Inadequacy of Modal Logic in Quantum Settings

TL;DR

This paper investigates the limitations of classical modal logic in quantum contexts, revealing fundamental incompatibilities and proposing new structures to better model quantum reasoning and trust among agents.

Contribution

It identifies the breakdown of classical modal logic axioms in quantum settings, clarifies the Frauchiger-Renner theorem's assumptions, and introduces trust and context constructs for quantum reasoning.

Findings

Classical modal logic fails in quantum scenarios.

A missing unitarity assumption affects the Frauchiger-Renner theorem.

New trust and context structures are proposed for quantum reasoning.

Abstract

We test the principles of classical modal logic in fully quantum settings. Modal logic models our reasoning in multi-agent problems, and allows us to solve puzzles like the muddy children paradox. The Frauchiger-Renner thought experiment highlighted fundamental problems in applying classical reasoning when quantum agents are involved; we take it as a guiding example to test the axioms of classical modal logic. In doing so, we find a problem in the original formulation of the Frauchiger-Renner theorem: a missing assumption about unitarity of evolution is necessary to derive a contradiction and prove the theorem. Adding this assumption clarifies how different interpretations of quantum theory fit in, i.e., which properties they violate. Finally, we show how most of the axioms of classical modal logic break down in quantum settings, and attempt to generalize them. Namely, we introduce constructions of trust and context, which highlight the importance of an exact structure of trust relations between agents. We propose a challenge to the community: to find conditions for the validity of trust relations, strong enough to exorcise the paradox and weak enough to still recover classical logic.