The Logic of Quantum Mechanics
1936/10/01 by Garrett Birkhoff, John Von Neumann, John von Neumann · 2,890 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebra over a field #Logic, Reasoning, and Knowledge #Mathematics #Physics #Pure mathematics #Quantum #Quantum Mechanics and Applications #Quantum computer #Quantum logic #Quantum mechanics
paper · doi:10.2307/1968621
published in Annals of Mathematics 37(4), 823 (Princeton University)
openalex publication_date 1936/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Abstract
One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations).
Cited by
- Quantum Mechanics Based on Real Numbers: A Consistent Description
- A Philosopher Looks at Quantum Mechanics (Again)
- Non-signalling Theories and Generalized Probability
- New scattering features of quaternionic point interaction in non-Hermitian quantum mechanics
- A state-space approach to quantum permutations
- The Logic of Dynamical Systems Is Relevant
- Explanation, Evolution and Subjective Probability in Everett Quantum Mechanics with Positive Preclusion
- Slice regular Besov spaces of hyperholomorphic functions and composition operators
- Is there life beyond Quantum Mechanics?
- Categorical Foundations of Quantum Logics and Their Truth Values Structures
- An algebraic theory of infinite classical lattices III: Theory of single measurements
- No master (key) No (measurement) problem
- Quantum mechanics without measurements
- A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations
- Functions of the infinitesimal generator of a strongly continuous quaternionic group
- Decoherence, einselection, and the quantum origins of the classical
- The Algebra of the Pseudo-Observables I: Why Quantum Mechanics is the ultimate description of Reality
- Reformulating and Reconstructing Quantum Theory
- Do quantum propositions obey the principle of excluded middle?
- Quantum theory and functional analysis
- Computational Complexity and the Nature of Quantum Mechanics
- Weakening the Born Rule - Towards a Stateless Formulation of Quantum Mechanics
- Complementary Observables and Non-Boolean Logic Outside Quantum Physics
- Bohrification of local nets of observables
- Mathematical Foundations of Quantum Mechanics: An Advanced Short Course
- The Minimal Modal Interpretation of Quantum Theory
- Holistic type extension for classical propositional logic in quantum computation
- Hilbert Space Quantum Mechanics Is Noncontextual
- On the Nature of Continuous Physical Quantities in Classical and Quantum Mechanics
- Eutactic quantum codes
- Quantum Mechanics interpreted in Quantum Real Numbers
- Automated Quantum Protocol Verification Based on Concurrent Dynamic Quantum Logic
- Logical Interpretation of a Reversible Measurement in Quantum Computing
- To What Type of Logic Does the "Tetralemma" Belong?
- Probabilities from Entanglement, Born's Rule from Envariance
- The Nine Lives of Schroedinger's Cat
- A topos for algebraic quantum theory
- Kindergarden quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus)
- Equal cost of computation for truth and falsity of experimental quantum propositions necessitates quantum parallel computing
- Logic Column 18: Alternative Logics: A Book Review
- Dynamic Logic of Quantum Field Theory
- Measuring Measuring
- Orthologics for Cones
- Meaning updating of density matrices
- Bohrification: From classical concepts to commutative algebras
- On the assumptions underlying KS-like contradictions
- Effectuses in Categorical Quantum Foundations
- Alternation in Quantum Programming: From Superposition of Data to Superposition of Programs
- О свойствах вещественной квантовой механики в пространстве Кэлера
- Epistemic nature of quantum reasoning
- Quantum logic. A brief outline
- Model Checking Applied to Quantum Physics
- Distribution can be dropped: Reply to Rumfitt
- Contextuality, Witness of Quantum Weirdness
- Algebraic hierarchy of logics unifying fuzzy logic and quantum logic
- Intrinsic and extrinsic properties of quantum systems
- Why Super-Quantum, Non-Signaling Correlations Cannot Exist
- The Significance of Information in Quantum Theory
- Neutral monism, perspectivism and the quantum dualism: An essay
- A Possible Mathematical Structure for Physics
- Non-Commutative Topology for Curved Quantum Causality
- Future Boundaries and the Black Hole Information Paradox
- The Logic of Consistent Histories: A Reply to Maudlin
- General Covariance in Algebraic Quantum Field Theory
- Probability over Plonka sums of Boolean algebras: states, metrics and topology
- Non-distributive logics: from semantics to meaning
- Test sets for tautologies in modular quantum logic
- From Quantum Metalanguage to the Logic of Qubits
- Classical limit of quantum propositions
- Dagger and Dilation in the Category of Von Neumann algebras
- Descriptive and Foundational Aspects of Quantum Cognition
- A Note on the "Third Life of Quantum Logic"
- On the Nature of Reality
- The spectral theorem for unitary operators based on the S-spectrum
- No-cloning implies unalterability of the past
- Toward a More Natural Expression of Quantum Logic with Boolean Fractions
- Towards a skew lattice approach to quantum (computational) logic
- QL(Cn) determines n
- QECV: Quantum Error Correction Verification
- What do we learn from computer simulations of Bell experiments?
