## Fibring Non-Truth-Functional Logics: Completeness Preservation (2000)

### Cached

### Download Links

- [logica.cle.unicamp.br]
- [www.cle.unicamp.br]
- [www.cs.math.ist.utl.pt]
- [www.cle.unicamp.br]
- [wslc.math.ist.utl.pt]
- [www.cs.math.ist.utl.pt]
- [www.cs.math.ist.utl.pt]
- [www.cs.math.ist.utl.pt]
- [www.cs.math.ist.utl.pt]
- [wslc.math.ist.utl.pt]
- [sqig.math.ist.utl.pt]
- [sqig.math.ist.utl.pt]
- [sqig.math.ist.utl.pt]
- [sqig.math.ist.utl.pt]
- [wslc.math.ist.utl.pt]
- DBLP

### Other Repositories/Bibliography

Venue: | Journal of Logic, Language and Information |

Citations: | 27 - 20 self |

### BibTeX

@ARTICLE{Caleiro00fibringnon-truth-functional,

author = {Carlos Caleiro and Walter Carnielli and Marcelo E. Coniglio and Amilcar Sernadas and Cristina Sernadas},

title = {Fibring Non-Truth-Functional Logics: Completeness Preservation},

journal = {Journal of Logic, Language and Information},

year = {2000},

volume = {12},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

Fibring has been shown to be useful for combining logics endowed with truth-functional semantics. One wonders if bring can be extended in order to cope with logics endowed with non-truth-functional semantics as, for example, paraconsistent logics. The rst main contribution of the paper is a positive answer to this question. Furthermore, it is shown that this extended notion of bring preserves completeness under certain reasonable conditions. This completeness transfer result, the second main contribution of the paper, generalizes the one established by Zanardo et al. and is obtained using a new technique exploiting the properties of the metalogic where the (possibly non-truth-functional) valuations are de ned. The modal paraconsistent logic of da Costa and Carnielli is obtained by bring and its completeness is so established.

### Citations

754 |
An Introduction to Modal Logic
- Hughes, Cresswell
- 1968
(Show Context)
Citation Context ...a generator in O satisfying: ( ! (>) = >); ( ! (u(x 1 ; x 2 )) = u((x 1 ); (x 2 ))); ( ! u((x 1 ); (( (x 1 ))) = (x 1 )); ( ! v(L y 1 ) = (v(y 1 ))). Example 2.7 Modal system KD [14, 15]: Object signature - C: { C 0 = fp n : n 2 Ng [ ft; f g; { C 1 = f:; Lg; { C 2 = f^; _; g. Meta-axioms - S: { Truth-values axioms: Further axioms in order to obtain a specication of the class ... |

419 |
Category Theory for Computer Science
- Barr, Wells
- 1990
(Show Context)
Citation Context ...rather than on the category theoretic details. Therefore, we believe that the paper can still be fully assessed by the reader not conversant with the elementary language of categories (MacLane, 1971; =-=Barr and Wells, 1990-=-). The paper is organized as follows. In Section 2, the notion of interpretation system presentation as a specification of the intended valuations within the meta-logic is introduced. The interpretati... |

193 | Fibring Logics
- Gabbay
- 1999
(Show Context)
Citation Context ... preservation results and/orsnding preservation counterexamples within the community of logicians working in the problem of combining logics. Among the dierent techniques for combining logics,sbring [=-=11, 12, 21, 22, 23]-=- deserves close attention. But what issbring? The answer can be given in a few paragraphs for the special case of logics with a propositional base, that is, with propositional variables and connective... |

159 | Membership algebra as a logical framework for equational specification
- Meseguer
- 1997
(Show Context)
Citation Context ...d valuations within a suitable meta-logic of equational nature is introduced. The interpretation structures appear as models (algebras) of the specication. We adopt CEQ (conditional equational logic [=-=13, 18-=-]) as the meta-logic. Section 3 denes the notions of unconstrained and constrainedsbring of interpretation system presentations. The main example,sbring the paraconsistent system C 1 and the modal sys... |

88 |
Lattice Theory
- Birkho¤
- 1967
(Show Context)
Citation Context ... ) for u()(x 1 ; x 2 ); )(x 2 ; x 1 )). The relation symbol denotes a partial order on truth-values. Furthermore, the partial order is a bounded lattice with meet u, join t, top > and bottom ? (cf. [=-=2-=-]). As expected, given an algebra A, a 1 A a 2 and ,A (a 1 ; a 2 ) are abbreviations of uA (a 1 ; a 2 ) = a 1 and uA ()A (a 1 ; a 2 ); )A (a 2 ; a 1 )), respectively. It is also well known that the He... |

