## A systematic proof theory for several modal logics (2005)

### Cached

### Download Links

- [www.textproof.com]
- [www.linearity.org]
- [www.aiml.net]
- [www.wv.inf.tu-dresden.de]
- DBLP

### Other Repositories/Bibliography

Venue: | Advances in Modal Logic, volume 5 of King’s College Publications |

Citations: | 24 - 1 self |

### BibTeX

@INPROCEEDINGS{Stewart05asystematic,

author = {Charles Stewart},

title = {A systematic proof theory for several modal logics},

booktitle = {Advances in Modal Logic, volume 5 of King’s College Publications},

year = {2005},

pages = {309--333}

}

### Years of Citing Articles

### OpenURL

### Abstract

abstract. The family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a Hilbert-Lewis style and conditions on the accessibility relation on frames. By contrast, the usual structural proof theory of modal logic, as given in Gentzen systems, is ad-hoc. While we can formulate several modal logics in the sequent calculus that enjoy cut-elimination, their formalisation arises through system-bysystem fine tuning to ensure that the cut-elimination holds, and the correspondence to the axioms of the Hilbert-Lewis systems becomes opaque. This paper introduces a systematic presentation for the systems K, D, M, S4, and S5 in the calculus of structures, a structural proof theory that employs deep inference. Because of this, we are able to axiomatise the modal logics in a manner directly analogous to the Hilbert-Lewis axiomatisation. We show that the calculus possesses a cut-elimination property directly analogous to cut-elimination for the sequent calculus for these systems, and we discuss the extension to several other modal logics. 1

### Citations

442 |
Extending Modal Logic
- Rijke
- 1993
(Show Context)
Citation Context ...tively natural extension of Gentzen-style sequent calculi provides cut-free characterisations of several important modal logics; however it fails 1 See Garson [12], and Blackburn, de Rijke and Venema =-=[3]-=- for readable accounts of this relationship.A Systematic Proof Theory for Several Modal Logics 311 to provide characterisations of others, and even in the cases where sequent systems can be provided,... |

240 | Communicating and Mobile Systems the Pi-Calculus - Milner - 1999 |

162 | Anytime, anywhere. Modal logics for mobile ambients - Cardelli, Gordon - 2000 |

162 | Basic Proof Theory - Troelstra, Schwichtenberg - 2000 |

123 |
Two-Dimensional Modal Logic
- Segerberg
- 1973
(Show Context)
Citation Context ...ful; • Most of the widely used modal logics being decidable; • The normal modal logics possession of an elementary model theory in the form of frame semantics (due to Kripke, Hintikka and Kanger, see =-=[8]-=-) that provides a systematic correspondence between their constitutive axioms as they are usually given in Hilbert style and conditions on the accessibility relation on frames 1 . Let’s look at the to... |

106 | Display logic - Belnap - 1982 |

99 | Proof Theory - Schütte - 1977 |

97 |
Quantification in modal logic
- Garson
- 1977
(Show Context)
Citation Context ...sis the situation is more fraught. A relatively natural extension of Gentzen-style sequent calculi provides cut-free characterisations of several important modal logics; however it fails 1 See Garson =-=[12]-=-, and Blackburn, de Rijke and Venema [3] for readable accounts of this relationship.A Systematic Proof Theory for Several Modal Logics 311 to provide characterisations of others, and even in the case... |

55 | Non-commutativity and MELL in the calculus of structures
- Guglielmi, Straßburger
- 2001
(Show Context)
Citation Context ... one obtains with deep inference with the traditional 7 Though there are logics expressible in the calculus of structures which it appears that display logic cannot express at all, such as system NEL =-=[13]-=-, due to the branching nature of proofs in display logic328 Charles Stewart and Phiniki Stouppa approach to proof analysis based on the subformula property. The result is that display logic leads a d... |

29 | Sequent Calculi for Normal Modal Propositional Logics - Wansing - 1994 |

26 | A local system for linear logic
- Straßburger
(Show Context)
Citation Context ...nded to deal with the case of SKS. Lastly, a very syntactically involved technique of permutability of rules can be applied to prove cut elimination; two proofs of this nature are due to Strassburger =-=[23]-=-, as well as a proof by Brünnler [6]. Of these approaches, a splitting proof would be the most valuable. A further issue relates to the role of proof analysis in the toolkit of the modal logician. The... |

24 | A purely logical account of sequentiality in proof search - Bruscoli - 2002 |

20 |
Deep Inference and Symmetry
- Brünnler
- 2004
(Show Context)
Citation Context ...19 2. The entire up-fragment, ie. the rules labelled with ↑, is admissible; that is, the full system is equivalent to the system obtained by removing the whole up-fragment. This is shown by Bruennler =-=[6]-=- and discussed in more detail below, by means of a translation from cut-free proofs of the sequent calculus into proofs of system KSg, that is, system SKSg without the rules of the up-fragment. Becaus... |

17 | Power and weakness of the modal display calculus
- Kracht
- 1996
(Show Context)
Citation Context ...to rules involving primitive tense formula, 8 By means of the translation − h described in section three.A Systematic Proof Theory for Several Modal Logics 329 following the results of Marcus Kracht =-=[15]-=-), and so the judgements appearing in the tree of a proof may be assertions not of S5 but of S5t, its tensed extension. By conservativity, we know that we have the right theorems, but conservativity o... |

