Logical Entailment Definition: "A set of sentences $\Delta$ logically entails a sentence $\varphi$ (written $\Delta \vDash \varphi$) if and only if every truth assignment that satisfies $\Delta$ also satisfies $\varphi$."
Question 1: If there is no truth assignment that satisfies a set of sentences $\Delta$, then what it means for $\varphi$?
Question 2: If a set of sentences $\Delta$ is empty,  then what it means for $\varphi$?
 A: 1) Then for any interpretation $I$, the following conditional is true (by virtue of having a false antecedent): if $I$ satisfies $\Delta$ it satisfies $\varphi$. So $\Delta \vDash \varphi$. But that doesn't settle the value of $\varphi$ on any given interpretation. [Example: $p, \neg p \vDash q$, but that doesn't settle the value of $q$!]
2) If $\Delta$ is empty, then  $\Delta \vDash \varphi$ is equivalent to  $\vDash \varphi$. [Why? Any interpretation $I$ makes true the wffs in $\Delta$ -- that's the null task! -- so $\Delta \vDash \varphi$ comes to this: every interpretation makes $\varphi$ true, i.e. $\vDash \varphi$.]
A: Question 1: $\phi$ can be anything. If no truth assignment satisfies $\Delta$, then there are no truth assignments to check whether they satisfy $\phi$. Another way to say this is that an unsatisfiable $\Delta$ logically implies any $\phi$.
Question 2: $\phi$ is a logical truth, for example, "$p$ or not $p$". Every truth assignment satisfies the empty set of propositions, so that if $p$ is entailed by the empty set, than $p$ holds under any truth assignment. So $p$ will hold no matter truth values we assign to propositional variables. In fact for a connection with philosophy, the logician Alfred Tarski in the early 20th century defined the philosophical concept of a "logical truth" using this model-theoretical definition of a statement which is implied the empty set.
The answers to questions 1 and 2 are two extremes. If we make $\Delta$ very large, the set of truth assignments satisfying it will be as small as possible, and then the set of $\phi$ following from $\Delta$ will be as large as possible. Conversely if we make $\Delta$ very small, the set of truth assignments satisfying it will be very large, and then the set of $\phi$ following from $\Delta$ will be as small as possible, the set of logical truths. Hope that helps! 
