Quantifiers for quantum logic

TL;DR

This paper explores the categorical logic framework for quantum logic within Hilbert spaces, characterizing subobjects and establishing the existence of an existential quantifier while proving the non-existence of a universal quantifier.

Contribution

It introduces a categorical logic approach to quantum logic, characterizes subobjects as orthomodular lattices, and defines an existential quantifier specific to quantum logic.

Findings

Closed subobjects form orthomodular lattices

Existential quantifier exists for quantum logic

Universal quantifier cannot exist in this framework

Abstract

We consider categorical logic on the category of Hilbert spaces. More generally, in fact, any pre-Hilbert category suffices. We characterise closed subobjects, and prove that they form orthomodular lattices. This shows that quantum logic is just an incarnation of categorical logic, enabling us to establish an existential quantifier for quantum logic, and conclude that there cannot be a universal quantifier.