Reductio ad Absurdum/Proof Rule

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.

As a proof rule it is expressed in the form:

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 $\neg \phi$.

It can be written:

$\ds {\begin{array}{|c|} \hline \neg \phi \\ \vdots \\ \bot \\ \hline \end{array} \over \phi} \textrm{RAA}$

Tableau Form

Let $\phi$ be a well-formed formula in a tableau proof.

The is invoked for $\neg \phi \vdash \bot$ in the following manner:

			Pool:					The pooled assumptions of $\bot$
			Formula:					$\phi$
			Description:
			Depends on:					The series of lines from where the assumption $\neg \phi$ was made to where $\bot$ was deduced
			Discharged Assumptions:					The assumption $\neg \phi$ is discharged
			Abbreviation:					$\text{RAA}$

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.

Also see

This is a rule of inference of the following proof systems:
- Definition:Natural Deduction

Linguistic Note

Reductio ad Absurdum is Latin for reduction to an absurdity.

Technical Note

When invoking in a tableau proof, use the {{Reductio}} template:

{{Reductio|line|pool|statement|start|end}}

where:

line is the number of the line on the tableau proof where the is to be invoked

pool is the pool of assumptions (comma-separated list)

statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters

start is the start of the block of the tableau proof upon which this line directly depends

end is the end of the block of the tableau proof upon which this line directly depends

Sources

1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 3$
2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.2$: Derived rules