# Tautology iff Negation is Unsatisfiable

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It has been suggested that this page or section be merged into Tautology is Negation of Contradiction.Maybe merge with Tautology is Negation of Contradiction? Can't imagine the look of the finalized product at this moment, thoughTo discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mergeto}}` from the code. |

## Theorem

Let $\mathbf A$ be a WFF of propositional logic.

Then $\mathbf A$ is a tautology if and only if its negation $\neg \mathbf A$ is unsatisfiable.

## Proof

### Necessary Condition

Let $\mathbf A$ be a tautology.

Let $v$ be a boolean interpretation of $\mathbf A$.

Then $\map v {\mathbf A} = \T$.

Hence, by definition of the boolean interpretation of negation:

- $\map v {\neg \mathbf A} = \F$

Since $v$ was arbitrary, it follows that $\neg \mathbf A$ is unsatisfiable.

$\Box$

### Sufficient Condition

Let $\neg \mathbf A$ be unsatisfiable.

Let $v$ be a boolean interpretation of $\neg \mathbf A$.

Then $\map v {\neg \mathbf A} = \F$.

Hence, by definition of the boolean interpretation of negation:

- $\map v {\mathbf A} = \T$

Since $v$ was arbitrary, it follows that $\mathbf A$ is a tautology.

$\blacksquare$

## Sources

- 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): $\S 2.5$: Theorem $2.39$