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The authors present two calculi for intuitionistic logic that eliminates the need for global metarules.

The authors provide direct proof for admissibility of cut and other rules for a sequent calculus in intuitionistic logic. They name this calculus G4i and show how to extend the proof with first-order syntax and multiple succedents.

The authors consider the modal-calculus in terms of interpolation, the Lyndon Theorem, and the Los-Tarski Theorem. All 3 classical theorems are proven for the mu-calculus.

A study of sequent calculus is presented in which the management of weakening and contraction is organized as in natural deduction with a comparison to natural deduction given through translation of derivations between the two systems. In the given calculus, only those cuts need to be eliminated that correspond to detour and permutation conversions in natural deduction are discussed.