The Complete Aristotle (eng.)

the following pairs as contradictory
propositions:
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It may be.
It cannot be.
It is contingent.
It is not contingent.
It is impossible.
It is not impossible.
It is necessary.
It is not necessary.
It is true.
It is not true.
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    Logical sequences follow in due course when we have arranged the
propositions thus. From the proposition ‘it may be’ it follows that
it is contingent, and the relation is reciprocal. It follows also
that it is not impossible and not necessary.
    From the proposition ‘it may not be’ or ‘it is contingent that
it should not be’ it follows that it is not necessary that it
should not be and that it is not impossible that it should not be.
From the proposition ‘it cannot be’ or ‘it is not contingent’ it
follows that it is necessary that it should not be and that it is
impossible that it should be. From the proposition ‘it cannot not
be’ or ‘it is not contingent that it should not be’ it follows that
it is necessary that it should be and that it is impossible that it
should not be.
    Let us consider these statements by the help of a table:
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A.
B.
It may be.
It cannot be.
It is contingent.
It is not contingent.
It is not impossible that it should be.
It is impossible that it should be.
It is not necessary that it should be.
It is necessary that it should not be.
C.
D.
It may not be.
It cannot not be.
It is contingent that it should not be.
It is not contingent that it should not be.
It is not impossible that it should not be.
It is impossible that it should not be.
It is not necessary that it should not be.
It is necessary that it should be.
    Now the propositions ‘it is impossible that it should be’ and
‘it is not impossible that it should be’ are consequent upon the
propositions ‘it may be’, ‘it is contingent’, and ‘it cannot be’,
‘it is not contingent’, the contradictories upon the
contradictories. But there is inversion. The negative of the
proposition ‘it is impossible’ is consequent upon the proposition
‘it may be’ and the corresponding positive in the first case upon
the negative in the second. For ‘it is impossible’ is a positive
proposition and ‘it is not impossible’ is negative.
    We must investigate the relation subsisting between these
propositions and those which predicate necessity. That there is a
distinction is clear. In this case, contrary propositions follow
respectively from contradictory propositions, and the contradictory
propositions belong to separate sequences. For the proposition ‘it
is not necessary that it should be’ is not the negative of ‘it is
necessary that it should not be’, for both these propositions may
be true of the same subject; for when it is necessary that a thing
should not be, it is not necessary that it should be. The reason
why the propositions predicating necessity do not follow in the
same kind of sequence as the rest, lies in the fact that the
proposition ‘it is impossible’ is equivalent, when used with a
contrary subject, to the proposition ‘it is necessary’. For when it
is impossible that a thing should be, it is necessary, not that it
should be, but that it should not be, and when it is impossible
that a thing should not be, it is necessary that it should be.
Thus, if the propositions predicating impossibility or
non-impossibility follow without change of subject from those
predicating possibility or non-possibility, those predicating
necessity must follow with the contrary subject; for the
propositions ‘it is impossible’ and ‘it is necessary’ are not
equivalent, but, as has been said, inversely connected.
    Yet perhaps it is impossible that the contradictory propositions
predicating necessity should be thus arranged. For when it is
necessary that a thing should be, it is possible that it should be.
(For if not, the opposite follows, since one or the other must
follow; so, if it is not possible, it is impossible, and it is thus
impossible that a thing should be, which must necessarily be; which
is absurd.)
    Yet from the proposition ‘it may be’ it follows that it is not
impossible, and from that it follows that it is not necessary; it
comes about therefore that the thing which must necessarily be need
not be; which is absurd. But again, the proposition ‘it is
necessary that it should be’ does not follow from the proposition
‘it may be’, nor does the proposition ‘it is necessary that it
should not be’. For the proposition ‘it may