We here present a number of results about the system GL. Some of these will be of direct interest for the study of provability in PA; others are simply independently interesting (we hope), and these occur toward the end of the chapter. The discussion here of letterless sentences and the notions of rank and trace will be particularly important in the next chapter, where we take up the fixed point theorem, certainly one of the most striking applications of modal logic ever made.

We begin with one of the oldest results of the subject of provability logic, the normal form theorem for letterless sentences. Recall that a modal sentence is called letterless if it contains no sentence letters, equivalently if it is a member of the smallest class containing ⊥ and containing (A→B) and □A whenever it contains A and B.

As ever, □0A = A and □i+1A = □□iA.

We shall say that a letterless sentence C is in normal form if it is a truth-functional combination of sentences of the form □i⊥.

The normal form theorem for letterless sentences

If B is a letterless sentence, there is a letterless sentence C in normal form such that GL⊢B↔C.

Proof. It clearly suffices to show how to construct a letterless sentence in normal form equivalent to □C from a letterless sentence C in normal form.