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Triangle algebras: a formal logic approach to interval-valued residuated lattices

Bart Van Gasse (UGent) , Chris Cornelis (UGent) , Glad Deschrijver (UGent) and Etienne Kerre (UGent)
(2008) Fuzzy Sets and Systems. 159(9). p.1042-1060
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Abstract
In this paper, we introduce triangle algebras: a variety of residuated lattices equipped with approximation operators, and with a third angular point u, different from 0 and 1. We show that these algebras serve as an equational representation of interval-valued residuated lattices (IVRLs). Furthermore, we present triangle logic (TL), a system of many-valued logic capturing the tautologies of IVRLs. Triangle algebras are used to cast the essence of using closed intervals of L as truth values into a set of appropriate logical axioms. Our results constitute a crucial first step towards solving an important research challenge: the axiomatic formalization of residuated t-norm based logics on L^I, the lattice of closed intervals of [0,1], in a similar way as was done for formal fuzzy logics on the unit interval.
Keywords
interval-valued fuzzy set theory, formal logic, residuated lattices

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Chicago
Van Gasse, Bart, Chris Cornelis, Glad Deschrijver, and Etienne Kerre. 2008. “Triangle Algebras: a Formal Logic Approach to Interval-valued Residuated Lattices.” Fuzzy Sets and Systems 159 (9): 1042–1060.
APA
Van Gasse, Bart, Cornelis, C., Deschrijver, G., & Kerre, E. (2008). Triangle algebras: a formal logic approach to interval-valued residuated lattices. Fuzzy Sets and Systems, 159(9), 1042–1060.
Vancouver
1.
Van Gasse B, Cornelis C, Deschrijver G, Kerre E. Triangle algebras: a formal logic approach to interval-valued residuated lattices. Fuzzy Sets and Systems. 2008;159(9):1042–60.
MLA
Van Gasse, Bart, Chris Cornelis, Glad Deschrijver, et al. “Triangle Algebras: a Formal Logic Approach to Interval-valued Residuated Lattices.” Fuzzy Sets and Systems 159.9 (2008): 1042–1060. Print.
@article{397598,
  abstract     = {In this paper, we introduce triangle algebras: a variety of residuated lattices equipped with approximation operators, and with a third angular point u, different from 0 and 1. We show that these algebras serve as an equational representation of interval-valued residuated lattices (IVRLs). Furthermore, we present triangle logic (TL), a system of many-valued logic capturing the tautologies of IVRLs. Triangle algebras are used to cast the essence of using closed intervals of L as truth values into a set of appropriate logical axioms. Our results constitute a crucial \unmatched{fb01}rst step towards solving an important research challenge: the axiomatic formalization of residuated t-norm based logics on L\^{ }I, the lattice of closed intervals of [0,1], in a similar way as was done for formal fuzzy logics on the unit interval.},
  author       = {Van Gasse, Bart and Cornelis, Chris and Deschrijver, Glad and Kerre, Etienne},
  issn         = {0165-0114},
  journal      = {Fuzzy Sets and Systems},
  language     = {eng},
  number       = {9},
  pages        = {1042--1060},
  title        = {Triangle algebras: a formal logic approach to interval-valued residuated lattices},
  url          = {http://dx.doi.org/10.1016/j.fss.2007.09.003},
  volume       = {159},
  year         = {2008},
}

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