Abstract
This paper presents Rasiowa–Sikorski deduction systems (R–S systems) for logics \(\mathsf {CPL}\), \(\mathsf {CLuN}\), \(\mathsf {CLuNs}\) and \(\mathsf {mbC}\). For each of the logics two systems are developed: an R–S system that can be supplemented with admissible cut rule, and a \(\mathbf {KE}\)-version of R–S system in which the non-admissible rule of cut is the only branching rule. The systems are presented in a Smullyan-like uniform notation, extended and adjusted to the aims of this paper. Completeness is proved by the use of abstract refutability properties which are dual to consistency properties used by Fitting. Also the notion of admissibility of a rule in an R–S-system is analysed.
Article PDF
Similar content being viewed by others
References
Agudelo-Agudelo, J.C., Translating Non-classical Logics into Classical Logic by Using Hidden Variables. Logica Universalis 11(2):205–224, 2017.
Batens, D., Paraconsistent extensional propositional logics. Logique et Analyse 90–91:195–234, 1980.
Batens, D., Inconsistency-adaptive logics. In E. Orłowska, (ed.), Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, Springer, Berlin, 1998, pp. 445–472.
Batens, D., and K. De Clercq, A Rich Paraconsistent Extension of Full Positive Logic. Logique et Analyse 185–188:227–257, 2005.
Batens, D., K. De Clercq, and N. Kurtonina, Embedding and Interpolation for Some Paralogics. The Propositional Case. Reports on Mathematical Logic 33:29–44, 1999.
Batens, D., and J. Meheus, A Tableau Method for Inconsistency-Adaptive Logics. In R. Dyckhoff, (ed.), Automated Reasoning with Analytic Tableaux and Related Methods, Lecture Notes in Artificial Intelligence, Springer, Berlin, 2000, pp. 127–142.
Batens, D., and J. Meheus, Shortcuts and Dynamic Marking in the Tableau Method for Adaptive Logics. Studia Logica 69:221–248, 2001.
Boolos, G., Don’t Eliminate Cut. Journal of Philosophical Logic 13(4):373–378, 1984.
Caleiro, C., J. Marcos, and M. Volpe, Bivalent semantics, generalized compositionality and analytic classic-like tableaux for finite-valued logics. Theoretical Computer Science 603:84–110, 2015.
Carnielli, W.A., and M.E. Coniglio, Logics of Formal Inconsistency. In F. Guenthner, and D.M. Gabbay, (eds.), Handbook of Philosophical Logic, vol. 14, Springer, Berlin, 2013, pp. 1–93.
Carnielli, W.A., and J. Marcos, A taxonomy of \(\mathbf{C}\)-systems. In I.M.L. D’Ottaviano, W.A. Carnielli, and M.E. Coniglio, (eds.), Paraconsistency—The Logical Way to the Inconsistent, Marcel Dekker, 2000, pp. 1–94.
Chlebowski, S.Z., Canonical and Dual Erotetic Calculi for First-Order Logic. Ph.D. Thesis, Adam Mickiewicz University, Poznań, 2018. (Unpublished manuscript, previously referred to as “The Method of Socratic Proofs for Classical Logic and Some Non-Classical Logics”).
Chlebowski, S.Z., A. Gajda, and M. Urbański, Abductive Question–Answer System for the Minimal Logic of Formal Inconsistency mbC. (Unpublished manuscript).
Chlebowski, S.Z., and D. Leszczyńska-Jasion, Dual Erotetic Calculi and the Minimal LFI. Studia Logica 103(6):1245–1278, 2015.
Coniglio, M.E., and T.G. Rodrigues, Some investigations on \({\sf mbC}\) and \({\sf mCi}\). In C.A. Mortari, (ed.), Tópicos de lógicas não clássicas, NEL/UFSC, 2014, pp. 11–70.
D’Agostino, M., Are tableaux an improvement on truth-tables? Journal of Logic, Language and Information 1(3):235–252, 1992.
D’Agostino, M., and M. Mondadori, The Taming of the Cut. Classical Refutations with Analytic Cut. Journal of Logic and Computation 4(3):285–319, 1994.
