What's the meaning of "$F \models \bot$" in propositional logic?

Question

What's the meaning of "$F \models \bot$" in propositional logic?

For example, in the demonstration that $F \vDash \bot \iff \forall G.F \vDash G $, how do I use the hypothesis $F \vDash \bot$?

Answer 1

$F \models \bot$ means that there is no valuation $v$ that satisfies the formula $F$.

If $F \models \bot$, then it is vacuously true that every valuation that satisfies $F$ satisfies $G$ as well (for every formula $G$), since the set of valuations that satisfy $F$ is empty.

Conversely, if $F$ is a formula such that $F \models G$ for any formula $G$, then in particular $F \models \lnot F$: this means that every valuation that satisfies $F$ satisfies $\lnot F$. Clearly, it is impossible that a valuation satisfies a formula and its negation. Therefore, there is no valuation that satisfies $F$, i.e. $F \models \bot$.