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RELEVANCE FOR THE CLASSICAL LOGICIAN

Published online by Cambridge University Press:  06 November 2018

ETHAN BRAUER
ETHAN BRAUER*
Affiliation:
Department of Philosophy, The Ohio State University
*
*DEPARTMENT OF PHILOSOPHY THE OHIO STATE UNIVERSITY 350 UNIVERSITY HALL 230 N. OVAL MALL COLUMBUS, OH 43210, USA E-mail: eebrauer@gmail.com
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Abstract

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Although much technical and philosophical attention has been given to relevance logics, the notion of relevance itself is generally left at an intuitive level. It is difficult to find in the literature an explicit account of relevance in formal reasoning. In this article I offer a formal explication of the notion of relevance in deductive logic and argue that this notion has an interesting place in the study of classical logic. The main idea is that a premise is relevant to an argument when it contributes to the validity of that argument. I then argue that the sequents which best embody this ideal of relevance are the so-called perfect sequents—that is, sequents which are valid but have no proper subsequents that are valid. Church’s theorem entails that there is no recursively axiomatizable proof-system that proves all and only the perfect sequents, so the project that emerges from studying perfection in classical logic is not one of finding a perfect subsystem of classical logic, but is rather a comparative study of classifying subsystems of classical logic according to how well they approximate the ideal of perfection.

Keywords

03A05relevanceperfect sequentsclassical logic
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 2 , June 2020 , pp. 436 - 457
Copyright
Copyright © Association for Symbolic Logic 2018 

