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Displaying the modal logic of consistency

Published online by Cambridge University Press:  12 March 2014

Heinrich Wansing
Heinrich Wansing*
Affiliation:
Dresden University of Technology, Institute of Philosophy, D-01062 Dresden, Germany, E-mail: Heinrich.Wansing@mailbox.tu-dresden.de
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Abstract

It is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.

Keywords

four-valued logicconstructive logicmodal logicmodal translationsnonmonotonic reasoning

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 4 , December 1999 , pp. 1573 - 1590
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1]Almukdad, A. and Nelson, D., Constructible falsity and inexact predicates, this Journal, vol. 49 (1984), pp. 231–233.Google Scholar
[2]Barba, J., A modal reduction for partial logic, Journal of Philosophical Logic, vol. 22 (1993), pp. 429–435.CrossRefGoogle Scholar
[3]Belnap, N., A useful four-valued logic, Modern uses of multiple-valued logic (Dunn, J. M. and Epstein, G., editors), Reidel, Dordrecht, 1977, pp. 8–37.Google Scholar
[4]Belnap, N., Display logic, Journal of Philosophical Logic, vol. 11 (1982), pp. 375–417.CrossRefGoogle Scholar
[5]Belnap, N., Linear logic displayed, Notre Dame Journal of Formal Logic, vol. 31 (1990), pp. 14–25.Google Scholar
[6]Belnap, N., The display problem, Proof theory of modal logic (Wansing, H., editor), Kluwer Academic Publishers, Dordrecht, 1996, pp. 79–93.CrossRefGoogle Scholar
[7]Clarke, M., Intuitionistic non-monotonic reasonig—further results, ECAI88, Proceedings of the 8th European conference on artificial intelligence (Kodratoff, Y., editor), Pitman, London, 1988, pp. 525–527.Google Scholar
[8]Clarke, M. and Gabbay, D., An intuitionistic basis for non-monotonic reasoning, Non-standard logics for automated reasoning (Smets, P.et al., editors), Academic Press, London, 1988, pp. 163–178.Google Scholar
[9]Dunn, M., The algebra of intensional logics, Ph.D. thesis, University of Pittsburgh, Ann Arbor, 1966, University Microfilms.Google Scholar
[10]Gabbay, D., Intuitionistic basis for non-monotonic logic, Proceedings of the 6th conference on automated deduction, Lecture Notes in Computer Science, no. 138, Springer-Verlag, Berlin, 1982, pp. 260–273.CrossRefGoogle Scholar
[11]Goré, R., Intuitionistic logic redisplayed, Technical Report TR-ARP-1-95, Australian National University, Canberra, 1995.Google Scholar
[12]Jaspars, J., Calculi for constructive communication, Ph.D. thesis, University of Tilburg, 1994.Google Scholar
[13]Kracht, M., Power and weakness of the modal display calculus, Proof theory of modal logic (Wansing, H., editor), Kluwer Academic Publishers, Dordrecht, 1996, pp. 95–122.Google Scholar
[14]Łukaszewicz, W., Non-monotonic reasoning. Formalization of commonsense reasoning, Ellis Horwood, Chichester, 1990.Google Scholar
[15]Mcdermott, D. and Doyle, J., Non-monotonic logic I, Journal of Artificial Intelligence, vol. 13 (1980), pp. 41–72.CrossRefGoogle Scholar
[16]Nelson, D., Constructible falsity, this Journal, vol. 14 (1949), pp. 16–26.Google Scholar
[17]Rautenberg, W., Klassische und nicht-klassische Aussagenlogik, Vieweg, Braunschweig, 1979.CrossRefGoogle Scholar
[18]Reiter, R., A logic for default reasoning, Artificial Intelligence, vol. 13 (1980), pp. 81–132.CrossRefGoogle Scholar
[19]Restall, G., Display logic and gaggle theory, Reports on Mathematical Logic, vol. 29 (1995), pp. 133–146, published in 1996.Google Scholar
[20]Restall, G., Displaying and deciding substructural logics 1: Logics with contraposition, Journal of Philosophical Logic, vol. 27 (1998), pp. 179–216.CrossRefGoogle Scholar
[21]Turner, R., Logics for artificial intelligence, Ellis Horwood, Chichester, 1984.Google Scholar
[22]Urquhart, A., Many-valued logic, Handbook of philosophical logic (Gabbay, D. and Guenthner, F., editors), vol. 3, 1986, pp. 71–116.CrossRefGoogle Scholar
[23]van Dalen, D., Intuitionistic logic, Handbook of philosophical logic (Gabbay, D. and Guenthner, F., editors), vol. 3, 1986, pp. 225–339.CrossRefGoogle Scholar
[24]Wansing, H., Sequent calculi for normal modal propositional logics, Journal of Logic and Computation, vol. 4 (1994), pp. 125–142.CrossRefGoogle Scholar
[25]Wansing, H., Semantics-based nonmonotonic inference, Notre Dame Journal of Formal Logic, vol. 36 (1995), pp. 44–54.CrossRefGoogle Scholar
[26]Wansing, H., Strong cut-elimination in display logic, Reports on Mathematical Logic, vol. 29 (1995), pp. 117–131, published in 1996.Google Scholar
[27]Wansing, H., A full-circle theorem for simple tense logic, Advances in intensional logic (de Rijke, M., editor), Kluwer Academic Publishers, Dordrecht, 1997, pp. 173–193.CrossRefGoogle Scholar
[28]Wansing, H., Negation as falsity: A reply to Tennant, What is negation? (Gabbay, D. and Wansing, H., editors), Kluwer Academic Publishers, Dordrecht, 1999, pp. 223–238.CrossRefGoogle Scholar
[29]Wansing, H., Predicate logics on display, Studia Logica, vol. 62 (1999), pp. 49–75.CrossRefGoogle Scholar