We now embark on a more thorough study of natural deduction, normalization and the structure of normal derivations. We describe a simple normalization strategy w.r.t. a specific set of conversions which transforms every deduction in Ni into a deduction in normal form; moreover, for →Nm we prove that deductions are in fact strongly normalizable, i.e. every sequence of normalization steps terminates in a normal deduction, which is in fact unique.

As in the case of cut elimination, there is a hyperexponential upper bound on the rate of growth under normalization. From a suitable example we also easily obtain a hyperexponential lower bound. This still leaves open the possibility that each theorem might have at least some cutfree deduction of “modest” length; but this possibility is excluded by an example, due to Orevkov, of a sequence of statements Cn, n ∈ ℕ, with deductions linearly bounded by n, for which the minimum depth of arbitrary normal proofs has a hyperexponential lower bound.

This points to the very important role of indirect proof in mathematical reasoning: without indirect reasoning, exemplified by non-normal proofs, we cannot present proofs of manageable size for the Cn.

Conversions and normalization

In this and the next section we shall study the process of normalization for Ni, which corresponds to cut elimination for intuitionistic sequent calculi.

We shall assume, unless stated otherwise, that applications of ⊥i have atomic conclusions in the deductions we consider.

In this chapter we study another form of inference, which forms the keystone of logic programming and certain theorem-proving systems. We do not aim at giving a complete introduction to the theory of logic programming; rather, we want to show how resolution is connected with other formalisms and to provide a proof-theoretic road to the completeness theorem for SLD-resolution.

The first three sections deal with propositional resolution, unification and resolution in predicate logic. The last two sections illustrate for Cp and Ip how deductions in a suitably chosen variant of the Gentzen systems can be directly translated into deductions based on resolution, which often permits us to lift strategies for proof search in Gentzen systems to resolution-based systems. The extension of these methods to predicate logic is more or less straightforward.

Introduction to resolution

Propositional linear resolution is a “baby example” of resolution methods, which is not of much interest in itself, but may serve as an introduction to the subject.

We consider programs consisting of finitely many sequents (clauses) of the form Γ ⇒ P, P a propositional variable and Γ a finite multiset of propositional variables (“definite clauses”, “Horn clauses” or “Horn sequents”). A goal or query Γ is a finite (possibly empty) set of propositional variables, and may be identified with the sequent Γ ⇒. [] is the empty goal.

The “applications of cut elimination” in the title of this chapter may perhaps be described more appropriately as “applications of cutfree systems”, since the applications are obtained by analyzing the structure of cutfree proofs; and in order to prove that the various cutfree systems are adequate for our standard logics all we need to know is that these systems are closed under Cut (that is to say, Cut is a an admissible rule). Nevertheless there are good reasons to be interested in the process of cut elimination, as opposed to semantical proofs of closure under Cut. True, the usual semantical proofs establish not only closure under Cut, but also completeness for the semantics considered. On the other hand, the proof of cut elimination for G3c is at least as efficient as the semantical proof (although G3cp permits a very fast semantical proof of closure under Cut), and in the case of logics with a more complicated semantics (such as intuitionistic logic, and the modal logic S4 in chapter 9) often more efficient. For linear logic in section 9.3, so far no semantical proof of closure under Cut has been published. Other reasons for being interested in the process of cut elimination will be found in certain results in sections 5.1 and 6.9, which describe bounds on the increase in size of deductions under cut elimination and normalization respectively.

Gentzen [1935] introduced his calculi LK, LJ as formalisms more amenable to metamathematical treatment than natural deduction. For these systems he developed the technique of cut elimination. Even if nowadays normalization as an “equivalent” technique is widely used, there are still many reasons to study calculi in the style of LK and LJ (henceforth to be called Gentzen calculi or Gentzen systems, or simply G-systems):

• Where normal natural deductions are characterized by a restriction on the form of the proof – more precisely, a restriction on the order in which certain rules may succeed each other – cutfree Gentzen systems are simply characterized by the absence of the Cut rule.

• Certain results are more easily obtained for cutfree proofs in G-systems than for normal proofs in N-systems.

