A Simple Proof that Super-Consistency Implies Cut Elimination

Abstract

We give a simple and direct proof that super-consistency implies the cut elimination property in deduction modulo. This proof can be seen as a simplification of the proof that super-consistency implies proof normalization. It also takes ideas from the semantic proofs of cut elimination that proceed by proving the completeness of the cut-free calculus. As an application, we compare our work with the cut elimination theorems in higher-order logic that involve V-complexes.

Deduction modulo theory

“…This semantic view on termination of proof reduction theorems also permits to relate these termination proofs to the so called semantic cut-elimination proofs that proceed by proving a completeness result for cut free provability. First, without proving the termination of proof-reduction, it is possible to prove directly that, in a super-consistent theory, each provable proposition has a cut free proof [36,10]. This completeness proof does not use the pre-Heyting algebra of reducibility candidates but a simpler one.…”

Semantic A-translations and Super-Consistency Entail Classical Cut Elimination

“…Whether we can widen the criterion and replace pseudo-Heyting algebras by Heyting algebras in Definition 10, the idea being to use cut-admissibility (through semantic completeness, in the mood of [17] for instance) instead of normalization in the proof of Theorem 2 is a conjecture. Analyzing [4,9] closely shows that cut-admissibility results crucially depend on finding in the interpretation of the atoms P a syntactical version of P in the model formed out of contexts/propositions. Super-consistency does not directly allows this, due to the abstract construction of a generic model.…”

Orthogonality and Boolean Algebras for Deduction Modulo

“…In this section, we exhibit a (non-trivial) Boolean algebra, similar but simpler to the one that can be found in [DH07], extracted from the pre-Boolean algebra of sequents of section 5.…”

“…The benefits of a direct proof would be an alternative proof of the cutelimination theorem, as it is done in [DH07], through the usual soundness theorem with respect to Boolean Algebras and strong completeness with respect to the particular Boolean Algebra we presented here.…”