Routley Star and Hyperintensionality


We compare the logic HYPE recently suggested by H. Leitgeb as a basic propositional logic to deal with hyperintensional contexts and Heyting-Ockham logic introduced in the course of studying logical aspects of the well-founded semantics for logic programs with negation. The semantics of Heyting-Ockham logic makes use of the so-called Routley star negation. It is shown how the Routley star negation can be obtained from Dimiter Vakarelov’s theory of negation and that propositional HYPE coincides with the logic characterized by the class of all involutive Routley star information frames. This result provides a much simplified semantics for HYPE and also a simplified axiomatization, which shows that HYPE is identical with the modal symmetric propositional calculus introduced by G. Moisil in 1942. Moreover, it is shown that HYPE can be faithfully embedded into a normal bi-modal logic based on classical logic. Against this background, we discuss the notion of hyperintensionality.

This is a preview of subscription content, access via your institution.


  1. 1.

    Białynicki-Birula, B., & Rasiowa, H. (1957). On the representation of quasi-Boolean algebras. Bulletin de l’Academie Polonaise des Sciences, 5, 259–261.

    Google Scholar 

  2. 2.

    Cabalar, P., Odintsov, S.P., & Pearce, D. (2006). Logical foundations of well-founded semantics. In Doherty, P. et al. (Eds.) Principles of knowledge representation and reasoning: proceedings of the 10-th international conference (KR2006) (pp. 25–36). Menlo Park: AAAI Press.

  3. 3.

    da Costa, N.C.A. (1974). On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic, 15, 497–510.

    Article  Google Scholar 

  4. 4.

    Cresswell, M. (1975). Hyperintensional logic. Studia Logica, 34, 25–38.

    Article  Google Scholar 

  5. 5.

    Došen, K. (1986). Negation as a modal operator. Reports on Mathematical Logic, 20, 15–27.

    Google Scholar 

  6. 6.

    Drobyshevich, S.A. (2015). Double negation operator in logic N. Journal of Mathematical sciences, 205(3), 389–402.

    Article  Google Scholar 

  7. 7.

    Drobyshevich, S.A., & Odintsov, S.P. (2013). Finite model property for negative modalities. Siberian Electronic Mathematical Reports, 10, 1–21. [Finite model property for negative modalities].

    Google Scholar 

  8. 8.

    Dunn, J.M. (1993). Star and Perp: Two Treatments of Negation. Philosophical Perspectives, 7, 331–357.

    Article  Google Scholar 

  9. 9.

    Font, J.M. (2016). Abstract algebraic logic an introductory textbook. London: College Publications.

    Google Scholar 

  10. 10.

    Horn, L., & Wansing, H. (2015). Negation, the Stanford encyclopedia of philosophy (Summer 2015 Edition). In EN Zalta (Ed.),

  11. 11.

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

    Google Scholar 

  12. 12.

    Humberstone, L. (2018). Sentence connectives in formal logic, the Stanford encyclopedia of philosophy (Summer 2018 Edition). In E.N. Zalta (Ed.),

  13. 13.

    Iturrioz, L. (1968). Sur une classe particulière d’algèbres de Moisil. C.R. Acad. Sc. Paris, 267, 585–588.

    Google Scholar 

  14. 14.

    Jago, M. (2014). The Impossible. An Essay on Hyperintensionality. Oxford University Press.

  15. 15.

    Johansson, J. (1937). Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus. Compositio Mathematica, 4, 119–136.

    Google Scholar 

  16. 16.

    Kamide, N., & Wansing, H. (2015). Proof theory of N4-related paraconsistent logics, studies in logic Vol. 54. London: College Publications.

    Google Scholar 

  17. 17.

    Leitgeb, H. (2019). HYPE: a system of hyperintensional logic (with an application to semantic paradoxes). Journal of Philosophical Logic, 48(2), 305–405.

    Article  Google Scholar 

  18. 18.

    Mares, E. (2004). Relevant logic: a philosophical interpretation. Cambridge: Cambridge University Press.

    Google Scholar 

  19. 19.

    Moisil, G.C. (1942). Logique modale. Disquisitiones Mathematicae et Physicae, 2, 3–98.

    Google Scholar 

  20. 20.

    Monteiro, A. (1969). Sur quelques extensions du calcul propositionnel intuitioniste. In IVème congrès des mathématiciens d’expression latine. Bucarest 17.–24.

  21. 21.

    Monteiro, A. (1980). Sur les algèbres de Heyting symétriques. Portugaliae Mathematica, 39(1–4), 1–237. Special Issue in honor of António Monteiro.

    Google Scholar 

  22. 22.

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

    Article  Google Scholar 

  23. 23.

    Odintsov, S.P. (2010). Combining intuitionistic connectives and Routley negation. Siberian Electronic Mathematical Reports, 7, 21–41.

    Google Scholar 

  24. 24.

    Odintsov, S.P., & Wansing, H. (2015). The logic of generalized truth values and the logic of bilattices. Studia Logica, 103, 91–112.

    Article  Google Scholar 

  25. 25.

