## A Computational Interpretation of Modal Proofs (1994)

Venue: | Proof Theory of Modal Logics |

Citations: | 29 - 2 self |

### BibTeX

@INPROCEEDINGS{Martini94acomputational,

author = {Simone Martini and Andrea Masini},

title = {A Computational Interpretation of Modal Proofs},

booktitle = {Proof Theory of Modal Logics},

year = {1994},

pages = {213--241},

publisher = {Kluwer}

}

### Years of Citing Articles

### OpenURL

### Abstract

The usual (e.g. Prawitz's) treatment of natural deduction for modal logics involves a complicated rule for the introduction of the necessity, since the naive one does not allow normalization. We propose natural deduction systems for the positive fragments of the modal logics K, K4, KT, and S4, extending previous work by Masini on a two-dimensional generalization of Gentzen's sequents (2-sequents). The modal rules closely match the standard rules for an universal quantifier and different logics are obtained with simple conditions on the elimination rule for 2. We give an explicit term calculus corresponding to proofs in these systems and, after defining a notion of reduction on terms, we prove its confluence and strong normalization. 1. Introduction Proof theory of modal logics, though largely studied since the fifties, has always been a delicate subject, the main reason being the apparent impossibility to obtain elegant, natural systems for intensional operators (with the excellent ex...

### Citations

1134 |
The Lambda calculus: Its syntax and semantics
- Barendregt
- 1984
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Citation Context ... i M h i [n] i N h . Proof. By induction on the lenght of the reduction, using lemma 6.4. 6.1. Confluence We prove the Church-Rosser property for i, using Tait's technique as formulated in [Gir87] or =-=[Bar84]-=-, here adapted to modal terms. We define first a new auxiliary notion of reduction ( ' ), corresponding to the (possible) parallel contraction of several non overlapping redexes; note that ' is not tr... |

125 |
Two-dimensional Modal Logic
- Segerberg
- 1973
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Citation Context ...eason being the apparent impossibility to obtain elegant, natural systems for intensional operators (with the excellent exception of intuitionistic logic). For example Segeberg, not earlier than 1984 =-=[BS84]-=-, observed that the Gentzen format, which works so well for truth functional and intuitionistic operators, cannot be a priori expected to remain valid for modal logics; carrying to the limit this obse... |

34 | Natural deduction for intuitionistic linear logic. Unpublished manuscript
- Troelstra
- 1993
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Citation Context ...proposed rule. In particular, a good introduction rule should construct its conclusion from all its premises and possibly some assumptions, a requirement clearly violated by the proposed 2I. Finally, =-=[Tro93]-=- shows a normal deduction in the positive fragment of the calculus in [BMdP92] which does not satisfy the subformula property. It is then necessary to add suitable permutative reductions, which are no... |

30 |
Sequent Calculi for Normal Modal Propositional Logics
- Wansing
- 1994
(Show Context)
Citation Context ...ferentiation between logics is delegated instead to the way (general) formulas are manipulated. A clear discussion on this topics (in the case of modal systems) may be found in recent work of Wansing =-=[Wan94]-=- and Dosen [Dos85]. This point of view (that [Wan94] calls Dosen principle, but that should be traced back to Gentzen, for the differentiation of intuitionistic from classical logic) may be stated as:... |

18 | Queiroz. Extending the Curry-Howard interpretation to linear, relevant and other resource logics - Gabbay, de - 1990 |

14 |
2-sequent calculus: A proof-theory of modalities
- Masini
- 1992
(Show Context)
Citation Context ...al generalization of the notion of sequent. Instead of asserting provability (`) between two sequences of formulas, provability is asserted between two-dimensional sequences of formulas. Developed in =-=[Mas92]-=- as a sequent calculus for classical KD, in [Mas93] the approach is tailored to the intuitionistic framework, for which it is also given an equivalent natural deduction system. The goal of the present... |

11 |
Lambda Calculus with Types
- Barendregt, Dekkers, et al.
- 2013
(Show Context)
Citation Context ... k )s(fi k ) (2ff k ) = (ff k+1 ) Proposition 6.14. \Gamma ` S4 M : oe k =) \Gamma `sM : (oe k ) Proof. Induction on the derivation, the modal rules becoming vacuous steps in . It is well known (e.g. =-=[Bar92]-=-) thatsenjoys strong normalization. It is now easy to obtain as a corollary the same result fors` S4. Theorem 6.15.s` S4 enjoys strong normalization. Proof. Let \Gamma `s` S4 M : oe k and suppose ther... |

11 |
A constructive presentation for the modal connective of necessity
- Benevides, Maibaum
- 1992
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Citation Context ...following deduction shows: 2oe 2IP 22oe Proofs in our system, however, are still enough for completeness and, moreover, seem to have a more direct interpretation as travelling in a Kripke model (cfr. =-=[BM92]-=-). Many consequences can be drawn from Corollary 4.2. Since any reduction step (see Section 6.) in our system becomes also a reduction step in `P under the ( ) [ translation, we immediately obtain nor... |

9 |
Sequent-systems for modal logic, The
- Došen
- 1985
(Show Context)
Citation Context ...en logics is delegated instead to the way (general) formulas are manipulated. A clear discussion on this topics (in the case of modal systems) may be found in recent work of Wansing [Wan94] and Dosen =-=[Dos85]-=-. This point of view (that [Wan94] calls Dosen principle, but that should be traced back to Gentzen, for the differentiation of intuitionistic from classical logic) may be stated as: The rules for the... |

9 |
2-sequent calculus: Intuitionism and natural deduction
- Masini
- 1993
(Show Context)
Citation Context ... of asserting provability (`) between two sequences of formulas, provability is asserted between two-dimensional sequences of formulas. Developed in [Mas92] as a sequent calculus for classical KD, in =-=[Mas93]-=- the approach is tailored to the intuitionistic framework, for which it is also given an equivalent natural deduction system. The goal of the present paper is to study the computational properties of ... |

1 |
Claudia Mer'e, and Valeria de Paiva. Intuitionistic necessity revisited
- Bierman
- 1992
(Show Context)
Citation Context ...ization. If one does not tackle normalization, why not using the simple, naive approaches outlined in the Introduction? 7.2. The intuitionistic calculus of Bierman, Mer' e and de Paiva Bierman et al. =-=[BMdP92]-=- present an approach to the 2 modality essentially based on a variation of Prawitz proposal. The problematic 2I rule has the form \Gamma 1 \Delta \Delta \Delta M 1 : 2oe 1 : : : \Gamma n \Delta \Delta... |

1 |
de Queiroz. An introduction to labelled natural deduction
- Gabbay, Ruy
- 1992
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Citation Context ... modal logic and it propositional counterpart when world-variables are introduced in the functional calculus of the labels (i.e. when a little of the semantics is brought to the syntax, so to speak). =-=[GdQ92b]-=- The calculus is a kind of second order system, where second order objects, which syntactically are only variables, correspond to possible worlds. The basic rules for introduction and elimination of 2... |

1 |
Natural Deduction. Acta Universitatis Stockholmiensis
- Prawitz
- 1965
(Show Context)
Citation Context ...` oe 2\Gamma ` oe ` 2 2\Gamma ` 2oe but, once again, problematic natural deduction. The sequent rules appear strictly related to those for the universal quantifier, and following this analogy Prawitz =-=[Pra65]-=- attempted the definition of a natural deduction system styled after the first order system. This naive approach, however, does not work, and in order to obtain normalization, Prawitz was forced to in... |