Operational meaning of the lattice meet and join of non-compatible experimental propositions, and a physical motivation for the modular law, in the Birkhoff–von Neumann quantum-logic program

In their 1936 founding paper of quantum logic, Birkhoff and von Neumann proposed to read the lattice of closed linear subspaces of Hilbert space as a calculus of experimental propositions, with meet as conjunction, join as disjunction, and orthocomplement as negation. They conceded that this rested on 'admittedly heuristic arguments' and posed two open questions: (1) what, physically, is the meet and what is the join of two experimental propositions as experimental propositions, given that the classical recipe of having independent observers read off both measurements and combining the results logically works in quantum mechanics 'only exceptionally – only when all the measurements involved commute'; and (2) what simple, physically plausible motivation can be given for the modular identity as a law of experimental propositions. Modularity itself fails for the closed-subspace lattice of infinite-dimensional Hilbert space, so the modern literature directs this demand at the orthomodular law. Both halves of the question remain open.

Background

A lattice is a partially ordered set in which every pair of elements $a, b$ has a greatest lower bound, called the meet ($a \wedge b$), and a least upper bound, called the join ($a \vee b$). In classical physics the propositions about a system form a Boolean algebra: a distributive lattice, in which conjunction is the meet, disjunction is the join, and negation is a complement. Birkhoff and von Neumann (1936) proposed that in quantum mechanics the propositions of a system are instead represented by the closed linear subspaces of a Hilbert space — intuitively, the propositions that a state lies in a given subspace, which are the closed ranges of the projection operators (the self-adjoint operators $P$ satisfying $P^2 = P$). This family of subspaces forms a lattice: the meet of two subspaces is their set-theoretic intersection, and the join is the closure of their linear span (the smallest closed subspace containing both). Orthogonality supplies a complement, $a^\perp$. An experimental proposition is a proposition whose truth can be settled by performing a specific measurement (a yes–no experiment). For two such propositions that are compatible — represented by commuting projections, equivalently simultaneously measurable — one can measure both and combine the outcomes with ordinary logic: this is the classical recipe, and it exactly reproduces meet and join. For non-compatible (non-commuting) propositions no joint measurement exists, and measuring one generally disturbs the statistics of the other, so the recipe fails: this is the content of question (1). Two further lattice identities organize the subject. The distributive identities, e.g. $a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)$, hold in every Boolean algebra but fail for subspace lattices of dimension at least 2; Birkhoff and von Neumann identified this failure, not any defect of negation, as the point where the logic of quantum mechanics departs from classical logic. The weaker modular identity, $a \subseteq c \Rightarrow a \vee (b \wedge c) = (a \vee b) \wedge c$, holds for the subspace lattice exactly when the Hilbert space is finite-dimensional, a limitation already noted in the 1936 paper; its further weakening, the orthomodular law $a \subseteq b \Rightarrow b = a \vee (b \wedge a^\perp)$, holds for the closed-subspace lattice of every Hilbert space and became the standard algebraic axiom of quantum logic. The demand, posed as question (2), for a simple physical motivation of such identities — rather than their derivation from the Hilbert-space model, which would be circular as a foundation — is the seed of the entire program of operational axiomatizations of quantum mechanics (Mackey's axioms on questions and states, the Geneva school's reconstruction from properties and states, and the later test-space and effect-algebra frameworks, which organize collections of yes–no experiments and model unsharp, measurement-outcome-based propositions), and of the representation theorems of Piron and Solèr that characterize when operationally defined structures force the standard Hilbert-space lattice.

Problem Statement

The question posed by the 1936 paper, restated in modern terms and faithful to its formulation: (1) Give an operational interpretation of the lattice meet $a \wedge b$ and join $a \vee b$ of two experimental propositions $a, b$ of a quantum-mechanical system — that is, specify an experiment or experimental procedure whose verification condition is exactly $a \wedge b$ (respectively $a \vee b$) for arbitrary, in particular non-compatible, propositions represented by closed subspaces of Hilbert space, the procedure being composed from the given yes–no experiments for $a$ and $b$ rather than by measuring the projection onto $a \wedge b$ (respectively $a \vee b$) itself — or prove that no such interpretation exists. The classical recipe of independent observers reading off both measurements and combining the results logically works only when the measurements commute, so a quantum recipe must be essentially new. Any interpretation proposed under the first alternative must reproduce the lattice operations on all pairs (not only compatible ones), coincide with the classical recipe in the commutative case, and be consistent with the quantum probability calculus (states); a negative answer must show that no interpretation meeting these three requirements exists. (2) Provide a simple, physically plausible motivation for the modular identity $a \subseteq c \Rightarrow a \vee (b \wedge c) = (a \vee b) \wedge c$ as a law governing experimental propositions — in its modern transposition, for the orthomodular law $a \subseteq b \Rightarrow b = a \vee (b \wedge a^\perp)$ and ultimately for the Hilbert-space structure of the lattice, since modularity itself fails for the closed-subspace lattice of infinite-dimensional Hilbert space — as a short list of operationally testable axioms from which the modular law in the finite-dimensional setting and the orthomodular law in general are deduced rather than postulated, together with an indication, though not necessarily a proof, of how the axioms yield the full Hilbert-space lattice.

