Equal and equivalent are about the same. (You could say they're equivalent.) Equal is used when the objects being compared are measurable: "There are equal numbers of apples in these two piles." It is also used when the things being compared are identical: "All citizens have equal rights." Equivalent is better when the things can perform the same function equally well, but may not be identical: "I replaced my Sony TV with an equivalent Panasonic TV." Contrary is can mean argumentative or contumacious: "A contrary child refuses to do what he is told." Contradictory has more to do with discussion: "I made a statement, and he made a contradictory statement." A paradox is an apparently false statement or condition that is nevertheless true. Paradoxicalness (which is a rarely used term) is the nature of that statement or condition that makes it a paradox.