Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809-838, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... Moreover, proofs are rei ed as objects which allows properties of or relations between proofs to be expressed within the framework [Pfe91] Representations of systems involving state remained cumbersome until the design of the linear logical framework LLF [CP98] and its close relative RLF [IP98]. For example, LLF allows an elegant representation of Mini ML with mutable references that rei es imperative computations as objects. LLF is a conservative extension of LF with the linear function type A B, the additive product type A B, and the additive unit type . This type theory ....

Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809-838, 1998.

....particular, we can achieve a bijection between hypothetical deductions of a judgment and canonical objects of the corresponding type. Representations in this style of systems involving state remained cumbersome until the design of the linear logical framework LLF [CP98] and its close relative RLF [IP98]. LLF is a conservative extension of LF with selected constructs from linear logic. The representation principles behind LLF are state as linear hypotheses and imperative computations as linear functions . Again, we can achieve a bijection between imperative computations of a program and ....

Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809-838, 1998.

....also the simply typed variant of the term language of LLF. Its theoretical relevance derives from the fact that it is the biggest linear calculus that admits unique long normal forms. shares similarities with the calculus proposed in [Bar96] and with the term language of the system RLF [IP98]. The implementation of a language based on linear type theories such as LLF and RLF raises new challenges that do not emerge neither for non linear languages such as Twelf [PS99] nor in linear logic programming languages featuring plain (non linear) terms such as Lolli [HM94] or Forum [Mil94] ....

....representation for unification and normalization over the linear expressions that can appear in an LLF specification. The adoption of linear term languages in LLF and RLF has been motivated by a number of applications. Linear terms provide a statically checkable notation for natural deductions [IP98] or sequent derivations [CP96] in substructural logics. In the realm of programming languages, linear terms naturally model computations in imperative languages [CP96] or sequences of moves in games [Cer96] When we want to specify, manipulate, or reason about such objects (which is common in ....

Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

.... Moreover, proofs are rei ed as objects which allows properties of or relations between proofs to be expressed within the framework [Pfe91] Representations of systems involving state remained cumbersome until the design of the linear logical framework LLF [CP98] and its close relative RLF [IP98]. For example, LLF allows an elegant representation of Mini ML with mutable references that rei es imperative computations as objects. LLF is a conservative extension of LF with the linear function type A B, the additive product type A B, and the additive unit type . This type theory ....

Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809-838, 1998.

.... Moreover, proofs are reified as objects which allows properties of or relations between proofs to be expressed within the framework [Pfe91] Representations of systems involving state remained cumbersome until the design of the linear logical framework LLF [CP98] and its close relative RLF [IP98]. For example, LLF allows an elegant representation of Mini ML with mutable references that reifies imperative computations as objects. LLF is a conservative extension of LF with the linear function type B, the additive product type A B, and the additive unit type #. This type theory ....

Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

.... Moreover, proofs are reified as objects which allows properties of or relations between proofs to be expressed within the framework [Pfe91] Representations of systems involving state remained cumbersome until the design of the linear logical framework LLF [CP98] and its close relative RLF [IP98]. For example, LLF allows an elegant representation of Mini ML with mutable references that reifies imperative computations as objects. LLF is a conservative extension of LF with the linear function type B, the additive product type A B, and the additive unit type #. This type theory ....

Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

....particular, we can achieve a bijection between hypothetical deductions of a judgment and canonical objects of the corresponding type. Representations in this style of systems involving state remained cumbersome until the design of the linear logical framework LLF [CP98] and its close relative RLF [IP98]. LLF is a conservative extension of LF with selected constructs from linear logic. The representation principles behind LLF are state as linear hypotheses and imperative computations as linear functions . Again, we can achieve a bijection between imperative computations of a program and ....

Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

....a small monoidal category, in which the intuitionistic dependent function space is described in the established way, but the linear dependent function space is described using Day s tensor product. 1 Introduction A long standing problem has been to combine type dependency and linearity. In [13], we introduced the lL calculus, a first order dependent type theory with a full linear dependent function space, as well as the usual intuitionistic dependent function space. The lL calculus can be seen to arise in two ways. Firstly, in logical frameworks [9, 18] in which it provides a language ....

....BI [15, 19] in which the antecedents of sequents are structured not as lists but as bunches, which have two combining operations, which admits Weakening and Contraction, and , which does not. The lL calculus stands in propositions as types correspondence with a fragment of BI [13, 12]. The purpose of this paper is to present the categorical semantics of the lL calculus. This is given by Kripke resource models, which are monoid indexed sets of functorial Kripke models. The indexing element can be seen as the resource able to realize the structure it indexes. We work with ....

