A New Solution to a Problem of Hosoi and Ono
Michael Zakharyaschev
Source: Notre Dame J. Formal Logic Volume 35, Number 3
(1994), 450-457.
Abstract
This paper gives a new, purely semantic proof of the following theorem: if an intermediate propositional logic L has the disjunction property then a disjunction free formula is provable in L iff it is provable in intuitionistic logic. The main idea of the proof is to use the well-known semantic criterion of the disjunction property for "simulating" finite binary trees (which characterize the disjunction free fragment of intuitionistic logic) by general frames.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1040511350
Mathematical Reviews number (MathSciNet): MR1326126
Digital Object Identifier: doi:10.1305/ndjfl/1040511350
Zentralblatt MATH identifier: 0837.03025
References
[1] Chagrov, A., and M. Zakharyaschev, The undecidability of the disjunction property of intermediate calculi, Preprint, Institute of Applied Mathematics, the USSR Academy of Sciences, 1989.
Mathematical Reviews (MathSciNet): MR1034339
[2] Chagrov, A., and M. Zakharyaschev, ``An essay in complexity aspects of intermediate calculi," pp. 26--29 in Proceedings of The Fourth Asian Logic Conference, Tokyo, 1990.
[3] Chagrov, A., and M. Zakharyaschev, ``The disjunction property of intermediate propositional logics," Studia Logica, vol. 50 (1991), pp. 189--216.
Mathematical Reviews (MathSciNet): MR93b:03040
Zentralblatt MATH: 0739.03016
Digital Object Identifier: doi:10.1007/BF00370182
[4] Chagrov, A., and M. Zakharyaschev, ``The undecidability of the disjunction property of propositional logics and other related problems," Journal of Symbolic Logic, vol. 58 (1993), pp. 967--1002.
Mathematical Reviews (MathSciNet): MR94i:03048
Zentralblatt MATH: 0799.03009
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.2307/2275108
[5] Fine, K., ``Logics containing K4. Part II," Journal of Symbolic Logic, vol. 50 (1985), pp. 619--651.
Mathematical Reviews (MathSciNet): MR87e:03030
Zentralblatt MATH: 0574.03008
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.2307/2274318
[6] Goldblatt, R., ``Metamathematics of modal logic," pp. 41--77 in Report on Mathematical Logic, vol. 6 (1976), and pp. 21--52 in Report on Mathematical Logic, vol. 7 (1976).
Mathematical Reviews (MathSciNet): MR58:27331a
Zentralblatt MATH: 0356.02016
[7] Gudovschikov, V., and V. Rybakov, ``The disjunction property in modal logics," pp. 35--36 in Proceedings of the 8th USSR Conference on Logic and Methodology of Science, Vilnius, 1982.
[8] Hosoi, T., and H. Ono, ``Intermediate propositional logics (A survey)," The Journal of Tsuda College, vol. 5 (1973), pp. 67--82.
Mathematical Reviews (MathSciNet): MR49:4753
[9] Maksimova, L., ``On maximal intermediate logics with the disjunction property," Studia Logica, vol. 45 (1986), pp. 69--75.
Mathematical Reviews (MathSciNet): MR88e:03038
Zentralblatt MATH: 0635.03019
Digital Object Identifier: doi:10.1007/BF01881550
[10] Minari, P., ``Intermediate logics with the same disjunctionless fragment as intuitionistic logic," Studia Logica, vol. 45 (1986), pp. 207--222.
Mathematical Reviews (MathSciNet): MR88d:03053
Zentralblatt MATH: 0634.03021
Digital Object Identifier: doi:10.1007/BF00373276
[11] Minari, P., Semantical investigations on intermediate propositional logics, Bibliopolis, Napoli, 1989.
[12] Segerberg, K., An essay in classical modal logic, Philosophical Studies, Uppsala, 1971.
Mathematical Reviews (MathSciNet): MR49:4756
Zentralblatt MATH: 0311.02028
[13] Segerberg, K., ``Proof of a conjecture of McKay," Fundamenta Mathematicae, vol. 81 (1974), pp. 267--270.
Mathematical Reviews (MathSciNet): MR50:1849
Zentralblatt MATH: 0287.02013
[14] Shehtman, V., ``On incomplete propositional logics," Soviet Mathematics Doklady, vol. 235 (1977), pp. 542--545.
Zentralblatt MATH: 0412.03011
Mathematical Reviews (MathSciNet): MR460075
[15] Shehtman, V., ``Topological models of propositional logics," Semiotic and Informatic, vol. 15 (1980), pp. 74--98.
Zentralblatt MATH: 0455.03013
Mathematical Reviews (MathSciNet): MR606571
[16] Zakharyaschev, M., ``On the disjunction property of superintuitionistic logics," p. 69 in Proceedings of the 7th USSR Conference for Mathematical Logic, Novosibirsk, 1984.
[17] Zakharyaschev, M., ``On the disjunction property of intermediate and modal logics," Matematicheskie Zametki, vol. 42 (1987), pp. 729--738.
Mathematical Reviews (MathSciNet): MR927453
[18] Zakharyaschev, M., ``Modal companions of intermediate logics: syntax, semantics and preservation theorems," Matematicheskii Sbornik, vol. 180, (1989), pp. 1415--1427.
Zentralblatt MATH: 0766.03015
[19] Zakharyaschev, M., ``Syntax and semantics of intermediate logics," Algebra i Logika, vol. 28 (1989), pp. 402--429.
Mathematical Reviews (MathSciNet): MR1086910
[20] Zakharyaschev, M., and S. Popov, On the complexity of countermodels for intuitionistic calculus, Preprint, Institute of Applied Mathematics, USSR Academy of Sciences, 1980.
Mathematical Reviews (MathSciNet): MR557997
Notre Dame Journal of Formal Logic
