A Semantic Analysis of Stone and Dual Stone Negations with Regularity

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10119)

Abstract

This article investigates whether a few well-known ‘negation’ operators may be termed as negations, using Dunn’s approach. The semantics of the Stone negation is investigated in perp frames, that of dual Stone negation in exhaustive frames, and that of Stone and dual Stone negations with the regularity property, in \(K_{-}\) frames. The study leads to new semantics for the logics corresponding to the classes of Stone algebras, dual Stone algebras and regular double Stone algebras.

Keywords

Perp semantics Regular double stone algebras 

References

  1. 1.
    Kripke, S.: Semantic analysis of intuitionistic logic I. In: Crossley, J., Dummett, M. (eds.) Formal Systems and Recursive Functions, pp. 92–129. North-Holland, Amsterdam (1963)Google Scholar
  2. 2.
    Dunn, J.: Star and Perp: two treatments of negation. In: Tomberlin, J. (ed.) Philosophical Perspectives, vol. 7, pp. 331–357. Ridgeview Publishing Company, Atascadero (1994)Google Scholar
  3. 3.
    Dunn, J.: Generalised ortho negation. In: Wansing, H. (ed.) Negation: A Notion in Focus, pp. 3–26. Walter de Gruyter, Berlin (1996)Google Scholar
  4. 4.
    Dunn, J.: A comparative study of various model-theoretic treatments of negation: a history of formal negations. In: Gabbay, D., Wansing, H. (eds.) What is Negation? pp. 23–51. Kluwer Academic Publishers, Netherlands (1999)Google Scholar
  5. 5.
    Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Banerjee, M., Chakraborty, M.K.: Algebras from rough sets. In: Pal, S.K., Polkowski, L., Skowron, A. (eds.) Rough-Neuro Computing: Techniques for Computing with Words. Cognitive Technologies, pp. 157–184. Springer, Berlin (2004)Google Scholar
  7. 7.
    Kumar, A.: A study of algebras and logics of rough sets based on classical and generalized approximation spaces. Doctoral dissertation, Indian Institute of Technology, Kanpur (2016)Google Scholar
  8. 8.
    Dunn, J.: Negation in the context of gaggle theory. Stud. Logica 80, 235–264 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Restall, G.: Defining double negation elimination. L. J. IGPL 8(6), 853–860 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dunn, J.: Positive modal logic. Stud. Logica 55, 301–317 (1995)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Varlet, J.: A regular variety of type (2,2,1,1,0,0). Algebra Univ. 2, 218–223 (1972)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dai, J.-H.: Logic for rough sets with rough double stone algebraic semantics. In: Ślęzak, D., Wang, G., Szczuka, M., Düntsch, I., Yao, Y. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 141–148. Springer, Heidelberg (2005). doi:10.1007/11548669_15 CrossRefGoogle Scholar
  13. 13.
    Banerjee, M., Khan, M.A.: Propositional logics from rough set theory. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI. LNCS, vol. 4374, pp. 1–25. Springer, Heidelberg (2007). doi:10.1007/978-3-540-71200-8_1 CrossRefGoogle Scholar
  14. 14.
    Comer, S.: Perfect extensions of regular double Stone algebras. Algebra Univ. 34(1), 96–109 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Institute of ScienceBanaras Hindu UniversityVaranasiIndia
  2. 2.Department of Mathematics and StatisticsIndian Institute of TechnologyKanpurIndia

Personalised recommendations