[Article contains additional citation context not shown here]

SS Ishtiaq and DJ Pym. A Relevant Analysis of Natural Deduction. J. Logic Computat., 8(6):809--838, 1998.

....BI [15, 19] in which the antecedents of sequents are structured not as lists but as bunches, which have two combining operations, which admits Weakening and Contraction, and , which does not. The lL calculus stands in propositions as types correspondence with a fragment of BI [13, 12]. The purpose of this paper is to present the categorical semantics of the lL calculus. This is given by Kripke resource models, which are monoid indexed sets of functorial Kripke models. The indexing element can be seen as the resource able to realize the structure it indexes. We work with ....

....function is the obvious one in which a term (type) is interpreted by the class of terms (types) definitionally equivalent to the term (type) in the appropriate component of the structure. The satisfaction relation is given by provability in the type theory. Details of the proof are in [12]. 4 A Class of Set theoretic Models We describe a class of set theoretic Kripke resource models, in which the Kripke resource lL structure fJ r : W ; C ; Cat] j r 2 Rg is given by BIFam: C ; Ctx ; Set] where C is a small monoidal category and Ctx is a small set theoretic category of ....

[Article contains additional citation context not shown here]

S Ishtiaq. A Relevant Analysis of Natural Deduction. PhD thesis, Queen Mary & Westfield College, University of London, 1999.

....the simply typed variant of the term language of LLF. Its theoretical relevance derives from the fact that it is the biggest linear calculus that admits unique long normal forms. shares similarities with the calculus proposed in [Bar96] and with the term language of the system RLF [IP98]. The implementation of a language based on linear type theories such as LLF and RLF raises new challenges that do not emerge neither for non linear languages such as Twelf [PS99] nor in linear logic programming languages featuring plain (non linear) terms such as Lolli [HM94] or Forum [Mil94] ....

....representation for unification and normalization over the linear expressions that can appear in an LLF specification. The adoption of linear term languages in LLF and RLF has been motivated by a number of applications. Linear terms provide a statically checkable notation for natural deductions [IP98] or sequent derivations [CP96] in substructural logics. In the realm of programming languages, linear terms naturally model computations in imperative languages [CP96] or sequences of moves in games [Cer96] When we want to specify, manipulate, or reason about such objects (which is common in ....

Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

....one 3 A CCC is bi cartesian closed if it is also bi cartesian, i.e. has co products as well as products. 4 X Y 6 F (X) X F (Y ) Y ( Fig. 2. Fibred Models of Proofs of which is bi cartesian. At the predicate level, the analysis is less clear. The calculus [14,13,15,16] is a dependently typed calculus which provides a partial analysis, being both in the spirit of BI and yet somewhat reliant on the presence of a form of Dereliction. Nevertheless, can be interpreted in the general bred framework sketched in Figure 2; Classical logic: the and ....

Ishtiaq, S., \A Relevant Analysis of Natural Deduction", Ph.D. thesis, Department of Computer Science, Queen Mary and Westeld College, University of London, 1999. 16

....This richer sequential structure allows additive and multiplicative implications to live side by side, without recourse to linear logic s exponentials. Propositional BI s proof objects are characterized by the calculus. Predicate BI s proof objects require a dependently typed calculus [86]. These examples illustrate that for a given logic, it is not always evident how to de ne proof representations that are type theoretic. In this setting, the proposals of new logics, to deal with new problems and applications, is strongly connected to the design of new calculi to express ....

.... More recently, the correspondence has been extended to classical propositional 8 and predicate logic by Parigot [123, 124, 101] to propositional intuitionistic linear logic (see [166] and to a bunched logic, combining linear and intuitionistic predicate logics [116, 147] by Ishtiaq and Pym [86]. A good view of the propositions as types correspondence for minimal intuitionistic logic is given by the cube [14] in which are represented eight calculi ( a la Church) i , covering the possible dependencies between terms and types (terms depending on terms or types and types depending ....

[Article contains additional citation context not shown here]

S.S. Ishtiaq and D.J. Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809-838, 1998.

....in logical frameworks enable a more exact representation of the semantics of languages with state, and it stands to reason that the same should hold true for program logics as well. Important advances include LLF [10] which incorporates linear (non dependent) function types, and the RLF system [21], which has even linear dependent function types. The problems in this area are dicult, but progress could prove especially useful in controlling sharing relationships between di erent parts of a speci cation or encoding. Interestingly, in a precursor of bunched typing, RLF was formulated using ....

