## A Relevant Analysis of Natural Deduction (1999)

Venue: | Journal of Logic and Computation |

Citations: | 24 - 7 self |

### BibTeX

@ARTICLE{Ishtiaq99arelevant,

author = {Samin Ishtiaq and Samin Ishtiaq},

title = {A Relevant Analysis of Natural Deduction},

journal = {Journal of Logic and Computation},

year = {1999},

volume = {8},

pages = {8--6}

}

### Years of Citing Articles

### OpenURL

### Abstract

Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and Plotkin's LF, with a representation mechanism: the language of RLF is the lL-calculus; the representation mechanism is judgements-as-types, developed for relevant logics. The lL-calculus type theory is a first-order dependent type theory with two kinds of dependent function spaces: a linear one and an intuitionistic one. We study a natural deduction presentation of the type theory and establish the required proof-theoretic meta-theory. The RLF framework is a conservative extension of LF. We show that RLF uniformly encodes (fragments of) intuitionistic linear logic, Curry's l I -calculus and ML with references. We describe the Curry-Howard-de Bruijn correspondence of the lL-calculus with a s...

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Citation Context ...behaved) normal-order reduction strategy.s3.6 Related systems In this section, we briefly compare thes-calculus to the appropriate fragments of other linear type theories. Abramsky's [1] and Benton's =-=[9]-=- linear type theories are in propositions-as-types correspondence with a propositional ILL. Our concern is with a predicate ILL. Consider a linear version of the Barendregt cube, displayed in so-calle... |

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Citation Context ...pes correspondence and a Gentzenization of thes-calculus. Secondly, a study of the semantics of thes-calculus would bring together and generalize the Kripke models of linear logic and typed -calculus =-=[2, 26]-=-. We note that many characteristics of functorial Kripke models of the \Pi-calculus [31] --- \Pi-formation as right adjoint to weakening, for instance --- are not immediately applicable in our case. W... |

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Citation Context ... \Xi into \Gamma and \Delta". An implementation of thes-calculus as a logic programming language would have to calculate such splittings, perhaps using techniques similar to those for Lolli and L=-=ygon [17, 20]-=-, 11 although it would be interesting to consider approaches in which [\Xi; \Gamma; \Delta] remained unevaluated for as long as possible during search. Such an approach would resemble matrix methods [... |

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Citation Context ...vative extension of the \Pi-calculus and RLF is a conservative extension of LF. 1 Introduction Linear and other relevant logics have been studied widely in mathematical [4, 16, 25, 36], philosophical =-=[3, 14, 33]-=- and computational [1, 20, 24, 32, 37] logic. We present a study of a logical framework, RLF, for defining natural deduction presentations of such logics. 1 RLF consists in a language together, in a m... |

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Citation Context ...gics. Thes-calculus is a conservative extension of the \Pi-calculus and RLF is a conservative extension of LF. 1 Introduction Linear and other relevant logics have been studied widely in mathematical =-=[4, 16, 25, 36]-=-, philosophical [3, 14, 33] and computational [1, 20, 24, 32, 37] logic. We present a study of a logical framework, RLF, for defining natural deduction presentations of such logics. 1 RLF consists in ... |

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Citation Context ...s the translation is faithful, the reduction in the simply-typed -calculus would be infinite too. But this cannot be so, as the simply-typed -calculus with pairing is known to be strongly normalizing =-=[15]-=-. So there cannot be an infinite reduction in thes-calculus. Predicativity arises as a corollary of Theorem 3.2. Finally, we have: Theorem 3.3 (Decidability) All assertions of thes-calculus are decida... |

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Citation Context ...pes correspondence and a Gentzenization of thes-calculus. Secondly, a study of the semantics of thes-calculus would bring together and generalize the Kripke models of linear logic and typed -calculus =-=[2, 26]-=-. We note that many characteristics of functorial Kripke models of the \Pi-calculus [31] --- \Pi-formation as right adjoint to weakening, for instance --- are not immediately applicable in our case. W... |

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Citation Context ...ariables in exactly the same manner. 48 Chapter 2. The lL-calculus and the RLF logical framework We refer also to Troelstra and Schwichtenberg for the technique of strong normalization by translation =-=[TS96]-=-. We will use this technique again when we remark on extending ChurchRosser to h-conversion, in Section 2.3.6 later. We note some minor technicalities to do with the translation. The translation of G ... |

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Citation Context ...ment. For example, the encoding of a fragment of intuitionistic linear logic requires the judgement of (J i = J =) proof . This is in contrast to the general multi-judgement representation techniques =-=[6]-=-. We conjecture that our studies can be applied to the general case, although we defer this development to another occasion. A certain class of uniform representations is identified by considering sur... |

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Citation Context ...tions: it must be preserved on the nose by any f and must behave well under quantification. Details of the treatment of the A:K-fragment in the case of contextual categories are in Streicher's thesis =-=[Str88]-=-. The analogous development in our setting is similar and we omit the details. Definition 53 Let S be a lL-calculus signature. A Kripke resource S-lL model is a 5-tuple hfJ r :[W ; [C op ; Cat]] j r 2... |

