Normalization, Cut-Elimination, and the Theory of Proofs (Center for the Study of Language and Information - Lecture Notes)

NORMALIZATION, CUT-ELIMINATION AND THE THEORY OF PROOFS

CSLI Lecture Notes No. 28

NORMALIZATION, CUT-ELIMINATION AND THE THEORY OF PROOFS

A. M. Ungar

V^OJUfA

CENTER FOR THE STUDY OF LANGUAGE AND INFORMATION

Copyright © 1992 Center for the Study of Language and Information Leland Stanford Junior University Printed in the United States CIP data and other information appear at the end of the book

Contents Introduction

1

1

Background

11

2

Comparing NJ with LJ

3

Natural Deduction Revisited

46

4

The Problem of Substitution

55

5

A Multiple-Conclusion Calculus

6

Reduction Procedures

7

Correspondence Results

8

Interpretations of Derivations

32

77

102 126 153

Appendices A

A Strong Cut-Elimination Theorem for LJ

B

A Formulation of the Classical Sequent Calculus

C

Proofs and Categories

List of Works Cited Index

220

229

234

v

186 200

Introduction The idea that proofs ate objects capable of being treated by a mathematical theory is due to Hilbert. His motivations, the resulting theory, its successes and failures are too well known to need rehearsing here. There is, however, one point to be emphasized about Hilbert's theory: because it was in part at least an attempt to reduce abstract reasoning to reasoning about concrete entities, proofs were identified with the derivations of a formal system and it was these latter which were the objects of study. It is equally well known that Hilbert's program found little favor with Brouwer who objected, on the one hand, that the foundational purpose of the program could not be achieved since the study of formal systems presupposed abstract mathematical principles of the kind it was intended to justify and, on the other, that Hilbert had falsified the nature of mathematics by identifying it with an activity which was essentially linguistic (or, at the very least, which was reducible to such an activity). Because Brouwer viewed mathematics as an activity taking place in the minds of individual mathematicians, a formal proof of the kind studied by Hilbert—or even an informal proof of the kind to be found in any account of mathematics—could serve only as a device to aid memory or communication. In fact, the same could be said of mathematical statements in general. As proofs describe certain kinds of mental activity, so assertions report that the activity in question has or may be carried out; more specifically, a mathematical statement asserts the execution (either as a possibility or as a fait accompli) of the mental construction described by its proof. This radical reinterpret at ion of mathematical statements so that, instead of describing properties of and relations between various kinds of objects, they refer to mental activity assigns to proofs a far more important role in the study of mathematics than being the subject matter of a small part of it. On this conception, the mental constructions described by proofs are the subject matter of every mathematical assertion. (Of course, the proofs themselves are secondary to the experiences or activities which they describe. It is the latter which validate them and there is no guaran1

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Normalization, Cut-Elimination and the Theory of Proofs

tee that every such experience can be given satisfactory expression in this form.) This emphasis upon the priority of non-linguistic activity concentrates attention upon what proofs describe and in the subsequent tradition even the meaning of the term proof has shifted from 'Interpreted linguistic structure" to "the interpretation of a certain linguistic structure," while a shift in the opposite direction takes place in the tradition beginning with Hilbert (proofs as interpreted linguistic structures are bled of their meaning and identified with derivations of a formal system). Brouwer's conception of mathematics may strike some as eccentric. Nevertheless, the criticisms he advances against Hilbert are well taken. Even if one doubts whether thought can be separated from language and believes that mathematical statements are indeed what they appear prima facie to be, namely, assertions about the properties of independently existing objects, a distinction can be drawn between proofs as linguistic objects and their interpretations, and it is the latter which are of primary interest. In fact, with the perspective of hindsight, it seems clear that Hilbert's proof theory is misnamed—on any interpretation of the term "proof: what he was interested in studying was not proof, but the combinatorial structure of particular representations of proofs. The interest of such a study depends of course upon the ability to translate knowledge about the latter into knowledge about the former. But, granted this possibility, the issue arises as to whether there are interesting questions about the former which cannot be approached through a study of the latter and, if so, whether a mathematical study of the former would be a profitable way to approach them. As far as the first question is concerned, it seems clear that the study of the interpretations of proofs has a general metaphysical interest of the same kind as the investigation of propositions—the interpretations of sentences— and is prompted by the same kinds of consideration. Sentences are tied to particular languages and are nothing more than conventional arrangements of signs unless they are thought of as expressing something significant; furthermore, whatever they express is not tied to a particular language and may indeed be independent of language altogether. The most fundamental problems about propositions concern their ontological status and their structure. At the very least, if there are to be propositions at all, we need to know when sentences express the same proposition, in other words, to be able to formulate identity criteria for propositions. The same can be said about the interpretations of proofs. Not only are proofs tied to a particular language but, if we consider formal derivations, they are constructed according to a particular set of rules whose choice is in some sense logically insignificant. (I am not referring here to the choice between first-order and higher-order rules, for example, or between classical and intuitionistic ones. These presumably are substantive ones motivated by the intended interpretation of the proofs in question. I am thinking rather of the choice that is made when one formulation of classical first-

Introduction

3

order logic, say, is preferred to another.) The recognition that two proofs constructed according to different rules (from the same or different calculi) are the same can only be explained by reference to their interpretations, the analogues of propositions, whose identity is being asserted. It might be thought that the problems associated with identifying and individuating propositions are sufficiently difficult that they become aggravated to the point of intractability when addressed in the context of proofs. It seems to me, however, that the opposite is in fact true, i.e., that they become more amenable to solution. It seems quite plausible after all to claim that logical operations can be represented by structurally similar linguistic ones, with the result that a connection can be established between the structure of a proof and that of its interpretation. On the other hand, there is little reason to believe in a general structural correspondence between sentences and the propositions they express. Furthermore, even if the interpretation of a mathematical proof should turn out to be a structure built up from propositions, at least these will be logical combinations of propositions of a standard, rather simple form (for example, the interpretations of equations whose terms range over a well-determined domain). In addition to questions of general philosophical interest, there are mathematically interesting ones which can be formulated in terms of proofs. On the view suggested above, proofs provide evidence for mathematical theorems. It is wrong, however, to think of any science, physical, mathematical or social, simply as a collection of putative truths the evidence for which, although perhaps essential to the determination of these truths, is not part of the subject matter of the science in question. The very idea of an organized body of knowledge implies that a theory is not to be described simply as a set of statements; these statements must be related to one another and, in some cases, to other data. Furthermore, in our scientific studies we seek not only to know what is true but also to understand why. In other words, the evidence for the claims of a particular science are themselves a part of it, and deserve to be considered as such. The application of this claim to mathematics is scarcely controversial. It boils down to saying that proofs are an integral part of mathematics on any conception of the subject. Even those mathematicians who have railed against them seem to have directed their barbs more against what they considered to be excessive rigor, or a mistaken view of the relationship between proofs and intuition, rather than against the notion of proof itself.1 The point is worth making, nevertheless, since the trend in logic during the last 60 years has been to confine serious interest in proofs to those 1 See, for example, the authors quoted on this subject by Morris Kline in The Loss of Certainty, Oxford, 1980.

Mathematics:

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Normalization, Cut-Elimination and the Theory of Proofs

favoring constructive or intuitionistic interpretations of mathematics. This is because of what might be called the model-theoretic view of mathematics, namely, that in mathematics one attempts to discover what statements are true in certain structures. Proofs matter solely because they provide a means of establishing such truths; hence, the importance of axiomatizability, etc. I am not claiming that this view is false, but it is surely incomplete. The model-theoretic view provides a useful classificatory framework for mathematics, but it neglects certain features of the subject. In particular, it suggests that axiomatizations are to be judged only by external criteria—for example, by the set of their consequences—and it neglects the explanatory function of proofs by judging them only by what they establish. In fact, how a theorem is established and the methods employed in its proof are matters of interest to all kinds of mathematicians, even though the nature of this interest, like the nature of the proofs themselves, may vary depending upon one's viewpoint. It is easy enough to illustrate this thesis from the history of mathematics, beginning with the ruler and compass constructions of the ancients and ending with the current interest in mechanical theorem proving and checking. More generally, it seems fair to claim that mathematicians are interested in why a theorem holds, not simply in its holding;2 they recognize that different proofs of the same result—for example, Cantor's and Liouiville's proofs of the existence of transcendental numbers—each have their own significance, and interest themselves in new proofs of familiar results. (The question here is not whether one proof is more reliable than another, but what additional understanding a particular proof provides.) Such an interest, however, presupposes an ability to distinguish between proofs of the same conclusion, in particular cases at least, and suggests more general questions about their identity and difference. Granted the mathematical interest of questions about proofs, it remains to argue that these are best formulated as questions about their interpretations, rather than about certain kinds of linguistic objects. It seems clear that the preceding discussion is not about the combinatorial properties of formal objects, but at the very least about interpreted proofs. Even a discussion of machine provability can only be carried out within a framework in which the reduction of proofs to formal objects is not presupposed. As for the advantages and disadvantages of interpretations as opposed to interpreted linguistic objects, the advantages are clear. Interpretations are not tied to particular languages, to particular formulations of the rules nor to particular concrete representations of the structure of proofs. Furthermore, they are fundamental to the questions raised above; our interest in linguistic objects arises from their role as representations and our evaluation of 2

T h e first chapter of The Book of Prime Number Records by Paulo Ribenboim (2nd. edition, New York, 1989), which is devoted to presenting nine and a half proofs of the existence of infinitely many primes, may serve to illustrate this fact—even though the theory of half proofs lies beyond the scope of the present work.

Introduction

5

them is in terms of what they represent. It seems perverse therefore, if not downright erroneous, to treat the representation as fundamental. The disadvantages, I suppose, have to do with clarity and metaphysics. It is easier to deal with concrete than with more abstract objects but, since it is the latter whose nature and properties we are attempting to understand, it seems better to face this problem directly, rather than attempting vainly to avoid it altogether (whatever success such tactics may have had in certain formal applications). As for the metaphysical issue, it seems to me that nothing is gained by denying existence to the interpretations of proofs. Once we acknowledge our interest in determining when two expressions have the same interpretation, we have as good as acknowledged it—no identity without an entity, one might say. Of course, the nature of these interpretations remains open—whether they are to be regarded as abstract objects, mental ones, or even equivalence classes of inscriptions—but such questions can be deferred for the time being. I wish to equivocate here not only about what constitutes such an interpretation, but even about the terminology appropriate to describe it. The fact is that usage provides little guidance in the matter. Phrases like "the denotation of a proof or "what is expressed by a proof" depend on analogies (like the one between proofs and sentences utilized above), and substantial issues may hinge upon which are deemed to be appropriate and, if more than one, whether significant distinctions can be formulated in terms of them. These issues should not be prejudged by the terminology used in discussing them. Some recent commentators have reserved the term "proof" for the interpretation of a certain kind of linguistic object called a "derivation". Although this terminology does not accord well with usage, it will be adopted henceforth for want of anything better (with the proviso that a derivation need not be formal). The precise relationship between derivations and proofs, in particular, whether the former can be said to express or denote the latter, remains to be investigated. Having argued for the interest of studying proofs (rather than derivations), it remains to argue for the suitability of a mathematical treatment of this subject. A conclusive argument for this claim would be a full blown theory of the kind in question. Lacking that, I hope the considerations advanced below make it seem worth pursuing. It should be said that a theory of proofs—in the sense explained above—is not a new idea. The attempt to utilize the methods so successfully applied to the foundations of classical mathematics for the study of intuitionistic foundations has resulted in efforts in this direction (for reasons which should be clear from my earlier discussion of Brouwer's opinions). I am thinking particularly of the attempts by Kreisel and some of his followers to formulate the basic properties of proofs, in other words, to provide an axiomatic theory of them. 3 3

These are described in a succession of papers, beginning with his "Foundations of

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Normalization, Cut-Elimination and the Theory of Proofs

It seems fair to say that the results of their efforts have been somewhat inconclusive, as Kreisel himself seemed to acknowledge when he remarked that "at present there do not seem to be any even mildly promising ideas for a systematic or fundamental science of proof."4 Another approach to formulating a mathematical theory is to regard it as a theory of familiar mathematical objects and attempt to specify which of these objects the theory is intended to describe. This too has been tried in the case of proofs, again in the constructive tradition for the most part, the familiar mathematical objects in this case being functions. The idea itself is an old one, and is already implicit in Heyting's interpretation of the logical operators. 5 More recently, it has been developed by Godel and exploited to prove the consistency of arithmetic. 6 I do not mean to suggest that these two approaches are independent of one another: the axioms for proofs are, one assumes, suggested by an intended interpretation and, conversely, the identification of proofs with mathematical objects of a certain kind needs to be supplemented by principles which pick out suitable ones of the kind in question. Nevertheless, the distinction between them is worth drawing if only for heuristic reasons. A mathematical theory may be regarded from a syntactic or a semantic point of view. Sometimes the semantic aspect is dominant in the sense that the truths of the theory are determined by a very clear conception of its intended interpretation, as in the case of arithmetic. In other cases, non-Euclidean geometry for example, the syntactic aspect predominates: we search for interpretations which make a certain set of statements true. These two possibilities are not exhaustive, however. There are surely cases where the intended interpretation is conceived too indistinctly to determine the truths of the theory even though there is no other source for them. Here it seems helpful to alternate between semantic and syntactic approaches. An initial body of data leads to the adoption of certain principles; the investigation of these principles sharpens our ideas about possible interpretations; these interpretations in turn suggest other principles, and so on. 7 The development of set theory seems to me to exemplify this general Intuitionistic Logic," pp. 198-210 of Logic, Methodology and Philosophy of Science, edited by E. Nagel, P. Suppes and A. Tarski, Stanford, 1962. 4 "On the kind of data needed for a theory of proof," Logic Colloquium 76, edited by R. Gandy and J. Hyland, Amsterdam, 1977, page 125. 5 See Jntuitionism: An Introduction by A. Heyting (3rd. edition, Amsterdam, 1971), although the interpretation was first formulated in his Mathematische Grundlagenforschung. Intuitionismus. Beweistheorie., Berlin, 1934. 6 "Uber eine bisher noch nicht beniitzte Erweiterung des finiten Standpunktes," Dialectica, Vol. 12, 1958. 7 This account accords well with Godel's views about the nature of axiomatic systems, as reported by Hao Wang in his book Reflections on Kurt Godel, Cambridge Mass., 1987. For example: "But Godel does not take an axiomatic system as an 'implicit definition' of the concept(s) in it, because it is supposed to be a report of our (generally incomplete) intuitions of the concept and can be revised and expanded. It does not

Introduction

7

schema, and my suggestion is that it also provides a helpful model for the development of a theory of proofs. The analysis of a certain part of our intellectual activity yields formal systems which purport to represent reasoning. The elements of these systems are derivations, related in various ways and with various properties. These systems of derivations themselves admit interpretations which impart significance to the aforementioned properties and relations, revealing difficulties and discrepancies between different formalizations which may then be resolved or reconciled by looking again at interpretations. Gentzen's systems of natural deduction—N systems, for short—seem to be the result of just such an analysis. I do not mean to suggest by this that they can be used to represent every kind of reasoning. Nevertheless, their derivations may plausibly be claimed to represent a certain class of proofs—and quite a significant class at that. Gentzen after all arrived at his rules by analyzing the actual practice 01 mathematicians, and our intuitions are sufficiently well developed to be able to recognize the success of his analysis. Furthermore, his rules offer a systematic treatment of the logical particles: the inferential behavior of each one is governed by a pair of symmetrical rules (an introduction and an elimination) which have a certain separability property. 8 This lends credibility to the claim, for its truth does not require that every step in a piece of reasoning correspond to a rule of the system, only that every such step can be expressed naturally in terms of these rules. An analysis like Gentzen's, therefore, which appears to break inference down into atomic steps, is well suited for this purpose. One of the distinguishing features of Gentzen's systems is that they admit a normal form theorem. This is a very general result about the structure of their derivations. Roughly speaking, it asserts that any derivation II can be converted into one of the same conclusion which has a particularly simple and direct form (and whose assumptions are included amongst those of IT). Now, when considering the derivations of a calculus as representations of proofs, it seems appropriate to concentrate upon those of their features which any object built up by means of its rules must possess, rather than upon those which depend on how the rules are interpreted as building up structures. In the case of natural deduction, the most conspicuous examples of the former are the reduction relations which hold between two derivations when one is obtained from the other by performing one of the transformations employed in the proof of the normal form theorem. As for the latter, although natural deduction derivations are usually thought of define (and fix completely) the concept, but rather invites improvements by comparison with the changing intuition." (page 247). 8 Let * be any connective, then its introduction and elimination rules make no mention of the other connectives and, if A contains no occurrences of *, then A is derivable in the system iff it is derivable in the system without these rules.