    Odintsov, S.P., & Wansing, H. (2016). On the methodology of paraconsistent logic. In Andreas, H., & Verdée, P. (Eds.) Logical studies of paraconsistent reasoning in science and mathematics (pp. 175–204): Springer.

  26. 26.

    Odintsov, S.P., & Wansing, H. (2017). Disentangling FDE-based paraconsistent modal logics. Studia Logica, 105(6), 1221–1254.

    Article  Google Scholar 

  27. 27.

    Omori, H., & Wansing, H. (2017). 40 years of FDE: an introductory overview. Studia Logica, 105, 1021–1049.

    Article  Google Scholar 

  28. 28.

    Priest, G. (2008). Many-valued modal logics: a simple approach. Review of Symbolic Logic, 1(2), 190–203.

    Article  Google Scholar 

  29. 29.

    Restall, G. (1999). Negation in relevant logics (how i stopped worrying and learned to love the Routley star). In Gabbay, D., & H Wansing, H (Eds.) What is negation? (pp. 53–76). Dordrecht: Springer.

  30. 30.

    Routley, R., & Meyer, R.K. (1973). The semantics of entailment I. In Leblanc, H. (Ed.) Truth, syntax and modality (pp. 199–243). North-Holland.

  31. 31.

    Routley, R., & Routley, V. (1972). Semantics of first-degree entailment. Noûs, 6, 335–359.

    Article  Google Scholar 

  32. 32.

    Sankappanavar, H.P. (1987). Heyting algebras with a dual lattice endomorphism. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 33, 565–573.

    Article  Google Scholar 

  33. 33.

    Shramko, Y., & Wansing, H. (2011). Truth and falsehood. An inquiry into generalized logical values, trends in logic Vol. 36. Berlin: Springer.

    Google Scholar 

  34. 34.

    Speranski, S. (2019). Negation as a modal operator in a quantified setting, submitted.

  35. 35.

    Sylvan, R. (1990). Variations on da Costa C systems and dual-intuitionistic logics I. Analyses of Cω and CCω. Studia Logica, 49, 47–65.

    Article  Google Scholar 

  36. 36.

    Urquhart, A. (1979). Distributive lattices with a dual homomorphic operation. Studia Logica, 38, 201–209.

    Article  Google Scholar 

  37. 37.

    Van Gelder, A., Ross, K.A., & Schlipf, J.S. (1991). The well founded semantics for general logic programs. Journal of the Association for Computing Machinery, 38, 620–650.

    Google Scholar 

  38. 38.

    Vakarelov, D. (1976). Theory of negation in certain logical systems. Algebraic and semantical approach. Ph.D. dissertation. University of Warsaw.

  39. 39.

    Vakarelov, D. (1976). Obobschennye reshetki Nelsona, In Chetvertaya Vsesoyuznaya Conferenciya po Matematicheskoy Logike, tezisy dokladov i soobschtenii, Kishinev. [Generalized Nelson’s lattices].

  40. 40.

    Vakarelov, D. (1989). Consistency, completeness and negation. In Priest, G. et al. (Eds.) Paraconsistent logic. Essays on the inconsistent (pp. 328–363). Münich: Philosophia Verlag.

  41. 41.

    Vakarelov, D. (2005). Nelson’s negation on the base of weaker versions of intuitionistic negation. Studia Logica, 80, 393–430.

    Article  Google Scholar 

  42. 42.

    Wansing, H. (2016). Falsification, natural deduction, and bi-intuitionistic logic. Journal of Logic and Computation, 26, 425–450. published online July 2013.

    Article  Google Scholar 

  43. 43.

    Wansing, H. (2016). On split negation, strong negation, information, falsification, and verification. In Bimbó, K., & Michael, J (Eds.) Dunn on information based logics (pp. 161–189): Springer.

  44. 44.

    Wansing, H. (2017). A more general general proof theory. Journal of Applied Logic, 25, 23–46.

    Article  Google Scholar 

  45. 45.

    Williamson, T. (2006). Indicative versus subjunctive conditionals, congruential versus non-hyperintensional contexts. Philosophical Issues, 16, 310–333.

    Article  Google Scholar 

  46. 46.

    Wójcicki, R. (1979). Referential matrix semantics for propositional calculi. Bulletin of the Section of Logic, 8, 170–176.

    Google Scholar 

Download references


The work reported in this paper has been carried out as part of the joint research project “FDE-based modal logics,” supported by the Russian Foundation for Basic Research, RFBR, grant No. 18-501-12019 and the Deutsche Forschungsgemeinschaft, DFG, grant WA 936/13-1. We gratefully acknowledge this support. Moreover, we would like to thank the two anonymous referees for their very helpful comments.

Author information



Corresponding author

Correspondence to Sergei Odintsov.

Additional information

Publisher’s Note

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

The work reported in this paper was supported by the Russian Foundation for Basic Research, RFBR, grant No. 18-501-12019 and the Deutsche Forschungsgemeinschaft, DFG, grant WA 936/13-1.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Odintsov, S., Wansing, H. Routley Star and Hyperintensionality. J Philos Logic 50, 33–56 (2021).

Download citation


  • Hyperintensional contexts
  • Routley star operation
  • Heyting-Ockham logic
  • Vakarelov’s theory of negation
  • HYPE