User:Lord Farin/Archive/Natural Deduction Axioms

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General note

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


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.


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.


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

\(\text {(\mathrm{ND}:\mathrm A)}: \quad\) \(\ds \) \(\) \(\ds q \vdash q\) Rule of Assumption
\(\text {(\mathrm{ND}:\lor \mathcal I_1)}: \quad\) \(\ds P \vdash p\) \(\, \Large\leadsto \,\) \(\ds P \vdash p \lor q\) Rule of Addition
\(\text {(\mathrm{ND}:\lor \mathcal I_2)}: \quad\) \(\ds P \vdash q\) \(\, \Large\leadsto \,\) \(\ds P \vdash p \lor q\) Rule of Addition
\(\text {(\mathrm{ND}:\lor \mathcal E)}: \quad\) \(\ds \left.{ \begin{align} P &\vdash p \lor q \\ Q, p &\vdash r \\ R, q &\vdash r \end{align} }\right\}\) \(\, \Large\leadsto \,\) \(\ds P, Q, R \vdash r\) Proof by Cases
\(\text {(\mathrm{ND}:\land \mathcal I)}: \quad\) \(\ds \left.{ \begin{align} P &\vdash p \\ Q &\vdash q \end{align} }\right\}\) \(\, \Large\leadsto \,\) \(\ds P, Q \vdash p \land q\) Rule of Conjunction
\(\text {(\mathrm{ND}:\land \mathcal E_1)}: \quad\) \(\ds P \vdash p \land q\) \(\, \Large\leadsto \,\) \(\ds P \vdash p\) Rule of Simplification
\(\text {(\mathrm{ND}:\land \mathcal E_2)}: \quad\) \(\ds P \vdash p \land q\) \(\, \Large\leadsto \,\) \(\ds P \vdash q\) Rule of Simplification
\(\text {(\mathrm{ND}:\implies \mathcal I)}: \quad\) \(\ds P, p \vdash q\) \(\, \Large\leadsto \,\) \(\ds P \vdash p \implies q\) Rule of Implication
\(\text {(\mathrm{ND}:\implies \mathcal E)}: \quad\) \(\ds \left.{ \begin{align} P &\vdash p \implies q \\ Q &\vdash p \end{align} }\right\}\) \(\, \Large\leadsto \,\) \(\ds P, Q \vdash q\) Modus Ponendo Ponens
\(\text {(\mathrm{ND}:\neg \mathcal I)}: \quad\) \(\ds P, p \vdash \bot\) \(\, \Large\leadsto \,\) \(\ds P \vdash \neg p\) Proof by Contradiction
\(\text {(\mathrm{ND}:\neg \mathcal E)}: \quad\) \(\ds \left.{ \begin{align} P &\vdash p \\ Q &\vdash \neg p \end{align} }\right\}\) \(\, \Large\leadsto \,\) \(\ds P, Q \vdash \bot\) Principle of Non-Contradiction
\(\text {(\mathrm{ND}:\bot \mathcal E)}: \quad\) \(\ds P \vdash \bot\) \(\, \Large\leadsto \,\) \(\ds P \vdash p\) Rule of Explosion


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.


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.