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Logic Journal of IGPL Advance Access originally published online on July 4, 2009
Logic Journal of IGPL 2009 17(4):413-419; doi:10.1093/jigpal/jzp023
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

HpsUL is not the logic of pseudo-uninorms and their residua

Sanmin Wang

Department of Mathematics, Shaanxi Normal University, Xi’an 710062 and Department of Computer Science, Nanchang University, Nanchang 330031, P.R. China.
E-mail: wangsanmin{at}hotmail.com

Bin Zhao

Department of Mathematics, Shaanxi Normal University, Xi’an 710062, P.R. China.
E-mail: bzhao{at}snnu.edu.cn

This paper presents several results on the non-commutative fuzzy logic HpsUL, a Hilbert system whose corresponding algebraic semantics is the variety of bounded representable residuated lattices. In particular, we prove that HpsUL is not complete with respect to algebras based on the real unit interval, which answers the question posed by Metcalfe, Olivetti and Gabbay and shows that HpsUL is not the logic of pseudo-uninorms and their residua.

MSC2000: 03B52, 03G10.

Key Words: Fuzzy logic • residuated lattices • pseudo-uninorm

Received for publication 21 October 2008.

References

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    [8]  Metcalfe G, Olivetti N, Gabbay D. Proof Theory for Fuzzy Logics (2009) Springer Series in Applied Logic (Vol.36).

    [9]  Ono H. Substructural logics and residuated lattices: an introduction. Trends in logic (2003) 20:177–212.

    [10]  San-min W, Cintula P. Logics with Disjunction and Proof by Cases. Archive for Mathematical Logic (2008) 47:435–446.[CrossRef][Web of Science]

    [11]  San-min W, Feng Q, Ming-yan W. Solutions to Cintula's open problems. Fuzzy Sets and Systems (2006.) 157:2091–2099.[CrossRef][Web of Science]

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This Article
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