## Toward problems for mathematical fuzzy logic (2006)

Venue: | PROC. OF IEEE INTERNATIONAL CONFERENCE ON FUZZY SYSTEMS |

Citations: | 1 - 0 self |

### BibTeX

@INPROCEEDINGS{Gottwald06towardproblems,

author = {Siegfried Gottwald},

title = {Toward problems for mathematical fuzzy logic},

booktitle = {PROC. OF IEEE INTERNATIONAL CONFERENCE ON FUZZY SYSTEMS},

year = {2006},

pages = {1649--1654},

publisher = {}

}

### OpenURL

### Abstract

The paper discusses some open problems in the field of mathematical fuzzy logic which may have a decisive influence for the future development of fuzzy logic within the next decade.

### Citations

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Citation Context ...t a cumulative set theory for fuzzy sets. Additionally there is a second approach toward a system of types offered by Novák [40]. This approach adopts the functional type theoretic approach of Church =-=[7]-=-. What remains open is to finish the first mentioned of these approaches, to clarify the relationships between both approaches, and to apply them to set theoretic as well as to linguistic settings. Bu... |

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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 =-=[1]-=- for the crisp case this may happen in the following steps, cf. [5]: 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 form i... |

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Citation Context ...fer some such problems of foundational nature. The last decade has seen an enormous development of propositional and firstorder mathematical fuzzy logics, with the important and influential monograph =-=[28]-=- as one of the early highlights. The development up to the recent state of the art is surveyed e.g. in [27, 24, 9]. We assume that the reader is aware of the basic facts concerning this development wh... |

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Citation Context ...gebras by Hájek [28] to get a semantics for the logic BL of all continuous t-norms. Completely independent of this development and particularly for applications to quantum theory Birkhoff/von Neumann =-=[3]-=- developed a particular logic, the quantum logic. Its aim is to allow to discuss validity relationships in the realm of quantum theoretical propositions. This logic has also an algebraic semantics, pr... |

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Citation Context ...ils are in [43]. 7sAnother combination of the ideas from fuzzy set theory and non-classical logic, the latter one often also understood as philosophical logic, has been offered by the work of Fitting =-=[12, 13]-=- which is explained also e.g. in [23]. In these papers graded modalities appear in the context of applications to multi-agent systems. And quite recently, again related to the field of computer scienc... |

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Citation Context ...e.g. in [23]. In these papers graded modalities appear in the context of applications to multi-agent systems. And quite recently, again related to the field of computer science logics, we see e.g. in =-=[47]-=- also approaches toward an integration of fuzzy sets and vague notions into the field of description logics. But the potential field of applications is much wider. Let us look at deontic notions like ... |

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Citation Context ...th degrees and thus a particular algebraic structure with this carrier as their (standard) logical matrix. However, all these systems proved to be algebraizable in the technical sense of Blok/Pigozzi =-=[4]-=-, which means that they also have algebraic semantics, provided by a class—and usually a variety—of algebraic structures, and which sometimes are defined immediately via an algebraic semantics. 1s2 Hi... |

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Citation Context ...ble algebraic structures—i.e. by classes, or preferably by varieties, of such structures. This development started with the introduction of the MV-algebras in a completeness proof for L∞ the by Chang =-=[6]-=- in 1958, continued much later with the product algebras of [31] which provide an algebraic semantics fort product logic Π, and culminated in the introduction of BL-algebras by Hájek [28] to get a sem... |

61 | A Treatise on Many-Valued Logics
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Citation Context ...ment of fuzzy logic within the next decade. 1 Introduction Based upon the recent development in mathematical fuzzy logics—understood as particular cases of infinite valued logics—as explained e.g. in =-=[23, 24, 27]-=-, and also based upon the development in the field of foundations of fuzzy set theory as explained e.g. in [25, 26], the present paper intends to locate areas which seem to provide fundamental open pr... |

55 | Quantum logic
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Citation Context ...cuss validity relationships in the realm of quantum theoretical propositions. This logic has also an algebraic semantics, provided by the class of all orthomodular lattices. A survey is given e.g. in =-=[10]-=-. Such an orthomodular lattice reflects the essential algebraic properties of the system of closed subspaces of a Hilbert space. It is an ortholattice, i.e. a bounded lattice with a universal upper bo... |

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Citation Context ...uitable singletons. And this happens not for global M-sets, but for M-sets, which generalize the global M-sets in that they refer to a notion of local existence as was done for the cHA valued case in =-=[44]-=-. Independent of the different technical tools both types of approaches—the axiomatic as well as the category theoretic ones—refer to, there seems to be a major difference between both, concerning the... |

