What is the set of models of the empty set?

Question

We have that :

$Mod (\emptyset) = \{ I \in Int_{At} / I \vDash \emptyset \}.$

My question is, what is the set of interpretations that satisfty the empty set?

Answer 1

If we are dealing with propositional logic, we usually write $v(\varphi)=1$ or $v(\varphi)=\text T$, where the valuation $v$ is a mapping from the set $\text{At}$ of atoms into $\{ 0, 1 \}$ ( or : $\{ \text F, \text T \}$), to mean that :

$v$ satisfies formula $\varphi$.

Having said that, we may write also : $v \vDash \varphi$.

What does it mean for a valuation $v$ to satisfy the empty set of formulas :

$v \vDash \emptyset$ ?

We have to agree that, in general :

$v \vDash \Gamma \text { iff } v \vDash \psi, \text { for every } \psi \in \Gamma$.

If so, the condition : $\text { for every } \psi \ [\psi \in \emptyset \Rightarrow v \vDash \psi]$, is vacuosulsy true.

Thus, we have that : for every $v : v \vDash \emptyset$.