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A modal formula is built from propositional variables and logical constants using Boolean connectives and modal operators, such as the box operator. In provability logic, arithmetic realization replaces propositional letters by arithmetic sentences and interprets a box as the corresponding arithmetic provability assertion. Modal validity, theoremhood and truth under a particular arithmetic realization are distinct notions.

A modal operator forms a new formula from a formula, such as $A\mapsto \square A$. In provability logic, it represents an assertion that a proof of $A$ exists, rather than an assertion that $A$ is true. A box is not interchangeable with an unboxed formula.

The Gödel-Löb logic $\mathrm{GL}$ uses propositional variables, $\perp$, implication, and a unary modal operator $\square$. Other Boolean connectives are abbreviations; for example $\neg A$ means $A\to \perp$ and $\top$ means $\perp \to \perp$.

Its axioms are all classical propositional tautologies and the two schemesIts rules are the modus ponens rule and the necessitation rule, $\mathrm{GL}⊢A\Rightarrow \mathrm{GL}⊢\square A$. Uniform propositional substitution is allowed, or equivalently built into the axiom schemes. necessitation rule applies to theorems; a temporary assumption cannot simply be boxed. In provability logic, $\square A$ is read as the formal assertion that $A$ is provable.