Quantum logic and physical modalities

Peter Mittelstaedt

Olímpia Lombardi, Matías Daniel Pasqualini, Juan Pablo Jorge +1 more

George Svetlichny

Kenji Tokuo

Abstract A modal logic based on quantum logic is formalized in its simplest possible form. Specifically, a relational semantics and a sequent calculus are provided, and the soundness and the completeness theorems connecting both notions are demonstrated. This framework is intended to serve as a basis for formalizing various modal logics over quantum logic, such as quantum alethic logic, quantum temporal logic, quantum epistemic logic, and quantum dynamic logic.

Osvaldo Frota Pessoa Junior

The question of which is the logic that underlies quantum physics does not have an absolute answer, but only in relation to a conventional choice of interpretation (in the sense used in the study of scientific theories). Most of the interpretations that have been offered work within the framework of classical logic. In contrast to these, we examine the corpuscular interpretation which is assumed in the application of non-distributive logic (section 3). The experiment in which single photons pass through a Mach-Zehnder interferometer is examined, indicating the difficulty of employing a realist corpuscular interpretation in this context. One way to save this interpretation would be to use non-distributive logic to analyze the experiment, but this is not satisfactory (section 4). However, the use of an alethic modal logic solves the problem, blocking the argument that put the aforementioned corpuscular interpretation into difficulty. In the discussion of the conceptual problems involved, we suggest that a stochastic corpuscular interpretation is well adapted to this logical description. The project of extending this modal logical approach to other experiments in quantum physics, and of providing a rigorous logical treatment, is left open.

Martin Strauss