### Abstract

In fuzzy set theory, the fuzzification of a crisp concept is not seldom rooted in a straightforward adjustment of the disjunctive or conjunctive Boolean normal form of the underlying mathematical expression. However, the fuzzified normal forms obtained can rarely be considered as true normal forms in an extended logic or algebra. They are to be considered as functions, defined on [0,1]n, for some n∈N0, and taking values in the support [0,1] of a BL-algebra ([0,1],∨,∧,T,IT,0,1), with T a continuous t-norm and IT its residual implicator. In this paper, we clear out some misunderstandings concerning fuzzified normal forms and explore for which continuous De Morgan triplets the disjunctive fuzzified normal form is smaller than or equal to the conjunctive fuzzified normal form. Furthermore, we figure out to what extent their mutual distance depends on the original Boolean function. Special attention is drawn to the Łukasiewicz triplet, as it is the only continuous De Morgan triplet for which the difference between both fuzzified normal forms is independent of the underlying Boolean function.

Original language | English |
---|---|

Pages (from-to) | 161-179 |

Number of pages | 19 |

Journal | Fuzzy Sets and Systems |

Volume | 153 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jul 16 2005 |

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### Keywords

- Łukasiewicz triplet
- Cauchy equation
- Conjunctive normal form
- De Morgan triplet
- Disjunctive normal form
- T-conorm
- T-norm

### ASJC Scopus subject areas

- Statistics and Probability
- Electrical and Electronic Engineering
- Statistics, Probability and Uncertainty
- Information Systems and Management
- Computer Vision and Pattern Recognition
- Computer Science Applications
- Artificial Intelligence

### Cite this

*Fuzzy Sets and Systems*,

*153*(2), 161-179. https://doi.org/10.1016/j.fss.2005.01.011

**The unique role of the Łukasiewicz-triplet in the theory of fuzzified normal forms.** / Maes, Koen; De Baets, Bernard; Fodor, J.

Research output: Contribution to journal › Article

*Fuzzy Sets and Systems*, vol. 153, no. 2, pp. 161-179. https://doi.org/10.1016/j.fss.2005.01.011

}

TY - JOUR

T1 - The unique role of the Łukasiewicz-triplet in the theory of fuzzified normal forms

AU - Maes, Koen

AU - De Baets, Bernard

AU - Fodor, J.

PY - 2005/7/16

Y1 - 2005/7/16

N2 - In fuzzy set theory, the fuzzification of a crisp concept is not seldom rooted in a straightforward adjustment of the disjunctive or conjunctive Boolean normal form of the underlying mathematical expression. However, the fuzzified normal forms obtained can rarely be considered as true normal forms in an extended logic or algebra. They are to be considered as functions, defined on [0,1]n, for some n∈N0, and taking values in the support [0,1] of a BL-algebra ([0,1],∨,∧,T,IT,0,1), with T a continuous t-norm and IT its residual implicator. In this paper, we clear out some misunderstandings concerning fuzzified normal forms and explore for which continuous De Morgan triplets the disjunctive fuzzified normal form is smaller than or equal to the conjunctive fuzzified normal form. Furthermore, we figure out to what extent their mutual distance depends on the original Boolean function. Special attention is drawn to the Łukasiewicz triplet, as it is the only continuous De Morgan triplet for which the difference between both fuzzified normal forms is independent of the underlying Boolean function.

AB - In fuzzy set theory, the fuzzification of a crisp concept is not seldom rooted in a straightforward adjustment of the disjunctive or conjunctive Boolean normal form of the underlying mathematical expression. However, the fuzzified normal forms obtained can rarely be considered as true normal forms in an extended logic or algebra. They are to be considered as functions, defined on [0,1]n, for some n∈N0, and taking values in the support [0,1] of a BL-algebra ([0,1],∨,∧,T,IT,0,1), with T a continuous t-norm and IT its residual implicator. In this paper, we clear out some misunderstandings concerning fuzzified normal forms and explore for which continuous De Morgan triplets the disjunctive fuzzified normal form is smaller than or equal to the conjunctive fuzzified normal form. Furthermore, we figure out to what extent their mutual distance depends on the original Boolean function. Special attention is drawn to the Łukasiewicz triplet, as it is the only continuous De Morgan triplet for which the difference between both fuzzified normal forms is independent of the underlying Boolean function.

KW - Łukasiewicz triplet

KW - Cauchy equation

KW - Conjunctive normal form

KW - De Morgan triplet

KW - Disjunctive normal form

KW - T-conorm

KW - T-norm

UR - http://www.scopus.com/inward/record.url?scp=18444374700&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18444374700&partnerID=8YFLogxK

U2 - 10.1016/j.fss.2005.01.011

DO - 10.1016/j.fss.2005.01.011

M3 - Article

VL - 153

SP - 161

EP - 179

JO - Fuzzy Sets and Systems

JF - Fuzzy Sets and Systems

SN - 0165-0114

IS - 2

ER -