## Labelled Tableaux for Multi-Modal Logics (1995)

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Venue: | Theorem Proving with Analytic |

Citations: | 17 - 9 self |

### BibTeX

@INPROCEEDINGS{Governatori95labelledtableaux,

author = {Guido Governatori},

title = {Labelled Tableaux for Multi-Modal Logics},

booktitle = {Theorem Proving with Analytic},

year = {1995},

pages = {79--94},

publisher = {Springer-Verlag}

}

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### Abstract

this paper we present a tableau-like proof system for multi-modal logics based on D'Agostino and Mondadori's classical refutation system KE [DM94]. The proposed system, that we call KEM , works for the logics S5A and S5P(n) which have been devised by Mayer and van der Hoek [MvH92] for formalizing the notions of actuality and preference. We shall also show how KEM works with the normal modal logics K45, D45, and S5 which are frequently used as bases for epistemic operators -- knowledge, belief (see, for example [Hoe93, Wan90]), and we shall briefly sketch how to combine knowledge and belief in a multi-agent setting through KEM modularity

### Citations

278 |
First-Order Logic
- Smullyan
(Show Context)
Citation Context ..., F } where S is the set of all the formulas of our language. The notion of L-model appropriate for the logic L can be obtained by restricting Ri to satisfy the conditions associated with L. As usual =-=[Smu68b] by a-=- signed formula (S-formula) we shall mean an expression of the form SA where A is a formula and S ∈ {T, F }. Thus T A if υ(A, x) = T ⋆ I would like to thank professor Alberto Artosi and Paola Cat... |

271 | Proof Methods for Modal and Intuitionistic Logics, volume 169 of Synthese Library - Fitting - 1983 |

87 | Automated Deduction in Nonclassical Logics - Wallen - 1990 |

50 | A generalization of analytic deduction via labelled deductive systems I: Basic substructural logics
- D’Agostino, Gabbay
- 1994
(Show Context)
Citation Context ..., Fit83, Wri85] and, more recently, in [Cat91, JR89, Tap87, Wal90] and also in the “translation” tradition of [AE92, Ohl91], and in Gabbay’s Discipline of Labelled Deductive Systems [Gab91] (see=-= also [DG94]-=- tableau extension with labels). KEM combines two kinds of rules: rules for processing the propositional part (which are the same for all modal logics), and rules for manipulating labels according to ... |

49 |
The taming of the cut
- D’Agostino, Mondadori
- 1994
(Show Context)
Citation Context ...ly, Fax +39(0)51-260782 E-mail: governat@cirfid.unibo.it In this paper we present a tableau-like proof system for multi-modal logics based on D’Agostino and Mondadori’s classical refutation system=-= KE [DM94]-=-. The proposed system, that we call KEM, works for the logics S5A and S5P (n) which have been devised by Mayer and van der Hoek [MvH92] for formalizing the notions of actuality and preference. We shal... |

47 | Modal theorem proving: An equational viewpoint - Auffray, Enjalbert - 1992 |

40 | Semantics-based translation methods for modal logics - Ohlbach - 1991 |

33 | TABLEAUX: a general theorem prover for modal logics - Catach - 1991 |

23 | Systems for knowledge and belief
- Hoek
- 1993
(Show Context)
Citation Context ...strictely related with the modal semantics; like axiomatic systems it is highly modular. To illustrate the last point we summarize how obtain the Kraus and Lehmann’s logic for knowledge and belief K=-=B [Hoe93].-=- Knowledge modalities act as S5 modalities whereas belief modalities act as D45 modalities, and it is well known that adding reflexivity to D45 leads to S5, therefore our basic high unification is σD... |

20 | Tableau methods of proof for modal logics - Fitting - 1972 |

17 | Don’t eliminate cut - Boolos - 1984 |

14 |
Labelled Deductive Systems, Part I
- Gabbay
- 1990
(Show Context)
Citation Context ...re found in [Fit72, Fit83, Wri85] and, more recently, in [Cat91, JR89, Tap87, Wal90] and also in the “translation” tradition of [AE92, Ohl91], and in Gabbay’s Discipline of Labelled Deductive Sy=-=stems [Gab91]-=- (see also [DG94] tableau extension with labels). KEM combines two kinds of rules: rules for processing the propositional part (which are the same for all modal logics), and rules for manipulating lab... |

14 |
A General Proof Method for Modal Predicate Logic
- Jackson, Reichgelt
- 1989
(Show Context)
Citation Context ...dality has its own high unification, and the various high unifications are combined into the low unification which models such logics. Remark. The modal proof system proposed by Jackson and Reichgelt =-=[JR89]-=- is the most closely related to ours. The index formalism is almost identical, but the unification algorithm used to resolve complementary formulas in the various modal logics does not work for the no... |

