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Sequent Calculi for Orthologic with Strict Implication

dc.contributor.authorKawano, Tomoaki
dc.date.accessioned2022-05-19T14:13:32Z
dc.date.available2022-05-19T14:13:32Z
dc.date.issued2021-11-09
dc.identifier.issn0138-0680
dc.identifier.urihttp://hdl.handle.net/11089/41869
dc.description.abstractIn this study, new sequent calculi for a minimal quantum logic (\(\bf MQL\)) are discussed that involve an implication. The sequent calculus \(\bf GO\) for \(\bf MQL\) was established by Nishimura, and it is complete with respect to ortho-models (O-models). As \(\bf GO\) does not contain implications, this study adopts the strict implication and constructs two new sequent calculi \(\mathbf{GOI}_1\) and \(\mathbf{GOI}_2\) as the expansions of \(\bf GO\). Both \(\mathbf{GOI}_1\) and \(\mathbf{GOI}_2\) are complete with respect to the O-models. In this study, the completeness and decidability theorems for these new systems are proven. Furthermore, some details pertaining to new rules and the strict implication are discussed.en
dc.language.isoen
dc.publisherWydawnictwo Uniwersytetu Łódzkiegopl
dc.relation.ispartofseriesBulletin of the Section of Logic;1en
dc.rights.urihttps://creativecommons.org/licenses/by-nc-nd/4.0
dc.subjectQuantum logicen
dc.subjectsequent calculusen
dc.subjectcompleteness theoremen
dc.subjectimplicationen
dc.subjectorthologicen
dc.titleSequent Calculi for Orthologic with Strict Implicationen
dc.typeOther
dc.page.number73-89
dc.contributor.authorAffiliationTokyo Institute of Technology, School of Computing, Department of Mathematical and Computing Science, 2-12-1 Okayama, Meguro-ku, Tokyo, Japanen
dc.identifier.eissn2449-836X
dc.referencesJ. C. Abbott, Orthoimplication Algebras, Studia Logica, vol. 35(2) (1976), pp. 173–177, DOI: https://doi.org/10.1007/BF02120879en
dc.referencesK. Bednarska, A. Indrzejczak, Hypersequent Calculi for S5 – The Methods of Cut-elimination, Logic and Logical Philosophy, vol. 24(3) (2015), pp. 277–311, DOI: https://doi.org/10.12775/LLP.2015.018en
dc.referencesG. Birkhoff, J. V. Neumann, The Logic of Quantum Mechanics, Annals of Mathematics, vol. 37(4) (1936), pp. 823–843, DOI: https://doi.org/10.2307/1968621en
dc.referencesI. Chajda, The axioms for implication in orthologic, Czechoslovak Mathematical Journal, vol. 58(1) (2008), pp. 735–744, DOI: https://doi.org/10.1007/s10587-008-0002-2en
dc.referencesI. Chajda, J. Cirulis, An Implicational Logic for Orthomodular Lattices, Acta Scientiarum Mathematicarum, vol. 82(34) (2016), pp. 383–394, DOI: https://doi.org/10.14232/actasm-015-813-6en
dc.referencesI. Chajda, R. Halaš, An Implication in Orthologic, International Journal of Theoretical Physics, vol. 44(7) (2005), pp. 735–744, DOI: https://doi.org/10.1007/s10773-005-7051-1en
dc.referencesI. Chajda, R. Halaš, H. Länger, Orthomodular Implication Algebras, International Journal of Theoretical Physics, vol. 40(11) (2001), pp. 1875–1884, DOI: https://doi.org/10.1023/A:1011933018776en
dc.referencesI. Chajda, H. Länger, Orthomodular Posets Can Be Organized as Conditionally Residuated Structures, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, vol. 53(2) (2014), pp. 29–33, URL: https://kma.upol.cz/data/xinha/ULOZISTE/ActaMath/2014/53-2-02.pdfen
dc.referencesI. Chajda, H. Länger, Orthomodular lattices can be converted into left-residuated l-groupoids, Miskolc Mathematical Notes, vol. 18(2) (2017), pp. 685–689, DOI: https://doi.org/10.18514/mmn.2017.1730en
dc.referencesI. Chajda, H. Länger, Residuation in orthomodular lattices, Topological Algebra and its Applications, vol. 5(1) (2017), pp. 1–5, DOI: https://doi.org/10.1515/taa-2017-0001en
dc.referencesI. Chajda, H. Länger, How to introduce the connective implication in orthomodular posets, Asian-European Journal of Mathematics, vol. 14(4) (2021), p. 2150066, DOI: https://doi.org/10.1142/S1793557121500662en
