Summary: We assert that the usual interpretation of what one means by “elasticity” is much too insular, and illustrate our thesis by introducing implicit constitutive theories that can describe the non-dissipative response of solids. There is another important aspect of the introduction of such an implicit approach to the non-dissipative response of solids, namely the development of a hierarchy of approximations wherein, while the strains are infinitesimal and the relationship between the stress and linearized strain is nonlinear. Such approximations would not be logically consistent within the context of explicit theories of Cauchy elasticity or Green elasticity that are currently popular.