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.

In the present chapter we are going to investigate the connections between modal logic and set theory, i.e., Zermelo–Fraenkel set theory, “ZF”, for short.

This chapter and the next, which deals with second-order arithmetic, or analysis as it is sometimes called, are, unfortunately, not self-contained. In order even to explain, let alone prove, their most interesting results we are forced to assume a level of knowledgeability about logical matters quite a bit higher than was necessary for the understanding of previous chapters. (Including the necessary background material would entail a lengthy exposition of matters largely irrelevant to the aims of this work.) The present chapter consists mainly of the proofs of two striking completeness theorems that concern interesting weakenings of the notion of provability: truth in all transitive models of set theory and truth in all models Vκ (alias Rκ), κ inaccessible. The theorems were discovered by Robert Solovay in the fall of 1975; their proofs have not hitherto appeared in print.

To understand the proofs of these results, one has to have a reasonable acquaintance with basic set theory, as well as some basic notions involved in proofs of independence à la Gödel and Cohen. An excellent source for this material is Kunen's Set Theory. The relevant system of modal logic for the notion: truth in all Vκ, κ inaccessible, is stronger than that for truth in all transitive models, which is itself stronger than GL.

Peano arithmetic (PA, or arithmetic, for short) is classical first-order arithmetic with induction. The aim of this chapter is to define the concepts mentioned in, and describe the proofs of, five important theorems about Bew(x), the standard “provability” or “theoremhood” predicate of PA:

1. (i) If ⊢ S, then ⊢ Bew(⌜S⌝),

2. (ii) ⊢ Bew(⌜(S → T)⌝) → (Bew(⌜S⌝) → Bew(⌜T⌝)),

3. (iii) ⊢ Bew(⌜S⌝) → Bew(⌜Bew(⌜S⌝)⌝),

4. (iv) Bew(⌜S⌝) is a Σ sentence, and

5. (v) if S is a Σ sentence, then ⊢ S → Bew(⌜S⌝)

(for all sentences S, T of Peano arithmetic).

‘⊢’ is, as usual, the sign for theoremhood; in this chapter we write ‘⊢S’ to mean that S is a theorem of PA. ┌S┐ is the numeral in PA for the Gödel number of sentence S, that is, if n is the Gödel number of S, then ┌S┐ is 0 preceded by n occurrences of the successor sign s. Bew(┌S┐) is therefore the result of substituting ┌S┐ for the variable x in Bew(x), and (iii) immediately follows from (iv) and (v). Bew(┌S┐) may be regarded as a sentence asserting that S is a theorem of PA. Σ sentences (often called Σ1 sentences) are, roughly speaking, sentences constructed from atomic formulas and negations of atomic formulas by means of conjunction, disjunction, existential quantification, and bounded universal quantification (“for all x less than y”), but not negation or universal quantification. A precise definition is given below.

Notice the distinction between ‘Bew(x)’ and ‘⊢’. ‘Bew(x)’ denotes a certain formula of the language of PA and thus Bew(x) is that formula; it is a formula that is true of (the Gödel numbers of) those formulas of PA that are provable in PA.

The semantical treatment of modal logic that we now present is due to Kripke and was inspired by a well-known fantasy often ascribed to Leibniz, according to which we inhabit a place called the actual world, which is one of a number of possible worlds. (It is a further part of the fantasy, which we can ignore, that because of certain of its excellences God selected the possible world that we inhabit to be the one that he would make actual. Lucky us.) Each of our statements is true or false in – we shall say at – various possible worlds. A statement is true at a world if it correctly describes that world and false if it does not. We sometimes call a particular statement true or false, tout court, but when we do, we are to be understood as speaking about the actual world and saying that the statement is true or false at it. Some of the statements we make are true at all possible worlds, including of course the actual world; these are the so-called necessary statements. A statement to the effect that another is necessary will thus be true if the other statement is true at all possible worlds. It follows that if a statement is necessary, then it is true. Some statements are true at at least one possible world; these are the possible statements.

Here and in our final chapter we study quantified (or predicate) provability logic. We consider translations of formulas of quantified modal logic (QML) into the language of arithmetic under which the box □ of modal logic is taken, as in earlier parts of this work, to represent provability in arithmetic. In the “pure” predicate calculus, function signs, the equals-sign =, and modal logical symbols such as □ and ◊ do not occur. We shall define an expression to be a formula of QML if and only if it can be obtained from a formula of the “pure” predicate calculus, by replacing (zero or more) occurrences of the negation sign ¬ with occurrences of □. Thus □ and ¬ have the same syntax in QML, as was the case in propositional modal logic.

Our results are negative: we show that there are no simple characterizations of the always provable or always true sentences of QML. Apart from the definition of the sentence D and Lemma 7 below, curiously little use is made of the quantificational–modal–logical properties of Bew(x). Indeed, the main definitions, techniques, and theorems that are to follow may seem to come from a branch of logic rather unrelated to the one we have been studying up to now.

