Advertisement

A Four-Valued Hybrid Logic with Non-dual Modal Operators

  • Diana CostaEmail author
  • Manuel A. Martins
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12005)

Abstract

Hybrid logics are an extension of modal logics where it is possible to refer to a specific state, thus allowing the description of what happens at specific states, equalities and transitions between them. This makes hybrid logics very desirable to work with relational structures.

However, as the amount of information grows, it becomes increasingly more common to find inconsistencies. Information collected about a particular hybrid structure is not an exception. Rather than discarding all the data congregated, working with a paraconsistent type of logic allows us to keep it and still make sensible inferences.

In this paper we introduce a four-valued semantics for hybrid logic, where contradictions are allowed both at the level of propositional variables and accessibility relations. A distinguishing feature of this new logic is the fact that the classical equivalence between modal operators will be broken. A sound and complete tableau system is also presented.

References

  1. 1.
    Arieli, O.: On the application of the disjunctive syllogism in paraconsistent logics based on four states of information. In: Proceedings of the Twelfth International Conference on Principles of Knowledge Representation and Reasoning, KR 2010, pp. 302–309. AAAI Press (2010)Google Scholar
  2. 2.
    Belnap, N.D.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-Valued Logic. D. Reidel (1977)Google Scholar
  3. 3.
    Besnard, P., Hunter, A.: Quasi-classical logic: non-trivializable classical reasoning from inconsistent information. In: Froidevaux, C., Kohlas, J. (eds.) ECSQARU 1995. LNCS, vol. 946, pp. 44–51. Springer, Heidelberg (1995).  https://doi.org/10.1007/3-540-60112-0_6CrossRefGoogle Scholar
  4. 4.
    Blackburn, P.: Representation, reasoning, and relational structures: a hybrid logic manifesto. Log. J. IGPL 8(3), 339–365 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Braüner, T.: Axioms for classical, intuitionistic, and paraconsistent hybrid logic. J. Logic Lang. Inform. 15(3), 179–194 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Braüner, T.: Hybrid Logic and its Proof-Theory. Springer, Dordrecht (2010).  https://doi.org/10.1007/978-94-007-0002-4CrossRefzbMATHGoogle Scholar
  7. 7.
    Chechik, M., Devereux, B., Easterbrook, S., Gurfinkel, A.: Multi-valued symbolic model-checking. ACM Trans. Softw. Eng. Methodol. 12(4), 371–408 (2003)CrossRefGoogle Scholar
  8. 8.
    Costa, D., Martins, M.A.: Para consistency in hybrid logic. J. Log. Comput. 27(6), 1825–1852 (2016)zbMATHGoogle Scholar
  9. 9.
    Fitting, M.: Fixpoint semantics for logic programming a survey. Theor. Comput. Sci. 278(1), 25–51 (2002). Mathematical Foundations of Programming Semantics 1996MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fitting, M.C.: Many-valued modal logics. Fundam. Inf. 15(3–4), 235–254 (1991)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hansen, J.U., Bolander, T., Braüner, T.: Many-valued hybrid logic. J. Log. Comput. 28(5), 883–908 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Odintsov, S.P., Wansing, H.: Disentangling FDE-based paraconsistent modal logics. Stud. Logica 105(6), 1221–1254 (2017) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rivieccio, U., Jung, A., Jansana, R.: Four-valued modal logic: Kripke semantics and duality. J. Log. Comput. (2015)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity College LondonLondonEngland
  2. 2.CIDMA – Department of MathematicsUniversity of AveiroAveiroPortugal

Personalised recommendations