65 | Completeness of many-sorted equational logic
- Goguen, Meseguer
- 1985
(Show Context)
Citation Context ...d valuations within a suitable meta-logic of equational nature is introduced. The interpretation structures appear as models (algebras) of the specication. We adopt CEQ (conditional equational logic [=-=13, 18-=-]) as the meta-logic. Section 3 denes the notions of unconstrained and constrainedsbring of interpretation system presentations. The main example,sbring the paraconsistent system C 1 and the modal sys... |

54 | A.Sernadas, Fibring of Logics as a Categorial Construction
- Caleiro
- 1999
(Show Context)
Citation Context ...nding preservation counterexamples within the community of logicians working in the problem of combining logics. Among the dierent techniques for combining logics,sbring (Gabbay, 1996; Gabbay, 1998; S=-=ernadas et al., 1999-=-; Sernadas et al., 2000; Zanardo et al., 2000) deserves close attention. But what issbring? The answer can be given in a few paragraphs for the special case of logics with a propositional base, that i... |

35 |
Fibred semantics and the weaving of logics: part 1
- Gabbay
- 1996
(Show Context)
Citation Context ... preservation results and/orsnding preservation counterexamples within the community of logicians working in the problem of combining logics. Among the dierent techniques for combining logics,sbring [=-=11, 12, 21, 22, 23]-=- deserves close attention. But what issbring? The answer can be given in a few paragraphs for the special case of logics with a propositional base, that is, with propositional variables and connective... |

25 |
forthcoming, ``Why combine logics?'' Studia Logica
- Blackburn, Rijke
- 1990
(Show Context)
Citation Context ...esides leading to very interesting applications whenever it is necessary to work with dierent logics at the same time, combinations of logics are also of great interest on purely theoretical grounds [3]. The practical impact of the problem is clear, at least from the point of view of those working in knowledge representation (within articial intelligence) and in formal specication and verication ... |

21 | A semantical analysis of the calculi Cn’, Notre Dame - Costa, A, et al. - 1977 |

15 |
Multiparadigm specification languages: a first attempt at foundations. Semantics of Specification Languages
- Astesiano, Cerioli
- 1992
(Show Context)
Citation Context ...with several formalisms at the same time is the rule rather than the exception. We refer the reader, for instance, to (Finger and Gabbay, 1992; Goguen and Burstall, 1992; Sannella and Tarlecki, 1993; =-=Astesiano and Cerioli, 1994-=-) for a discussion of this and related problems, including some early attempts at their solution. Obviously, an approach to the combination of logics will be of significance only if general preservati... |

13 | M.E.Coniglio. A categorial approach to the combination of logics
- Carnielli
- 1999
(Show Context)
Citation Context ...gatory". In order to recover C D 1 at the semantic level in our setting we have to add one additional meta-axiom for valuations: (v(y 1 ) = > ! v((L y 1 ) ) = >): Using the terminology introduced=-= in (Carnielli and Coniglio, 1999-=-), this procedure can be seen as a splitting of C D 1 in the components KD and C 1 . 5. Logic systems This section is devoted to extending thesbring techniques to the syntactical counterpart of isp's.... |

12 |
Théorie de la valuation”, Logique et Analyse
- Costa, Béziau
- 1994
(Show Context)
Citation Context ...functional logics. This approach is more widely applicable as it appears to be: a large family of logics (including many-valued and intuitionistic) admits bivalued non-truth-functional semantics (cf. =-=[1-=-]). Whenever that happens, the completeness preservation theorem can then be used for establishing the completeness of the result as long as the given logics are complete and fulll the requirements of... |

8 |
The "Lemmon Notes": An Introduction to Modal Logic
- Lemmon
- 1977
(Show Context)
Citation Context ...a generator in O satisfying: ( ! (>) = >); ( ! (u(x 1 ; x 2 )) = u((x 1 ); (x 2 ))); ( ! u((x 1 ); (( (x 1 ))) = (x 1 )); ( ! v(L y 1 ) = (v(y 1 ))). Example 2.7 Modal system KD [14, 15]: Object signature - C: { C 0 = fp n : n 2 Ng [ ft; f g; { C 1 = f:; Lg; { C 2 = f^; _; g. Meta-axioms - S: { Truth-values axioms: Further axioms in order to obtain a specication of the class ... |

6 | Every quotient algebra for C1 is trivial - Mortensen - 1980 |

5 |
A categorial approach to the combination of logics. Manuscrito
- Carnielli, Coniglio
- 1999
(Show Context)
Citation Context ...atory". In order to recover C D 1 at the semantic level in our setting we have to add one additional meta-axiom for valuations: (v(y 1 ) = > ! v((L y 1 ) ) = >): Using the terminology introduced=-= in [4]-=-, this procedure can be seen as a splitting of C D 1 in the components KD and C 1 . 5 Logic systems This section is devoted to extending thesbring techniques to the syntactical counterpart of isp's. A... |

