This paper investigates the logical interrelations between four properties that may be useful in understanding different aspects of sustainability and making it a more operational and useful concept. These properties are system stability, continuation, longevity, and health (integrity). The principal findings are as follows: (1) Stability is necessary but not sufficient for sustainability, continuation, longevity, and health. (2) Continuation is (a) sufficient but not necessary for sustainability, stability, and health and (b) both sufficient and necessary for longevity. (3) Longevity is (a) sufficient and necessary for sustainability and continuation and (b) sufficient but not necessary for stability and health. (4) Health is (a) necessary but not sufficient for sustainability, continuation, and longevity and (b) sufficient but not necessary for stability. (5) Sustainability itself is (a) sufficient but not necessary for stability and health; (b) necessary but not sufficient for continuation; and (c) both sufficient and necessary for longevity. These logical interrelations indicate that sustainability is not a simple concept but is related to others, some of which may provide useful measures for different applications. It seems important to explore formally the many dimensions of sustainability to build a precise concept for scientific use. The four attributes investigated here do not exhaust the possibilities.