If F satisfiable then ¬F is unsatisfiable.

Question

If F satisfiable then ¬F is unsatisfiable. I know this is false and to show this I need to show a contradiction, this is my attempted answer, any ideas where I'm going wrong, this is revision for an exam.

[A]α = ¬(¬[A]))
     = ¬([A])
     = ¬ (0)
     = 1 

Answer 1

In propositional logic, a formula $F$ is satisfiable when there is a valuation (or : truth-assignment) $v$ such that $v(F)=$ t (in first-order logic : when there is an interpretation that makes the formula true).

This does not implies that its negation is unsatisfiable.

Consider the simple example of a formula $F := p$, where $p$ is a sentential letter.

Clearly $F$ is sat, and also $\lnot F$ is sat.

But there are many more : $p \lor q, p \to q, p \land q, \ldots$

---

The interesting relation is :

if a formula $F$ is unsatisfiable, then $\lnot F$ is a tautology (i.e. always true).