The classification scheme of strong and fragile liquid relaxation behavior introduced by Angell is examined, and the relations between the different fragility parameters are derived and discussed. The relation between fragility and non-linear relaxation behavior within and below the glass transition temperature range is explored. It is suggested that the minimum value of the fragility parameter, m, is proportional to the quotient of the glassy state activation enthalpy and T-g, and implications of this interpretation are discussed. It is argued that T-g Delta C-p(T-g) is an inappropriate measure of the thermodynamic contribution to the Angell strength parameter, D, because it contains the kinetic quantity, T-g.