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.
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