Generalising Propositional Modal Logic Using Labelled Deductive Systems

  • Alessandra Russo
Part of the Applied Logic Series book series (APLS, volume 3)

Abstract

A family of labelled deductive systems called Propositional Modal Labelled Deductive (PMLD) systems is described. These logics combine the standard syntax of propositional modal logic with a simple subset of first-order predicate logic, called a labelling algebra, to allow syntactic reference to a Kripke-like structure of possible worlds. PMLD systems are a generalisation of normal propositional modal logic in that they facilitate reasoning about what is true at different points in a (possibly singleton) structure of actual worlds, called a configuration. A model-theoretic semantics (based on first-order logic) is provided and its equivalence to Kripke semantics for normal propositional modal logics is shown whenever the initial configuration is a single point. A sound and complete natural deduction style proof system is also described. Unlike traditional proof systems for modal logics, this system is uniform in that every deduction rule is applicable to (the generalisation of) each normal modal logic extension of K obtained by adding combinations of the axiom schemas T, 4,5, D and B

Keywords

Modal Logic Actual World Inference Rule Proof System Natural Deduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.R.F. Benevides. Multiple Database Logic. Ecsqaru’95, Lecture Notes in Artificial Intelligence, 946, 1995Google Scholar
  2. T. Borghuis. Interpreting Modal Natural Deduction in Type Theory. In Maarten de Rijke editor, Diamonds and Defaults, Springer Science+Business Media New York , 67–102, 1993Google Scholar
  3. M. Fitting. Proof Methods for Modal and Intuitionistic Logics. D. Reidel, Dordrecht, 1983CrossRefGoogle Scholar
  4. A. M. Frisch and R. B. Scherl. A General Framework for Modal Deduction. In Proceedings of the 2nd Conference on Principles of Knowledge Representation and Reasoning. Morgan-Kaufmann, 1991Google Scholar
  5. D.M. Gabbay. How to Construct a Logic for Your Application. Gwai’92, Lecture Note in Artificial Intelligence, 671:1–30, 1992MathSciNetGoogle Scholar
  6. D.M. Gabbay. LDS - Labelled Deductive Systems, Volume 1 - Foundations.Technical Report MPI-I-94/223, Max-Planck-Institut Fur Informatik, 1994Google Scholar
  7. I. Gent. Theory Matrices (for Modal Logics) using Alphabetical Monotonicity. Studia Logica, 52(2):233–257,1993CrossRefMATHMathSciNetGoogle Scholar
  8. G.E. Hughes M.J. Cresswell. An Introduction to Modal Logic. Methuen and Co. Ltd, 1968. Reprinted by Routledge, 1989Google Scholar
  9. F. Massacci. Strong Analytic Tableaux for Normal Modal Logics. In Proceedings of CADE-12, LNAI 814 Springer, 1994Google Scholar
  10. R.C. Moore. Reasoning About Knowledge and Action. MIT, Cambridge, 1980Google Scholar
  11. H.J. Ohlbach. Semantics-based Translation Methods for Modal Logics. Journal of Logic and Computation, l(5):691–746,1991CrossRefMATHMathSciNetGoogle Scholar
  12. D.Prawitz. Natural Deduction: a Proof-Theoretical Study. Almqvist and Wiksell, 1965Google Scholar
  13. A. Russo. Modal Labelled Deductive Systems. Technical Report 95/7, Imperial College, Department of Computing, 1995.Available at http://theory.doc.ic.ac.uk/~ar3/PMLDSreport.psGoogle Scholar
  14. A. Sympson. The Proof Theory and Semantics of Intuitionistic Modal Logics. PhD Thesis,University of Edinburgh, 1993Google Scholar
  15. J. van Benthem. Correspondence Theory. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume II, Extensions of Classical Logics. D. Reidel Publishing Company, 1983Google Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Alessandra Russo
    • 1
  1. 1.Department of ComputingImperial College of Science Technology and MedicineLondonUnited Kingdom

Personalised recommendations