14 |
Displaying modal logic
- Wansing
- 1998
(Show Context)
Citation Context ...ssing is to say that a calculus is focussed if each rule that deals with a connective is only about that connective. As such this is related to the properties such as separation as defined by Wansing =-=[27]-=-, but neither strong nor weak focussedness is expressible in terms of Wansing’s properties, and our interest here is technical, concerned with permutability of cut, rather that the meaning theoretic i... |

8 |
A cut-free Gentzen formulation of the modal logic S5
- Braüner
- 2000
(Show Context)
Citation Context ...formulae have � as their main operator, which in each case is the principal conclusion of the appropriate �i rule. □ While cut-free sequent systems for S5 have appeared in the literature; for example =-=[4]-=-, these formulations do not take the simple form of the sequent systems treated in this paper, possessing either sophisticated rules that expose proof structure (either explicitly as in Braüner’s conn... |

8 | Sequent-Systems for modal logic - Doˇsen - 1985 |

4 |
Gentzen method in modal calculi, parts I and II
- Ohnishi, Matsumoto
- 1957
(Show Context)
Citation Context ...he cut-elimination proof is more subtle because many of the usual permutations on cuts fail, a fact that is unfortunately rather glossed over in the literature. Cf. Ohnishi and Matsumoto, and Valenti =-=[17, 24, 25]-=-. □ Our axiomatisation in the calculus of structures proceeds by defining the inference rules: ⋄2 d S{�R} S{⋄R} S{⋄ ⋄ R} 4 ↓ S{⋄R} Then we obtain the equivalent systems: 1. KSg-K is KSg-M without the ... |

2 | Avron The method of hypersequents in the proof theory of propositional nonclassical logics - unknown authors - 1996 |

2 |
Locality for classical logic. Submitted to Archive for Mathematical Logic, 2003. Preprint available http://www.wv.inf.tu-dresden.de/ kai/LocalityClassical.pdf
- Brünnler
(Show Context)
Citation Context ...ram of research in the calculus of structures. A thorough mathematical and conceptual examination of these properties and their importance is given in Bruennler’s [6]; a shorter discussion appears in =-=[5]-=-. We can extend this calculus to obtain the system M by allowing formulae of the form �R, and ⋄R, extending the equivalences defining structures with tt = �tt and ff = ⋄ff and extending the set of inf... |

2 |
A System of Interaction and Structure. Submitted to
- Guglielmi
- 2002
(Show Context)
Citation Context ...al proof theory in texts such as Blackburn, de Rijke and Venema [3]. In this paper we will provide a new proof theoretic basis for modal logic using the calculus of structures introduced by Guglielmi =-=[14]-=-. This proof calculus provides a structural proof theory in the sense of Gentzen for logical systems because it possesses a notion of cut-free proof directly analogous to that for the sequent calculus... |

1 |
The modal aether
- Forster
(Show Context)
Citation Context ...sonably be considered to be about possibility or contingency, and further we agree with Thomas Forster that the language of possible world semantics is intoxicating to the careless and is best avoided=-=[11]-=-. Hence we call Hintikka/Kripke/Kanger style semantics, frame semantics; we avoid formalising these but will need to talk about the frame accessibility relation, which is given over nodes.314 Charles... |

1 |
Proof theory of intuitionistic modal logic
- Simpson
- 1994
(Show Context)
Citation Context ...e problem of cut elimination for modal logic becomes mostly a solved problem: we simply use calculi that embed modal logics in sequent calculi for geometric theories, such as pioneered by Alex Simpson=-=[20]-=-. Certainly there is room for dispute over the relative merits of the two approaches to providing a structural proof theory for modal logic; however we think it is important to recognise firstly that ... |

1 |
Which 2-sided sequent systems are equivalent to 1-sided systems? Unpublished manuscript
- Stewart
- 2004
(Show Context)
Citation Context ...nts on the form of the sequent calculus, 2-sided (Gentzen style) systems and 1-sided (Schütte style) systems characterise exactly the same consequence relations in the presence of De Morgan dualities =-=[21]-=-, and this demonstration generalises to hypersequents.A Systematic Proof Theory for Several Modal Logics 315 Axiom and cut: ⊢ Γ, A, A ax ⊢ Γ, A ⊢ ∆, A cut ⊢ Γ, ∆ Contraction and weakening: ⊢ Γ, A, A ... |

1 | The design of modal proof theories. MSc dissertation, Technische Universität Dresden - Stouppa |

1 |
Cut-elimination in a modal sequent calculus for K
- Valenti
- 1982
(Show Context)
Citation Context ...he cut-elimination proof is more subtle because many of the usual permutations on cuts fail, a fact that is unfortunately rather glossed over in the literature. Cf. Ohnishi and Matsumoto, and Valenti =-=[17, 24, 25]-=-. □ Our axiomatisation in the calculus of structures proceeds by defining the inference rules: ⋄2 d S{�R} S{⋄R} S{⋄ ⋄ R} 4 ↓ S{⋄R} Then we obtain the equivalent systems: 1. KSg-K is KSg-M without the ... |

1 |
The sequent calculus for the modal logic D
- Valenti
- 1993
(Show Context)
Citation Context ...he cut-elimination proof is more subtle because many of the usual permutations on cuts fail, a fact that is unfortunately rather glossed over in the literature. Cf. Ohnishi and Matsumoto, and Valenti =-=[17, 24, 25]-=-. □ Our axiomatisation in the calculus of structures proceeds by defining the inference rules: ⋄2 d S{�R} S{⋄R} S{⋄ ⋄ R} 4 ↓ S{⋄R} Then we obtain the equivalent systems: 1. KSg-K is KSg-M without the ... |