3 |
Sistemas Formais Inconsistentes. Cathedra Thesis
- Costa
- 1963
(Show Context)
Citation Context ...ofsbring have assumed that the logics at hand are truth-functional and, therefore, have not considered non-truth-functional logics like paraconsistent logics. Paraconsistent logics were introduced in =-=[6]-=- and since then have been the object of continued attention, because of their theoretical and practical significance. In particular, the paraconsistent systems C n of [6] are subsystems of proposition... |

2 |
Fibring (para)consistent logics. Research report
- Coniglio, Martins, et al.
- 2000
(Show Context)
Citation Context ...er than the exception. For instance, in a knowledge representation problem it may be necessary to work with both temporal and deontic aspects. This work is a natural sequel of the work presented in [=-=5-=-], which started during a visit by the third author to CMA at IST, entirely supported by Fundac~ao para a Ci^encia e a Tecnologia, Portugal. A subsequent visit to CMA by the second and third authors w... |

2 |
A semantical analysis of the calculi C n . Notre Dame Journal of Formal Logic
- Costa, Alves
- 1977
(Show Context)
Citation Context ...ding the use of the that is not in the signature (C). 4 Returning to thesrst example (paraconsistent system C 1 ), it is straightforward to verify that every paraconsistent valuation introduced in [7=-=-=-] has a counterpart, up to isomorphism because of the built-in freedom in the truthvalues, in Int(S). Furthermore, the additional interpretation structures do not change the semantic entailment (as de... |

2 |
Algebraization of paraconsistent logic
- Lewin, Mikenberg, et al.
- 1990
(Show Context)
Citation Context ...y 1 y 2 : y 1 y 2 . But, paraconsistent negation y 1 : : y 1 is not truth-functional. 1 The question of algebraizing paraconsistent logic is a separate issue and we refer the interested reader to [19, 16]. 6 In the following isp, all derived connectives are truth-functional, but the modality y 1 : L y 1 would require in (C) the extra generator in O satisfying: ( ! (>) = >); ( ! (u(x 1 ; ... |

2 |
The semantics of the systems C n of da Costa
- Loparic, Alves
- 1980
(Show Context)
Citation Context ...semantic entailment (as dened below). Note that it is easy to extend this example in order to set up the isp's for the whole hierarchy C n by specifying the paraconsistent n-valuations introduced in [=-=17]-=-. 7 With respect to the second example (modal system KD), it is also straightforward to verify that every Kripke model has a counterpart in Int(S): consider the algebra of truth-values given by the po... |

2 |
Kantian and non-Kantian logics'. Logique et Analyse 31(121/122
- Puga, Costa, et al.
- 1988
(Show Context)
Citation Context ...assical logic in which the principle of Pseudo Scotuss; :s`sdoes not hold. Some systems of paraconsistent modal logic were investigated in (Deutsch, 1979; Deutsch, 1984; da Costa and Carnielli, 1988; =-=Puga et al., 1988-=-). One might wonder if we should try to obtain such a mixed logic as thesbring of the underlying modal and paraconsistent logics. Thesrst main contribution of this paper is a positive answer to this q... |

1 |
Paraconsistent deontic logic. Philosophia - The Philosophical Quarterly of Israel
- Costa, Carnielli
- 1988
(Show Context)
Citation Context ...sistent systems C n of [6] are subsystems of propositional classical logic in which the principle of Pseudo Scotuss; :s`sdoes not hold. Some systems of paraconsistent modal logic were investigated in =-=[9, 10, 8, 20]-=-. One might wonder if we should try to obtain such a mixed logic as thesbring of the underlying modal and paraconsistent logics. Thesrst main contribution of this paper is a positive answer to this qu... |

1 | The completeness of S - Deutsch - 1979 |

1 |
Paraconsistent analytic implication
- Deutsch
- 1984
(Show Context)
Citation Context ...sistent systems C n of [6] are subsystems of propositional classical logic in which the principle of Pseudo Scotuss; :s`sdoes not hold. Some systems of paraconsistent modal logic were investigated in =-=[9, 10, 8, 20]-=-. One might wonder if we should try to obtain such a mixed logic as thesbring of the underlying modal and paraconsistent logics. Thesrst main contribution of this paper is a positive answer to this qu... |

1 | Rijke: 1997, `Why combine logics - Blackburn, de |

1 | 2000, `Fibring (Para)consistent Logics - Coniglio, Martins, et al. |

1 | Mossakowski: 2000, `Categorial Fibring of Logics with Terms and Binding Operators - Sernadas, Sernadas, et al. |

1 | Multiparadigm Speci Languages: a First Attempt at Foundations - Astesiano, Cerioli - 1994 |