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

Venue: | Information Sciences |

Citations: | 10 - 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

### Citations

3043 | Fuzzy Sets
- Zadeh
- 1965
(Show Context)
Citation Context ... degree set [0, 1]. And this logic has to have, besides symbols for fuzzy sets and elements of the corresponding universe of discourse, connectives as well as quantifiers. Already Zadeh’s first paper =-=[73]-=- (implicitly) offers different proposals to understand the above mentioned and-operation: as taking the minimum or as taking the usual, algebraic product. 1 And similarly he proposes as versions of th... |

713 |
On the logic of theory change: Partial meet contraction and revision functions
- Alchourrón, Gärdenfors, et al.
- 1985
(Show Context)
Citation Context ...as B ∈ FL(F). The revision information (ϕ/a), understood as the fuzzy singleton of ϕ with membership degree a, tells that a “new” formula ϕ should be integrated with degree a. As in the AGM framework =-=[2]-=- for the crisp case this may happen in the following steps, cf. [6,7]: (1) Form the family B ⊥ (ϕ/a) of all maximal X ∈ FL(B) consistent with (ϕ/a). (2) Select a subset γ(B ⊥ (ϕ/a)) ⊆ B ⊥ (ϕ/a) and fo... |

549 |
Outline of a new approach to the analysis of complex systems and decision process
- Zadeh
- 1973
(Show Context)
Citation Context ...d 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 transformation into a fuzzy relation, and the application of the compositional rule of inference, is to transform such a list... |

417 |
An experiment in linguistic synthesis with a fuzzy logic controller
- Mamdani, Assilian
- 1975
(Show Context)
Citation Context ...nd discusses 25sparticularly two of them: RS = n� i=1 RMA = n� i=1 {(x, y) | x ∈ Ai → y ∈ Bi} , (44) {(x, y) | x ∈ Ai & y ∈ Bi} , (45) resulting out of approaches by Sanchez [67] and Mamdani/Assilian =-=[59]-=-. Here we will, following [60], discuss an embedding of these considerations into provability questions inside fuzzy logic, at the same time generalizing the context to the logical system BL of Sectio... |

381 |
Metamathematics of Fuzzy Logics
- Hájek
- 1998
(Show Context)
Citation Context ...ogic of continuous t-norms to extend the class of all divisible t-norm algebras in a moderate way to get a variety. And indeed this idea works: it was developed by P. Hájek and in detail explained in =-=[39]-=-. The core points are that one considers instead of the divisible t-norm algebras, which are linearly ordered integral monoids as mentioned previously, now lattice ordered integral monoids which are d... |

270 | A course in Universal Algebra
- Burris, Sankappanavar
- 1981
(Show Context)
Citation Context ... a class K of algebraic structures which is closed under the formations of subalgebras, of homomorphic images, and of direct products. For the algebraic details the interested reader may e.g. consult =-=[9,17,37]-=-. For the Gödel logic such a class of structures is, according to the completeness proof given by Dummett [19], the class of all Heyting algebras, i.e. of all rel5satively pseudo-complemented lattices... |

234 |
Triangular norms
- Klement, Mesiar, et al.
- 2000
(Show Context)
Citation Context ... ⊗ y) ⊗ z, (T2) x ⊗ y = y ⊗ x, (T3) if x ≤ y then x ⊗ z ≤ y ⊗ z, (T4) x ⊗ 1 = x. Such binary operations are known as t-norms and have been used in the context of probabilistic metric spaces, cf. e.g. =-=[55]-=-. At the same time they are considered as natural candidates for truth degree functions of conjunction connectives. And from such a t-norm one is able to derive (essentially) all the other truth degre... |

97 |
Universal Algebra
- Gratzer
- 1980
(Show Context)
Citation Context ... a class K of algebraic structures which is closed under the formations of subalgebras, of homomorphic images, and of direct products. For the algebraic details the interested reader may e.g. consult =-=[9,17,37]-=-. For the Gödel logic such a class of structures is, according to the completeness proof given by Dummett [19], the class of all Heyting algebras, i.e. of all rel5satively pseudo-complemented lattices... |

88 |
Zum intuitionistischen Aussagenkalkül, Akademie des Wissenschaften in
- Gödel
- 1932
(Show Context)
Citation Context ...e 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 extensions, however, there is a u... |

