## On Triangular Norm-Based Propositional Fuzzy Logics (1995)

Venue: | Fuzzy Sets and Systems |

Citations: | 16 - 3 self |

### BibTeX

@ARTICLE{Butnariu95ontriangular,

author = {Dan Butnariu and Erich Peter Klement and Samy Zafrany},

title = {On Triangular Norm-Based Propositional Fuzzy Logics},

journal = {Fuzzy Sets and Systems},

year = {1995},

volume = {69},

pages = {241--255}

}

### OpenURL

### Abstract

Fuzzy logics based on triangular norms and their corresponding conorms are investigated. An affirmative answer to the question whether in such logics a specific level of satisfiability of a set of formulas can be characterized by the same level of satisfiability of its finite subsets is given. Tautologies, contradictions and contingencies with respect to such fuzzy logics are studied, in particular for the important cases of min-max and Lukasiewicz logics. Finally, fundamental t-norm-based fuzzy logics are shown to provide a gradual transition between minmax and Lukasiewicz logics. Key words: Fuzzy Logics, Min-max logic, Lukasiewicz Logic, Triangular Norms, Satisfiability. AMS-Classification: 03B52, 03B50, 03B05 0

### Citations

3133 | Fuzzy sets
- Zadeh
- 1965
(Show Context)
Citation Context ...words: Fuzzy Logics, Min-max logic, Lukasiewicz Logic, Triangular Norms, Satisfiability. AMS-Classification: 03B52, 03B50, 03B05 0 Introduction Fuzzy logic emerged as a necessary extension of Zadeh's =-=[13]-=- theory of fuzzy sets. Its development during the last two decades was stimulated by its applications in fields like artificial intelligence and heuristic decision making (see Zadeh [16]). Fuzzy logic... |

600 |
A Mathematical Introduction to Logic
- Enderton
- 2001
(Show Context)
Citation Context ...ruth assignment in " '2\Gamma C ' . By definition of C ' , t(') 2 K for every ' 2 \Gamma, and therefore \Gamma is K-satisfiable. The compactness property of bivalued propositional logic (see Ende=-=rton [4]-=-, page 59) is a special case of Theorem 2.3 with K = f1g (the choice of S is irrelevant in this case since every t-conorm coincides with the truth table ofsin bivalued logic). The case when K is a sin... |

277 |
Topology: A First Course
- Munkres
- 1975
(Show Context)
Citation Context ...= 1; 2; : : : ; n, we have t(' i ) 2 K, i.e., n " i=1 C ' i 6= ;. Therefore, fC ' g '2\Gamma is a collection of closed sets 7 having the finite intersection property. Tychonoff's Theorem (see Mun=-=kres [8], page 229-=-) ensures that [0; 1] P is a compact topological space. Hence from the finite intersection property we deduce that " '2\Gamma C ' 6= ;. Let t be a truth assignment in " '2\Gamma C ' . By def... |

252 |
Probabilistic Metric Spaces
- Schweizer, Sklar
- 1983
(Show Context)
Citation Context ...]). In this work, by the term "fuzzy logic" we mean a specific [0; 1]-valued propositional logic whose connectives are interpreted via triangular norms. Recall (see, for instance, Schweizer =-=and Sklar [12]-=-, Butnariu and Klement [1]) that a triangular norm (t-norm for short) is a function T : [0; 1]\Theta[0; 1] ! [0; 1] which is symmetric (T (x; y) = T (y; x)), associative (T (x; T (y; z)) = T (T (x; y)... |

149 |
A Theory of Approximate Reasoning
- Zadeh
- 1979
(Show Context)
Citation Context ... fields like artificial intelligence and heuristic decision making (see Zadeh [16]). Fuzzy logical systems formalize the rules of reasoning with vague statements in various settings (see, e.g., Zadeh =-=[15]-=- and Dubois and Prade [3]). The variety of fuzzy logical systems discussed in the literature reflects the diversity of ways in which vagueness can be approached from a logical point of view as well as... |

68 |
Knowledge Representation in Fuzzy Logic
- Zadeh
- 1989
(Show Context)
Citation Context ...on of Zadeh's [13] theory of fuzzy sets. Its development during the last two decades was stimulated by its applications in fields like artificial intelligence and heuristic decision making (see Zadeh =-=[16]-=-). Fuzzy logical systems formalize the rules of reasoning with vague statements in various settings (see, e.g., Zadeh [15] and Dubois and Prade [3]). The variety of fuzzy logical systems discussed in ... |

51 |
On fuzzy logic
- Pavelka
- 1979
(Show Context)
Citation Context ...relate the syntax and the semantics of the logics based on t-norms which were studied separately in different contexts, and under various aspects by many people (see, for instance, Novak [9], Pavelka =-=[10]-=-, Dubois and Prade [3], and the references therein). Section 3 is devoted to satifiability problems in fuzzy logics. The question there is whether a specific level of satifiability (see Definition 2.4... |

