User:Lord Farin/Backup/Definition:Natural Deduction

Definition
Natural deduction is a technique for deducing valid sequents from other valid sequents by applying precisely defined proof rules, each of which themselves are either "self-evident" axioms or themselves derived from other valid sequents, by a technique called logical inference.

Proof Rules
The following rules are often treated as the axioms of PropLog. Some of them are "obvious", but they still need to be stated formally. Others are more subtle.

This is not the only valid analysis of this subject. There are other systems which use other proof rules, but these ones are straightforward and are easy to get to grips with. It needs to be pointed out that the axioms described in this section do not constitute a minimal set by any means. However, the fewer the axioms, the more complicated the arguments are, and the more difficult they are to establish the truth of them.

Also note that premises of an argument are considered to be assumptions themselves.

Axioms of Natural Deduction
In most treatments of PropLog various subsets of the following rules are treated as the axioms. Some of them are obvious. Others are more subtle.

These rules are not all independent, in that it is possible to prove some of them using sequents constructed from combinations of others. However, when a set of proof rules is selected as the axioms for any particular treatment of this subject, those rules are usually selected carefully so that they are independent.

Also see
Certain schools of logic have investigated the situation of what happens when certain of the above proof rules are disallowed.


 * Johansson's Minimal Calculus allows all the above axioms except the Rule of Bottom-Elimination and the Law of the Excluded Middle.


 * Intuitionist Propositional Calculus allows all the above axioms except the Law of the Excluded Middle.


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

Technical Note
In order to make the use of the above proof rules in a tableau proof, the following templates have been developed:


 * Template:Premise, for using the Rule of Assumption for a premise
 * Template:Assumption, for using the Rule of Assumption for an assumption which is not a premise
 * Template:Conjunction, for using the Rule of Conjunction
 * Template:Simplification, for using the Rule of Simplification
 * Template:Addition, for using the Rule of Addition
 * Template:OrElimination, for using the Rule of Or-Elimination
 * Template:ModusPonens, for using the Modus Ponendo Ponens
 * Template:ModusTollens, for using the Modus Tollendo Tollens
 * Template:Implication, for using the Rule of Implication
 * Template:DoubleNegIntro, for using the Double Negation Introduction
 * Template:DoubleNegElimination, for using the Double Negation Elimination
 * Template:NonContradiction, for using the Principle of Non-Contradiction
 * Template:Contradiction, for using Proof by Contradiction
 * Template:BottomElimination, for using the Rule of Bottom-Elimination
 * Template:ExcludedMiddle, for using the Law of Excluded Middle
 * Template:Reductio, for using the Reductio Ad Absurdum

For convenience, other templates are also available, for the following derived rules:


 * Template:Commutation, for using the Rule of Commutation
 * Template:DeMorgan, for using De Morgan's Laws (Logic)
 * Template:Idempotence, for using the Rule of Idempotence
 * Template:IdentityLaw for using the Law of Identity

For the general Rule of Sequent Introduction and Rule of Theorem Introduction, there exist the following templates:


 * Template:SequentIntro
 * Template:TheoremIntro