### Abstract

Contrapositive symmetry of R- and QL-implications defined from t-norms, t-conorms and strong negations is studied. For R-implications, characterizations of contrapositive symmetry are proved when the underlying t-norm satisfies a residuation condition. Contrapositive symmetrization of R-implications not having this property makes it possible to define a conjunction so that the residuation principle is preserved. Cases when this associated conjunction is a t-norm are characterized. As a consequence, a new family of t-norms (called nilpotent minimum) owing several attractive properties is discovered. Concerning QL-implications, contrapositive symmetry is characterized by solving a functional equation. When the underlying t-conorm is continuous and the t-norm is Archimedean, the t-conorm must be isomorphic to the Lukasiewicz one, while the t-norm must be isomorphic to a member from the well-known Frank family of t-norms. Finally, contrapositive symmetry for some new families of fuzzy implications is investigated.

Original language | English |
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Pages (from-to) | 141-156 |

Number of pages | 16 |

Journal | Fuzzy Sets and Systems |

Volume | 69 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 27 1995 |

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### Keywords

- Contrapositive symmetry
- Fuzzy implications
- Nilpotent minimum
- Strong negations
- t-conorms
- t-norms

### ASJC Scopus subject areas

- Artificial Intelligence
- Logic

### Cite this

**Contrapositive symmetry of fuzzy implications.** / Fodor, J.

Research output: Contribution to journal › Article

*Fuzzy Sets and Systems*, vol. 69, no. 2, pp. 141-156. https://doi.org/10.1016/0165-0114(94)00210-X

}

TY - JOUR

T1 - Contrapositive symmetry of fuzzy implications

AU - Fodor, J.

PY - 1995/1/27

Y1 - 1995/1/27

N2 - Contrapositive symmetry of R- and QL-implications defined from t-norms, t-conorms and strong negations is studied. For R-implications, characterizations of contrapositive symmetry are proved when the underlying t-norm satisfies a residuation condition. Contrapositive symmetrization of R-implications not having this property makes it possible to define a conjunction so that the residuation principle is preserved. Cases when this associated conjunction is a t-norm are characterized. As a consequence, a new family of t-norms (called nilpotent minimum) owing several attractive properties is discovered. Concerning QL-implications, contrapositive symmetry is characterized by solving a functional equation. When the underlying t-conorm is continuous and the t-norm is Archimedean, the t-conorm must be isomorphic to the Lukasiewicz one, while the t-norm must be isomorphic to a member from the well-known Frank family of t-norms. Finally, contrapositive symmetry for some new families of fuzzy implications is investigated.

AB - Contrapositive symmetry of R- and QL-implications defined from t-norms, t-conorms and strong negations is studied. For R-implications, characterizations of contrapositive symmetry are proved when the underlying t-norm satisfies a residuation condition. Contrapositive symmetrization of R-implications not having this property makes it possible to define a conjunction so that the residuation principle is preserved. Cases when this associated conjunction is a t-norm are characterized. As a consequence, a new family of t-norms (called nilpotent minimum) owing several attractive properties is discovered. Concerning QL-implications, contrapositive symmetry is characterized by solving a functional equation. When the underlying t-conorm is continuous and the t-norm is Archimedean, the t-conorm must be isomorphic to the Lukasiewicz one, while the t-norm must be isomorphic to a member from the well-known Frank family of t-norms. Finally, contrapositive symmetry for some new families of fuzzy implications is investigated.

KW - Contrapositive symmetry

KW - Fuzzy implications

KW - Nilpotent minimum

KW - Strong negations

KW - t-conorms

KW - t-norms

UR - http://www.scopus.com/inward/record.url?scp=0002118960&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002118960&partnerID=8YFLogxK

U2 - 10.1016/0165-0114(94)00210-X

DO - 10.1016/0165-0114(94)00210-X

M3 - Article

AN - SCOPUS:0002118960

VL - 69

SP - 141

EP - 156

JO - Fuzzy Sets and Systems

JF - Fuzzy Sets and Systems

SN - 0165-0114

IS - 2

ER -