## Fibring: Completeness Preservation (2000)

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Venue: | Journal of Symbolic Logic |

Citations: | 46 - 23 self |

### BibTeX

@ARTICLE{Zanardo00fibring:completeness,

author = {Alberto Zanardo and Amilcar Sernadas and Cristina Sernadas},

title = {Fibring: Completeness Preservation},

journal = {Journal of Symbolic Logic},

year = {2000},

volume = {66},

pages = {2001}

}

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### Abstract

A completeness theorem is established for logics with congruence endowed with general semantics (in the style of general frames). As a corollary, completeness is shown to be preserved by bring logics with congruence provided that congruence is retained in the resulting logic. The class of logics with equivalence is shown to be closed under bring and to be included in the class of logics with congruence. Thus, completeness is shown to be preserved by bring logics with equivalence and general semantics. An example is provided showing that completeness is not always preserved by bring logics endowed with standard (non general) semantics. A categorial characterization of bring is provided using coproducts and cocartesian liftings. 1 Introduction Much attention has been recently given to the problems of combining logics and obtaining transference results. Besides leading to very interesting applications whenever it is necessary to work with dierent logics at the same time, ...

### Citations

754 |
An Introduction to Modal Logic
- Hughes, Cresswell
- 1968
(Show Context)
Citation Context ...otion of general interpretation systems generalizes the notion of interpretation system as adopted in [SSC99]. This generalization follows the style of general frames for modal logic (see for example =-=[HC96]-=-), herein in a more general context. In Section 4, we bring together Hilbert calculi and general interpretation systems in order to establish the appropriate notion of general logic system where we ca... |

419 |
Category Theory for Computer Science
- Barr, Wells
- 1990
(Show Context)
Citation Context ...egory theory may skip the subsections on categorial constructions without any loss of continuity. Anyway, only elementary notions are borrowed from category theory. They are presented for instance in =-=[BW90]-=-. 2 Hilbert calculi Since we work only with propositional-based logics, the following notion of signature is enough: a signature is a family C = fC k g k2N where each C k is a set. The elements of C k... |

193 | Fibring Logics
- Gabbay
- 1999
(Show Context)
Citation Context ...9]). However, thesbring of intuitionistic logic with classical logic still leads to the collapsing of all connectives into classical logic. This is a well known problem with current accounts ofsbring =-=[Gab-=-99]. At the proof-theoretic level it is possible to avoid the collapse by constraining the use of the axiom ( 1 ! ( 2 ! 1 )) as proposed in [dCH96]. But at the model-theoretic level it is an open pro... |

187 | Modal Logic - Chagrov, Zakharyaschev - 1996 |

187 |
Completeness in the theory of types
- Henkin
- 1950
(Show Context)
Citation Context ... the signature C. The use of a subset of }U as the set of admissible valuations has a long tradition which starts with the Henkin general semantics for second order logic and for the theory of types (=-=[Hen50-=-]) and passes through the general frames in modal and temporal logics [Ben83, Ben85, HC96, CZ97]. If, in a C-structure S = hU; B; i, the set B is }U , we say that S is standard. Such standard C-struct... |

144 | The Logic of Time - Benthem - 1991 |

90 | Properties of independently axiomatizable bimodal logics
- Kracht, Wolter
- 1991
(Show Context)
Citation Context ...e present two examples from modal logic, using the standard (non general) semantics. Thesrst example shows a case of preservation of weak completeness bysbring as a direct corollary of the results in =-=[KW91]-=-. The second example, based upon [Wol96], shows a case where weak completeness is not preserved. In Section 5, we obtain a (strong) completeness theorem for logic systems with full general semantics a... |

54 | A.Sernadas, Fibring of Logics as a Categorial Construction
- Caleiro
- 1999
(Show Context)
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35 | Fibred semantics and the weaving of logics: part 1 - Gabbay - 1996 |

29 | Combining classical and intuitionistic logic, or: intuitionistic implication as a conditional
- Cerro, Herzig
- 1996
(Show Context)
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25 |
forthcoming, ``Why combine logics?'' Studia Logica
- Blackburn, Rijke
- 1990
(Show Context)
Citation Context ...ce results. Besides leading to very interesting applications whenever it is necessary to work with dierent logics at the same time, combination of logics is of interest on purely theoretical grounds [=-=Bd-=-R97]. This paper was mainly developed during a three month visit of thesrst author to the Centro de Matematica Aplicada of the Instituto Superior Tecnico of the Universidade Tecnica de Lisboa. The vi... |

24 | Fibred semantics and combinations of logics - Gabbay - 1992 |

21 | Rijke. Why combine logics
- Blackburn, de
- 1997
(Show Context)
Citation Context ...e results. Besides leading to very interesting applications whenever it is necessary to work with different logics at the same time, combination of logics is of interest on purely theoretical grounds =-=[BdR97]-=-. ∗ This paper was mainly developed during a three month visit of the first author to the Centro de Matemática Aplicada of the Instituto Superior Técnico of the Universidade Técnica de Lisboa. The vis... |

17 | Basic logic: reflection, symmetry, visibility
- Sambin, Battilotti, et al.
- 2000
(Show Context)
Citation Context ...s are logics with equivalence 35sand hence they belong to this class. With respect to this, it is important to observe that these logics can be viewed as extensions of the logic called basic logic in =-=[SBF]-=-, which turns out to be a logic with equivalence as well (where ⇔ is defined on the basis of ⇒ and of &). This implies that other extensions of basic logic (like, e.g., linear logic and quantum logic)... |

14 | Categorical fibring of logics with terms and binding operators
- Sernadas, Sernadas, et al.
(Show Context)
Citation Context ...pose requirements in inference rules in order to be able to constrain their use. Therefore, the solution proposed in [dCH96] implies a more complex notion of Hilbert calculus, along the same lines in =-=[SSCM99]-=- where inference rules with requirements were introduced for another reason (for dealing with the problems of logics with variables, terms and binding operators). Future work is planned at extending t... |

12 | Importing logics - Rasga, Sernadas, et al. |

9 | Synchronization of logics with mixed rules: Completeness preservation - Sernadas, Sernadas, et al. - 1997 |

9 |
Properties of tense logics
- Wolter
- 1996
(Show Context)
Citation Context ..., using the standard (non general) semantics. Thesrst example shows a case of preservation of weak completeness bysbring as a direct corollary of the results in [KW91]. The second example, based upon =-=[Wol96]-=-, shows a case where weak completeness is not preserved. In Section 5, we obtain a (strong) completeness theorem for logic systems with full general semantics and with congruence, using an adapted Hen... |

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

4 | Synchronization of logics. Studia Logica - Sernadas, Sernadas, et al. - 1997 |

3 |
Basic logic: re symmetry, visibility
- Sambin, Battilotti, et al.
(Show Context)
Citation Context ...s are logics with equivalence 35 and hence they belong to this class. With respect to this, it is important to observe that these logics can be viewed as extensions of the logic called basic logic in =-=[SBF-=-], which turns out to be a logic with equivalence as well (where , is dened on the basis of ) and of &). This implies that other extensions of basic logic (like, e.g., linear logic and quantum logic) ... |

1 |
Categorial bring of logics with terms and binding operators
- Sernadas, Sernadas, et al.
- 1999
(Show Context)
Citation Context ...pose requirements in inference rules in order to be able to constrain their use. Therefore, the solution proposed in [dCH96] implies a more complex notion of Hilbert calculus, along the same lines in =-=[SSCM99]-=- where inference rules with requirements were introduced for another reason (for dealing with the problems of logics with variables, terms and binding operators). Future work is planned at extending t... |