Skip to main content
SpringerLink
Log in

Reasoning with Incomplete Information in Generalized Galois Logics Without Distribution: The Case of Negation and Modal Operators

  • Chapter
  • First Online:
Book cover J. Michael Dunn on Information Based Logics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 8))

Abstract

We extend Dunn’s treatment of various forms of negation developed in the context of his theory of generalized Galois logics (known as gaggle theory), by dropping the assumption of distribution. We also study modal operators of possibility and impossibility in a non-distributive context and in standard Kripke semantics, thus improving significantly over existing approaches developed in the last decade or so on the semantics of modalities when distribution of conjunction over disjunction and conversely is dropped. We prove representation and completeness theorems for the related logical calculi in appropriate Kripke frames. Without distribution, the points of the frame (we call them information sites) appear as possessing incomplete only information, supporting the truth of a disjunction \(\varphi \vee \psi \) without necessarily supporting the truth of either \(\varphi \) or \(\psi \). Our approach is based on and extends past results we have obtained on the (topological) representation (and Stone type duality) of non-distributive lattices with additional operators.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Allwein, G., & Hartonas, C. (1993). Duality for bounded lattices, Preprint series, IULG-93-25, Indiana University Logic Group.

    Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  • Conradie, W., & Palmigiano, A. (2015). Algorithmic correspondence and canonicity for non-distributive logics. Journal of Logic and Computation. Forthcoming.

    Google Scholar 

  • Dalla Chiara, M. L., & Giuntini, R. (2001). Qantum logics. arXiv:quant-ph/0101028.

  • Dalla Chiara, M. L., & Giuntini, R. (2002). Quantum logics. In D. M. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (pp. 129–228). Dordrecht: Kluwer Academic Publishers.

    Chapter  Google Scholar 

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

    Google Scholar 

  • Došen, K. (1999). Negation in the light of modal logic. In D. Gabbay & H. Wansing (Eds.), What is negation? (pp. 77–86). Dordrecht: Kluwer.

    Google Scholar 

  • Dunn, J. M. (1991). Gaggle theory: An abstraction of Galois connections and residuation with applications to negation, implication, and various logical operators. In J. van Eijck (Ed.), Logics in AI: European workshop JELIA ’90, Lecture notes in computer science (Vol. 478, pp. 31–51). Berlin: Springer.

    Google Scholar 

  • Dunn, J. M. (1993). Star and perp: Two treatments of negation, Philosophical Perspectives 7, 331–357. (Language and Logic, J. E. Tomberlin (Ed.)).

    Google Scholar 

  • Dunn, J. M. (1996). Generalised ortho negation. In H. Wansing (Ed.), Negation: A notion in focus (pp. 3–26). New York, NY: Walter de Gruyter.

    Google Scholar 

  • Dunn, J. M. (1999). A comparative study of various model-theoretic treatments of negation: A history of formal negation. In D. M. Gabbay & H. Wansing (Eds.), What is negation? (pp. 23–51). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Dunn, J. M., Moss, L. S., & Wang, Z. (2013). The third life of quantum logic: Quantum logic inspired by quantum computing. Journal of Philosophical Logic, 42, 443–459.

    Article  Google Scholar 

  • Dunn, J. M., & Zhou, C. (2005). Negation in the context of gaggle theory. Studia Logica, 80, 235–264.

    Article  Google Scholar 

  • Düntsch, I., Orlowska, E., Radzikowska, A. M., & Vakarelov, D. (2004). Relational representation theorems for some lattice-based structures, Preprint CS-04-07.

    Google Scholar 

  • Gehrke, M. (2006). Generalized Kripke frames. Studia Logica, 84(2), 241–275.

    Article  Google Scholar 

  • Gerhke, M., & Harding, J. (2001). Bounded lattice expansions. Journal of Algebra, 238, 345–371.

    Article  Google Scholar 

  • Gerhke, M., & van Gool, S. J. (2014). Distributive envelopes and topological duality for lattices via canonical extensions. Order, 31(3), 435–461.

    Article  Google Scholar 

  • Goldblatt, R. (1974). Semantic analysis of orthologic. Journal of Philosophical Logic, 3, 19–35.

    Article  Google Scholar 

  • Goldblatt, R. (1984). Orthomodularity is not elementary. Journal of Symbolic Logic, 49, 401–404.

    Article  Google Scholar 

  • Goldblatt, R. (1993). Mathematics of modality, CSLI lecture notes (Vol. 43). Stanford, CA: CSLI Publications.

    Google Scholar 

  • Hartonas, C. (1996). Order-duality, negation and lattice representation. In H. Wansing (Ed.), Negation: A notion in focus (pp. 27–36). W. de Gruyter.

