## Heterogeneous fibring of deductive systems via abstract proof systems (2005)

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@TECHREPORT{Cruz-filipe05heterogeneousfibring,

author = {Luís Cruz-filipe and Amílcar Sernadas and Cristina Sernadas},

title = {Heterogeneous fibring of deductive systems via abstract proof systems},

institution = {},

year = {2005}

}

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### Abstract

Fibring is a meta-logical constructor that applied to two logics produces a new logic whose formulas allow the mixing of symbols. Homogeneous fibring assumes that the original logics are presented in the same way (e.g via Hilbert calculi). Heterogeneous fibring, allowing the original logics to have different presentations (e.g. one presented by a Hilbert calculus and the other by a sequent calculus), has been an open problem. Herein, consequence systems are shown to be a good solution for heterogeneous fibring when one of the logics is presented in a semantic way and the other by a calculus and also a solution for the heterogeneous fibring of calculi. The new notion of abstract proof system is shown to provide a better solution to heterogeneous fibring of calculi namely because derivations in the fibring keep the constructive nature of derivations in the original logics. Preservation of compactness and semi-decidability is investigated.

### Citations

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(Show Context)
Citation Context ...ce relation is also the union of the original consequence relations). More interesting combination mechanisms can be defined when describing logics in a more concrete way. In the case of fibring (see =-=[11, 12]-=-), it is necessary to indicate the signatures (set of symbols) of the (usually two) original logics. The set of formulas is the free algebra generated by the set of symbols in the signature and a set ... |

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Citation Context ... [25] is the best understood in what concerns preservation of properties as soundness, weak completeness, uniform Craig interpolation (for theoremhood) and decidability via finite model property (see =-=[27, 19, 13]-=-). Further results on preservation of weak completeness can be seen in [10]. Preservation results were also obtained in the context of temporalization (adding a temporal dimension to an original logic... |

89 |
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Citation Context ...ass of modal logics while fibring is an operation on the class of logics). For a nice and gentle motivation of the topic see [2], and for an early example of the combination of tense and modality see =-=[25]-=-. The different methods for combination depend upon and impose different presentations of the original logics ranging, on one hand, from very abstract to more concrete ones and, on the other hand, fro... |

61 | Temporalizing description logics
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Citation Context ...er results on preservation of weak completeness can be seen in [10]. Preservation results were also obtained in the context of temporalization (adding a temporal dimension to an original logic) as in =-=[28, 18]-=-. Some interesting preservation properties can also be found for the product of (modal) logics, see [14, 15, 16, 20]. Fibring is a more general mechanism since it goes beyond modal logic. For instance... |

53 |
A Course
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Citation Context ...1:c(ϕ1, . . . , ϕk) and a negative rule with conclusion 0:c(ϕ1, . . . , ϕk). 8Rule EM is not always included in the definition of a tableau calculus, but it can easily be shown to be admissible (see =-=[2]-=- for details) if the tableau system is known to be complete, otherwise the proof of admissibility is not straightforward (coinciding with cut-elimination in sequent calculi). We chose to include it in... |

48 |
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Citation Context ... in the context of temporalization (adding a temporal dimension to an original logic) as in [28, 18]. Some interesting preservation properties can also be found for the product of (modal) logics, see =-=[14, 15, 16, 20]-=-. Fibring is a more general mechanism since it goes beyond modal logic. For instance, we can think of fibring relevance logic with intuitionistic logic, or modal logic and first-order logic. Although ... |

47 | Fusions of modal logics revisited
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Citation Context ... [25] is the best understood in what concerns preservation of properties as soundness, weak completeness, uniform Craig interpolation (for theoremhood) and decidability via finite model property (see =-=[27, 19, 13]-=-). Further results on preservation of weak completeness can be seen in [10]. Preservation results were also obtained in the context of temporalization (adding a temporal dimension to an original logic... |

46 | Fibring: Completeness preservation
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Citation Context ...tuitionistic logic, or modal logic and first-order logic. Although preservation of soundness, completeness and interpolation has been already investigated in the context of propositional-based logics =-=[29, 24, 4]-=-, first-order quantification [23], higher-order quantification [6], non truth-functional semantics [3], sequent calculus and other deductive systems [17, 21], other forms of preservation are still to ... |

