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Implicational F-structures and implicational relevance logics

Published online by Cambridge University Press:  12 March 2014

A. Avron
A. Avron*
Affiliation:
Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel Aviv University, Ramat Aviv 69978, Israel
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Abstract

We describe a method for obtaining classical logic from intuitionistic logic which does not depend on any proof system, and show that by applying it to the most important implicational relevance logics we get relevance logics with nice semantical and proof-theoretical properties. Semantically all these logics are sound and strongly complete relative to classes of structures in which all elements except one are designated. Proof-theoretically they correspond to cut-free hypersequential Gentzen-type calculi. Another major property of all these logics is that the classical implication can faithfully be translated into them.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 2 , June 2000 , pp. 788 - 802
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[AB75]Anderson, A. R. and Belnap, N. D., Entailment, vol. 1, Princeton University Press, Princeton, N. J., 1975.Google Scholar
[ABD92]Anderson, A. R., Belnap, N. D., and Dunn, J. M., Entailment, vol. 2, Princeton University Press, Princeton, N. J., 1992.Google Scholar
[Av84]Avron, A., Relevant entailment—Semantics and formal systems, This Journal, vol. 49 (1984), pp. 334–342.Google Scholar
[Av87]Avron, A., A constructive analysis of RM, This Journal, vol. 52 (1987), pp. 939–951.Google Scholar
[Av90]Avron, A., Relevance and paraconsistency — A new approach, This Journal, vol. 55 (1990), pp. 707–732. Part II (the Formal systems): Notre Dame Journal of Formal Logic, vol. 31 (1990), pp. 169–202.Google Scholar
[Av95]Avron, A., The method of hypersequents in proof theory of propositional non-classical logics, Logic: Foundations to applications (Hodges, W., Hyland, M., Steinhorn, C., and Truss, J., editors), Oxford Science Publications, 1996, pp. 1–32.Google Scholar
[Av97]Avron, A., Multiplicative conjunction as an extensional conjunction, Logic Journal of the IGPL, vol. 5 (1997), pp. 181–208.CrossRefGoogle Scholar
[Av98]Avron, A., Multiplicative conjunction and the algebraic meaning of contraction and weakening, This Journal, vol. 63 (1998), pp. 831–859.Google Scholar
[Ch51]Church, A., The weak theory of implication, Kontrolliertes Denken, Munich, 1951.Google Scholar
[Do93]Došen, K., A historical introduction to substructural logics, Substructural logics (Schroeder-Heister, P. and Došen, K., editors), Oxford University Press, 1993, pp. 1–30.Google Scholar
[Du70]Dunn, J. M., Algebraic completeness results for R-mingle and its extensions This Journal, vol. 35 (1970), pp. 1–13.Google Scholar
[Du86]Dunn, J. M., Relevant logic and entailment, Handbook of philosophical logic (Gabbay, D. and Guenthner, F., editors), vol. III, Reidel, Dordrecht, Holland; Boston, U.S.A., 1986.Google Scholar
[MP72]Meyer, R. K. and Parks, R. Z., Independent axioms for the implicational fragment of Sobocinski's three-valued logic, Zeitschrift fur Mathematische Logic und Grundlagen der Mathematik, vol. 18 (1972), pp. 291–295.CrossRefGoogle Scholar
[MS92]Meyer, R. K. and Slaney, J., A structurally complete fragment of relevant logic, Notre-Dame Journal of Formal Logic, vol. 33 (1992), pp. 561–566.CrossRefGoogle Scholar
[So52]Sobociński, B., Axiomatization of partial system of three-valued calculus of propositions, The Journal of Computing Systems, vol. 1 (1952), pp. 23–55.Google Scholar