In　experimentally　measured　temperature-composition　ferroelectric　phase　diagrams　of　BaTiO3-based　binary　systems,　a　quadruple　point　where　cubic　(C),　tetragonal　(T),　orthorhombic　(O),　and　rhombohedral　(R)　phases　converge　has　been　frequently　reported　in　previous　work.　More　interestingly,　the　quadruple　points　are　experimentally　found　to　behave　as　a　critical　point　with　large　enhancement　in　properties.　However,　it　has　remained　a　fundamental　question　as　to　whether　a　quadruple　point　in　a　binary　ferroelectric　system　defies　the　thermodynamic　phase　rule　and　whether　such　a　point　necessarily　goes　critical.　In　this　study,　it　is　demonstrated　by　Landau　theory　that　a　C-T-O-R　quadruple　point　in　a　binary　ferroelectric　system　can　only　exist　in　the　form　of　a　unique　type　of　critical　point　at　which　two　first-order　transition　lines　and　two　second-order　ones　meet,　and　such　critical　quadruple　points　do　not　defy　the　thermodynamic　phase　rule.　It　is　further　shown　that　at　such　a　critical　C-T-O-R　quadruple　point,　the　system　exhibits　infinitely　large　piezoelectric　coefficients,　which　agrees　with　the　high　piezoelectricity　observed　at　the　C-T-O-R　quadruple　point　in　a　number　of　BaTiO3-based　binary　ferroelectric　systems　and　also　helps　to　explain　the　large　piezoelectricity　obtained　at　the　morphotropic　phase　boundaries　of　these　quadruple　point　based　systems.　©　2021　American　Physical　Society.