Foundations of Science

, Volume 11, Issue 4, pp 399–418 | Cite as

The Generalised Liar Paradox: A Quantum Model and Interpretation



The formalism of abstracted quantum mechanics is applied in a model of the generalized Liar Paradox. Here, the Liar Paradox, a consistently testable configuration of logical truth properties, is considered a dynamic conceptual entity in the cognitive sphere (Aerts, Broekaert, & Smets, [Foundations of Science 1999, 4, 115–132; International Journal of Theoretical Physics, 2000, 38, 3231–3239]; Aerts and colleagues[Dialogue in Psychology, 1999, 10; Proceedings of Fundamental Approachs to Consciousness, Tokyo ’99; Mind in Interaction]. Basically, the intrinsic contextuality of the truth-value of the Liar Paradox is appropriately covered by the abstracted quantum mechanical approach. The formal details of the model are explicited here for the generalized case. We prove the possibility of constructing a quantum model of the m-sentence generalizations of the Liar Paradox. This includes (i) the truth–falsehood state of the m-Liar Paradox can be represented by an embedded 2m-dimensional quantum vector in a (2m) m -dimensional complex Hilbert space, with cognitive interactions corresponding to projections, (ii) the construction of a continuous ‘time’ dynamics is possible: typical truth and falsehood value oscillations are described by Schrödinger evolution, (iii) Kirchoff and von Neumann axioms are satisfied by introduction of ‘truth-value by inference’ projectors, (iv) time invariance of unmeasured state.


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  1. Aerts, D. 1982Description of Many Physical Entities Without the Paradoxes Encountered in Quantum MechanicsFoundations of Physics1211311170CrossRefGoogle Scholar
  2. Aerts, D. 1983aClassical Theories and Non Classical Theories as a Special Case of a More General TheoryJournal of Mathematical Physics2424412453CrossRefGoogle Scholar
  3. Aerts, D.: 1983b, The Description of One and Many Physical Systems. In C. Gruber (eds.), Foundations of Quantum Mechanics. Lausanne: A.V.C.P.63–148Google Scholar
  4. Aerts, D. 1986A Possible Explanation for the Probabilities of Quantum MechanicsJournal of Mathematical Physics27202210CrossRefGoogle Scholar
  5. Aerts, D. 1992The Construction of Reality and its Influence on the Understanding of Quantum StructuresInternational Journal of Theoretical Physics3118151837CrossRefGoogle Scholar
  6. Aerts, D. 1994Quantum Structures, Separated Physical Entities and ProbabilityFoundations of Physics2412271259CrossRefGoogle Scholar
  7. Aerts, D. 1999Foundations of Quantum Physics: A General Realistic and Operational ApproachInternational Journal of Theoretical Physics38289358CrossRefGoogle Scholar
  8. Aerts, D., J. Broekaert and L. Gabora (1999). Nonclassical Contextuality in Cognition: Borrowing From Quantum Mechanical Approaches to Indeterminism and Observer Dependence, Dialogues in Psychology 10.Google Scholar
  9. Aerts, D., J. Broekaert and L. Gabora (2000). Intrinsic Contextuality as the Crux of Consciousness. In K. Yasue, M. Jibu and T. Della Senta (eds.), Proceedings of Fundamental Approaches to Consciousness, Tokyo ’99. Amsterdam: John Benjamins Publishing Company.Google Scholar
  10. Aerts, D., J. Broekaert and L. Gabora (2002). A Case for Applying an Abstracted Quantum Formalism to Cognition. In M.H. Bickhard and R.L. Campbell (eds.), Mind in Interaction. Amsterdam: John Benjamins Publishing Company.Google Scholar
  11. Aerts, D., J. Broekaert and S. Smets (1999). The Liar-Paradox in a Quantum Mechanical Perspective, Foundations of Science 4:115–132. Preprint at
  12. Aerts, D., J. Broekaert and S. Smets (2000). A Quantum Structure Description of the Liar-Paradox, International Journal of Theoretical Physics 38:3231–3239. Preprint at
  13. Pitowsky, I. (1989). Quantum Probability-Quantum Logic. Lecture Notes in Physics, vol. 321: Wien: Springer-Verlag.Google Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Jan Broekaert
    • 1
  • Diederik Aerts
    • 1
  • Bart D’Hooghe
    • 1
  1. 1.Center Leo Apostel for Interdisciplinary Studies (CLEA) Foundations of the Exact Sciences (FUND) Department of MathematicsVrije Universiteit BrusselBrusselsBelgium

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