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.
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