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Choice in Dynamic Linking
 IN FOSSACS’04  FOUNDATIONS OF SOFTWARE SCIENCE AND COMPUTATION STRUCTURES 2004, LECTURE NOTES IN COMPUTER SCIENCE
, 2004
"... We introduce a computational interpretation for Hilbert's choice operator (#). This interpretation yields a typed foundation for dynamic linking in software systems. The use of choice leads to interesting difficultiessome known from proof theory and others specific to the programminglanguage ..."
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We introduce a computational interpretation for Hilbert's choice operator (#). This interpretation yields a typed foundation for dynamic linking in software systems. The use of choice leads to interesting difficultiessome known from proof theory and others specific to the programminglanguage perspective that we develop. We therefore emphasize an important special case, restricting the nesting of choices. We define
Hilbert’s “Verunglückter Beweis,” the first epsilon theorem and consistency proofs. History and Philosophy of Logic
"... Abstract. On the face of it, Hilbert’s Program was concerned with proving consistency of mathematical systems in a finitary way. This was to be accomplished by showing that that these systems are conservative over finitistically interpretable and obviously sound quantifierfree subsystems. One propo ..."
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Abstract. On the face of it, Hilbert’s Program was concerned with proving consistency of mathematical systems in a finitary way. This was to be accomplished by showing that that these systems are conservative over finitistically interpretable and obviously sound quantifierfree subsystems. One proposed method of giving such proofs is Hilbert’s epsilonsubstitution method. There was, however, a second approach which was not refelected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert’s first epsilon theorem and a certain “general consistency result. ” An analysis of this socalled “failed proof ” lends further support to an interpretation of Hilbert according to which he was expressly concerned with conservatitvity proofs, even though his publications only mention consistency as the main question. §1. Introduction. The aim of Hilbert’s program for consistency proofs in the 1920s is well known: to formalize mathematics, and to give finitistic consistency proofs of these systems and thus to put mathematics on a “secure foundation.” What is perhaps less well known is exactly how Hilbert thought this should be carried out. Over ten years before Gentzen developed sequent calculus formalizations
The computational content of classical arithmetic ∗
, 2009
"... Dedicated to Grigori Mints in honor of his seventieth birthday. Almost from the inception of Hilbert’s program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various m ..."
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Dedicated to Grigori Mints in honor of his seventieth birthday. Almost from the inception of Hilbert’s program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various methods of extracting computational information from proofs in classical firstorder arithmetic, and reflects on some of the relationships between them. Variants of the GödelGentzen doublenegation translation, some not so well known, serve to provide canonical and efficient computational interpretations of that theory. 1
SPECIMENS: “MOST OF ” GENERIC NPS IN A CONTEXTUALLY FLEXIBLE TYPE THEORY
"... Overview This paper proposes to compute the meanings associated to sentences with generic NPs corresponding to the most of generalized quantifier. We call these generics specimens and they resemble stereotypes or prototypes in lexical semantics. The meanings are viewed as logical formulae that can b ..."
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Overview This paper proposes to compute the meanings associated to sentences with generic NPs corresponding to the most of generalized quantifier. We call these generics specimens and they resemble stereotypes or prototypes in lexical semantics. The meanings are viewed as logical formulae that can be thereafter interpreted in your favorite models. We rather depart from the dominant Fregean single untyped universe and go for type theory with hints from Hilbert ε calculus [8, 3] and from medieval philosophy see e.g. [9]. Our type theoretic analysis bears some resemblance with on going work in lexical semantics. [2, 4] Our model also applies to classical examples involving a class (or a generic element of this class) which is provided by the context. An outcome of this study is that, in
Ch. Retoré / Variable types for meaning assembly 1
"... Variable types for meaning assembly: a logical syntax for generic noun phrases introduced by most 1. Overview This paper proposes a way to compute the meanings associated with sentences with generic noun phrases corresponding to the generalized quantifier most. We call these generics specimens and t ..."
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Variable types for meaning assembly: a logical syntax for generic noun phrases introduced by most 1. Overview This paper proposes a way to compute the meanings associated with sentences with generic noun phrases corresponding to the generalized quantifier most. We call these generics specimens and they resemble stereotypes or prototypes in lexical semantics. The meanings are viewed as logical formulae that can thereafter be interpreted in your favourite models. To do so, we depart significantly from the dominant Fregean view with a single untyped universe. Indeed, our proposal adopts type theory with some hints from Hilbert εcalculus (Hilbert, 1922; Avigad and Zach, 2008) and from medieval philosophy, see e.g. de Libera (1993, 1996). Our typeCh. Retoré / Variable types for meaning assembly 2 theoretic analysis bears some resemblance with ongoing work in lexical semantics (Asher 2011; Bassac et al. 2010; Moot, Prévot and Retoré 2011). Our model also applies to classical examples involving a class, or a