Paradox, antinomy, anomaly are comparable terms that involve the idea of expressing or revealing an inner or inherent contradiction and are therefore not always clearly distinguished.
A paradox is primarily a statement or proposition which contains a contradiction yet which, absurd as it seems to be, may still be true and in accordance with the facts and common sense.
By extension paradox may apply to something which is known to exist, yet which when described or put in words seems incredible because it involves a logical contradiction.
An antinomy, in philosophical use, is a contradiction between two laws, principles, or conclusions, both of which are held on good grounds or are correctly inferred from the same facts or premises; thus, the conclusions that every material thing can be explained by mechanical causes and that some material things cannot be explained unless a final cause is postulated, present an antinomy, but in the opinion of Kant both can be accepted as rules regulative of experience. In more general use the term is often applied to one thing that contradicts another thing and is irreconcilable with it or it may apply to a conflict (as of principles, beliefs, forces, tendencies, or aspirations) that is irresolvable in the light of present knowledge.
An anomaly is something that is contrary to what it should be. For example, it may be an exception or a contradiction to a rule; it may be a freak, a monster, a sport, or a contradiction to a type; it may be an anachronism or solecism, irreconcilable with its surroundings or conditions; it may be an action, a practice, or a mood, that is in effect a denial of what one believes or teaches.