Rejection of divisibility principle in quantum logic with respect to uncertainty principle and EPR paradox

From the time physicists have proposed the quantum logic, this logic is formed somehow in relation with quantum mechanics and experiences based on it. In fact, quantum mechanics and experiences gained from it assumed an approval to this logic. One of the highlights of the quantum mechanics, is uncertainty principle, which is a doctrine to reject the divisibility in quantum logic. Also EPR is assumed as a doctrine to reject the quantum mechanics. In the case of rejection of quantum mechanics, does the quantum logic also be questioned? In this paper it is shown that the uncertainty principle is rejecting the divisibility principle and EPR The aim of this study is to show that the uncertainty principal rejects divisibility principal, and hidden-variable theory which comes after EPR paradox, known as a rival theory, (even in the case of rejection of standard quantum mechanics) doesn’t reject the quantum logic. The outcome of this is that in practice quantum logic is independent of quantum mechanics, and it might be applied in areas other than quantum mechanics.