Number of Boolean Interpretations for Finite Set of Variables

Theorem
Let $\mathcal P_0$ be the vocabulary of language of propositional logic.

Let $S \subseteq \mathcal P_0$ be a finite set of $n$ letters from $\mathcal P_0$.

Then there are $2^n$ different partial boolean interpretations for $S$.

Proof
A partial boolean interpretation for $S$ is a mapping from $S$ 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.