In quantum mechanics, there are many counter-intuitive phenomena comparing with classical mechanics. In this book, we consider the two methods to compare the classical and the quantum systems. One is to quantitatively characterize the counter-factual phenomena in quantum mechanics related to the classical logic. We show that the weak value, which was initiated and developed by Yakir Aharonov and his colleagues and is an experimental accessible quantity, can characterize them by a simple example. We propose that the weak value is interpreted as a fundamental quantity to reformulate the quantum...