## Many-Valued Modal Logics II (1992)

Venue: | Fundamenta Informaticae |

Citations: | 25 - 0 self |

### BibTeX

@ARTICLE{Fitting92many-valuedmodal,

author = {Melvin Fitting},

title = {Many-Valued Modal Logics II},

journal = {Fundamenta Informaticae},

year = {1992},

volume = {17}

}

### OpenURL

### Abstract

Suppose there are several experts, with some dominating others (expert A dominates expert B if B says something is true whenever A says it is). Suppose, further, that each of the experts has his or her own view of what is possible --- in other words each of the experts has their own Kripke model in mind (subject, of course, to the dominance relation that may hold between experts). How will they assign truth values to sentences in a common modal language, and on what sentences will they agree? This problem can be reformulated as one about many-valued Kripke models, allowing many-valued accessibility relations. This is a natural generalization of conventional Kripke models that has only recently been looked at. The equivalence between the many-valued version and the multiple expert one will be formally established. Finally we will axiomatize many-valued modal logics, and sketch a proof of completeness.

### Citations

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(Show Context)
Citation Context ...c logic, there is a finite Kripke intuitionistic model, 〈E, D,v〉, and an e ∈E, with v(e, A)=false. This is simply the completeness theorem for intuitionistic logic plus the finite model property=-=. See [4, 2, 1]-=- for proofs. Next we turn to algebraic notions. Definition 2.2 A Heyting algebra is a lattice T with a bottom element false, in which relative pseudo-complements exist. The pseudo-complement of a rela... |

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(Show Context)
Citation Context ...was investigated. There, the ordering of experts was one of increasing sharpness of perception, which amounts to the converse of the ordering considered here. In the predecessor of the present paper, =-=[3]-=-, two families of many-valued modal logics were examined. Semantically speaking, one family allowed formulas to take on values in a many-valued logic at possible worlds, but otherwise the general stru... |

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(Show Context)
Citation Context ...c logic, there is a finite Kripke intuitionistic model, 〈E, D,v〉, and an e ∈E, with v(e, A)=false. This is simply the completeness theorem for intuitionistic logic plus the finite model property=-=. See [4, 2, 1]-=- for proofs. Next we turn to algebraic notions. Definition 2.2 A Heyting algebra is a lattice T with a bottom element false, in which relative pseudo-complements exist. The pseudo-complement of a rela... |

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Citation Context ...nd {1, 2}. We still have closure under intersection and union, but complementation is no longer available to us. In fact, the appropriate structure now is that of a pseudo-Boolean or Heyting algebra (=-=[11]-=- and Section 2 below). But, in order to see the issue more clearly, it is best to reconsider the experts directly again. If we assume some experts dominate others, constraints are placed on behavior t... |

31 |
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(Show Context)
Citation Context ...ion of truth at a possible world may be different for each expert, subject, of course, to the dominance relation among them. Incidentally, the ideas presented here are somewhat related to the work of =-=[9]-=-, but there the emphasis is on intuitionistic logic, while here we think of a particular set of experts as furnishing us with a many-valued logic, of interest for its own sake. Definition 3.1 A multip... |

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Citation Context ... family allowed formulas to take on values in a many-valued logic at possible worlds, but otherwise the general structure of a Kripke frame was not altered. Such logics, in fact, have a long history, =-=[13, 14, 12, 5, 7, 6, 8]-=-. The other family considered in [3] allowed the accessibility relation itself to be many-valued; something apparently new. In this paper the investigation of the second family of logics is continued.... |

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Citation Context ... family allowed formulas to take on values in a many-valued logic at possible worlds, but otherwise the general structure of a Kripke frame was not altered. Such logics, in fact, have a long history, =-=[13, 14, 12, 5, 7, 6, 8]-=-. The other family considered in [3] allowed the accessibility relation itself to be many-valued; something apparently new. In this paper the investigation of the second family of logics is continued.... |

7 |
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Citation Context ... family allowed formulas to take on values in a many-valued logic at possible worlds, but otherwise the general structure of a Kripke frame was not altered. Such logics, in fact, have a long history, =-=[13, 14, 12, 5, 7, 6, 8]-=-. The other family considered in [3] allowed the accessibility relation itself to be many-valued; something apparently new. In this paper the investigation of the second family of logics is continued.... |

6 |
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Citation Context ... experts), the set of modal formulas valid in all T valued modal models can be given a proof-theoretic formulation, in a sequent calculus, and we prove soundness and completeness of this calculus. In =-=[10]-=- a closely related system of logic, also based on multiple experts, was investigated. There, the ordering of experts was one of increasing sharpness of perception, which amounts to the converse of the... |

3 |
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