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On implicational intermediate logics axiomatizable by formulas minimal in classical logic: a counter-example to the Komori-Kashima problem. (English) Zbl 07450676

Summary: The Komori-Kashima problem, that asks whether (or not) the implicational intermediate logics axiomatizable by formulas minimal in classical logic are only intuitionistic logic and classical logic, has stood for over a decade. In this paper, we give a counter-example to this problem. Additionally, we also give some open problems derived from this result.

MSC:

03-XX Mathematical logic and foundations
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[1] Barendregt, H., W. Dekkers, and R. Statman, Lambda Calculus with Types, Cambridge University Press, 2013. · Zbl 1347.03001
[2] Dummett, Michael, A Propositional Calculus with Denumerable Matrix, The Journal of Symbolic Logic, 24, 2, 97-106 (1959) · Zbl 0089.24307
[3] Dyckhoff, Roy, Contraction-Free Sequent Calculi for Intuitionistic Logic, The Journal of Symbolic Logic, 57, 3, 795-807 (1992) · Zbl 0761.03004
[4] Hirai, Y., Personal Communication (Aug 9 2018), 2018.
[5] Jankov, VA, On the extension of the intuitionist propositional calculus to the classical calculus, and the minimal calculus to the intuitionist calculus, Mathematics of the USSR-Izvestiya, 2, 1, 205-208 (1968) · Zbl 0191.28602
[6] Jankov, V. A., The Calculus of the Weak “Law of Excluded Middle”, Mathematics of the USSR-Izvestiya 2(5):997-1004, 1968. · Zbl 0187.26306
[7] Kashima, R., On non-generality of axioms of intermediate propositional logics (in Japanese), available at http://www.is.titech.ac.jp/
[8] Kashima, R., Problems on Axiomatization of Intermediate Propositional Logics, in the 39th MLG meeting in 2005, pp.59-62 (available at http://www.st.nanzan-u.ac.jp/info/sasaki/2005mlg/59-62.pdf), 2005.
[9] Komori, Y., BCK algebras and lambda calculus, in Proceedings of 10th Symposium on Semigroups, Sakado 1986, 1987, pp. 5-11.
[10] Komori, Y., A problem on logics axiomatized with formulas minimal in classical logic, and more (in Japanese), in the Mathematical Society of Japan Autumn Meeting 2005 (available at http://komoriyuichi.web.fc2.com/gakkai/05-09/kyokusyou/gakkai.pdf), 2005.
[11] Komori, Y., Independent Axiom Systems of Minimal formulas for Classical Logic, in The 39th MLG meeting in 2005 (available at http://www.st.nanzan-u.ac.jp/info/sasaki/2005mlg/56-58.pdf), 2005, pp. 56-58.
[12] Komori, Y., Propositional logics revisited - deployments from misunderstanding and mistakes (in Japanese), in The Mathematical Society of Japan Autumn Meeting 2007 (available at doi:10.11429/emath1996.2007.autumn-meeting1_82), The Mathematical Society of Japan, 2007,pp. 82-94.
[13] Nakamura, Y., Coq Files for “On the Axiomatization of Implicational Intermediate Logics with Formulas Minimal in Classical Logic: A Counter-Example to the Komori-Kashima Problem”, available at https://bitbucket.org/yoshikinakamura/komori-kashima-coq, 2020.
[14] The Coq Development Team, The Coq Proof Assistant, Version 8.10.0, available at doi:10.5281/zenodo.3476303, 2019.
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