How to prove tautology from entail?

Question

If $\emptyset$ $\models$ $\phi$, can we say $\phi$ is a tautology because we can entail $\phi$ from nothing?

Answer 1

Yes. To say :

$\Gamma \vDash \phi$

means that :

for every valuation $v$, if $v$ satisfy every formula in $\Gamma$, then $v$ satisfy also $\phi$.

Thus, when we have : $\emptyset \vDash \phi$, this means that :

for every valuation $v$, if $v$ satisfy all the formulae in $\emptyset$, then $v$ satisfy also $\phi$.

But there are no formulae in $\emptyset$ and thus the conditional in the definition of $\vDash$ is vacuously true.

Conclusion : every valuation $v$ satisfy $\phi$, and this amounts to say that $\phi$ is a tautology.