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Citation Context ...in mind that for each cHA H the Higgs topos SET(H) of H-valued sets (and H-set morphisms) is categorically equivalent to the category sh(H) of sheaves over H. Furthermore one knows that Fourman/Scott =-=[14]-=- have shown that every presheaf determines an H-set, and that the sheaves correspond to the 5scomplete H-sets (which have all their singleton subsets determined by a single element). His generalizatio... |

40 |
Fuzzy Sets and Fuzzy Logic: Foundations of Application—from a Mathematical Point of View
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Citation Context ...se and the other sort for fuzzy sets. The advantage of this choice is that (i) this logic is well understood, and that (ii) it has sufficiently high expressive power such that former approaches, like =-=[22]-=-, 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... |

38 | What the lottery paradox tells us about default reasoning
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Citation Context ...ilar theoretical results as in the crisp case, as can be seen from [42]. Also another tool from non-monotonic reasoning has a natural generalization to a fuzzy setting: Poole systems as introduced in =-=[41]-=-. 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... |

34 |
Fuzzy Logic: Mathematical Tools for Approximate Reasoning
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Citation Context ...z logic L∞, and gives some basic properties of the non-monotonic inference operator defined via minimal models. Interesting new ideas, based upon the abstract approach toward fuzzy logic discussed in =-=[15]-=-, have quite recently been offered in [5, 42]. 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} , (5) and to define t... |

31 |
New trends in Quantum Structures
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Citation Context ...the orthomodular lattices also other algebraic structures have been studied which are designed to model (basic) algebraic properties of quantum theoretical propositions. A survey offers the monograph =-=[11]-=- which discusses in detail all the structures we are going to mention now. As a common generalization of the MV-algebras and the orthomodular lattices one has on the one hand defined the QMV-algebras ... |

31 | A complete many-valued logic with product-conjunction, Archive for Mathematical Logic 35
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Citation Context ...eties, of such structures. This development started with the introduction of the MV-algebras in a completeness proof for L∞ the by Chang [6] in 1958, continued much later with the product algebras of =-=[31]-=- which provide an algebraic semantics fort product logic Π, and culminated in the introduction of BL-algebras by Hájek [28] to get a semantics for the logic BL of all continuous t-norms. Completely in... |

24 |
residuated l-monoids”; Non-Classical Logics and Their Applications to Fuzzy Subsets
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Citation Context ...-sets. A more general non-intuitionistic, particularly a monoidal context together with a graded notion of identity for fuzzy sets is the core problem for the approach of Höhle as explained mainly in =-=[33, 34, 35, 36]-=-. Instead of totally fuzzy sets, i.e. instead of H-sets, he considers M-sets, with M = (L, ≤, ∗) an integral, divisible, commutative completely lattice-ordered monoid with zero, i.e. a complete divisi... |

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Citation Context ...ituation in crisp set theory with axiomatic approaches in the standard sense, and with category theoretic approaches. The first who introduced a category of fuzzy sets was Goguen [17]. A bit later he =-=[19]-=- gave a categorical characterization of fuzzy sets. For the development that arose out of these approaches, a nice unifying survey is given in [49]. The basic logic in any case is the intuitionistic o... |

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Citation Context ...rly days of fuzzy set theory, to identify the membership degrees of fuzzy sets with truth degrees of a suitable many-valued logic. In different forms, this idea has been offered and explained e.g. in =-=[16, 18, 21]-=-. 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 ...m [46] and resulted—after a series of intermediate steps mentioned e.g. in [23]—in a proof theoretic proof (in the realm of L∞) of the consistency of naive set theory with comprehension only by White =-=[48]-=-. Two equality predicates come into consideration here—Leibniz equality =l and extensional equality =e with definitions x =l y =def ∀z(x ∈ z ↔ y ∈ z) , x =e y =def ∀z(z ∈ x ↔ z ∈ y) . Leibniz equality... |

17 |
On fuzzy type theory
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Citation Context ...d even more one has to wait whether on this way one will be able to arrive at a cumulative set theory for fuzzy sets. Additionally there is a second approach toward a system of types offered by Novák =-=[40]-=-. This approach adopts the functional type theoretic approach of Church [7]. What remains open is to finish the first mentioned of these approaches, to clarify the relationships between both approache... |

16 |
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Citation Context ...rly days of fuzzy set theory, to identify the membership degrees of fuzzy sets with truth degrees of a suitable many-valued logic. In different forms, this idea has been offered and explained e.g. in =-=[16, 18, 21]-=-. 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 ...-sets. A more general non-intuitionistic, particularly a monoidal context together with a graded notion of identity for fuzzy sets is the core problem for the approach of Höhle as explained mainly in =-=[33, 34, 35, 36]-=-. Instead of totally fuzzy sets, i.e. instead of H-sets, he considers M-sets, with M = (L, ≤, ∗) an integral, divisible, commutative completely lattice-ordered monoid with zero, i.e. a complete divisi... |

