Satisfiable iff Negation is Falsifiable

From ProofWiki
Jump to: navigation, search

Theorem

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


Then $\mathbf A$ is satisfiable iff 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:

$v \left({\mathbf A}\right) = T$

Hence, by definition of the boolean interpretation of negation:

$v \left({\neg \mathbf A}\right) = F$

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:

$v \left({\neg \mathbf A}\right) = F$

Hence, by definition of the boolean interpretation of negation:

$v \left({\mathbf A}\right) = T$

It follows that $\mathbf A$ is satisfiable.

$\blacksquare$


Sources