....could prove especially useful in controlling sharing relationships between di erent parts of a speci cation or encoding. Interestingly, in a precursor of bunched typing, RLF was formulated using two forms of context extension, one corresponding to linear assumptions and the other to non linear [21]. It did this, however, with an elimination rule for additive function types that ts more closely with linear typing than the bunched typing considered here and in [34, 39] there were required to be no linear assumptions in the the argument for additive function elimination (compare to the rule ....

S.S. Ishtiaq and D.J. Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809-838, 1998.

....[CHP96] In particular, the instantiation of logical variables relies on the traditional unification algorithms, in their first or higher order variants, depending on the language. More recent proposals, such as the language of the linear logical framework LLF [Cer96, CP96] and the system RLF [IP96], introduce linearity not only at the level of formulas, but also within terms. Consequently, implementations of these languages must solve higher order equations on linear terms in order to instantiate existential variables. In this paper we present a complete algorithm for pre unification in a ....

....would be unnecessary in LLF . The problem representation would therefore be more direct and compact in this language. The introduction of linear term languages in LLF and RLF has been motivated by a number of applications. Linear terms provide a statically checkable notation for natural deductions [IP96] or sequent derivations [CP96] in substructural logics. In the realm of programming languages, linear terms naturally model computations in imperative languages [CP96] or sequences of moves in games [Cer96] When we want to specify, manipulate, or reason about such objects (which is common in ....

Samin Ishtiaq and David Pym. A relevant analysis of natural deduction, December 1996. Manuscript.

....formulas [3] In particular, the instantiation of logical variables relies on the traditional unification algorithms, in their first or higherorder variants, depending on the language. More recent proposals, such as the language of the linear logical framework LLF [2, 4] and the system RLF [17], introduce linearity not only at the level of formulas, but also within terms. Consequently, implementations of these languages must solve higher order equations on linear terms in order to instantiate existential variables. In this paper we present a complete algorithm This work was supported ....

....on rewriting contexts in the logical framework in LF is unsound, while it would be adequately represented in LLF . The introduction of linear term languages in LLF and RLF has been motivated by a number of applications. Linear terms provide a statically checkable notation for natural deductions [17] or sequent derivations [4] in substructural logics. In the realm of programming languages, linear terms naturally model computations in imperative languages [4] or sequences Types: A : a Terms: M : c j x j A 1 A 2 j x : A: M j M 1 M 2 (intuitionistic functions) j A 1 Gammaffi A 2 j ....

S. Ishtiaq and D. Pym. A relevant analysis of natural deduction, Dec. 1996. Manuscript.

....allocation and deallocation in the previous section is also from [46, 22] but for a simplified model where locations or names do not have associated contents. A related example, presented from the point of view of a dependently typed calculus which is intimately related to BI, can be found in [23]. The inclusion of pointers brings out several issues, most importantly sharing. That is, data structures are often constructed so that there are two or more pointers to the same cell, as happens when considering graphs or circular or doubly linked lists. When this happens, there are multiple ....

S.S. Ishtiaq and D.J. Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809-- 838, 1998.

....and deallocation 18 in the previous section is also from [47, 22] but for a simplified model where locations or names do not have associated contents. A related example, presented from the point of view of a dependently typed calculus which is intimately related to BI, can be found in [23]. The inclusion of pointers brings out several issues, most importantly sharing. That is, data structures are often constructed so that there are two or more pointers to the same cell, as happens when considering graphs or circular or doubly linked lists. When this happens, there are multiple ....

S.S. Ishtiaq and D.J. Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809-- 838, 1998.

....in the established way, but the linear dependent function space is described using Day s tensor product. Keywords: dependent type theory, categorical semantics, logical frameworks, sub structural logics 1 Introduction A long standing problem has been to combine type dependency and linearity. In [16], the present authors introduced the calculus, a rst order dependent type theory with a full linear dependent function space, as well as the usual intuitionistic dependent function space. The calculus can be seen to arise in two ways: in Logical Frameworks and in Bunched Logic. 1 Logical ....

....is X ; ILL M :J( By uniformity, the latter is the image of an object logic consequence (X) ILL 0 :J( which implies weakening in linear logic, a contradiction. This structural strength excludes from LF s scope object logics involving the notions of intension and state. In [16], the present authors present a language in which such weakening and contraction are not forced. The connectives of such a language are motivated by studying the natural deduction form of rules for relevant logics. This is done quite generally, by considering Prawitz s general form of schematic ....

