For grant a minor term of which it
is true to predicate man-even if it be also true to predicate
not-man of it--still grant simply that man is animal and not
not-animal, and the conclusion follows: for it will still be true to
say that Callias--even if it be also true to say that
not-Callias--is animal and not not-animal. The reason is that the
major term is predicable not only of the middle, but of something
other than the middle as well, being of wider application; so that the
conclusion is not affected even if the middle is extended to cover the
original middle term and also what is not the original middle term.
The law that every predicate can be either truly affirmed or truly
denied of every subject is posited by such demonstration as uses
reductio ad impossibile, and then not always universally, but so far
as it is requisite; within the limits, that is, of the genus-the
genus, I mean (as I have already explained), to which the man of
science applies his demonstrations. In virtue of the common elements
of demonstration-I mean the common axioms which are used as
premisses of demonstration, not the subjects nor the attributes
demonstrated as belonging to them-all the sciences have communion with
one another, and in communion with them all is dialectic and any
science which might attempt a universal proof of axioms such as the
law of excluded middle, the law that the subtraction of equals from
equals leaves equal remainders, or other axioms of the same kind.
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