## Modulated Fibring and the Collapsing Problem (2001)

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### BibTeX

@MISC{Sernadas01modulatedfibring,

author = {Cristina Sernadas and João Rasga and Walter A. Carnielli},

title = {Modulated Fibring and the Collapsing Problem},

year = {2001}

}

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### Abstract

Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse to one of them, indicating that bring imposes unwanted interconnections between the given logics. Modulated bring allows a ner control of the combination, solving the collapsing problem both at the semantic and deductive levels. Main properties like soundness and completeness are shown to be preserved, comparison with bring is discussed, and some important classes of examples are analyzed with respect to the collapsing problem. 1

### Citations

184 | Fibring Logics
- Gabbay
- 1999
(Show Context)
Citation Context ... combined, the way such combinations can be applied and understanding the general properties (cf. [1]). Among the several approaches for spelling out such combinations, the techniques of fibring (cf. =-=[9, 10, 11, 16, 18]-=-) have been the most auspicious: the fibring of logics leads to a new logic where not only constructors are mixed, but proof methods are combined. Although the fibring techniques can be defined in the... |

52 |
Systematization of the finite many-valued logics through the method of tableaux
- Carnielli
- 1987
(Show Context)
Citation Context ...s, terms and quantifiers, thus extending the work in [17]. Another issue is to define modulated fibring for other kinds of deductive 32ssystems such as labeled deduction systems extending the work in =-=[4]-=- and [14]. We would also like to investigate the case where we have several designated truth-values instead of just one: we would thus be able to deal, for instance, with the logic LF I1 considered in... |

51 | Fibring of logics as a categorial construction
- Sernadas, Sernadas, et al.
- 1999
(Show Context)
Citation Context ... combined, the way such combinations can be applied and understanding the general properties (cf. [1]). Among the several approaches for spelling out such combinations, the techniques of fibring (cf. =-=[9, 10, 11, 16, 18]-=-) have been the most auspicious: the fibring of logics leads to a new logic where not only constructors are mixed, but proof methods are combined. Although the fibring techniques can be defined in the... |

45 | Fibring: Completeness preservation
- Zanardo, Sernadas, et al.
(Show Context)
Citation Context ... combined, the way such combinations can be applied and understanding the general properties (cf. [1]). Among the several approaches for spelling out such combinations, the techniques of fibring (cf. =-=[9, 10, 11, 16, 18]-=-) have been the most auspicious: the fibring of logics leads to a new logic where not only constructors are mixed, but proof methods are combined. Although the fibring techniques can be defined in the... |

41 | A taxonomy of C-systems
- Carnielli, Marcos
- 2002
(Show Context)
Citation Context ...nce, with the logic LF I1 considered in [6]. Of course also related with this work is the extension to logics of contradiction [12] and to paraconsistent logics [5] and logics of formal inconsistency =-=[7]-=- in general. Finally, we also intend to investigate bi-Heyting algebras [15] in the context of modulated fibring. Acknowledgments We are particularly grateful to Amílcar Sernadas for many interesting ... |

35 |
Fibred semantics and the weaving of logics: part 1
- Gabbay
- 1996
(Show Context)
Citation Context |

29 | Combining classical and intuitionistic logic, or: intuitionistic implication as a conditional
- Cerro, Herzig
- 1996
(Show Context)
Citation Context ... the collapsing problem, which consists of the unexpected collapse of two logics when combined even by unconstrained fibring (no symbols are shared). A simple yet paradigmatic example was provided in =-=[8, 10]-=-, where it is shown that, in our terms, the unconstrained fibring (sharing nothing!) of classical and intuitionistic logic collapses into classical logic. The result is that the original notion of fib... |

26 | Fibring Non-Truth-Functional Logics: Completeness Preservation. Submitted for publication. Available at http://www.cle.unicamp.br/prof/coniglio/publications.htm
- Caleiro, Carnielli, et al.
(Show Context)
Citation Context ...ring has been restricted to logics endowed with truth-functional semantics, some steps have been taken towards encompassing logics (like paraconsistent logics [5]) with non truth-functional semantics =-=[3]-=-. Fibring can be presented from a proof-theoretical or from a model-theoretical perspective. From the proof-theoretical point of view, fibring is treated in a natural way over logic systems with a Hil... |

25 | forthcoming, ``Why combine logics?'' Studia Logica - Blackburn, Rijke - 1990 |

