Abstract
Complete deductive systems are constructed for the non-valid (refutable) formulae and sequents of some propositional modal logics. Thus, complete syntactic characterizations in the sense of Lukasiewicz are established for these logics and, in particular, purely syntactic decision procedures for them are obtained. The paper also contains some historical remarks and a general discussion on refutation systems.
Similar content being viewed by others
References
J. F. A. K. van Benthem,Modal logic and classical logic, Bibliopolis, Napoli, 1986.
G. Bryll andM. Maduch,Rejected axioms for Łukasiewicz's many-valued logics, (in Polish), Zeszyty Naukowe Wyższej Szkoly Pedagogicznej w Opolu,Mathematyka VI, Logika i algebra (1969), pp. 3 – 19.
R. Dutkiewicz,The method of axiomatic rejection for the intuitionistic propositional logic,Studia Logica,XLVIII,4, (1989), pp. 449–460.
G. G. Hughes, M. J. Cresswell,A companion to modal logic, Methuen, London, 1984.
T. Inoue,On Ishimoto's theorem in axiomatic rejection — the philosophy of unprovability (in Japanese),Philosophy of Science, 22 (Waseda University Press, Tokyo, 1989), pp. 77–93.
T. Inoue, A. Ishimoto andM. Kobayashi,Axiomatic rejection for the propositional fragment of Leśniewski's ontology, Soviet-Japan Symposium “Leśniewski's ontology and its applications”, Moscow 1990.
G. Kreisel andH. Putnam,Unableitbarkeitbeweismethode für den intuitionistischen Aussagenkalkul,Archiv Math. Logik Grundlagenforsch, 3, (1957), pp. 74–78.
J. Łukasiewicz,On the intuitionistic theory of deduction,Indag. Math., 14, (1952), pp. 202–212.
J. Łukasiewicz,Aristotle's syllogistic from the standpoint of modern formal logic, Oxford 1957.
D. Scott,Completeness proofs for the intuitionistic sentential calculus, Summaries of talks presented at the Summer Institute for Symbolic Logic, Inst. for defense Analyses, Cornell University 1957, 2nd. ed., Communications Research division, Princeton 1960, pp. 231–241.
K. Segerberg,An essay in classical modal logic Filosofiska Studier 13, Uppsala, 1971.
T. Skura,A complete syntactical characterization of the intuitionistic logic,Reports on mathematical logic,23 (1989), pp. 75–80.
T. Skura,On pure refutation formulations of sentential logics,Bull. of Sect. of Logic, PAS, vol. 19, 3 (1990), pp. 102–107.
T. Skura,On decision procedures for sentential logics,Studia Logica, L 2 (1991), pp. 173–179.
T. Skura,Refutation rules for three modal logics,Bull. of Sect. of Logic, vol.21 1 (1992), pp. 31–32.
J. Słupecki andG. Bryll,Proof of Ł-decidability of Lewis system S5,Studia Logica,XXXII (1973), pp. 99–105.
J. Słupecki, G. Bryll andU. Wybraniec-Skardowska,Theory of rejected propositions, I,Studia Logica XXIX (1971), pp. 75–115;II,Studia Logica, XXX (1972), pp. 97 – 115.
W. Staszek,On proofs of rejection,Studia Logica,XXIX (1971), pp. 17–23.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Goranko, V. Refutation systems in modal logic. Stud Logica 53, 299–324 (1994). https://doi.org/10.1007/BF01054714
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01054714