Abstract: The pseudo-additive relation satisfied by the Tsallis entropy is not linked with the super- or sub-additive properties of the entropy. These latter properties, like concavity and convexity, are fundamentally geometric inequalities and cannot be reduced to equalities. The pseudo-additivity relation is, rather, a functional equation that determines the functional forms of the random entropies. A similar pseudo-additive relation is satisfied by the Arimoto entropy which is a first-order homogeneous form. No conclusions, based on the pseudo-additive functional equation, may be drawn about the extensive nature of systems from either the Tsallis or Arimoto entropy. Further, it is shown that Tsallis` statistical thermodynamic formulation of the non-additive entropy of degree-α is neither correct nor self-consistent.