Proof by Contradiction/Proof Rule

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Proof Rule

Proof by contradiction is a valid argument in types of logic dealing with negation $\neg$ and contradiction $\bot$.

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


As a proof rule it is expressed in the form:

If, by making an assumption $\phi$, we can infer a contradiction as a consequence, then we may infer $\neg \phi$.
The conclusion $\neg \phi$ does not depend upon the assumption $\phi$.


It can be written:

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


Tableau Form

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

The Proof by Contradiction is invoked for $\phi \vdash \bot$ in the following manner:

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


Explanation

Proof by Contradiction can be expressed in natural language as follows:

If we know that by making an assumption $\phi$ we can deduce a contradiction, then it must be the case that $\phi$ cannot be true.

Thus it provides a means of introducing a negation into a sequent.


Also known as

Proof by Contradiction is also known as not-introduction, and can be seen abbreviated as $\neg \II$ or $\neg_i$.

However, there are technical reasons why this form of abbreviation are suboptimal on this website, and PBC (if abbreviation is needed at all) is to be preferred.


It is also known as proof of negation, that is, proof that some (positive!) assumption is not true.


Some sources do not explicitly distinguish between Proof by Contradiction and Reductio ad Absurdum, which starts with a negative assumption ($\neg \phi$).

Both can be referred to as indirect proof, but Reductio ad Absurdum is rejected by the intuitionistic school.


Also see


Technical Note

When invoking Proof by Contradiction in a tableau proof, use the {{Contradiction}} template:

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

where:

line is the number of the line on the tableau proof where the Proof by Contradiction 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

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