[Article contains additional citation context not shown here]

SS Ishtiaq and DJ Pym. A Relevant Analysis of Natural Deduction. Journal of Logic and Computation, 8(6):809-838, 1998. 46

....is given by the following table: BI ( 8 8 new : One view of this correspondence is, then, that the RLF meta logic uses this fragment of BI, just as the LF meta logic uses the f ; 8g fragment of Intuitionistic Logic. A detailed account of the correspondence is given in [12, 15]. It remains a challenging and open problem to give a systematic analysis of the relationship between substructural logics and dependent type theories. In particular, it remains to formulate a dependent type theory in correspondence to a proper fragment of BI. 2.8 An algebraic presentation In ....

S Ishtiaq. A Relevant Analysis of Natural Deduction. PhD thesis, Queen Mary and Westeld College, University of London, 1999.

....one 3 A CCC is bi cartesian closed if it is also bi cartesian, i.e. has co products as well as products. 4 X Y 6 F (X) X F (Y ) Y ( Fig. 2. Fibred Models of Proofs of which is bi cartesian. At the predicate level, the analysis is less clear. The calculus [14,13,15,16] is a dependently typed calculus which provides a partial analysis, being both in the spirit of BI and yet somewhat reliant on the presence of a form of Dereliction. Nevertheless, can be interpreted in the general bred framework sketched in Figure 2; Classical logic: the and ....

Ishtiaq, S. and D. Pym, A Relevant Analysis of Natural Deduction, Journal of Logic and Computation 8(6) 809-839 1998.

....the semantics of storage, where sharing rather than duplication is the major concern [12, 13, 14, 15] The demands of the specific models there require a logic different from extant linear or relevant logics. A similar view, from a type theoretic, or logical frameworks, perspective, can be seen in [16, 17]. The other, more foundational, motivation comes from a proposal of resource as a primitive concept, among others, in the emerging discipline of informatics, which can be defined as the science of the structure, complexity and communication of information [3] The treatment of resource in our ....

....pointers to other cons cells. The model presented in this section is from work on using BI to reason about pointers [31] which builds on work of Reynolds [32] A related example, presented from the point of view of a dependently typed calculus which is intimately related to BI, can be found in [16]. The inclusion of pointers brings out several issues, most importantly sharing. That is, data structures are often constructed so that there are two or more pointers to the same cell, as happens when considering graphs or circular or doubly linked lists. When this happens, there are multiple ....

[Article contains additional citation context not shown here]

S.S. Ishtiaq and D.J. Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

....of Computer Science Queen Mary and West eld College University of London fsi,pymg dcs.qmw.ac. uk October 12, 2000 Abstract This document contains corrections to errors discovered to date in, and also some remarks upon, both Samin Ishtiaq s thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. 1 Introduction This document contains corrections to errors discovered to date in both Samin Ishtiaq s thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, ....

....A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. 1 Introduction This document contains corrections to errors discovered to date in both Samin Ishtiaq s thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. The postscript version of the thesis, available http: www.dcs.qmw.ac.uk si, is the source of the hard bound copies submitted to the libraries of the University of London (Senate House) and Queen Mary and ....

[Article contains additional citation context not shown here]

S Ishtiaq. A Relevant Analysis of Natural Deduction. PhD thesis, Queen Mary and Westeld College, University of London, 1999.

....Queen Mary and West eld College University of London fsi,pymg dcs.qmw.ac. uk October 12, 2000 Abstract This document contains corrections to errors discovered to date in, and also some remarks upon, both Samin Ishtiaq s thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. 1 Introduction This document contains corrections to errors discovered to date in both Samin Ishtiaq s thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, ....

....of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. 1 Introduction This document contains corrections to errors discovered to date in both Samin Ishtiaq s thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. The postscript version of the thesis, available http: www.dcs.qmw.ac.uk si, is the source of the hard bound copies submitted to the libraries of the University of London (Senate House) and Queen Mary and West eld College. A copy of ....

[Article contains additional citation context not shown here]

SS Ishtiaq and DJ Pym. A Relevant Analysis of Natural Deduction. Journal of Logic and Computation, 8(6):809-838, 1998.

