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 $\varphi$?
Best Answer
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. $\vDash \varphi$.]