## A TERM ASSIGNMENT FOR DUAL INTUITIONISTIC LOGIC.

Citations: | 5 - 3 self |

### BibTeX

@MISC{Bellin_aterm,

author = {Gianluigi Bellin and Nicola Gambino and Maria Emilia Maietti and Kurt Ranalter and Edmund Robinson},

title = {A TERM ASSIGNMENT FOR DUAL INTUITIONISTIC LOGIC.},

year = {}

}

### OpenURL

### Abstract

Abstract. We study the proof-theory of co-Heyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a single-assumption multiple-conclusions Natural Deduction system NJ � � for this logic: unlike the best-known treatments of multiple-conclusion systems (e.g., Parigot’s λ−µ calculus, or Urban and Bierman’s term-calculus) here the term-assignment is distributed to all conclusions, and exhibits several features of calculi for concurrency, such as remote capture of variable and remote substitution. The present construction can be regarded as the construction of a free co-Cartesian

### Citations

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(Show Context)
Citation Context ...lism, to be discussed below. From the graphs of Natural Deduction derivations for minimal logic, decorated with lambda terms, one constructs the free Cartesianclosed category generated by the syntax (=-=[16]-=-, pp. 55) in a familiar way. We would like to dualize this construction, in particular, to define a term assignment suitable for the construction of free coCartesian closed categories. However, the fo... |

224 |
Natural Deduction. A Proof-Theoretical Study. Almquist and Wiksell
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Citation Context ...s methods are forced to go back to the basic definitions. It is important to notice, not just as an historical point but also in order to put current research in perspective, that in Prawitz’s thesis =-=[18]-=- two distinct representations of Natural Deduction derivations are suggested, which may be called sequent-style (or type-theoretic) and graphical, respectively. On one hand, natural deductions may be ... |

123 |
Lambda-calcul types et modèles
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- 1990
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Citation Context ...n open at that edge; correct derivations are inductively generated from assumptions of the form A, Γ − A according to Prawitz’s deduction rules. This view is common in type theory (see, e.g., Krivine =-=[15]-=-, p. 34) where the assignment of variables to assumptions makes the discharging operations completely explicit (otherwise, the use of labelled formulas to characterize assumption classes is necessary)... |

46 |
On intuitionistic linear logic
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(Show Context)
Citation Context .... . . ,?C◦ n (namely, ?E ◦ ⊢ ?C ◦ 1℘ . . .℘?C ◦ n) Spelling out the term assignment for the linearization of LJ �� and the categorical term model means to dualize the construction of Bierman’s thesis =-=[7]-=- and then extract the term model for LJ �� itself; we shall not do this here. We simply outline the definition of the term assignment which appears to provide the required duality with the simply type... |

23 | A formulae-as-types interpretation of subtractive logic
- Crolard
(Show Context)
Citation Context ...ypes, on one hand, and between derivations in NJ and λ-terms, on the other, resulting in the abstract characterization of the Curry-Howard correspondence in terms of Cartesian closed categories. 4 In =-=[11]-=- Tristan Crolard presents a Natural Deduction system and a term assignment for Subtractive Logic, called λµ →+×− -calculus, in the tradition of Parigot’s λµ-calculus for classical Natural Deduction. R... |

23 | Semi-boolean algebras and their applications to intuitionistic logic with dual operators - Rauszer - 1974 |

23 |
Basic Proof Theory. Cambridge Tracts in Theoretical Computer Science 43
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(Show Context)
Citation Context ... Υ,C D ⇒ Υ � −R E ⇒ Υ,C � D Θ ⇒ A B,Θ ⇒ F ⊃ −R A ⊃ B,Θ ⇒ F C ⇒ D,Υ � −L C � D ⇒ Υ A,Θ ⇒ B ⊃ −R Θ ⇒ A ⊃ B and similarly for disjunction. The rules of the sequent calculi LJ ⊃∩ and LJ �� in a G3 system =-=[21]-=- are given in the Appendix, Table 2. We present a corresponding natural deduction system NJ �� for this logic: as Natural Deduction NJ ⊃∩ for minimal logic is a multipleassumptions single-conclusion s... |

20 | Subtractive logic
- Crolard
(Show Context)
Citation Context .... We would like to dualize this construction, in particular, to define a term assignment suitable for the construction of free coCartesian closed categories. However, the following fact (spelt out in =-=[10]-=-, Proposition 1.15) reminds us that not every mathematical model is suitable for such a dualization: Fact. In the category of sets, the coexponent BA of two sets A and B is defined if and only if A = ... |

