Semantics of sequent calculi

The aim of this study is to provide a generic fuzzy semanticsof propositional sequent calculi - both two-side and signed ones -with basic structural rules: Enlargement, Permutation and Contraction. As a representative application, we explore substructural (more precisely, Cut and/or Sharing free) versionsof Gentzen's original calculus LK with basic structural rulessupplemented by the rules inverse to the logical ones andprove that they are equivalent to certain subclassicalextensions of the bounded version of Belnap's four-valued logic such as Priest's logic of paradox and Kleene's three-valued logic, the notorious Cut Elimination property of LK appearing justa consequence of the well-known regularity of the operations of Kleene'sthree-valued logic