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## A Judgmental Reconstruction of Modal Logic (1999)

Venue: | Mathematical Structures in Computer Science |

Citations: | 183 - 42 self |

### Citations

865 | Notions of computation and monads
- Moggi
- 1991
(Show Context)
Citation Context ...lax modality fl A as 32A and lax implication A ) B as (2A) oe B. Through a computational interpretation of proofs in modal logic we further obtain a new formulation of Moggi's computational -calculus =-=[Mog91]-=-, combining and systematizing previous work by S. Kobayashi [Kob97] and Benton, Bierman, and de Paiva [BBdP98]. At the level of judgments, the above development requires surprisingly few primitive not... |

694 |
Untersuchungen über das Logische Schließen I und II
- Gentzen
- 1935
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Citation Context ... for both A and B. As a rule of inference: A true B true I AsB true where we presuppose that A and B are already known to be propositions. This is known as an introduction rule, a term due to Gentzen =-=[Gen35]-=- who first formulated a system of natural deduction. Conversely, what do we know if we know that AB is true? Since a verification of AB consists of verifications for both A and B, we know that A must ... |

500 | Computational lambda-calculus and monads - Moggi - 1989 |

340 | Logic programming in a fragment of intuitionistic linear logic.
- Hodas, Miller
- 1994
(Show Context)
Citation Context ...ntroduction rule for necessity. Local soundness and completeness can be verified using the substitution principle for validity from Section 4. These rules are well-known from linear logic programming =-=[HM94]-=-, because in linear logic with a modal operator ! (which corresponds to 2 in our setting), goal-directed search is incomplete. Replacing it by the analogue of lax implication avoids this problem and a... |

316 |
Constructive mathematics and computer programming.
- Martin-Löf
- 1979
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Citation Context ...for a proof of A. Proofs are essential in our use of logic in computer science, since they contain constructions and algorithms with computational contents. Therefore Martin-Lof bases his type theory =-=[ML80] on severa-=-l analytical judgments. Again, we simplify 2 and consider "M is a proof term for A" (written M : A). It is important that M contain enough information to reconstruct the evidence for A true ... |

212 | A modal analysis of staged computation. - Davies, Pfenning - 2001 |

158 |
Ideas and Results in Proof Theory. In
- Prawitz
- 1971
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Citation Context ...ht to left the theorem follows by case analysis on local or commuting reductions. 24 The calculus further satisfies a normalization theorem. This can be proven either directly via Tait's method as in =-=[Pra71]-=-, by a detour via cut elimination as in [Pfe99], via CPS translation as in [dG99], or via an interpretation into a simply-typed lambda-calculus with disjunction [BBdP98]. The latter is in many ways th... |

143 |
On the meaning of the logical constants and the justification of the logical laws.
- Martin-Lof
- 1996
(Show Context)
Citation Context ...ce Carnegie Mellon University December 1999 1 Introduction In this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions =-=[ML85]-=-. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deduction for intuitionistic modal logic which does not e... |

129 | Primitive recursion for higher-order abstract syntax
- Despeyroux, Pfenning, et al.
- 1997
(Show Context)
Citation Context ...o retain extensionality as given by the local expansions, we can obtain a calculus suitable as a meta-logical framework, that is, a logical framework in which we can reason about the specified logics =-=[DPS97]-=-. The modal operator here serves to avoid the usual paradoxes which would arise if we incorrectly identify an expression with its denotation. In this paper we have also shown how lax logic can be embe... |

93 | Using Typed Lambda Calculus to Implement Formal Systems on a Machine. - Avron, Honsell, et al. - 1992 |

93 | Natural Deduction - Prawitz - 1965 |

66 | Propositional lax logic
- Fairtlough, Mendler
- 1997
(Show Context)
Citation Context ...exercise yields a simple and uniform system of natural deduction for intuitionistic modal logic which does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic =-=[FM97]-=- and find that it is already contained in modal logic, using the decomposition of the lax modality fl A as 32A and lax implication A ) B as (2A) oe B. Through a computational interpretation of proofs ... |

64 | A reflection on call-by-value - Sabry, Wadler - 1997 |

61 | Computational types from a logical perspective.
- Benton, Bierman, et al.
- 1998
(Show Context)
Citation Context ...oofs in modal logic we further obtain a new formulation of Moggi's computational -calculus [Mog91], combining and systematizing previous work by S. Kobayashi [Kob97] and Benton, Bierman, and de Paiva =-=[BBdP98]-=-. At the level of judgments, the above development requires surprisingly few primitive notions. In particular, we only need hypothetical judgments to explain implication, and categorical judgments to ... |

47 |
Labelled Non-classical Logics
- Viganò
- 2000
(Show Context)
Citation Context ...rbose and requiring explicit reasoning about worlds, this approach is also more flexible in that various traditional modal logics can be expressed simply by varying the reachability judgment. Vigan`o =-=[Vig97]-=- has conducted a systematic study of modal logic via Kripke structures from the point of view of logical frameworks. In certain cases this can be simplified to obtain a formulation of natural deductio... |

