Concepts of consistency have long played a key role in constraint programming but never developed in integer programming (IP). Consistency nonetheless plays a role in IP as well. For example, cutting planes can reduce backtracking by achieving various forms of consistency as well as by tightening the linear programming (LP) relaxation. We introduce a type of consistency that is particularly suited for 0-1 programming and develop the associated theory. We define a 0-1 constraint set as LP-consistent when any partial assignment that is consistent with its linear programming relaxation is consistent with the original 0-1 constraint set. We prove basic properties of LP-consistency, including its relationship with Chvatal-Gomory cuts and the integer hull. We show that a weak form of LP-consistency can reduce or eliminate backtracking in a way analogous to k-consistency but is easier to achieve. In so doing, we identify a class of valid inequalities that can be more effective than traditional cutting planes at cutting off infeasible 0-1 partial assignments.