12 | Mathematical fuzzy logic as a tool for the treatment of vague information
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Citation Context ...ment of fuzzy logic within the next decade. 1 Introduction Based upon the recent development in mathematical fuzzy logics—understood as particular cases of infinite valued logics—as explained e.g. in =-=[23, 24, 27]-=-, and also based upon the development in the field of foundations of fuzzy set theory as explained e.g. in [25, 26], the present paper intends to locate areas which seem to provide fundamental open pr... |

10 |
Fuzzy class theory, Fuzzy Sets and Systems 154
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Citation Context ... 1〉, of course with unions at limit stages. 1 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 =-=[2]-=- 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 |
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Citation Context ...l, and finally also a kind of Henkin-style fuzzy type theory and higher order fuzzy logics, formally treated as a many-sorted first-order logic. Actually this is work in progress, partly contained in =-=[8]-=-, which seems to tend toward a type hierarchy which is not cumulative, i.e. which has strongly to distinguish and to separate the different levels of the type hierarchy. One has to wait whether this a... |

9 |
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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 =-=[30]-=- and the subsequent Ph.D. Thesis [32] use the (firstorder) logic BL of continuous t-norms, extended with the △-operator, to develop a ZF-like axiomatization for a formalized fuzzy set theory together ... |

9 |
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Citation Context ... over L∞. That older approach toward a consistency proof of naive set theory, i.e. of set theory with comprehension and extensionality only, in the realm of ̷Lukasiewicz logic was initiated by Skolem =-=[46]-=- and resulted—after a series of intermediate steps mentioned e.g. in [23]—in a proof theoretic proof (in the realm of L∞) of the consistency of naive set theory with comprehension only by White [48]. ... |

6 |
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Citation Context ...]] ∗ △[[y � x]] . And this last condition which is formally necessary that the approach does not collapse to the crisp case forces the equality to be crisp. Somehow similar, most recently again Hájek =-=[29]-=- was going back to an older approach and considered a Cantorian set theory over L∞. That older approach toward a consistency proof of naive set theory, i.e. of set theory with comprehension and extens... |

6 |
Presheaves over GL-monoids. Non-Classical Logic and Their Applications to Fuzzy Subsets
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Citation Context ...-sets. A more general non-intuitionistic, particularly a monoidal context together with a graded notion of identity for fuzzy sets is the core problem for the approach of Höhle as explained mainly in =-=[33, 34, 35, 36]-=-. Instead of totally fuzzy sets, i.e. instead of H-sets, he considers M-sets, with M = (L, ≤, ∗) an integral, divisible, commutative completely lattice-ordered monoid with zero, i.e. a complete divisi... |

6 |
VUber einen Ansatz zur mehrwertigen Mengenlehre, Monatsb
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Citation Context ... identity The different approaches toward set theories for fuzzy sets which have been offer through the years, and surveyed in [25, 26], differ in their way to treat identity. Quite early, already in =-=[37, 38, 20]-=- the idea has been present that the identity between fuzzy sets should itself be a graded property. But these approaches have been semi-formalized only, from an axiomatic point of view. On the other h... |

5 |
On revising fuzzy belief bases, Studia Logica
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Citation Context ...s of the non-monotonic inference operator defined via minimal models. Interesting new ideas, based upon the abstract approach toward fuzzy logic discussed in [15], have quite recently been offered in =-=[5, 42]-=-. 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} , (5) and to define the theory of a class K ⊆ M of models as which... |

5 |
Triangular norm based predicate fuzzy logics
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Citation Context ...ional and firstorder mathematical fuzzy logics, with the important and influential monograph [28] as one of the early highlights. The development up to the recent state of the art is surveyed e.g. in =-=[27, 24, 9]-=-. We assume that the reader is aware of the basic facts concerning this development which has led to a wealth of formal systems of mathematical fuzzy logic, which mainly arose out of the desire to axi... |

5 |
Fuzzy sets theory: Some aspects of the early development, Aspects of vagueness
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Citation Context ...rly days of fuzzy set theory, to identify the membership degrees of fuzzy sets with truth degrees of a suitable many-valued logic. In different forms, this idea has been offered and explained e.g. in =-=[16, 18, 21]-=-. 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... |

4 |
Universes of fuzzy sets and axiomatizations of fuzzy set theory. Part I: Model-based and axiomatic approaches
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Citation Context ...cs—understood as particular cases of infinite valued logics—as explained e.g. in [23, 24, 27], and also based upon the development in the field of foundations of fuzzy set theory as explained e.g. in =-=[25, 26]-=-, the present paper intends to locate areas which seem to provide fundamental open problems for mathematical fuzzy logic and to offer some such problems of foundational nature. The last decade has see... |

