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and this in turn would inhere in C and D through a third； so that
between two terms an infinity of intermediates would fall…an
impossibility。 Thus it need not always be in virtue of a common middle
term that a single attribute inheres in several subjects； since
there must be immediate intervals。 Yet if the attribute to be proved
common to two subjects is to be one of their essential attributes； the
middle terms involved must be within one subject genus and be
derived from the same group of immediate premisses； for we have seen
that processes of proof cannot pass from one genus to another。
It is also clear that when A inheres in B； this can be
demonstrated if there is a middle term。 Further； the 'elements' of
such a conclusion are the premisses containing the middle in question；
and they are identical in number with the middle terms； seeing that
the immediate propositions…or at least such immediate propositions
as are universal…are the 'elements'。 If； on the other hand； there is
no middle term； demonstration ceases to be possible： we are on the way
to the basic truths。 Similarly if A does not inhere in B； this can
be demonstrated if there is a middle term or a term prior to B in
which A does not inhere： otherwise there is no demonstration and a
basic truth is reached。 There are； moreover； as many 'elements' of the
demonstrated conclusion as there are middle terms； since it is
propositions containing these middle terms that are the basic
premisses on which the demonstration rests； and as there are some
indemonstrable basic truths asserting that 'this is that' or that
'this inheres in that'； so there are others denying that 'this is
that' or that 'this inheres in that'…in fact some basic truths will
affirm and some will deny being。
When we are to prove a conclusion； we must take a primary
essential predicate…suppose it C…of the subject B； and then suppose
A similarly predicable of C。 If we proceed in this manner； no
proposition or attribute which falls beyond A is admitted in the
proof： the interval is constantly condensed until subject and
predicate become indivisible； i。e。 one。 We have our unit when the
premiss becomes immediate； since the immediate premiss alone is a
single premiss in the unqualified sense of 'single'。 And as in other
spheres the basic element is simple but not identical in all…in a
system of weight it is the mina； in music the quarter…tone； and so
onso in syllogism the unit is an immediate premiss； and in the
knowledge that demonstration gives it is an intuition。 In
syllogisms； then； which prove the inherence of an attribute； nothing
falls outside the major term。 In the case of negative syllogisms on
the other hand； （1） in the first figure nothing falls outside the
major term whose inherence is in question； e。g。 to prove through a
middle C that A does not inhere in B the premisses required are； all B
is C； no C is A。 Then if it has to be proved that no C is A； a
middle must be found between and C； and this procedure will never
vary。
（2） If we have to show that E is not D by means of the premisses；
all D is C； no E； or not all E； is C； then the middle will never
fall beyond E； and E is the subject of which D is to be denied in
the conclusion。
（3） In the third figure the middle will never fall beyond the limits
of the subject and the attribute denied of it。
24
Since demonstrations may be either commensurately universal or
particular； and either affirmative or negative； the question arises；
which form is the better？ And the same question may be put in regard
to so…called 'direct' demonstration and reductio ad impossibile。 Let
us first examine the commensurately universal and the particular
forms； and when we have cleared up this problem proceed to discuss
'direct' demonstration and reductio ad impossibile。
The following considerations might lead some minds to prefer
particular demonstration。
（1） The superior demonstration is the demonstration which gives us
greater knowledge （for this is the ideal of demonstration）； and we
have greater knowledge of a particular individual when we know it in
itself than when we know it through something else； e。g。 we know
Coriscus the musician better when we know that Coriscus is musical
than when we know only that man is musical； and a like argument
holds in all other cases。 But commensurately universal
demonstration； instead of proving that the subject itself actually
is x； proves only that something else is x… e。g。 in attempting to
prove that isosceles is x； it proves not that isosceles but only that
triangle is x… whereas particular demonstration proves that the
subject itself is x。 The demonstration； then； that a subject； as such；
possesses an attribute is superior。 If this is so； and if the
particular rather than the commensurately universal forms
demonstrates； particular demonstration is superior。
（2） The universal has not a separate being over against groups of
singulars。 Demonstration nevertheless creates the opinion that its
function is conditioned by something like this…some separate entity
belonging to the real world； that； for instance； of triangle or of
figure or number； over against particular triangles； figures； and
numbers。 But demonstration which touches the real and will not mislead
is superior to that which moves among unrealities and is delusory。 Now
commensurately universal demonstration is of the latter kind： if we
engage in it we find ourselves reasoning after a fashion well
illustrated by the argument that the proportionate is what answers
to the definition of some entity which is neither line； number； solid；
nor plane； but a proportionate apart from all these。 Since； then； such
a proof is characteristically commensurate and universal； and less
touches reality than does particular demonstration； and creates a
false opinion； it will follow that commensurate and universal is
inferior to particular demonstration。
We may retort thus。 （1） The first argument applies no more to
commensurate and universal than to particular demonstration。 If
equality to two right angles is attributable to its subject not qua
isosceles but qua triangle； he who knows that isosceles possesses that
attribute knows the subject as qua itself possessing the attribute； to
a less degree than he who knows that triangle has that attribute。 To
sum up the whole matter： if a subject is proved to possess qua
triangle an attribute which it does not in fact possess qua
triangle； that is not demonstration： but if it does possess i