• The treatment of classical logic in Gentzen systems is more elegant than in N-systems.

The Gentzen systems for M, I and C have many variants. There is no reason for the reader to get confused by this fact. Firstly, we wish to stress that in dealing with Gentzen systems, no particular variant is to be preferred over all the others; one should choose a variant suited to the purpose at hand. Secondly, there is some method in the apparent confusion.

As our basic system we present in the first section below a slightly modified form of Gentzen's original calculi LJ and LK for intuitionistic and classical logic respectively: the Gl-calculi. In these calculi the roles of the logical rules and the so-called structural rules are kept distinct.

Until we come to chapter 9, we shall concentrate on our three standard logics: classical logic C, intuitionistic logic I and minimal logic M. The informal interpretation (semantics) for C needs no explanation here. The logic I was originally motivated by L. E. J. Brouwer's philosophy of mathematics (more information in Troelstra and van Dalen [1988, chapter 1]); the informal interpretation of the intuitionistic logical operators, in terms of the primitive notions of “construction” and “constructive proof”, is known as the “Brouwer–Heyting–Kolmogorov interpretation” (see 1.3.1, 2.5.1). Minimal logic M is a minor variant of I, obtained by rejection of the principle “from a falsehood follows whatever you like” (Latin: “ex falso sequitur quodlibet”, hence the principle is often elliptically referred to as “ex falso”), so that, in M, the logical symbol for falsehood ⊥ behaves like some unprovable propositional constant, not playing a role in the axioms or rules.

This chapter opens with a precise description of N-systems for the full first-order language with proofs in the form of deduction trees, assumptions appearing at top nodes. After that we present in detailthe corresponding term system for the intuitionistic N-system, an extension of simple type theory. Once a precise formalism has been specified, we are ready for a section on the Godel-Gentzen embedding of classical logic into minimal logic. This section gives some insight into the relations between C on the one hand and M, I on the other hand.

This chapter is devoted to two topics: the rate of growth of deductions under the process of cut elimination, and permutation of rules.

It is not hard to show that there is a hyperexponential upper bound on the rate of growth of the depth of deductions under cut elimination. For propositional logic much better bounds are possible, using a clever strategy for cut elimination. This contrasts with the situation for normalization in the case of N-systems (chapter 6), where propositional logic is as bad as predicate logic in this respect.

In contrast to the case of normalization for N-systems, it is not easy to extract direct computational content from the process of cut elimination for G-systems, since as a rule the process is non-deterministic, that is to say the final result is not a uniquely defined “value”. Recent proof-theoretical studies concerning linear logic (9.3) lead to a more or less satisfactory analysis of the computational content in cut elimination for C (and I); in these studies linear logic serves to impose a “fine structure” on sequent deductions in classical and linear logic (some references are in 9.6.5).

We also show that in a GS-system for Cp with Cut there are sequences of deduction with proofs linearly increasing in size, while the size of their cutfree proofs has exponentially increasing lower bounds.

These results indicate that the use of “indirect proof”, i.e. deductions that involve some form of Cut play an essential role in formalized versions of proofs of theorems from mathematical practice, since otherwise the length of proofs would readily become unmanageable.

For this chapter preliminary knowledge of some basic notions of category theory (as may be found, for example, in Mac Lane [1971], Blyth [1986], McLarty [1992], Poigné [1992]) will facilitate understanding, but is not necessary, since our treatment is self-contained. Familiarity with chapter 6 is assumed.

In this chapter we introduce another type of formal system, inspired by notions from category theory. The proofs in formalisms of this type may be denoted by terms; the introduction of a suitable equivalence relation between these terms makes it possible to interpret them as arrows in a suitable category.

In particular, we shall consider a system for minimal →∧⊤-logic connected with a special cartesian closed category, namely the free cartesian closed category over a countable discrete graph, to be denoted by CCC(PV). In this category we have a decision problem: when are two arrows from A to B the same?

This problem will be solved by establishing a correspondence between the arrows of CCC(PV) and the terms of the extensional typed lambda calculus. For this calculus we can prove strong normalization, and the decision problem is thereby reduced to computing and comparing normal forms of terms of the lambda calculus.

