Polyadic algebras over nonclassical logics

Don Pigozzi; Antonino Salibra

Banach Center Publications (1993)

  • Volume: 28, Issue: 1, page 51-66
  • ISSN: 0137-6934

Abstract

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The polyadic algebras that arise from the algebraization of the first-order extensions of a SIC are characterized and a representation theorem is proved. Standard implicational calculi (SIC)'s were considered by H. Rasiowa [19] and include classical and intuitionistic logic and their various weakenings and fragments, the many-valued logics of Post and Łukasiewicz, modal logics that admit the rule of necessitation, BCK logic, etc.

How to cite

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Pigozzi, Don, and Salibra, Antonino. "Polyadic algebras over nonclassical logics." Banach Center Publications 28.1 (1993): 51-66. <http://eudml.org/doc/262551>.

@article{Pigozzi1993,
abstract = {The polyadic algebras that arise from the algebraization of the first-order extensions of a SIC are characterized and a representation theorem is proved. Standard implicational calculi (SIC)'s were considered by H. Rasiowa [19] and include classical and intuitionistic logic and their various weakenings and fragments, the many-valued logics of Post and Łukasiewicz, modal logics that admit the rule of necessitation, BCK logic, etc.},
author = {Pigozzi, Don, Salibra, Antonino},
journal = {Banach Center Publications},
keywords = {lambda calculus; modal logic; intuitionistic logic; many-valued logic; BCK logic; implicational calculi; polyadic algebras; functional representation},
language = {eng},
number = {1},
pages = {51-66},
title = {Polyadic algebras over nonclassical logics},
url = {http://eudml.org/doc/262551},
volume = {28},
year = {1993},
}

TY - JOUR
AU - Pigozzi, Don
AU - Salibra, Antonino
TI - Polyadic algebras over nonclassical logics
JO - Banach Center Publications
PY - 1993
VL - 28
IS - 1
SP - 51
EP - 66
AB - The polyadic algebras that arise from the algebraization of the first-order extensions of a SIC are characterized and a representation theorem is proved. Standard implicational calculi (SIC)'s were considered by H. Rasiowa [19] and include classical and intuitionistic logic and their various weakenings and fragments, the many-valued logics of Post and Łukasiewicz, modal logics that admit the rule of necessitation, BCK logic, etc.
LA - eng
KW - lambda calculus; modal logic; intuitionistic logic; many-valued logic; BCK logic; implicational calculi; polyadic algebras; functional representation
UR - http://eudml.org/doc/262551
ER -

References

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  1. [1] H. P. Barendregt, The Lambda Calculus. Its Syntax and Semantics, revised edition, Stud. Logic Found. Math. 103, North-Holland, Amsterdam 1985. 
  2. [2] W. J. Blok and D. Pigozzi, Algebraizable logics, Mem. Amer. Math. Soc. 396 (1989). Zbl0664.03042
  3. [3] Z. B. Diskin, Polyadic algebras for non-classical logics, I,II,III,IV, Latv. Mat. Ezhegodnik (in Russian), to appear. 
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  5. [5] G. Georgescu, A representation theorem for tense polyadic algebras, Mathematica (Cluj), 21 (1979), 131-138. Zbl0448.03050
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  10. [10] L. Henkin, An algebraic characterization of quantifiers, Fund. Math. 37 (1950), 63-74. Zbl0041.34804
  11. [11] L. Henkin, J. D. Monk and A. Tarski, Cylindric Algebras, Parts I and II, North-Holland, Amsterdam 1971 and 1985. 
  12. [12] J. Kotas and A. Pieczkowski, On a generalized cylindrical algebra and intuitionistic logic, Studia Logica 17 (1966), 73-80. Zbl0304.02012
  13. [13] A. R. Meyer, What is a model of the lambda calculus?, Inform. Control 52 (1982), 87-122. Zbl0507.03002
  14. [14] D. Monk, Polyadic Heyting algebras, Notices Amer. Math. Soc. 7 (1960), 735. 
  15. [15] A. Mostowski, Proofs of non-deducibility in intuitionistic functional calculus, J. Symbolic Logic 13 (1948), 204-207. Zbl0031.19304
  16. [16] I. Németi, Algebraizations of quantifier logics. An introductory overview, Studia Logica 50 (1991), 485-569. Zbl0772.03033
  17. [17] D. Pigozzi and A. Salibra, The abstract variable-binding calculus, manuscript. Zbl0836.03038
  18. [18] D. Pigozzi and A. Salibra, An introduction to lambda abstraction algebras, manuscript. 
  19. [19] H. Rasiowa, An Algebraic Approach to Non-Classical Logics, North-Holland, Amsterdam 1974. Zbl0299.02069
  20. [20] H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics, PWN, Warszawa 1963. 
  21. [21] D. Schwartz, Polyadic MV-algebras, Z. Math. Logik Grundlag. Math. 26 (1980), 561-564. Zbl0488.03035
  22. [22] A. Wroński, BCK-algebras do not form a variety, Math. Japon. 28 (1983), 211-213. Zbl0518.06014

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