73 |
Monoidal t-norm based logic: towards a logic for left-continuous t-norms
- Esteva, Godo
(Show Context)
Citation Context ... 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 deletes the continuity condition, i.e. the divisibility condition (23). ... |

68 |
A propositional calculus with denumerable matrix
- Dummett
- 1959
(Show Context)
Citation Context ...of direct products. For the algebraic details the interested reader may e.g. consult [9,17,37]. For the Gödel logic such a class of structures is, according to the completeness proof given by Dummett =-=[19]-=-, the class of all Heyting algebras, i.e. of all rel5satively pseudo-complemented lattices, which satisfy the prelinearity condition (u ↣ v) ⊔ (v ↣ u) = 1 . (11) Here ⊔ is the lattice join and ↣ the r... |

59 |
C.: 1957, Algebraic analysis of many valued logics
- Chang
(Show Context)
Citation Context ...oin and ↣ the relative pseudo-complement. 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 a... |

53 |
Algebraic foundations of many-valued reasoning
- Cignoli, D’Ottaviano, et al.
- 2000
(Show Context)
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... |

52 | A Treatise on Many-valued Logics
- Gottwald
- 2001
(Show Context)
Citation Context ...that paper did not explicitly consider his “algebraic product” of fuzzy sets as a kind of intersection. 2sAs a reference for these and also other many-valued logics in general, the reader may consult =-=[33]-=-. 2.1 Gödel logic The simplest one of these logics is the Gödel logic G which has a conjunction ∧ and a disjunction ∨ defined by the minimum and the maximum, respectively, of the truth degrees of the ... |

52 |
Intuitionistic fuzzy logic and intuitionistic fuzzy sets theory
- Takeuti, Titani
- 1984
(Show Context)
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... |

50 |
On fuzzy logic I
- Pavelka
(Show Context)
Citation Context ... these considerations to the case that one starts from fuzzy sets of formulas, and that one gets from them as consequence hulls again fuzzy sets of formulas. This problem was first treated by Pavelka =-=[62]-=-. The basic monograph elaborating this approach is [61]. We discuss in the present section this kind of approach, because it uses graded relations of entailment and of provability. However, it should ... |

47 |
Resolution of composite fuzzy relation equation
- Sanchez
- 1976
(Show Context)
Citation Context ...f the problem, cf. [34], and discusses 25sparticularly two of them: RS = n� i=1 RMA = n� i=1 {(x, y) | x ∈ Ai → y ∈ Bi} , (44) {(x, y) | x ∈ Ai & y ∈ Bi} , (45) resulting out of approaches by Sanchez =-=[67]-=- and Mamdani/Assilian [59]. Here we will, following [60], discuss an embedding of these considerations into provability questions inside fuzzy logic, at the same time generalizing the context to the l... |

44 |
Infinite-valued Gödel logics with 0-1-projections and relativizations
- Baaz
- 1996
(Show Context)
Citation Context ...extension or the modification of the expressive power of these logical systems. A first, quite fundamental addition to the standard vocabulary of the languages of t-norm based systems was proposed in =-=[3]-=-: a unary propositional 16soperator △ which has for t-norm algebras the semantics △(x) = 1 for x = 1 , △(x) = 0 for x �= 1 . (32) This unary connective can be added to the systems BL and MTL via the a... |

38 | What the lottery paradox tells us about default reasoning
- Poole
- 1989
(Show Context)
Citation Context ...r theoretical results as in the crisp case, as can be seen from [65]. 28sAlso another tool from non-monotonic reasoning has a natural generalization to a fuzzy setting: Poole systems as introduced in =-=[64]-=-. Such a crisp Poole system P is determined by a pair (D, C) of sets of sentences understood as the relevant defaults and constraints. For each set Σ of formulas and a suitably chosen closure operator... |

36 |
Fuzzy Sets and Fuzzy Logic — Foundations of Applications from a Mathematical Point of View
- Gottwald
- 1993
(Show Context)
Citation Context ...r 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 way within the language of ̷LΠ, again with the primitive predicates ∈, =. So one can e.g. expres... |

32 |
G.,Fuzzy Logic: Mathematical Tools for Approximate Reasoning
- Gerla
- 2001
(Show Context)
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... |

