that of which both the
predicates are predicated； by extremes I mean the predicates； by the
major extreme that which is further from the middle； by the minor that
which is nearer to it。 The middle term stands outside the extremes；
and is last in position。 A syllogism cannot be perfect in this
figure either； but it may be valid whether the terms are related
universally or not to the middle term。
If they are universal； whenever both P and R belong to S； it follows
that P will necessarily belong to some R。 For； since the affirmative
statement is convertible； S will belong to some R： consequently
since P belongs to all S； and S to some R； P must belong to some R：
for a syllogism in the first figure is produced。 It is possible to
demonstrate this also per impossibile and by exposition。 For if both P
and R belong to all S； should one of the Ss； e。g。 N； be taken； both
P and R will belong to this； and thus P will belong to some R。
If R belongs to all S； and P to no S； there will be a syllogism to
prove that P will necessarily not belong to some R。 This may be
demonstrated in the same way as before by converting the premiss RS。
It might be proved also per impossibile； as in the former cases。 But
if R belongs to no S； P to all S； there will be no syllogism。 Terms
for the positive relation are animal； horse； man： for the negative
relation animal； inanimate； man。
Nor can there be a syllogism when both terms are asserted of no S。
Terms for the positive relation are animal； horse； inanimate； for
the negative relation man； horse； inanimate…inanimate being the middle
term。
It is clear then in this figure also when a syllogism will be
possible and when not； if the terms are related universally。 For
whenever both the terms are affirmative； there will be a syllogism
to prove that one extreme belongs to some of the other； but when
they are negative； no syllogism will be possible。 But when one is
negative； the other affirmative； if the major is negative； the minor
affirmative； there will be a syllogism to prove that the one extreme
does not belong to some of the other： but if the relation is reversed；
no syllogism will be possible。 If one term is related universally to
the middle； the other in part only； when both are affirmative there
must be a syllogism； no matter which of the premisses is universal。
For if R belongs to all S； P to some S； P must belong to some R。 For
since the affirmative statement is convertible S will belong to some
P： consequently since R belongs to all S； and S to some P； R must also
belong to some P： therefore P must belong to some R。
Again if R belongs to some S； and P to all S； P must belong to
some R。 This may be demonstrated in the same way as the preceding。 And
it is possible to demonstrate it also per impossibile and by
exposition； as in the former cases。 But if one term is affirmative；
the other negative； and if the affirmative is universal； a syllogism
will be possible whenever the minor term is affirmative。 For if R
belongs to all S； but P does not belong to some S； it is necessary
that P does not belong to some R。 For if P belongs to all R； and R
belongs to all S； then P will belong to all S： but we assumed that
it did not。 Proof is possible also without reduction ad impossibile；
if one of the Ss be taken to which P does not belong。
But whenever the major is affirmative； no syllogism will be
possible； e。g。 if P belongs to all S and R does not belong to some
S。 Terms for the universal affirmative relation are animate； man；
animal。 For the universal negative relation it is not possible to
get terms； if R belongs to some S； and does not belong to some S。
For if P belongs to all S； and R to some S； then P will belong to some
R： but we assumed that it belongs to no R。 We must put the matter as
before。' Since the expression 'it does not belong to some' is
indefinite； it may be used truly of that also which belongs to none。
But if R belongs to no S； no syllogism i