User:Lord Farin/Backup/Definition:Natural Deduction

Definition
Natural deduction is a proof system for propositional calculus.

As such, it consists of certain axioms, which together constitute a collection of theorems.

This can be specified as follows.

Notation
To indicate that a collection of WFFs $P$, the pool of assumptions, entails a particular WFF $p$, the conclusion, we agree to write a sequent:


 * $P \vdash p$

To make exposition more natural, we also agree to omit the brackets of explicit set definition and use commas in a suggestive way, so that:


 * $p, q \vdash p \land q$
 * $P, Q \vdash p \land q$
 * $P, p \vdash q$

are used instead of the formally more correct:


 * $\left\{{p, q}\right\} \vdash p \land q$
 * $P \cup Q \vdash p \land q$
 * $P \cup \left\{{p}\right\} \vdash q$

Furthermore, in the last expression it is implicitly understood that $p \notin P$.

The $\Large\leadsto$ symbol indicates that given the sequent(s) on the left-hand side, the sequent on the right-hand side may be inferred.

Axioms
Natural deduction has the following twelve axioms, explained in more detail on their own pages:

{{eqn|n = \mathrm{ND}:\lor \mathcal E    |l = \left.{ \begin{align} P &\vdash p \lor q \\ Q, p &\vdash r \\ R, q &\vdash r \end{align} }\right\} |o = \, \Large\leadsto \, |r = P, Q, R \vdash r     |c = Rule of Or-Elimination }} {{eqn|n = \mathrm{ND}:\land \mathcal I    |l = \left.{ \begin{align} P &\vdash p \\ Q &\vdash q \end{align} }\right\} |o = \, \Large\leadsto \, |r = P, Q \vdash p \land q     |c = Rule of Conjunction }}

{{eqn|n = \mathrm{ND}:\implies \mathcal E    |l = \left.{ \begin{align} P &\vdash p \implies q \\ Q &\vdash p \end{align} }\right\} |o = \, \Large\leadsto \, |r = P, Q \vdash q     |c = Modus Ponendo Ponens }}

{{eqn|n = \mathrm{ND}:\neg \mathcal E    |l = \left.{ \begin{align} P &\vdash p \\ Q &\vdash \neg p \end{align} }\right\} |o = \, \Large\leadsto \, |r = P, Q \vdash \bot |c = Rule of Not-Elimination }}

Theorems
The theorems of natural deduction are those WFFs $p$ allowing a sequent $\vdash p$, i.e., that may be derived with an empty pool of assumptions.

Proofs
Although it is satisfying to find a (formal) proof of a theorem using the above rules, it is advisable to cast such a proof in a standard framework.

On ProofWiki, the framework chosen is that of a tableau proof, which most easily lends itself for a MediaWiki architecture.

However, many other notations exist and are used by various authors.

Also see
There are many other proof systems for propositional calculus, see here.

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.

Different logical schools
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.