A note on cut-elimination for classical propositional logic

Atomic Cut Elimination for classical Logic

System SKS is a set of rules for classical propositional logic presented in the calculus of structures. Like sequent systems and unlike natural deduction systems, it has an explicit cut rule, which is admissible. In contrast to sequent systems, the cut rule can easily be restricted to atoms. This allows for a very simple cut elimination procedure based on plugging in parts of a proof, like norm...

Herbrand-Confluence for Cut Elimination in Classical First Order Logic

We consider cut-elimination in the sequent calculus for classical first-order logic. It is well known that this system, in its most general form, is neither confluent nor strongly normalizing. In this work we take a coarser (and mathematically more realistic) look at cut-free proofs. We analyze which witnesses they choose for which quantifiers, or in other words: we only consider the Herbrand-d...

Notes on Classical Propositional Logic

On Cut-elimination for Provability Logic GL

In 1983, Valentini presented a syntactic proof of cut-elimination for a sequent calculus GLS for the provability logic GL. The sequents in GLS were built from sets, as opposed to multisets, thus permitting implicit contractions. Recently it has been claimed that Valentini’s transformations to eliminate cut do not terminate when applied to GLS in a multiset-sequent setting with a rule of contrac...

Call-by-name reduction and cut-elimination in classical logic

We present a version of Herbelin’s λμ-calculus in the call-byname setting to study the precise correspondence between normalization and cut-elimination in classical logic. Our translation of λμ-terms into a set of terms in the calculus does not involve any administrative redexes, in particular η-expansion on μ-abstraction. The isomorphism preserves β, μ-reduction, which is simulated by a local-...