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102
A Judgmental Analysis of Linear Logic
, 2003
"... We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives ..."
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Cited by 61 (33 self)
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We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives, extending dual intuitionistic linear logic but differing from both classical linear logic and Hyland and de Paiva's full intuitionistic linear logic. We also provide a corresponding sequent calculus that admits a simple proof of the admissibility of cut by a single structural induction. Finally, we show how to interpret classical linear logic (with or without the MIX rule) in our system, employing a form of doublenegation translation.
What's so special about Kruskal's Theorem AND THE ORDINAL Γ0? A SURVEY OF SOME RESULTS IN PROOF THEORY
 ANNALS OF PURE AND APPLIED LOGIC, 53 (1991), 199260
, 1991
"... This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, an ..."
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Cited by 54 (2 self)
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This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Veblen hierarchies, some subsystems of secondorder logic, slowgrowing and fastgrowing hierarchies including Girard’s result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the “tree theorem”, as well as a “finite miniaturization ” of Kruskal’s theorem due to Harvey Friedman. These versions of Kruskal’s theorem are remarkable from a prooftheoretic point of view because they are not provable in relatively strong logical systems. They are examples of socalled “natural independence phenomena”, which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal’s tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of KnuthBendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γ0 and Kruskal’s theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence.
Focusing the inverse method for linear logic
 Proceedings of CSL 2005
, 2005
"... 1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10 ..."
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1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10
Typed lambdacalculus in classical ZermeloFraenkel set theory
 ARCHIVE OF MATHEMATICAL LOGIC
, 2001
"... In this paper, we develop a system of typed lambdacalculus for the ZermeloFraenkel set theory, in the framework of classical logic. The first, and the simplest system of typed lambdacalculus is the system of simple types, which uses the intuitionistic propositional calculus, with the only connect ..."
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Cited by 47 (12 self)
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In this paper, we develop a system of typed lambdacalculus for the ZermeloFraenkel set theory, in the framework of classical logic. The first, and the simplest system of typed lambdacalculus is the system of simple types, which uses the intuitionistic propositional calculus, with the only connective #. It is very important, because the well known CurryHoward correspondence between proofs and programs was originally discovered with it, and because it enjoys the normalization property : every typed term is strongly normalizable. It was extended to second order intuitionistic logic, in 1970, by J.Y. Girard[4], under the name of system F, still with the normalization property. More recently, in 1990, the CurryHoward correspondence was extended to classical logic, following Felleisen and Griffin [6] who discovered that the law of Peirce corresponds to control instructions in functional programming
Extracting constructive content from classical logic via controllike reductions
 In Bezem and Groote [12
, 1993
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Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 24 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
Atranslation and Looping Combinators in Pure Type Systems
 Journal of Functional Programming
, 1994
"... We present here a generalization of Atranslation to a class of Pure Type Systems. We apply this translation to give a direct proof of the existence of a looping combinator in a large class of inconsistent type systems, class which includes type systems with a type of all types. This is the first no ..."
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Cited by 12 (1 self)
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We present here a generalization of Atranslation to a class of Pure Type Systems. We apply this translation to give a direct proof of the existence of a looping combinator in a large class of inconsistent type systems, class which includes type systems with a type of all types. This is the first nonautomated solution to this problem.
Intuitionistic Validity in TNormal Kripke Structures
 Ann. Pure Appl. Logic
, 1991
"... Let T be a firstorder theory. A Tnormal Kripke structure is one in which every world is a classical model of T. This paper gives a characterization of the intuitionistic theory T/T of sentences intuitionistically valid (forced) in all Tnormal Kripke structures and proves the corresponding sou ..."
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Cited by 11 (0 self)
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Let T be a firstorder theory. A Tnormal Kripke structure is one in which every world is a classical model of T. This paper gives a characterization of the intuitionistic theory T/T of sentences intuitionistically valid (forced) in all Tnormal Kripke structures and proves the corresponding soundness and completeness theorems. For Peano arithmetic (PA), the theory TtPA is a proper subtheory of Heyting arithmetic (HA), so HA is complete but not sound for PA normal Kripke structures.