8

Normalization, Cut-Elimination and the Theory of Proofs

as trees whose nodes are labelled by formulae, there is nothing about the rules which forces us to interpret them in this way. They could equally well be sequences of formulae, trees labelled by sequents or, for that matter, a variety of physical structures. For this reason, properties distinctive to these labelled trees fall into the latter category. In contrast to the relations mentioned earlier, they are connected only very indirectly with the content of the rules and seem less likely, therefore, to be significant from the point of view of a theory of proofs. (In fact, a likely benefit of such a theory would be to obliterate distracting distinctions between different realizations of derivations.) These considerations lend credence to the suggestion that, although formulated as a result about the combinatorial properties of derivations, the normal form theorem really tells us something about the structure of the underlying proofs. The relationship between a derivation and its normal form, so it has been claimed, reflects a relationship between the proofs which they represent, and the operations by which the former is transformed into the latter can be interpreted either as operations on the proofs themselves or as preserving certain properties of these proofs. In addition, the formal analogy between logical calculi and theories of functions (the A-calculus, for example), according to which derivations of the former can be correlated with terms of the latter, provides a suggestive interpretation of the theorem. This is because the reduction steps which effect the conversion referred to in its statement correspond on this analogy to the computation rules for terms which are used to analyze the equality relation between them. It seems reasonable, therefore, to hope that the theorem casts some light on the question of the identity of proofs: just as any model of the Acalculus must respect the equivalence generated by the computation rules for its terms, so any interpretation of what might be called the N theory of proofs (i.e., the theory whose axioms are inequalities expressing the reduction steps between natural deduction derivations) would have to assign the same proof to interconvertible derivations. The terms of this theory are derivations, and the proofs which they are supposed to denote can be thought of as functions. It has even been conjectured that interconvertibility characterizes the identity relation between proofs.9 The preceding seems to provide an attractive and coherent framework within which to investigate a certain conception of proofs and their identity criteria. Notice incidentally that proofs are treated here as the denotations of derivations. Although this appears to conflict with a widely held view of proofs as intensional objects (whatever that phrase may mean), it seems to me that such a treatment poses no new problems for those who have at9 T h e conjecture is defended in a number of papers by Dag Prawitz, who attributes it to Martin-Ldf and the influence of some related ideas of Tait.

Introduction

9

tempted to explain meaning in proof-theoretic terms. 10 There is, however, a striking obstacle in the path of developing the N theory along the lines suggested above, namely, what might be called by analogy the L theory of proofs. As is well known, Gentzen abandoned his systems of natural deduction as a tool for metamathematical investigation in favor of sequent calculi—L systems, for short. The latter are a reformulation of the former which constitute little more than a notational variant in the case of the introduction rules, but replace the eliminations by a more restrictive set of rules whose equivalence to them is most easily demonstrated with the aid of the cut rule. Corresponding to the normal form theorem for N derivations is a cut-elimination theorem for L ones, whose proof proceeds by systematically transforming a derivation with cut into a derivation of the same conclusion without cut. There is an obvious similarity between derivations in the two kinds of calculus which extends in a rough and ready way to a correlation between normal and cut-free derivations and, in an even rougher way, to the steps by which derivations are converted into normal or cut-free form. In view of this, it is possible to interpret sequent derivations too as terms denoting functions and the cut-elimination steps as analyzing equality. This is what I mean by the L theory of proofs. Unfortunately, on closer inspection, discrepancies emerge between the two theories. These were noticed first by Zucker in his paper "The Correspondence between Cut-Elimination and Normalization." 11 He showed that, although a very satisfactory correspondence obtains when attention is restricted to fragments without disjunction or the existential quantifier, the full theories as they stand are incompatible with one another. At the end of his paper, he considers a modification of the L theory which conforms better to the N theory but, as he himself admits, it is at best ad hoc. As a result, he seems to give up the idea of reconciling the two theories and concludes that there may be meaningful properties of proofs which are preserved by all reductions of the N theory, but not by those of the L theory. I believe, however, that this assessment is both too optimistic and too pessimistic: too optimistic because, if the two theories cannot be reconciled, this detracts from the plausibility of any claim to the effect that the structure of derivations in either calculus tells us very much about properties of proofs, and too pessimistic because it may yet be possible to reconcile them. The chapters which follow are intended to defend this last claim and expand on some of the issues raised above. Chapter 1 contains a brief introduction to Gentzen's N and L systems, including a sketch of the proofs of the normal form and cut-elimination theorems. In Chapter 2, I outline the correspondence between the two kinds of calculus and discuss Zucker's results, both positive and negative. See Chapter 8 below for a fuller discussion of these and related matters. Annals of Mathematical Logic, Vol. 7, 1974.

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Normalization, Cut-Elimination and the Theory of Proofs

Chapter 3 is concerned with the issue of whether natural deduction derivations can be said to represent proofs adequately and suggests that they might be deficient in some respects; these deficiencies, I argue, play a role in the breakdown of the correspondence. The main result of Chapter 4 is that almost any formal system whose derivations are supposed to represent proofs will inevitably share some of these defects; this leads me to conclude that there is little hope of explaining the usual procedures for reducing a derivation to its normal form solely in terms of proofs and their properties. In Chapter 5, I present a system of natural deduction whose derivations have more than one conclusion. Systems of this kind, I argue, have some advantages over more traditional calculi; in particular, they provide a suitable framework within which to compare the different reduction procedures used for N and L derivations. Chapter 6 contains a description and analysis of reduction steps for the derivations of this system and, in Chapter 7, I show in some detail how they can be related to reduction procedures for the classical sequent calculus. The results of this chapter suggest that proofs may have less structure than is usually thought. Chapter 8 opens with a critical examination of some proposals to interpret the normal form theorem along the lines suggested above. The conclusions arrived at earlier are used to argue that obstacles in the way of establishing a correspondence between various reduction procedures and of justifying them in terms of proofs do not rule out the possibility of such an interpretation, but rather indicate that traditional procedures are too restrictive and traditional ideas about proofs too narrow. To substantiate this opinion, I suggest how proofs, regarded as denotations of formal derivations, may be incorporated into a general account of justification and how, by applying the methodological principle advocated above, the interplay between theory and interpretation may be exploited to help clarify their properties. The resulting notion of proof, although not entirely standard, has some virtues. It avoids the more annoying obstacles in the way of a unified treatment of cut-elimination and normalization. Furthermore, by diminishing the significance of some structural features of derivations, it opens up the possibility that a general theory of proofs can afford to disregard the minutiae of syntax. This seems to me to be a helpful idea. Proof theory is usually identified with a detailed study of the syntactic features of formal systems. Such a view of the subject, however appropriate when it is regarded as a metamathematical tool, has not provided many answers to general questions about the nature of proofs. Although the conclusions of Chapter 8 are speculative, the approach advocated there, I suggest, may yet do so. Acknowledgments: I am grateful to Solomon Feferman for instruction and advice on the topics discussed in this book. I would also like to thank the publications staff at CSLI for their painstaking work in producing the final copy.

1

Background It is a commonplace of the textbooks that logic can be regarded from two points of view. Here, for example, is Copi's account of the matter: "On the one hand, logic is an instrument or organon for appraising the correctness of reasoning; on the other hand, the principles and methods of logic used as organon are interesting and important topics to be themselves systematically investigated." 1 Furthermore, it has become customary to associate this distinction with that between formal system and metatheory—as Copi appears to do. Such early pioneers of formalization as Russell, however, seem not to have thought in quite these terms. Not only was their view of logic somewhat broader than the one taken above but, more to the present point, they tended to blur the distinction between logic as organon and logic as systematic investigation. In part, this was because they did not regard formal systems simply as instruments for the appraisal of reasoning. Consider, for example, how Russell introduces his axiomatization of propositional logic. But the subject to be treated in what follows is the theory of how one proposition can be inferred from another. Now in order that one proposition may be inferred from another, it is necessary that the two should have that relation which makes one a consequence of the other. When a proposition q is a consequence of a proposition p, we say that p implies q. Thus deduction depends upon the relation of implication, and every deductive system must contain among its premises as many of the properties of implication as are necessary to legitimate the ordinary procedure of deduction. 2 The system Russell presents is to be a theory of the relation which holds between premises and conclusion of a valid argument; it is intended to legitimate the ordinary procedure of deduction. All this sounds a little strange 1 2

Symbolic Logic by I. M. Copi, fifth edition, New York, 1979, p. vii. Principia Mathematical p. 90. 11

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Normalization,

Cut-Elimination

and the Theory

of Proofs

to our ears, accustomed as we are to enforcing a sharp distinction between relations and connectives. Clearly, however, it is Russell's conception of a formal system of logic which underlies the theory of strict implication, and the tradition lives on in current work on the logic of entailment. Nevertheless, it seems fair to say that it is no longer the orthodox one. In fact, we are so used to thinking of logical systems as attempts to replicate ordinary reasoning in a formal context that, unless we keep in mind the different aims which they were originally intended to serve,3 it will seem incomprehensible that the obvious disparity between formal deduction and informal proof was ignored for so long. The first person to pay attention to it appears to have been Lukasiewicz. In 1926 Professor J. Lukasiewicz called attention to the fact that mathematicians in their proofs do not appeal to the theses of the theory of deduction, but make use of other methods of reasoning. The chief means employed in their method is that of an arbitrary supposition. The problem raised by Mr. Lukasiewicz was to put those methods under the form of structural rules and to analyze their relation to the theory of deduction. 4 Russell did not suppose that his particular axioms and rules coincided with those ordinarily employed in deduction, but he did claim that they were in some sense "sufficient for all common forms of inference."5 As Lukasiewicz points out, however, there is a difference in kind between the derivations of an axiomatic theory of deduction on the one hand and ordinary proofs on the other. The logical content of a mathematical proof is contained, for the most part at least, in the steps which lead from statements to their consequences. (The qualification is needed because an appeal to some logical principle—the law of excluded middle, for example—may occasionally be made as well.) Furthermore, these statements need not themselves be accepted as true, except for the sake of argument, nor are they in general valid propositions about logical relationships. Jaskowski, in the paper quoted above, proposed a solution to Lukasiewicz's problem; it took the form of a novel kind of logical calculus, one distinguished from axiomatic systems in allowing the introduction of arbitrary statements into a derivation—as 3

This is not to suggest that there were clearly articulated rival views about the nature of formal systems, only that at one time logicians had a variety of different ambitions for them, not all of which were clearly distinguished, and including some that we are more likely to attempt to realize through the study of formal systems, rather than through the systems themselves. 4 "On the Rules of Suppositions in Formal Logic" by Stanislaw Jaskowski, Studia Logica, Vol. 1, 1934; reprinted in Polish Logic: 1920-1939, edited by Storrs McCall, Oxford, 1967. 5 Principia, p. 90. He seems to have meant by this phrase only that, if C is commonly (and correctly) inferred from 5 i , . . . , 5 n , then a certain implication—roughly speaking, one equivalent to (Si A . . . A Sn) —* C—will be a theorem of Principia Mathematica.

Background

13

assumptions, hypotheses or suppositions—as well as by the use of rules which could discharge assumptions. These are the characteristic properties of so-called natural deduction calculi, as opposed to Hilbert-style formalizations of logic. The latter term is the one usually employed to describe any axiomatic treatment of logic interpreted as a deductive engine, i.e., as an instrument for deriving the consequences of a set of statements. If Hilbert did not originate such axiomatic treatments, he did at least encourage this sort of interpretation of them. 6 In particular, he seems to have thought that any step in an (informal) argument could be made to correspond to a series of steps in a formal derivation. The transition from statements to their logical consequences, as occurs in the drawing of conclusions, is analyzed into its primitive elements, and appears as the formal transformation of the initial formulas in accordance with certain rules. 7 Seen in this light, derivations are simply the formal counterparts of informal proofs and discrepancies between the two are reduced to matters of style. A formal derivation, because it makes explicit the "primitive elements" that combine to make up each step of the proof, will be less natural than its informal counterpart. Hilbert acknowledges as much when, at the conclusion of a general discussion about different axiomatizations of logic, he comments on a calculus devised by one of his assistants. Finally, we mention, as a system which occupies a special place, the "Calculus of Natural Inferences," as set forth by G. Gentzen, which constitutes an attempt to make out of the formal deduction of formulas something more similar to the usual method of proof . . . , such as is customary, e.g., in mathematics. The calculus contains no logical axioms, but only figures of inference which indicate which inferences can be drawn from given assumptions, as well as figures which yield formulas in which the dependence upon the assumptions is eliminated. 8 As is apparent from the brief descriptions given above, Gentzen's calculus is similar in kind to that of Jaskowski. It is presented in a paper entitled "Untersuchungen uber das logische Schliefien" which appeared in Mathematische Zeitschrift for 1935.9 6

See, for example, Chapter II, Section 11 of Principles of Mathematical Logic by D. Hilbert and W. Ackermann, New York, 1950. This is a revised translation of the second edition of their Grundziige der theoretischen Logik. 7 Ibid., p. 1. 8 Ibid., p. 30. 9 Gentzen's work appears to have been done independently of Jaskowski. The latter's was published in 1934 although, according to Prawitz, it represents a revision of results obtained and announced in the late twenties. Gentzen's paper was submitted for publi-

14

Normalization, Cut-Elimination and the Theory of Proofs

Gentzen's describes the initial motivation for his work in the following terms. My starting point was this: The formalization of logical deduction, especially as it had been developed by Frege, Russell and Hilbert, is rather far removed from the forms of deduction used in practice in mathematical proofs. Considerable formal advantages are achieved in return. In contrast, I intended first to set up a formal system which comes as close as possible to actual reasoning. The result was a calculus of natural deduction (NJ for intuitionist, NK for classical predicate logic).10 Although Gentzen begins his paper with an account of this system of natural deduction and concludes it with a proof of its equivalence to more conventional formulations of first-order logic, the intervening parts are devoted to other topics. Here again is Gentzen. A closer investigation of the specific properties of the natural calculus finally led me to a very general theorem which will be referred to below as the 'Hauptsatz'. The Hauptsatz says that every purely logical proof can be reduced to a definite, though not unique, normal form. Perhaps we may express the essential properties of such a normal proof by saying: it is not roundabout. . . . In order to be able to enunciate and prove the Hauptsatz in a convenient form, I had to provide a logical calculus especially suited to the purpose. For this the natural calculus proved unsuitable. For, although it already contains the properties essential to the validity of the Hauptsatz, it does so only with respect to its intuitionist form, . . . n The calculus for which Gentzen proved his Hauptsatz is called the calculus of sequents (LJ for intuitionist, LK for classical predicate logic). I shall be concerned in what follows with the relationship between these two kinds of calculus and, in particular, with the significance of the Hauptsatz as it applies to both of them. As a preliminary, the present chapter contains a brief introduction to those parts of Gentzen's work relevant to this inquiry. Assume a first-order language, say £, containing the usual logical symbols (V, A, —•, ->, V and 3) as well as a constant, _L, for falsity. In addition, C contains n-place predicate letters, for each n, individual variables and parameters, and whatever punctuation devices are needed. (For simplicity cation in July of 1933. An English translation appears on pp. 68-131 of The Collected Papers of Gerhard Gentzen, edited by M.E. Szabo, Amsterdam, 1969. All subsequent page references will be to this translation. 10 Op. cit, page 68. 11 Ibid., pp. 68-69.

