Modelchecking on Automata, $\phi$ not SAT and $\phi \models$ False

Question

Given a formula $\phi$

Is $\phi \models FALSE$ equivalent to $\phi$ not SAT?

Or does $\phi \models FALSE$ means that $\phi$ is never $TRUE$ and $\phi$ not SAT means, that there existst at least one counter example?
How can I express "not SAT" then?

Answer 1

$\phi \models \text{FALSE}$ means that every model of $\phi$ is also a model of FALSE. Since FALSE does not have models (at last in any sane form of logic), this implies that $\phi$ has no model, i.e., it is not satisfiable.

If you wish to express that $\phi$ has at least one counterexample, try $\text{TRUE} \not\models \phi$.