Skip to main content
SpringerLink
Log in

Substructural Nuclear (Image-Based) Logics and Operational Kripke-Style Semantics

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

This paper deals with substructural nuclear (image-based) logics and their algebraic and Kripke-style semantics. More precisely, we first introduce a class of substructural logics with connective N satisfying nucleus property, called here substructural nuclear logics, and its subclass, called here substructural nuclear image-based logics, where N further satisfies homomorphic image property. We then consider their algebraic semantics together with algebraic characterizations of those logics. Finally, we introduce operational Kripke-style semantics for those logics and provide two sorts of completeness results for them, one of which is based on algebraic completeness for all the logics and the other of which is based on set-theoretic one for the nuclear image-based 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. Ávila, F., G. Bezhanishvili, P. J. Morandi, and A. Zaldívar, When is the frame of nuclei spatial: A new approach. ArXiv:1906.03636v3 [math.GN] 2 Dec 2019.

  2. Benton, P. N., G. M. Bierman, and V. C. V. de Paiva, Computational types from a logical perspective, Journal of Functional Programming 8:177–193, 1998.

    Article  Google Scholar 

  3. Beth, E. W., Semantic construction of intuitionistic logic, Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, 19:357–388, 1956.

    Google Scholar 

  4. Beth, E. W., The Foundations of Mathematics, Harper & Row, New York, 1964.

    Google Scholar 

  5. Bezhanishvili, G. and W. H. Holliday, Locales, nuclei, and Dragalin frames, in L. Beklemishev, S. Demri, and A. Máté, (eds.), Advances in Modal Logic, Vol. 11. College Publications, London, 2016. pp. 177–196.

    Google Scholar 

  6. Bezhanishvili, G. and W. H. Holliday, A semantic hierarchy for intuitionistic logic, Indagationes Mathematicae 30:403–469, 2019.

    Article  Google Scholar 

  7. Cintula, P., Weakly implicative (fuzzy) logics I: Basic properties, Archive for Mathematical Logic 45:673–704, 2006.

    Article  Google Scholar 

  8. Cintula, P., R. Horčík, and C. Noguera, Non-associative substructural logics and their semilinear extensions: Axiomatization and completeness properties, Review of Symbolic Logic 6:394–423, 2013.

    Article  Google Scholar 

  9. Cintula, P., R. Horčík, and C. Noguera, The quest for the basic fuzzy logic, in F. Montagna, (ed.), Petr Hájek on Mathematical Fuzzy Logic, Springer, Dordrecht, 2015, pp. 245–290.

    Chapter  Google Scholar 

  10. Cintula, P. and C. Noguera, A general framework for mathematical fuzzy logic, in P. Cintula, P. Hájek, and C. Noguera, (eds.), Handbook of Mathematical Fuzzy Logic, vol 1, College Publications, London, 2011, pp. 103–207.

    Google Scholar 

  11. van Dalen, D., An interpretation of intuitionistic analysis, Annals of Mathematical Logic 13:1–43, 1978.

    Article  Google Scholar 

  12. Došen, K., Sequent systems and groupoid models I, Studia Logica 47:353–385, 1988.

  13. Došen, K., Sequent systems and groupoid models II, Studia Logica 48:41–65, 1989.

  14. Dragalin, A. G., Matematicheskii Intuitsionizm: Vvedenie v Teoriyu Dokazatelstv, in Matematicheskaya Logika i Osnovaniya Matematiki, “Nauka”, Moscow, 1979.

  15. Dragalin, A. G., Mathematical Intuitionism: Introduction to Proof Theory, vol. 67 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1988.

    Google Scholar 

  16. Dummett, M., Elements of Intuitionism, second ed., Clarendon Press, Oxford, 2000.

    Google Scholar 

  17. Dunn, J. M. Relevance logic and entailment, in D. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, vol. III, Reidel, Dordrecht, 1986, pp. 117–224.

  18. Dunn, J. M. and G. Hardegree, Algebraic Methods in Philosophical Logic, Oxford Univ Press, Oxford, 2001.

  19. Fairtlough, M. and M. Mendler, An intuitionistic modal logic with applications to the formal verification of hardware, in L. Pacholski, and J. Tiuryn, (eds.), Computer Science Logic Proceedings 1994, vol. 933 of Lecture Notes in Computer Science, Springer, 1995, pp. 354–368.

