# 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 $\psi$.

### 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 $\neg \left({\phi \land \psi}\right)$, 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 $\psi \implies \phi$.

### 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 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 $\phi$.

### 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 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:

- 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.

## 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): $3.1$: 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 - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**natural deduction** - 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 - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**natural deduction**