Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-02-09T01:49:10.278Z Has data issue: false hasContentIssue false
  • English
  • Français

A SUBSTRUCTURAL GENTZEN CALCULUS FOR ORTHOMODULAR QUANTUM LOGIC

Part of: Algebraic logic General logic

Published online by Cambridge University Press:  27 January 2022

DAVIDE FAZIO ,
ANTONIO LEDDA ,
FRANCESCO PAOLI  and
GAVIN ST. JOHN Open the ORCID record for GAVIN ST. JOHN [Opens in a new window]
DAVIDE FAZIO
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA UNIVERSITÀ DI CAGLIARI CAGLIARI, ITALY E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
ANTONIO LEDDA
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA UNIVERSITÀ DI CAGLIARI CAGLIARI, ITALY E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
FRANCESCO PAOLI*
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA UNIVERSITÀ DI CAGLIARI CAGLIARI, ITALY E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
GAVIN ST. JOHN
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA UNIVERSITÀ DI CAGLIARI CAGLIARI, ITALY E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
*
E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one recovers a calculus for classical logic.

Keywords

orthomodular latticequantum logicGentzen systemsubstructural logicleft residuated ℓ-groupoid

MSC classification

Primary: 03G12: Quantum logic
Secondary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 4 , December 2023 , pp. 1177 - 1198
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Avron, A. (1988). The semantics and proof theory of linear logic. Theoretical Computer Science, 57, 161–184.Google Scholar
Beran, L. (1980). Central and exchange properties of orthomodular lattices. Mathematische Nachrichten, 97, 247–251.Google Scholar
Beran, L. (1985). Orthomodular Lattices: Algebraic Approach. Dordrecht: Reidel.Google Scholar
Bimbo, K. (2014). Proof Theory: Sequent Calculi and Related Formalisms. London: Routledge.Google Scholar
Blok, W. J., & Jónsson, B. (2006). Equivalence of consequence operations. Studia Logica, 83, 91–110.Google Scholar
Blount, K., & Tsinakis, C. (2003). The structure of residuated lattices. International Journal of Algebra and Computation, 13, 437–461.Google Scholar
Bruns, G., & Harding, J. (2000). Algebraic aspects of orthomodular lattices. In Coecke, B., Moore, D., and Wilce, A., editors. Current Research in Operational Quantum Logic: Algebras, Categories, Languages. Dordrecht: Springer Netherlands, pp. 37–65.Google Scholar
Buszkowski, W. (2017). Involutive nonassociative Lambek calculus: Sequent systems and complexity. Bulletin of the Section of Logic, 46, 75–91.Google Scholar
Chajda, I., & Länger, H. (2017). Orthomodular lattices can be converted into left residuated $l$ -groupoids. Miskolc Mathematical Notes, 18, 685–689.Google Scholar
Chajda, I., & Radelecki, S. (2016). Involutive right-residuated l-groupoids. Soft Computing, 20, 119–131.Google Scholar
Cintula, P., Horčík, R., & Noguera, C. (2013). Nonassociative substructural logics and their semilinear extensions: Axiomatization and completeness properties. The Review of Symbolic Logic, 6, 394–423.Google Scholar
Coecke, B., & Smets, S. (2004). The Sasaki hook is not a [static] implicative connective but induce a backward [in time] dynamic one that assigns causes. International Journal of Theoretical Physics, 43, 1705–1736.Google Scholar
Cutland, N., & Gibbins, P. (1982). A regular sequent calculus for quantum logic in which $\wedge$ and $\vee$ are dual. Logique et Analyse, 25, 221–248.Google Scholar
Czelakowski, J. (2010). Protoalgebraic Logics. Trends in Logic. Dordrecht: Kluwer.Google Scholar
Dalla Chiara, M., Giuntini, R., & Greechie, R. (2004). Reasoning in Quantum Theory. Trends in Logic, Dordrecht: Kluwer.Google Scholar
Font, J. (2016). Abstract Algebraic Logic: An Introductory Textbook. London: College Publications.Google Scholar
Galatos, N., Jipsen, P., Kowalski, T., & Ono, H. (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Studies in Logic and the Foundations of Mathematics, Vol. 151. Amsterdam: Elsevier.Google Scholar
Galatos, N., & Ono, H. (2010). Cut elimination and strong separation for substructural logics: An algebraic approach. Annals of Pure and Applied Logic, 161, 1097–1133.Google Scholar
Galatos, N., & Tsinakis, C. (2009). Equivalence of closure operators: An order-theoretic and categorical perspective. The Journal of Symbolic Logic, 74, 780–810.Google Scholar
Goldblatt, R. (1974). Semantic analysis of orthologic. Journal of Philosophical Logic, 19, 19–35.Google Scholar
Gudder, S., & Schelp, R. (1980). Coordinatization of orthocomplemented and orthomodular posets. Proceedings of the American Mathematical Society, 25, 229–237.Google Scholar
Hardegree, G. (1981). Material implication in orthomodular (and Boolean) lattices. Notre Dame Journal of Formal Logic, 22, 168–182.Google Scholar
Humberstone, L. (2011). The Connectives. Cambridge: MIT Press.Google Scholar
Kalmbach, G. (1974). Orthomodular logic. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 20, 395–406.Google Scholar
Kornell, A. (2021). A natural deduction system for orthomodular quantum logic. Preprint. arXiv:2109.05383 [math.LO].Google Scholar
Metcalfe, G., Paoli, F., & Tsinakis, C. (2010). Ordered algebras and logic. In Hosni, H. and Montagna, F., editors. Uncertainty and Rationality, Publications of the Scuola Normale Superiore di Pisa, Pisa, Vol. 10, pp. 1–85.Google Scholar
Moot, R., & Retoré, C. (2012). The non-associative Lambek calculus. In Moot, R., & Retoré, C., editors. The Logic of Categorial Grammars. Berlin: Springer, pp. 101–147.Google Scholar
Nishimura, H. (1980). Sequential method in quantum logic. The Journal of Symbolic Logic, 45, 339–352.Google Scholar
Paoli, F. (2002). Substructural Logics: A Primer. Trends in Logic. Dordrecht: Kluwer.Google Scholar
Fazio, D., Ledda, A., & Paoli, F. (2021). Residuated structures and orthomodular lattices. Studia Logica, 109, 1201–1239. Google Scholar
Raftery, J. G. (2006). Correspondences between Gentzen and Hilbert systems. The Journal of Symbolic Logic, 71, 903–957.Google Scholar