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Works by Aristotle
Pages of Metaphysics

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contrarieties and differences of being), and things of this sort can
fall under one science, the difficulty we stated at the beginning
appears to be solved,-I mean the question how there can be a single
science of things which are many and different in genus.

Since even the mathematician uses the common axioms only in a
special application, it must be the business of first philosophy to
examine the principles of mathematics also. That when equals are taken
from equals the remainders are equal, is common to all quantities, but
mathematics studies a part of its proper matter which it has detached,
e.g. lines or angles or numbers or some other kind of quantity-not,
however, qua being but in so far as each of them is continuous in
one or two or three dimensions; but philosophy does not inquire
about particular subjects in so far as each of them has some attribute
or other, but speculates about being, in so far as each particular
thing is.-Physics is in the same position as mathematics; for
physics studies the attributes and the principles of the things that
are, qua moving and not qua being (whereas the primary science, we
have said, deals with these, only in so far as the underlying subjects
are existent, and not in virtue of any other character); and so both
physics and mathematics must be classed as parts of Wisdom.

There is a principle in things, about which we cannot be deceived,
but must always, on the contrary recognize the truth,-viz. that the
same thing cannot at one and the same time be and not be, or admit any
other similar pair of opposites. About such matters there is no
proof in the full sense, though there is proof ad hominem. For it is
not possible to infer this truth itself from a more certain principle,
yet this is necessary if there is to be completed proof of it in the
full sense. But he who wants to prove to the asserter of opposites
that he is wrong must get from him an admission which shall be
identical with the principle that the same thing cannot be and not
be at one and the same time, but shall not seem to be identical; for
thus alone can his thesis be demonstrated to the man who asserts
that opposite statements can be truly made about the same subject.
Those, then, who are to join in argument with one another must to some
extent understand one another; for if this does not happen how are
they to join in argument with one another? Therefore every word must
be intelligible and indicate something, and not many things but only
one; and if it signifies more than one thing, it must be made plain to
which of these the word is being applied. He, then, who says 'this
is and is not' denies what he affirms, so that what the word
signifies, he says it does not signify; and this is impossible.
Therefore if 'this is' signifies something, one cannot truly assert
its contradictory.
Further, if the word signifies something and this is asserted
truly, this connexion must be necessary; and it is not possible that
that which necessarily is should ever not be; it is not possible
therefore to make the opposed affirmations and negations truly of
the same subject. Further, if the affirmation is no more true than the
negation, he who says 'man' will be no more right than he who says
'not-man'. It would seem also that in saying the man is not a horse
one would be either more or not less right than in saying he is not
a man, so that one will also be right in saying that the same person
is a horse; for it was assumed to be possible to make opposite
statements equally truly. It follows then that the same person is a
man and a horse, or any other animal.
While, then, there is no proof of these things in the full
sense, there is a proof which may suffice against one who will make
these suppositions. And perhaps if one had questioned Heraclitus
himself in this way one might have forced him to confess that opposite
statements can never be true of the same subjects. But, as it is, he

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