The truth is that the geometer does not
draw any conclusion from the being of the particular line of which
he speaks, but from what his diagrams symbolize. A further distinction
is that all hypotheses and illegitimate postulates are either
universal or particular, whereas a definition is neither.
11
So demonstration does not necessarily imply the being of Forms nor a
One beside a Many, but it does necessarily imply the possibility of
truly predicating one of many; since without this possibility we
cannot save the universal, and if the universal goes, the middle
term goes witb. it, and so demonstration becomes impossible. We
conclude, then, that there must be a single identical term
unequivocally predicable of a number of individuals.
The law that it is impossible to affirm and deny simultaneously
the same predicate of the same subject is not expressly posited by any
demonstration except when the conclusion also has to be expressed in
that form; in which case the proof lays down as its major premiss that
the major is truly affirmed of the middle but falsely denied. It makes
no difference, however, if we add to the middle, or again to the minor
term, the corresponding negative.
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