Abstract
This paper recounts the history and solution of the problem of admissibility of the rule \(\gamma \) in the context of E and other relevance logics.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ackermann, W. (1956). Begründung einer strengen Implikation. Journal of Symbolic Logic, 21, 113–128.
Aczel, P. (1968). Saturated intuitionistic theories, Contributions to Mathematical Logic. Proceedings of the Logic Colloquium, Hannover 1966 (pp. 1–11), Vol. 50 of Studies in Logic and the Foundations of Mathematics, North Holland.
Anderson, A. R. (1963). Some open problems concerning the system E of entailment. Proceedings of a Colloquium on Modal and Many-Valued Logics, Helsinki. Retrieved August 23–26, 1962, number Fasc. 16. in Acta Philosophica Fennica, pp. 9–18.
Anderson, A. R., & Belnap, N. D. (1975). Entailment: The logic of relevance and necessity (Vol. I). Princeton, NJ: Princeton University Press.
Bimbó, K., & Dunn, J. M. (2008). Generalized Galois Logics. Relational Semantics of Nonclassical Logical Calculi, Vol. 188 of CSLI Lecture Notes. Stanford, CA: CSLI Publications.
Dunn, J. M. (1966). The Algebra of Intensional Logics, PhD thesis, University of Pittsburgh, Ann Arbor (UMI).
Dunn, J. M. (1970). Algebraic completeness results for R-Mingle and its extensions. Journal of Symbolic Logic, 35, 1–13.
Dunn, J. M., & Hardegree, G. M. (2001). Algebraic methods in philosophical logic, Vol. 41 of Oxford Logic Guides. Oxford, UK: Oxford University Press.
Dunn, J. M., & Meyer, R. K. (1989). Gentzen’s cut and Ackermann’s gamma. In J. Norman & R. Sylvan (Eds.), Directions in Relevant Logic (pp. 229–240). Dordrecht: Kluwer.
Dunn, J. M., & Restall, G. (2002). Relevance logic. In D. Gabbay & F. Guenthner (Eds.), Handbook of Philosophical Logic (2nd ed., Vol. 6, pp. 1–128). Amsterdam: Kluwer.
Harrop, R. (1956). On disjunctions and existential statements in intuitionistic systems of logic. Mathematische Annalen, 132, 342–361.
Harrop, R. (1960). Concerning formulas of the types \(A\rightarrow B\vee C\), \(A\rightarrow (\exists x)Bx\) in intuitionistic formal systems. Journal of Symbolic Logic, 25, 27–32.
Kleene, S. C. (1962). Disjunction and existence under implication in elementary intuitionistic formalisms. Journal of Symbolic Logic, 27, 11–17.
Meyer, R. K. (1971a). On coherence in modal logics. Logique et Analyse, 14, 658–668.
Meyer, R. K. (1971b). R-mingle and relevant disjunction, (abstract). Journal of Symbolic Logic, 36, 366.
Meyer, R. K. (1976a). Ackermann, Takeuti and Schnitt: \(\gamma \) for higher-order relevant logics. Bulletin of the Section of Logic, 138–144.
Meyer, R. K. (1976b). Metacompleteness. Notre Dame Journal of Formal Logic, 17(4), 501–516.
Meyer, R. K., & Dunn, J. M. (1968). Entailment logics and material implication, (abstract). Notices of the American Mathematical Society, 15, 1021–1022.
Meyer, R. K., & Dunn, J. M. (1969). E, R and \(\gamma \). Journal of Symbolic Logic, 34, 460–474.
Meyer, R. K., Dunn, J. M., & Leblanc, H. (1974). Completeness of relevant quantification theories. Notre Dame Journal of Formal Logic, 15(1), 97–121.
Pudlák, P. (1998). The lengths of proofs. In S. R. Buss (Ed.), Handbook of Proof Theory (pp. 547–637). Amsterdam: North Holland.
Routley, R., & Meyer, R. K. (1973). The semantics of entailment. In H. Leblanc (Ed.), Truth, Syntax and Modality. Proceedings of the Temple University Conference on Alternative Semantics, (pp. 199–243). Amsterdam: North-Holland.
Statman, R. (1978). Bounds for proof-search and speed-up in the predicate calculus. Annals of Mathematical Logic, 15, 225–287.
Sugihara, T. (1955). Strict implication free from implicational paradoxes. Memoirs of the Faculty of Liberal Arts, Fukui University, Series, 1(4), 55–59.
Urquhart, A. (1996). Duality for algebras of relevant logics. Studia Logica, 56, 263–276.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Urquhart, A. (2016). The Story of \(\gamma \) . In: Bimbó, K. (eds) J. Michael Dunn on Information Based Logics. Outstanding Contributions to Logic, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-29300-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-29300-4_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-29298-4
Online ISBN: 978-3-319-29300-4
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)