Number of Boolean Interpretations for Finite Set of Variables

Theorem
Let $\mathcal P_0$ be a finite signature for the language of propositional logic.

That is, let $\mathcal P_0$, the vocabulary of propositional logic, be a finite set.

Let $n$ denote the number of letters in $\mathcal P_0$.

Then there are $2^n$ different boolean interpretations for $\mathcal P_0$.

Proof
A boolean interpretation is a mapping from $\mathcal P_0$ to the set of truth values $\left\{{T, F}\right\}$.

By Cardinality of Set of All Mappings, the total number of mappings from $S$ to $T$ is:


 * $\left|{T^S}\right| = \left|{T}\right| ^ {\left|{S}\right|}$

The result follows directly.