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A Classification of Logics over FLew and Almost Maximal Logics

  • Hiroakira Ono
  • Masaki Ueda
Chapter
Part of the Synthese Library book series (SYLI, volume 320)

Abstract

Let FL ew be the logic obtained from the intuitionistic propositional logic by deleting contraction rule if we formulate it in a sequent system. Sometimes, this logic is called intuitionistic affine logic. The class of logics over FL ew , i.e. logics stronger than or equal to FL ew , includes many interesting logics, e.g., intermediate logics, Łukasiewicz’s many-valued logics, Grišin’s logic and product logic, etc. (See, e.g., Cignoli et al., 2000; Grišin, 1976; Hájek, 1998.) The study of logics over FL ew , will enable us to discuss these different kinds of logics within a uniform framework (see Ono and Komori, 1985; Ono, 1999 for the detail).

Keywords

Classical Logic Residuated Lattice Intuitionistic Logic Heyting Algebra Product Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Blok, W. J. and Ferreirim, I. M. A. (2000). On the structure of hoops. Algebra Universalis, 43: 233–257.CrossRefGoogle Scholar
  2. Blok, W. J. and R,aftery, J. G. (1995). On the quasivariety of BCKalgebras and its subvarieties. Algebra Universalis, 33: 68–90.CrossRefGoogle Scholar
  3. Cignoli, R., D’Ottaviano, I. M. L., and Mundici, D. (2000). Algebraic Foundations of Many-valued Reasoning, volume VII of Trends in Logic, Studia Logica Library. Kluwer Academic Publishers, Dordrecht.CrossRefGoogle Scholar
  4. Cignoli, R. and Torrens, A. (2000). An algebraic analysis of product logic. Multiple Valued Logic, 5: 45–65.Google Scholar
  5. Cornish, W. H. (1980). Varieties generated by finite BCK-algebras. Bulletin of Australian Mathematical Society, 22: 411–430.CrossRefGoogle Scholar
  6. Gris’in, V. N. (1976). On algebraic semantics for logic without the contraction rule (in Russian), Isslédovanija po formalizovannym jazykam i néklassicéskim logikam, pp. 247–264. Nauka, Moscow.Google Scholar
  7. Hâjek, P. (1998). Metamathematics of Fuzzy Logic, volume IV of Trends in Logic, Studia Logica Library. Kluwer Academic Publishers, Dordrecht.CrossRefGoogle Scholar
  8. Hosoi, T. (1967). On intermediate logics I. Journal of the Faculty of Science, University of Tokyo Sec. I, 14: 293–312.Google Scholar
  9. Kowalski, T. and Ono, H. Splittings in the variety of residuated lattices, Algebra Universalis,forthcoming.Google Scholar
  10. Kowalski, T. and Ueda, M. (2000). Almost minimal varieties of residuated lattices, draft of a talk presented at the 6th Barcelona Logic Meeting, Barcelona, Spain, July, 2000.Google Scholar
  11. McKenzie, R. (1972). Equational bases and non-modular lattice varieties. Transactions of the American Mathematical Society, 174: 1–43.CrossRefGoogle Scholar
  12. Okada, M. and Terui, K. (1999). The finite model property for various fragments of intuitionistic linear logic. Journal of Symbolic Logic, 64: 790–802.CrossRefGoogle Scholar
  13. Ono, H. (1999). Logics without contraction rule and residuated lattices I. Unpublished.Google Scholar
  14. Ono, H. and Komori, Y. (1985). Logics without the contraction rule. Journal of Symbolic Logic, 50: 169–201.CrossRefGoogle Scholar
  15. Ueda, M. (2000). A Study of a Classification of Residuated Lattices and Logics without Contraction Rule. Master Thesis, Japan Advanced Institute of Science and Technology.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • Hiroakira Ono
    • 1
  • Masaki Ueda
    • 2
  1. 1.School of Information ScienceJAISTIshikawaJapan
  2. 2.Japan Patent OfficeTokyoJapan

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