Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T22:13:53.575Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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