# Reductio ad Absurdum

## Proof Rule

**Reductio ad Absurdum** is a valid argument in certain types of logic dealing with negation $\neg$ and contradiction $\bot$.

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

### Proof Rule

- 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 $\neg \phi$.

### Sequent Form

The Reductio ad Absurdum can be symbolised by the sequent:

- $\paren {\neg p \vdash \bot} \vdash p$

## Explanation

**Reductio ad Absurdum** can be expressed in natural language as:

- If, by making an assumption that a statement is false, a contradiction can be deduced, that statement must in fact be true.

## Variants

The following forms can be used as variants of this theorem:

### Variant 1

- $\neg p \implies \bot \vdash p$

### Variant 2

- $\neg p \implies \paren {q \land \neg q} \vdash p$

## Law of the Excluded Middle

This theorem depends on the Law of the Excluded Middle.

This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.

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

This in turn invalidates this theorem from an intuitionistic perspective.

## Also see

- Indirect Proof, of which this is an instance

- Proof by Contradiction, often treated as another aspect of the same thing.

From the point of view of purely classical logic, this is acceptable. However, in the context of intuitionistic logic, it is **essential** to bear in mind that only the Proof by Contradiction is valid.

This is because Proof by Contradiction starts with a positive assumption $\phi$.

As a result, it does not depend on the Law of Excluded Middle.

## Linguistic Note

**Reductio ad Absurdum** is Latin for **reduction to an absurdity**.

## Sources

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Axiom systems - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{I}$: The Logic of Statements $(1): \ 14$: A Short Cut to Truth-tables - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**indirect proof** - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if... - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**indirect proof**