Antinomy (Greek anti-, against, plus nomos, law) is a term used in logic and epistemology, which, loosely, means a paradox or unresolvable contradiction.
Immanuel Kant believed that when reason goes beyond possible experience it often falls into various antinomies, or equally rational but contradictory views. Reason cannot here play the role of establishing rational truths because it goes beyond possible experience and becomes transcendent. For example, Kant thought that one could reason from the assumption that the world had a beginning in time to the conclusion that it did not, and vice versa. This was part of Kant's critical program of determining limits to science and philosophical inquiry.
In fact, antinomies do not highlight limitations in the power of logical reasoning, as is often thought. This is because the conclusion that there is a limitation is (supposedly) derived from the antinomy by logical reasoning; therefore any limitation in the validity of logical reasoning imposes a limitation on the conclusion that there is a limitation on logical reasoning. (This is an argument by self-reference.)
In short, in terms of the validity of logical reasoning as a whole, antinomies are self-isolating: they are like scattered discontinuities within the field of logic, incapable of casting doubt on anything else but themselves.
Article based on antinomy at FOLDOC, used with permission.