## Constructive CK for Contexts (2005)

Venue: | In Proceedings of the First Workshop on Context Representation and Reasoning, CONTEXT’05 |

Citations: | 5 - 2 self |

### BibTeX

@INPROCEEDINGS{Mendler05constructiveck,

author = {Michael Mendler and Valeria De Paiva},

title = {Constructive CK for Contexts},

booktitle = {In Proceedings of the First Workshop on Context Representation and Reasoning, CONTEXT’05},

year = {2005}

}

### OpenURL

### Abstract

Abstract. This note describes possible world semantics for a constructive modal logic CK. The system CK is weaker than other constructive modal logics K as it does not satisfy distribution of possibility over disjunctions, neither binary (✸(A ∨ B) → ✸A ∨ ✸B) nor nullary (✸ ⊥ → ⊥). We are interested in this version of constructive K for its application to contexts in AI [dP03]. However, our previous work on CK described only a categorical semantics [BdPR01] for the system, while most logicians interested in contexts prefer their semantics possible worlds style. This note fills the gap by providing the possible worlds model theory for the constructive modal system CK, showing soundness and completeness of the proposed semantics, as well as the finite model property and (hence) decidability of the system. Wijesekera [Wij90] investigated possible worlds semantics of a system similar to CK, without the binary distribution, but satisfying the nullary one. The semantics presented here for CK is new and considerably simpler than the one of Wijesekera. 1

### Citations

348 | Notes on formalizing context
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- 1993
(Show Context)
Citation Context ...cated to formally modeling contexts, in several disciplines, is quite staggering. In previous work ([dP03]) we surveyed several of the logical systems that arose out of McCarthy’s original intuitions =-=[McC93]-=- that context ought to be a first-class object in a logical system devised to reason using common sense. Using purely prooftheoretical criteria we concluded in [dP03] that it would be worth investigat... |

216 | L.: “First-Order Modal Logic
- Fitting, Mendelsohn
- 1999
(Show Context)
Citation Context ...re valid, Hilbert deduction ⊢CK would enjoy the Deduction Theorem. But it does not. For we have A ⊢CK ✷A by Nec, while from soundness of ⊢CK it will follow that �⊢CK A → ✷A. As pointed out by Fitting =-=[Fit94]-=- and Simmons [Pop94] it is useful in modal logics to distinguish between local and global notions of validity, and local and global assumptions. We use the following terminology to make this precise: ... |

185 | A modal analysis of staged computation - Davies, Pfenning |

141 |
Manydimensional modal logics: theory and applications
- Gabbay, Kurucz, et al.
- 2003
(Show Context)
Citation Context ...ite Model Property and Decidability We now show that CK has the finite model property, which implies decidability. Both results can be obtained also from general work on many-dimensional modal logics =-=[GKWZ03]-=- by encoding CK into a classical bi-modal (S4,K) system, thus making the underlying intuitionistic accessibility explicit. We find it instructive, nevertheless, to give a direct proof in order to shed... |

123 | Primitive recursion for higherorder abstract syntax - Despeyroux, Pfenning, et al. - 1997 |

60 | Propositional lax logic
- Fairtlough, Mendler
- 1997
(Show Context)
Citation Context ...he formulae in ∆ are falsified at w and the formulae in Θ are falsified at every world R-reachable from w. This representation of worlds has been introduced originally for propositional lax logic PLL =-=[FM97]-=-. 3 Definition 2. A theory (Γ, ∆, Θ) is consistent if for every choice of formulae N1, N2, . . . , Nn in ∆ and K1, K2, . . . Kk in Θ such that n + k ≥ 1 it is not the case that ∅; Γ ⊢CK N1 ∨ N2 . . . ... |

57 |
First Steps in Modal Logic
- Popkorn
- 1994
(Show Context)
Citation Context ...duction ⊢CK would enjoy the Deduction Theorem. But it does not. For we have A ⊢CK ✷A by Nec, while from soundness of ⊢CK it will follow that �⊢CK A → ✷A. As pointed out by Fitting [Fit94] and Simmons =-=[Pop94]-=- it is useful in modal logics to distinguish between local and global notions of validity, and local and global assumptions. We use the following terminology to make this precise: Let Γ1 and Γ2 be set... |

23 | Categorical and Kripke Semantics for Constructive S4 Modal Logic
- Alechina, Mendler, et al.
- 2001
(Show Context)
Citation Context ...it. We hoped that a direct adaptation of Wisejekera’s results would work for us. The adaptation chosen meant that the proof of completeness could be streamlined and made similar to our previous work (=-=[AMdPR01]-=-) on a constructive and categorical version of modal S4, known as CS4, which is just a special axiomatic theory of CK. The work on CS4 has had many applications within computer science (for examples s... |

19 | On an intuitionistic modal logic - Bierman, Paiva |

19 |
Constructive modal logics I
- Wijesekera
- 1990
(Show Context)
Citation Context ...odel theory for the constructive modal system CK, showing soundness and completeness of the proposed semantics, as well as the finite model property and (hence) decidability of the system. Wijesekera =-=[Wij90]-=- investigated possible worlds semantics of a system similar to CK, without the binary distribution, but satisfying the nullary one. The semantics presented here for CK is new and considerably simpler ... |

18 | A basic logic for textual inference - Bobrow, Condoravdi, et al. - 2005 |

15 | Packed rewriting for mapping semantics to KR
- Crouch
(Show Context)
Citation Context ... notions of contextscould and would be useful can be found in [CCS + 01,CCS + 03]. Some discussion on how we are already implementing some version of contexts using a rewriting system can be found in =-=[Cro05]-=-. A preliminary logical discussion of what kinds of inferences this desired system of contexts needs to perform is presented in [BCC + 05]. But the mathematical connection between CK and the ideas and... |

13 | Entailment, intensionality and text understanding - Condoravdi, Crouch, et al. - 2003 |

12 | Preventing existence - Condoravdi, Crouch, et al. - 2001 |

12 | ML Systems: A Proof Theory for Contexts - Serafini, Giunchiglia |

10 | Extended curry-howard correspondence for a basic constructive modal logic
- Bellin, Paiva, et al.
- 2001
(Show Context)
Citation Context ...A ∨ ✸B) nor nullary (✸⊥ → ⊥). We are interested in this version of constructive K for its application to contexts in AI [dP03]. However, our previous work on CK described only a categorical semantics =-=[BdPR01]-=- for the system, while most logicians interested in contexts prefer their semantics possible worlds style. This note fills the gap by providing the possible worlds model theory for the constructive mo... |

8 | Towards constructive hybrid logic - Braüner, Paiva - 2003 |

8 |
Natural Deduction and Context as (Constructive) Modality
- Paiva
- 2003
(Show Context)
Citation Context ...isfy distribution of possibility over disjunctions, neither binary (✸(A ∨ B) → ✸A ∨ ✸B) nor nullary (✸⊥ → ⊥). We are interested in this version of constructive K for its application to contexts in AI =-=[dP03]-=-. However, our previous work on CK described only a categorical semantics [BdPR01] for the system, while most logicians interested in contexts prefer their semantics possible worlds style. This note f... |

1 | Modalities in constructive logics and type theories - Paiva, Goré, et al. |