Abstract
Gentzen’s singular sequential system of first-order logic was an alternative notation for his system of natural deductions. His multiple sequential system was his symmetric generalization that was more appropriate to classical logic. Beth’s tableaus system was a system that was derived directly from the semantic analysis of connectives and quantifiers. It was soon realized that the Beth’s system and the Gentzen’s multiple system were only notational variants of each other. Kneale’s system of multiple natural deductions was a generalization of Gentzen’s system of natural deductions. We prove that Kneale’s natural deductions are also a notational variant of Beth’s tableaus.
Similar content being viewed by others
References
Beth, E.W.: Semantic Entailment and Formal Derivability, Mededelingen van de Koninklijke Nederlandse Akademie van Wetenschappen. Afd. Letterkunde n.s 18, 309–342 (1955)
Beth, E.W.: The Foundations of Mathematics, Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam (1959)
Fitting, M.: Proof Methods for Modal and Intuitionistic Logics. D. Reidel Publishing Co., Dordrecht (1983)
Gentzen, G.: Untersuchungen über das logische Schließen I. Mathematische Zeitschrift 39(2), 176–210 (1935)
Gentzen, G.: Untersuchungen über das logische Schließen II. Mathematische Zeitschrift 39(3), 405–431 (1935)
Kneale, W., Kneale, M.: The Development of Logic. Clarendon Press, Oxford (1962)
Maretić, M.: On multiple conclusion deductions in classical logic. Math. Commun. 23(1), 79–95 (2018)
Shoesmith, D.J., Smiley, T.J.: Multiple Conclusion Logic. Cambridge University Press, Cambridge (1978)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the late Kosta Došen who asked me to write it.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The paper was awarded the Georgius Beningus Logic Prize (Croatia) and will be presented at the universal logic contest at UNILOG 2022 in Crete.
Rights and permissions
About this article
Cite this article
Šikić, Z. Kneale’s Natural Deductions as a Notational Variant of Beth’s Tableaus. Log. Univers. 16, 11–26 (2022). https://doi.org/10.1007/s11787-021-00279-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11787-021-00279-0