\begin{definition}[Definition:Polish Notation/Formal Definition]
Let $\AA$ be an alphabet.
Let each $s \in \AA$ be assigned a natural number called its arity.
The formal grammar for Polish notation is given by the single bottom-up rule:
::If $s$ has arity $n$ and $\phi_1, \ldots, \phi_n$ are well-formed formulas, then:
:::$s \phi_1 \cdots \phi_n$
::is also a well-formed formula.
Notably, in the case where $s$ has arity $0$, this is a vacuous truth, so any such $s$ constitutes a well-formed formula.
\end{definition}