Antinomy:
Contradiction; what is against the law.
The second definition is a literal
translation. In logic, mathematics and philosophy, "antinomy" is
contrasted with "paradox", that also has the sense of
contradiction, but that means literally what is beyond the teaching, or
beyond expectation.
The reason to have both terms is that
not all paradoxes are antinomies, in the sense that something may be
unexpected, quaint, odd, and thus contradicts what one believes would
or should be the case, but nevertheless may not stand in logical
contradiction with something one can prove.
If one can prove so, one has a real
antinomy, that accordingly requires some proof. Accordingly, logically
speaking some antinomies  such as "it rains and it does not
rain"  are not paradoxical, since one knows them to be contradictory,
but then it is generally better to speak simply of contradiction.
Outside logic, mathematics and
philosophy, antinomies and paradoxes are often confused, and are then
usually both called "paradox", and then normally in the strong sense of
logical contradiction.
This is somewhat unfortunate, in that
there are quite a few mathematical theorems that prove things that one
would not expect or that do not conform with one's intuitions about the
subject, yet can be proved.
