## Fibring: Completeness Preservation (2000)

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Venue: | Journal of Symbolic Logic |

Citations: | 45 - 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, ...

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Citation Context ... Gab96b, Gab99] deserves close study. Whensbring two given logics we produce a logic where we allow the free mixing of the connectives from both logics and we use the proof rules from both logics. In =-=[SSC99]-=- an explicit semantics is provided forsbring. Therein, soundness is shown to be preserved bysbring, but the preservation of completeness is left as an open problem. Herein, we concentrate on the probl... |

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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)... |

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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... |

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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... |

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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... |