Satisfiable iff Negation is Falsifiable

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Theorem

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


Then $\mathbf A$ is satisfiable if and only if its negation $\neg \mathbf A$ is falsifiable.


Proof

Necessary Condition

Let $\mathbf A$ be satisfiable.

Then there exists a boolean interpretation $v$ of $\mathbf A$ such that:

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

Hence, by definition of the boolean interpretation of negation:

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

It follows that $\neg \mathbf A$ is falsifiable.

$\Box$


Sufficient Condition

Let $\neg \mathbf A$ be falsifiable.

Then there exists a boolean interpretation $v$ of $\neg \mathbf A$ such that:

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

Hence, by definition of the boolean interpretation of negation:

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

It follows that $\mathbf A$ is satisfiable.

$\blacksquare$


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