23 |
An Overview of Fibred Semantics and The Combination of Logics
- Gabbay
- 1996
(Show Context)
Citation Context |

22 |
A Treatise on Many-Valued Logics. Research Studies
- Gottwald
- 2001
(Show Context)
Citation Context ... m ′, etc. Prop/Definition 2.8 Interpretation systems and their morphisms constitute the category Int. Some of the examples we consider are many-valued logics. For more details about these logics see =-=[2, 13]-=-. In all examples the signature is as follows: Σ = 〈C, &, Ξ, S〉 where t ∈ C0 (in general in C0 we also have propositional symbols), C1 = {¬}, C2 = {∧, ∨, ⇒}, Ck = ∅ for all k ≥ 3, & is ∧, Ξ = {ξi : i ... |

21 | Rijke. Why combine logics
- Blackburn, de
- 1997
(Show Context)
Citation Context ...cteristics of particular logic systems towards a general manner of investigating the way logics can be combined, the way such combinations can be applied and understanding the general properties (cf. =-=[1]-=-). Among the several approaches for spelling out such combinations, the techniques of fibring (cf. [9, 10, 11, 16, 18]) have been the most auspicious: the fibring of logics leads to a new logic where ... |

13 | Fibring labelled deduction systems
- Rasga, Sernadas, et al.
(Show Context)
Citation Context ...ce sometimes requiring Gentzen (sequent) or tableau style presentation. An appropriate framework for fibring natural deduction systems by means of labeled (or annotated) deduction systems is given in =-=[14]-=-. Usually, soundness is preserved under the process of fibring in the sense that the fibring of a family of logics is sound, provided that the components are sound. However, a more difficult problem i... |

13 |
Bi-Heyting algebras, Toposes and Modalities
- Reyes, Zolfaghari
- 1996
(Show Context)
Citation Context ...is work is the extension to logics of contradiction [12] and to paraconsistent logics [5] and logics of formal inconsistency [7] in general. Finally, we also intend to investigate bi-Heyting algebras =-=[15]-=- in the context of modulated fibring. Acknowledgments We are particularly grateful to Amílcar Sernadas for many interesting discussions on fibring and for many timely questions, suggestions and remark... |

12 | Fibring modal first-order logics: Completeness preservation
- Sernadas, Sernadas, et al.
(Show Context)
Citation Context ...ring of logics leads to a new logic where not only constructors are mixed, but proof methods are combined. Although the fibring techniques can be defined in the context of quantificational logic (cf. =-=[17]-=-), even if restricted to the propositional level (avoiding variables, terms, binding operators such as quantifiers, and the subtleties therein), ∗ This work started during a visit by Walter A. Carniel... |

11 | Limits for paraconsistent calculi
- Carnielli, Marcos
- 1999
(Show Context)
Citation Context ...hermore, although most of the work on fibring has been restricted to logics endowed with truth-functional semantics, some steps have been taken towards encompassing logics (like paraconsistent logics =-=[5]-=-) with non truth-functional semantics [3]. Fibring can be presented from a proof-theoretical or from a model-theoretical perspective. From the proof-theoretical point of view, fibring is treated in a ... |

11 | Tableau systems for logics of formal inconsistency
- Carnielli, Marcos
- 2001
(Show Context)
Citation Context ...and [14]. We would also like to investigate the case where we have several designated truth-values instead of just one: we would thus be able to deal, for instance, with the logic LF I1 considered in =-=[6]-=-. Of course also related with this work is the extension to logics of contradiction [12] and to paraconsistent logics [5] and logics of formal inconsistency [7] in general. Finally, we also intend to ... |

8 | Systematization of many-valued logics through the method of tableaux - Carnielli - 1987 |

6 | An overview of semantics and the combination of logics - Gabbay - 1996 |

6 | Fibring modal logics: Completeness preservation - Sernadas, Sernadas, et al. - 2001 |

4 |
The logic of contradiction
- Goodman
- 1981
(Show Context)
Citation Context ...ruth-values instead of just one: we would thus be able to deal, for instance, with the logic LF I1 considered in [6]. Of course also related with this work is the extension to logics of contradiction =-=[12]-=- and to paraconsistent logics [5] and logics of formal inconsistency [7] in general. Finally, we also intend to investigate bi-Heyting algebras [15] in the context of modulated fibring. Acknowledgment... |

2 | A taxonomy of C-systems. Submitted to the - Carnielli, Marcos |