4 |
T-norm based mathematical fuzzy logics
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(Show Context)
Citation Context ...ment of fuzzy logic within the next decade. 1 Introduction Based upon the recent development in mathematical fuzzy logics—understood as particular cases of infinite valued logics—as explained e.g. in =-=[23, 24, 27]-=-, and also based upon the development in the field of foundations of fuzzy set theory as explained e.g. in [25, 26], the present paper intends to locate areas which seem to provide fundamental open pr... |

4 |
Mathematical and Metamathematical Properties of Fuzzy Logic
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Citation Context ... 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 [30] and the subsequent Ph.D. Thesis =-=[32]-=- use the (firstorder) logic BL of continuous t-norms, extended with the △-operator, to develop a ZF-like axiomatization for a formalized fuzzy set theory together with a kind of standard model constru... |

4 |
Über einen zweiten Ansatz zur mehrwertigen
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- 1966
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Citation Context ... identity The different approaches toward set theories for fuzzy sets which have been offer through the years, and surveyed in [25, 26], differ in their way to treat identity. Quite early, already in =-=[37, 38, 20]-=- the idea has been present that the identity between fuzzy sets should itself be a graded property. But these approaches have been semi-formalized only, from an axiomatic point of view. On the other h... |

3 |
A Cumulative System of Fuzzy Sets
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Citation Context ... identity The different approaches toward set theories for fuzzy sets which have been offer through the years, and surveyed in [25, 26], differ in their way to treat identity. Quite early, already in =-=[37, 38, 20]-=- the idea has been present that the identity between fuzzy sets should itself be a graded property. But these approaches have been semi-formalized only, from an axiomatic point of view. On the other h... |

3 |
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Citation Context |

3 |
On fuzzy circumscription
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Citation Context ...se, i.e. to the case in which either the knowledge comes e.g. with degrees of vagueness, or of confidence, or in which e.g. the defaults are accepted only to some degrees. A first idea was offered in =-=[39]-=-. This paper generalizes the circumscription approach in a straightforward way from classical logic to the infinite-valued ̷Lukasiewicz logic L∞, and gives some basic properties of the non-monotonic i... |

3 | Nonmonotonic inference operators for fuzzy logic
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(Show Context)
Citation Context ...s of the non-monotonic inference operator defined via minimal models. Interesting new ideas, based upon the abstract approach toward fuzzy logic discussed in [15], have quite recently been offered in =-=[5, 42]-=-. 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} , (5) and to define the theory of a class K ⊆ M of models as which... |

3 | Nonmonotonic Fuzzy Inference
- Richter
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(Show Context)
Citation Context ...n of fuzzy theories is not as straightforward as in the crisp case, but can also be handled sufficiently well with some extra care regarding the moment for taking (deductive) closures. Details are in =-=[43]-=-. 7sAnother combination of the ideas from fuzzy set theory and non-classical logic, the latter one often also understood as philosophical logic, has been offered by the work of Fitting [12, 13] which ... |

2 |
Tableaus for many-valued modal logic, Studia Logica 55
- Fitting
- 1995
(Show Context)
Citation Context ...ils are in [43]. 7sAnother combination of the ideas from fuzzy set theory and non-classical logic, the latter one often also understood as philosophical logic, has been offered by the work of Fitting =-=[12, 13]-=- which is explained also e.g. in [23]. In these papers graded modalities appear in the context of applications to multi-agent systems. And quite recently, again related to the field of computer scienc... |

1 |
Proposals for future developments in fuzzy set technology
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(Show Context)
Citation Context ...ble fuzzifications of areas of philosophical logic is immediately directed toward a deeper understanding of commonsense reasoning, but it is also of deeper philosophical significance, as explained in =-=[45]-=-. And our fourth problem of relationships between fuzzy and quantum logic aims to support the—partly rather old—efforts to understand the importance of the notion of fuzziness for quantum physics, and... |

1 |
Fuzzy logic and categories of fuzzy sets. In: Non-Classical Logics and Their Appl. to Fuzzy Subsets
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(Show Context)
Citation Context ...tegory of fuzzy sets was Goguen [17]. A bit later he [19] gave a categorical characterization of fuzzy sets. For the development that arose out of these approaches, a nice unifying survey is given in =-=[49]-=-. The basic logic in any case is the intuitionistic one, i.e. reference is to complete Heyting algebras (= cHA’s) H as structures of membership and of truth degrees. But graded identities come into co... |