The modal logic of quantum logic

Abstract

Modal logic is concerned with the concepts of necessity and possibility and a certain class of object propositions. In this paper we develop the basic concepts of a modal logic which is related to propositions about quantum physical objects. Since the object logic of quantum mechanical propositions is given by the calculi of quantum logic, the structure investigated in this paper will be called the modal logic of quantum logic. The object language and logic of quantum physical propositions is developed here within the dialogic approach to quantum logic [1]. On the basis of these object-linguistic structures we investigate the language of meta-propositions which state the material or formal truth of objectpropositions. Applying again the dialogic technique to meta-propositions the important notion of a formally true meta-proposition can be defined. Using this concept it turns out that the formal logic of meta-propositions is equivalent to the corresponding structure in ordinary logic, i.e., to the effective (intuitionistic) logic. The modalities "necessary" and "possible" are introduced here in the framework of a meta-linguistic interpretation, which considers the modalities as statements about the object-propositions under discussion [9]. On this basis it is found that meta-propositions which state the material truth of a special class of object propositions may be considered as quantum logical modalities (Section 2). A detailed investigation then shows that these quantum logical modalities are intimately related to the important quantum mechanical concepts of commensurability and objectivity. An illustration of the quantum logical modalities by relations between projection operators in Hilbert space concludes this part. In the third section we introduce the concept of a formally true modality which leads to the modal logic of quantum logic. It is shown that this logic of quantum logical modalities contains all the quantum logical restrictions which come from the possible incommensurability of quantum physical