Background

15

and definiteness, assume it contains neither equality nor function symbols.) The well-formed formulas of C are defined in the usual way, with the proviso that variables may only occur bound in them. (The parameters and variables are supposed to form disjoint sets.) There will be no occasion to use the language £, only to talk about it and its expressions. So, the symbols used above should be thought of as metalinguistic names for the corresponding symbols of C. Upper-case letters, A,B,C,..., sometimes with natural numbers as superscripts, will range over formulas of £, lower-case letters a, 6, c , . . . from the beginning of the alphabet over its parameters, and lower-case letters x, y, z,... from the end of the alphabet over its variables. If a formula A contains occurrences of the parameter a, it may be written as A(a) and the (ill-formed) expression which results from replacing one or more occurrences of a in A by the variable x can be written as A(x)—provided that x is free for a in A(a). Names for the operators will be used to form complex expressions which range over various classes of formula in the usual way: A A B will range over conjunctions, etc. Furthermore, parentheses will be used where necessary to disambiguate these expressions. Gentzen's natural calculi are systems for constructing derivations from assumptions. These are supposed to be tree-like structures whose root is their conclusion and whose topmost branches are assumptions. An occurrence of an assumption may be open or closed. The idea here is that the conclusion of a derivation will depend only on its open assumptions; the closed assumptions, having been introduced as auxiliary (open) assumptions in the course of the derivation, will have been discharged before the final conclusion is reached. An isolated assumption is treated as a trivial derivation whose root and only branch are the same, i.e., the assumption A is a derivation of the conclusion A which depends on itself. In addition, there are rules of inference of two kinds. The first kind allows the construction of a derivation of its conclusion which depends upon the same assumptions as the derivation(s) of its premise(s). Consider, for example, the rule which sanctions the construction of a derivation of the conjunction AAB from derivations of A and B: given two such derivations, their roots are transformed into lower branches and a new root, A A B, is joined to them from below.12 The second kind of rule is more interesting; it allows assumptions to be closed in the process of transforming the derivation(s) of its premise(s) into a derivation of its conclusion. 12 It was Gentzen who originally described these derivations as having a tree form. Students of biology may object that, since they may lack a trunk, they would be better described as shrubs. There will be no occasion to consider any other kind of vegetation in the sequel, however, so this choice of terminology should not cause confusion. In graph theory a tree is usually defined more generally to be a connected graph without circuits. Derivation trees satisfy the additional requirement that, when a direction (up and down the page) is imposed upon them, they have exactly one root.

16

Normalization,

Cut-Elimination

and the Theory

of Proofs

Conditional Proof is a good example of such a rule: it transforms a derivation of the conclusion B from the assumption A into one of the conclusion A —• B which no longer depends on A. Because of their tree structure, the derivations of a natural calculus may contain more than one occurrence of a given assumption. The question then arises as to whether all or only some of these occurrences should be discharged and, indeed, whether any assumptions need be discharged at all. The answer to this seemingly unimportant question, that the application of such a rule may discharge some, none, or all occurrences of the appropriate assumptions, is of some significance.13 It is convenient, therefore, to suppose that the occurrences of an assumption in a derivation are grouped into one or more classes (not excluding the empty one) and to stipulate that the application of a rule of the second kind discharges all the members of an assumption class. The basic rules of inference for natural deduction are as follows: (1)

(3) a.

A B AAB A AVB

b.

(2) a.

AAB

b. A A B

A B AVB

B

(4) AV B A

[A] [B] C C C A^B B

(5)

[A) B A^B

(6)

(7)

[A]

(8)

A

(9)

->A A(a)

(10)

VxA(x) A(t)

VxA(x) (11)

A(t) 3xA(x)

^A

(12)

[A(a)\ C

3xA(x)

c In each case, the premise(s) of the rule is (are) separated from its conclusion by a horizontal line. The formulae in square brackets stand for the assumption occurrences, if any, discharged by the inference. (They are written above the premise in whose derivation they may occur.) In rules (10)#and (11) t, like a in rules (9) and (12), is supposed to be any parame13

See "Assumption Classes in Natural Deduction" by D.Leivant, Zeitschrift fiir mathematische Logik und Grundlagen der Mathematik, Vol. 25, 1979, pp. 1-4. The issue is not one of deductive strength, but has to do with properties of the procedure by which a derivation can be reduced to normal form. These will be discussed in the next chapter.

Background

17

ter of C. Given a language with function symbols, however, t could be any term, whereas a would still have to be a parameter, a is sometimes called the proper parameter or eigenvariable of the inference. It must satisfy the following restrictions: in (9), a cannot occur in the conclusion of the inference nor in any assumption on which the premise depends; in (12), a cannot appear in either premise of the inference, nor in any assumption on which the derivation of the premise C depends except for those of the form A(a). It is easy to generate invalid inferences by ignoring these restrictions on the proper parameter. Writing the rules in two columns, as in the preceding paragraph, exhibits their symmetry. There is a pair of rules for each operator, * : the conclusion of the left member of the pair is a formula with * as its principal operator; the right member has such a formula as a premise. (This statement is slightly inaccurate, since (2) and (3) each comprise a pair of rules. For ease of expression, however, I shall continue referring to each as though it were a single rule.) The rule on the left introduces * into its conclusion; the rule on the right eliminates * from its premise. Following Gentzen, therefore, the odd numbered rules are usually called Introductions, and the even numbered ones Eliminations. When an elimination rule has a multiplicity of premises, the one whose principal operator is being eliminated is called its major premise; the remainder are minor premises. For example, (5) is called —•-Introduction, and (6) —•-Elimination—rather than conditional proof and modus ponens, respectively—and the major premise of (6) is the one of the form A —• B. It is on this symmetry that Gentzen's "very general theorem" depends. Note that, if -> is treated as a defined symbol with -u4 an abbreviation for A —• -L, the introduction and elimination rules for -i become special cases of the corresponding —• rules. The system characterized by the rules given above is called minimal logic. It is weaker than intuitionistic logic as formalized by Heyting. The system NJ is obtained from it by adding the intuitionistic negation rule _L_ A that any conclusion follows from a falsehood.14 To obtain NK there are a number of different possibilities. Gentzen adds as axioms all instances of A V ->A, the law of excluded middle. He also considers adding instead a rule of double negation, A 14 A is assumed to be distinct from ±. Nothing much depends upon this restriction, but it simplifies the statement of certain results below.

18

Normalization, Cut-Elimination and the Theory of Proofs

Prawitz, in his monograph, 15 favors the classical negation rule i A And, finally, there is

[A] hA] J? B__ B This last is preferable in some respects. 16 NK, without further qualification, will refer to any system obtained from NJ by adding one of the four principles or rules just described. Unfortunately, the addition of a negation rule, whether classical or intuitionistic, spoils the symmetry of the calculus. NJ and NK can be shown to be deductively equivalent to any of the familiar axiomatic formulations of intuitionistic and classical logic, respectively, in the sense that: (1) If there is an NJ [NK] derivation of the conclusion C from the assumptions A1,..., An, then (A1 A . . . A An) —> C is a theorem of intuitionistic [classical] logic. (2) If T is a theorem of intuitionistic [classical] logic, then there is an NJ [NK] derivation of T from no assumptions. These equivalences were first established by Gentzen for particular axiomatic systems due to Glivenko, and Hilbert and Ackermann.17 The argument in both directions is a straightforward induction on the length of derivations. The relationship between introductions and eliminations is explained by Prawitz in the following terms. Observe that an elimination rule is, in a sense, the inverse of the corresponding introduction rule: by an application of an elimination rule one essentially only restores what had already been established if the major premise of the application was inferred by an application of an introduction rule. 18 Consider, for example, the derivation iii n 2 _A B_ AAB A Natural Deduction: a proof-theoretical study, Stockholm, 1965. See Chapter 7 below. Gentzen, op. cit, pp. 115-131. Prawitz, op. cit.., p. 33.

Background

19

Its conclusion A is obtained by applying A-elimination to the premise A A B, which itself has been inferred by A-introduction It accomplishes no more, however, than the derivation III of A, and that in a roundabout way. A slightly more complicated example is provided by: [A]

n2 III A

B A^B

B This may be replaced by the derivation IIi A

n2 B obtained from n 2 by replacing each occurrence of the assumption A discharged by the indicated application of —•-introduction by a copy of the derivation IIi of A (or by II2 if no assumption was discharged). In each case, one derivation is replaced by another of the same conclusion from the same or fewer assumptions which is more direct in the sense that a detour, introduction followed by elimination, has been removed. Another kind of detour may be introduced into a derivation by the intutionistic negation rule. This is so when the conclusion of the rule is itself the major premise of an elimination. Consider, for example, the derivations

n

n

_L IT -L AAB A A^B B B In either case, the simplification of the conclusion of the negation rule effected by eliminating its principal operator is more straightforwardly accomplished by inferring the simpler conclusion directly. In other words, either derivation could just as well be replaced by

n B Call an occurrence of a formula in a derivation maximal if it serves both as the conclusion of an introduction or the intuitionistic negation rule and as the major premise of an elimination. The above examples show how a maximal formula occurrence of the form A A B or A —• B may be removed from a derivation without changing its conclusion or increasing the number of assumptions on which that conclusion depends. Using these as a model, it is not hard to formulate similar procedures for each of the other logi-

20

Normalization, Cut-Elimination and the Theory of Proofs

cal operators, i.e., for removing maximal formula occurrences of the forms i V 5 , -\A, VXA{X) and 3xA(x).19 Note that a derivation terminating in a redundant application of 3- or V-elimination (i.e., one which discharges no occurrence of an assumption) whose major premise is maximal will be replaced by the derivation of (one of) its minor premise(s). A derivation whose last inference has a maximal major premise is sometimes called a redex, and the derivation by which it is replaced is its contractum. Ill is said to be a subderivation of II2 whenever the latter can be constructed from the former using the rules of inference, and EEi is said to reduce in one step to II2 if the latter can be obtained from the former by replacing a subderivation having the form of a redex by its contractum. Roughly speaking, a derivation is said to be in normal form if it contains no maximal formula occurrences. The normal form theorem for NJ asserts that every derivation can be reduced to normal form (where the reduction relation is the transitive, reflexive closure of one-step reduction). The proof proceeds by systematically removing maximal formula occurrences using the techniques illustrated above. Both the statement and proof of the theorem have been reconstructed by Prawitz from passing remarks to be found in Gentzerrs original paper. 20 This deliberately simplified account of the matter will be elaborated in the chapters which follow. There are however a couple of points which deserve mention here. The first is that, in order to remove a maximal occurrence of A —• B or -1 A, it is necessary to know which occurrences of the assumption A are discharged by the relevant introduction inference. Similarly, to remove maximal occurrences of disjunctions or existential formulae, one needs to know which assumption occurrences are discharged by their respective elimination rules. Each derivation tree must therefore be such that this information can be read off from it. There are a number of ways to accomplish this. The one adopted in the sequel is to replace the formulae occurring at each vertex by indexed formulae; these will be expressions of £ with a numerical subscript. Assumption occurrences of the same formula with the same index are said to belong to the same assumption class; each application of —•-introduction, -^-introduction or 3-elimination will discharge all the members of at most one class, and V-elimination all the members of at most two. Classes will be identified by their numerical indices, so that a natural number (or two) beside the conclusion of an inference which discharges a particular assumption class will be sufficient to provide the requisite information. Unfortunately, this necessary device complicates the structure of the derivations. The second point is more fundamental. The preceding account of reduction and normal form is adequate only for NJ~, the fragment of NJ See Prawitz, Natural Deduction, pp. 35-38 for details. Prawitz, ibid., especially Chapter IV.

Background

21

without V and 3. (NJ~ is called the negative fragment of NJ.) Even a cursory inspection reveals that the eliminations for these two operators differ from the other rules: an application of either one yields a conclusion of the same form as its minor premise(s), albeit dependent on different assumptions. Hence, a derivation of NJ may contain a sequence of occurrences of the same formula—say, A\,... ,An—such that, for 1 < i < n, Ai+i is the conclusion of an inference with minor premise Ai. A sequence of this kind which is not properly contained in another such sequence is called a segment. It is useful to distinguish between four kinds of segment: (1) If A\ is the conclusion of an introduction or the intuitionistic negation rule and a. ^*-JI i s the major premise of an elimination, then it is called a maximal segment, b. otherwise it is called an introduction segment. (2) If A\ is an assumption or the conclusion of an elimination and a. An is the major premise of an elimination, then it is called an elimination segment, b. otherwise it is called a minimal segment. Suppose now that Ai,... ,An is a maximal segment, then (unless n = 1) none of its members is itself a maximal formula occurrence, even though the segment as a whole constitutes a detour of the same kind as a maximal formula would. It is desirable, therefore, that a normal derivation should contain no maximal segment. This necessitates the addition of reduction procedures for their removal. Prawitz's solution to this problem utilizes another distinctive feature of V- and 3-elimination, the fact that they can be permuted with other inferences without necessarily destroying the structure of a derivation. He introduces permutative reductions which diminish the length of maximal segments by permuting the elimination inference whose major premise is An with the application of V- or 3-elimination whose conclusion is An. For example, the derivation on the left below—assuming the lower occurrence of A A B belongs to a maximal segment—reduces to the one on the right [C(a)]

iii

[C(a)\

n2

3xC(x) AAB AAB A

n2 III 3xC{x)

AAB A A

and the other permutative reductions follow the same pattern. 21 The result of n - 1 applications of this procedure will be a maximal segment of 21 See Prawitz, op. cit., p. 51. There is a minor technical problem which may arise when an elimination rule with more than one premise is permuted with 3-elimination: the proper parameter of the inference may no longer satisfy the restrictions placed on it.

