This paper presents Separable Sequent Calculus as an extension of Gentzen’s first-order classical sequent calculus with Herband-style structural rules (subformula contraction and weakening). Every proof can be transformed into separated form, in which all logical rules precede all structural rules, a result in the spirit of Gentzen’s midsequent theorem or sharpened Hauptsatz (where all proposit...

Bi-intuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for bi-intuitionistic propositional logic: (1) a basic standard-style sequent calculus that restricts the premises of implication-right and exclusio...

A labelled sequent calculus is proposed for Priorean linear time logic, the rules of which reflect a natural closure algorithm derived from the fixed-point properties of the temporal operators. All the rules of the system are finitary, but proofs may contain infinite branches. Soundness and completeness of the calculus are stated with respect to a notion of provability based on a condition on d...

We extend Parigot's-calculus to form a system of realizers for classical logic which reeects the structure of Gentzen's cut-free, multiple-conclusioned, sequent calculus LK when used as a system for proof-search. Speciically, we add (i) a second binding operator, , which realizes classical, multiple-conclusioned disjunction, and (ii) explicit substitutions, , which provide suucient term-structu...

It is well-known [7] that the standard formalizations of classical and intuitionistic logic based on Hilbert calculi, sequent calculi and natural deduction are equivalent. In spite of this, proof-search has been mainly developed around the notion of sequent calculi almost neglecting the cases of natural deduction and Hilbert calculi. This is primarily motivated by the fact that the latters lack...

− A central aspect of proof search is the identification and control over various forms of redundancies in the search space. We investigate systematic techniques for managing some redundancies in proof search in sequent calculi. This paper is a summary of some results of our investigation. In particular we have enriched inference rules with some additional information about status the search in...

Which modal logics can be ‘naturally’ captured by a sequent system? Clearly, this question hinges on what one believes to be natural, i.e. which format of sequent rules one is willing to accept. This paper studies the relationship between the format of sequent rules and the corresponding syntactical shape of axioms in an equivalent Hilbert-system. We identify three different such formats, the m...

Motivated by the fact that nearly all conditional logics are axiomatised by so-called shallow axioms (axioms with modal nesting depth ≤ 1) we investigate sequent calculi and cut elimination for modal logics of this type. We first provide a generic translation of shallow axioms to (one-sided, unlabelled) sequent rules. The resulting system is complete if we admit pseudo-analytic cut, i.e. cuts o...