This mainly refers to rule by a majority of voters. We generally assume that this majority will be determined in a fair and accurate way, so that also ties into the promise.
Democracy also promises that everyone is entitled to an equal voice in government.
IRV does not actually fulfill this promise, because it only works for single-winner elections. Only reforms like proportional or direct representation can address this. IRV should be used for any election at the state or local level that chooses one candidate. These include U.S. Senator, Governor, Lt. Governor, Treasurer, Attorney General, Secretary of State, Controller, Superintendent, and Insurance Commissioner at the state level. Legislative elections should also use IRV, preferably as a transition to proportional representation by multi-member districts that help ensure that everyone's voice is heard.