## A Relevant Analysis of Natural Deduction (1999)

Venue: | Journal of Logic and Computation |

Citations: | 23 - 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...

### Citations

1115 |
The Lambda Calculus: Its Syntax and Semantics
- Barendregt
- 1981
(Show Context)
Citation Context ... not include the units, but ? and 1 can be added to the type theory with little difficulty. We can define the notion of linear occurrence by extending the general idea of occurrence for the -calculus =-=[7]-=-, though we note that other definitions may be possible. Definition 3.1 (linear occurrence in U) 1. x linearly occurs in x; 2. If x linearly occurs in U or V (or both), then x linearly occurs insy2U :... |

696 | A Framework for Defining Logics
- Harper, Honsel, et al.
- 1993
(Show Context)
Citation Context ...4, 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 manner similar to that of LF =-=[5, 18, 28]-=-, with a representation mechanism. The language of RLF, thes-calculus, is a system of first-order linear dependent function types which uses a functionsto describe the degree of sharing of variables b... |

518 | Lambda calculi with types
- Barendregt
- 1992
(Show Context)
Citation Context ...ng lemma analyses how a type assignment for an abstraction can be obtained. It is a specific (part 4) and specialized (for linearity) case of Barendregt's Generation Lemma for Pure Type Systems (PTS) =-=[8]-=-. Lemma 3.2 (Inversion) If \Gamma ` \Sigma x2A :U :sx2A :V , then \Gamma; x2A ` \Sigma U :V . Proof Consider a derivation of x2A :U :sx2A :V . The conversion rules do not change the term x2A :U . We f... |

303 | D.: Logic programming in a fragment of intuitionistic linear logic
- Hodas, Miller
- 1994
(Show Context)
Citation Context ...lculus 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 manner similar to that of LF [5, 18, 28... |

281 | Computational interpretations of linear logic
- Abramsky
- 1993
(Show Context)
Citation Context ...lculus 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 manner similar to that of LF [5, 18, 28... |

215 | Logical frameworks
- Pfenning
- 2001
(Show Context)
Citation Context ... The case for ? would be similar. 5.2 ML with references Our second encoding is that of the programming language ML extended with references (MLR), a reworking of an example in Cervesato and Pfenning =-=[11, 12]-=-. In our reworking, we exploit the use of theswhich is not available to Cervesato and Pfenning. Consequently, we are in the fulls-calculus type theory, in which 's action is non-trivial. The basic MLR... |

153 |
Entailment: The Logic of Relevance and Necessity
- Anderson, Belnap, et al.
- 1975
(Show Context)
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... |

125 |
On the meanings of the logical constants and the justifications of the logical laws
- Martin-Löf
- 1996
(Show Context)
Citation Context ...s section. 2 In the literature, the terms "sub-structural" and "weak" are sometimes used in this way. 2 One representation mechanism is that of judgements-as-types, which originate=-=s from Martin-Lof's [23]-=- development 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, respectiv... |

121 |
Logic and structure
- Dalen
- 1994
(Show Context)
Citation Context ...ll set of connectives, including disjunctions and existential quantifiers. The language of the logic BI is defined over a structure. The following definition is a slight generalization of van Dalen's =-=[vD94]-=- and of Barendregt's [Bar92]. Definition 33 A structure is a tuple consisting of non-empty sets called sorts, relations, typed functions and constants. Barendregt works exclusively with the example st... |

107 | An algorithm for testing conversion in type theory - Coquand - 1991 |

101 |
Relevant Logic and Entailment
- Dunn
- 1986
(Show Context)
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... |

94 | A Mixed Linear and Non-Linear Logic: Proofs, Terms and Models
- Benton
- 1995
(Show Context)
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... |

87 |
Automated Deduction in Nonclassical Logics
- Wallen
- 1990
(Show Context)
Citation Context ...ygon [HPW96, HM94], although it would be interesting to consider approaches in which [X;G;D] remained unevaluated for as long as possible during search. Such an approach would resemble matrix methods =-=[Wal90]-=-. 2.3.2 Multiple occurrences Consider the multiple occurrences idea from a proposition-as-types reading. Then x:A;x:A can be understood as two uses of the same proof of the same proposition, as oppose... |

83 | Using typed lambda calculus to implement formal systems on a machine
- Avron, Honsell, et al.
- 1987
(Show Context)
Citation Context ...4, 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 manner similar to that of LF =-=[5, 18, 28]-=-, with a representation mechanism. The language of RLF, thes-calculus, is a system of first-order linear dependent function types which uses a functionsto describe the degree of sharing of variables b... |

69 | A Uniform Proof-Theoretic Investigation of Linear Logic Programming
- Harland, Pym
- 1994
(Show Context)
Citation Context ...lculus 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 manner similar to that of LF [5, 18, 28... |

