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of demonstration…I mean the common axioms which are used as
premisses of demonstration； not the subjects nor the attributes
demonstrated as belonging to them…all the sciences have communion with
one another； and in communion with them all is dialectic and any
science which might attempt a universal proof of axioms such as the
law of excluded middle； the law that the subtraction of equals from
equals leaves equal remainders； or other axioms of the same kind。
Dialectic has no definite sphere of this kind； not being confined to a
single genus。 Otherwise its method would not be interrogative； for the
interrogative method is barred to the demonstrator； who cannot use the
opposite facts to prove the same nexus。 This was shown in my work on
the syllogism。
12
If a syllogistic question is equivalent to a proposition embodying
one of the two sides of a contradiction； and if each science has its
peculiar propositions from which its peculiar conclusion is developed；
then there is such a thing as a distinctively scientific question； and
it is the interrogative form of the premisses from which the
'appropriate' conclusion of each science is developed。 Hence it is
clear that not every question will be relevant to geometry； nor to
medicine； nor to any other science： only those questions will be
geometrical which form premisses for the proof of the theorems of
geometry or of any other science； such as optics； which uses the
same basic truths as geometry。 Of the other sciences the like is true。
Of these questions the geometer is bound to give his account； using
the basic truths of geometry in conjunction with his previous
conclusions； of the basic truths the geometer； as such； is not bound
to give any account。 The like is true of the other sciences。 There
is a limit； then； to the questions which we may put to each man of
science； nor is each man of science bound to answer all inquiries on
each several subject； but only such as fall within the defined field
of his own science。 If； then； in controversy with a geometer qua
geometer the disputant confines himself to geometry and proves
anything from geometrical premisses； he is clearly to be applauded； if
he goes outside these he will be at fault； and obviously cannot even
refute the geometer except accidentally。 One should therefore not
discuss geometry among those who are not geometers； for in such a
company an unsound argument will pass unnoticed。 This is
correspondingly true in the other sciences。
Since there are 'geometrical' questions； does it follow that there
are also distinctively 'ungeometrical' questions？ Further； in each
special science…geometry for instance…what kind of error is it that
may vitiate questions； and yet not exclude them from that science？
Again； is the erroneous conclusion one constructed from premisses
opposite to the true premisses； or is it formal fallacy though drawn
from geometrical premisses？ Or； perhaps； the erroneous conclusion is
due to the drawing of premisses from another science； e。g。 in a
geometrical controversy a musical question is distinctively
ungeometrical； whereas the notion that parallels meet is in one
sense geometrical； being ungeometrical in a different fashion： the
reason being that 'ungeometrical'； like 'unrhythmical'； is
equivocal； meaning in the one case not geometry at all； in the other
bad geometry？ It is this error； i。e。 error based on premisses of
this kind…'of' the science but false…that is the contrary of
science。 In mathematics the formal fallacy is not so common； because
it is the middle term in which the ambiguity lies； since the major
is predicated of the whole of the middle and the middle of the whole
of the minor （the predicate of course never has the prefix 'all'）； and
in mathematics one can； so to speak； see these middle terms with an
intellectual vision； while in dialectic the ambiguity may escape
detection。 E。g。 'Is every circle a figure？' A diagram shows that
this is so； but the minor premiss 'Are epics circles？' is shown by the
diagram to be false。
If a proof has an inductive minor premiss； one should not bring an
'objection' against it。 For since every premiss must be applicable
to a number of cases （otherwise it will not be true in every instance；
which； since the syllogism proceeds from universals； it must be）； then
assuredly the same is true of an 'objection'； since premisses and
'objections' are so far the same that anything which can be validly
advanced as an 'objection' must be such that it could take the form of
a premiss； either demonstrative or dialectical。 On the other hand；
arguments formally illogical do sometimes occur through taking as
middles mere attributes of the major and minor terms。 An instance of
this is Caeneus' proof that fire increases in geometrical
proportion： 'Fire'； he argues； 'increases rapidly； and so does
geometrical proportion'。 There is no syllogism so； but there is a
syllogism if the most rapidly increasing proportion is geometrical and
the most rapidly increasing proportion is attributable to fire in
its motion。 Sometimes； no doubt； it is impossible to reason from
premisses predicating mere attributes： but sometimes it is possible；
though the possibility is overlooked。 If false premisses could never
give true conclusions 'resolution' would be easy； for premisses and
conclusion would in that case inevitably reciprocate。 I might then
argue thus： let A be an existing fact； let the existence of A imply
such and such facts actually known to me to exist； which we may call
B。 I can now； since they reciprocate； infer A from B。
Reciprocation of premisses and conclusion is more frequent in
mathematics； because mathematics takes definitions； but never an
accident； for its premisses…a second characteristic distinguishing
mathematical reasoning from dialectical disputations。
A science expands not by the interposition of fresh middle terms；
but by the apposition of fresh extreme terms。 E。g。 A is predicated
of B； B of C； C of D； and so indefinitely。 Or the expansion may be
lateral： e。g。 one major A； may be proved of two minors； C and E。
Thus let A represent number…a number or number taken
indeterminately； B determinate odd number； C any particular odd
number。 We can then predicate A of C。 Next let D represent determinate
even number； and E even number。 Then A is predicable of E。
13
Knowledge of the fact differs from knowledge of the reasoned fact。
To begin with； they differ within the same science and in two