....the semantics of storage, where sharing rather than duplication is the major concern [11, 12, 13, 14] The demands of the specific models there require a logic different from extant linear or relevant logics. A similar view, from a type theoretic, or logical frameworks, perspective, can be seen in [15, 16]. In this paper, the spatial view of semantics arises in the more abstract setting of topological models of BI (q.v. x 5) The other, more foundational, motivation comes from a proposal of resource as a primitive concept, among others, in the emerging discipline of informatics, which can be ....

....pointers to other cons cells. The model presented in this section is from work on using BI to reason about pointers [23] which builds on work of Reynolds [24] A related example, presented from the point of view of a dependently typed calculus which is intimately related to BI, can be found in [15]. The inclusion of pointers brings out several issues, most importantly sharing. That is, data structures are often constructed so that there are two or more pointers to the same cell, as happens when considering graphs or circular or doubly linked lists. When this happens, there are multiple ....

[Article contains additional citation context not shown here]

S.S. Ishtiaq and D.J. Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

....very challenging: structural control is required on the level of individuals rather than just propositions. A more principled logical approach would be most welcome, and seems a good test problem for the multiplicative approaches to dependent types and predication described by Ishtiaq and Pym [14, 25]. The reader might have been bemused by our apparently devious use of dangling pointers in the treatment of frame axioms. The local nature of a speci cation fPgCfQg comes about from the fact that if C tried to alter a cons cell not guaranteed to exist by P then we could contradict the speci ....

Ishtiaq, S., and Pym, D. A relevant analysis of natural deduction. Journal of Logic and Computation 8(6) (1998), 809-838.

....over a small monoidal category, in which the intuitionistic dependent function space is described in the established way, but the linear dependent function space is described using Day s tensor product. 1 Introduction A long standing problem has been to combine type dependency and linearity. In [13], the present authors introduced the calculus, a first order dependent type theory with a full linear dependent function space, as well as the usual intuitionistic dependent function space. The calculus can be seen to arise in two ways: in Logical Frameworks and in Bunched Logic. Logical ....

....M ffi :J(OE) By uniformity, the latter is the image of an object logic consequence (X ) Delta; Theta ILL ffi 0 :J(OE) which implies weakening in linear logic, a contradiction. This structural strength excludes from LF s scope object logics involving the notions of intension and state. In [13], the present authors present a language in which such weakening and contraction are not forced. The connectives of such a language are motivated by studying the natural deduction form of rules for relevant logics. This is done quite generally, by considering Prawitz s general form of schematic ....

[Article contains additional citation context not shown here]

SS Ishtiaq and DJ Pym. A Relevant Analysis of Natural Deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

....following table: BI ( 8 Gamma Gamma 8 new Gamma : Gamma One view of this correspondence is, then, that the RLF meta logic uses this fragment of BI, just as the LF meta logic uses the f ; 8g fragment of Intuitionistic Logic. A detailed account of the correspondence is given in [12]. 2.8 An algebraic presentation In preparation for our presentation of a categorical semantics of the calculus in general and, in particular, for the completeness argument later, we give an algebraic presentation of the calculus type theory. The idea is to consider provably well formed ....

S Ishtiaq. A Relevant Analysis of Natural Deduction. PhD thesis (submitted), Queen Mary and Westfield College, University of London, 1999.

....This richer sequential structure allows additive and multiplicative implications to live side by side, without recourse to linear logic s exponentials. Propositional BI s proof objects are characterized by the ff calculus. Predicate BI s proof objects require a dependently typed calculus [86]. These examples illustrate that for a given logic, it is not always evident how to define proof representations that are type theoretic. In this setting, the proposals of new logics, to deal with new problems and applications, is strongly connected to the design of new calculi tp express ....

.... [14] More recently, the correspondence has been extended to classical propositional and predicate logic by Parigot [123, 124, 101] to propositional intuitionistic linear logic (see [166] and to a bunched logic, combining linear and intuitionistic predicate logics [116, 147] by Ishtiaq and Pym [86]. A good view of the propositions as types correspondence for minimal intuitionistic logic is given by the cube [14] in which are represented eight calculi ( a la Church) i , covering the possible dependencies between terms and types (terms depending on terms or types and types depending on ....

[Article contains additional citation context not shown here]

S.S. Ishtiaq and D.J. Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

....established way, but the linear dependent function space is described using Day s tensor product. Keywords: Type theory, categorical models and logics, linear logic, relevant logic, Logical Frameworks. 1 Introduction A long standing problem has been to combine typedependency and linearity. In [12], we introduced the calculus, a first order dependent type theory with a full linear dependent function space, as well as the usual intuitionistic dependent function space. The calculus can be seen to arise in two ways. Logical frameworks. Logical frameworks are formal meta logics which, inter ....

