The Calculus of Full Quantum Logic

Abstract

In this chapter, we establish the calculus Q of full quantum logic which is obtained from Q _eff by addition of the principle of excluded middle V ⩽ A v ⌉A for all propositions A. In Section 6.1, we investigate the value-definiteness of material propositions. Starting from the value-definiteness of elementary propositions we show that the availability propositions k and $\bar k$ are, in general, not value-definite. In Section 6.2, we demonstrate that in contrast to ordinary effective logic the value-definiteness of the elementary propositions in Q _eff is not inherited by finitely compound propositions. It turns out that the reason for the missing principle of excluded middle for compound propositions is the lack of value-definiteness of the commensurability propositions k and $\bar k$ (Section 6.3). If, however, by a rather weak assumption concerning the measurability of k and $\bar k$, the value-definiteness of the availability propositions is re-established, it follows that the principle of excluded middle which is valid for elementary propositions, is, in fact, inherited by all finitely compound propositions. In this way, the calculus Q of full quantum logic can be justified (Section 6.4).