We shall suppose that the variables, v0,v1,…, are common to the languages of QML and of arithmetic. The first n variables are, of course, v0,…,vn−1.

We shall now present a method for constructing modal-logical models. The method enables us to construct from each consistent normal system L of propositional modal logic a model ML, = 〈WL, RL, VL〉, called the canonical model for L, in which all and only the theorems of L are valid. Although canonical models are of great interest in the study of systems of modal logic other than GL, the canonical model for GL is not particularly useful for the study of GL itself. (Outside this chapter, the notion of a canonical model is used to prove only one theorem in this book, Theorem 3 of Chapter 13.)

We shall begin by defining the canonical model for a consistent normal system L and then prove a completeness theorem for each member of a quite large family of systems that includes K, K4, T, S4, B, and S5 – but not GL, alas.

Let L be a consistent system of normal modal propositional logic. Thus L⊬⊥.

A set X of arbitrary modal sentences is called (L-) consistent iff for no finite subset Y of X, L⊢¬ ∧ Y. If X is consistent, at most one of A and ¬A belongs to X; otherwise, evidently, L⊢¬(A ∧ ¬A), and X is not consistent.

One of the principal aims of this study is to investigate the effects of interpreting the box of modal logic to mean “it is provable (in a certain formal theory) that…”. When modal logic is viewed in this way, a question immediately comes to mind: Which principles of modal logic are correct when the box is interpreted in this way? The answer is not evident; near the end of this chapter we shall say what the answer is, and in Chapter 9, when we prove the arithmetical completeness theorems of Solovay, we shall show that it is the answer.

In order to express our question precisely, we make two definitions:

A realization is a function that assigns to each sentence letter a sentence of the language of Peano arithmetic. It is standard practice to use “*” as a variable over interpretations; we shall use “#” as well.

The translation A* of a modal sentence A under a realization * is defined inductively:

1. ⊥ = ⊥

2. p* = *(p) (p a sentence letter)

3. (A → B)* = (A* → B*)

4. ⌜(A)* = Bew[⌜A*⌝]

(Bew[A*] = Bew(┌A*┐), as A* is a sentence.)

We have taken ⊥ and → to be among the primitive logical symbols of PA, and therefore the translation of any modal sentence under any realization is a sentence of the language of PA. Clauses (1) and (3) guarantee that the translation (under *) of a truthfunctional combination of sentences is that same truth-functional combination of the translations of those sentences.

The beautiful fixed point theorem for GL, due independently to Dick de Jongh and Giovanni Sambin, is the most striking application of modal logic to the study of the concept of provability in formal systems.

We recall the two definitions necessary for the statement of the theorem.

⊡A is the sentence (□A ∧ A).

A sentence A is said to be modalized in p if every occurrence of the sentence letter p in A is in the scope of an occurrence of □; equivalently, A is modalized in p if and only if A is a truth-functional compound of sentences of the form □D and sentence letters other than p.

The fixed point theorem then reads: For every sentence A modalized in p, there is a sentence H containing only sentence letters contained in A, not containing the sentence letter p, and such that GL⊢⊡(p↔A)↔⊡(p↔H).

Any such sentence H is called a fixed point of A.

If GL⊢H↔I, then GL⊢⊡(H↔I), and therefore GL⊢⊡(p↔H)↔⊡(p↔I). And if GL⊢⊡(p↔H)↔⊡(p↔I) and neither H nor I contains p, then substituting H for p yields GL⊢⊡(H↔H)↔⊡(H↔I), whence GL⊢H↔I. It follows that any sentence equivalent in GL to a fixed point of A and containing only sentence letters in A other than p is itself a fixed point of A, and that all fixed points of A are equivalent in GL.

We are now going to establish a completeness theorem for each of the seven modal systems we have considered. We call a frame (or a model) appropriate to K4, T, B, S4, S5, or GL if and only if it is transitive, reflexive, symmetric and reflexive, transitive and reflexive, euclidean and reflexive, or transitive and converse well founded, respectively of course. All frames are appropriate to K. In Chapter 11 we shall give a general definition of a frame's being appropriate to a normal system, but as yet we have only defined the notion with respect to seven particular normal systems.

We are going to show that a modal sentence A is a theorem of one of our seven systems L if A is valid in all finite frames that are appropriate to L – equivalently, if A is valid in all finite models appropriate to L.

Thus, e.g., we shall show that if A is valid in all finite transitive and reflexive frames, then A is a theorem of S4. When we have done so, we shall have established the coextensiveness of the conditions:

validity in all transitive and reflexive frames;

validity in all finite transitive and reflexive frames;

provability in S4.

For, as we saw in Chapter 4, if A is a theorem of S4, A is valid in all transitive and reflexive frames, and thus certainly valid in all finite transitive and reflexive frames.