35 |
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Citation Context ...ce relation is also the union of the original consequence relations). More interesting combination mechanisms can be defined when describing logics in a more concrete way. In the case of fibring (see =-=[11, 12]-=-), it is necessary to indicate the signatures (set of symbols) of the (usually two) original logics. The set of formulas is the free algebra generated by the set of symbols in the signature and a set ... |

31 |
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Citation Context ...act and closed for substitution. Example 2.6 A sequent calculus GS4 for modal logic S4 (characterized by reflexive and transitive frames) has the following specific rules, as presented in page 287 of =-=[26]-=-, where (�∆) is {(�δ) : δ ∈ ∆} and (♦∆) is {(♦δ) : δ ∈ ∆}: ∆1, ξ1 → ξ2, ∆2 ∆1 → (ξ1 ⇒ ξ2), ∆2 ∆1, ξ1 → ∆2 ∆1 → (¬ ξ1), ∆2 (�∆1) → ξ1, (♦∆2) R ⇒ ∆1 → ξ1, ∆2 ∆1, ξ2 → ∆2 ∆1, (ξ1 ⇒ ξ2) → ∆2 R ¬ ∆ ′ 1 , (... |

29 | The unrestricted combination of temporal logic systems
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Citation Context ...ponents since in the same formula we can have symbols from both signatures). Signatures are also needed for the fusion of modal logics, see [25], as well as when combining temporal logic systems, see =-=[9]-=-. Also the consequence relation can be more concrete, for instance, when it is generated from a Hilbert calculus (with axioms and rules). The induced consequence relation is then defined in a construc... |

27 | Fibring non-truth-functional logics: Completeness preservation
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Citation Context ... interpolation has been already investigated in the context of propositional-based logics [29, 24, 4], first-order quantification [23], higher-order quantification [6], non truth-functional semantics =-=[3]-=-, sequent calculus and other deductive systems [17, 21], other forms of preservation are still to be fully understood, namely the ones related to compactness, decidability and complexity. The main obj... |

24 |
Formal definitions in the theory of ordinal numbers
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Citation Context ...y semi-decidable consequence systems and let C = 〈C, ⊢〉 be their fibring. Assume that, for each recursively enumerable Γ ⊆ L(C), Γ⊢ = Γ⊢α for some α < ωCT 1 , where ωCT 1 is the Church–Kleene ordinal =-=[5]-=- (in other words, α is recursively enumerable). Then C is a strongly semi-decidable consequence system. 15sProof: Let Γ ⊆ L(C ′ ) ∪ L(C ′′ ) be a recursively enumerable set. By hypothesis there is a r... |

21 | Rijke. Why combine logics
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Citation Context ... finite number of logics (for instance, fusion is an operation on the subclass of modal logics while fibring is an operation on the class of logics). For a nice and gentle motivation of the topic see =-=[2]-=-, and for an early example of the combination of tense and modality see [25]. The different methods for combination depend upon and impose different presentations of the original logics ranging, on on... |

21 | Modulated fibring and the collapsing problem
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(Show Context)
Citation Context ...tuitionistic logic, or modal logic and first-order logic. Although preservation of soundness, completeness and interpolation has been already investigated in the context of propositional-based logics =-=[29, 24, 4]-=-, first-order quantification [23], higher-order quantification [6], non truth-functional semantics [3], sequent calculus and other deductive systems [17, 21], other forms of preservation are still to ... |

17 | Temporalising tableaux
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- 2004
(Show Context)
Citation Context ...er results on preservation of weak completeness can be seen in [10]. Preservation results were also obtained in the context of temporalization (adding a temporal dimension to an original logic) as in =-=[28, 18]-=-. Some interesting preservation properties can also be found for the product of (modal) logics, see [14, 15, 16, 20]. Fibring is a more general mechanism since it goes beyond modal logic. For instance... |

14 | L.: Fibring labelled deduction systems
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(Show Context)
Citation Context ...e context of propositional-based logics [29, 24, 4], first-order quantification [23], higher-order quantification [6], non truth-functional semantics [3], sequent calculus and other deductive systems =-=[17, 21]-=-, other forms of preservation are still to be fully understood, namely the ones related to compactness, decidability and complexity. The main objective of this paper is to provide solutions to open pr... |

