The Complete Aristotle (eng.)

or necessity: for
the subject is not without difficulty.
    We admit that of composite expressions those are contradictory
each to each which have the verb ‘to be’ its positive and negative
form respectively. Thus the contradictory of the proposition ‘man
is’ is ‘man is not’, not ‘not-man is’, and the contradictory of
‘man is white’ is ‘man is not white’, not ‘man is not-white’. For
otherwise, since either the positive or the negative proposition is
true of any subject, it will turn out true to say that a piece of
wood is a man that is not white.
    Now if this is the case, in those propositions which do not
contain the verb ‘to be’ the verb which takes its place will
exercise the same function. Thus the contradictory of ‘man walks’
is ‘man does not walk’, not ‘not-man walks’; for to say ‘man walks’
merely equivalent to saying ‘man is walking’.
    If then this rule is universal, the contradictory of ‘it may be’
is may not be’, not ‘it cannot be’.
    Now it appears that the same thing both may and may not be; for
instance, everything that may be cut or may walk may also escape
cutting and refrain from walking; and the reason is that those
things that have potentiality in this sense are not always actual.
In such cases, both the positive and the negative propositions will
be true; for that which is capable of walking or of being seen has
also a potentiality in the opposite direction.
    But since it is impossible that contradictory propositions
should both be true of the same subject, it follows that’ it may
not be’ is not the contradictory of ‘it may be’. For it is a
logical consequence of what we have said, either that the same
predicate can be both applicable and inapplicable to one and the
same subject at the same time, or that it is not by the addition of
the verbs ‘be’ and ‘not be’, respectively, that positive and
negative propositions are formed. If the former of these
alternatives must be rejected, we must choose the latter.
    The contradictory, then, of ‘it may be’ is ‘it cannot be’. The
same rule applies to the proposition ‘it is contingent that it
should be’; the contradictory of this is ‘it is not contingent that
it should be’. The similar propositions, such as ‘it is necessary’
and ‘it is impossible’, may be dealt with in the same manner. For
it comes about that just as in the former instances the verbs ‘is’
and ‘is not’ were added to the subject-matter of the sentence
‘white’ and ‘man’, so here ‘that it should be’ and ‘that it should
not be’ are the subject-matter and ‘is possible’, ‘is contingent’,
are added. These indicate that a certain thing is or is not
possible, just as in the former instances ‘is’ and ‘is not’
indicated that certain things were or were not the case.
    The contradictory, then, of ‘it may not be’ is not ‘it cannot
be’, but ‘it cannot not be’, and the contradictory of ‘it may be’
is not ‘it may not be’, but cannot be’. Thus the propositions ‘it
may be’ and ‘it may not be’ appear each to imply the other: for,
since these two propositions are not contradictory, the same thing
both may and may not be. But the propositions ‘it may be’ and ‘it
cannot be’ can never be true of the same subject at the same time,
for they are contradictory. Nor can the propositions ‘it may not
be’ and ‘it cannot not be’ be at once true of the same subject.
    The propositions which have to do with necessity are governed by
the same principle. The contradictory of ‘it is necessary that it
should be’, is not ‘it is necessary that it should not be,’ but ‘it
is not necessary that it should be’, and the contradictory of ‘it
is necessary that it should not be’ is ‘it is not necessary that it
should not be’.
    Again, the contradictory of ‘it is impossible that it should be’
is not ‘it is impossible that it should not be’ but ‘it is not
impossible that it should be’, and the contradictory of ‘it is
impossible that it should not be’ is ‘it is not impossible that it
should not be’.
    To generalize, we must, as has been stated, define the clauses
‘that it should be’ and ‘that it should not be’ as the
subject-matter of the propositions, and in making these terms into
affirmations and denials we must combine them with ‘that it should
be’ and ‘that it should not be’ respectively.
    We must consider