The Complete Aristotle (eng.)

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