Abstract
Two new multilattice logics called submultilattice logic and indexed multilattice logic are introduced as a monosequent calculus and an indexed monosequent calculus, respectively. The submultilattice logic is regarded as a monosequent calculus version of Shramko’s original multilattice logic, which is also known as the logic of logical multilattices. The indexed multilattice logic is an extension of the submultilattice logic, and is regarded as the logic of multilattices. A completeness theorem with respect to a lattice-valued semantics is proved for the submultilattice logic, and a completeness theorem with respect to a multilattice-valued semantics is proved for the indexed multilattice logic.
Similar content being viewed by others
References
Almukdad, A., and D. Nelson, Constructible falsity and inexact predicates, Journal of Symbolic Logic 49(1):231-233, 1984.
Aoyama, H., On a weak system of sequent calculus, Journal of Logical Philosophy 3:29–37, 2003.
Aoyama, H., Dual-intuitionistic logic and some other logics, Journal of Logical Philosophy 6:34–56, 2009.
Arieli, O., and A. Avron, Reasoning with logical bilattices, Journal of Logic, Language and Information 5:25–63, 1996.
Arieli, O., and A. Avron, The value of the four values, Artificial Intelligence 102(1):97–141, 1998.
Belnap, N., A useful four-valued logic, in G. Epstein, J.M. Dunn, (eds.) Modern Uses of Multiple-Valued Logic, Reidel, Dordrecht, 1977, pp. 5–37.
Belnap, N., How a computer should think, in G. Ryle, (ed.) Contemporary Aspects of Philosophy, Oriel Press, Stocksfield, 1977, pp. 30–56
Béziau, J.-Y., Monosequent proof systems, in C. Caleiro, F. Dionisio, P. Gouveia, P. Mateus and J. Rasga (eds.), Logic and Computation – Essays in Honor of Amilcar Sernadas, College Publication, London, 2017, pp. 111–137.
Bimbó, K., and J.M. Dunn, Four-valued logic, Notre Dame Journal of Formal Logic 42(3):171–192, 2001.
Birkhoff, G., and J. von Neumann, The logic of quantum mechanics, Annals of Mathematics 37:823–843, 1936.
Cockett, J.R.B., and R.A.G. Seely, Finite sum-product logic, Theory and Applications of Categories 8(5):63–99, 2001.
Craig, W., Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory, Journal of Symbolic Logic 22(3):269–285, 1957.
Dalla Chiara, M.L., and R. Giuntini, Paraconsistent quantum logics, Foundations of Physics 19(7):891–904, 1989.
Dunn, J.M., Intuitive semantics for first-degree entailment and ‘coupled trees’, Philosophical Studies 29:149–168, 1976.
Dunn, J.M., and G.M. Hardegree, Algebraic Methods in Philosophical Logic, Clarendon Press, Oxford University Press, Oxford, New York, 2001.
Faggian, C., and G. Sambin, From basic logic to quantum logics with cut-elimination, International Journal of Theoretical Physics 37(1):31–37, 1998.
Ginsberg, M., Multi-valued logics, in Proceedings of the 5th National Conference on Artificial Intelligence (AAAI-86), Morgan Kaufman Publishers, Los Altos, 1986, pp. 243–247.
Ginsberg, M., Multivalued logics: a uniform approach to reasoning in artificial intelligence, Computational Intelligence 4:256–316, 1988.
Gurevich, Y., Intuitionistic logic with strong negation, Studia Logica 36:49–59, 1977.
Hartonas, C., Modal and temporal extensions of non-distributive propositional logics, Logic Journal of the IGPL 24(2):156–185, 2016.
Hartonas, C., Order-dual relational semantics for non-distributive propositional logics, Logic Journal of the IGPL 25(2):145–182, 2017
Humberstone, L., The Connectives, MIT Press, Cambridge, 2011.
Kamide, N., Proof theory of paraconsistent quantum logic, Journal of Philosophical Logic 47(2):301–324, 2018.
Kamide, N., Extending paraconsistent quantum logic: A single-antecedent/ succedent system approach, Mathematical Logic Quarterly 64 (4–5):371–386, 2018.
Kamide, N., Some properties for first-order Nelsonian paraconsistent quantum logic, Journal of Applied Logics - IfCoLog Journal of Logics and their Applications 7(1):59–88, 2020.
Kamide, N., Lattice logic, bilattice logic and paraconsistent quantum logic: A unified framework based on monosequent systems, Journal of Philosophical Logic, Online First, 2021.
Kamide, N., and Y. Shramko, Embedding from multilattice logic into classical logic and vice versa, Journal of Logic and Computation 27(5):1549–1575, 2017.
Mönting, J.S., Cut elimination and word problems for varieties of lattices, Algebra Universalis 12:290–321, 1981.
Nelson, D., Constructible falsity, Journal of Symbolic Logic 14:16–26, 1949.
Ołowska, E., and D. Vakarelov, Lattice-based modal algebras and modal logics, in: Logic, Methodology and Philosophy of Science, Proceedings of the 12th International Congress, King’s College Publications, 2005, pp. 147–170.
Rautenberg, W., Klassische und nicht-klassische Aussagenlogik, Vieweg, Braunschweig, 1979.
Restall, G., and F. Paoli, The geometry of nondistributive logics, Journal of Symbolic Logic 70(4):1108–1126, 2005.
Sambin, G., C. Battilotti and C. Faggian, Basic logic: Reflection, symmetry, visibility, Journal of Symbolic Logic 65(3):979–1013, 2000.
Shramko, Y., Truth, falsehood, information and beyond: The American plan generalized, in K. Bimbo (ed.), J. Michael Dunn on Information Based Logics, vol 8 of Outstanding Contributions to Logic, Springer, 2016, pp. 191–212.
Shramko, Y., J.M. Dunn, and T. Takenaka, The trilattice of constructive truth values, Journal of Logic and Computation 11(6):761–788, 2001.
Shramko, Y., and H. Wansing, Some useful sixteen-valued logics: How a computer network should think, Journal of Philosophical Logic 34:121–153, 2005.
Shramko, Y., D. Zaitsev, and A. Belikov, The FMLA-FMLA axiomatizations of the exactly true and non-falsity logics and some of their cousins, Journal of Philosophical Logic 48(5):787–808, 2019.
Vorob’ev, N.N., A constructive propositional calculus with strong negation (in Russian), Doklady Akademii Nauk SSSR 85:465–468, 1952.
Zaitsev, D., A few more useful 8-valued logics for reasoning with tetralattice $EIGHT_4$, Studia Logica 92(2):265–280, 2009.
Acknowledgements
We would like to thank the anonymous referees for their valuable comments and suggestions. This research was supported by JSPS KAKENHI Grant Numbers JP18K11171 and JP16KK0007.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Presented by Heinrich Wansing
Rights and permissions
About this article
Cite this article
Kamide, N. Alternative Multilattice Logics: An Approach Based on Monosequent and Indexed Monosequent Calculi. Stud Logica 109, 1241–1271 (2021). https://doi.org/10.1007/s11225-020-09939-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-020-09939-6