References

1. G. Birkhoff and J. von Neumann, "The Logic of Quantum Mechanics", Annals of Mathematics 37(4):823–843 (1936). DOI: 10.2307/1968621, https://doi.org/10.2307/1968621
2. S. Kochen and E. P. Specker, "The Problem of Hidden Variables in Quantum Mechanics", Indiana University Mathematics Journal 17(1):59–87 (1967). DOI: 10.1512/iumj.1968.17.17004, https://doi.org/10.1512/iumj.1968.17.17004
3. H. Putnam, "Is Logic Empirical?" (originally published 1968), reprinted in C. A. Hooker (ed.), The Logico-Algebraic Approach to Quantum Mechanics, pp. 181–206, Reidel (1979). DOI: 10.1007/978-94-009-9351-8_9, https://doi.org/10.1007/978-94-009-9351-8_9
4. M. Friedman, "Quantum Logic, Conditional Probability, and Interference", Dialectica 32(3/4):305–315 (1978). DOI: 10.1111/j.1746-8361.1978.tb01319.x, https://doi.org/10.1111/j.1746-8361.1978.tb01319.x
5. J. M. Jauch and C. Piron, "On the Structure of Quantal Proposition Systems", in C. A. Hooker (ed.), The Logico-Algebraic Approach to Quantum Mechanics, pp. 427–436, Reidel (1975). DOI: 10.1007/978-94-010-1795-4_24, https://doi.org/10.1007/978-94-010-1795-4_24
6. C. Piron, "On the Foundations of Quantum Physics", in M. Flato et al. (eds.), Quantum Mechanics, Determinism, Causality, and Particles, pp. 105–116, Reidel (1976). DOI: 10.1007/978-94-010-1440-3_7, https://doi.org/10.1007/978-94-010-1440-3_7
7. C. H. Randall and D. J. Foulis, "Properties and Operational Propositions in Quantum Mechanics", Foundations of Physics 13(8):843–857 (1983). DOI: 10.1007/BF01906272, https://doi.org/10.1007/BF01906272
8. F. Herbut, "On a Possible Empirical Meaning of Meets and Joins for Quantum Propositions", Letters in Mathematical Physics 8(5):397–402 (1984). DOI: 10.1007/BF00418115, https://doi.org/10.1007/BF00418115
9. D. J. Foulis and M. K. Bennett, "Effect Algebras and Unsharp Quantum Logics", Foundations of Physics 24(10):1331–1352 (1994). DOI: 10.1007/BF02283036, https://doi.org/10.1007/BF02283036
10. M. P. Solèr, "Characterization of Hilbert Spaces by Orthomodular Spaces", Communications in Algebra 23(1):219–243 (1995). DOI: 10.1080/00927879508825218, https://doi.org/10.1080/00927879508825218
11. S. S. Holland Jr., "Orthomodularity in Infinite Dimensions; a Theorem of M. Solèr", Bulletin of the American Mathematical Society 32(2):205–234 (1995). DOI: 10.1090/s0273-0979-1995-00593-8, arXiv: math/9504224, https://doi.org/10.1090/s0273-0979-1995-00593-8
12. S. Smets, "The Modes of Physical Properties in the Logical Foundations of Physics", Logic and Logical Philosophy 14:37–53 (2005). DOI: 10.12775/llp.2005.004, https://doi.org/10.12775/llp.2005.004
13. A. Baltag and S. Smets, "Quantum Logic as a Dynamic Logic", Synthese 179(2):285–306 (2011). DOI: 10.1007/s11229-010-9783-6, https://doi.org/10.1007/s11229-010-9783-6
14. J. Pykacz and P. Frąckiewicz, "The Problem of Conjunction and Disjunction in Quantum Logics", International Journal of Theoretical Physics 56(12):3963–3970 (2017). DOI: 10.1007/s10773-017-3402-y, https://doi.org/10.1007/s10773-017-3402-y
15. P. Kazakopoulos and G. Regkas, "Infinite Permutation Groups and the Origin of Quantum Mechanics", preprint (2023). arXiv:2307.13044, https://arxiv.org/abs/2307.13044
16. I. Chajda and H. Länger, "Derived Operators on Skew Orthomodular and Strong Skew Orthomodular Posets", preprint (2025). arXiv:2502.01266, https://arxiv.org/abs/2502.01266
17. H. Emori, A. Iriiki, A. Khrennikov, and K. Kondo, "Quantum Logic as the Logic of Contexts", preprint (2026). arXiv:2607.09032, https://arxiv.org/abs/2607.09032
18. A. Wilce, "Quantum Logic and Probability Theory", in E. N. Zalta and U. Nodelman (eds.), The Stanford Encyclopedia of Philosophy, substantive revision of 10 August 2021. https://plato.stanford.edu/entries/qt-quantlog/