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 ways:
(1) when the premisses of the syllogism are not immediate (for then
the proximate cause is not contained in them-a necessary condition
of knowledge of the reasoned fact): (2) when the premisses are
immediate, but instead of the cause the better known of the two
reciprocals is taken as the middle; for of two reciprocally predicable
terms the one which is not the cause may quite easily be the better
known and so become the middle term of the demonstration. Thus (2) (a)
you might prove as follows that the planets are near because they do
not twinkle: let C be the planets, B not twinkling, A proximity.
Then B is predicable of C; for the planets do not twinkle. But A is
also predicable of B, since that which does not twinkle is near--we
must take this truth as having been reached by induction or
sense-perception. Therefore A is a necessary predicate of C; so that
we have demonstrated that the planets are near. This syllogism,
then, proves not the reasoned fact but only the fact; since they are
not near because they do not twinkle, but, because they are near, do
not twinkle. The major and middle of the proof, however, may be
reversed, and then the demonstration will be of the reasoned fact.
Thus: let C be the planets, B proximity, A not twinkling. Then B is an
attribute of C, and A-not twinkling-of B. Consequently A is predicable
of C, and the syllogism proves the reasoned fact, since its middle
term is the proximate cause. Another example is the inference that the
moon is spherical from its manner of waxing. Thus: since that which so
waxes is spherical, and since the moon so waxes, clearly the moon is
spherical. Put in this form, the syllogism turns out to be proof of
the fact, but if the middle and major be reversed it is proof of the
reasoned fact; since the moon is not spherical because it waxes in a
certain manner, but waxes in such a manner because it is spherical.
(Let C be the moon, B spherical, and A waxing.) Again (b), in cases
where the cause and the effect are not reciprocal and the effect is