42 | Structural cut elimination in linear logic
- Pfenning
- 1994
(Show Context)
Citation Context ...ween natural deduction and the sequent calculus as in [Pfe99], together with a cut elimination argument for modal logic in a formulation based on judgments very similar to the one for linear logic in =-=[Pfe94]-=-. At present, we have not verified the details of such a construction. 10 Conclusion We have presented a judgmental reconstruction of the modal logic of necessity and possibility, leading to a clean a... |

42 | Natural Deduction. Almquist - Prawitz - 1965 |

31 | Masini,“A computational interpretation of modal proofs”, - Martini, A - 1994 |

25 | Intuitionistic necessity revisited”, - Bierman, Paiva - 1992 |

24 | Modal types as staging specifications for run-time code generation. ACM computing surveys - Wickline, Lee, et al. - 1998 |

20 |
An intuitionistic modal logic with applications to the formal veri of hardware
- Fairtlough, Mendler
- 1995
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Citation Context ...let boxu = x in dia (let dia z = y in u z) : 2(A oe B) oe(3A oe 3B) 7 Lax Logic Lax logic [FM97] is an intuitionistic logic with a single modal operator fl . It was motivated by hardware verification =-=[FM94]-=- and has found applications in the foundations of constraint logic programming [FMW97]. It has also been related to the computational -calculus [BBdP98], which we will examine in the next section, and... |

19 |
Monad as modality
- Kobayashi
- 1997
(Show Context)
Citation Context ...rough a computational interpretation of proofs in modal logic we further obtain a new formulation of Moggi's computational -calculus [Mog91], combining and systematizing previous work by S. Kobayashi =-=[Kob97]-=- and Benton, Bierman, and de Paiva [BBdP98]. At the level of judgments, the above development requires surprisingly few primitive notions. In particular, we only need hypothetical judgments to explain... |

18 |
Analytic and synthetic judgements in type theory
- Martin-Löf
- 1994
(Show Context)
Citation Context ...A oe 3A true ` 33A oe 3A true ` 2(A oe B) oe(3A oe 3B) true Natural deductions for these axioms are given in abbreviated form as proof terms in the next section. 6 Analytic and Synthetic Judgments In =-=[ML94]-=-, Martin-Lof reviews the notions of analytic and synthetic judgments as analyzed by Kant. He states: [: : :] an analytic judgement is one which is evident in virtue of the meanings of the terms that o... |

17 | and Valeria de Paiva. Intuitionistic necessity revisited - Bierman - 1992 |

15 | The Russel-Prawitz modality
- Aczel
- 1999
(Show Context)
Citation Context ...nstraint logic programming [FMW97]. It has also been related to the computational -calculus [BBdP98], which we will examine in the next section, and to higher-order definitions of logical connectives =-=[Acz99]-=-. We develop here the fragment of lax logic containing implication A ) B and the lax modality fl A. We use a different notation for implication than in modal logic, so we may later give the connective... |

14 | First-order lax logic as a framework for constraint logic programming
- Fairtlough, Mendler, et al.
- 1997
(Show Context)
Citation Context ...ogic [FM97] is an intuitionistic logic with a single modal operator fl . It was motivated by hardware verification [FM94] and has found applications in the foundations of constraint logic programming =-=[FMW97]-=-. It has also been related to the computational -calculus [BBdP98], which we will examine in the next section, and to higher-order definitions of logical connectives [Acz99]. We develop here the fragm... |

14 | On a modal -calculus for S4 - Pfenning, Wong - 1995 |

6 |
On the strong normalization of natural deduction with permutation-conversions
- Groote
- 2005
(Show Context)
Citation Context ...24 The calculus further satisfies a normalization theorem. This can be proven either directly via Tait's method as in [Pra71], by a detour via cut elimination as in [Pfe99], via CPS translation as in =-=[dG99]-=-, or via an interpretation into a simply-typed lambda-calculus with disjunction [BBdP98]. The latter is in many ways the simplest and easily extends to additional connectives. We map both 2A and 3A as... |

5 | On computational interpretations of the modal logic S4, part IIIb - Goubault-Larrecq - 1997 |

3 | Relating categorical and Kripke semantics for intuitionistic modal logics - Alechina, Paiva, et al. - 1998 |

1 | Relating categorical and kripke semantics for intuitionistic modal logics - Alechine, Paiva, et al. - 1998 |

1 | A modal analysis of staged computation. Submitted. Available as - Davies, Pfenning - 1999 |

1 |
Automated theorem proving. Unpublished lecture notes
- Pfenning
- 1999
(Show Context)
Citation Context ... on local or commuting reductions. 24 The calculus further satisfies a normalization theorem. This can be proven either directly via Tait's method as in [Pra71], by a detour via cut elimination as in =-=[Pfe99]-=-, via CPS translation as in [dG99], or via an interpretation into a simply-typed lambda-calculus with disjunction [BBdP98]. The latter is in many ways the simplest and easily extends to additional con... |

1 | Encoding Logical Theories of Programs. PhD thesis TD-7/97, Dipartimento di Informatica, Universit`a di - Miculan - 1997 |