My textbook says:
I can't understand how $\vDash p$ is the same as $\emptyset \vDash p$. How many models $t$ are there on the $\Sigma = \emptyset$? Zero! So there is no "truth assigment" that can actually verify that $t(p) = 1$. I don't understand why I can't have $\emptyset \vDash \bot$ (which is nonsense: falsum is not a tautology by the definition): one has to show that there exists some $t$ and that $t(\bot) \neq 1$... but there are no such $t$.
What am I missing here?
A model of $\varnothing$ is a truth assignment such that $t(p)=1$ for all $p\in\varnothing$. This condition is always vacuously true, so every truth assignment is a model of $\varnothing$.