12 | Labelled model modal logic
- Artosi, Governatori
- 1994
(Show Context)
Citation Context ...s: f(SA, i) = g(i), υ(A, g(h(i))) = S. Lemma 1. For any i, k ∈ ℑ, L = K45,D45,S5,S5A,S5P (n), if X, i and (i, k)σ L , then X, k. We only give the proofs for S5, S5A, S5P (n); for the other logic=-=s see [AG94] Proof L -=-= S5. Let us suppose that X, i, X C , k and (i, k)σ S5 , but (i, k)σ S5 ⇐⇒ (h(i), h(k))σ S5 . From the definition of f we have and f(X, i) = g(i), υ(A, g(h(i))) = S, f(X C , k) = g(k), υ(A, g... |

5 | An Automated Approach to Deontic Reasoning - Artosi, Cattabriga, et al. |

4 | A simplified natural deduction approach to certain modal systems - Tapscott - 1987 |

3 |
Labelled Modal Proofs
- Artosi, Governatori
- 1993
(Show Context)
Citation Context ...f labels h(i), h(k) are, we get a contradiction.sTheorem 2. ⊢L A ⇔⊢ KEM(L) A for L = K45, D45, S5, S5A, S5P (n). Proof ⇒. The modus ponens and the characteristic axioms for L are provable in K=-=EM (see [AG93] f-=-or the proof of the axioms and [DM94] for a proof that modus ponens is a derived rule in KE). We shall give a KEM-proof of the rule of necessitation. Let us assume that ⊢ KEM(L) A. Then the followin... |

3 |
Tree Proofs in Modal Logic (abstract
- Fitch
- 1966
(Show Context)
Citation Context ...s are unifiable, will stand for formulas which are contradictory “in the same world”.sRemark. The idea of using a label scheme to bookkeep “world” paths in modal theorem proving goes back at l=-=east to [Fi66]. Simi-=-lar, or related, ideas are found in [Fit72, Fit83, Wri85] and, more recently, in [Cat91, JR89, Tap87, Wal90] and also in the “translation” tradition of [AE92, Ohl91], and in Gabbay’s Discipline ... |

3 | Non-Classical Logic Theorem Proving - Wrightson - 1985 |

2 |
A Modal Theorem Prover. Forthcoming in Annali dell’Università di
- KEM
(Show Context)
Citation Context ...each closed KEM-tree a closed canonical KEM-tree exists. Let φ be the function which deletes the modal operators from given formulas. Lemma 3. �⊢KE φA ⇒�⊢ KEM(L) A Proof. Obvious. For the =-=details see [ACG95].-=- This lemma gives a first termination check for the canonical KEM-trees; in fact a KEM-tree finds out whether complementary formulas exist and it verifies (through the σL-unifications) whether the pa... |

2 |
A Modal Logic for Nonmonotonic Reasoning
- Mayer, Hoeck
- 1992
(Show Context)
Citation Context ...ased on D’Agostino and Mondadori’s classical refutation system KE [DM94]. The proposed system, that we call KEM, works for the logics S5A and S5P (n) which have been devised by Mayer and van der H=-=oek [MvH92] f-=-or formalizing the notions of actuality and preference. We shall also show how KEM works with the normal modal logics K45, D45, and S5 which are frequently used as bases for epistemic operators – kn... |

1 |
An Automated Modal Framework for Nonmonotonic Reasoning
- Artosi, Cattabriga, et al.
(Show Context)
Citation Context ...f” property (i.e. the transitivity property of proof, see [DM94]). As regards the axioms of S5A and S5P (n) we shall show example proofs of △A → ✷△A and ¬Pi⊥ → (PiPjA ≡ PjA) (for the =-=other axioms see [ACG94b]). ⊢KES5A △A → ✷△A ⊢KES5P (n) ¬Pi-=-⊥ → (PiPjA ≡ PjA) 1. F △A → ✷△A w1 2. T △A w1 3. F ✷△A w1 4. T A (a, w1) 5. F △A (w2, w1) 6. F A (a, (w2, w1)) 7. × a 1. F ¬Pi⊥ → (PiPjA ≡ PjA) w1 2. T ¬Pi⊥ w1 3. F Pi... |

1 |
First-Order Logic and Automathed Theorem Prover
- Fitting
- 1990
(Show Context)
Citation Context ...nsrules, then the branch is closed, (vi) we repeat the procedure in each branch generated by P B. Remark. As is well known [Smu68a], what destroys analyticity is losing the (weak) subformula property =-=[Fit90], an-=-d not having a cut rule restricted to subformulas. Otherwise each tableau system is not analytic, since from the formula ¬(A → B) we obtain two branches containing respectively ¬A and B, but, obvi... |

1 |
Analityc Cut
- Smullyan
- 1968
(Show Context)
Citation Context ...label exists or can be built using already existing labels and the unificationsrules, then the branch is closed, (vi) we repeat the procedure in each branch generated by P B. Remark. As is well known =-=[Smu68a]-=-, what destroys analyticity is losing the (weak) subformula property [Fit90], and not having a cut rule restricted to subformulas. Otherwise each tableau system is not analytic, since from the formula... |

1 | A General Possible Words Framework for Reasoning about Knowledge and Belief - Wansing - 1990 |