## Notre Dame Journal of Formal Logic

### Shortest Axiomatizations of Implicational S4 and S5

#### Abstract

Shortest possible axiomatizations for the strict implicational fragments of the modal logics S4 and S5 are reported. Among these axiomatizations is included a shortest single axiom for implicational S4—which to our knowledge is the first reported single axiom for that system—and several new shortest single axioms for implicational S5. A variety of automated reasoning strategies were essential to our discoveries.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 43, Number 3 (2002), 169-179.

Dates
First available in Project Euclid: 16 January 2004

https://projecteuclid.org/euclid.ndjfl/1074290715

Digital Object Identifier
doi:10.1305/ndjfl/1074290715

Mathematical Reviews number (MathSciNet)
MR2034744

Zentralblatt MATH identifier
1045.03021

#### Citation

Ernst, Zachary; Fitelson, Branden; Harris, Kenneth; Wos, Larry. Shortest Axiomatizations of Implicational S4 and S5. Notre Dame J. Formal Logic 43 (2002), no. 3, 169--179. doi:10.1305/ndjfl/1074290715. https://projecteuclid.org/euclid.ndjfl/1074290715

#### References

• Anderson, A. R., and N. D. Belnap, "The pure calculus of entailment", The Journal of Symbolic Logic, vol. 27 (1962), pp. 19–52.
• Anderson, A. R., and N. D. Belnap, Entailment. Volume I. The Logic of Relevance and Necessity, Princeton University Press, Princeton, 1975.
• Curry, H. B., "Review of H"acking's `What is strict implication?', 1963.
• Ernst, Z., B. Fitelson, K. Harris, and L. Wos, "A concise axiomatization of $\mathit{RM}_{\rightarrow}$", University of \Lódz. Department of Logic. Bulletin of the Section of Logic, vol. 30 (2001), pp. 191–94.
• Hacking, I., “What is strict implication?” The Journal of Symbolic Logic, vol. 28 (1963), pp. 51–71.
• Kalman, J. A., "Condensed detachment as a rule of inference", Studia Logica, vol. 42 (1983), pp. 443–51 (1984).
• Kripke, S., "The problem of entailment", The Journal of Symbolic Logic, vol. 24 (1959), p. 324.
• Lemmon, E. J., C. A. Meredith, D. Meredith, A. N. Prior, and I. Thomas, Calculi of Pure Strict Implication, Canterbury University College, Christchurch, 1957.
• McCune, W., " Otter" 3.0 Reference Manual and Guide", Technical Report ANL-94/6, Argonne National Laboratory, Argonne, IL, 1994.
• McCune, W., R. Veroff, and R. Padmanabhan, "Yet another single law for lattices". forthcoming in Algebra Universalis.
• Meredith, C. A., and A. N. Prior, "Investigations into implicational S5", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 10 (1964), pp. 203–20.
• Meyer, R. K., and R. Z. Parks, "Independent axioms for the implicational fragment of S"obociński's three-valued logic, Zeitschrift für mathematische Logik Grundlagen Mathematik, vol. 18 (1972), pp. 291–95.
• Parks, R. Z., "A note on R"-Mingle and Soboci\' nski's three-valued logic, Notre Dame Journal of Formal Logic, vol. 13 (1972), pp. 227–28.
• Prior, A. N., Formal Logic, Clarendon Press, Oxford, 1962.
• Slaney, J., " MaGIC": Matrix Generator for Implication Connectives", Technical Report TR-ARP-11-95, Research School of Information Science and Engineer and Centre for Information Science Research, Australian National University, 1995.
• Ulrich, D., "Strict implication in a sequence of extensions of S4", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 27 (1981), pp. 201–12.
• Veroff, R., "Using hints to increase the effectiveness of an automated reasoning program: C"ase studies, Journal of Automated Reasoning, vol. 16 (1996), pp. 223–39.
• Wos, L., "The resonance strategy", Computers & Mathematics with Applications. An International Journal, vol. 29 (1995), pp. 133–78.
• Wos, L., "Searching for circles of pure proofs", Journal of Automated Reasoning, vol. 15 (1995), pp. 279–315.
• Wos, L., " Otter" and the Moufang identity problem", Journal of Automated Reasoning, vol. 17 (1996), pp. 215–57.
• Wos, L., "The strategy of cramming", preprint ANL/MCS-P898-0801, Argonne National Laboratory, Argonne, 2001.
• Wos, L., and G. W. Pieper, A Fascinating Country in the World of Computing: Your Guide to Automated Reasoning, World Scientific Publishing Company, 2000.
• Wos, L., and G. W. Pieper, Automated Reasoning and the Discovery of Missing and Elegant Proofs, Rinton Press, forthcoming.
• Zhang, J., and H. Zhang, "S"EM: A System for Enumerating Models", Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI-95), (1995), pp. 298–303.