In the previous chapter we developed linear logic in the form of natural deduction , which is appropriate for many applications of linear logic. It is also highly economical, in that we only needed one basic judgment (A true) and two judgment forms (linear and unrestricted hypothetical judgments) to explain the meaning of all connectives we have encountered so far. However, it is not well-suite...

Focusing is a general technique for transforming a sequent proof system into one with a syntactic separation of non-deterministic choices without sacrificing completeness. This not only improves proof search, but also has the representational benefit of distilling sequent proofs into synthetic normal forms. We show how to apply the focusing technique to nested sequent calculi, a generalization ...

2 Natural Deduction and Sequent Calculus 1 2.1 Natural Deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 Sequent Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

We introduce a sequent calculus for bilattice-based annotated logic (BAL). We show that this logic can be syntactically and semantically translated into a fragment MSL∗ of conventional many-sorted logic MSL. We show deductive equivalence of sequent calculus for BAL and sequent calculus for MSL∗.

Linear logic enjoys strong symmetries inherited from classical logic while providing a constructive framework comparable to intuitionistic logic. However, the computational interpretation of sequent calculus presentations of linear logic remains problematic, mostly because of the many rule permutations allowed in the sequent calculus. We address this problem by providing a simple interpretation...

A succession of works have contributed to the understanding of the computational content of Gentzen-style classical sequent calculus. Especially, it has been shown that the left-right duality of sequent calculus expresses a syntactic duality between programs and their evaluation contexts and that sequent calculus has two dual syntactic restrictions that respectively correspond to callby-name ev...

This paper positively solves an open problem if it is possible to provide a Hilbert system to Epistemic Logic of Friendship (EFL) by Seligman, Girard and Liu. To find a Hilbert system, we first introduce a sound, complete and cut-free tree (or nested) sequent calculus for EFL, which is an integrated combination of Seligman’s sequent calculus for basic hybrid logic and a tree sequent calculus fo...

Proofs Without Syntax [37] introduced polynomial-time checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. The paper lifts a simple, strongly normalising cut elimination from combinatori...