The probabilistic interpretation was developed by Bohr³ and others^12-15 by using operator formation. Von Neumann's¹⁶ logical structure of quantum mechanics is extended to statistical theory of quantum mechanics. In quantum mechanics, the elementary of the observed object is the property which does not allow to neglect the influence of the measuring devices. The fruitfulness of the deterministic point of view in the classical mechanics give rise to the illusion of its universally observed facts¹². However with the penetration of experimental physics into the atomic domain the inapplicability of the classical approach and the relevance of the statistical concepts in this domain became more evident. The behavior of the atomic and subatomic objects is essentially probabilistic. A quantum version of entropy has been developed by Von Neumann¹⁶. In the present study, we have examined the conjectures of Bohr³ and Von Neumann¹⁶ using the theory of harilially ordered vector spaces. Mackey¹² and Lolmogorov¹¹ developed axiomatic approach related to probability distri-butions and operator theory in Hilbert⁷ spaces. Von-Neumann¹⁶ derived some significant results on quantum logic approach to the foundations of quantum theory.