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CUT AND GAMMA I: PROPOSITIONAL AND CONSTANT DOMAIN R

Published online by Cambridge University Press:  29 August 2019

YALE WEISS
YALE WEISS*
Affiliation:
The Saul Kripke Center, The Graduate Center, CUNY
*
†THE SAUL KRIPKE CENTER THE GRADUATE CENTER, CUNY 365 FIFTH AVE., ROOM 7118 NEW YORK, NY 10016, USA E-mail: yweiss@gradcenter.cuny.edu
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Abstract

The main object of this article is to give two novel proofs of the admissibility of Ackermann’s rule (γ) for the propositional relevant logic R. The results are established as corollaries of cut elimination for systems of tableaux for R. Cut elimination, in turn, is established both nonconstructively (as a corollary of completeness) and constructively (using Gentzen-like methods). The extensibility of the techniques is demonstrated by showing that (γ) is admissible for RQ* (R with constant domain quantifiers). The status of the admissibility of (γ) for RQ* was, to the best of the author’s knowledge, an open problem. Further extensions of these results will be explored in the sequel(s).

Keywords

03F0303F0503B47propositional relevant logicconstant domain relevant logicgammacut
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 4 , December 2020 , pp. 887 - 909
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Copyright
Copyright © Association for Symbolic Logic 2019 

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References

BIBLIOGRAPHY

Ackermann, W. (1956). Begründung einer strengen implikation. Journal of Symbolic Logic, 21(2), 113–128.CrossRefGoogle Scholar
Anderson, A. R. (1963). Some open problems concerning the system E of entailment. Acta Philosophica Fennica, 16, 7–18.Google Scholar
Anderson, A. R. & Belnap, N. D. Jr. (1975). Entailment: The Logic of Relevance and Necessity, Vol. I. Princeton: Princeton University Press.Google Scholar
Anderson, A. R., Belnap, N. D. Jr., & Dunn, J. M. (1992). Entailment: The Logic of Relevance and Necessity, Vol. II. Princeton: Princeton University Press.Google Scholar
Dunn, J. M. & Meyer, R. K. (1989). Gentzen’s cut and Ackermann’s gamma. In Norman, J. and Sylvan, R., editors. Directions in Relevant Logic. Dordrecht: Springer, pp. 229–240.CrossRefGoogle Scholar
Dunn, J. M. & Restall, G. (2002). Relevance logic. In Gabbay, D. and Guenthner, F., editors. Handbook of Philosophical Logic, Vol. 6. Dordrecht: Kluwer Academic Publishers, pp. 1–128.Google Scholar
Dyckhoff, R. & Negri, S. (2012). Proof analysis in intermediate logics. Archive for Mathematical Logic, 51(1–2), 71–92.CrossRefGoogle Scholar
Fine, K. (1974). Models for entailment. Journal of Philosophical Logic, 3(4), 347–372.CrossRefGoogle Scholar
Fine, K. (1988). Semantics for quantified relevance logic. Journal of Philosophical Logic, 17(1), 27–59.CrossRefGoogle Scholar
Fine, K. (1989). Incompleteness for quantified relevance logics. In Norman, J. and Sylvan, R., editors. Directions in Relevant Logic. Dordrecht: Springer, pp. 205–225.CrossRefGoogle Scholar
Fitting, M. (1996). First-Order Logic and Automated Theorem Proving (second edition). New York: Springer-Verlag.10.1007/978-1-4612-2360-3CrossRefGoogle Scholar
Fitting, M. & Mendelsohn, R. L. (1998). First-Order Modal Logic. Dordrecht: Kluwer Academic Publishers.10.1007/978-94-011-5292-1CrossRefGoogle Scholar
Kripke, S. A. (1965). Semantical analysis of intuitionistic logic I. In Crossley, J. N. and Dummett, M. A. E., editors. Formal Systems and Recursive Functions: Proceedings of the Eighth Logic Colloquium, Oxford, July 1963, Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland, pp. 92–130.CrossRefGoogle Scholar
Kripke, S. A. Logical Troubles, Chapter A Proof of Gamma. Oxford University Press, forthcoming.Google Scholar
McRobbie, M. A. & Belnap, N. D. Jr. (1979). Relevant analytic tableaux. Studia Logica, 38(2), 187–200.CrossRefGoogle Scholar
Méndez, J. M. (2009). A Routley-Meyer semantics for Ackermann’s logics of “strenge implikation”. Logic and Logical Philosophy, 18(3–4), 191–219.Google Scholar
Meyer, R. K. (1976). Ackermann, Takeuti, and schnitt: γ for higher-order relevant logic. Bulletin of the Section of Logic, 5(4), 138–144.Google Scholar
Meyer, R. K. & Dunn, J. M. (1969). E, R and γ. Journal of Symbolic Logic, 34(3), 460–474.CrossRefGoogle Scholar
Meyer, R. K., Dunn, J. M., & Leblanc, H. (1974). Completeness of relevant quantification theories. Notre Dame Journal of Formal Logic, 15(1), 97–121.CrossRefGoogle Scholar
Negri, S. (2005). Proof analysis in modal logic. Journal of Philosophical Logic, 34(5/6), 507–544.CrossRefGoogle Scholar
Negri, S. (2014). Proofs and countermodels in non-classical logics. Logica Universalis, 8(1), 25–60.CrossRefGoogle Scholar
Priest, G. (2008). An Introduction to Non-Classical Logic (second edition). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Priest, G. & Sylvan, R. (1992). Simplified semantics for basic relevant logics. Journal of Philosophical Logic, 21(2), 217–232.CrossRefGoogle Scholar
Restall, G. (1993). Simplified semantics for relevant logics (and some of their rivals). Journal of Philosophical Logic, 22(5), 481–511.CrossRefGoogle Scholar
Routley, R. & Meyer, R. K. (1973). The semantics of entailment. In Leblanc, H., editor. Truth, Syntax and Modality: Proceedings of the Temple University Conference on Alternative Semantics, Studies in Logic and the Foundations of Mathematics, Vol. 68. Amsterdam: North-Holland, pp. 199–243.CrossRefGoogle Scholar
Smullyan, R. M. (1968). First-Order Logic. New York: Dover Publications.CrossRefGoogle Scholar
Urquhart, A. (2016). The story of γ. In Bimbó, K., editor. J. Michael Dunn on Information Based Logics, Outstanding Contributions to Logic, Vol. 8. Cham: Springer, pp. 93–105.CrossRefGoogle Scholar