## Recovering a logic from its fragments by meta-fibring. Logica Universalis

Venue: | In print. Preliminary version available at CLE e-Prints 5(4), 2005. URL = http://www.cle.unicamp.br/e-prints/vol 5,n 4,2005.html |

Citations: | 4 - 3 self |

### BibTeX

@INPROCEEDINGS{Coniglio_recoveringa,

author = {Marcelo E. Coniglio},

title = {Recovering a logic from its fragments by meta-fibring. Logica Universalis},

booktitle = {In print. Preliminary version available at CLE e-Prints 5(4), 2005. URL = http://www.cle.unicamp.br/e-prints/vol 5,n 4,2005.html},

year = {}

}

### OpenURL

### Abstract

In this paper we address the question of recovering a logic system by combining two or more fragments of it. We show that, in general, by fibring two or more fragments of a given logic the resulting logic is weaker than the original one, because some meta-properties of the connectives are lost after the combination process. In order to overcome this problem, the categories Mcon and Seq of multiple-conclusion consequence relations and sequent calculi, respectively, are introduced. The main feature of these categories is the preservation, by morphisms, of meta-properties of the consequence relations, which allows, in several cases, to recover a logic by fibring of its fragments. The fibring in this categories is called meta-fibring. Several examples of well-known logics which can be recovered by metafibring its fragments (in opposition to fibring in the usual categories) are given. Finally, a general semantics for objects in Seq (and, in particular, for objects in Mcon) is proposed, obtaining a category of logic systems

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Citation Context ...ng the different methods for combining logics, fibring has been revealed as a very valuable tool for combining logic systems, and it was successfully applied in different contexts (see, for instance, =-=[14, 16, 18, 22, 10, 6]-=-). One of the most outstanding features of fibring is the obtainment, under certain conditions, of preservation of metaproperties of the given logic systems through fibring; in particular, the preserv... |

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Citation Context ...ng the different methods for combining logics, fibring has been revealed as a very valuable tool for combining logic systems, and it was successfully applied in different contexts (see, for instance, =-=[14, 16, 18, 22, 10, 6]-=-). One of the most outstanding features of fibring is the obtainment, under certain conditions, of preservation of metaproperties of the given logic systems through fibring; in particular, the preserv... |

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Citation Context |

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(Show Context)
Citation Context ... the proof is done by induction on the length of a derivation in A1 of A; Γ ≻ ∆; B from {〈A1; Γ1|∆1; B1〉, . . . , 〈An; Γn|∆n; Bn〉}. We left the details to the reader (a detailed proof can be found in =-=[7]-=-). � The last theorem guarantees that a morphism between consequence relations preserves intrinsic characteristics of the source system. In particular, if we consider the inclusion morphism (the canon... |

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(Show Context)
Citation Context ...rently simpler) solution to the collapsing problem is found in [5], using a relaxed fibring technique called cryptofibring. A related form of the collapsing problem was also observed in [2] (see also =-=[3]-=-) where it is shown that, if we join up the usual sequent rules for (classical) conjunction with the rules for (classical) disjunction, the resulting sequent calculus will prove the distributivity bet... |

4 |
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Citation Context ...ng multiple-conclusion relations In this section the concept of (categorial) fibring multiple-conclusion relations is introduced. As usual, this construction can be characterized as a coproduct. From =-=[13]-=- the following result is known: Proposition 4.1 The category Sig has finite coproducts. Given signatures C1 and C2 , the coproduct of C1 and C2 will be denoted by C1 ⊕C 2 , with canonical injections i... |

3 |
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Citation Context |

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2 |
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Citation Context ...ther (and apparently simpler) solution to the collapsing problem is found in [5], using a relaxed fibring technique called cryptofibring. A related form of the collapsing problem was also observed in =-=[2]-=- (see also [3]) where it is shown that, if we join up the usual sequent rules for (classical) conjunction with the rules for (classical) disjunction, the resulting sequent calculus will prove the dist... |

2 |
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