Quine's under-determination thesis has various formulations. One of them states that, for any theory formulation, there is another formulation that is empirically equivalent to it but logically incompatible with it, and cannot be rendered logically equivalent to it by any re-construal of predicates. This definition is coherent with the following simpler one: we may face several theories (possibly not only two of them) which are empirically equivalent but conflicting. I introduce and discuss a variant in which a possible way for two theories to conflict is to exhibit different ontologies (say, furniture's of the world). This defines what I call the ontological under-determination (possibly with other terminological variants). In this paper, the thesis of ontological under-determination is discussed by relying on three examples pertaining to physical sciences (i)one pertaining to classical mechanics : Newton's formulation of classical mechanics versus Hamilton-Jacobi's formulation (ii) causal interpretations of quantum mechanics (mainly : pilot wave theory) versus Copenhagen interpretation and (iii) Mie's theory versus Lorenz' theory, describing the electromagnetic interaction between an illuminating plane wave and a scattering sphere defined by its complex refractive index and by its diameter. Discriminations between ontological under-determined theories are afterward achieved by using implicative arguments.