Results 1  10
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13
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a pla ..."
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
Rescorla, Quarterly
 Journal of Experimental Psychology
, 2003
"... Abstract. We show that Shoenfield’s functional interpretation of Peano arithmetic can be factorized as a negative translation due to J. L. Krivine followed by Gödel’s Dialectica interpretation. Mathematics Subject Classification: 03F03, 03F10. ..."
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Abstract. We show that Shoenfield’s functional interpretation of Peano arithmetic can be factorized as a negative translation due to J. L. Krivine followed by Gödel’s Dialectica interpretation. Mathematics Subject Classification: 03F03, 03F10.
A Note On The GödelGentzen Translation
, 1998
"... . We give a variant of the GodelGentzennegative translation, and a syntactic characterization which entails conservativity result for formulas. The GodelGentzennegative translation, and its variants, of classical logic into intuitionistic logic are well known (see [5], [4], [3] and [6]). Danie ..."
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. We give a variant of the GodelGentzennegative translation, and a syntactic characterization which entails conservativity result for formulas. The GodelGentzennegative translation, and its variants, of classical logic into intuitionistic logic are well known (see [5], [4], [3] and [6]). Daniel Leivant gave a systematization of the conservative extension results, not only for predicate logic but also for mathematical theories (see [7]; see [9, 2.3] for a comprehensive exposition of the negative translation as well as conservativity results). He gave a simple syntactic characterization of certain theories and formula A, for which ` c A implies `m A (or ` i A), where ` c , ` i and `m denote derivabilities in classical, intuitionistic and minimal logics, respectively. We give here a variant of the GodelGentzennegative translation, and another syntactic characterization which entails conservativity result for formulas. We refer to Gentzen's natural deduction systems for classical...
Scattered Toposes
"... A class of toposes is introduced and studied, suitable for semantical analysis of an extension of the Heyting predicate calculus admitting Godel's provability interpretation. ..."
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A class of toposes is introduced and studied, suitable for semantical analysis of an extension of the Heyting predicate calculus admitting Godel's provability interpretation.
Proofs, Lambda Terms and Control Operators
, 1995
"... ed M : V and typed by M A : V :A ffi ) and context unwrapping (denoted V E and typed by requiring V to be of type :B ffi and the evaluation context E[] to be of type B with the `hole' of type A). Here we essentially give an exposition of Griffin's result, with some simplifications and exte ..."
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ed M : V and typed by M A : V :A ffi ) and context unwrapping (denoted V E and typed by requiring V to be of type :B ffi and the evaluation context E[] to be of type B with the `hole' of type A). Here we essentially give an exposition of Griffin's result, with some simplifications and extensions based on work of Sabry and Felleisen [18]. In particular we stress its connection with questions of termination of different normalization strategies for minimal, intuitionistic and classical logic, or more precisely their fragments in implicational propositional logic. We also give some examples (due to Hirokawa) of derivations in minimal and classical logic which reproduce themselves under certain reasonable conversion rules. This work clearly owes a lot to other people. Robert Const
Interpolation via translations
"... A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties betwe ..."
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A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a KolmogorovGentzenGödel style translation, and a new translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a KiriyamaOno style translation with features of a KolmogorovGentzenGödel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics. AMS Classification: 03C40, 03F03, 03B22
Première partie
"... 1 Interprétation calculatoire de la logique soustractive..................... 7 1.1 Présentation du premier article..................................... 7 1.2 À propos des systèmes avec relations de dépendance...................... 13 1.3 Logique à domaine constant et arithmétique................... ..."
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1 Interprétation calculatoire de la logique soustractive..................... 7 1.1 Présentation du premier article..................................... 7 1.2 À propos des systèmes avec relations de dépendance...................... 13 1.3 Logique à domaine constant et arithmétique............................ 14
Dependent Type Systems for delimited control operators
, 2010
"... • How do you prove the correctness of this program from [Wadler, 1994]? let g = (reset (if (shift λf.f) then 2 else 3)) in (g True) + (g False) ..."
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• How do you prove the correctness of this program from [Wadler, 1994]? let g = (reset (if (shift λf.f) then 2 else 3)) in (g True) + (g False)