32 |
Theory of logical calculi. Basic theory of consequence operations, vol. 199 of Synthese
- Wójcicki
- 2004
(Show Context)
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... |

31 |
Boolean-Valued Models and Independence Proofs in Set Theory
- Bell
- 1985
(Show Context)
Citation Context ...lop a ZF-like axiomatization for a formalized fuzzy set theory together with a kind of standard model constructed in the style of Boolean valued models for (standard) set theory, as explained e.g. in =-=[5]-=-. The axioms are suitable versions of the axioms of extensionality, pairing, union, powerset, ∈-induction (i.e. foundation), separation, collection (i.e. comprehen23ssion), and infinity, together with... |

30 |
Basic Fuzzy Logic is the logic of continuous t-norms and their residua, Soft Computing 4
- Cignoli, Esteva, et al.
- 2000
(Show Context)
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... |

30 |
A proof of standard completeness for Esteva and Godo’s logic MTL, Studia Logica 70
- Jenei, Montagna
- 2002
(Show Context)
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... |

28 | A complete many-valued logic with product conjunction, Archive for
- L, Esteva
- 1996
(Show Context)
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... |

26 | Residuated fuzzy logics with an involutive negation
- Esteva, Godo, et al.
- 2000
(Show Context)
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... |

24 |
Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften. Monatshefte für Mathematik und Physik
- Tarski
- 1930
(Show Context)
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... |

20 | Lukasiewicz logic and fuzzy set theory - Giles - 1976 |

17 | Basic fuzzy logic and BL-algebras
- Hájek
- 1996
(Show Context)
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... |

17 |
Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküs, Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie
- ̷Lukasiewicz
- 1930
(Show Context)
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... |

16 | On triangular norm-based propositional fuzzy logics
- Butnariu, Klement, et al.
- 1995
(Show Context)
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... |

15 | The logic of inexact concepts, Synthese 19 - Goguen - 1968 |

12 |
Equational Characterization of the Subvarieties of BL Generated by t-norm Algebras
- Esteva, Godo, et al.
(Show Context)
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 |
Fuzzy points, fuzzy relations and fuzzy functions
- Klawonn
- 2000
(Show Context)
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)... |

10 |
Hoops and fuzzy logic
- Esteva, Godo, et al.
(Show Context)
Citation Context ...s. And for this kind of algebraic semantics one can find adequate axiomatizations for corresponding hoop logics quite similar to the approaches of Sections 7 and 8. The details have been developed in =-=[23]-=-. And a fifth stream discusses the generalization of the algebraic semantics from the case of abelian lattice ordered minoids with residuation to the case of non-commutative lattice ordered semigroups... |

9 |
Fuzzy class theory, Fuzzy Sets and Systems
- Běhounek, Cintula
- 2005
(Show Context)
Citation Context ...st condition forces the equality to be crisp. Besides this “global” approach toward a generalization of the idea of the cumulative set universe for fuzzy sets, there is also a recent more “local” one =-=[4]-=- which only aims to give a unified treatment of a theory of fuzzy subsets of a given universe of discourse, i.e. which–in a suitable sense–restricts the considerations to the first level of the transf... |

9 |
Fuzzy logic and fuzzy set theory
- Takeuti, Titani
- 1992
(Show Context)
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 |
Basic hoops: an algebraic study of continuous t-norms
- Aglianó, Ferreirim, et al.
(Show Context)
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 |
From fuzzy logic to fuzzy mathematics
- Cintula
- 2004
(Show Context)
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 |
A development of set theory in fuzzy logic
- Hájek, Haniková
- 2003
(Show Context)
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 |
Cintula P. Product ̷Lukasiewicz logic
- Horčík
(Show Context)
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 |
Two approaches to fuzzy propositional logics
- Hekrdla, Klement, et al.
- 2003
(Show Context)
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 |

6 | Pseudo t-norms and pseudo BLalgebras, Soft Computing 5 no - Flondor, Georgescu, et al. |

6 |
Rational Pavelka predicate logic is a conservative extension of lukasiewicz predicate logic
- Hájek, Paris, et al.
(Show Context)
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
- Booth, Richter
- 2005
(Show Context)
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... |