  20. Fairtlough, M. and M. Mendler, Propositional lax logic, Information and Computation 137:1–33, 1997.

  21. Fairtlough, M., M. Mendler, and M. Walton, First-order lax logic as a framework for constraint logic programming, technical report MIPS-9714, University of Passau, 1997.

  22. Fairtlough, M. and M. Walton, Quantified lax logic, technical report CS-97-11, Department of Computer Science, University of Sheffield, 1997.

    Google Scholar 

  23. Fine, K., Models for entailment, Journal of Philosophical Logic 3:347–372, 1974.

    Article  Google Scholar 

  24. Galatos, N., Non-associative residuated lattices, in AMS Sectional Meeting, Special Session on Universal Algebra and Lattice Theory, Vanderbilt University, 2004.

    Google Scholar 

  25. Galatos, N., P. Jipsen, T. Kowalski, and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier, Amsterdam, 2007.

  26. Galatos, N. and H. Ono, Non-associative substructural logics: Algebraization, cut elimination and separation, in Conference on Residuated Structures and Many-Valued Logics, Patras University, Greece, 2004.

  27. Galatos, N. and H. Ono, Cut elimination and strong separation for substructural logics, Annals of Pure and Applied Logic 161:1097–1133, 2010.

    Article  Google Scholar 

  28. Galatos, N. and P. Jipsen, Residuated frames with applications to decidability, Transactions of the American Mathematical Society 365:1219–1249, 2013.

    Article  Google Scholar 

  29. Girard, J.-Y., Linear logic, Theoretical Computer Science 50:1–102, 1987.

    Article  Google Scholar 

  30. Goldblatt, R., Semantic analysis of orthologic, Journal of Philosophical Logic 3:19–35, 1974.

    Article  Google Scholar 

  31. Goldblatt, R., Grothendieck topology as geometric modality, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 27:495–529, 1981.

    Article  Google Scholar 

  32. Goldblatt, R., Cover semantics for quantified lax logic, Journal of Logic and Computation 21:1035–1063, 2011.

    Article  Google Scholar 

  33. Howe, J. M. Proof search in lax logic, Mathematical Structures in Computer Science 11:573–588, 2001.

    Article  Google Scholar 

  34. Isbell, J. R., Atomless parts of spaces, Mathematica Scandinavica 31:5–32, 1972.

    Article  Google Scholar 

  35. Kripke, S., Semantic analysis of modal logic I: Normal modal propositional calculi, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 9:67–96, 1963.

    Article  Google Scholar 

  36. Kripke, S., Semantic analysis of intuitionistic logic I, in J. Crossley, and M. Dummett, (eds.), Formal systems and Recursive Functions, North-Holland Publ. Co., Amsterdam, 1965, pp. 92–129.

    Chapter  Google Scholar 

  37. Kripke, S., Semantic analysis of modal logic II, in J. Addison, L. Henkin, and A. Tarski, (eds.), The Theory of Models, North-Holland Publ. Co., Amsterdam, 1965, pp. 206–220.

    Google Scholar 

  38. Litak, T., A continuum of incomplete intermediate logics, Reports on Mathematical Logic 36:131–141, 2002.

    Google Scholar 

  39. Mendler, M., Constrained proofs: A logic for dealing with behavioural constraints in formal hardware verification, in G. Jones, and M. Sheeran, (eds.), Designing Correct Circuits (DCC ’90), Springer-Verlag, Berlin/New York, 1990, pp. 1–28.

    Google Scholar 

  40. Mendler, M., A Modal Logic for Handling Behavioural Constraints in Formal Hardware Verification, Ph.D. thesis, Department of Computer Science, ECS-LFCS-93-255. Edinburgh University, 1993.

  41. Moggi, E., Notions of computation and monads, Information and Computation 93:55–92, 1991.

    Article  Google Scholar 

  42. Montagna, F. and H. Ono, Kripke semantics, undecidability and standard completeness for Esteva and Godo’s Logic MTL\(\forall \), Studia Logica 71:227–245, 2002.

    Article  Google Scholar 

  43. Montagna, F. and L. Sacchetti, Kripke–style semantics for many-valued logics, Mathematical Logic Quarterly 49:629–641, 2003.

