Skip to main content
SpringerLink
Log in

Lattice Logic, Bilattice Logic and Paraconsistent Quantum Logic: a Unified Framework Based on Monosequent Systems

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

Lattice logic, bilattice logic, and paraconsistent quantum logic are investigated based on monosequent systems. Paraconsistent quantum logic is an extension of lattice logic, and bilattice logic is an extension of paraconsistent quantum logic. Monosequent system is a sequent calculus based on the restricted sequent that contains exactly one formula in both the antecedent and succedent. It is known that a completeness theorem with respect to a lattice-valued semantics holds for a monosequent system for lattice logic. A completeness theorem with respect to a lattice-valued semantics is proved for paraconsistent quantum logic, and a completeness theorem with respect to a bilattice-valued semantics is proved for bilattice logic. Some syntactical properties, including cut-elimination and duality, are also investigated for the monosequent systems for these logics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Almukdad, A., & Nelson, D. (1984). Constructible falsity and inexact predicates. Journal of Symbolic Logic, 49(1), 231–233.

    Article  Google Scholar 

  2. Aoyama, H. (2003). On a weak system of sequent calculus. Journal of Logical Philosophy, 3, 29–37.

    Google Scholar 

  3. Aoyama, H. (2009). Dual-intuitionistic logic and some other logics. Journal of Logical Philosophy, 6, 34–56.

    Google Scholar 

  4. Arieli, O., & Avron, A. (1996). Reasoning with logical bilattices. Journal of Logic, Language and Information, 5, 25–63.

    Article  Google Scholar 

  5. Arieli, O., & Avron, A. (1998). The value of the four values. Artificial Intelligence, 102(1), 97–141.

    Article  Google Scholar 

  6. Belnap, N. (1977). A useful four-valued logic. In Epstein, G., & Dunn, J.M. (Eds.) Modern uses of multiple-valued logic (pp. 5–37). Dordrecht: Reidel.

  7. Belnap, N. (1977). How a computer should think. In Ryle, G. (Ed.) Contemporary aspects of philosophy (pp. 30–56). Stocksfield: Oriel Press.

  8. Béziau, J.-Y. (2017). Monosequent proof systems. In Caleiro, C., Dionisio, F., Gouveia, P., Mateus, P., & Rasga, J. (Eds.) Logic and computation – essays in honor of Amilcar Sernadas (pp. 111–137). London: College Publication.

  9. Birkhoff, G., & von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823–843.

    Article  Google Scholar 

  10. Cockett, J.R.B., & Seely, R.A.G. (2001). Finite sum-product logic. Theory and Applications of Categories, 8(5), 63–99.

    Google Scholar 

  11. Dalla Chiara, M.L., & Giuntini, R. (1989). Paraconsistent quantum logics. Foundations of Physics, 19(7), 891–904.

    Article  Google Scholar 

  12. Dunn, J.M. (1976). Intuitive semantics for first-degree entailment and ‘coupled trees’. Philosophical Studies, 29, 149–168.

    Article  Google Scholar 

  13. Dunn, J.M. (2000). Partiality and its dual. Studia Logica, 65, 5–40.

    Article  Google Scholar 

  14. Dunn, J.M., & Hardegree, G.M. (2001). Algebraic methods in philosophical logic. Oxford, New York: Clarendon Press, Oxford University Press.

  15. Faggian, C., & Sambin, G. (1998). From basic logic to quantum logics with cut-elimination. International Journal of Theoretical Physics, 37(1), 31–37.

    Article  Google Scholar 

  16. Fitting, M. (1991). Bilattices and the semantics of logic programming. Journal of Logic Programming, 11(1&2), 91–116.

    Article  Google Scholar 

  17. Fitting, M. (2006). Bilattices are nice things. In Bolander, T., Hendricks, V., & Pedersen, S.A. (Eds.) Self-reference (pp. 53–77). Stanford: CSLI Publications.

  18. Ginsberg, M. (1986). Multi-valued logics. In Proceedings of the 5th national conference on artificial intelligence (AAAI 1986) (pp. 243–247). Los Altos: Morgan Kaufman Publishers.

