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the difficulty we have just stated： in addition their theory reduces
to the mere statement that if a thing exists； then it does exist…an
easy way of proving anything。 That this is so can be clearly shown
by taking three terms； for to constitute the circle it makes no
difference whether many terms or few or even only two are taken。
Thus by direct proof； if A is； B must be； if B is； C must be；
therefore if A is； C must be。 Since then…by the circular proof…if A
is； B must be； and if B is； A must be； A may be substituted for C
above。 Then 'if B is； A must be'='if B is； C must be'； which above
gave the conclusion 'if A is； C must be'： but C and A have been
identified。 Consequently the upholders of circular demonstration are
in the position of saying that if A is； A must be…a simple way of
proving anything。 Moreover； even such circular demonstration is
impossible except in the case of attributes that imply one another；
viz。 'peculiar' properties。
Now； it has been shown that the positing of one thing…be it one
term or one premiss…never involves a necessary consequent： two
premisses constitute the first and smallest foundation for drawing a
conclusion at all and therefore a fortiori for the demonstrative
syllogism of science。 If； then； A is implied in B and C； and B and C
are reciprocally implied in one another and in A； it is possible； as
has been shown in my writings on the syllogism； to prove all the
assumptions on which the original conclusion rested； by circular
demonstration in the first figure。 But it has also been shown that
in the other figures either no conclusion is possible； or at least
none which proves both the original premisses。 Propositions the
terms of which are not convertible cannot be circularly demonstrated
at all； and since convertible terms occur rarely in actual
demonstrations； it is clearly frivolous and impossible to say that
demonstration is reciprocal and that therefore everything can be
demonstrated。
4
Since the object of pure scientific knowledge cannot be other than
it is； the truth obtained by demonstrative knowledge will be
necessary。 And since demonstrative knowledge is only present when we
have a demonstration； it follows that demonstration is an inference
from necessary premisses。 So we must consider what are the premisses
of demonstration…i。e。 what is their character： and as a preliminary；
let us define what we mean by an attribute 'true in every instance
of its subject'； an 'essential' attribute； and a 'commensurate and
universal' attribute。 I call 'true in every instance' what is truly
predicable of all instances…not of one to the exclusion of
others…and at all times； not at this or that time only； e。g。 if animal
is truly predicable of every instance of man； then if it be true to
say 'this is a man'； 'this is an animal' is also true； and if the
one be true now the other is true now。 A corresponding account holds
if point is in every instance predicable as contained in line。 There
is evidence for this in the fact that the objection we raise against a
proposition put to us as true in every instance is either an
instance in which； or an occasion on which； it is not true。
Essential attributes are （1） such as belong to their subject as
elements in its essential nature （e。g。 line thus belongs to
triangle； point to line； for the very being or 'substance' of triangle
and line is composed of these elements； which are contained in the
formulae defining triangle and line）： （2） such that； while they belong
to certain subjects； the subjects to which they belong are contained
in the attribute's own defining formula。 Thus straight and curved
belong to line； odd and even； prime and compound； square and oblong；
to number； and also the formula defining any one of these attributes
contains its subject…e。g。 line or number as the case may be。
Extending this classification to all other attributes； I distinguish
those that answer the above description as belonging essentially to
their respective subjects； whereas attributes related in neither of
these two ways to their subjects I call accidents or 'coincidents'；
e。g。 musical or white is a 'coincident' of animal。
Further （a） that is essential which is not predicated of a subject
other than itself： e。g。 'the walking ＇thing＇' walks and is white in
virtue of being something else besides； whereas substance； in the
sense of whatever signifies a 'this somewhat'； is not what it is in
virtue of being something else besides。 Things； then； not predicated
of a subject I call essential； things predicated of a subject I call
accidental or 'coincidental'。
In another sense again （b） a thing consequentially connected with
anything is essential； one not so connected is 'coincidental'。 An
example of the latter is 'While he was walking it lightened'： the
lightning was not due to his walking； it was； we should say； a
coincidence。 If； on the other hand； there is a consequential
connexion； the predication is essential； e。g。 if a beast dies when its
throat is being cut； then its death is also essentially connected with
the cutting； because the cutting was the cause of death； not death a
'coincident' of the cutting。
So far then as concerns the sphere of connexions scientifically
known in the unqualified sense of that term； all attributes which
（within that sphere） are essential either in the sense that their
subjects are contained in them； or in the sense that they are
contained in their subjects； are necessary as well as
consequentially connected with their subjects。 For it is impossible
for them not to inhere in their subjects either simply or in the
qualified sense that one or other of a pair of opposites must inhere
in the subject； e。g。 in line must be either straightness or curvature；
in number either oddness or evenness。 For within a single identical
genus the contrary of a given attribute is either its privative or its
contradictory； e。g。 within number what is not odd is even； inasmuch as
within this sphere even is a necessary consequent of not…odd。 So；
since any given predicate must be either affirmed or denied of any
subject； essential attributes must inhere in their subjects of
necessity。
Thus； then； we have established the distinction between the
attribute which is 'true in every instance' and the 'essential'
attribute。
I term 'commensurately universal' an attribute which belongs to
every instance of its subject； and to every instance essentially and
as such； from which it clearly follows that all commensurate