# 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.

This article is complete as far as it goes, but it could do with expansion.In particular: Define exactly what these restrictions are and preferably do this on a subpageYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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 $\psi$.

### Rule of Addition

- $(1): \quad$ 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 $\psi \implies \phi$.

### Principle of Non-Contradiction

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

### Proof by 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 $\phi$.

### Reductio ad Absurdum

- 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

**Natural deduction** 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:

- Johansson's Minimal Logic disallows the Rule of Explosion and the Law of Excluded Middle.

- Intuitionistic Propositional Logic disallows the Law of Excluded Middle.

- Classical Propositional Logic is the school of propositional logic which allows
*all*the above rules.

- Results about
**natural deduction**can be found**here**.

## 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:

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 |

## Sources

- 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 4.1$: The Purpose of the Axiomatic Method - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.1$: Formal Proof of Validity - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 1$: Decision procedures and proofs - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.2$: Natural Deduction

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- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**natural deduction** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**natural deduction**