We 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.

We prove the strong normalization of full classical natural deduction by using a translation into the simply typed λμ-calculus. We also extend Mendler’s result on recursive equations to this system.

This paper proves the strong normalization of classical natural deduction with disjunction and permutative conversions, by using CPS-translation and augmentations. Using them, this paper also proves the strong normalization of classical natural deduction with general elimination rules for implication and conjunction, and their permutative conversions. This paper also proves that natural deduction can be embedded into natural deduction with general elimination rules, strictly preserving proof normalization.

Termination for classical natural deduction is difficult in the presence of commuting/permutative conversions for disjunction. An approach based on reducibility candidates is presented that uses non-strictly positive inductive definitions.It covers second-order universal quantification and also the extension of the logic with fixed points of non-strictly positive operators, which appears to be a new result.Finally, the relation to Parigot’s strictly positive inductive definition of his set of reducibility candidates and to his notion of generalized reducibility candidates is explained.

In this paper, we present the -calculus which at the typed level corresponds to the full classical propositional natural deduction system. The Church–Rosser property of this system is proved using the standardisation and the finiteness developments theorem. We also define the leftmost reduction and prove that it is a winning strategy.

This paper gives new confluence proofs for several lambda calculi with permutation-like reduction, including lambda calculi corresponding to intuitionistic and classical natural deduction with disjunction and permutative conversions, and a lambda calculus with explicit substitutions. For lambda calculi with permutative conversion, naïve parallel reduction technique does not work, and traditional notion of residuals is required as Ando pointed out. This paper shows that the difficulties can be avoided by extending the technique proposed by Dehornoy and van Oostrom, called the Z (...) theorem: existence of a mapping on terms with the Z property concludes the confluence. Since it is still hard to directly define a mapping with the Z property for the lambda calculi with permutative conversions, this paper extends the Z theorem to compositional functions, called compositional Z, and shows that we can adopt it to the calculi. (shrink)