He wrote important papers on free logic, general modal logic, and natural deduction systems.

Between the natural deduction system and the lambda calculus there are the following correspondences:

Now that the deduction system of HM is at hand, one could present an algorithm and validate it with respect to the rules.

This can be ensured (along with stronger conditions) by, e.g., placing certain restrictions on the rules of a natural deduction system.

It is common to include in a Hilbert-style deduction system only axioms for implication and negation.

It describes (among others) a part of the Hilbert-style deduction system (restricted to propositional calculus).

So there was only a difference between the prorated deduction system and the "one-year rule" for the first and last year of the lease term.

The sequent calculus was developed to study the properties of natural deduction systems.

The preceding alternative calculus is an example of a Hilbert-style deduction system.

Proof theory is the study of formal proofs in various logical deduction systems.