22

Normalization, Cut-Elimination and the Theory of Proofs

length 1, i.e., a maximal formula occurrence, which can then be removed in the usual way. Although the proof of the normal form theorem is not much complicated by their presence, these reductions introduce problems of another kind which will be discussed in the next chapter. The virtue of normal derivations is that, for the most part, their elimination inferences come before their introductions. More precisely, define an ordering relation on the formula occurrences in a derivation as follows: (1) A premise of an introduction inference or of an application of the intuitionistic negation rule precedes its conclusion. (2) The premise of an application of A- or V-elimination precedes its conclusion. (3) A minor premise of an application of V- or 3-elimination precedes its conclusion. (4) The major premise of an application of —>- or -"-elimination precedes its conclusion. (5) The major premise of an application of V- or 3-elimination precedes every assumption occurrence discharged by that inference. (6) If A precedes B and B precedes C, then A precedes C. According to the above, nothing precedes an open assumption occurrence and nothing is preceded by the conclusion of the derivation. Note also that nothing is preceded by the minor premise of an application of -i- or —•-elimination. Inferences can now be ordered by the precedence relation on their premises, the major premise in the case of an elimination. It then follows that, for normal derivations, no elimination is preceded by an application of the intuitionistic negation rule, no application of the intuitionistic negation rule is preceded by an introduction, and no elimination is preceded by an introduction. Using these facts, it is not difficult to verify that normal derivations have the suhformula property in the sense that each formula (not of the form ±) occurring in a normal derivation is a subformula of an open assumption or of the conclusion. If -> is defined in the manner suggested earlier, then the parenthetical qualification involving _L can be dropped. An alternative description of the structure of normal derivations, which is more in the spirit of Prawitz, 22 runs as follows. For any derivation II, let T\i be the family of subsets of vertices of n which are linearly ordered by the precedence relation, and call the maximal elements of Tu23 routes through II; routes containing the conclusion of II are called main routes. In such a case, before permuting the inferences, one can simply replace it by a more suitable choice throughout the derivation of the minor premise. 22 See Natural Deduction, Chapter IV, Section 2. 23 These are maximal with respect to the inclusion relation on Tn'} they are not to be confused with maximal formula occurrences.

Background

23

Then each route through a normal derivation consists of three successive parts (some of which may be empty): (1) an analytic part comprising formulae which belong to some elimination segment (2) a minimal segment (3) a synthetic part comprising formulae which belong to some introduction segment. With the exception of the rules for negation, the conclusion of an elimination is always a subformula of its major premise and a premise of an introduction is always a subformula of its conclusion, it should be obvious therefore that every formula (other than _L, perhaps) appearing in a route through a normal derivation is a subformula of the first or last formula in the route. It follows that normal derivations have the subformula property described above, even though a route need not begin with an open assumption nor terminate with the conclusion. To see this, consider the other possibilities: (1) The route begins with an assumption discharged by an application of -,_ o r —^-introduction. In this case, the initial formula will be a subformula of the conclusion of the inference and, hence, of the last formula in the route. (2) The route terminates with the minor premise of an application of ->or —^-elimination. In this case, the terminal formula will be a subformula of the major premise of the inference and, hence, of the initial formula in any route on which that premise lies. (3) The route terminates with the major premise of a redundant application of V- or 3-elimination. In this case the route can contain only an analytic part and the terminal formula will be a subformula of the initial formula in the route. Of course, in cases (2) and (3), the initial formula in question may have been discharged by an application of -i- or —>- introduction and case (1) will apply. On the other hand, in case (1), the last formula in the route may be the minor premise of an application of ->- or —•-elimination and case (2) will apply. Nevertheless, it is easy to see that no circularity can arise, and a straightforward induction on any reasonable measure of the complexity of a normal derivation (e.g., the number of inferences it contains) establishes that it has the subformula property. Not only did Prawitz reconstruct Gentzen's original result for NJ, he also succeeded in extending it to classical logic, albeit at the cost of some artificiality. Utilizing the definability of V and 3, he was able to restrict attention to the fragment of NK without these operators and to prove a

24

Normalization, Cut-Elimination and the Theory of Proofs

version of the normal form theorem for the resulting system. The first step in the proof is to show that each application of the classical negation rule

hA] J_ A can be replaced by one or more applications in which A is atomic. This is accomplished by means of a set of reductions which lower the complexity of A. For example, the figure on the left below reduces to the one on the right (provided that a is not the proper parameter of any inference in II): [VxB(x)] B(a) [-.VxB(ar)] n j. MxB(x)

hB(a)} J. -.Vxfl(x) n X Bja) VxB{x)

The assumption occurrences VxB(x) are discharged by -"-introduction; the classical negation rule is used only to discharge -*B(a), a formula of lower complexity than -WxB(x). The reductions for A of the form -«B, B —• C and B A C are similar.24 (It is this step in the proof which fails when V and 3 are included.) Maximal formula occurrences can then be removed as before, and the resulting normal derivations will still have a form of the subformula property: each formula occurrence which is not of the form _L, nor an assumption discharged by an application of the classical negation rule, is a subformula of an open assumption or the conclusion. Gentzen was less sanguine about the prospects for a useful normal form theorem for NK and, for this reason, abandoned natural deduction in favor of sequent calculi. He describes his aim as "to formulate a deductive calculus (for predicate logic) which is logistic on the one hand, i.e., in which the derivations do not, as in the calculus NJ, contain assumption formulae, but which, on the other hand, takes over from the calculus NJ the division of forms of inference into introductions and eliminations of the various logical symbols." 25 Natural deduction systems have an inconvenient feature not shared by axiomatic (logistic) calculi: some of their rules— notably, those like —^-introduction which discharge assumptions and those like V-introduction which place restrictions on the proper parameter of the inference—are properly applied to the derivations of their premises, rather than to the premises by themselves. This introduces some additional complexity into the structure of derivations, and makes axiomatic calculi easier 24 25

See Prawitz, op. cit, p. 40, for details. Gentzen, p. 82.

Background

25

to work with in some contexts. Nevertheless, this is not the crucial difference between N and L systems. Gentzen turned to sequent calculi because they enabled him to give a uniform treatment of classical and intuitionistic logic; in particular, no additional rules were required to obtain the former from the latter. There are axiomatic systems related to NJ and NK in a very direct way, namely, those for deriving statements of the form "C is deducible by the rules of NJ (or NK) from the assumptions A1, . . . , An" By introducing a two-place relation symbol h (for deducibility) and interpreting the rules of natural deduction as clauses in an inductive definition of this relation, it is possible to transform NJ and NK into logistic calculi. The axioms will be of the form A h A, A is deducible from the assumption A. A-introduction will translate into ThA Ahff r,AhAA5 ' and A-elimination into the pair T\-AAB _____

, and

T\-AAB _____

Translation of the remaining rules is equally straightforward. Notice that —•-introduction becomes r,A\- B rh A-+ B so, to allow the requisite flexibility in discharging assumptions, 26 it is better not to read T and A simply as sets of formulae. They are most conveniently handled as sets of indexed formulae, and will be so in subsequent chapters. For Gentzen, however, T and A were sequences of formulae, and it is his treatment that will be followed here. (I\ A is the concatenation of T with A, and likewise for I\ A.) He called expressions of the form F h A, and even T h A, sequents. The calculi sketched in the previous paragraph, although related to Gentzen's LJ and LK, differ from them in a significant respect. The theorems of the L calculi are sequents, and their rules are inspired by those of natural deduction. In fact, each introduction rule of NJ is translated into a sequent rule in the manner described above. The elimination rules are treated differently however. Consider, for example, the A-elimination rule which asserts that A follows from A A B. This becomes inverted in the sequent calculus into a rule which asserts that whatever can be deduced from A can also be deduced from A A B:

A,r\-c AAB,T\-C See note 13 above.

26

Normalization,

Cut-Elimination

and the Theory

of Proofs

All the logical rules of a sequent calculus will be introduction rules, but the symmetry between introductions and eliminations will be preserved in the symmetry between rules which introduce an operator into a formula on the right of h and those which introduce the same operator into a formula on the left. It is not immediately apparent that a left rule has the same deductive strength as the elimination which inspired it. Consider an N derivation of the conclusion C from the assumptions A1,... ,An which contains an application of A-elimination. If that inference were to be removed, the derivation would split into two parts: the subderivation terminating in the premise of the A-elimination (AAS, say) and a derivation terminating in C, with the conclusion of the missing A-elimination (A, say) as an additional open assumption. In other words, A A B is deducible from Al,...,An l n ( r h i A B), and C is deducible from A, A ,...,A (A, A h C—where the terms of T and A are drawn from among A1,... ,An). The problem then is to put the derivation back together again, i.e., to infer the sequent r , A h C, using A-left. This rule will allow us to infer A A B, A h C from A, A h C, but something more is needed. It is for this reason that Gentzen introduced the cut rule: $hP

P, 9 h Q

Applying cut to T \- A A B and A A B, A r- C as premises immediately yields the desired conclusion.27 A sequent calculus will consist therefore of (1) Axioms (2) Right (introduction) rules for the logical operators—which are nothing more than the translations of the introduction rules into sequent notation (3) Left (introduction) rules for the operators—which are related to the elimination rules and from which their sequent translations can be derived with the aid of cut (4) Cut and, in addition, some structural rules for manipulating the sequences to the left and right of h; these comprise rules for padding a sequence (thinning), rearranging its terms (interchange) and identifying occurrences of the same formula (contraction). Here is a list of the axioms and rules for LK: Axioms:

AY- A

27

In fact, it is even easier to derive directly the translation of A-elimination into sequent notation, using cut together with A-left, as the following shows: A\- A AAB\-A

T\-AAB

r\-

A

Background

27

Logical Rules: Right

Left

T\-A,A n-A,s TI-A,ylA£ TI-A,A r\-A,A\/B

TI-A,B T\-A,AVB

A,T\r\-

B,r\-A

A,T\-A AAB,T\-A

A/\B,T\-A

A,ri-A BJ^rA A\ZB,T\- A r\-A,A

A,B

5,$h$

A ^ B,r,} (the calculus which is like LJ without the rules for disjunction, except that vacuous applications of 3-left are disallowed) to those of Nj(~v)f. If the definition of = is extended in such a way that permuting cuts upwards past applications of 3-right and -left are included among the permutative reductions, 8 we can show that for all d, df in LJ^ d = d' iff -i n ;

for some II, IT such that i/j(Ti) = a and ^(IT) = r. Finally, there is an intermediate procedure which requires that maximal formulae-occurrences be removed one by one, except in the case of existential formulae all of whose a-connected occurrences can be removed at the same time (where a is supposed to be the proper parameter of the elimination). How close a correspondence obtains between normalization in Nj(~v^' and in the other calculi depends, of course, upon the particular normalization procedure chosen. But for all those sketched above we have at least the following: (1) For all 11,11' in 7Vj(~v), n yx II' implies ^(11) y (II').9 (2) For all a, r in Nj(~v^, a ^ T implies that there exist II, II' such that

a = (n), r y ip{W) and n yx n 7 . 10

(3) For all dj in L J ( " V \ d yx d' implies 0 o tp(d) y o ip{df). (4) For all a, r in Nj(~^', a y± r implies that there exists d, d' such that a = <j> o ip(d), r y <j> o i/>(d') and d >-i d'. Such differences as exist between reduction procedures in the three calculi are not unexpected, given the nature of the correspondence between their derivations. In the passage from Lj(~v^ to NJ^~V^ and from the latter to AfJ^"^', a single formula occurrence in a derivation may be associated with a number of occurrences of that same formula. Now, generally speaking, a reduction step eliminates a single maximal occurrence (or its analogue in the sequent calculus). So, one such step may correspond to a succession of steps under the mappings (f> and ^ (as in (1) and (3)) 9 Since vacuous applications of 3-elimination are not allowed in Nj(~v\ the normalization procedure for this calculus will be like the usual one for NJ except that these must be removed—by what Prawitz calls immediate simplifications—as they arise in the course of the reduction process. In addition, of course, we allow the permutations which generate ~ . 10 Here, and in (4), r may range over quasi-derivations when the first or third reduction procedure for J V J ^ - ^ ' is being considered.

54

Normalization,

Cut-Elimination

and the Theory

of Proofs

whereas, going in the opposite direction, a single step may need to be combined with some others before it corresponds to one step (as in (2) and (4)). Furthermore, the correspondence is close enough to allow results to be translated from one calculus to another. In particular, strong normalization and uniqueness of normal forms in Nj(~w^f translate via 0 and ip into the corresponding results for Nj(~v^ and LJ^~y\ Although normal forms may not be unique in NJ^~V^ or LJ^~y\ they will be equivalent under ~ and = , respectively. As for strong normalization, it can be translated into the assertion that every non-repeating reduction sequence terminates. Because equivalence classes of derivations under ~ are finite, permutations can be included among the reduction steps for NJ^~V\ In the case of LJ^~y\ however, equivalence classes under = are not finite, as Zucker already observed for the negative fragment. Nevertheless, by utilizing the asymmetry of the reduction steps, he was able to show that any sequence of reductions in LJ~ must either terminate or have infinite (The translat ion will also go repetitions and, the same holds for Lj(~vln the other way, if we adopt the second of our normalization procedures for ATj(-v)', for it allows y to be replaced by = in (2) and (4). 12 ) Although it is possible to define the class of Nj(~y>}' derivations directly, using a rule of existential specification, and to normalize them by removing one maximal formula occurrence at a time, such an approach is neither as natural nor as convenient as one would wish. Nevertheless, Nj(~y^f does provide what might be called a permutation-free representation of proofs (albeit for a restricted set of connectives) and a reasonable framework within which to compare normalization and cut-elimination procedures. To some extent, therefore, it vindicates my earlier remarks about the desirability of a revised treatment of V- and 3-elimination. Unfortunately, when it comes to full predicate logic, there is no satisfactory analogue of Nj(~v>}'. The problems encountered in attempting to extend the above treatment to disjunction form the subject of the next chapter.

11 A detailed account of how these results can be translated between the negative fragments of NJ and LJ is to be found in Sections 5 and 6 of his paper. 12 This will be so, however, only if Zucker's indexing conventions and contraction conversions are adopted as well. It does not hold for the versions of the sequent calculus presented in the appendices. This is because (4), even with y replaced by = , does not guarantee that reduction sequences in Nj(~v^f can be translated into reduction sequences in Lj(~w\ If we were to add the clause: Furthermore, for all d" such that d" = )n2x.

