In this work we study two non-classical features of quantum compound systems, namely, entanglement and indistinguishability using logical and algebraic techniques. First, we study improper mixtures from a quantum logical and geometrical point of view. This is done by extending the von Neumann lattice of propositions in order to include improper mixtures as atoms of the new lattice. Then, we study the problem of quantum non-individuality. We use a quantum structure which is a modification of Zermelo-Frenkel set-theory based on quantum mechanics, namely, Quasi-set Theory (Q). Using Q we develop a new formulation of quantum mechanics which does not uses first order identity on its logical bases. These constructions answer interesting discussions posed in the literature.