41 | Fuzzy Logic and Approximate Reasoning. Synthese - Zadeh - 1975 |

35 |
Triangular norm-based measures and games with fuzzy coalitions
- Butnariu, Klement
- 1993
(Show Context)
Citation Context ...and x 2 [0; 1] let c(s; x) = 8 ? ? ! ? ? : x if ss1; s x \Gamma1 s\Gamma1 if s ! 1: Then T s (a; T s (a; b))sc(s; a)\DeltaT s (a; b); (5.9) 20 for all a; b 2 [0; 1]. According to Butnariu and Klement =-=[2]-=- (Proposition 1.12), if ss1, then T s (a; T s (a; b))sT 1 (a; T s (a; b)) = a\DeltaT s (a; b) for all a; b 2 [0; 1], and (5.9) holds in this case. Suppose that s ! 1. Denote c = c(s; a) for some fixed... |

26 |
D.: Mthematical Logic
- Monk
- 1976
(Show Context)
Citation Context ...= fH ' (t) j t 2 [0; 1] P g = H ' i [0; 1] P j ; 9 the proof of the proposition is complete. We can be more specific about the nature of V ' . To this end we recall the following definition (see Monk =-=[7]-=-). Let us abbreviate the formula :ffsfi by ff ! fi. Definition 3.2 (see Monk [7]) A set \Gamma ` F is said to be closed under Modus Ponens, if / 2 \Gamma whenever ' 2 \Gamma and ' ! / 2 \Gamma. We say... |

25 |
Fragments of many-valued statement calculi, Transactions of the American Mathematical Society 87
- Rose, Rosser
- 1958
(Show Context)
Citation Context ...ia infinitary disjunctions. We show that whenever a fuzzy logic has a fundamental t-norm-based interpretation, it automatically incorporates the / Lukasiewicz [0; 1]-valued logic (see Rose and Rosser =-=[11]-=-) as well as the min-max fuzzy logic (see Definition 1.1). Precisely, Proposition 5.4 says that, given a fundamental t-norm-based fuzzy logic P, 2 any well-formed formula ' in / Lukasiewicz logic or i... |

15 |
On the syntactico-semantical completeness of first-order fuzzy logic, parts
- Novák
- 1990
(Show Context)
Citation Context ...empt to interrelate the syntax and the semantics of the logics based on t-norms which were studied separately in different contexts, and under various aspects by many people (see, for instance, Novak =-=[9]-=-, Pavelka [10], Dubois and Prade [3], and the references therein). Section 3 is devoted to satifiability problems in fuzzy logics. The question there is whether a specific level of satifiability (see ... |

13 |
Fuzzy logics and the generalized modus ponens revisited
- Dubois, Prade
- 1984
(Show Context)
Citation Context ...telligence and heuristic decision making (see Zadeh [16]). Fuzzy logical systems formalize the rules of reasoning with vague statements in various settings (see, e.g., Zadeh [15] and Dubois and Prade =-=[3]-=-). The variety of fuzzy logical systems discussed in the literature reflects the diversity of ways in which vagueness can be approached from a logical point of view as well as the heterogeneous nature... |

10 |
Triangular-norm-based measures and their Markov kernel representation
- Butnariu, Klement
(Show Context)
Citation Context ...al point of view as well as the heterogeneous nature of the vagueness concepts appearing in real world problems (see R. L. De Mantaras [6]). In this work, by the term "fuzzy logic" we mean a=-= specific [0; 1]-=--valued propositional logic whose connectives are interpreted via triangular norms. Recall (see, for instance, Schweizer and Sklar [12], Butnariu and Klement [1]) that a triangular norm (t-norm for sh... |

9 |
On the simultaneous Associativity of F (x; y) and x + y \Gamma F (x; y
- Frank
- 1979
(Show Context)
Citation Context ...); (5.12) where ffl n g 1 n=1 is the sequence of formulas in F(P s ) associated to ' and / by the rules (5.7) and (5.8). Denote a = G('), b = G(/), and a n = t s (ff n ); b n = t s (fi n ) From Frank =-=[5]-=- it follows that T s (x; y) + S s (x; y) = x + y; for every x; y 2 [0; 1]. This implies a n + b n = a + b; (5.13) 21 by induction on n, because t s is the natural extension of t to F(P s ). Let q := S... |

8 |
Approximate Reasoning Models
- Mantaras, R
- 1990
(Show Context)
Citation Context ...versity of ways in which vagueness can be approached from a logical point of view as well as the heterogeneous nature of the vagueness concepts appearing in real world problems (see R. L. De Mantaras =-=[6]). In this-=- work, by the term "fuzzy logic" we mean a specific [0; 1]-valued propositional logic whose connectives are interpreted via triangular norms. Recall (see, for instance, Schweizer and Sklar [... |