The two unquestionable senses of abdicate are (1) to give up power (especially a throne or national leadership), and (2) to cut oneself off from a responsibility. But the word is also commonly used in the sense to give up or set aside a privilege. Here it is almost synonymous with abnegate, which means to deny oneself something or to surrender a privilege. Because abdicate has taken over much of abnegate’s territory, the latter is now rare.
A third word, abrogate, is also in the mix. It means to formally repeal or abolish. Where what’s repealed or abolished is a responsibility, the word comes close to abdicate. Unlike abnegate, though, it has not yielded much of its turf and is still widely used.
Etymology
The roots of the three words are separate—-dicate comes from a Latin word for proclaim, –rogate comes from a Latin word for ask, and –negate of course means to nullify or to deny—but they share the suffix ab-, which means away. So, considered this way, abdicate means to proclaim away, abrogate means to ask away, and abnegate means to nullify away or to deny away. This might not be useful, however, as all three words have been in English about five centuries and have taken multiple senses over the years.
Examples
Only if Assad assesses that Western intervention is a real threat might he abdicate and make room for leadership with better prospects for halting the violence. [Independent]
With us descendants of the Puritans especially, these weather-competitions supply the abnegated excitement of the race-course. [A Moosehead journal, James Russell Lowell (1913)]
Earlier this month AMR successfully petitioned the bankruptcy-court judge for permission to abrogate the pilots’ current contract. [Wall Street Journal]
Paul Ryan launched a withering attack … , charging the US president with abdicating responsibility for the economy. [Financial Times]
But it is extremely doubtful if there is any intelligent being that has utterly abnegated his freedom. [The philosophic function of value, Nathan Blechman (1918)]
[A]brogating the Anti-Ballistic Missile Treaty with Russia was much more of a factor than Saddam Hussein and Iraq. [Time]