## Interpolation via translations

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@MISC{Carnielli_interpolationvia,

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

title = {Interpolation via translations},

year = {}

}

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### Abstract

A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a Kolmogorov-Gentzen-Gödel style translation, and a new translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a Kiriyama-Ono style translation with features of a Kolmogorov-Gentzen-Gödel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics. AMS Classification: 03C40, 03F03, 03B22

### Citations

219 |
Mathematical logic
- Shoenfield
- 1967
(Show Context)
Citation Context ...on symbols appearing in the formula. 3We observe that Maehara interpolation is also known in the literature as CraigRobinson interpolation property or Strong interpolation property (see for instance =-=[8, 27, 6]-=-). Interesting relationships can be established between these interpolation properties when the underlying consequence system enjoys some additional characteristics. For instance, when the consequence... |

46 | Fibring: Completeness preservation
- Zanardo, Sernadas, et al.
(Show Context)
Citation Context ...usually shown in a case by case basis, although recently there can be seen a widely and growing effort to generalize preservation results so that they can be applied once to a whole class of entities =-=[4, 5, 26, 28]-=-. Herein we propose a new method for checking whether Craig interpolation and Maehara interpolation hold in a logic. This method is based on the existence of a special kind of translation between an a... |

45 |
Zur intuitionistischen Arithmetik und Zahlentheorie, Ergebnisse eines Mathematischen Kolloquiums 4
- Gödel
- 1933
(Show Context)
Citation Context ...ncy of classical logic by reducing it to the consistency of intuitionistic logic (see the works of Kolmogorov [18] and Glivenko [11] for the propositional case and the works of Gentzen [10] and Gödel =-=[12]-=- for the predicate case). Subsequently, special designed translations were used to show that other properties besides consistency, like for instance cut-elimination [16] and undecidability [17], hold ... |

23 | Proof-theoretic methods for nonclassical logic - an introduction
- Ono
- 1998
(Show Context)
Citation Context ...3.2 we can now establish the following corollary, capitalizing on the fact that FLe enjoys Craig interpolation and Maehara interpolation over its global consequence relation with respect to vare, see =-=[8, 23]-=-. Corollary 5.14 The logic FLew⊕MLL enjoys Craig interpolation and Maehara interpolation with respect to varw+m over its global consequence relation. 5.3 Interpolation in IL⊕CL In this subsection we i... |

20 | Modulated fibring and the collapsing problem
- Sernadas, Rasga, et al.
(Show Context)
Citation Context ...usually shown in a case by case basis, although recently there can be seen a widely and growing effort to generalize preservation results so that they can be applied once to a whole class of entities =-=[4, 5, 26, 28]-=-. Herein we propose a new method for checking whether Craig interpolation and Maehara interpolation hold in a logic. This method is based on the existence of a special kind of translation between an a... |

19 |
Amalgamation and interpolation in abstract algebraic logic’, Models, algebras, and proofs
- Czelakowski, Pigozzi
- 1995
(Show Context)
Citation Context ...on symbols appearing in the formula. 3We observe that Maehara interpolation is also known in the literature as CraigRobinson interpolation property or Strong interpolation property (see for instance =-=[8, 27, 6]-=-). Interesting relationships can be established between these interpolation properties when the underlying consequence system enjoys some additional characteristics. For instance, when the consequence... |

17 |
quelques points de la logique de M. Brouwer, Bulletin Academie des Sciences de Belgique 15
- Glivenko, Sur
- 1929
(Show Context)
Citation Context ...o intuitionistic logic, were introduced as a way of proving the consistency of classical logic by reducing it to the consistency of intuitionistic logic (see the works of Kolmogorov [18] and Glivenko =-=[11]-=- for the propositional case and the works of Gentzen [10] and Gödel [12] for the predicate case). Subsequently, special designed translations were used to show that other properties besides consistenc... |

15 | On intuitionistic arithmetic and number theory - Gödel - 1965 |

15 | Intuitionistische Untersuchungen derformalistischen Logik - KURODA - 1951 |

14 |
C.: Analysis and synthesis of logics: how to cut and paste reasoning systems. Applied logic series
- Carnielli, Coniglio, et al.
(Show Context)
Citation Context ...s system can be seen as the combination of full Lambek calculus with exchange and weakening (an extended BCK logic), FLew, and multiplicative linear logic, MLL, and so is represented as FLew⊕MLL, see =-=[17, 28, 3, 16]-=-. The deductive system Dw+m is such that, its language, Lw+m, is generated by the signature Cw+m with: • Cw+m0 contains ⊥, ⊥m, 0, 1, 1m and a denumerable set of propositional symbols P , • Cw+m1 • Cw+... |

14 | Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL, Studia Logica 83
- Galatos, Ono
- 2006
(Show Context)
Citation Context ...on symbols appearing in the formula. 3We observe that Maehara interpolation is also known in the literature as CraigRobinson interpolation property or Strong interpolation property (see for instance =-=[8, 27, 6]-=-). Interesting relationships can be established between these interpolation properties when the underlying consequence system enjoys some additional characteristics. For instance, when the consequence... |

14 | On the principle of excluded middle - Kolmogorov - 1977 |

11 | Transfers between logics and their applications
- Coniglio, Carnielli
(Show Context)
Citation Context ...usually shown in a case by case basis, although recently there can be seen a widely and growing effort to generalize preservation results so that they can be applied once to a whole class of entities =-=[4, 5, 26, 28]-=-. Herein we propose a new method for checking whether Craig interpolation and Maehara interpolation hold in a logic. This method is based on the existence of a special kind of translation between an a... |

