# Definition:Natural Deduction

## Definition

Natural deduction is a technique for deducing valid sequents from other valid sequents by applying precisely defined proof rules, by a technique called logical inference.

As such, natural deduction forms a proof system, which is focused on practical applicability.

### Motivation

In its practical applicability, natural deduction differs from most proof systems in literature, which are more pedantic and formalistic, but therefore also more rigorous.

One may interpret natural deduction as the full-fledged proof system to actually use, once a formalistic alternative has been proved to satisfy all the rules of inference of natural deduction.

In doing so, it no longer matters which exact formalism was employed, and one can focus on the mathematical content itself.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, we use natural deduction for both propositional logic and predicate logic.

Additionally, natural deduction may be applied not only for classical propositional logic, but also for more limited forms such as intuitionistic propositional logic, by employing the appropriate restrictions.

## Rules of Inference

The proof system called natural deduction deals exclusively with the notion of provable consequence.

As such, it does not contain any axioms.

Practically, this means that any proof of natural deduction will start with premises or the Rule of Assumption.

The complete list of rules of inference of natural deduction is as follows:

### Rule of Assumption

An assumption $\phi$ 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 $\map \neg {\phi \land \psi}$, and we can also conclude $\phi$, then we may infer $\neg \psi$.
$(2): \quad$ If we can conclude $\map \neg {\phi \land \psi}$, 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 $\phi \implies \psi$.
The conclusion $\phi \implies \psi$ does not depend on the assumption $\phi$, which is thus discharged.

### 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 $\neg \phi$ does not depend upon the assumption $\phi$.

### Rule of Explosion

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

### 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 $\phi$ does not depend upon the assumption $\neg \phi$.

## Derived Rules

In practically working with natural deduction, the following derived rules are useful.

### Rule of Substitution

Let $S$ be a sequent of propositional logic that has been proved.

Then we may infer any sequent $S'$ resulting from $S$ by substitutions for letters.

### 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 sequent for which we already have a proof.

Then we may infer, at any stage of a proof (citing SI), the conclusion $Q$ of the sequent already proved.

This conclusion depends upon the pool of assumptions upon which $P_1, P_2, \ldots, P_n$ rest.

### Rule of Theorem Introduction

We may infer, at any stage of a proof (citing $\text {TI}$), a theorem already proved, together with a reference to the theorem that is being cited.

## 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