Double Negation/Double Negation Introduction/Proof Rule

Proof Rule

The rule of double negation introduction is a valid argument in types of logic dealing with negation $\neg$.

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

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:

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

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

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 $\neg \neg \II$

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

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.

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): $\S 1.2.1$: Rules for natural deduction