## Mathematical fuzzy logic as a tool for the treatment of vague information (2005)

Venue: | Information Sciences |

Citations: | 11 - 1 self |

### BibTeX

@ARTICLE{Gottwald05mathematicalfuzzy,

author = {Siegfried Gottwald},

title = {Mathematical fuzzy logic as a tool for the treatment of vague information},

journal = {Information Sciences},

year = {2005},

volume = {172},

pages = {41--71}

}

### OpenURL

### Abstract

The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by a calculus for the derivation of formulas. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon these theoretical considerations. Key words: mathematical fuzzy logic, algebraic semantics, continuous t-norms, left-continuous t-norms, Pavelka-style fuzzy logic, fuzzy set theory, non-monotonic fuzzy reasoning 1

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Citation Context ...plement. For the ̷Lukasiewicz logic the corresponding class of structures is the class of all MV-algebras, first introduced again within a completeness proof by Chang [10], and extensively studied in =-=[12]-=-. And for the product logic the authors of [44] introduce a class of lattice ordered semigroups which they call product algebras. It is interesting to recognize that all these structures—prelinear Hey... |

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Citation Context ...truth degrees of a suitable many-valued logic. In different forms, this idea has been offered and explained e.g. in [28,30–32,53]. And it has since been the topic of occasional investigations like in =-=[68,69]-=-. This point of view toward fuzzy set theory has been one of the motivations behind the development of mathematical fuzzy logics. Therefore one may expect that the recent results in this field of math... |

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Citation Context ...ted e.g. in [72]. This approach treats consequence operations as closure operations. And this type of approach has been generalized to closure operations in classes of fuzzy sets of formulas by Gerla =-=[27]-=-. It shall be discussed in Section 11. The Pavelka-style approach has to deal with fuzzy sets Σ ∼ of formulas, i.e. besides formulas ϕ also their membership degrees Σ ∼ (ϕ) in Σ ∼ . And these membersh... |

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Citation Context ...wever, it should be mentioned that there is also another, more algebraically oriented approach toward consequence operations for the classical case, originating from Tarski [70] and presented e.g. in =-=[72]-=-. This approach treats consequence operations as closure operations. And this type of approach has been generalized to closure operations in classes of fuzzy sets of formulas by Gerla [27]. It shall b... |

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Citation Context ...rovable and leads back to the starting point of the whole approach: the logical calculus KBL characterizes just those formulas which hold true w.r.t. all divisible t-norm algebras. This was proved in =-=[11]-=-. Theorem 9 (Standard Completeness) The class of all formula which are provable in the system BL coincides with the class of all formulas which are logically valid in all t-norm algebras with a contin... |

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Citation Context ... is provable: the logical cal15sculus KMTL characterizes just these formulas which hold true w.r.t. all those t-norm based logics which are determined by a left continuous t-norm. A proof is given in =-=[51]-=-. Theorem 14 (Standard Completeness) The class of all formulas which are provable in the logical calculus KMTL coincides with the class of all formulas which are logically valid in all t-norm algebras... |

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Citation Context ...eded as the underlying logic for a theory of fuzzy sets, one finds three main systems: • the ̷Lukasiewicz logic L as explained in [58]; • the Gödel logic G from [29]; • the product logic Π studied in =-=[44]-=-. In their original presentations, these logics look rather different, regarding their propositional parts. For the first order extensions, however, there is a unique strategy: one adds a universal an... |

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Citation Context ...lculations show that) this non-idempotent Gödel negation is the standard negation of all those t-norm algebras with a t-norm ⊗ which does not have zero-divisors. 2 A very general approach is given in =-=[22]-=-, and a more particular axiomatization problem discussed in [35]. Another stream of papers, partly related to the previously mentioned one, is devoted to the problem of a unified treatment of differen... |

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Citation Context ...ment and of provability. However, it should be mentioned that there is also another, more algebraically oriented approach toward consequence operations for the classical case, originating from Tarski =-=[70]-=- and presented e.g. in [72]. This approach treats consequence operations as closure operations. And this type of approach has been generalized to closure operations in classes of fuzzy sets of formula... |

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Citation Context ...early ordered MV-algebras, or linearly ordered product algebras, such 13sthat (iii) each such ordinal summand is locally embedable into a t-norm based residuated lattice with a continuous t-norm, cf. =-=[11,38]-=- and again [33]. This is a lot more of algebraic machinery as necessary for the proof of the General Completeness Theorem 7 and thus offers a further indication that the extension of the class of divi... |

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Citation Context ...Logics If one looks for infinite valued logics of the kind which is needed as the underlying logic for a theory of fuzzy sets, one finds three main systems: • the ̷Lukasiewicz logic L as explained in =-=[58]-=-; • the Gödel logic G from [29]; • the product logic Π studied in [44]. In their original presentations, these logics look rather different, regarding their propositional parts. For the first order ex... |

