Exhibiting Wide Families of Maximal Intermediate Propositional Logics with the Disjunction Property
We provide results allowing to state, by the simple inspection of suitable classes of posets (propositional Kripke frames), that the corresponding intermediate propositional logics are maximal among the ones which satisfy the disjunction property. Starting from these results, we directly exhibit, wi...
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| Published in | Mathematical logic quarterly Vol. 42; no. 1; pp. 501 - 536 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Berlin
WILEY-VCH Verlag Berlin GmbH
1996
WILEY‐VCH Verlag Berlin GmbH |
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| ISSN | 0942-5616 1521-3870 |
| DOI | 10.1002/malq.19960420141 |
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| Abstract | We provide results allowing to state, by the simple inspection of suitable classes of posets (propositional Kripke frames), that the corresponding intermediate propositional logics are maximal among the ones which satisfy the disjunction property. Starting from these results, we directly exhibit, without using the axiom of choice, the Kripke frames semantics of 2No maximal intermediate propositional logics with the disjunction property. This improves previous evaluations, giving rise to the same conclusion but made with an essential use of the axiom of choice, of the cardinality of the set of the maximal intermediate propositional logics with the disjunction property.
Mathematics Subject Classification: 03B55, 03C90. |
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| AbstractList | We provide results allowing to state, by the simple inspection of suitable classes of posets (propositional Kripke frames), that the corresponding intermediate propositional logics are maximal among the ones which satisfy the disjunction property. Starting from these results, we directly exhibit, without using the axiom of choice, the Kripke frames semantics of 2
No
maximal intermediate propositional logics with the disjunction property. This improves previous evaluations, giving rise to the same conclusion but made with an essential use of the axiom of choice, of the cardinality of the set of the maximal intermediate propositional logics with the disjunction property.
Mathematics Subject Classification: 03B55, 03C90. We provide results allowing to state, by the simple inspection of suitable classes of posets (propositional Kripke frames), that the corresponding intermediate propositional logics are maximal among the ones which satisfy the disjunction property. Starting from these results, we directly exhibit, without using the axiom of choice, the Kripke frames semantics of 2No maximal intermediate propositional logics with the disjunction property. This improves previous evaluations, giving rise to the same conclusion but made with an essential use of the axiom of choice, of the cardinality of the set of the maximal intermediate propositional logics with the disjunction property. Mathematics Subject Classification: 03B55, 03C90. |
| Author | Silvestrini, Daniela Miglioli, Pierangelo Bertolotti, Guido |
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| References | Levin, L. A., Some syntactic theorems on the calculus of finite problems of Ju. T. Medvedev. Soviet Math. Dokl. 10 (1969), 288-290. Šhethman, V. B., Rieger-Nishimura lattices. Doklady Acad. Nauk. SSSR 241 (1978), 1288-1291 (in Russian). Šhethman, V. B., On incomplete propositional logics. Soviet Math. Dokl. 18 (1977), 985-989. Miglioli, P., An infinite class of maximal intermedate logics with the disjunction property. Archive Math. Logic 31 (1992), 415-432. Medvedev, Ju. T., Finite problems. Soviet Math. Dokl. 3 (1962), 227-230. Miglioli, P., U. Moscato, M. Ornaghi, S. Quazza, and G. Usberti, Some results on intermediate constructive logics. Notre Dame J. Formal Logic 30 (1989), 543-562. Chagrov, A. V., and M. V. Zacharyaschev, The disjunction property of intermediate propositional logics. Studia Logica 50 (1991), 189-216. Kirk, R. E., A result on propositional logics having the disjunction property. Notre Dame J. Formal Logic 23 (1982), 71-74. Ferrari, M., and P. Miglioli, A method to single out maximal propositional logics with the disjunction property I. Annals Pure Applied Logic 76 (1995), 1-46. Ferrari, M., and P. Miglioli, Counting the maximal intermediate constructive logics. J. Symbolic Logic 58 (1993), 1365-1401. Ferrari, M., and P. Miglioli, A method to single out maximal propositional logics with the disjunction property II. Annals Pure Applied Logic 76 (1995), 117-168. Maksimova, L. L., On the maximal intermediate logics with the disjunction property. Studia Logica 45 (1986), 69-75. Bellissima, F., Finite and finitely separable intermediate propositional logics. J. Symbolic Logic 53 (1988), 403-420. Chagrov, A. V., The cardinality of the set of maximal itermediate logics with the disjunction property is of continuum. Mat. Zametki 51 (1992), 117-123 (in Russian). 1982; 23 1993; 58 1989; 30 1990 1986; 45 1962; 3 1977; 18 1991; 50 1969; 10 1995; 76 1978; 241 1984 1973 1988; 53 1992; 31 1992; 51 Chagrov A. V. (e_1_2_1_3_2) 1992; 51 Šhethman V. B. (e_1_2_1_18_2) 1978; 241 Levin L. A. (e_1_2_1_10_2) 1969; 10 e_1_2_1_6_2 e_1_2_1_7_2 e_1_2_1_4_2 e_1_2_1_5_2 e_1_2_1_2_2 e_1_2_1_11_2 Šhethman V. B. (e_1_2_1_17_2) 1977; 18 e_1_2_1_12_2 e_1_2_1_15_2 e_1_2_1_16_2 e_1_2_1_14_2 Medvedev Ju. T. (e_1_2_1_13_2) 1962; 3 e_1_2_1_19_2 e_1_2_1_8_2 e_1_2_1_9_2 |
