# Double Negation

## Theorem

## Double Negation Introduction

The **rule of double negation introduction** is a valid argument in types of logic dealing with negation $\neg$.

This includes propositional logic and predicate logic, and in particular natural deduction.

### Proof Rule

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

### Sequent Form

- $p \vdash \neg \neg p$

## Double Negation Elimination

The **rule of double negation elimination** is a valid argument in certain types of logic dealing with negation $\neg$.

This includes classical propositional logic and predicate logic, and in particular natural deduction, but for example not intuitionistic propositional logic.

### Proof Rule

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

### Sequent Form

- $\neg \neg p \vdash p$

## Combined Double Negation Law

These are often combined into one law:

### Formulation 1

- $p \dashv \vdash \neg \neg p$

### Formulation 2

- $\vdash p \iff \neg \neg p$

## Double Negation from Intuitionistic Perspective

The intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom.

This in turn invalidates the Law of Double Negation Elimination from the system of intuitionistic propositional logic.

Hence a difference is perceived between Double Negation Elimination and Double Negation Introduction, whereby it can be seen from the Principle of Non-Contradiction that if a statement is true, then it is not the case that it is false.

However, if all we know is that a statement is not false, we can not be certain that it *is* actually true without accepting that there are only two possible truth values.

Such distinctions may be important when considering, for example, multi-value logic.

However, when analysing logic from a purely classical standpoint, it is common and acceptable to make the simplification of taking just one Double Negation rule:

- $p \dashv \vdash \neg \neg p$

## Also see

## Sources

- 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 3$ - 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $2$ Conditionals and Negation - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 3$: Statements and conditions; quantifiers - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction