Quantifiers for quantum logic
Chris Heunen

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.
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Taxonomy
TopicsAdvanced Algebra and Logic · Logic, Reasoning, and Knowledge · Logic, programming, and type systems