dc.referencesM. L. D. Chiara, R. Giuntini, Quantum Logics, [in:] D. M. Gabbay, F. Guenthner (eds.), Handbook of Philosophical Logic, Springer Netherlands, Dordrecht (2002), pp. 129–228, DOI: https://doi.org/10.1007/978-94-017-0460-1_2en
dc.referencesP. D. Finch, Quantum logic as an implication algebra, Bulletin of the Australian Mathematical Society, vol. 2(1) (1970), pp. 101–106, DOI: https://doi.org/10.1017/S0004972700041642en
dc.referencesG. M. Hardegree, Material Implication in Orthomodular (and Boolean) Lattices, Notre Dame Journal of Formal Logic, vol. 22(2) (1981), pp. 163–182, DOI: https://doi.org/10.1305/ndjfl/1093883401en
dc.referencesG. M. Hardegree, Quasi-implication algebras, Part I: Elementary theory, Algebra Universalis, vol. 12(1) (1981), pp. 30––47, DOI: https://doi.org/10.1007/BF02483861en
dc.referencesG. M. Hardegree, Quasi-implication algebras, Part II: Structure theory, Algebra Universalis, vol. 12(1) (1981), pp. 48–65, DOI: https://doi.org/10.1007/BF02483862en
dc.referencesL. Herman, E. L. Marsden, R. Piziak, Implication Connectives in Orthomodular Lattices, Notre Dame Journal of Formal Logic, vol. 16(3) (1975), pp. 305–328, DOI: https://doi.org/10.1305/ndjfl/1093891789en
dc.referencesA. Indrzejczak, Two Is Enough – Bisequent Calculus for S5, [in:] A. Herzig, A. Popescu (eds.), Frontiers of Combining Systems, Springer International Publishing, Cham (2019), pp. 277–294, DOI: https://doi.org/10.1007/978-3-030-29007-8_16en
dc.referencesA. Indrzejczak, Sequents and Trees. An Introduction to the Theory and Applications of Propositional Sequent Calculi, Studies in Universal Logic, Birkhäuser, Basel (2021), DOI: https://doi.org/10.1007/978-3-030-57145-0en
dc.referencesK. Ishii, R. Kashima, K. Kikuchi, Sequent Calculi for Visser’s Propositional Logics, Notre Dame Journal of Formal Logic, vol. 42(1) (2001), pp. 1–22, DOI: https://doi.org/10.1305/ndjfl/1054301352en
dc.referencesG. Kalmbach, Orthomodular Lattices, Academic Press, New York (1983).en
dc.referencesT. Kawano, Advanced Kripke Frame for Quantum Logic, [in:] L. S. Moss, R. de Queiroz, M. Martinez (eds.), Logic, Language, Information, and Computation, Springer Berlin Heidelberg, Berlin, Heidelberg (2018), pp. 237–249, DOI: https://doi.org/10.1007/978-3-662-57669-4_14en
dc.referencesT. Kawano, Labeled Sequent Calculus for Orthologic, Bulletin of the Section of Logic, vol. 47(4) (2018), pp. 217–232, DOI: https://doi.org/10.18778/0138-0680.47.4.01en
dc.referencesS. Negri, Proof Theory for Modal Logic, Philosophy Compass, vol. 6(8) (2011), pp. 523–538, DOI: https://doi.org/10.1111/j.1747-9991.2011.00418.xen
dc.referencesH. Nishimura, Sequential method in quantum logic, Journal of Symbolic Logic, vol. 45(2) (1980), pp. 339–352, DOI: https://doi.org/10.2307/2273194en
dc.referencesH. Nishimura, Proof Theory for Minimal Quantum Logic I, International Journal of Theoretical Physics, vol. 33(1) (1994), pp. 103–113, DOI: https://doi.org/10.1007/BF00671616en
dc.referencesH. Nishimura, Proof Theory for Minimal Quantum Logic II, International Journal of Theoretical Physics, vol. 33(7) (1994), pp. 1427–1443, DOI: https://doi.org/10.1007/BF00670687en
dc.referencesH. Ozawa, H. Odera, S. Chitani, Formal System QL for Quantum Logic, The Japan Association for Logical Philosophy, Logical Philosophy Research, vol. 3 (2003).en
dc.referencesM. Ozawa, Operational Meanings of Orders of Observables Defined through Quantum Set Theories with Different Conditionals, Electronic Proceedings in Theoretical Computer Science, vol. 236 (2017), pp. 127–144, DOI: https://doi.org/10.4204/eptcs.236.9en
dc.referencesF. Poggiolesi, Gentzen Calculi for Modal Propositional Logic, vol. 32 of Trends in Logic, Springer, Dordrecht (2011).en
dc.referencesA. Visser, A Propositional Logic with Explicit Fixed Points, Studia Logica, vol. 40(2) (1981), pp. 155–175, DOI: https://doi.org/10.1007/BF01874706en
dc.contributor.authorEmailkawano.t.af@m.titech.ac.jp
dc.identifier.doi10.18778/0138-0680.2021.22
dc.relation.volume51


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