12 | Fibring modal first-order logics: Completeness preservation
- Sernadas, Sernadas, et al.
(Show Context)
Citation Context ...irst-order logic. Although preservation of soundness, completeness and interpolation has been already investigated in the context of propositional-based logics [29, 24, 4], first-order quantification =-=[23]-=-, higher-order quantification [6], non truth-functional semantics [3], sequent calculus and other deductive systems [17, 21], other forms of preservation are still to be fully understood, namely the o... |

11 | Fibring logics with topos semantics
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(Show Context)
Citation Context ...ation of soundness, completeness and interpolation has been already investigated in the context of propositional-based logics [29, 24, 4], first-order quantification [23], higher-order quantification =-=[6]-=-, non truth-functional semantics [3], sequent calculus and other deductive systems [17, 21], other forms of preservation are still to be fully understood, namely the ones related to compactness, decid... |

9 | Preservation of interpolation features by fibring
- Carnielli, Rasga, et al.
(Show Context)
Citation Context ...tuitionistic logic, or modal logic and first-order logic. Although preservation of soundness, completeness and interpolation has been already investigated in the context of propositional-based logics =-=[29, 24, 4]-=-, first-order quantification [23], higher-order quantification [6], non truth-functional semantics [3], sequent calculus and other deductive systems [17, 21], other forms of preservation are still to ... |

9 | On Fibring Semantics for BDI Logics
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- 2002
(Show Context)
Citation Context ...e context of propositional-based logics [29, 24, 4], first-order quantification [23], higher-order quantification [6], non truth-functional semantics [3], sequent calculus and other deductive systems =-=[17, 21]-=-, other forms of preservation are still to be fully understood, namely the ones related to compactness, decidability and complexity. The main objective of this paper is to provide solutions to open pr... |

7 | Fibred tableaux for multi-implication logics
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- 1996
(Show Context)
Citation Context ...ted by Hilbert calculi is a logic presented by a Hilbert calculus having the axioms and the rules of the component Hilbert calculi. Also some results on the fibring of tableau systems can be found in =-=[7, 1]-=-. In the examples above, we are implicitly thinking of combining logics in a homogeneous scenario: both logics are presented in the same way either by Hilbert calculi or by Kripke structures. However,... |

7 |
Theory of Numberings
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Citation Context ...at ϕ ∈ Γ⊢a . (c) β is a limit ordinal: straightforward. ⋄ 4 Computability In this small section we define more rigorously notions related to computability. These concepts have been introduced e.g. in =-=[8]-=-. We extend some concepts of computability to generic sets (namely sets of formulas and sets of derivations). The central notion is the one of a Gödel domain. A Gödelization or a Gödel map for A is a ... |

6 |
Fibring semantic tableaux. In Automated reasoning with analytic tableaux and related methods
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(Show Context)
Citation Context ...ted by Hilbert calculi is a logic presented by a Hilbert calculus having the axioms and the rules of the component Hilbert calculi. Also some results on the fibring of tableau systems can be found in =-=[7, 1]-=-. In the examples above, we are implicitly thinking of combining logics in a homogeneous scenario: both logics are presented in the same way either by Hilbert calculi or by Kripke structures. However,... |

3 |
Products of modal logics. II. Relativised quantifiers in classical logic
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Citation Context ... in the context of temporalization (adding a temporal dimension to an original logic) as in [28, 18]. Some interesting preservation properties can also be found for the product of (modal) logics, see =-=[14, 15, 16, 20]-=-. Fibring is a more general mechanism since it goes beyond modal logic. For instance, we can think of fibring relevance logic with intuitionistic logic, or modal logic and first-order logic. Although ... |

3 | C.Sernadas. Combining logic systems: Why, how, what for
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Citation Context ...he other semantically presented by Kripke structures. Note that heterogeneous combination was never dealt with any of the existing combination mechanisms. The heterogeneous scenario was dealt with in =-=[22]-=- for fibring logics presented semantically but with different semantic domains, for instance fibring modal logic presented by Kripke structures with first-order logic with the usual first-order struct... |

2 |
Products of modal logics. III. Products of modal and temporal logics
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(Show Context)
Citation Context ... in the context of temporalization (adding a temporal dimension to an original logic) as in [28, 18]. Some interesting preservation properties can also be found for the product of (modal) logics, see =-=[14, 15, 16, 20]-=-. Fibring is a more general mechanism since it goes beyond modal logic. For instance, we can think of fibring relevance logic with intuitionistic logic, or modal logic and first-order logic. Although ... |

1 |
Analytic calculi for product logics
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(Show Context)
Citation Context |