# Double Negation/Double Negation Introduction/Proof Rule

## Proof Rule

The rule of double negation introduction is a valid deduction sequent in propositional logic.

As a proof rule it is expressed in the form:

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

It can be written:

$\displaystyle {\phi \over \neg \neg \phi} \neg \neg_i$

### Tableau Form

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

The Law of Double Negation Introduction is invoked for $\phi$ as follows:

 Pool: The pooled assumptions of $\phi$ Formula: $\neg \neg \phi$ Description: Double Negation Introduction Depends on: The line containing the instance of $\phi$ Abbreviation: $\text{DNI}$ or $\neg \neg \mathcal I$

## Technical Note

When invoking Double Negation Introduction in a tableau proof, use the {{DoubleNegIntro}} template:

{{DoubleNegIntro|line|pool|statement|depends}}

or:

{{DoubleNegIntro|line|pool|statement|depends|comment}}

where:

line is the number of the line on the tableau proof where Double Negation Introduction 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
depends is the line of the tableau proof upon which this line directly depends
comment is the (optional) comment that is to be displayed in the Notes column.