## Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity (2003)

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Venue: | LOGIC JOURNAL OF THE IGPL |

Citations: | 7 - 5 self |

### BibTeX

@ARTICLE{Mateus03modalsequent,

author = {P. Mateus and A. Sernadas and C. Sernadas and L. Vigano},

title = {Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity},

journal = {LOGIC JOURNAL OF THE IGPL},

year = {2003},

volume = {13},

pages = {173--199}

}

### OpenURL

### Abstract

Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural two-sorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result

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Citation Context ...of this di#culty. With labelled deduction it is possible to keep the rules of the core calculus, adding for each modal system some extra rules about the labels (worlds), as proposed in, for instance, =-=[8, 11, 3, 6, 16]-=-. 2 The idea of labelled deduction also appeared outside the context of modal logic, namely for finite many-valued logics where it is natural to use the truth values as labels [7, 2, 12]. More recentl... |

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Citation Context ...of this di#culty. With labelled deduction it is possible to keep the rules of the core calculus, adding for each modal system some extra rules about the labels (worlds), as proposed in, for instance, =-=[8, 11, 3, 6, 16]-=-. 2 The idea of labelled deduction also appeared outside the context of modal logic, namely for finite many-valued logics where it is natural to use the truth values as labels [7, 2, 12]. More recentl... |

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Citation Context ... for a specific modal system one has to start from scratch (or almost). Usually, the rules needed for the modal system at hand are quite and subtly di#erent from those of system K (see, for instance, =-=[10, 17]-=-). Labelled deduction opened a way out of this di#culty. With labelled deduction it is possible to keep the rules of the core calculus, adding for each modal system some extra rules about the labels (... |

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Citation Context ..., in the worst case, of a hyperexponential increase in the logical depth, see [5]. Contrasting with first order logic where the complexity of cut elimination has been deserving a continuous attention =-=[30, 42, 2]-=-, and with some subsystems of linear logic where the complexity of the cut elimination problem has being studied [16], in modal logic we do not know of any work besides our own on the complexity of cu... |

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Citation Context ... As we shall see in Subsection 3.3, the sequent calculus CK is sound with respect to general Kripke structures as we will see in Theorem 3.17 (recall that a general Kripke structure, see for instance =-=[5]-=-, is a tuple hW, , B, V i where W is the non-empty set of worlds,sis the accessibility relation between worlds, B ` ""W is the set of admissible truth values, and the valuation V maps each proposition... |

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Citation Context ...bring [31] which is a refinement of fibring preventing collapses. Then, in the application front, it would be very interesting to use those results to prove properties about probabilistic logics like =-=[18]-=- and also about quantum logics [19]. Acknowledgements The authors are deeply grateful to Am'ilcar Sernadas and Luca Vigan`o for relevant observations and contributions on the project of this paper. Th... |

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Citation Context ...c (like intuitionistic, relevance, many-valued logics) and develop a theory of fibring such calculi (where the general algebraic completeness result obtained in this paper can be used in the style of =-=[18, 15]-=- for establishing the preservation of completeness by fibring). Acknowledgments This work was partially supported by Fundacao para a Ciencia e a Tecnologias(FCT, Portugal), namely via the FEDER Projec... |

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Citation Context ...g a natural representation and simultaneously enjoying properties like symmetry, explicitness, and elimination of cut [40]. Curry seems to have been the first to seek sequent systems for modal logics =-=[9]-=- soon followed by Ohnishi and Matsumoto [23] and Kanger [15]. More recently, several modal sequent calculi answering some of the problems faced by the initial presentations have been proposed [20, 40,... |

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Citation Context ... fibring preventing collapses. Then, in the application front, it would be very interesting to use those results to prove properties about probabilistic logics like [18] and also about quantum logics =-=[19]-=-. Acknowledgements The authors are deeply grateful to Am'ilcar Sernadas and Luca Vigan`o for relevant observations and contributions on the project of this paper. The authors thank the anonymous refer... |

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Citation Context ...ly enjoying properties like symmetry, explicitness, and elimination of cut [40]. Curry seems to have been the first to seek sequent systems for modal logics [9] soon followed by Ohnishi and Matsumoto =-=[23]-=- and Kanger [15]. More recently, several modal sequent calculi answering some of the problems faced by the initial presentations have been proposed [20, 40, 4, 12, 21, 1, 39, 8, 17, 35, 29]. For examp... |

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Citation Context ...he kind of analysis necessary to deal with plausible reasoning and other kinds of incomplete proofs as described by Polya [25], and can be used to construct elementary proofs from non-elementary ones =-=[36]-=-. Automated logical reasoning is also a motivation for studying cut elimination since inference rules like the cut rule make efficient proof search infeasible. Moreover, the extraction of programs fro... |

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Citation Context ...l and other nonclassical logics. A detailed discussion of such rules and of the semantic and proof-theoretic properties of the resulting labelled sequent calculi (e.g. a form of correspondence theory =-=[18]-=- or the eliminability of cut) is out of the scope of this paper and we leave it as future work. 2.7 Towards a hybrid version of CK The discussion above (about, for instance, the non inter-derivability... |

