User:Lord Farin/Archive/Natural Deduction Axioms

= General note =

The below page is a copy of the revision 96983 of Definition:Natural Deduction, accessible here.

Definition
Natural deduction is a proof system for propositional logic.

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 = Proof by Cases }} {{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 logic, see here.