Reductio ad Absurdum/Proof Rule

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Proof Rule

The reductio ad absurdum is a valid deduction sequent in 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 does not depend upon the assumption $\neg \phi$.


It can be written:

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


Tableau Form

Let $\phi$ be a propositional formula in a tableau proof.

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

Pool:    The pooled assumptions of $\bot$             
Formula:    $\phi$             
Description:    Reductio ad Absurdum             
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.


Linguistic Note

Reductio ad Absurdum is Latin for reduction to an absurdity.


Technical Note

When invoking Reductio ad Absurdum 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 Reductio ad Absurdum 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