Double Negation/Double Negation Introduction/Proof Rule

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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 $\neg \neg \mathcal I$             


Also see


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.


Sources