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Modal logic

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Abstract

This paper surveys the main concepts and systems of modal logic. It shows how the tree or tableau method provides a simple and easily comprehensible decision procedure for systems such as K, T, S4 and S5. It also shows how the formal techniques of modal logic can be used to analyse several informal problems involving modal concepts, including cases combining modality with quantification. Although much of the discussion is in terms of the possible worlds interpretation of the modalities, provability interpretations are also examined. One of the central applications of modal operators within Artificial Intelligence (AI) is then discussed: the analysis of non-monotonic reasoning.

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References

  • Boolos, G. (1979) The Unprovability of Consistency, Cambridge University Press.

  • Cargile, J. & Thomas, G. (1968) A Note on ‘Time, Truth and Modalities’, Mind, LXXVII, 572–574.

    Google Scholar 

  • Farinas-del-Cerro, L. (1985) Resolution Modal Logic, Logique et Analyse, 28, 153–172.

    Google Scholar 

  • Frege, G. (1879) Begriffsschrift Chapter 1. In Translations from the Philosophical Writings of Gottlob Frege (eds P. Geach & M. Black), Blackwell, Oxford.

    Google Scholar 

  • Gabbay, D. M. (1989) Modal Provability Foundations for Negation By Failure 1, Report, Imperial College London (submitted for publication).

  • Gabbay, D. & Guenthner, F. (eds.) (1984) Handbook of Philosophical Logic. Volume II: Extensions of Classical Logic, Reidel, Dordrecht.

    Google Scholar 

  • Garson, J. (1984) Quantification in Modal Logic. In Handbook of Philosophical Logic (eds D. Gabbay & F. Guenthner).

  • Geissler, C. & Konolige, K. (1986) A Resolution Method for Quantified Modal Logics of Knowledge and Belief. In Theoretical Aspects of Reasoning About Knowledge (ed. J. Halpern), Morgan Kaufmann, Los Altos.

    Google Scholar 

  • Gödel, K. (1933) An Interpretation of the Intuitionistic Sentential Logic, Ergebnisse eines mathematischen Kolloquiums, Vol. 4, trans Hintikka (ed.) The Philosophy of Mathematics, Oxford University Press, Oxford.

    Google Scholar 

  • Hilpinen, R. (ed.) (1971) Deontic Logic: Introductory and Systematic Readings, Reidel, Dordrecht.

    Google Scholar 

  • Hodges, W. (1977) Logic, Penguin, Harmondsworth.

    Google Scholar 

  • Hughes, G. & Cresswell, M. (1968) An Introduction to Modal Logic, Methuen, London.

    Google Scholar 

  • Israel, D. (1980) What's Wrong with Non-Monotonic Logic?. In Proceedings of the First National Conference on Artificial Intelligence.

  • Kripke, S. (1963) Semantical Analysis of Modal Logic I, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 9, 67–96.

    Google Scholar 

  • Kripke, S. (1972) Naming and Necessity. In Semantics of Natural Language (eds D. Davidson & G. Harman), Reidel, Dordrecht.

    Google Scholar 

  • Lehrer, K. & Taylor, R. (1965) Time, Truth and Modalities, Mind, LXXIV, 390–398.

    Google Scholar 

  • Lemmon, E. J. (1959) Is There Only One Correct System of Modal Logic? Proceedings of the Aristotelian Society, Supplementary Volume, XXXIII, 23–40.

    Google Scholar 

  • McDermott, D. (1982) Non-monotonic Logic II: Non-monotonic modal theories, Journal of the ACM, 29, 33–57.

    Google Scholar 

  • McDermott, D. & Doyle, J. (1980) Non-monotonic Logic I, Artificial Intellience, 13, 41–72.

    Google Scholar 

  • Moore, R. (1983) Semantical Considerations on Non-monotonic Logic, Proceedings of the 8th IJCAI.

  • Prior, A. (1967) Past, Present and Future, Clarendon, Oxford.

    Google Scholar 

  • Quine, W.V.O. (1953) Three Grades of Modal Involvement, in The Ways of Paradox and Other Essays, Harvard University Press, Cambridge.

    Google Scholar 

  • Quine, W.V.O. (1960) Word and Object, MIT Press, Cambridge.

    Google Scholar 

  • Reeves, Steve (1988) Semantic Tableaux as Implementation Devices for Theorem-Provers. In Proceedings of the Workshop on Programming for Logic Teaching, (eds J. Derrick & H.A. Lewis), Centre for Theoretical Computer Science, University of Leeds.

  • Reiter, R. (1980) A Logic for Default Reasoning, Artificial Intelligence, 13, 81–132.

    Google Scholar 

  • Richards, B., Bethke, I., van der Does, J. & Oberlander, J. (1989) Temporal Representation and Inference. Academic Press, London.

    Google Scholar 

  • Thomason, R. (1984) Combinations of Tense and Modality. In Handbook of Philosophical Logic (eds D. Gabbay & F. Guenthner).

  • Wallen, L. A. & Wilson, G. V. (1987) A Computationally Efficient Proof System for S5 Modal Logic. In Advances in Artificial Intelligence (eds) J. Hallam & C. Mellish, Wiley, Chichester. pp. 141–153.

    Google Scholar 

  • Wright, C. (1980) Wittgenstein on the Foundations of Mathematics, Duckworth, London.

    Google Scholar 

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Authors and Affiliations

  1. A.I. Group, Middlesex Polytechnic, Queensway, EN3 4SF, Enfield, UK

    Richard Spencer-Smith

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  1. Richard Spencer-Smith
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Spencer-Smith, R. Modal logic. Artif Intell Rev 5, 5–34 (1991). https://doi.org/10.1007/BF00129533

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  • DOI: https://doi.org/10.1007/BF00129533

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