Definition:Natural Deduction

Definition
Natural deduction is a technique for deducing valid sequents from other valid sequents by applying precisely defined proof rules, by a technique called logical inference.

As such, natural deduction forms a proof system, which is focused on practical applicability.

Motivation
In its practical applicability, natural deduction differs from most proof systems in literature, which are more pedantic and formalistic, but therefore also more rigorous.

One may interpret natural deduction as the full-fledged proof system to actually use, once a formalistic alternative has been proved to satisfy all the rules of inference of natural deduction.

In doing so, it no longer matters which exact formalism was employed, and one can focus on the mathematical content itself.

On, we use natural deduction for both propositional logic and predicate logic.

Additionally, natural deduction may be applied not only for classical propositional logic, but also for more limited forms such as intuitionistic propositional logic, by employing the appropriate restrictions.

Rules of Inference
Natural deduction deals exclusively with the notion of provable consequence.

As such, it does not contain any axioms.

Practically, this means that any proof of natural deduction will start with premises or the.

Also known as
This technique is seen under various less precise names, for example decision procedure or decision method.

Some sources call it the axiomatic method.

Also see
Certain schools of logic have investigated the situation of what happens when certain of the above proof rules (and their equivalents) are disallowed:


 * Johansson's Minimal Logic disallows the and the.


 * Intuitionistic Propositional Logic disallows the.


 * Classical Propositional Logic is the school of propositional logic which allows all the above rules.