Another interesting application of this correspondence will be a proof of a certain “coherence theorem” for CCC(PV). (A coherence theorem is a theorem of the form: “between two objects satisfying certain conditions there is at most one arrow”.)

Another possible title for this chapter might have been “some non-standard logics”, since its principal aim is to illustrate how the methods we introduced for the standard logics M, I and C are applicable in different settings as well.

For the illustrations we have chosen two logics which are of considerable interest in their own right: the wellknown modal logic S4, and linear logic. For a long time modal logic used to be a fairly remote corner of logic. In recent times the interest in modal and tense logics has increased considerably, because of their usefulness in artificial intelligence and computer science. For example, modal logics have been used (1) in modelling epistemic notions such as belief and knowledge, (2) in the modelling of the behaviour of programs, (3) in the theory of non-monotonic reasoning.

The language of modal logics is an extension of the language of first-order predicate logic by one or more propositional operators, modal operators or modalities. Nowadays modal logics are extremely diverse: several primitive modalities, binary and ternary operators, intuitionistic logic as basis, etc. We have chosen S4 as our example, since it has a fairly well-investigated proof theory and provides us with the classic example of a modal embedding result: intuitionistic logic can be faithfully embedded into S4 via a so-called modal translation, a result presented in section 9.2.

Linear logic is one of the most interesting examples of what are often called “substructural logics” – logics which in their Gentzen systems do not have all the structural rules of Weakening, Exchange and Contraction.

Proof theory may be roughly divided into two parts: structural proof theory and interpretational proof theory. Structural proof theory is based on a combinatorial analysis of the structure of formal proofs; the central methods are cut elimination and normalization.

In interpretational proof theory, the tools are (often semantically motivated) syntactical translations of one formal theory into another. We shall encounter examples of such translations in this book, such as the Gödel-Gentzen embedding of classical logic into minimal logic (2.3), and the modal embedding of intuitionistic logic into the modal logic S4 (9.2). Other wellknown examples from the literature are the formalized version of Kleene's realizability for intuitionistic arithmetic and Gödel's Dialectica interpretation (see, for example, Troelstra [1973]).

The present text is concerned with the more basic parts of structural proof theory. In the first part of this text (chapters 2–7) we study several formalizations of standard logics. “Standard logics”, in this text, means minimal, intuitionistic and classical first-order predicate logic. Chapter 8 describes the connection between cartesian closed categories and minimal conjunctionimplication logic; this serves as an example of the applications of proof theory in category theory. Chapter 9 illustrates the extension to other logics (namely the modal logic S4 and linear logic) of the techniques introduced before in the study of standard logics. The final two chapters deal with first-order arithmetic and second-order logic respectively.

Preface to the first edition

The discovery of the set-theoretic paradoxes around the turn of the century, and the resulting uncertainties and doubts concerning the use of high-level abstractions among mathematicians, led D. Hilbert to the formulation of his programme: to prove the consistency of axiomatizations of the essential parts of mathematics by methods which might be considered as evident and reliable because of their elementary combinatorial (“finitistic”) character.

Although, by Gödel's incompleteness results, Hilbert's programme could not be carried out as originally envisaged, for a long time variations of Hilbert's programme have been the driving force behind the development of proof theory. Since the programme called for a complete formalization of the relevant parts of mathematics, including the logical steps in mathematical arguments, interest in proofs as combinatorial structures in their own right was awakened. This is the subject of structural proof theory; its true beginnings may be dated from the publication of the landmark-paper Gentzen [1935].

Nowadays there are more reasons, besides Hilbert's programme, for studying structural proof theory. For example, automated theorem proving implies an interest in proofs as combinatorial structures; and in logic programming, formal deductions are used in computing.

There are several monographs on proof theory (Schütte [1960,1977], Takeuti [1987], Pohlers [1989]) inspired by Hilbert's programme and the questions this engendered, such as “measuring” the strength of subsystems of analysis in terms of provable instances of transfinite induction for definable wellorderings (more precisely, ordinal notations).