  19. Ginsberg, M. (1988). Multivalued logics: a uniform approach to reasoning in artificial intelligence. Computational Intelligence, 4, 256–316.

    Google Scholar 

  20. Gurevich, Y. (1977). Intuitionistic logic with strong negation. Studia Logica, 36, 49–59.

    Article  Google Scholar 

  21. Hartonas, C. (2016). Modal and temporal extensions of non-distributive propositional logics. Logic Journal of the IGPL, 24(2), 156–185.

    Article  Google Scholar 

  22. Hartonas, C. (2017). Order-dual relational semantics for non-distributive propositional logics. Logic Journal of the IGPL, 25(2), 145–182.

    Google Scholar 

  23. Humberstone, L. (2011). The connectives. Cambridge: MIT Press.

    Book  Google Scholar 

  24. Kamide, N. (2018). Proof theory of paraconsistent quantum logic. Journal of Philosophical Logic, 47(2), 301–324.

    Article  Google Scholar 

  25. Kamide, N. (2018). Extending paraconsistent quantum logic: a single-antecedent/succedent system approach. Mathematical Logic Quarterly, 64(4-5), 371–386.

    Article  Google Scholar 

  26. Kamide, N. (2019). First-order Nelsonian paraconsistent quantum logic. Proceedings of the 49th IEEE International Symposium on Multiple-Valued Logic, (ISMVL 2019), 176–181.

    Google Scholar 

  27. Kamide, N. (2019). Gentzen-type sequent calculi for extended Belnap-Dunn logics with classical negation: a general framework. Logica Universalis, 13(1), 37–63.

    Article  Google Scholar 

  28. Kamide, N. (2020). Some properties for first-order Nelsonian paraconsistent quantum logic. Journal of Applied Logics - IfCoLoG Journal of Logics and their Applications, 7(1), 59–88.

    Google Scholar 

  29. Mönting, J.S. (1981). Cut elimination and word problems for varieties of lattices. Algebra Universalis, 12, 290–321.

    Article  Google Scholar 

  30. Nelson, D. (1949). Constructible falsity. Journal of Symbolic Logic, 14, 16–26.

    Article  Google Scholar 

  31. Ołowska, E., & Vakarelov, D. (2005). Lattice-based modal algebras and modal logics. In Logic, methodology and philosophy of science: Proceedings of the 12th international congress (pp. 147–170): College Publications.

  32. Rautenberg, W. (1979). Klassische und nicht-klassische Aussagenlogik. Vieweg, Braunschweig.

  33. Restall, G., & Paoli, F. (2005). The geometry of nondistributive logics. Journal of Symbolic Logic, 70(4), 1108–1126.

    Article  Google Scholar 

  34. Sambin, G., Battilotti, C., & Faggian, C. (2000). Basic logic: reflection, symmetry, visibility. Journal of Symbolic Logic, 65(3), 979–1013.

    Article  Google Scholar 

  35. Vorob’ev, N.N. (1952). A constructive propositional calculus with strong negation (in Russian). Doklady Akademii Nauk SSSR, 85, 465–468.

    Google Scholar 

Download references

Acknowledgments

We would like to thank the anonymous referee for his or her valuable comments and suggestions. We would also like to thank Prof. Mitio Takano for his valuable comments on an early version of this paper. This research was supported by JSPS KAKENHI Grant Numbers JP18K11171 and JP16KK0007.

Author information

Authors and Affiliations

  1. Department of Information and Electronic Engineering, Faculty of Science and Engineering, Teikyo University, Toyosatodai 1-1, Utsunomiya, Tochigi, 320-8551, Japan

    Norihiro Kamide

Authors
  1. Norihiro Kamide
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Norihiro Kamide.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kamide, N. Lattice Logic, Bilattice Logic and Paraconsistent Quantum Logic: a Unified Framework Based on Monosequent Systems. J Philos Logic 50, 781–811 (2021). https://doi.org/10.1007/s10992-020-09585-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-020-09585-2

Keywords

Search

Navigation