### Аннотация

The relationships between various modal logics based on Belnap and Dunn’s paraconsistent four-valued logic FDE are investigated. It is shown that the paraconsistent modal logic BK^{□}, which lacks a primitive possibility operator ◊, is definitionally equivalent with the logic BK, which has both and ◊ as primitive modalities. Next, a tableau calculus for the paraconsistent modal logic KN4 introduced by L. Goble is defined and used to show that KN4 is definitionally equivalent with BK^{□} without the absurdity constant. Moreover, a tableau calculus is defined for the modal bilattice logic MBL introduced and investigated by A. Jung, U. Rivieccio, and R. Jansana. MBL is a generalization of BK that in its Kripke semantics makes use of a four-valued accessibility relation. It is shown that MBL can be faithfully embedded into the bimodal logic BK^{□} × BK^{□} over the non-modal vocabulary of MBL. On the way from BK^{□} to MBL, the Fischer Servi-style modal logic BK^{FS} is defined as the set of all modal formulas valid under a modified standard translation into first-order FDE, and BK^{FS} is shown to be characterized by the class of all models for BK^{□} × BK^{□}. Moreover, BK^{FS} is axiomatized and this axiom system is proved to be strongly sound and complete with respect to the class of models for BK^{□} x BK^{□}. Moreover, the notion of definitional equivalence is suitably weakened, so as to show that BK^{FS} and BK^{□} × BK^{□} are weakly definitionally equivalent.

Язык оригинала | английский |
---|---|

Страницы (с-по) | 1221-1254 |

Число страниц | 34 |

Журнал | Studia Logica |

Том | 105 |

Номер выпуска | 6 |

DOI | |

Состояние | Опубликовано - 23 сен 2017 |

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## Цитировать

*Studia Logica*,

*105*(6), 1221-1254. https://doi.org/10.1007/s11225-017-9753-9