Question about Logical implication Suppose $A$ is a set of propositional formulas, and suppose $\varphi$ is a propositional formula. In my textbook they write $A \models \varphi$, if for every truth assignment $w$ such that $w(\psi) = 1$ for every $\psi \in A$, we also have $w(\varphi)=1$.
They now ask me to show that $\emptyset\models \varphi$ if and only if $\varphi$ is a tautology. But how can I show this? The definition tells us nothing about empty $A$?

Suppose $A = \varnothing$. Then, for any truth value assignment $w$, $w(\psi) = 1$ (vacuously) for each and every $\psi \in A = \varnothing$  
Why? 
Because there are no $\psi \in A = \varnothing$. So the hypothesis is satisfied vacuously, because it is not possible for there to exist any $\psi \in \varnothing$ such that $w(\psi) \neq 1$.
Hence $\varnothing \models \varphi$ if and only if for each truth value assignment $w$, $w(\varphi) =  1$, too. This holds if and only if $\varphi$ is a tautology.