    Google Scholar 

  • Hartonas, C. (1997). Duality for lattice-ordered algebras and for normal algebraizable logics. Studia Logica, 58, 403–450.

    Article  Google Scholar 

  • Hartonas, C. (2015). Elementary (first-order) frames for orthomodularity and the semantics of orthomodular quantum logic, Preprint CS-2015-3, University of Applied Sciences of Thessaly (TEI of Thessaly).

    Google Scholar 

  • Hartonas, C., & Dunn, J. M. (1997). Stone duality for lattices. Algebra Universalis, 37, 391–401.

    Article  Google Scholar 

  • Hartung, G. (1992). A topological representation for lattices. Algebra Universalis, 29, 273–299.

    Article  Google Scholar 

  • Järvinen, J., & Orlowska, E. (2005). Relational correspondences for lattices with operators. In I. D. Wendy MacCaull & M. Winter (Eds.), Relational Methods in Computer Science. Amsterdam.

    Google Scholar 

  • Jónsson, B., & Tarski, A. (1952). Boolean algebras with operators, II. American Journal of Mathematics, 74(1), 127–162.

    Article  Google Scholar 

  • Kamide, N. (2002). Kripke semantics for modal substructural logics. Journal of Logic, Language and Information, 11, 453–470.

    Article  Google Scholar 

  • Moshier, M. A., & Jipsen, P. (2014a). Topological duality and lattice expansions, I: A topological construction of canonical extensions. Algebra Universalis, 71, 109–126.

    Article  Google Scholar 

  • Moshier, M. A., & Jipsen, P. (2014b). Topological duality and lattice expansions, II: Lattice expansions with quasi-operators. Algebra Universalis, 71, 221–234.

    Article  Google Scholar 

  • Priestley, H. A. (1970). Representation of distributive lattices by means of ordered Stone spaces. Bulletin of the London Mathematical Society, 2, 186–190.

    Article  Google Scholar 

  • Routley, R., & Meyer, R. K. (1973). The semantics of entailment. In H. Leblanc (Ed.), Truth, syntax and modality. Proceedings of the Temple University Conference on Alternative Semantics (pp. 199–243). North-Holland, Amsterdam.

    Google Scholar 

  • Routley, R., & Routley, V. (1972). The semantics of first degree entailment. Noûs, 6(4), 335–359.

    Article  Google Scholar 

  • Stone, M. H. (1937–38). Topological representations of distributive lattices and Brouwerian logics. Časopis pro pěstování matematiky a fysiky, Čast matematická, 67, 1–25.

    Google Scholar 

  • Stone, M. H. (1938). The representation of Boolean algebras. Bulletin of the American Mathematical Society, 44, 807–816.

    Article  Google Scholar 

  • Suzuki, T. (2010). Bi-approximation semantics for substructural logic at work. Advances in Modal Logic, 8, 411–433.

    Google Scholar 

  • Suzuki, T. (2012). Morphisms on bi-approximation semantics. Advances in Modal Logic, 9, 494–515.

    Google Scholar 

  • Suzuki, T. (2014). On polarity frames: Applications to substructural and lattice-based logics. Advances in Modal Logic, 10, 533–552.

    Google Scholar 

  • Urquhart, A. (1978). A topological representation theorem for lattices. Algebra Universalis, 8, 45–58.

    Article  Google Scholar 

  • Vakarelov, D. (1977). Theory of Negation in Certain Logical Systems: Algebraic and Semantic Approach, PhD thesis, University of Warsaw.

    Google Scholar 

  • Vakarelov, D. (1989). Consistency, completeness and negations. In G. Priest, R. Routley, & J. Norman (Eds.), Paraconsistent logic: Essays on the inconsistent (pp. 328–368). Munich: Philosophia Verlag.

    Google Scholar 

  • Wille, R. (1987). Bedeutungen von Begriffsverbänden. In R. W. B. Ganter & K. E. Wolff (Eds.), Beiträge zur Begriffsanalyse (pp. 161–211). Mannheim: B. L.-Wissenschafts Verlag.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science & Engineering, University of Applied Sciences of Thessaly (TEI of Thessaly), Larissa, Greece

    Chrysafis Hartonas

Authors
  1. Chrysafis Hartonas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chrysafis Hartonas .

Editor information

Editors and Affiliations

  1. Department of Philosophy, University of Alberta, Edmonton, Alberta, Canada

    Katalin Bimbó

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Hartonas, C. (2016). Reasoning with Incomplete Information in Generalized Galois Logics Without Distribution: The Case of Negation and Modal Operators. In: Bimbó, K. (eds) J. Michael Dunn on Information Based Logics. Outstanding Contributions to Logic, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-29300-4_14

Download citation

Publish with us

Policies and ethics

Search

Navigation