# 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 Rule

A proof rule is a rule in natural deduction which allows one to infer the validity of propositional formulas from other propositional formulas.

### Rule of Substitution

Let $S$ be a sequent that has been proved.

Then a proof can be found for any substitution instance of $S$.

### Rule of Sequent Introduction

Let the statements $P_1, P_2, \ldots, P_n$ be conclusions in a proof, on various assumptions.

Let $P_1, P_2, \ldots, P_n \vdash Q$ be a substitution instance of a sequent for which we already have a proof.

Then we may introduce, at any stage of a proof (citing SI), either:

The conclusion $Q$ of the sequent already proved

or:

A substitution instance of such a conclusion, together with a reference to the sequent that is being cited.

This conclusion depend upon the pool of assumptions upon which $P_1, P_2, \ldots, P_n \vdash Q$ rests.

### Rule of Theorem Introduction

We may introduce, at any stage of a proof (citing TI), a theorem already proved, or a substitution instance of such a theorem, together with a reference to the theorem that is being cited.

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.

## Elementary Valid Argument Forms

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.

### Rule of Assumption

An assumption may be introduced at any stage of an argument.

### Rule of Conjunction

If we can conclude both $\phi$ and $\psi$, we may infer the compound statement $\phi \land \psi$.

### Rule of Simplification

$(1): \quad$ If we can conclude $\phi \land \psi$, then we may infer $\phi$.
$(2): \quad$ If we can conclude $\phi \land \psi$, then we may infer $\psi$.

$(1): \quad$ If we can conclude $\phi$, then we may infer $\phi \lor \psi$.
$(2): \quad$ If we can conclude $\psi$, then we may infer $\phi \lor \psi$.

### Proof by Cases

If we can conclude $\phi \lor \psi$, and:
$(1): \quad$ By making the assumption $\phi$, we can conclude $\chi$
$(2): \quad$ By making the assumption $\psi$, we can conclude $\chi$
then we may infer $\chi$.

### Modus Ponendo Ponens

If we can conclude $\phi \implies \psi$, and we can also conclude $\phi$, then we may infer $\psi$.

### Modus Tollendo Tollens

If we can conclude $\phi \implies \psi$, and we can also conclude $\neg \psi$, then we may infer $\neg \phi$.

### Modus Tollendo Ponens

$(1): \quad$ If we can conclude $\phi \lor \psi$, and we can also conclude $\neg \phi$, then we may infer $\psi$.
$(2): \quad$ If we can conclude $\phi \lor \psi$, and we can also conclude $\neg \psi$, then we may infer $\phi$.

### Modus Ponendo Tollens

$(1): \quad$ If we can conclude $\neg \left({\phi \land \psi}\right)$, and we can also conclude $\phi$, then we may infer $\neg \psi$.
$(2): \quad$ If we can conclude $\neg \left({\phi \land \psi}\right)$, and we can also conclude $\psi$, then we may infer $\neg \phi$.

### Rule of Implication

If, by making an assumption $\phi$, we can conclude $\psi$ as a consequence, we may infer $\phi \implies \psi$.

### Double Negation Introduction

If we can conclude $\phi$, then we may infer $\neg \neg \phi$.

### Double Negation Elimination

If we can conclude $\neg \neg \phi$, then we may infer $\phi$.

### Biconditional Introduction

If we can conclude both $\phi \implies \psi$ and $\psi \implies \phi$, then we may infer $\phi \iff \psi$.

### Biconditional Elimination

$(1): \quad$ If we can conclude $\phi \iff \psi$, then we may infer $\phi \implies \psi$.
$(2): \quad$ If we can conclude $\phi \iff \psi$, then we may infer $\psi \implies \phi$.

If we can conclude both $\phi$ and $\neg \phi$, we may infer a contradiction.

If, by making an assumption $\phi$, we can infer a contradiction as a consequence, then we may infer $\neg \phi$.
The conclusion does not depend upon the assumption $\phi$.

### Rule of Explosion

If a contradiction can be concluded, it is possible to infer any statement.

### Law of Excluded Middle

$\phi \lor \neg \phi$ for all statements $\phi$.

If, by making an assumption $\neg \phi$, we can infer a contradiction as a consequence, then we may infer $\phi$.
The conclusion does not depend upon the assumption $\neg \phi$.

## 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:

## Historical Note

The first system of rules for natural deduction was devised by Gerhard Gentzen in $1934$.

## Technical Note: Templates

In order to make the use of the proof rules of natural deduction in a tableau proof on $\mathsf{Pr} \infty \mathsf{fWiki}$, the following templates have been developed:

 Template:Premise to invoke the Rule of Assumption for a premise Template:Assumption to invoke the Rule of Assumption for a non-premise assumption Template:Conjunction to invoke the Rule of Conjunction Template:Simplification to invoke the Rule of Simplification Template:Addition to invoke the Rule of Addition Template:ProofByCases to invoke Proof by Cases Template:ModusPonens to invoke Modus Ponendo Ponens Template:ModusTollens to invoke Modus Tollendo Tollens Template:ModusPonendoTollens to invoke Modus Ponendo Tollens Template:ModusTollendoPonens to invoke Modus Tollendo Ponens Template:Implication to invoke the Rule of Implication Template:DoubleNegIntro to invoke Double Negation Introduction Template:DoubleNegElimination to invoke Double Negation Elimination Template:BiconditionalIntro to invoke Biconditional Introduction Template:BiconditionalElimination to invoke Biconditional Elimination Template:NonContradiction to invoke the Principle of Non-Contradiction Template:Contradiction to invoke Proof by Contradiction Template:Explosion to invoke the Rule of Explosion Template:ExcludedMiddle to invoke the Law of Excluded Middle Template:Reductio to invoke Reductio ad Absurdum

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

 Template:Commutation to invoke the Rule of Commutation Template:DeMorgan to invoke an instance of De Morgan's Laws Template:Idempotence to invoke the Rule of Idempotence Template:IdentityLaw to invoke the Law of Identity

For the other general proof rules, there exist the following templates:

 Template:SequentIntro to invoke the Rule of Sequent Introduction Template:TheoremIntro to invoke the Rule of Theorem Introduction Template:Substitution to invoke the Rule of Substitution