The logic of Quantum Mechanics - Revisited: From quantum to classical to intuitionistic to classical

Bob Coecke

Rosella Hermens

Within the many versions of the Copenhagen interpretation numerous hints can be found about how to reason about quantum mechanics. However, strict guidelines are often missing. The standard quantum logic is sometimes opted as providing such a guideline but, in itself, it is again puzzling due to the failure of the law of distributivity. In this paper I take an instrumentalist approach inspired by the Copenhagen interpretation towards the interpretation of propositions in a physical theory. Starting with so-called elementary propositions for a physical theory I introduce logical connectives in accordance with this approach and relate the thereby expanding set of propositions to mathematical objects within the theory in a consistent way. This is first done for classical mechanics and then for quantum mechanics. In both cases this approach leads to the development of intuitionistic logics rather than classical ones. It also follows that the obtained intuitionistic quantum logic is a proper generalization of the obtained intuitionist logic for classical mechanics. Finally, some hints are provided in how these logics are to be used in practical situations.

Stanley Gudder

Heng Zhou, Yongjun Wang, Baoshan Wang +2 more

The classical belief revision framework, as proposed by Alchourron, Gardenfors, and Makinson, involves the revision of a theory based on eight postulates. In this paper, we focus on the exploration of a revision theory grounded in quantum mechanics, referred to as the natural revision theory. There are two reasoning modes in quantum systems: static intuitionistic reasoning, which incorporates contextuality, and dynamic reasoning, which is achieved through projection measurement. We combine the advantages of two intuitionistic quantum logic frameworks, as proposed by D{ö}ring and Coecke, respectively. Our goal is to establish a truth-value assignment for intuitionistic quantum logic that not only aligns with the inherent characteristics of quantum mechanics but also supports truth-value reasoning. The natural revision theory is then investigated based on this approach. We introduce two types of revision operators that correspond to the two reasoning modes in quantum systems: static and dynamic revision. Furthermore, we highlight the distinctions between these two operators. Shifting away from classical revision paradigms, we consider the revision of consequence relations in intuitionistic quantum logic. We demonstrate how, within the natural revision theory framework, both revision operators collectively influence the consequence relations. Notably, the outcomes of revision process are impacted by the sequence in which these interweaved operators are deployed.

Mituo Taketani, Masayuki Nagasaki

Jerome Manyahi