Majorities with a quorum

Based on a general model of "quaternary" voting rule, sensitive to voters' choices between four different options (abstaining, voting "yes", voting "no" and staying home), we systematically study different types of majority and quorum. The model allows for a precise formulation of majority rules and quorum constraints. For such rules four types of majority can be defined. We also consider four types of quorum. Then we study the possible combinations of a majority system with a type of quorum and provide examples from rules actually used in parliaments.