Everyone knows what an antonym is. Or do they?
Hot and cold? Fine... But do big and small exclude big and little, large and small or great and small? What about large and little? These matches are decided by how we use the words on a day-to-day basis -- by culture and usage.
Over and under, above and below, up and down are clearly antonyms. But if Fred lives over the hill, is the opposite of over still under? If "the game is over", the opposite must be that "the game has begun," not that "the game is under." So what is going on here? Clearly, a particular antonym of a word relates to a particular sense of that word. Context tells us which sense that should be.
So, if the "up" sense of over has the antonym under, why doesn't above also have under as its antonym? -- it means the same thing (has the same sense) in this context. George Miller, the cognitive scientist and founder of the lexical database WordNet, realised that antonymy is a relation between words, not their senses -- so synonyms don't (usually, at least) share their "antonymy" -- hence above and over don't share their antonyms.
This does not prevent the synonyms of antonyms from being shared:
And of course there are exceptions, especially in natural language, for every rule...