11 |
Interpolation and definability: Modal and Intuitionistic Logics (Oxford Logic Guides), Oxford science publications
- Gabbay, Maksimova
(Show Context)
Citation Context ...erpolation and Maehara interpolation over local consequence are not so well studied in the literature, as far as we know, as Craig interpolation and Maehara interpolation over global consequence (see =-=[14, 7, 1, 21]-=- mainly about Craig interpolation and Maehara interpolation over global consequence). Interesting relationships can be established between the local consequence and the global consequence induced by a... |

9 | Preservation of interpolation features by fibring
- Carnielli, Rasga, et al.
(Show Context)
Citation Context |

9 | Sufficient conditions for cut elimination with complexity analysis - Rasga |

8 |
Definability and interpolation: model-theoretic investigations, Ph. D. Dissertation DS-2001-05
- Hoogland
- 2001
(Show Context)
Citation Context ...erpolation and Maehara interpolation over local consequence are not so well studied in the literature, as far as we know, as Craig interpolation and Maehara interpolation over global consequence (see =-=[14, 7, 1, 21]-=- mainly about Craig interpolation and Maehara interpolation over global consequence). Interesting relationships can be established between the local consequence and the global consequence induced by a... |

8 |
Completeness proofs for linear logic based on the proof search method (preliminary report
- Okada, Terui
- 1998
(Show Context)
Citation Context ...er the question of whether the linear logic associated with the deductive system having as rules the rules of MALL (additive and multiplicative rules) and the rules of NLL (intuitionistic rules), see =-=[22, 15]-=-, enjoys Craig interpolation and Maehara interpolation over the global and the local consequence relation. (Interpolation for this logic was not studied in [25].) Using the results of this work we red... |

7 | Über das Verhältnis zwischen intuitionistischer und klassischer Arithmetik. Galley proof, Mathematische Annalen. English translation (Gentzen - Gentzen - 1933 |

5 | The Craig interpolation theorem for schematic systems
- Carbone
- 1996
(Show Context)
Citation Context ...erpolation and Maehara interpolation over local consequence are not so well studied in the literature, as far as we know, as Craig interpolation and Maehara interpolation over global consequence (see =-=[14, 7, 1, 21]-=- mainly about Craig interpolation and Maehara interpolation over global consequence). Interesting relationships can be established between the local consequence and the global consequence induced by a... |

5 |
The contraction rule and decision problems for logics without structural rules’, Studia Logica
- Kiriyama, Ono
- 1991
(Show Context)
Citation Context ... Gödel [12] for the predicate case). Subsequently, special designed translations were used to show that other properties besides consistency, like for instance cut-elimination [16] and undecidability =-=[17]-=-, hold on a deductive system relative to another, that is, hold on a deductive system whenever they hold on another deductive system. Similar results can be found in other scientific areas, like for i... |

3 | A note on the Gödel-Gentzen translation
- Ishihara
- 2000
(Show Context)
Citation Context ...er the question of whether the linear logic associated with the deductive system having as rules the rules of MALL (additive and multiplicative rules) and the rules of NLL (intuitionistic rules), see =-=[22, 15]-=-, enjoys Craig interpolation and Maehara interpolation over the global and the local consequence relation. (Interpolation for this logic was not studied in [25].) Using the results of this work we red... |

3 |
On the principle “tertium non datur
- Kolmogorov
- 1924
(Show Context)
Citation Context ...lassical logic into intuitionistic logic, were introduced as a way of proving the consistency of classical logic by reducing it to the consistency of intuitionistic logic (see the works of Kolmogorov =-=[18]-=- and Glivenko [11] for the propositional case and the works of Gentzen [10] and Gödel [12] for the predicate case). Subsequently, special designed translations were used to show that other properties ... |

3 |
On the interpolation theorem of Craig
- Maehara
(Show Context)
Citation Context |

2 |
Interpolation in fragments of classical linear logic
- Roorda
- 1994
(Show Context)
Citation Context ... of NLL (intuitionistic rules), see [22, 15], enjoys Craig interpolation and Maehara interpolation over the global and the local consequence relation. (Interpolation for this logic was not studied in =-=[25]-=-.) Using the results of this work we reduce this problem to the same problem over its intuitionistic (NLL) fragment. We denote this linear logic by NLL⊕MALL. The only difference of NLL with respect to... |

1 |
Appendix on Interpolation via translations: proofs as expected
- Carnielli, Rasga, et al.
- 2008
(Show Context)
Citation Context ...ver interesting relationships between those consequence relations are also established in Section 4. Finally, in Section 6, we draw the conclusions. For the benefit of the reader there is an Appendix =-=[2]-=- to this work where we provide the details of the proofs that are just sketched or not proved at all herein (proofs that follow in an expected way are just sketched or not proved at all in order to si... |

1 |
Synthesized substructural logics
- Kamide
(Show Context)
Citation Context ...orks of Gentzen [10] and Gödel [12] for the predicate case). Subsequently, special designed translations were used to show that other properties besides consistency, like for instance cut-elimination =-=[16]-=- and undecidability [17], hold on a deductive system relative to another, that is, hold on a deductive system whenever they hold on another deductive system. Similar results can be found in other scie... |