- Consistent Quantum Measurements
- Topos Theoretical Reference Frames on the Category of Quantum Observables
- E. Specker: "The logic of non-simultaneously decidable propositions" (1960)
- The Conformal Universe III: Basic Mechanisms of Matter Generation
- Eigenlogic: a Quantum View for Multiple-Valued and Fuzzy Systems
- Quantum logics close to Boolean algebras
- How to play macroscopic quantum game
- Concrete Foundations for Categorical Quantum Physics
- "Haunted" quantum contextuality
- Operational General Relativity: Possibilistic, Probabilistic, and Quantum
- Reforming Takeuti's Quantum Set Theory to Satisfy De Morgan's Laws
- Sheaf-Theoretic Methods in Quantum Mechanics and Quantum Information Theory
- Quantum-Like Uncertain Conditionals for Text Analysis
- The Entropic Dynamics of Relativistic Quantum Fields in Curved Space-time
- The H^∞ functional calculus based on the S-spectrum for quaternionic operators and for n-tuples of noncommuting operators
- Probabilistic logic of quantum observations
- Combining Bohm and Everett: Axiomatics for a Standalone Quantum Mechanics
- Algebraic assignments of truth values to experimental quantum propositions
- Modal logic approach to preferred bases in the quantum universe
- Bell-type inequalities for bivariate maps on orthomodular lattices
- Fields and Quantum Mechanics
- Editors’ Introduction: The Third Life of Quantum Logic: Quantum Logic Inspired by Quantum Computing
- The C*-Algebras of Three-Valued Logic
- Superlogic Manifolds and Geometric approach to Quantum Logic
- Quantum Temporal Logic
- Discrimination against or in favor of qubits in quantum theory
- From the problem of Future Contingents to Peres-Mermin square experiments: An introductory review to Contextuality
- Squeezed states in the quaternionic setting
- Generalised Proof-Nets for Compact Categories with Biproducts
- Algebraic aspects of quantum indiscernibility
- Non-associativity, supersymmetry and hidden variables
- Quantum Mechanics: From Realism to Intuitionism
- Chu spaces: automata with quantum aspects
- De-linearizing Linearity: Projective Quantum Axiomatics from Strong Compact Closure
- Between classical and quantum
- Understanding Quantum Mechanics Through the Complementary Descriptions Approach
- Basic logic and quantum entanglement
- Compatible Quantum Theory
- Value-indefinite observables are almost everywhere
- Model-Theoretic Investigations into Consequence Operation (Cn) in Quantum Logics: An Algebraic Approach
- Complex Projection of Quasianti-Hermitian Quaternionic Hamiltonian Dynamics
- Relativistic quaternionic wave equation
- Quantum scholasticism: On quantum contexts, counterfactuals, and the absurdities of quantum omniscience
- Probabilistic theories with purification
- Relating toy models of quantum computation: comprehension,\n complementarity and dagger mix autonomous categories
- Sections in orthomodular structures of decompositions
- Quantum logic and the histories approach to quantum theory
- Quantum Mechanics Unscrambled
- From classical to quantum mechanics: “How to translate physical ideas into mathematical language”
- Logical independence and quantum randomness
- Floyd--hoare logic for quantum programs
- Contextuality, Fine-Tuning, and Teleological Explanation
- Quantum information-flow, concretely, and axiomatically
- Discrete quantum structures
- Conformal Invariance and Electrodynamics: Applications and General Formalism
- A Dynamic-Logical Perspective on Quantum Behavior
- Modal‐type orthomodular logic
- Survey of an approach to quantum measurement, classical properties and realist interpretation problems
- Operational Meanings of Orders of Observables Defined through Quantum Set Theories with Different Conditionals
- Confíe en mí, soy científico
- How probability arises in quantum mechanics
- Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead)
- The Neumann Compendium
- “unsolved problems in mathematics” J. von Neumann’s address to the international congress of mathematicians, amsterdam, september 2-9, 1954
- John von Neumann 1903-1957
- Compendium of Quantum Physics
- Quantum picturalism
- CONTINUOUS SLICE FUNCTIONAL CALCULUS IN QUATERNIONIC HILBERT SPACES
- Spectral representations of normal operators via Intertwining Quaternionic Projection Valued Measures
- Foundations of Quaternion Quantum Mechanics
- Quantum temporal logic and decoherence functionals in the histories approach to generalized quantum theory
- Quantum theory and determinism
- George David Birkhoff and John von Neumann: A Question of Priority and the Ergodic Theorems, 1931–1932
- Foundations for Near-Term Quantum Natural Language Processing
- Conformal partial waves and the operator product expansion
- The internal logic of Bell's states
- Introduction to the book "Quantum Theory: Informational Foundations and Foils"
- Quantum Hoare Type Theory
- Classical and quantum probability
- Quantum logic revisited
- No-cloning theorem on quantum logics
- A Topos Model for Loop Quantum Gravity
- A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity
- On regular ^*-algebras of bounded linear operators: A new approach towards a theory of noncommutative Boolean algebras
- On quantum vs. classical probability
- Quantum causality, stochastics, trajectories and information
- Exotic smooth R4, noncommutative algebras and quantization
- Epistemic Horizons and the Foundations of Quantum Mechanics
- Physical Properties as Modal Operators in the Topos Approach to Quantum Mechanics
- A Presentation of Quantum Logic Based on an and then Connective
- Quantum Information: What Is It All About?