(4.9)

Applying (4.5) twice to the expression within the outermost brackets yields

(4.io)

n = ((((n"/c)n 1 x /A)(n;7c)nr/B)(n^7c)nr/D)n 2 x

and, using (4.5) again to distribute Il 2 , we get (4.11) 11 =

((((n'7c)nr/D)n x M)((n' 1 7c)n 1 x /D)n x /s)((n 2 7c)n 1 x / J D)n x But, writing

n' AM B A B

66

Normalization, Cut-Elimination and the Theory of Proofs

for ((n"/C)n i x /Z))n 2 x and writing Ifc for ( ( n ^ 7 C ) n x / D ) n 2 x , where i = 1 or 2, (4.11) becomes

n' n=

AVB

A

IIi/£

n2

= AVB A B

ni n2 2. a is part of II* or n j . We might as well assume cr is part of II x (If it is part of IIJ, the argument is the same.) Here the various subcases (according to whether C occurs as an assumption in II*, n^ or Uf2) do not require separate treatment because it follows from (4.1)-(4.3) that, when A is not among the assumptions of C

,

{Ap} C

A

which may reduce to

ir n

n" n

B,C B,C (assuming that A —> B and A —> C are of the same degree and that 11', II" contain only maximal formulae of lesser degree). These are some of the problems involved in finding a correspondence between normalization in a calculus like Nf and normalization in NJ. None of this is strictly relevant to the matter at hand, however, although I will return to the topic later. Normal derivations have a particularly simple structure—more so even than in NJ. If a branch of a derivation is defined to be any subset linearly ordered by the edge relation which does not pass through (although it may terminate with) the minor premise of an application of rule (6), then every branch begins with a (possibly empty) series of eliminations, ends with a (possibly empty) series of introductions and, in between, (perhaps) an application of rule (7). It is obvious, therefore, that normal derivations have the subformula property. L e m m a 4.6 The formula *(A,B) A *(C, D) is not derivable in Nf where *(X, Y) abbreviates {X V Y) -+ ((X V Y -> X) V (X V Y -> Y)). Proof. Assume A, B, C, and D are all atomic. This makes matters a little

The Problem of Substitution

73

simpler, although it is not essential to do so. If the formula is derivable, it has a normal derivation, whose last rule of inference must therefore be an introduction, A-introduction, in fact. So, consider what form a derivation of *(A, B), say, can take. Again, the last inference must be an introduction, and the derivation will consist of an application of V-elimination followed immediately by introductions. An inspection of the rules reveals that it must look like the following: Ay B A (AvB)-+A (AVB^A)V{AVB^B) *(A,B)

B {AvB)-*B {AV B -» A)V (A V B -» B) *(A,B)

where the assumption AWB is discharged by any or all of the applications of —•-introduction. Similar considerations apply to the derivation of *(C, D). What is interesting about these derivations is that they cannot have fewer than two conclusions. It is apparent, therefore, that however many times A-introduction is applied there will always be a pair of conjuncts left over. Consequently, there can be no normal derivation of *(A, B) A *(C, D), only of *(4, B) A *(C, D), *(A B) or *(4, B) A *(C, £>), * ( C D). • Theorem 1, whose proof is now complete, indicates that there is a difficulty about defining substitution for the derivations of the propositional calculus. If this operation is to be well-defined, derivations which differ only in the order of their construction cannot be distinguished from one another. Yet I have just shown that there is no way to obliterate such distinctions entirely—at least not for any class of derivations sufficient to yield all valid formulae. The difficulty could be avoided, of course, by weakening the properties required of substitution. Unless we are prepared to change our understanding of this operation in more familiar contexts, however, there seems to be little doubt that they are appropriate ones. (This claim becomes even more plausible when substitution is considered in the abstract framework of category theory.) Anyway, although such an expedient would allow substitution to be defined for derivations of the kind considered above, it would leave their structural properties unaffected. In particular, there would still be distinct derivations which differed only in the order of their construction and, as a result, permutative reductions or their analogues would be required for a satisfactory normalization procedure. This circumstance, regrettable though it may be, seems unavoidable given the treatment of derivations and rules of inference adopted above. My aim, however, was to make it as general as possible. Permutative reductions will be needed not just for this particular multiple-conclusion calculus, but for all those systems (like Gentzen's N and L calculi—in both their intuitionistic and classical forms) whose rules can be interpreted as

74

Normalization,

Cut-Elimination

and the Theory of Proofs

generating a sufficiently large subset of its derivations. 11 Derivations are taken to be arrays of formulae structured by a logically significant relation and rules of inference are supposed to operate on their premises in a uniform way, but it is difficult even to imagine what would be entailed by doing without these assumptions. The same cannot be said for the stipulation that the two premises of an application of rule (1) or (6) be from separate derivations. If this restriction is relaxed and, more generally, even the conclusion of an inference is allowed to belong to the same derivation as its premise(s), application can be reduced to a single basic operation: connecting a pair of vertices, whether from the same or different derivations. It then becomes necessary to place some global restrictions on the structure of a derivation to ensure its correctness. This approach to multiple-conclusion logic is discussed extensively by Smiley and Shoesmith, who devote Part II of their book to the issues involved in piecing together derivations in this way. The flexibility gained by such a piecemeal method of construction does facilitate the definition of an associative substitution operation—several of them in fact, but this turns out to be a relatively minor advantage. It is hard to attach much intuitive significance to the joining of individual vertices, especially when they belong to the same derivation. Furthermore the resulting derivations, although they may not strictly speaking allow permutations of inference, do display structural differences which appear to be no more logically significant. (These will depend on whether different applications of rules with the same conclusion connect to one or more occurrences of it.) More importantly, our basic problem remains unsolved because we are looking for a homomorphism (with respect to the logical rules) from a familiar Gentzen calculus to one of this kind which will preserve substitution, and the image of A-introduction, say, under such a mapping must be a derived rule which transforms separate derivations of A and B into a derivation of A A B. Girard accepts the impossibility of finding a homomorphism of this kind. 12 He is led therefore to take a more radical approach to multiple11 Since Lemmas 4.2 and 4.3 depend upon the special form of rule (5), it might seem that the result could be avoided by replacing this rule with a more traditional version of —^-introduction, but this is not so. These lemmas will still hold for derivations in which no application of —•-introduction discharges an assumption in the derivation of one of the premises of an application of V- elimination lying above it, and it is only reasonable to require the structural effect of applying a rule in these special circumstances to be the same as in all others. 12 He too is interested in how to represent the proofs of classical logic by derivations whose inferences cannot be permuted with each other and which can be reduced to normal form without permutative reductions, and remarks in a number of places that it cannot be done. See, for example, page 9 of "Linear Logic" (Theoretical Computer Science, Vol. 50, 1987): "It seems that the problem is hopeless in usual classical logic and the accumulation of several inconclusive attempts is here to back up this impression." The results of the present chapter may also be seen as backing it up.

The Problem of Substitution

75

conclusion calculi, albeit one which may also be described as a piecemeal method of derivation construction. His rules operate by connecting pairs of individual formula occurrences, and the resulting figures are called proofnets if they satisfy a certain global soundness condition. Unlike the situation described in the previous paragraph, however, each occurrence must be the conclusion of exactly one application of a rule (axioms are treated as conclusions of a 0 premise rule) and the premise of at most one. What distinguishes Girard's approach above all is that he abandons the traditional logical vocabulary and studies a variety of novel connectives. In terms of these he is able to characterize fragments of classical logic with nice properties. There is no space here to do justice to his ideas and results, but they do provide further evidence for the interest of derivations with multiple conclusions.13 Another possibility would be to relax the requirement that a derivation must be an array of formulae. There are no doubt a number of ways to do this, but what I have in mind here is to treat them as sets of such arrays—sets of trees, in fact. For example, the derivation [A]

n AyB

[B]

nx n2 c

c

c

is to be interpreted as the union of

n

»°

A

B

"e? "c

with some notation to indicate that the occurrences of A V B shown are not among the conclusions of the resulting derivation, nor A and B among its (open) assumptions. This results in a rather unfamiliar notion of derivation which is difficult to reconcile with the idea of a proof as a determinate procedure for arriving at a conclusion. By removing the connection between the major premise of an application of V-elimination and the assumptions discharged by that application, it would appear that an essential component of the derivation has been lost. Furthermore, some justification is needed for treating A B AAB as an entirely different structure from AVB A B 13

His recent book Proofs and Types (Cambridge, 1989), written with Yves Lafont and Paul Taylor, includes a readable sketch of these ideas. A more detailed account is to be found in his paper "Linear Logic."

76

Normalization, Cut-Elimination and the Theory of Proofs

(After all, if the idea of a structural similarity between proofs and derivations is to be taken seriously, inferences have to be represented in a uniform way unless there is some compelling logical reason for not doing so.) Despite these objections, however, if the derivations of such a calculus could be shown to characterize the equivalence relation on natural deduction derivations generated by permutations of inference, they would at the very least be of formal interest. The impossibility of defining an associative substitution operator on derivations does not vitiate entirely this particular approach to logic since the derivations in question may still be (and indeed are) closed under substitution in the sense that, given derivations of A, A from T and of A' from T',^4, we can find a derivation of A, A' from r , r ' . Nonetheless, it is disappointing that there appears to be no straightforward extension of the negative fragment of NJ which preserves its distinctive combinatorial properties. This has consequences for both of the issues raised earlier. Granted that, whatever notion of proof is captured by NJ, it cannot be one in which the order of (permutable) inferences is important, there now seems to be little hope of representing this feature conveniently in a formal derivation. As for the correspondence between cut-elimination and normalization, had it been possible to exhibit a calculus C which needed no permutative reductions, together with homomorphisms from NJ and L J to C which preserved the proper reduction steps, this would have sufficed not only to establish a correspondence between reduction procedures for these calculi but also to make it plausible that permutative reductions are logically insignificant. Under the circumstances, it seems best to consider equivalence classes of derivations (the equivalence being generated by permutations of inferences) and try to interpret these within the context of a general discussion of the identity of proofs. If it can be successfully argued that equivalence in this sense and interreducibility (factored out by this equivalence) represent significant relations on proofs, it should be possible to make better sense of permutative reductions and reestablish a correspondence between cutelimination and normalization procedures.

5

A Multiple-Conclusion Calculus Before pursuing the line of investigation suggested at the end of the previous chapter, I think it worth considering multiple-conclusion systems of logic in a little more detail. I am less concerned with their intrinsic interest than with the fact that they seem to be the natural analogues of sequent calculi—for classical logic at least. As such, they provide a convenient framework for the comparison of N and L calculi in general, and a treatment of classical natural deduction which is superior to the conventional one. I shall attempt to substantiate these claims below after having first outlined a usable version of natural deduction with multipleconclusions. In the previous chapter, I discussed the relationship between consequence and derivability relative to a set of multiple-conclusion rules—for propositional logic at least—but failed to describe an adequate notion of derivation. In order to do so, I must specify what operation on graphs is to represent the application of a rule of inference. It is apparent that Lemma 4.3 cannot be expected to hold for such an operation, and derivations containing circuits will have to be allowed. I do, however, want to exclude circuits formed by joining a number of occurrences of the premise of a rule (or a number of occurrences of each premise in the cases of rules (1) and (6)) to a single occurrence of its conclusion (or a single occurrence of each conclusion in the case of rule (4)). My objection to this procedure is that its effect is to reintroduce the kind of structural features which prompted the search for alternatives to Gentzen's TV calculi in the first place. Consider for simplicity any one premise rule A R(A)R and suppose that the result of applying R to

n A,B,... 77

78

Normalization, Cut-Elimination and the Theory of Proofs

is the derivation

n R(AY

' •"•

obtained from II by adding a new vertex labelled R(A) below all the bottom vertices of II labelled A. (There is no loss of generality here since it is easy to restrict the application of the rule to fewer occurrences of A by appropriate use of subscripts.) Now, the obvious mapping from NJ derivations to multiple-conclusion ones constructed by rules which operate in this way, call it T, is clearly an isomorphism between its range and the derivations of NJ. So, for example, the difference between

n CV D

[C] [D] rii n 2 A A A

and

[C] ni A n CV D R(A)

R(A)

[D]

n2

A R(A)

R(A)

will be reflected exactly by the difference between the multiple-conclusion derivations CVD C

CVD D

.F(ni) T(u2) A A R(A)

C

and

D

^(no A R(A)

T(ii2) A R(A)

(It is conceivable that there is some alternative to T which would avoid these consequences, but it is hard to imagine what it would look like. On the whole, this particular line of inquiry seems not worth pursuing.) These considerations lie behind the development which follows. Since (4) is the only rule with more than one conclusion, a derivation having exactly two conclusions, both labelled ^4, can be represented by a figure of the form

nx CVD n = c D n2 n3 A A where IIi,Il2 and II3 are all single-conclusion derivations. II specifies two ways of reaching the conclusion A, depending on how the disjunction C V D is decided. (C V D is decided when IIi, or the result of substituting derivations of the appropriate kinds for the assumptions of IIi, contains a proof of C or of D in the sense that it reduces to a derivation whose last inference . C D , lS —.D °r C - D - }

A Multiple-Conclusion Calculus

79

Similarly, a derivation having exactly three conclusions, all labelled B, can be represented by a figure of the form

ni

ni

EVF E W

n'2

=

EV F E F

n;

or

GVH G H

n2

n"

B

n3

G\/H G H

K n'5

B B B B Suppose now that we want to derive A A B from II and II'. The resulting derivation should specify six ways of reaching the conclusion A A B, depending upon how the disjunctions C V D , E\/ F and G V H are decided. These can be represented in a single derivation by taking three copies of II and two of II7, and joining their conclusions to six new vertices labelled A A B in the following manner. (5.1) CVD C D

ni CVD C D

rii CVD C D

A

A

EWF E

n2 n 3 n2 n 3 n2 n 3 A

AAB

A

A/\B

A

AAB

A

AAB

ni F

EWF E

F

n2

n3

n'2

n^

GV#

B

GVif

B

AAB

AAB

This is not simply an arbitrary arrangement. It is designed to ensure that, no matter how each disjunction in II and II' is decided, there will always be at least one way to reach the conclusion .AAB. (Of course, a disjunction can be decided in only one way in all copies of a single derivation.) Furthermore, in view of the preceding discussion, each such way should be represented by a different path. If we add the (quite reasonable) requirement that each one should be represented by at most one path, then the above arrangement is the only possibility.

80

Normalization, Cut-Elimination and the Theory of Proofs

Guided by this example, I define below an operation of combination on graphs. The application of a rule of inference will then be interpreted as the combination of graphs, one of which has a special form. So, the above derivation is obtained by combining II and II' with the graph A B AAB (This notation is explained in the next paragraph.) In the case of a onepremise rule, for example (3a), its application to U should result in a derivation of A \JB obtained by adding two new vertices with this label, one below each occurrence of the conclusion A. Such a figure can be obtained by combining II with the graph A AVB Before proceeding further I need to introduce a few conventions: (1) I will write, for n > 0, A!... An A for the graph comprising n vertices labelled A\, . . . , An, respectively, which are joined to a single vertex labelled A below them. A Al...An is the graph obtained from Ax... An A by reversing the direction of the edge relation. Finally, I will use A to denote the graph consisting of a single vertex labelled A. (It should be obvious from the context when A is being used to denote a formula, and when a graph.) (2) As mentioned earlier, formulae occurring in a derivation will be assigned natural numbers as subscripts. For the present, these subscripts are to be considered part of the formalism of the calculus. The use of subscripts is simply a bookkeeping device. It corresponds to the use of sequences of formulae on the left and right of a sequent, and generalizes the idea of equivalence classes of assumptions which is routinely employed nowadays in the treatment of natural deduction. When derivations are regarded as instances of valid argument forms which can be combined together to produce new forms, subscripts make it possible to preserve distinctions which would otherwise be lost. They serve as place holders in much the same way as variables do in the usual notation for functions and terms. There is also a strong reason to use subscripts if one is interested in the strong

A Multiple-Conclusion Calculus

81

normalization theorem since this is known to fail for the version of NJ (in fact, even for its pure implicational fragment) in which either all assumptions of the appropriate kind are discharged by an application of —•-introduction or none are. Subscripts are the means of distinguishing between different occurrences of a particular assumption when it is desirable to do so. Although not absolutely necessary, matters are simplified if a graph comprising a single vertex is labelled by a formula with a pair of subscripts, its subscript as an assumption and its subscript as a conclusion. The advantages of this modification are twofold. It facilitates the comparison of multiple-conclusion calculi with the familiar Gentzen ones, and it simplifies various definitions below by eliminating the need to treat one element graphs as special cases. I will write the subscript as an assumption above the subscript as a conclusion so that, for example, the one element graph labelled by A whose subscript as an assumption is i and as a conclusion is j will be denoted by Aj. A one element graph labelled in this way will be called an axiom. It is convenient to be able to describe the axiom A1- as having a top vertex labelled Ai and a bottom one labelled Aj. I propose to adopt this manner of speaking henceforth, even though it suggests erroneously that the graph in question has at least two vertices. (3) Any axiom and any directed graph with at least two vertices, each of which is labelled by a subscripted formula, will be called a quasiderivation. The labels of the top vertices of a quasi-derivation are called assumptions, and those of its bottom vertices conclusions, but I will also use these terms to refer to the vertices themselves. Again, it should be obvious when a formula occurrence is meant, and when a vertex. (4) Graphs which are identical except for their vertices are said to be copies of one another. There is no need to distinguish between different copies of the same quasi-derivation, and I will always assume that distinct graphs and distinct copies of the same graph have disjoint set of vertices. (5) If II is a quasi-derivation which is not an axiom, I will write (^4^)11 for the quasi-derivation obtained from II by using Ai to relabel all assumptions of the form Aj. Similarly, U(Ai/j) is the quasi-derivation obtained by using Ai to relabel all conclusions of the form A3. If II is an axiom, say C^, (Ai/j)H is C^ if C = A and n = j , and II otherwise. Similarly, U(Ai/j) is Cf if C — A and m = j , and II otherwise. Anticipating the comparison with LK, I will describe (Ai/j)U as having been obtained from II by left contraction and U(Ai/j) by right contraction.

