Do quantum propositions obey the principle of excluded middle?

TL;DR

This paper investigates the failure of the principle of excluded middle in quantum logic, specifically in the lattice of subspaces of a Hilbert space, challenging traditional assumptions about quantum propositions.

Contribution

It demonstrates that PEM does not hold in quantum logic for qubits, raising questions about the semantics and interpretation of quantum propositions.

Findings

PEM fails in the lattice of quantum propositions for qubits

Quantum propositions can both be false, contradicting classical logic

The paper discusses implications for quantum semantics

Abstract

The present paper demonstrates the failure of the principle of excluded middle (PEM) in the lattice of all closed linear subspaces of a Hilbert space (usually defined as quantum logic). Namely, it is shown that for a qubit, a proposition and its negation can be both false. Since PEM is the assumed theorem of quantum logic, this raises the question: If PEM holds in the orthocomplemented lattice of all propositions of the quantum system, then how the failure of PEM in quantum logic can be explained? Alternatively, if the propositions relating to the quantum system do not obey PEM, then what is the semantics of those propositions? Possible answers to these questions are analyzed in the present paper.