| References_xml | – reference: Ferrari, M., and P. Miglioli, Counting the maximal intermediate constructive logics. J. Symbolic Logic 58 (1993), 1365-1401. – reference: Medvedev, Ju. T., Finite problems. Soviet Math. Dokl. 3 (1962), 227-230. – reference: Šhethman, V. B., On incomplete propositional logics. Soviet Math. Dokl. 18 (1977), 985-989. – reference: Ferrari, M., and P. Miglioli, A method to single out maximal propositional logics with the disjunction property II. Annals Pure Applied Logic 76 (1995), 117-168. – reference: Ferrari, M., and P. Miglioli, A method to single out maximal propositional logics with the disjunction property I. Annals Pure Applied Logic 76 (1995), 1-46. – reference: Šhethman, V. B., Rieger-Nishimura lattices. Doklady Acad. Nauk. SSSR 241 (1978), 1288-1291 (in Russian). – reference: Miglioli, P., U. Moscato, M. Ornaghi, S. Quazza, and G. Usberti, Some results on intermediate constructive logics. Notre Dame J. Formal Logic 30 (1989), 543-562. – reference: Miglioli, P., An infinite class of maximal intermedate logics with the disjunction property. Archive Math. Logic 31 (1992), 415-432. – reference: Chagrov, A. V., The cardinality of the set of maximal itermediate logics with the disjunction property is of continuum. Mat. Zametki 51 (1992), 117-123 (in Russian). – reference: Levin, L. A., Some syntactic theorems on the calculus of finite problems of Ju. T. Medvedev. Soviet Math. Dokl. 10 (1969), 288-290. – reference: Bellissima, F., Finite and finitely separable intermediate propositional logics. J. Symbolic Logic 53 (1988), 403-420. – reference: Maksimova, L. L., On the maximal intermediate logics with the disjunction property. Studia Logica 45 (1986), 69-75. – reference: Chagrov, A. V., and M. V. Zacharyaschev, The disjunction property of intermediate propositional logics. Studia Logica 50 (1991), 189-216. – reference: Kirk, R. E., A result on propositional logics having the disjunction property. Notre Dame J. Formal Logic 23 (1982), 71-74. – start-page: 000 year: 1984 end-page: 000 – start-page: 41 year: 1990 – year: 1984 – volume: 45 start-page: 69 year: 1986 end-page: 75 article-title: On the maximal intermediate logics with the disjunction property publication-title: Studia Logica – volume: 53 start-page: 403 year: 1988 end-page: 420 article-title: Finite and finitely separable intermediate propositional logics publication-title: J. Symbolic Logic – volume: 18 start-page: 985 year: 1977 end-page: 989 article-title: On incomplete propositional logics publication-title: Soviet Math. Dokl. – volume: 3 start-page: 227 year: 1962 end-page: 230 article-title: Finite problems publication-title: Soviet Math. Dokl. – volume: 76 start-page: 1 year: 1995 end-page: 46 article-title: A method to single out maximal propositional logics with the disjunction property I publication-title: Annals Pure Applied Logic – volume: 10 start-page: 288 year: 1969 end-page: 290 article-title: Some syntactic theorems on the calculus of finite problems of Ju publication-title: T. Medvedev. Soviet Math. Dokl. – volume: 23 start-page: 71 year: 1982 end-page: 74 article-title: A result on propositional logics having the disjunction property publication-title: Notre Dame J. Formal Logic – volume: 30 start-page: 543 year: 1989 end-page: 562 article-title: Some results on intermediate constructive logics publication-title: Notre Dame J. Formal Logic – volume: 51 start-page: 117 year: 1992 end-page: 123 article-title: The cardinality of the set of maximal itermediate logics with the disjunction property is of continuum publication-title: Mat. Zametki – volume: 76 start-page: 117 year: 1995 end-page: 168 article-title: A method to single out maximal propositional logics with the disjunction property II publication-title: Annals Pure Applied Logic – start-page: 324 year: 1973 end-page: 391 – volume: 58 start-page: 1365 year: 1993 end-page: 1401 article-title: Counting the maximal intermediate constructive logics publication-title: J. Symbolic Logic – volume: 241 start-page: 1288 year: 1978 end-page: 1291 article-title: Rieger‐Nishimura lattices publication-title: Doklady Acad. Nauk. SSSR – volume: 50 start-page: 189 year: 1991 end-page: 216 article-title: The disjunction property of intermediate propositional logics publication-title: Studia Logica – volume: 31 start-page: 415 year: 1992 end-page: 432 article-title: An infinite class of maximal intermedate logics with the disjunction property publication-title: Archive Math. Logic – ident: e_1_2_1_7_2 doi: 10.1016/0168-0072(94)00052-5 – ident: e_1_2_1_9_2 doi: 10.1305/ndjfl/1093883567 – ident: e_1_2_1_11_2 – volume: 51 start-page: 117 year: 1992 ident: e_1_2_1_3_2 article-title: The cardinality of the set of maximal itermediate logics with the disjunction property is of continuum publication-title: Mat. Zametki – ident: e_1_2_1_5_2 doi: 10.2307/2275149 – ident: e_1_2_1_15_2 doi: 10.1305/ndjfl/1093635238 – volume: 18 start-page: 985 year: 1977 ident: e_1_2_1_17_2 article-title: On incomplete propositional logics publication-title: Soviet Math. Dokl. – volume: 10 start-page: 288 year: 1969 ident: e_1_2_1_10_2 article-title: Some syntactic theorems on the calculus of finite problems of Ju publication-title: T. Medvedev. Soviet Math. 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| SubjectTerms | Constructive logic Intermediate logic Irreductible frame Kripke frame semantics Kripke model Maximal constructive logic Maximal nonstandard constructive logic Nonstandard constructive logic Nonstandard intermediate logic Regular Kripke model Weak canonical model |
| Title | Exhibiting Wide Families of Maximal Intermediate Propositional Logics with the Disjunction Property |
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