82

Normalization,

Cut-Elimination

and the Theory of Proofs

Definition 5.1 Given Ak, let II be a quasi-derivation with m bottom vertices labelled Ak and II' be a quasi-derivation with n top vertices labelled Ak,(m,n > 0). Suppose also that these vertices are enumerated in some arbitrary way, and let I I I , . . . , II n be n copies of II and U[,..., 11^ be m copies of IT. [II, j4fc,n/], the result of combining the conclusions Ak of n with the assumptions Ak of IT, is defined as follows: (1) If n is an axiom A[, [II, Ak, IT] = {Ai/k)Il'. (2) If IT is an axiom A\, [II, Ak, IT] = Tl(Ai/k). (3) If neither II nor IT is an axiom, [II, Ak, IT] is the graph obtained from the union of I I i , . . . , II n ,!![,... 11^ by identifying the vertices V(p^ and y(p'q) for each p, q (1 < p < n, 1 < q < m), where V(p,g) is the qth bottom vertex of Up labelled Ak and v^p^ is the pth top vertex ofn^ labelled .4*. It is easy to verify that, because copies are not distinguished from one another, [II, a, II'] does not depend upon the particular enumerations chosen for the conclusions a of II and the assumptions a of IT. Also, suppose for a moment that combination has been defined for graphs labelled by unsubscripted formulae as well, then figure (5.1) above can be written as

n',s,

TI,A

A B A/\B

(provided that B does not occur among the assumptions of II) or n,yi, IT,JS,

A B] AAB

(provided that A does not occur among the assumptions of II'). The next task is to explain what it means to be a derivation in this calculus. For this I need some additional notation and terminology: (1)

a. Two quasi-derivations are said to be congruent if they are obtainable from one another by a one-one mapping T between labels which satisfies the condition: For all A, i, T{Ai) = Aj for some j . b. Call a subscript occurrence intermediate if it is not part of the label of an (open) assumption 1 or conclusion. Quasi-derivations which are identical once all intermediate subscript occurrences have been deleted are said to be almost alike. I would like to be able to claim that derivations which are almost alike or congruent (i.e., related by the transitive closure of the union of (a) and (b)) are indistinguishable. As will emerge, however, only derivations which are both can be identified. Such derivations are

A formal distinction between open and closed assumptions is drawn below.

A Multiple-Conclusion Calculus

83

said to be alike, and in what follows a derivation will be considered well-defined if it has been specified uniquely up to likeness. (2) The quasi-derivations II and II' are said to be compatible if no subscript with an intermediate occurrence in II occurs anywhere in II', and vice versa. For the remainder of this work, I will tacitly assume that all quasiderivations II satisfy the following condition: no subscript which occurs on an assumption or conclusion of II has any intermediate occurrences in II. This guarantees that, for any II and II', it will always be possible to find a pair of mutually compatible quasi-derivations which are like them. (3) I will write

n

ii Bk for n*(An/i),An

Bk where n occurs nowhere in II*, and II* is a quasi-derivation like II which contains no intermediate occurrences of k. Similarly,

n Bk

Cj

will denote

|n**(^n/i),An,^An n c j where n occurs nowhere in II**, which is like II except that it contains no intermediate occurrences of k or j . (4) Let 111 and n 2 be quasi-derivations with conclusions of the form A{ and S j , respectively. I will write U

as

\An/i),An,

A {£>m/j)i m, n**(n ^ R

11

n

n

X-Y

Ft 1 m

^

\

J-

ni n 2 Ai

B3

Ck where II* and II** are like III and n 2 , respectively, except that neither contains an intermediate occurrence of k and they are compatible with one another; furthermore, m and n are distinct subscripts which occur nowhere in II* or II**.

84

Normalization,

Cut-Elimination

and the Theory of Proofs

(5) The notations Bk A{

Bk

n

Cj

Az

Ai

Bk

n

ni n2

Cj

are dual to the above—the duality being between assumptions and conclusions, and as such do not require separate explanation. Granted that the application of a rule is to be interpreted in terms of combination, there are still a number of different ways to read the axioms and rules of Chapter 4 as the clauses of a definition of (propositional) derivation, depending upon how these rules are to be applied. It is most natural to think of them as being applied downwards so that the definition becomes: Axioms: For all A, n and m, A7^ is a derivation of Am from An? Rules: (1) If II is a derivation of A, Am from T and II' is a derivation of A', Bn from r", then

n

ir

AABP is a derivation of A, A , A A Bp from T, Ff for all p. (I do not intend to exclude the possibility that A A Bp is already a member of A or A'.) (2a) If n is a derivation of A, A A Bp from T, then 7

n AABP is a derivation of A, Am from T. (Again, Am may be a member of A.) With the possible exception of rule (5), it should be obvious how the remaining clauses are to be formulated, so there is no need to list them here. As for (5): (5) Let II be a derivation of # m , A from T, then

(4,/»)n Bm A->BP

M

is a derivation of A —• £?p, A from T — An, where q is a subscript (distinct from p) which occurs nowhere in II, and A —• Bp ^ is the label A —> Bp augmented by some notation which indicates that any 2 I have changed the form of the axioms slightly so that they will satisfy the definition of quasi-derivation given above.

A Multiple-Conclusion Calculus vertex so labelled discharges all assumptions of the form Aq. write

85 I will

n Bm

A-*BP

{An}

for

(Aq/n)n Bm A->BP



Two remarks: (1) It is apparent that the derivability relation characterized by this set of derivations coincides with the consequence relation determined by the rules of Chapter 4. (2) If no restrictions had been placed on subscripts, or indeed if subscripts had been omitted entirely, the result would have been equally satisfactory from the point of view of derivability. I mention this to emphasize that the complications these involve have less to do with the multiple-conclusion approach than with the use I want to make of it—in particular, to study normalization and compare the derivations of different calculi. Formulations of more conventional systems suitable for these purposes would be no less complicated. As mentioned above, there are other ways in which the class of derivations might be defined. One possibility is to apply the rules upwards. A slight complication arises because rule (5) will now allow closed assumptions to be introduced into a derivation. Furthermore, when this happens, it is inconvenient to insist that there are no other closed assumptions of the form in question already present in the derivation. For the purposes of this paragraph, therefore, let the subscripts on closed assumptions not be classified as intermediate. (Of course, this alters slightly the meaning of compatibility, as well as the various notations explained in terms of it.) With this proviso, the clauses of the definition are easy to state. For example, clause (2a) reads: If II is a derivation of A from T, Am then

AABp

n is a derivation of A from F if every occurrence of the assumption Am 3 It is not hard to make this description more precise and to spell out a particular labelling procedure, but the above should be sufficient. Also, it goes without saying that the definition of quasi-derivation is now modified to include labels of this kind, and that an assumption is closed if it is discharged by some vertex.

86

Normalization, Cut-Elimination and the Theory of Proofs in II lies above a vertex labelled (A A B) —• Cq ^ and from T, A A Bp otherwise.

(for some C, Bp and n appears neither as a subscript nor as an annotation in II, 4 then Bm A-+Bp

W A is classically valid. The following conditions suffice for this purpose: (1) / i s one-one. (2) There is an enumeration {va)a 3v'A{v')n (6) 3, 3v'A(v% (10 3 ) ^ A(f(3vA(v))) J B -> A(f(3vA(v)))r {Bm} } ^ 3v'(B - A{v'))8

It seems reasonable to claim that (10 3 ) itself is intuitionistically valid and that rule (5) is problematic. Certainly, when the rules are applied downwards, the only way to obtain intuitionistic logic seems to be by restricting the latter—and, of course, (9V)—rather than (10 3 ). A purist might argue that the individual asserted to exist on the basis of an intuitionistic proof of 3vA(v) depends on the derivation of this conclusion and not simply on its form. Even a version of (10 3 ) modified to meet this objection, however, would not prevent the derivation of 3v(B —• A(v)) from B —• 3vA(v). It may seem anomalous that a rule which has traditionally been claimed to express the meaning of intuitionistic implication turns out to be invalid. This is so, however, only when it is applied to 1) and that [ I I / ^ ] ! ! " has been defined for all II" such at c(II // ) < n. There are a number of cases to consider, depending upon the form of II', but these fall into four groups. Cases 2, 3, 7, 9^ and 10^: IT is of the form IT

zr~ R where R is any single-premise, single-conclusion rule except (5). Let p be an index which occurs nowhere in n or II", then

[nMt](n"(cp/n)) \a/Ai]n' = Case 4:

cp

II' is of the form

n" C\tDn This is just a notational variant of the preceding cases. Cases 1 and 6: IT is of the form iii n2 D where R is rule (1) or (6). Let r and s be indices which occur nowhere in II or 11', then

[nM,](n2(c,/n))

rnM Cm

{Bp}

The foregoing definition may fairly be claimed to express the meaning of substitution in the context of multiple-conclusion derivations. Nevertheless, it might be thought that substitution could simply have been identified with combination. D, however, is not closed under arbitrary combinations. It is, of course, immediate that D is closed under substitution as defined above. Furthermore, if n is a derivation of A, Ai from T and IT is a derivation of A ; from T\Ai, then [ I I / ^ i T is a derivation of A, A' from r , H . It only remains to show: Theorem 5.1 The operation of substitution is well-defined. Proof. The problem here is that the various cases of the definition are not exclusive. They are easily seen to be exhaustive. The only doubt that may arise is on account of Case 5. Recall however that, if IIi A->Bm

{Aq}

and p is any subscript not occurring in II, then (Ap/qWx

n=

An A

• JDm

\Ap)

as well. In other words, if II is obtained by an application of rule (5), it can always be written in the form required by Case 5. The proof is by induction on c(lT), utilizing the fact that combination satisfies

(5.3)

p i , A,,n2],B7,n3] = [[ni,B„n3],A„n2]

provided that Ai is not among the conclusions of II3, nor Bj among those of n 2 . 1 0 The basis, c(II') = 1, is trivial. As for the induction step, there are Cf. the remarks following the definition of combination.

92

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and the Theory of Proofs

ten kinds of case to consider—some divided into subcases—corresponding to pairs of the four kinds of case in the definition. By way of example, I will sketch the case in which both representations of II' fall under the third of the four inductive clauses of the definition, i.e., the case in which

n' =

nx n2 n3 n4 B^ c_ = £ F_ D

G

(I omit subscripts to simplify the notation.) There are four subcases to consider.11

n"

Subcase 1: For some „ „ o hi

n" n4

ux n"

U2 = C E F G

and

n3 = B

C E D

n"

Given a derivation ~ „ let G , hi

~ = C

[n/411" [u/A]u4 E F and G

,-, = E

\n/A]U! [u/A]u" B C D

11

The point to be emphasized is that these are the only possibilities. It was to ensure this that I insisted on distinguishing between derivations which are almost alike and did not simply identify the application of a rule with combination. If IT is thought of as having been constructed by two different sequences of operations, say (ai)i Bj

iff

i > j , or i — j and A comes after B in e.

For U € D, replace all free occurrences of variables in the assumptions or conclusions of II by their images under g~l, then place the formulae thus obtained from the assumptions in decreasing order (with respect to > ) to the left of h, and those which result from the conclusions in increasing order to the right. Finally, delete all subscripts. The rules of LK can now be interpreted as generating the equivalence classes of members of D obtained by identifying congruent derivations associated with the same sequent. One need only ensure that a formula occurrence introduced by the application of a rule is assigned a sufficiently large subscript, and allow for the necessary resubscripting in the case of the two premise rules. Interchange is taken care of by one or more changes of subscript. If the axioms for _L are replaced by left and right rules for negation, there are two possibilities. One is to define negation in terms of _L and treat its rules as special cases of the corresponding ones for implication. The drawback to this approach is that LJ cannot be obtained from LK by allowing no more than one formula to appear on the right of a sequent; in addition, sequents of the form 0 \-L,A or 0 h A, _L are needed. 15 The 15 T h e resulting system [when restricted in the manner indicated in the text] is an adequate formalization of classical [intuitionistic] logic in the sense that @ h ^ is provable

A Multiple-Conclusion Calculus

99

other alternative is perhaps more natural. It involves treating negation as primitive and augmenting the multiple-conclusion rules of Chapter 4 by the analogues of ->-left and -±) - ± i Aj IS

[An] AV±K Aj

ip A->J. g {n}

n (A^±)->±j

in The resulting extension of NJD, however, is just a subset of NK2D' As a matter of fact, none of these extensions seem particularly natural. What seems to be essential to each of them is the use of the configuration [Am] AV1P Ai

JL2A-+±n{m}

in constructing derivations. But the obvious way to do this, while preserving the single-conclusion character of NK, is to admit the rule [A]

hA]

n n' c c c

whose interpretation will be

[A, AV ±p A.i

n

ck

-Lj

A ^l„{m}

n' ck

The resulting extension of NJ&, although obviously a proper subset of LKD, properly includes NK\D U NK2DThe uniform interpretation of Gentzen's calculi in terms of D helps us better understand the similarities and differences between their respective rules. Furthermore, it allows relationships between the derivations of the various calculi to be expressed in a rather satisfactory way. In particular, it is a routine matter to verify (by induction on the rules in each case) that NJD = LJD C NKD C LKD C D . For the propositional parts of LKD and D, this last inclusion can be replaced by an equality. The reason why LKD ^ D is that the restrictions on

A Multiple-Conclusion Calculus

101

(9V) and (10 3 ) are more generous than those placed on V-right and 3-left (or on V-introduction and 3-elimination, for that matter). Clearly, there is no sequent derivation corresponding to A(f(VvA(v)))n WA(v')m or 3v'A(v')n A(f(3vA(v))) m However, even members of D which contain no quasi-formulae among their (open) assumptions or conclusions may not be in LKp. Consider, for example,

ni 3v{A(v)AB{v))n A(x) A B(x)p A(x)q

n2

BvijAdv) A B(vi))m A(x)AB{x)r B(x)a A{x) A B(x)t 3v2(A(2v) A B(v2))u where x = f(3v3(A(v3) A Bfa))) for some V3.

