Can there be a vacuous tautological consequence $F\vDash F$?

Question

Can there be a vacuous tautological consequence $F\vDash F$?

Since $α⊨φ \iff ⊨α→φ$ then is: $(k∧¬k)⊨(p∧¬p)$ for example considered a tautological consequence?

Answer 1

Yes. This is a consequence of the definition of $\models$ between two formulas:

$\phi \models \psi$ iff $M \models \phi$ implies $M \models \psi$ for all models $M$.

(In the case of propositional logic, the models are the lines of a truth table.) Now since $M \models F$ can never occur, it vacuously follows that $F \models F$, and indeed that $F \models \phi$ for every formula $\phi$.