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 invokedpool
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$ ... $
delimitersdepends
is the line of the tableau proof upon which this line directly dependscomment
is the (optional) comment that is to be displayed in the Notes column.
Sources
- 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