Here’s a description of the language of modal logic from Boolos’s The Logic of Provability:

Modal sentences. Fix a countably infinite sequence of distinct objects, of which the first five are `⊥`, `→`, `□`, `(`, `)`, and the others are the sentence letters… Modal sentences will be certain finite sequences of these objects. We shall use `A`, `B`, … as variables over modal sentences. Here is an inductive definition of modal sentence:

(1) `⊥` is a modal sentence;

(2) each sentence letter is a modal sentence;

(3) if `A` and `B` are modal sentences, so is `(A → B)`; and

(4) if `A` is a modal sentence, so is `□(A)`.

[We shall very often write: `(A → B)` and: `□(A)` as: `A → B` and: `□A`.]

Sentences that do not contain sentence letters are letterless. For example, `⊥`, `□⊥`, and `□⊥ → ⊥` are letterless sentences.

It’s easy to translate this into a definition in Reazon miniKanren:

And given such a definition, it’s even easier to generate instances:

• _0
• (□ ⊥)
• (□ _0)
• (⊥ → ⊥)
• (⊥ → _0)
• (□ (□ ⊥))
• (□ (□ _0))
• (0 → ⊥)
• (0 → _1)

All possible modal logic sentences would eventually be generated from that definition.

It’s also easy to add new syntax, like the negation operator `¬`.

We can also tweak the definition to generate specific subsets of sentences. For example, the letterless sentences mentioned above contain some interesting cases, like `¬□⊥ → ¬□¬□⊥`. Under the provability interpretation of modal logic, this sentence says that if there is no proof of a falsehood then there is no proof that there is no proof of a falsehood, i.e. Goedel’s second incompleteness theorem.

To get the letterless sentences, simply don’t include the sentence letters.

• (¬ ⊥)
• (¬ (¬ ⊥))
• (□ ⊥)
• (⊥ → ⊥)
• (¬ (¬ (¬ ⊥)))
• (□ (¬ ⊥))
• (¬ (□ ⊥))
• (¬ (⊥ → ⊥))
• (⊥ → (¬ ⊥))

We’re getting some interesting sentences, like `(¬ (□ ⊥))`, but also a lot of junk, like `(⊥ → ⊥)` and `(¬ (¬ (¬ ⊥)))`. We can fix this by applying `¬` only to sentences beginning with `□` and by applying `→` only to compound sentences.

• (¬ (□ ⊥))
• (□ ⊥)
• (¬ (□ (¬ (□ ⊥))))
• (¬ (□ (□ ⊥)))
• (□ (¬ (□ ⊥)))
• (¬ (□ (¬ (□ (¬ (□ ⊥))))))
• (□ (□ ⊥))
• (¬ (□ (¬ (□ (□ ⊥)))))
• (¬ (□ (□ (¬ (□ ⊥)))))
• (□ (¬ (□ (¬ (□ ⊥)))))
• (¬ (□ (¬ (□ (¬ (□ (¬ (□ ⊥))))))))
• ((¬ (□ ⊥)) → (¬ (□ ⊥)))
• (¬ (□ (□ (□ ⊥))))
• (□ (¬ (□ (□ ⊥))))
• (¬ (□ (¬ (□ (¬ (□ (□ ⊥)))))))
• ((¬ (□ ⊥)) → (□ ⊥))
• (□ (□ (¬ (□ ⊥))))
• (¬ (□ (¬ (□ (□ (¬ (□ ⊥)))))))
• (¬ (□ (□ (¬ (□ (¬ (□ ⊥)))))))
• (□ (¬ (□ (¬ (□ (¬ (□ ⊥)))))))
• (¬ (□ (¬ (□ (¬ (□ (¬ (□ (¬ (□ ⊥))))))))))
• ((¬ (□ ⊥)) → (¬ (□ (¬ (□ ⊥)))))

The twenty-third statement generated from this definition is `((¬ (□ ⊥)) → (¬ (□ (¬ (□ ⊥)))))`, the modal logic statement of the second incompleteness theorem. Further on can be found its converse `((□ ⊥) → (□ (¬ (□ ⊥))))`, and further still can be found `((□ ⊥) → (□ (□ ⊥)))`, an instance of one of the Hilbert-Bernays derivability conditions.

We can also generate substitution instances. Take the modal distribution axiom `((□ (A → B)) → ((□ A) → (□ B)))`.

• ((□ (⊥ → ⊥)) → ((□ ⊥) → (□ ⊥)))
• ((□ (⊥ → (¬ (□ ⊥)))) → ((□ ⊥) → (□ (¬ (□ ⊥)))))
• ((□ ((¬ (□ ⊥)) → ⊥)) → ((□ (¬ (□ ⊥))) → (□ ⊥)))
• ((□ (⊥ → (□ ⊥))) → ((□ ⊥) → (□ (□ ⊥))))
• ((□ (⊥ → (¬ (□ (¬ (□ ⊥)))))) → ((□ ⊥) → (□ (¬ (□ (¬ (□ ⊥)))))))
• ((□ ((¬ (□ ⊥)) → (¬ (□ ⊥)))) → ((□ (¬ (□ ⊥))) → (□ (¬ (□ ⊥)))))
• ((□ ((□ ⊥) → ⊥)) → ((□ (□ ⊥)) → (□ ⊥)))
• ((□ (⊥ → (¬ (□ (□ ⊥))))) → ((□ ⊥) → (□ (¬ (□ (□ ⊥))))))
• ((□ ((¬ (□ ⊥)) → (□ ⊥))) → ((□ (¬ (□ ⊥))) → (□ (□ ⊥))))
• ((□ (⊥ → (□ (¬ (□ ⊥))))) → ((□ ⊥) → (□ (□ (¬ (□ ⊥))))))

Notice that the third result, `((□ ((¬ (□ ⊥)) → ⊥)) → ((□ (¬ (□ ⊥))) → (□ ⊥)))`, contains the contrapositive of the second incompleteness theorem as its consequent.