....only logics which also admit these structurals, thereby excluding from its scope object logics involving the notions of intension and state. Whilst it is true that systems such as linear logic and certain program logics can be coded in the Pi calculus, they cannot be represented uniformly [12, 10]. By analysing the form of the rules of relevant natural deduction, we obtain a meta language, the calculus, with the f ; Pi; g set of connectives which is complete for defining this class of inference rules. The RLF logical framework consists of the calculus together with the ....

[Article contains additional citation context not shown here]

SS Ishtiaq and DJ Pym. A Relevant Analysis of Natural Deduction. Journal of Logic and Computation, 8(6):809-- 838, 1998.

....as an infinite additive conjunction. Rather, 8new is closely allied to the multiplicative implication Gamma , although a version allied to multiplicative conjunction is possible under restricted circumstances. It would be interesting to formulate a dependent function type, along the lines of [21], which generalizes both of them. We now extend the Kripke Resource Semantics to quantifiers. To do this, we must first define a notion of environment , which specifies a binding of variables to individuals. For this, we suppose that we are given a functor D in Set M ; this functor is the ....

....have not been able to locate a worked out predicate logic which has multiplicative quantifiers. That is, apart from Ambler s system for the existential only [2] the formulation of which is somewhat more complex than ours. Also relevant is the theory of multiplicative dependent function types in [21], together with its fibrational semantics [20] incidentally, it can be regarded as relying on a version of bunches appropriate to dependent type theory. Acknowledgments We are grateful to Pablo Armelin, Guy McCusker, Peter Schroeder Heister and Edmund Robinson for useful discussions about this ....

S.S. Ishtiaq and D.J. Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

....as an infinite additive conjunction. Rather, 8 new is closely allied to the multiplicative implication Gamma , although a version allied to multiplicative conjunction is possible under restricted circumstances. It would be interesting to formulate a dependent function type, along the lines of [10], which generalizes both of them. For the additive universal quantifier one might have expected here that the elimination rule would be formulated as (X) Gamma 8x : OE X t : Term (X) Gamma OE[t=x] 8E Such a rule is admissible if we have Contraction for ; on the level of terms. Similar ....

S.S. Ishtiaq and D.J. Pym. A relevant analysis of natural deduction. To appear (1998) in Journal of Logic and Computation.

....as an infinite additive conjunction. Rather, 8new is closely allied to the multiplicative implication Gamma , although a version allied to multiplicative conjunction is possible under restricted circumstances. It would be interesting to formulate a dependent function type, along the lines of [15], which generalizes both of them. For the additive universal quantifier, one might expect the elimination rule to be (X) Gamma 8x : OE X t : Term (X) Gamma OE[t=x] 8E Such a rule is admissible if we have Contraction for ; on the level of terms. Similar considerations apply to the ....

....the years, we have not been able to locate a worked out predicate logic with multiplicative quantifiers, apart from Ambler s system for the existential only [2] the formulation of which is somewhat more complex than ours. Also relevant is the theory of multiplicative dependent function types in [15], together with its fibrational semantics [14] it can be regarded as relying on a version of bunches appropriate to dependent type theory. Linear logic has a number of computational readings, including: the number of uses reading, based on the original coherence space model; an eager and lazy ....

S.S. Ishtiaq and D.J. Pym. A relevant analysis of natural deduction. J. Logic Computat. 8(6):809-838, 1998.

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809-838, 1998.

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

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Ishtiaq, S. and D. Pym, A relevant analysis of natural deduction, Journal of Logic and Computation 8 (1998), pp. 809--838.

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998. 32

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction, December 1996. Unpublished.

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S. Ishtiaq and D. Pym. A relevant analysis of natural deduction, Dec. 1996. Manuscript.

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998. 32

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction, December 1996. Manuscript.

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction, December 1996. Manuscript.

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

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Ishtiaq, S. and D. Pym, A relevant analysis of natural deduction, Journal of Logic and Computation 8 (1998), pp. 809--838.

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

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S.S. Ishtiaq and D. J. Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809-838, 1998.

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998. 32

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Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809--838, 1998.

No context found.

Samin Ishtiaq and David Pym. A relevant analysis of natural deduction. Journal of Logic and Computation, 8(6):809-838, 1998.

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