Finding Missing Proofs with Automated Reasoning
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
This article features long-sought proofs with intriguing properties (such as the absence of double negation and the avoidance of lemmas that appeared to be indispensable), and it features the automated methods for finding them. The theorems of concern are taken from various areas of logic that include two-valued sentential (or propositional) calculus and infinite-valued sentential calculus. Many of the proofs (in effect) answer questions that had remained open for decades, questions focusing on axiomatic proofs. The approaches we take are of added interest in that all rely heavily on the use of a single program that offers logical reasoning, William McCune's automated reasoning program OTTER. The nature of the successes and approaches suggests that this program offers researchers a valuable automated assistant. This article has three main components. First, in view of the interdisciplinary nature of the audience, we discuss the means for using the program in question (OTTER), which flags, parameters, and lists have which effects, and how the proofs it finds are easily read. Second, because of the variety of proofs that we have found and their significance, we discuss them in a manner that permits comparison with the literature. Among those proofs, we offer a proof shorter than that given by Meredith and Prior in their treatment of Łukasiewicz's shortest single axiom for the implicational fragment of two-valued sentential calculus, and we offer a proof for the Łukasiewicz 23-letter single axiom for the full calculus. Third, with the intent of producing a fruitful dialogue, we pose questions concerning the properties of proofs and, even more pressing, invite questions similar to those this article answers.
- Fitelson, B., and L. Wos, 'missing proofs found', J. Automated Reasoning 27, no. 2 (2001), in press.
- Harris, K., and B. Fitelson, 'Distributivity in Lw and other sentential logics', J. Automated Reasoning 27, no. 2 (2001), in prcss.
- Kalman, J.., 'Condensed detachment as a rule of inference', Studia Logica 42(1983), 443-451.
- Lukasiewicz, J., Elements of Mathematical Logic, Macmillan, New York, 1963.
- Lukasiewicz, J., Selected Works, edited by L. Borkowski, North Holland, Amsterdam, 1970.
- Mccune, W., OTTER 3.0 Reference Manual and Guide, Tcch. Report ANL-94/6, Argonne National Laboratory, Argonne, IL, January 1994.
- Meredith, C. A., 'Single axioms for the systems (C,N), (C,O), and (A,N) of the two-valued propositional calculus', J. Computing Systems 1, no. 3 (1953), 155-164.
- Meredith, C. A., 'The dependence of an axiom of Łukasiewicz', Trans. AMS 87, no. 1(1958), 54.
- Meredith, C. A., and A. Prior, 'Notes on the axiomatics of the propositional calculus', Notre Dame J. Formal Logic 4, no. 3 (1963), 171-187.
- Rose, A., and J. B. Rosser, 'Fragments of many-valued statement calculi', Trans. AMS87 (1958), 1-53.
- Wajsberg, M., Logical Works, Polish Academy of Sciences, Warsaw, 1977.
- Wos, L., and G. W. Pieper, A Fascinating Country in the World of Conputing: Your Guide to Automated Reasoning, Singapore, World Scientific, 1999.
- Wos, L., and G. W. Pieper, The Collected Works of Larry Wos, Singapore, World Scientific, 2000
- Wos, L., 'Conquering thc Meredith single axiom', J. Automated Reasoning 27, no. 2 (2001), in press.
- Finding Missing Proofs with Automated Reasoning
Volume 68, Issue 3 , pp 329-356
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- missing proofs
- axiomatic proofs
- logic calculi
- condensed detachment
- term-avoidance proofs
- automated reasoning
- Author Affiliations
- 1. Department of Philosophy, University of Wisconsin, Madison, WI, 53706
- 2. Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 60439-4801
- 3. Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 60439-4801