An algebraic completeness proof for Kleene's 3-valued logic

Maurizio Negri

Bollettino dell'Unione Matematica Italiana (2002)

  • Volume: 5-B, Issue: 2, page 447-467
  • ISSN: 0392-4041

Abstract

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We introduce Kleene's 3-valued logic in a language containing, besides the Boolean connectives, a constant n for the undefined truth value, so in developing semantics we can switch from the usual treatment based on DM-algebras to the narrower class of DMF-algebras (De Morgan algebras with a single fixed point for negation). A sequent calculus for Kleene's logic is introduced and proved complete with respect to threevalent semantics. The completeness proof is based on a version of the prime ideal theorem that is typical of DMF-algebras. Only for the weak completeness theorem the proof is fully algebrical, because in the proof of strong completeness we have been compelled to use topological methods (Tychonoff theorem on the product of compact spaces).

How to cite

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Negri, Maurizio. "An algebraic completeness proof for Kleene's 3-valued logic." Bollettino dell'Unione Matematica Italiana 5-B.2 (2002): 447-467. <http://eudml.org/doc/195535>.

@article{Negri2002,
abstract = {We introduce Kleene's 3-valued logic in a language containing, besides the Boolean connectives, a constant $n$ for the undefined truth value, so in developing semantics we can switch from the usual treatment based on DM-algebras to the narrower class of DMF-algebras (De Morgan algebras with a single fixed point for negation). A sequent calculus for Kleene's logic is introduced and proved complete with respect to threevalent semantics. The completeness proof is based on a version of the prime ideal theorem that is typical of DMF-algebras. Only for the weak completeness theorem the proof is fully algebrical, because in the proof of strong completeness we have been compelled to use topological methods (Tychonoff theorem on the product of compact spaces).},
author = {Negri, Maurizio},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {extension of Kleene's 3-valued logic; De Morgan algebras with a single fixed-point negation; sequent calculus; completeness},
language = {eng},
month = {6},
number = {2},
pages = {447-467},
publisher = {Unione Matematica Italiana},
title = {An algebraic completeness proof for Kleene's 3-valued logic},
url = {http://eudml.org/doc/195535},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Negri, Maurizio
TI - An algebraic completeness proof for Kleene's 3-valued logic
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/6//
PB - Unione Matematica Italiana
VL - 5-B
IS - 2
SP - 447
EP - 467
AB - We introduce Kleene's 3-valued logic in a language containing, besides the Boolean connectives, a constant $n$ for the undefined truth value, so in developing semantics we can switch from the usual treatment based on DM-algebras to the narrower class of DMF-algebras (De Morgan algebras with a single fixed point for negation). A sequent calculus for Kleene's logic is introduced and proved complete with respect to threevalent semantics. The completeness proof is based on a version of the prime ideal theorem that is typical of DMF-algebras. Only for the weak completeness theorem the proof is fully algebrical, because in the proof of strong completeness we have been compelled to use topological methods (Tychonoff theorem on the product of compact spaces).
LA - eng
KW - extension of Kleene's 3-valued logic; De Morgan algebras with a single fixed-point negation; sequent calculus; completeness
UR - http://eudml.org/doc/195535
ER -

References

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  1. BLAMEY, S., Partial logic, in D. Gabbay and F. Guenthner (eds.) Handbook of Philosophical Logic, Vol. III, D. Reidel, Dordrecht, 1986, 1-70. Zbl0875.03023
  2. CLEAVE, J. P., A Study of Logics, Oxford, 1991. Zbl0763.03003MR1149599
  3. DAVEY, B. A.- PRIESTLEY, H. A., Introduction to Lattices and Order, Cambridge, 1990. Zbl0701.06001MR1058437
  4. MALINOWSKI, G., Many Valued Logics, Oxford, 1992. Zbl0807.03010MR1269112
  5. NEGRI, M., Three valued semantics and DMF-algebras, Boll. Un. Mat. Ital. (7), 10-B (1996), 733-60. Zbl0882.03019MR1411525
  6. NEGRI, M., DMF-algebras: Representation and Topological Characterization, Boll. Un. Mat. Ital. (8), 10-B (1998), 369-90. Zbl0907.06011MR1638155

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