Quantum logic, Hilbert space, revision theory

Sylvia Pulmannová

David W. Cohen

David W. Cohen, George Svetlichny

José Alejandro Fernández Cuesta, Juan Pablo Jorge

Abstract Standard quantum logics are typically defined over the orthomodular lattice associated with Hilbert space, yet there is no consensus on how this structure should be interpreted logically or ontologically. This paper offers a unified perspective on these issues by combining an epistemological three-layer model for scientific theories with quantum set theory. We argue that quasi-set theory is best understood not as an alternative semantics for quantum logics, but as an ontological framework operating at a different epistemological level: the ontological one. From this standpoint, the distinctive non-classical features of quantum logic are explained by representational assumptions about identity and individuality, rather than by positing any direct revisions of the logical principles themselves.

Allen Stairs

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.