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Citation Context ...gics [9] soon followed by Ohnishi and Matsumoto [23] and Kanger [15]. More recently, several modal sequent calculi answering some of the problems faced by the initial presentations have been proposed =-=[20, 40, 4, 12, 21, 1, 39, 8, 17, 35, 29]-=-. For example, in [39] a sequent style proof theory, based on Belnap's [3] display logic, is presented for any modal logic obtained by combination of D, T, K4, K5, B, Tr, V and Alt1. In [20], sequent ... |

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Citation Context ...ibility relation but we refrain from going into details. The sequent calculus C @ K represents a first step towards a hybrid version of CK combining the ideas in this paper and those of hybrid logics =-=[5, 4, 1]-=-. 2.8 Local and global reasoning In the context of a modal sequent calculus, local and global notions of prooftheoretic consequence can be defined as follows: . # 1 , . . . , # k # g R # i# #R # # # 1... |

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Citation Context ...erein we show that cuts can be eliminated in sequent calculi labelled with truth values, based on [20], for a wide class of modal logics. The great advantage of considering labelled deduction systems =-=[10, 38, 28, 20, 32, 27]-=- is that they allow to present in a natural, uniform and modular way families of logic systems. In order to accomplish that desiderata, the basic declarative unit in these presentations is a formula p... |

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Citation Context ...m, Craig interpolation, and Beth definability. Cut elimination provides also the kind of analysis necessary to deal with plausible reasoning and other kinds of incomplete proofs as described by Polya =-=[25]-=-, and can be used to construct elementary proofs from non-elementary ones [36]. Automated logical reasoning is also a motivation for studying cut elimination since inference rules like the cut rule ma... |

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Citation Context ... for a specific modal system one has to start from scratch (or almost). Usually, the rules needed for the modal system at hand are quite and subtly di#erent from those of system K (see, for instance, =-=[10, 17]-=-). Labelled deduction opened a way out of this di#culty. With labelled deduction it is possible to keep the rules of the core calculus, adding for each modal system some extra rules about the labels (... |

9 |
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Citation Context ..., in the worst case, of a hyperexponential increase in the logical depth, see [5]. Contrasting with first order logic where the complexity of cut elimination has been deserving a continuous attention =-=[30, 42, 2]-=-, and with some subsystems of linear logic where the complexity of the cut elimination problem has being studied [16], in modal logic we do not know of any work besides our own on the complexity of cu... |

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Citation Context ...for a specific modal system one has to start from scratch (or almost). Usually, the rules needed for the modal system at hand are quite and subtly different from those of system K (see, for instance, =-=[12, 20]-=-). Labelled deduction opened a way out of this difficulty. With labelled deduction it is possible to keep the rules of the core calculus, adding for each modal system some extra rules about the labels... |

8 |
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Citation Context ...gics [9] soon followed by Ohnishi and Matsumoto [23] and Kanger [15]. More recently, several modal sequent calculi answering some of the problems faced by the initial presentations have been proposed =-=[20, 40, 4, 12, 21, 1, 39, 8, 17, 35, 29]-=-. For example, in [39] a sequent style proof theory, based on Belnap’s [3] display logic, is presented for any modal logic obtained by combination of D, T, K4, K5, B, Tr, V and Alt1. In [20], sequent ... |

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Citation Context ...the complexity of cut elimination has been deserving a continuous attention [30, 42, 2], and with some subsystems of linear logic where the complexity of the cut elimination problem has being studied =-=[16]-=-, in modal logic we do not know of any work besides our own on the complexity of cut elimination in a modal sequent calculus. There are instead a widely variety of results on complexity 24sof decision... |

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Citation Context ...ogic we do not know of any work besides our own on the complexity of cut elimination in a modal sequent calculus. There are instead a widely variety of results on complexity 24sof decision procedures =-=[33, 14]-=-. Various reasons contribute to this happening. One is certainly the fact that modal logics are usually presented by Hilbert systems and not by sequent systems. Another reason is that when the modal l... |

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Citation Context ...tance, [8, 11, 3, 6, 16]. 2 The idea of labelled deduction also appeared outside the context of modal logic, namely for finite many-valued logics where it is natural to use the truth values as labels =-=[7, 2, 12]-=-. More recently, labelled deduction has been applied in the context of research on combining logics. Namely, as discussed in [14], combining deduction systems requires a su#ciently general notion of l... |

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Citation Context ...gics [9] soon followed by Ohnishi and Matsumoto [23] and Kanger [15]. More recently, several modal sequent calculi answering some of the problems faced by the initial presentations have been proposed =-=[20, 40, 4, 12, 21, 1, 39, 8, 17, 35, 29]-=-. For example, in [39] a sequent style proof theory, based on Belnap's [3] display logic, is presented for any modal logic obtained by combination of D, T, K4, K5, B, Tr, V and Alt1. In [20], sequent ... |

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Citation Context ... Rovisco Pais, 1049-001, Lisboa, Portugal 1slogic has faced difficulties in finding a natural representation and simultaneously enjoying properties like symmetry, explicitness, and elimination of cut =-=[40]-=-. Curry seems to have been the first to seek sequent systems for modal logics [9] soon followed by Ohnishi and Matsumoto [23] and Kanger [15]. More recently, several modal sequent calculi answering so... |