- Partial and unsharp quantum logics
- How device-independent approaches change the meaning of physical theory
- A quantum logical and geometrical approach to the study of improper mixtures
- Towards probing for hypercomplex quantum mechanics in a waveguide interferometer
- On the Representation of Boolean and Real Functions as Hamiltonians for Quantum Computing
- Constraining the state space in any physical theory with the principle of information symmetry
- Interpolation Methods for Binary and Multivalued Logical Quantum Gate Synthesis
- Language Heedless of Logic - Philosophy Mindful of What? Failures of Distributive and Absorption Laws
- Orthogonal Vector Computations
- The Logos Categorical Approach to Quantum Mechanics: I. Kochen-Specker Contextuality and Global Intensive Valuations
- Quantization via Linear homotopy types
- Analysis of Wallace’s Proof of the Born Rule in Everettian Quantum Mechanics II: Concepts and Axioms
- The complex and quaternionic quantum bit from relativity of simultaneity on an interferometer
- Quantum state change in light of changes in valuational entropies
- The Born rule and logical inference in quantum mechanics
- Interpreting Quantum Logic as a Pragmatic Structure
- Topos Theory and ‘Neo-Realist’ Quantum Theory
- Choice of consistent family, and quantum incompatibility
- Duality (projective geometry) [wikipedia]
- John von Neumann [wikipedia]
- Quantum logic [wikipedia]
- Sesquilinear form [wikipedia]
- Timeline of quantum mechanics [wikipedia]
- Detecting event-related recurrences by symbolic analysis: applications to human language processing. [europepmc]
- Quantum-like behavior without quantum physics I : Kinematics of neural-like systems. [europepmc]
- The complex and quaternionic quantum bit from relativity of simultaneity on an interferometer. [europepmc]
- Cognitive Structures of Space-Time. [europepmc]
- Developments in Quantum Probability and the Copenhagen Approach. [europepmc]
- Probabilities and Epistemic Operations in the Logics of Quantum Computation. [europepmc]
- Closing the Door on Quantum Nonlocality. [europepmc]
- Logical Structures Underlying Quantum Computing. [europepmc]
- Adapting Logic to Physics: The Quantum-Like Eigenlogic Program. [europepmc]
- What Is So Special about Quantum Clicks? [europepmc]
- Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics. [europepmc]
- Foundations of the Quaternion Quantum Mechanics. [europepmc]
- Kolmogorovian versus Non-Kolmogorovian Probabilities in Contextual Theories. [europepmc]
- Generalized Greenberger-Horne-Zeilinger Arguments from Quantum Logical Analysis. [europepmc]
- Quantum Epistemology and Falsification. [europepmc]
- Transitivity and homogeneity of orthosets and inner-product spaces over subfields of R. [europepmc]
- Extending Kolmogorov's Axioms for a Generalized Probability Theory on Collections of Contexts. [europepmc]
- On the Complete Description of Entangled Systems Part II: The (Meta)Physical Status and Semantic Aspects. [europepmc]
- A third transition in science? [europepmc]
- Quantum Circuit Components for Cognitive Decision-Making. [europepmc]
- Open Systems, Quantum Probability, and Logic for Quantum-like Modeling in Biology, Cognition, and Decision-Making. [europepmc]
- Rényi Entropy, Signed Probabilities, and the Qubit. [europepmc]
- 'The agency of observation not to be neglected': complementarity, causality and the arrow of events in quantum and quantum-like theories. [europepmc]
- A New Logic, a New Information Measure, and a New Information-Based Approach to Interpreting Quantum Mechanics. [europepmc]
- Symbolic model checking quantum circuits in Maude. [europepmc]
Related