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Journal of Logic and Computation Advance Access published online on September 11, 2009
Journal of Logic and Computation, doi:10.1093/logcom/exp050
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© The Author, 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Fuzzy Logic Corner

On White's Expansion of Lukasiewicz Logic

Petr Hájek

Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodárenskou vezí 2, Prague, 182 07 Czech Republic.
E-mail: hajek{at}cs.cas.cz

Received 2 February 2009.

Avariant of Lukasiewicz logic defined in a remark in White's 1979 paper on the consistency of the axiom of comprehension in the infinite-valued predicate logic of Lukasiewicz is studied; it is shown that the logic in question is an interesting expansion of Lukasiewicz logic which deserves further investigation together with its further expansions. Several formulas are proved in it and some model theory is offered. In the Appendix A, it is shown that a set theory with full comprehension over this logic (claimed consistent by White) is contradictory.

Keywords: Lukasiewicz fuzzy logic; Cantor-Lukasiewicz set theory; full comprehension



References

  1. Cintula P, Hájek P. Triangular norm based predicate logics. Fuzzy Sets and Systems (2009) (to appear).
  2. Hájek P. Metamathematics of Fuzzy Logic (1998) Kluwer.
  3. Hájek P. Arithmetical complexity of fuzzy predicate logics - a survey. Soft Computing (2005) 9:935–941.[CrossRef][Web of Science]
  4. Hájek P. Fuzzy logic and arithmetical hierarchy IV. In: First-order Logic Revisited—Hendricks FV, et al, eds. (2004) Logos Verlag Berline. 107–115.
  5. Hájek P. On Arithmetic in the Cantor-Lukasiewicz Fuzzy Set Theory. Archive for Mathematical Logic (2005) 44:763–782.[CrossRef][Web of Science]
  6. Hájek P. Mathematical fuzzy logic and natural numbers. Fundamenta Informaticae (2007) 81:155–163.[Web of Science]
  7. Hájek P. Some hedges for continuous t-norms. Neural Network World (2002) 12:159–164.
  8. Hájek P, Cintula P. On theories and models in fuzzy predicate logics. Journal of Symbolic Logic (2006) 71:863–880.[CrossRef][Web of Science]
  9. White RB. The consistency of the axiom of comprehension in the infinite-valued predicate logic of Lukasiewicz. Journal of Philosophical Logic (1979) 8:509–534.[Web of Science]
  10. White RB. A consistent theory of attributes in a logic without contraction. Studia Logica (1993) 52:113–142.[CrossRef]
  11. Yatabe S. Distinguishing non-standard natural numbers in a set theory within Lukasiewicz logic. Archive for Mathematical Logic (2007) 46:281–287.[CrossRef][Web of Science]

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This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
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Right arrow Articles by Hájek, P.
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