= Unsatisfiable Boolean formula
{title2=$\nexists v:\ v\models F$}
An <unsatisfiable Boolean formula> has no <satisfying assignment>. For
$$
G=x\land\neg x,
$$
the <Boolean valuation> $x=0$ falsifies the first <Boolean clause>, and $x=1$ falsifies the second. Both possible <truth values> fail, proving `UNSAT`. Each <Boolean clause> alone has a <satisfying assignment>, but their <logical conjunction> does not: compatibility is the central issue. By contrast, finding just one failed <Boolean valuation> for a general <Boolean formula> only rejects that candidate. Finding one successful <Boolean valuation> proves `SAT`; proving `UNSAT` requires excluding every candidate, possibly with a compact argument rather than a full <truth table>.
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