6

Reduction Procedures The interpretation discussed at the end of the previous chapter can, of course, be extended to the various reduction procedures for TV and L calculi. I shall not spell out how this is to be done since it is a purely mechanical matter to translate Prawitz's reduction steps for NJ, for example, into operations on the members of NJD- I shall, however, assume such a translation in the discussion which follows. In addition, the rules for generating D themselves suggest a method of normalization. The possibility seems to exist, therefore, for a uniform treatment of reduction in all five calculi. It is to this topic that I now turn. Suppose U £ D and A is any formula. There is no need to refer to the rules in order to explain what it means for an occurrence of A to be maximal in II. Definition 6.1 An occurrence of A is maximal in a derivation II if there is a subformula B of A such that B occurs as one of its immediate predecessors and successors in II. In other words, a maximal occurrence is one which appears in a configuration of the form B

i

A

I

B (I will not bother with subscripts for the moment.) A reduction step deletes the maximal occurrence of A and identifies the two occurrences of its subformula B. This is essentially what all the familiar reduction procedures accomplish. Unfortunately, there are various complications which obscure somewhat the basic picture. These are discussed in (l)-(4) below. (i) Reduction steps are thought of as operations on derivations. This means, in particular, that the result of applying one to a derivation must 102

Reduction Procedures

103

itself be a derivation. The procedure described above does not satisfy this condition, however, since matters have been arranged so that the application of a rule may involve more than one occurrence of a formula. By itself, there is no reason why this should lead to different reduction procedures for the different calculi—except in the case of NK. (This is because NKp is the only class under consideration which is not closed under all the usual reductions.) In addition, however, reductions are required to preserve as far as possible the local character of the above procedure and this means, in effect, defining them relative to a set of rules: a reduction removes a certain kind of inference step from a derivation, namely all the occurrences of a formula which figure as the premise of a particular application of a rule together with all the occurrences of the conclusion of this application. But different sets of rules do suggest different reductions when the latter are conceived in this way. For example, suppose II, II' G NJp and IT results from II by removing an inference step of this kind while leaving the rest of the derivation intact; even though n and IT must also be in Up, there is no reason to suppose that the relationship between them can be expressed in such simple terms relative to the rules of L J. This dependence on the rules seems to be the major source of the differences between various reduction procedures. (2) Because there are rules with more than one premise or conclusion, the configuration displayed above may occur as part of a larger configuration having one of the forms B

B

B

i

i

\/

C

C

B

A

A

A

A

/\

S\

I

I

\S

B C C B B B Once the vertex labelled A has been deleted and the two vertices labelled B have been identified, the question arises as to what should be done with the vertex labelled C and that part of the derivation connected to it. There are basically two alternatives: one is to delete these as well, the other is to retain them while ensuring that C does not appear as an additional assumption or conclusion in the resulting derivation. For the latter an operation is needed which allows redundant formula occurrences or derivations to be adjoined to a given derivation. Out of deference to tradition I propose to call this operation thinning. Although the most satisfactory procedure would seem to be one which pruned as much of the derivation as possible, there are a number of reasons for adopting the latter alternative, or something approximating to it. In the first place, the inclusion of thinning may actually simplify reduction in a calculus. This is because, if we consider the sequence (or possibly, tree) of

104

Normalization, Cut-Elimination and the Theory of Proofs

operations by which a derivation is constructed, V- and 3-elimination (in NJ and NK), left rules (in LJ and LK) and right rules (in LK) are all such that applications of them may be made redundant by the removal of an earlier step of the construction. So, without thinning, reduction steps cannot operate simply on some initial subsequence (or subtree) of the construction. (In terms of the traditional representations of the derivations of Gentzen's calculi, the situation can be described by saying that applications of these rules which lie below a given inference step may be made redundant by its removal. As a result, a reduction step cannot simply operate on the subderivation terminating with the configuration to be removed, while the rest of the derivation remains unchanged, unless redundant applications of at least some of the rules are allowed.) Considerations of this kind are probably sufficient to explain Gentzen's treatment of thinning. It does not seem to be based on any general principles but, on the contrary, to be rather ad hoc and designed simply to facilitate the proof of the normal-form or cut-elimination theorem. 1 A second reason to allow some thinning is the well-known fact that, in its absence, normal forms are not in general unique. More precisely, they are not unique in any fragment which contains both conjunction and disjunction, as the following configuration illustrates: A Ay B A B C BAC C Although the uniqueness of normal forms is an important desideratum, it did not become one until relatively recently and, therefore, cannot properly be used to explain features of the traditional reduction procedures. Finally, without at least thinning on the right, the cut-elimination theorem will not hold for LK; there will, for example, be no cut-free derivation

of\-A,A-+B. There is no doubt that the treatment of thinning is another source 1

In the case of natural deduction, the fact that redundant applications of only two of the rules are allowed suffices to justify the claim that Gentzen's treatment is ad hoc. In the case of the sequent calculus, it is the use of thinning in the cut-elimination procedure which is disturbing. Neither alternative mentioned in the text is employed consistently; instead both of them are permitted, as well as everything in between—that is to say, using the example in the text, the whole subderivation connected to C may be deleted, selectively pruned, or left entirely intact. Its final form is determined by the position of the inferences from which it is constructed relative to the cut being eliminated. The principle is that, when an inference becomes redundant as a result of a cut-elimination step, it is deleted if it lies above the cut in question and retained if it lies below it. One consequence of this is that derivations which differ from one another in what are usually thought to be insignificant ways ( i e . , when one results from the other by a trivial permutation of inferences) may reduce to entirely different cut-free forms.

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105

of differences between reduction procedures. From the present point of view, however, it should not be. We can hardly claim to have understood the significance of reduction if we are unable to decide on general grounds what to do with those parts of a derivation made redundant by the removal of maximal formula occurrences. Furthermore, our decision on the matter should not simply reflect what is convenient given the format of a particular set of rules. (3) A third complication arises from the fact that the removal of an inference may add open assumptions or conclusions to the non-redundant part of a derivation. For the members of D and its various sub-classes, this can only occur in the reduction of maximal occurrences of an implication. There is no need to dwell on this case, however, since complete unanimity exists on how it is to be handled: such a reduction involves the removal of an application a of rule (5) followed by an application (3 of rule (6), so the derivation of the minor premise of /? can be substituted for all the assumption occurrences reopened by the removal of a. (Negation treated as a primitive and governed by rules analogous to -i-left and —>-right provides another example of a connective which requires a reduction step involving this kind of complication.) Notice that, were it not for this feature of the step for implication, reduction would be a trivial matter. This is especially evident for the members of D since all the other reduction steps actually diminish the size of the derivation to which they are applied. (4) The last point I want to raise is connected with the first. Call a collection X of occurrences of a formula in a derivation a premise or conclusion occurrence if its members constitute together one of the premises or conclusions, respectively, of a single application of some rule. If X is a premise occurrence, it is clear that its members need not all belong to the same conclusion occurrence (even if they are all intermediate) which means, in particular, that X may contain both maximal and non-maximal occurrences. The procedure outlined in (1) must, therefore, be supplemented if it is to succeed in removing all maximal formula occurrences from a derivation. For example, suppose X = X\ U X2 where X\ and X2 are conclusion occurrences of different rules, the members of X\ are maximal while those of X2 are not, and X is a premise occurrence of an application a of rule R. Clearly, what is needed is a way to replace a by successive applications a i and 0:2 of R which differ from it only by using the premises X\ and X2, respectively, instead of X. (X\ together with the conclusion of ot\ can then be removed in the manner described in (1) above.) In case R is a one-premise rule, this is accomplished by replacing the derivation in question by one which is almost alike. In general, however, two-premise

106

Normalization, Cut-Elimination and the Theory of Proofs

rules require something more complicated. In any event, the above demonstrates the need to incorporate some method of splitting up inferences into the reduction procedure. This need is another source of differences between reduction in N and reduction in L. In both kinds of system it is met by steps which allow the permutation of inferences. The effect of the permutative conversions in N, however, is to allow a premise occurrence to be split only when it contains some maximal members. On the other hand, there is no such restriction in L, where the fact that cuts may be permuted upwards past any other inference legitimizes every kind of split. As in the case of thinning, these differences are most plausibly explained by reference to the format of the rules and what is convenient for the proof of the normal-form or cutelimination theorem. This state of affairs, however, is no more satisfactory than the corresponding one in (2). We ought to be able to decide on the basis of general principles, and independently of any particular formalism, which inferences may be split (without affecting whatever properties are preserved by reduction) and which, if any, may not.

Let me conclude this discussion with a couple of observations. All the problems associated with establishing a correspondence between cutelimination and normalization derive from the issues raised in (2) and (4) above, i.e., from the need for thinning and permutative reductions. Because these are not treated in an altogether satisfactory manner in the traditional accounts of reduction, it would be desirable to provide a treatment which is in some sense more natural and, in addition, applies equally to both N and L calculi. By itself, this does not seem to be an especially difficult task; what is harder is to accomplish it in such a way that uniqueness of normal forms and the strong normalization property are preserved. I turn now to a description of possible reduction procedures for the members of D. In view of my previous remarks, a necessary preliminary step is to define an operation corresponding to thinning. To this end, I propose to augment rules (l)-(7) of Chapter 4 by the following:

It would seem more natural to formulate these rules with Ak replaced by A{. The resulting class of derivations would not be closed under contractions however. Thinning is essentially a method of combining derivations together, while at the same time dispensing with some assumptions or conclusions. There are at least two ways to represent it. The one I have chosen is to attach the thinned derivation (or formula) to the occurrences of a conclusion. The drawback to this approach is that there is a measure of artificiality in-

Reduction Procedures

107

volved in interpreting the usual thinning rules since these make no reference to a particular assumption or conclusion by means of which the thinned formula is attached to the rest of the derivation. The alternative is to do without attachments of this kind altogether and allow derivations to be disconnected graphs; roughly speaking, derivations will be constructed by taking sets of derivations in the old sense and discharging some of their assumptions or conclusions. Although superficially attractive, this approach is fraught with difficulties. It becomes necessary to redefine such basic notions as likeness, combination and substitution. Substitution, however, is not easily defined for derivations of this kind without sacrificing some of its basic properties. It must either be defined relative to some construction tree of the derivation being substituted into, or distinctions must be made between a variety of derivations all of which are essentially alike. (An example of the kind of undesirable distinction I have in mind is that between {II} and {II, IT}, where II and IT are alike and compatible.) It is technical problems of this kind which have led me to prefer the thinning rules given above. Let DT denote the class of derivations which is like D except that it is generated by the rules (LT) and (RT) in addition to (l)-(7), (9Q) and ( 1 0 Q ) . The definition of substitution is easily extended to the members of DT for, although new cases arise, these fall within the groups specified in the definition. In particular, case (RT) can be treated with Cases 1 and 6, and case (LT) can be treated with Case 4. A similar remark applies to the argument that substitution is well defined: new cases arise, but no new kinds of case. In view of this, I shall not bother to rewrite the definition, but will take it for granted that substitution into a member of DT has been properly defined. Furthermore, the notation introduced earlier in the context of members of D will be used unchanged for the derivations of DTThe rules of Gentzen's N and L calculi can be interpreted in DT without the restrictions required by D. In the cases of NJ and NK, this involves extending the earlier interpretation to cover applications of V- and 3-elimination which do not discharge any assumptions in the derivation of the minor premise (or which do not discharge assumptions in both of them in the case of V-elimination). The idea is simply to use (LT) to add assumptions of the appropriate form when these are lacking. I consider the case of 3-elimination below. It should be obvious how to apply the same technique to the troublesome cases of V-elimination: Using the same notation as before,

n z\ Af IV \

M ))q

^k

t r

corresponds to

A(x)q

108

Normalization,

Cut-Elimination

and the Theory of Proofs

where IIJ is n^ if rii contains occurrences of A(g(x))q as an assumption, and U{ is A[X)g

Cr

otherwise. To interpret LJ and LK, provision must be made for left and right thinning, which I take to be the following pairs of rules: . (°)

A. a. r u A Ai,Bj,r\-A

Bk,T\-A

(°)

r u A p. A. r\-A,Bi,Ai

... r\-A,Bk (v6)' Ai.ri-A.B,-

and 7 (bv>

Bj,T\-A,Ai

Then, A*

n Bfc, T h A AuBj,T\-&

Bk

corresponds to

Bj T

n

A

r n

n r h A, 5fe

corresponds to

^,rhA,B,

Aj

r h A, 5fe

corresponds to

A

Bk

w^-

B,

n Bfc, T h A B.-.ri-AMi

A

r n

n n-A,*,,*

Bk

corresponds to

rs fc ^ n A

When k = j m the above, these rules are obviously equivalent to the familiar thinning rules for the sequent calculus. Their (b) versions do not add to the deductive strength of the calculus, for it is easy to demonstrate that without them h A AihA is a derived rule. In other words, given

n I-A

Reduction Procedures

109

it is always possible to construct IT without using the (b) versions of left or right thinning. The argument is by induction on the number of inference steps which follow the last application of —»-right in II and is routine. 2 As for their resubscripting function, it can be duplicated by the instances r h A

'**

and

**'

r h A

of the (a) versions. (The right-hand side of a sequent can never be empty in this formulation of L.) These (b) rules are not even necessary for cut-elimination to hold. They do, however, make it possible to give a more systematic and, I hope, more rational treatment of thinning in the cut-elimination procedure, and that is why I have chosen to include them. The thinning rules of an L calculus for deriving sequents composed of sequences of unindexed formulae are not hard to interpret in DT* Right thinning is thought of as operating on the right most formula of a sequent, and left thinning on the left most one. I mention this only to emphasize once more that all variants of Gentzen's calculi can be interpreted within this multiple-conclusion framework without the need to tinker with their rules. To substantiate this claim further, let us briefly consider how to interpret -i-left and -i-right in a multiple-conclusion calculus which treats negation as primitive. There are a number of ways to add rules for negation to DT. Perhaps the most convenient is to follow Kneale and allow rules which have zero premises or conclusions.3 Then rule (7) can be replaced by: (7')

An

"Am

and

(8')

—

(Here * is supposed to indicate that (7') has no conclusion and (8') no premises. It can be thought of as an auxiliary symbol whose function is to close certain assumptions or conclusions. It is convenient to stipulate that, when * occurs as a result of applying (7')> this occurrence cannot be part of an application of (8') as well.) Let D'T be the class of derivations generated by this modified set of rules. Corresponding to D'T is a sequent calculus obtained from LK by removing the axioms for _L and adding the usual negation rules. It is easy to see 2 3

"last" here means "having no application of —+-right below or to the left of it." The Development of Logic, page 542.

110

Normalization, Cut-Elimination and the Theory of Proofs

how this calculus can be interpreted in D'T.

r n

n r h A, An r,-AmhA

corresponds to

r , An h A

corresponds to

AAn

-.A

and

r h A, -,Am Everything else is as before. Notice, however, that the (6) version of thinning on the right is necessary in this formulation of the sequent calculus if r h Ai is to be derivable from V K Let A/rj£,T be the class of derivations obtained by interpreting the rules of NJ as instructions for constructing members of DT, and similarly for NKDT1 LJDT and LKDT. It is an easy matter to establish relationships between these classes similar to those described at the end of Chapter 5. Now, however, NJDT

C NKDDT

C LKDT

C

DT

and NJDT

C LJDT

C LKDT

C

DT.

