# Definition:Natural Deduction/Elementary Valid Argument Forms

## Contents

- 1 Definition
- 1.1 Rule of Assumption
- 1.2 Rule of Conjunction
- 1.3 Rule of Simplification
- 1.4 Rule of Addition
- 1.5 Proof by Cases
- 1.6 Modus Ponendo Ponens
- 1.7 Modus Tollendo Tollens
- 1.8 Modus Tollendo Ponens
- 1.9 Modus Ponendo Tollens
- 1.10 Rule of Implication
- 1.11 Double Negation Introduction
- 1.12 Double Negation Elimination
- 1.13 Biconditional Introduction
- 1.14 Biconditional Elimination
- 1.15 Principle of Non-Contradiction
- 1.16 Proof by Contradiction
- 1.17 Rule of Explosion
- 1.18 Law of Excluded Middle
- 1.19 Reductio ad Absurdum

- 2 Also known as
- 3 Sources

## Definition

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

Some sources refer to **elementary valid argument forms** as **axioms of natural deduction**.

They are also seen referred to as **rules of inference**.

## Sources

- 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 - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.2.3$: Natural Deduction in summary