scholarly journals A note on strong normalization in classical natural deduction

2016 ◽  
Vol 213 ◽  
pp. 41-51
Author(s):  
José Espírito Santo
Keyword(s):  
Natural Deduction ◽  
Strong Normalization

scholarly journals Reduction in first-order logic compared with reduction in implicational logic

2007 ◽  
Vol 5 ◽  
Author(s):  
Tigran M. Galoyan
Keyword(s):  
Natural Deduction ◽  
Order Logic ◽  
First Order Logic ◽  
Strong Normalization ◽  
First Order ◽  
Reduction Sequence

In this paper we discuss strong normalization for natural deduction in the →∀-fragment of first-order logic. The method of collapsing types is used to transfer the result (concerning strong normalization) from implicational logic to first-order logic. The result is improved by a complement, which states that the length of any reduction sequence of derivation term r in first-order logic is equal to the length of the corresponding reduction sequence of its collapse term rc in implicational logic.

Normalization theorems for full first order classical natural deduction

1991 ◽  
Vol 56 (1) ◽  
pp. 129-149 ◽  
Author(s):  
Gunnar Stålmarck
Keyword(s):  
Normal Form ◽  
Natural Deduction ◽  
Full System ◽  
Strong Normalization ◽  
Logical Constants ◽  
Restricted Version ◽  
First Order ◽  
Form Theorem ◽  
Normalization Theorem ◽  
Normal Form Theorem

In this paper we prove the strong normalization theorem for full first order classical N.D. (natural deduction)—full in the sense that all logical constants are taken as primitive. We also give a syntactic proof of the normal form theorem and (weak) normalization for the same system.The theorem has been stated several times, and some proofs appear in the literature. The first proof occurs in Statman [1], where full first order classical N.D. (with the elimination rules for ∨ and ∃ restricted to atomic conclusions) is embedded in a system for second order (propositional) intuitionistic N.D., for which a strong normalization theorem is proved using strongly impredicative methods.A proof of the normal form theorem and (weak) normalization theorem occurs in Seldin [1] as an extension of a proof of the same theorem for an N.D.-system for the intermediate logic called MH.The proof of the strong normalization theorem presented in this paper is obtained by proving that a certain kind of validity applies to all derivations in the system considered.The notion “validity” is adopted from Prawitz [2], where it is used to prove the strong normalization theorem for a restricted version of first order classical N.D., and is extended to cover the full system. Notions similar to “validity” have been used earlier by Tait (convertability), Girard (réducibilité) and Martin-Löf (computability).In Prawitz [2] the N.D. system is restricted in the sense that ∨ and ∃ are not treated as primitive logical constants, and hence the deductions can only be seen to be “natural” with respect to the other logical constants. To spell it out, the strong normalization theorem for the restricted version of first order classical N.D. together with the well-known results on the definability of the rules for ∨ and ∃ in the restricted system does not imply the normalization theorem for the full system.

scholarly journals Strong Normalization for Truth Table Natural Deduction

2019 ◽  
Vol 170 (1-3) ◽  
pp. 139-176
Author(s):  
Herman Geuvers ◽  
Iris van der Giessen ◽  
Tonny Hurkens
Keyword(s):  
Natural Deduction ◽  
Truth Table ◽  
Strong Normalization

scholarly journals Strong normalization of classical natural deduction with disjunctions

2008 ◽  
Vol 153 (1-3) ◽  
pp. 21-37 ◽  
Author(s):  
Koji Nakazawa ◽  
Makoto Tatsuta
Keyword(s):  
Natural Deduction ◽  
Strong Normalization

Strong normalization proof with CPS-translation for second order classical natural deduction

2003 ◽  
Vol 68 (3) ◽  
pp. 851-859 ◽  
Author(s):  
Koji Nakazawa ◽  
Makoto Tatsuta
Keyword(s):  
Natural Deduction ◽  
Second Order ◽  
Strong Normalization

AbstractThis paper points out an error of Parigot's proof of strong normalization of second order classical natural deduction by the CPS-translation, discusses erasing-continuation of the CPS-translation, and corrects that proof by using the notion of augmentations.

scholarly journals A short proof of the strong normalization of classical natural deduction with disjunction

2003 ◽  
Vol 68 (4) ◽  
pp. 1277-1288 ◽  
Author(s):  
René David ◽  
Karim Nour
Keyword(s):  
Natural Deduction ◽  
Elementary Proof ◽  
Short Proof ◽  
Cut Elimination ◽  
Strong Normalization ◽  
Elimination Procedure

AbstractWe give a direct, purely arithmetical and elementary proof of the strong normalization of the cut-elimination procedure for full (i.e., in presence of all the usual connectives) classical natural deduction.

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