14 | On the pi-Calculus and Linear Logic
- Bellin, Scott
- 1994
(Show Context)
Citation Context ... analysis of branches is robust and has clear counterparts in the lambda-calculus, in game semantics and in other representations of proofs. See also the applications of this concept to proof-nets in =-=[2]-=- and in F. Lamarche’s essential nets, fruitful both for the abstract theory [3] and for applications to complexity theory [17].TERM ASSIGNMENT 11 In the full system NJ, which includes the (ordinary a... |

13 |
Towards a logic for pragmatics. assertions and conjectures
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Citation Context ...btractive Logic and its term calculus. Clearly we work in a much more restricted environment, in which a duality result with the simply typed lambda calculus holds. But even extending our logic as in =-=[4]-=- and introducing negations, the crucial difference of our approach is that our terms are distributed over the conclusions (with the possible exception of postpone terms), and therefore there is no foc... |

13 | Bi-Heyting algebras, Toposes and Modalities - Reyes, Zolfaghari - 1996 |

11 | A pragmatic interpretation of substructural logics
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(Show Context)
Citation Context ... results, as our work in dual intuitionistic logic is also part of investigations on the logic for pragmatics. This research project, introduced by Dalla Pozza and Garola in [12, 13] and developed in =-=[4, 6, 5]-=-, aims at a formal characterization of the logical properties of illocutionary operators: it is concerned, e.g., with the operations by which we performs the act of asserting a proposition as true, ei... |

11 |
A pragmatic interpretation of intuitionistic propositional logic
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- 1995
(Show Context)
Citation Context ... tell” about our technical results, as our work in dual intuitionistic logic is also part of investigations on the logic for pragmatics. This research project, introduced by Dalla Pozza and Garola in =-=[12, 13]-=- and developed in [4, 6, 5], aims at a formal characterization of the logical properties of illocutionary operators: it is concerned, e.g., with the operations by which we performs the act of assertin... |

11 | Dual intuitionistic logic revisited - Goré - 2000 |

10 | Extension de l’isomorphisme de CurryHoward au traitement des exceptions - Crolard - 1996 |

9 | Categorical proof theory of classical propositional calculus - Bellin, Hyland, et al. |

8 | A Kripke-style semantics for the intuitionistic logic of pragmatics ILP
- Ranalter
- 2003
(Show Context)
Citation Context ... results, as our work in dual intuitionistic logic is also part of investigations on the logic for pragmatics. This research project, introduced by Dalla Pozza and Garola in [12, 13] and developed in =-=[4, 6, 5]-=-, aims at a formal characterization of the logical properties of illocutionary operators: it is concerned, e.g., with the operations by which we performs the act of asserting a proposition as true, ei... |

5 | Chu’s Construction: A Proof-theoretic Approach, Logic for Concurrency and Synchronisation
- Bellin
- 2003
(Show Context)
Citation Context ...us, in game semantics and in other representations of proofs. See also the applications of this concept to proof-nets in [2] and in F. Lamarche’s essential nets, fruitful both for the abstract theory =-=[3]-=- and for applications to complexity theory [17].TERM ASSIGNMENT 11 In the full system NJ, which includes the (ordinary assertive) disjunction ∨, the analysis of branches is subsumed in that of paths ... |

5 |
Pozza. Una logica prammatica per la concezione “espressiva” delle norme
- Dalla
- 1997
(Show Context)
Citation Context ... tell” about our technical results, as our work in dual intuitionistic logic is also part of investigations on the logic for pragmatics. This research project, introduced by Dalla Pozza and Garola in =-=[12, 13]-=- and developed in [4, 6, 5], aims at a formal characterization of the logical properties of illocutionary operators: it is concerned, e.g., with the operations by which we performs the act of assertin... |

4 | Verso una logica degli operatori prammatici asserzioni e congetture, Tesi di Laurea, Facoltà di Scienze, Università di - Biasi - 2003 |

2 |
Fast verification of MLL proof nets via IMLL
- Murawski, Ong
(Show Context)
Citation Context ...ions of proofs. See also the applications of this concept to proof-nets in [2] and in F. Lamarche’s essential nets, fruitful both for the abstract theory [3] and for applications to complexity theory =-=[17]-=-.TERM ASSIGNMENT 11 In the full system NJ, which includes the (ordinary assertive) disjunction ∨, the analysis of branches is subsumed in that of paths ([18], pp. 52-3). Paths extend branches but in ... |