Abstract
In this chapter, the calculus of effective (intuitionistic) quantum logic will be established. Starting from the observation that there are some material propositions which can be defended within the material dialog-game D* m irrespective of the elementary propositions contained in it, the concept of a formally true proposition will be defined in Section 4.1 by the respective argument rules. Eliminating elementary propositions but not commensurabilities in D m the semi-formal quantum dialog-game D f (m) is formulated in Section 4.2. Following this, the material availability propositions are also eliminated, thus achieving the formal quantum dialog-game D f (Section 4.3). The totality of propositions which can be defended in this formal dialog-game is comprehended in the calculus Q eff of effective quantum logic, which is established in Section 4.4. This calculus Q eff can be shown to be consistent and complete with respect to those propositions which can be defended in a formal dialog.
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