Fitting, M., Proof Methods for Modal and Intuitionistic Logics. Springer, Netherlands, 1983.
Fitting, M., First-order Logic and Automated Theorem Proving. Springer, Berlin, 1990.
Golińska-Pilarek, J., T. Huuskonen, and E. Muñoz-Velasco, Relational dual tableau decision procedures and their applications to modal and intuitionistic logics. Annals of Pure and Applied Logic 165(2):409–427, 2014.
Golińska-Pilarek, J., and E. Orłowska. Tableaux and dual tableaux: transformation of proofs. Studia Logica 85:283–302, 2007.
Grzelak, A., and D. Leszczyńska-Jasion, Automatic proof generation in an axiomatic system for \({\sf CPL}\) by means of the method of Socratic proofs. Logic Journal of the IGPL 26(1):109–148, 2018.
Hählne, R., Tableaux and Related Methods. In A. Robinson, and A. Voronkov, (eds.), Handbook of Automated Reasoning, Chapter 3, Elsevier Science Publishers, Amsterdam, 2001, pp. 101–175.
Ignaszak, M., Dual erotetic version of system KE. Master’s Thesis, Department of Logic and Cognitive Science, Institute of Psychology, Adam Mickiewicz University, 2017.
Konikowska, B., Rasiowa–Sikorski deduction system: a handy tool for Computer Science logic. In Proceedings WADT98, Springer Lecture Notes in Computer Science, volume 1589, Springer, Berlin, 1999, pp. 183–197.
Konikowska, B., Rasiowa–Sikorski deduction systems in computer science applications. Theoretical Computer Science 286(2):323–366, 2002.
Leszczyńska-Jasion, D., The Method of Socratic Proofs for Normal Modal Propositional Logics. Adam Mickiewicz University Press, Poznań, 2007.
Leszczyńska-Jasion, D., The Method of Socratic Proofs for Modal Propositional Logics: K5, S4.2, S4.3, S4M, S4F, S4R and G. Studia Logica 89(3):371–405, 2008.
Neto, A.G.S.S., and M. Finger, Effective prover for minimal inconsistency logic. In M. Bramer, (ed.), IFIP International Federation for Information Processing, Springer, Berlin, 2006, pp. 465–474.
Orłowska, E., and J. Golińska-Pilarek, Dual Tableaux: Foundations, Methodology, Case Studies, volume 33 of Trends in Logic. Springer, Dordrecht, 2011.
Pudlák, P., The Lengths of Proofs. In S.R. Buss, (ed.), Handbook of Proof Theory, chapter VIII, Elsevier, 1998, pp. 547–637.
Rasiowa, H., and R. Sikorski, On the Gentzen theorem. Fundamenta Mathematicae 48:57–69, 1960.
Rasiowa, H., and R. Sikorski, The Mathematics of Metamathematics. Polish Scientific Publishers, Warsaw, 1963.
Smullyan, R.M., First-Order Logic. Springer, Berlin, 1968.
Smullyan, R.M., A Beginner’s Guide to Mathematical Logic. Dover Books on Mathematics. Dover Publications, 2014.
Urbański, M., Tabele syntetyczne a logika pytań. Wydawnictwo UMCS, Lublin, 2002.
Wiśniewski, A., Socratic Proofs. Journal of Philosophical Logic 33(3):299–326, 2004.
Wiśniewski, A., Questions, Inferences, and Scenarios. College Publications, London, 2013.
Wiśniewski, A., and V. Shangin, Socratic proofs for quantifiers. Journal of Philosophical Logic 35(2):147–178, 2006.
Wiśniewski, A., G. Vanackere, and D. Leszczyńska, Socratic Proofs and Paraconsistency: A Case Study. Studia Logica 80(2-3):433–468, 2005.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Andrzej Indrzejczak
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Leszczyńska-Jasion, D., Ignaszak, M. & Chlebowski, S. Rasiowa–Sikorski Deduction Systems with the Rule of Cut: A Case Study. Stud Logica 107, 313–349 (2019). https://doi.org/10.1007/s11225-018-9795-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-018-9795-7