46 |
On intuitionistic linear logic
- Bierman
- 1993
(Show Context)
Citation Context ...ject x!A:M is an inhabitant of the typesx!A :B, which amounts to the Martin-Lof-style \Pi x:A :B. The notion of linear free- and bound-variables (LFV, LBV) and substitution may be defined accordingly =-=[10]-=-. When x is not free in B we write A ( B and A ! B forsx:A :B andsx!A :B, respectively. Our basic study does not include the units, but ? and 1 can be added to the type theory with little difficulty. ... |

46 |
Generating Plans in Linear Logic
- Masseron, Tollu, et al.
- 1991
(Show Context)
Citation Context |

44 | Kripke-style models for typed lambda calculus
- Mitchell, Moggi
- 1991
(Show Context)
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... |

40 | Programming in lygon: An overview
- Harland, Pym, et al.
- 1996
(Show Context)
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 [... |

34 |
The Language Theory of Automath
- Daalen
- 1980
(Show Context)
Citation Context ...of idea, as strengthening trivially holds for the linear parts of the context. However, the argument is complicated and relies on the equivalence between several different type theories of van Daalen =-=[vD80]-=-. The translation argument above is more than adequate for our purposes. 2.3.7 Related systems In this section, we briefly compare the lL-calculus to the appropriate fragments of other linear type the... |

33 |
Semantics for relevant logics
- Urquhart
- 1972
(Show Context)
Citation Context ...n of comparison v between resources, indicating when one resource is better than another, in that it may prove more propositions. Similar ideas can be seen, post hoc, in the relevant logic literature =-=[Urq72]-=-. A resource semantics elegantly explains the difference between the linear and intuitionistic connectives in that the action, or computation, of the linear connectives can be seen to consume resource... |

31 | The semantics and proof theory of linear logic
- Avron
- 1988
(Show Context)
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 ... |

30 |
Relevant logic: A philosophical examination of inference
- Read
- 1988
(Show Context)
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... |

29 |
Proofs of strong normalization
- Gandy
- 1980
(Show Context)
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... |

21 |
Kripke models for linear logic
- Allwein, Dunn
- 1993
(Show Context)
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... |

21 | Logical Frameworks - Huet, Plotkin, et al. - 1991 |

18 |
and Computation in General Logic
- Proofs
- 1990
(Show Context)
Citation Context ...4, 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 manner similar to that of LF =-=[5, 18, 28]-=-, with a representation mechanism. The language of RLF, thes-calculus, is a system of first-order linear dependent function types which uses a functionsto describe the degree of sharing of variables b... |

17 |
Basic Proof Theory,” Number 43
- Troelstra, Schwichtenberg
- 1996
(Show Context)
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 ... |

14 |
Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions
- Streicher
- 1989
(Show Context)
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... |

14 | Encoding modal logics in logical frameworks
- Avron, Honsell, et al.
- 1998
(Show Context)
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... |

10 |
Proofs and the meaning and completeness of the logical constants
- Prawitz
- 1979
(Show Context)
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... |

8 | Generalized rules for quantifiers and the completeness of the intuitionistic operators - Schroeder-Heister - 1984 |

8 | Structural frameworks, substructural logics, and the role of elimination inferences - Schroeder-Heister - 1991 |

8 |
The Logic of Information Structures. Number 681
- Wansing
- 1993
(Show Context)
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... |

4 |
On the adequacy of representing higher order intuitionistic logic as a pure type system
- Tonino, Fujita
- 1992
(Show Context)
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 ... |

4 |
A proof of the Church-Rosser property for the Edinburgh LF with j-conversion
- Salvesen
- 1990
(Show Context)
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... |

3 |
Modal deduction
- Benthem, D'Agostino, et al.
- 1996
(Show Context)
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... |

3 |
A note on representation and semantics in logical frameworks
- Pym
- 1996
(Show Context)
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... |

2 |
A Tarlecki. Structured theory representations and logic representations
- Harper, Sannella
- 1994
(Show Context)
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... |

2 |
A relevant analysis of natural deduction. Lecture at Workshop, EU Espirit Basic Research Action 3245, Logical Frameworks: Design, Implementation and Experiment
- Pym
- 1992
(Show Context)
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 |
Functorial Kripke models of the \Pi-calculus
- Pym
- 1997
(Show Context)
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... |

1 |
Combinators, l-terms and Proof Theory
- Stenlund
- 1972
(Show Context)
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... |

1 |
Immanuel Kants Logik (Edited by G.B
- Kant
- 1988
(Show Context)
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 |
Relevant arithmetic. Polish Academy of Sciences, Institute of Philosophy and Bulletin of the Section of logic, 5:133--137
- Meyer
- 1976
(Show Context)
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 ... |