    Article  Google Scholar 

  44. Montagna, F. and L. Sacchetti, Corrigendum to Kripke–style semantics for many-valued logics, Mathematical Logic Quarterly 50:104–107, 2004.

    Article  Google Scholar 

  45. Mulvey, C. J., &, Supplemento ai Rendiconti del Circolo Matematico di Palermo II 12:99–104, 1986.

  46. Ono, H., Semantics for substructural logics, in P. Schroeder-Heister, and K. Došen, (eds.), Substructural Logics, Clarendon Press, Oxford, 1993, pp. 259–291.

    Google Scholar 

  47. Ono, H., Closure operators and complete embeddings of residuated lattices, Studia Logica 74:427–449, 2003.

    Article  Google Scholar 

  48. Ono, H., Completions of algebras and completeness of modal and substructural logics, in P. Balbiani, N.-Y. Suzuki, F. Wolter, and M. Zakharyaschev, (eds.), Advances in Modal Logic, vol. 4, College publications, London, pp. 335–353.

  49. Ono, H. and Y. Komori, Logics without the contraction rule, Journal of Symbolic Logic 50:169–201, 1985.

    Article  Google Scholar 

  50. Punčochář, V., Substructural inquisitive logics. Review of Symbolic Logic 12, 2019, 296-330.

    Article  Google Scholar 

  51. Restall, G., An Introduction to Substructural Logics, Routledge, New York, 2000.

  52. Rogozin, D., Cover Semantics for Predicate Extensions of Intuitionistic Modal Logics. Institute for Information Transmission Problems, RAS, Moscow, Russia, 2020.

  53. Rosenthal, K. I. Quantales and Their Applications, vol. 234 of Pitman Research Notes in Mathematics, Addison Wesley Longman, 1990.

  54. Routley, R. and R. K. Meyer, The semantics of entailment (II), Journal of Philosophical Logic 1:53–73, 1972.

    Article  Google Scholar 

  55. Routley, R. and R. K. Meyer, The semantics of entailment (III), Journal of Philosophical Logic 1:192–208, 1972.

  56. Routley, R. and R. K. Meyer, The semantics of entailment (I), in H. Lebranc, (ed.), Truth, Syntax, and Modality, North-Holland Publ. Co., Amsterdam, 1973, pp. 199–243.

  57. Sambin, G., Pretopologies and completeness proofs, Journal of Symbolic Logic 60:861–878, 1995.

    Google Scholar 

  58. Shehtman, V. B., Incomplete propositional logics, Doklady Akademii Nauk SSSR 235:542–545, 1977, (Russian).

    Google Scholar 

  59. Urquhart, A., Semantics for relevant logics, Journal of Symbolic Logic 37:159–169, 1972.

    Article  Google Scholar 

  60. Wansing, H., The Logic of Information Structures, Springer, Berlin, 1993.

    Book  Google Scholar 

  61. Yang, E., Kripke–style semantics for UL, Korean Journal of Logic 15:1–15, 2012.

  62. Yang, E., Algebraic Kripke-style semantics for weakening-free fuzzy logics, Korean Journal of Logic 17:181–195, 2014.

  63. Yang, E., Algebraic Kripke-style semantics for substructural fuzzy logics, Korean Journal of Logic 19:295–322, 2016.

    Google Scholar 

  64. Yang, E., Algebraic relational semantics for basic substructural logics, Logique et Analyse 252:415–441, 2020.

    Google Scholar 

  65. Yetter, D. N., Quantales and (noncommutative) linear logic, Journal of Symbolic Logic 55:41–64, 1990.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy and Institute of Critical Thinking and Writing, Jeonbuk National University, Rm 417 Center for Humanities and Social Sciences, Baekje-daero 567, Jeonju, Korea

    Eunsuk Yang

Authors
  1. Eunsuk Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eunsuk Yang.

Additional information

Publisher's Note

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

Presented by Jacek Malinowski

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, E. Substructural Nuclear (Image-Based) Logics and Operational Kripke-Style Semantics. Stud Logica (2023). https://doi.org/10.1007/s11225-023-10069-y

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11225-023-10069-y

Keywords

Search

Navigation