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Citation Context ...iring uniformity over all "presentations" of a given logic. Such concerns are beyond our present scope. 3 Prawitz's general form of schematic introductions from a more relevant point of view=-=. Prawitz [27]-=- gives these for intuitionistic logic. A schematic introduction rule for an n-ary sentential operator # is represented by an introduction rule of the form below. In the rule, only the bound assumption... |

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Citation Context ... or none, bound in one action). Relevance logic is defined by adding the combinator S (distribution) to the above set and abstraction is allowed as long as it is "non-vacuous". Independently=-=, Wansing [Wan93]-=- covers the same subject as Gabbay and de Queiroz. Wans3.6. Summary 89 ing's logic is presented in a Hilbert-style and the analysis is more in the spirit of Howard in that it covers the correspondence... |

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Citation Context ...ition-as-types embedding is extended to the entire lcube. Guevers also shows that completeness fails at the lPw--PREDw node of the cube. Barendregt 's completeness theorem refers to Tonino and Fujita =-=[TF92]-=-, who show the correspondence between higher order predicate logic and type theory (Tonino and Fujita's type theory is a subsystem of CC). Abramsky [Abr93], Benton [Ben94] and Barber [Bar97] show the ... |

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Citation Context ... (! M1 ) 1 hM; Ni ! N 0 Table 4: Parallel nested reduction We remark that while fi-conversion is sufficient for our current purposes, we forsee little difficulty (other than that for the \Pi-calculus =-=[13, 34]-=-) in strengthening the definitional equality relationship by the j-rule. 15 3.5 Basic properties of thes-calculus In this section, we summarize the basic properties of thes-calculus, the proofs of whi... |

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Citation Context ...the field can be seen as a programme in van Benthem's notion of logical pluralism, investigating connections between meta-properties of proof calculi and structural properties of systems of deduction =-=[vB93]-=-. This thesis began by stating that logical frameworks provide a foundation for the development of a generic proof assistant, and then went on to develop a logical framework for linear and other relev... |

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Citation Context ...r object-logics. The above prescription can conveniently be summarized by the slogan F ramework = Language + Representation: We remark that these components are not entirely independent of each other =-=[30]. We will point out -=-some interdependencies later in this section. 2 In the literature, the terms "sub-structural" and "weak" are sometimes used in this way. 2 One representation mechanism is that of j... |

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Citation Context ...other occasion. A certain class of uniform representations is identified by considering surjective encodings between consequences of the object-logic ` L and consequences of the meta-logic ` \Sigma L =-=[19]-=-. 3 So, all judgements in the meta-logic have corresponding judgements in the object-logic. The judgement-as-types methodology has the property that encoded systems inherit the structural properties o... |

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Citation Context ...conservative extension of LF; the notion of conservative extension takes account of the representation mechanism as well as the type theory. The work reported here builds on ideas presented by Pym in =-=[29]. An expla-=-nation regarding our use of the word "relevant" is in order. Following Read [33], we use the term relevant for the family of logics which have weaker structural properties than intuitionisti... |

2 |
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Citation Context ...s of thes-calculus would bring together and generalize the Kripke models of linear logic and typed -calculus [2, 26]. We note that many characteristics of functorial Kripke models of the \Pi-calculus =-=[31]-=- --- \Pi-formation as right adjoint to weakening, for instance --- are not immediately applicable in our case. We comment that, besides proof- and model-theoretic semantics, it is also important to st... |

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Citation Context ...(M 0 [N 0 =y])[V 0 =x].s34 Chapter 2. The lL-calculus and the RLF logical framework The proof of confluency for the parallel reduction relation ! is an adaption of the combinatorial proof by Stenlund =-=[Ste72]-=-, who traces it back to Tait and Martin-Lof [ML75]. The combinatorial approach works as the reduction relation (b for Kinds, Types and Objects) is defined disjointly for each of the three levels of te... |

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Citation Context ...ure, the terms "sub-structural" and "weak" are sometimes used in this way. 2 One representation mechanism is that of judgements-as-types, which originates from Martin-Lof's [23] de=-=velopment of Kant's [22]-=- notion of judgement. The two higher-order judgements, the hypothetical J ` J 0 and the general V x2C : J(x), correspond to ordinary and dependent function spaces, respectively. The methodology of jud... |

1 |
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Citation Context ...gics. Thes-calculus is a conservative extension of the \Pi-calculus and RLF is a conservative extension of LF. 1 Introduction Linear and other relevant logics have been studied widely in mathematical =-=[4, 16, 25, 36]-=-, philosophical [3, 14, 33] and computational [1, 20, 24, 32, 37] logic. We present a study of a logical framework, RLF, for defining natural deduction presentations of such logics. 1 RLF consists in ... |