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

A completeness proof for the implicative fragment of intuitionistic calculus with respect to semantics of computability logic, which understands intuitionist implication as interactive algorithmic reduction, is provided.

A direct proof of completeness of multiplicative proof-nets, MLL starting from a set of simple generators is provided by examining 'weak-distributivity' as a rewriting rule.

Intensional models for the classical theory of types are defined and an intensional type logic, ITL is resulted. A cut-free sequent calculus for type theory is presented and the completeness of this calculus with respect to the class of intensional models through a model existence theorem is shown.

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.

The partition calculus of ordinals is discussed, with a focus on complementary nonarrow consistency proofs. The negative partition relation proved by Galvin is shown to be consistent; ccc forcing is also examined.

The author develops a witnessing theorem and employs it to explain the different fragments of the quantified propositional calculus proof system. The formulas of the quantified propositional proof system are shown to be closely related to the different theorems of Buss's theory.

The article discusses a new approach, which employs Klop's l-translation and perpetual reductions for proving the strong cut-elimination in the intuitionistic sequent calculus. The method is shown to be extremely effective and accurate and has been used for a large number of systems.

Both S4 and the Intuitionistic propositional calculus Int are P-SPACE complete, enabling a polynomial translation between the systems. A solution for a sound and faithful polynomial translation from S4 into Int is presented, which is based on Kripke semantics and which describes model checking for S4 using formulas of Int.

Natural deductions systems with indefinite and definite descriptions are presented, and interpreted in Martin-Lof's intensional type theory. The interpretations are formalizations of ideas which are implicit in the literature of constructive mathematics.

A study conducted showed that axiomatic extensions of intuitionistic propositional calculus, which define univocally new connectives including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. Facts relativizing to connectives over intermediate logics along with proof of theorems with examples are presented.

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.