# Definition:Polish Notation/Formal Definition

## Definition

Let $\mathcal A$ be an alphabet.

Let each $s \in \mathcal A$ 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.