References

BIBLIOGRAPHY

Abramsky, S. (1993). Computational interpretations of linear logic. Theoretical Computer Science, 111, 3–57.CrossRefGoogle Scholar
Anderson, A. R. & Belnap, N. D. (1962). Tautological entailments. Philosophical Studies, 13 (1), 9–24.CrossRefGoogle Scholar
Anderson, A. R. & Belnap, N. D. (1975). Entailment: The Logic of Relevance and Entailment, Vol. I. Princeton: Princeton University Press.Google Scholar
Anderson, A. R., Belnap, N. D., & Dunn, J. M. (1992). Entailment: The Logic of Relevance and Entailment, Vol. II. Princeton: Princeton University Press.Google Scholar
Avron, A. (1984). Relevant entailment—Semantics and formal systems. Journal of Symbolic Logic, 49(2), 334–342.CrossRefGoogle Scholar
Avron, A. (2014). What is a relevance logic? Annals of Pure and Applied Logic, 165, 26–48.CrossRefGoogle Scholar
Bennett, J. (1965). Review: T. J. Smiley, Entailment and Deducibility and Alan Ross Anderson and Nuel D. Belnap, The Calculus of Pure Entailment. Journal of Symbolic Logic, 30(2), 240–241.CrossRefGoogle Scholar
Boolos, G., Burgess, J., & Jeffrey, R. (2007). Computability and Logic (fifth edition). New York: Cambridge University Press.CrossRefGoogle Scholar
Burgess, J. (1981). Relevance: A fallacy? Notre Dame Journal of Formal Logic, 22, 97–104.CrossRefGoogle Scholar
Chang, C. & Keisler, H. J. (2012). Model Theory (third edition). Mineola, NY: Dover.Google Scholar
Conan Doyle, A. (1976). The adventure of the priory school. In The Original Illustrated Sherlock Holmes. Edison, NJ: Castle Books, pp. 508–526.Google Scholar
Copeland, B. (1980). The trouble Anderson and Belnap have with relevance. Philosophical Studies, 37, 325–334.CrossRefGoogle Scholar
Copeland, B. (1984). Horseshoe, hook, and relevance. Theoria, 50, 148–164.CrossRefGoogle Scholar
Diaz, M. R. (1981). Topics in the Logic of Relevance. Munich: Philosophia Verlag.Google Scholar
Dunn, J. M. & Restall, G. (2002). Relevance logic. In Gabbay, D. and Geunther, F., editors. Handbook of Philosophical Logic, Vol. 6. Dordrecht: Kluwer Academic, pp. 1–136.Google Scholar
Feferman, S. (1960). Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae, 49(1), 35–92.CrossRefGoogle Scholar
Frege, G. (1879). Begriffsschrift. In van Heijenoort, J., editor. From Frege to Gödel, Cambridge, MA: Harvard University Press, pp. 1–82.Google Scholar
Friedman, H. & Flagg, R. C. (1990). A framework for measuring the complexity of mathematical concepts. Advances in Applied Mathematics, 11, 1–34.CrossRefGoogle Scholar
Geach, P. (1958). Entailment. Proceedings of the Aristotelian Society, Supplementary Volumes, 32, 157–172.Google Scholar
Geach, P. (1970). Entailment. Philosophical Review, 79(2), 237–239.CrossRefGoogle Scholar
Gemes, K. (2007). Irrelevance: Strengthening the bayesian requriements. Synthese, 157(2), 161–166.CrossRefGoogle Scholar
Krämer, S. (2017). A hyperintensional criterion of irrelevance. Synthese, 194(8), 2917–2930.CrossRefGoogle Scholar
Krämer, S. & Roski, S. (2017). Difference-making grounds. Philosophical Studies, 174, 1191–1215.CrossRefGoogle Scholar
Lambek, J. (1958). The mathematics of sentence structure. American Mathematical Monthly, 65(3), 154–170.CrossRefGoogle Scholar
Lapara, N. (1976). Semantics for a natural notion of entailment. Philosophical Studies, 29(2), 91–113.CrossRefGoogle Scholar
Lehrer, K. (1973). Relevant deduction and minimally inconsistent sets. Philosophia, 3(3), 153–165.CrossRefGoogle Scholar
Lewis, C. I. & Langford, C. H. (1959). Symbolic Logic. New York: Dover.Google Scholar
Makinson, D. (2017). Relevance via decomposition: A project, some results, an open question. Australasian Journal of Logic, 14(3), 356–377.CrossRefGoogle Scholar
Mares, E. (2014). Relevance logic. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Spring 2014 edition). Available at: https://plato.stanford.edu/archives/spr2014/entries/logic-relevance/.Google Scholar
Meyer, R. (1971). Entailment. Journal of Philosophy, 68(21), 808–818.CrossRefGoogle Scholar
Prawitz, D. (2006). Natural Deduction: A Proof-Theoretical Study. Mineola, NY: Dover. Original published 1965.Google Scholar
Read, S. (1988). Relevant Logic. Oxford: Blackwell.Google Scholar
Restall, G. (2000). An Introduction to Substructural Logic. New York: Routledge.CrossRefGoogle Scholar
Shapiro, L. (2017). Lp, k3, and fde as substructural logics. In Arazim, P. and Lavic˘ka, T., editors. The Logica Yearbook 2016. London: College Publications.Google Scholar
Smiley, T. (1959). Entailment and deducibility. Proceedings of the Aristotelian Society, 59, 233–254.CrossRefGoogle Scholar
Tennant, N. (1979). Entailment and proofs. Proceedings of the Aristotelian Society, 79, 167–189.CrossRefGoogle Scholar
Tennant, N. (1984). Perfect validity, entailment and paraconsistency. Studia Logica, 43, 179–198.CrossRefGoogle Scholar
Tennant, N. (1987). Anti-Realism and Logic. Oxford: Clarendon Press.Google Scholar
Tennant, N. (2005). Relevance in reasoning. In Shapiro, S., editor. The Oxford Handbook of Philosophy of Logic and Mathematics. New York: Oxford University Press, pp. 696–726.CrossRefGoogle Scholar
Tennant, N. (2015). The relevance of premises to conclusions of core proofs. Review of Symbolic Logic, 8, 743–784.CrossRefGoogle Scholar
Verdée, P. & de Bal, I. (2015). A new approach to classical relevance. Studia Logica, 103, 919–954.CrossRefGoogle Scholar
von Wright, G. (1957). The concept of entailment. In Logical Studies. London: Routledge and Kegan Paul, pp. 166–191.Google Scholar
Woods, J. (1964). Relevance. Logique et Analyse, 7(27), 130–137.Google Scholar