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Citation Context ...ion, and had to add in any case a generalized negation. In this case the implication, defined according to one of the equivalences in (19), is often coined “S-implication”. This has been done e.g. in =-=[8,49]-=-, but this approach does not really become simpler as the former one because one needs to fix either, besides the basic t-norm, an additional negation, or one has to fix a negation together with a dis... |

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Citation Context ... t-norm ⊗ led to (finite) axiomatizations of those t-norm based logics which have a standard semantics determined just by this continuous t-norm algebra. These results have recently been presented in =-=[25]-=-. 8 The Logic of Left Continuous T-Norms The guess of Esteva/Godo [21] has been that one should arrive at the logic of left continuous t-norms if one starts from the logic of continuous t-norms and de... |

12 |
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Citation Context ... & ϱS(x, y)) ↔ ψi(y) � . Finally we take a look at a necessary and sufficient condition that the Mamdani-Assilian fuzzy relation (45) is a solution of the system of fuzzy relation equations, given in =-=[54]-=-. This can again be proved inside BL purely syntactically. Theorem 22 Suppose T ⊢ ∃xϕi(x) for all i = 1, . . . , m. Then the provability conditions (46) are satisfied for ϱMA(x, y) (instead of ϱ(x, y)... |

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9 |
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Citation Context ...truth degrees of a suitable many-valued logic. In different forms, this idea has been offered and explained e.g. in [28,30–32,53]. And it has since been the topic of occasional investigations like in =-=[68,69]-=-. This point of view toward fuzzy set theory has been one of the motivations behind the development of mathematical fuzzy logics. Therefore one may expect that the recent results in this field of math... |

8 |
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Citation Context ...easoning 1 Introduction The standard membership degrees of fuzzy sets, i.e. the reals from the unit interval, can in a natural way be understood as the truth degrees of an infinite valued logic, with =-=[0, 1]-=- as its truth degree set. With this understanding, the membership function µA of a fuzzy set A becomes the truth degree function of a graded membership predicate “. . . ε A”, in the sense that µA(x) i... |

8 |
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Citation Context ...s to develop a machinery to discuss also fuzzy sets of higher level, and finally also a kind of fuzzy type theory and higher order fuzzy logics. Actually this is work in progress, partly contained in =-=[16]-=-. 12.2 Discussing relation equations via BL-provability One of the mathematical ways to understand Zadeh’s idea [74] of a fuzzy control approach via a list of linguistic control rules, their transform... |

8 | Pseudo-t-norms and pseudoBL algebras - Flondor, Georgescu, et al. |

8 | Fuzzy logic with non-commutative conjunctions - Hàjek |

8 |
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Citation Context ...s in this field of mathematical fuzzy logics give rise to a return to this starting point to use the new insights e.g. for a coherent development of a (formalized) fuzzy set theory. Indeed, the paper =-=[45]-=- and the subsequent Ph.D. Thesis [48] use the (firstorder) logic BL of continuous t-norms, extended with the △-operator mentioned in (32), to develop a ZF-like axiomatization for a formalized fuzzy se... |

8 |
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Citation Context ...sorted language with one sort of variables for objects of the universe of discourse and the other sort for fuzzy sets. The advantage of this choice is that (i) this logic is well understood, cf. e.g. =-=[50]-=-, and that (ii) it has sufficiently high expressive power such that former approaches, like [32], which used a mixture of object and metalanguage considerations, can be unified and given in a uniform ... |

7 | The ̷LΠ and ̷LΠ 1 2 logics: two complete fuzzy systems joining ̷Lukasiewicz and product logics - Esteva, Godo, et al. - 2001 |

7 |
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Citation Context ...ion, and had to add in any case a generalized negation. In this case the implication, defined according to one of the equivalences in (19), is often coined “S-implication”. This has been done e.g. in =-=[8,49]-=-, but this approach does not really become simpler as the former one because one needs to fix either, besides the basic t-norm, an additional negation, or one has to fix a negation together with a dis... |

6 | The ̷LΠ and ̷LΠ 1 2 propositional and predicate logics, Fuzzy Sets and Systems - Cintula - 2001 |

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6 |
Rational Pavelka predicate logic is a conservative extension of lukasiewicz predicate logic
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Citation Context ...+ L instead of KL, becomes already provable if one adds truth degree constants only for all the rationals in [0, 1], as was shown in [39]. And this extension of L is even only a conservative one, cf. =-=[46]-=-, i.e. K + L proves only such constant-free formulas of the language with rational constants which are already provable in the standard infinite-valued ̷Lukasiewicz logic L. For more details the reade... |

5 |
On revising fuzzy belief bases, Studia Logica
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Citation Context ...he non-monotonic inference operator defined via minimal models. Interesting new ideas, based upon the abstract approach toward fuzzy logic discussed in Section 11, have quite recently been offered in =-=[7,66]-=-. It is possible to define for abstract fuzzy semantics M the model class of a fuzzy set Σ of formulas as modM(Σ) = {M ∈ M | M |=M Σ} , (49) and to define the theory of a class K ⊆ M of models as th(K... |