It is not the case that NJ&T — LJryT C NKJJT, because in the TV calculi thinning is only used when it is needed for an application of V- or 3elimination, whereas arbitrary left thinnings are allowed in LJ. I now want to list some reduction steps for the members of DT- In order to simplify matters, let DT be modified so that An in rule (7) is always atomic and A itself is distinct from _L I assume that the corresponding modifications have also been made to both the classical and intuitionistic negation rules in natural deduction, and to the axioms for ± in the sequent calculi. It is well known that such changes do not affect deductive strength. I propose to describe these reductions without first specifying what it means for a derivation to be in normal form. My reason for so doing is that I am unwilling to commit myself in advance to a particular notion of normal form. Clearly, however, a normal derivation should contain no maximal formula occurrences and should possess the subformula property—or something approximating to it. In addition, I would like to give a sufficiently comprehensive and general list of reduction steps so that the familiar reduction procedures can be interpreted in terms of them. This is my primary concern here, and I will not worry about whether all reduction sequences terminate or about the uniqueness of normal forms until later. To state the reductions I must introduce a further piece of notation. Consider for a moment the simplest kind of proper reduction step in any Gentzen calculus. Roughly speaking, it can be translated as follows:

Reduction Procedures

111

If II has some subderivations of the form II' A

Introduction

f(A)

Elimination

then II reduces to the result of replacing these by derivations of the form IT A The problem is to specify what constitutes a subderivation of II, and what it means to replace a subderivation in II by some other figure. In the case of tree derivations the answers to both these questions are familiar and obvious. The matter is a little more complicated here, however. I want to say that III is a subderivation of II2 if the latter can be obtained from the former by applying a series of rules of inference, and that the result of replacing II1 by II3 in II2 is the figure obtained by applying this same series to II3 rather than III. (In fact, I am really describing a special kind of subderivation here, namely one whose assumptions are also assumptions of II2. Call it an initial subderivation. A more general notion is obtained if, in addition to applications of rules, substitution for the assumptions of IIi is allowed.) The step described above cannot always be expressed in the form

n' A

f(A)

reduces to

n'

A A n" U" even if it is simply the translation of a reduction in NJ or LJ. Suppose, for example, that I want to remove a maximal occurrence of f(A) in the derivation of the left minor premise of an application of V-elimination. This will correspond, on the multiple-conclusion interpretation of the rules of NJ, to the removal of one or more occurrences of f{A) from a derivation II* in NJDT- It may not be possible, however, to represent II* in the form shown on the left above, where the occurrences of f(A) displayed in the figure are those to be removed by the reduction. II* might have the form ni

c ( n2 \ A f(A) A

V n3 / E

D

n4 E

112

Normalization, Cut-Elimination and the Theory of Proofs

or be the result of substituting this latter for the assumptions E in some II5, and so on. (Again, it is the failure of generalized associativity, and in particular of condition (4.4), which is responsible for the present difficulty.) Another possibility is that II* results from the figure shown by applying rule (5) to its conclusions of the form E. In this case, there may not even be a II5 such that II* is of the form {[[[U1/C}Il+/A}U3/D}Ui/E}U5 where n + is

n2

A f(A) A. These considerations underlie the following definition. Definition 6.2 (1) a. For n e DT, the subset S(I1) of DT is defined by induction as follows: i. II € S(n). ii. If IT € S(I1) and II" results from II' by applying any one premise rule, then II" € S(II). iii. If III n 2 eDT € S(II) and R then

rii

n2

Bm

Cn

and

n 2 II! c_n^ _ B_ m_ Dp

are both in S(II), where R is any two premise rule, b. S'(II) is defined like S(II) except for the additional clause: iv. If IT G S'(II) and II" G DT has conclusions of the form A%, then \n"/Ai]n' E S'(n). II is said to be a subderivation of II* if II* G S'(II), and an initial subderivation of II* if II* G S(II). (2) a. So(II) is defined like S(II) above except that (i) is replaced by the clause: i ; . If II' G DT, and Ai appears among both the conclusions of II and the open assumptions of II', then [ n / ^ ] l T G So(II). II' G S„+i(n) iff W G So(n /; ) for some II" G S n (II). b. S;0(II) is defined like S 0 (II) except that S'(II) replaces S(II) in its definition. S^ + 1 (n) is defined like S n + i(II) with S^(II) and S'0(U) playing the roles of S n (II) and So(II), respectively. II* is said to have subderivations of the form II if II* G S^(II) for some

Reduction Procedures

113

rc, and to have initial subderivations of the form II if II* G S n (II) for some n. If II* has initial subderivations of the form II, then II* is the result of applying a sequence of operations to II. These can be taken to consist of substitutions and applications of rule (5), since applications of all the other rules can be treated as special cases of substitution. If E abbreviates such a sequence, II* may be written as y. (Notice that, although the latter denotes a derivation, E by itself merely stands for a sequence of operations.) Suppose that II* E S m (II) and that IT is a derivation with the same conIT elusions as II. It is obvious that y the result of applying the operations in E to IT, 4 is a member of S m (II / ) with the same conclusions as II*. If, in addition, the open assumptions of II' are included amongst those of II, IT then those of v will also be included amongst those of II*. I propose to use E, E', E i , . . . to stand for sequences of such operations in general. Using this notation, the reduction step (6.1) above can be written as IT

-Af(A)

II' reduces to

v

E It is just the special case in which E consists of a single substitution. I will write II* as ,y. to indicate that II* has subderivations of the form II. (/.e., the parentheses around E mean that it may contain operations which require substitution for the assumptions of the derivation under construction.) It might be argued that, since the notion of initial subderivation does not correspond to anything very natural when applied to the derivations of N and L, the reduction steps for these calculi could be better formulated in terms of subderivations. The above would then become IT

-Af(A)

n' reduces to

,v.

The former, however, seems more appropriate in the present context. Both 4 W h a t is intended by this phrase should be sufficiently clear for the purposes of the present discussion. See definition 3 below for a proper account of it and of some related notions.

114

Normalization, Cut-Elimination and the Theory of Proofs

formulations are, of course, equivalent because, if

n' A f(A) A (S) it is always possible to find II" and E' such that IT =

n" A

—— n* = f{A) E'

rr n" and

,£. =

E,

(I expand on this remark below.) The differences between the reduction steps for NJ, LJ and LK cannot be expressed simply by placing conditions on E. From this perspective, (proper) reductions in all three calculi are essentially of the same sort. To represent them we must in general allow the derivation on the left to be a member of S n (II + ), for any n, where IT A

n+ = f(A) This kind of reduction step does not seem particularly natural, however, when the rules for generating D? are considered. (The criterion of naturalness employed here is that a single reduction should remove exactly those maximal formula occurrences which constitute the conclusion of a single application of an introduction and the major premise of a single application of an elimination.) The reason is that substitution is not a basic rule of DT- In view of this, it would be more appropriate to restrict the step described above by requiring that the derivation on the left be a member of S ( n + ) (where II + is as above). There is a sense, therefore, in which a proper reduction step in any of Gentzen's calculi corresponds to a series of one or more natural reductions in DTDefinition 6.3

n IT (1) Given _, and II', _ is defined by induction on the length n of E as follows: II IT a. If n = 0, y is just II and y = If.

Reduction Procedures b. If n = m + 1, ^ results from

yn

115

where E' has length m, by

performing one of the following operations: i. Substituting the conclusions Ai of y, for the assumptions Ai of II" (for some A{ and n") In this case y = yn

and

II" if Ai is among the conclusions

L,/Ai

= _,, otherwise,

ii. Applying rule (5) to the conclusions Ai of _,,

n;. In this case y is the result of applying rule (5) to the conIT . n' IT elusions Ai of yf if such there be, and = , otherwise.

n n' IT (2) Given ,y, and II', ,y, is defined in the same way as ~ except for the additional clause: iii. Substituting the conclusions Ai of II" for the assumptions n

A of

IT IT In this case ,„v = [W'/Ai], ,y (3) Given ,~. and II' such that II, II' G LJDT

and LJDT is closed under

IT . IT the operations in E, / v n is defined in the same way as ( , except that (iii) is replaced by the clause: iii'. Substituting the conclusions Ai of II" for the assumptions n

A-of A% 0 t

(E')

n' In this case

/ v l

IT = [ n / ; / i 4 i ] / v n if

IT /v,x

has assumptions

n' of the form Ai, and

/ v l

= [II"/.4 2 ]n* otherwise, where

n' IE'}

IT s

IT = ,

D • (-^j * the- conclusion of

/V/V)

Bj Notice that, even if the open assumptions and conclusions of II' are

116

Normalization, Cut-Elimination and the Theory of Proofs

n' included in those of II,

y

may have open assumptions which are closed

in v . A similar remark applies to the relationship between , , and , v v Notice also that, IT if II = i y

then

IT II = . ,.

for some £'. Before listing the reduction steps, I introduce one last convention. As remarked earlier, a reduction step is taken to have the general form: If II has initial subderivations of the form IIi, then II reduces to the result of replacing IIi by II2 in II. For ease of writing, however, I will simply display the subderivations directly involved in the reduction, so that the above will be written IIi reduces to II2 rather than *

reduces to §

for any E.

The reductions themselves fall into three groups: I. Proper Reductions These remove maximal formula occurrences.

ni

(i)

n2 -Dm

An

ni reduces to

AABP Bq

n2 -Om

An

Bq

and similarly for Aq in place of Bq.

(2)

n

n

An

An

A V Bm

reduces to

V

Q

and similarly for Bn in place of An. (3)

a. II An

n' Bm A->BP W Br

n(A,/n) reduces to

Aq

n,(Br/m) Br

if Aq is among the assumptions of IT. (Notice that, because II and II' have to be compatible, II cannot have conclusions of the form Aq.)

Reduction Procedures b. II An

II' Bm A^BP Br

reduces to W

117

n n' A

BT

R

otherwise. (4)

n A(v')n VvA(v)m A{x)p

reduces to

n'{A(x)p/n) A{x)p

provided that the figure on the right is a derivation with the same open assumptions and conclusions as the one on the left, where II' is obtained from II by substituting x for each occurrence of v' connected to those in conclusions of the form A(v')n.5 (5)

U{A(x)p/n) A{x)p

n

A(x)n 3vA(v)n A{v')p

reduces to

n"

ir provided that the figure on the right is a derivation with the same open assumptions and conclusions as the one on the left, where II" is obtained from IT by substituting x for each occurrence of v' connected to those in assumptions of the form A(v')p. Comment: The restrictions on (4) and (5) are required because DT is not closed under transformations of the kind employed in these steps. More disturbing than this fact is the impossibility of eliminating all maximal occurrences of universal and existential formulae by means of these reductions, even from derivations without quasi-formulae among their (open) assumptions or conclusions. This can be seen from the following rather trivial examples: 5 Connectedness is defined in footnote 5 of Chapter 3. In the presence of (LT) and {RT), it seems best to stipulate that for the purposes of Clause 2 of this definition Bk does not lie immediately below Ai in a configuration of the form

Ai

£*

nor does Ax lie immediately below Bk in Bk B3

A, •

118

Normalization, Cut-Elimination and the Theory of Proofs

A

\/v{A{v)VA{v))n (A{v')\/A{vf))m A{v')p

\fvA{v)r A(b)t

A(v')q

W'A(v")s

(b)t 3vA(v)r A V

( ')P

and

3v"A(v")8 A V

( ')Q

A(v') A A(v')m 3v(A{v) A A{v))n

(where v' = f(VvA{v)))

( w h e r e In the special case where II has only Bk as its conclusion, the figure on the right is to be interpreted as U'(Cq/i). (2)

n Op

n Ci Cq Bk

—n—7*

reduces to

Op

ir

B

C

i,

P

L/p

Cq

BTm

where A\x,..., Afn are all the open assumptions of IT with the exception of Bk, and B} . . . . , BY1 are all its conclusions.6

n(B fc/j )

n

(3) Ai

reduces to

Bj

Bk

Bk

n

(4)

n(sfe/j) reduces to

Bj Bk

Bk

Ai

n

(5)

n n' 11'

reduces3

-^71

tO

\^T

Ck

ck (6)

n An

n ^„

n' reduces to

/l

m

1

n' r

Og

c~k where r occurs nowhere in 11 or 11'. 6 In Dip (the version of DT with rules for -i, rather than axioms for _L) the possibility arises that IT may have no conclusions. In this case, the figure on the right is taken to be

n(c,„) whenever Bk is the only open assumption of IT.

122

Normalization, Cut-Elimination and the Theory of Proofs

Comments: (a) (1), (2), and (3) are sufficient for interpreting the reductions employed in the usual normalization/cut-elimination procedures for the various Gentzen calculi. Unless they are restricted in some way, however, normal forms will not be unique. (b) Notice that (1), (2), and (6) have the property that the figure on the left has the same open assumptions and conclusions as the one on the right. In (5), on the other hand, the figure on the right may have additional open assumptions. B. Thinning Permutations (i)

iii Ai

n2 Bj

reduces to

~W^~

n2 II i Bj

At C~qR

Cd Cd and vice versa, where R is any single-premise, single-conclusion rule. If the step is read from left to right, I assume that the only occurrences of Bk as a conclusion in

ni n 2 Ai

Bj Bk are the ones displayed and q is supposed to be a new index; furthermore, if R is an application of rule (5), I assume that no assumptions in Iii are discharged by it. If the step is read from right to left, I assume that the only occurrences of Cq as a conclusion in

n2 Ei

cq are the ones displayed and k is supposed to be a new index; furthermore, if R is an application of rule (5), I assume that no assumptions in rix become closed by it. (2) a. iii n 2 n2 Ai Bj reduces to IIi Bj

and vice versa, where R is any two conclusion rule. If the step is read from left to right, I assume that the only occurrences of Bk as a conclusion in

ni n 2 Ai

Bj

Bk are the ones displayed and q is supposed to be a new index. If the step

Reduction Procedures

123

is read from right to left, I assume t h a t the only occurrences of Cq as a conclusion in

n2 Bj Cq

Dn

are the ones displayed and k is supposed to be a new index. (2)

b.

iii

n2 Bj Bk

n2 Bi

reduces to R

R

Cq Cn and vice versa. Again, R is any two conclusion rule and the same assumptions about Bk and Cq are made as in (2)a. "m

(3)

a.

^n

Dn

n2 n3

Eh n 2 At

At

reduces to

Bj n 3 Bk Cn R Dm

Iii Bj Cn R Ai Dq Dm

and vice versa. (3)

iii

n2

n 3 At

B3

b. Cn

Bk Dn

n3 n2 reduces to

}

Iii Cn Bj R M Dg

D„

and vice versa. In b o t h cases, R is supposed to be any two premise rule; furthermore, if the step is read from left to right, I assume t h a t the only occurrences of Bk as a conclusion in

ni n2 Ai

BL

Bk are the ones displayed and q is supposed to be a new index. If the step is read from right to left, I assume t h a t the only occurrences of Dq as a conclusion in

n2

n3

Bj

Cn Dn

or in

n3 n2 Cn

Bj Dg

124

Normalization, Cut-Elimination and the Theory of Proofs

in the case of (3)b, are the ones displayed and k is supposed to be a new index.

n

(4) Ak

n reduces to

Bi

R cJ C Bj n

where R is any single-premise, single-conclusion rule, and vice versa. Also, the usual assumptions (i.e., those made in (l)-(3) above, adapted in the obvious way to the present case) about Ak and Cq apply. (5)

a.

n

n

Ai

Ai

Ak Cn

Dr

reduces to

Bi

R

Cn

R

Da Dm

Bj

and vice versa. (5)

b.

n

n

Ai

Ai

Ak

reduces to

Bi ^n R

Urn

3

Dn

&n

Dm

Bj

and vice versa. In each case, k is supposed to be any two conclusion rule, and the usual assumptions apply to Ak and Dq. (6)

a.

M

Cn

Ak^

Bj

^n

Ai

Dm

Bj

reduces to

R

R

~DZ~

and vice versa. (6)

n2 nx

ni

n2

nx n 2

b. ni At C

Ak

n

Bi

reduces to

Dg

0

n

i /

-m

B.

and vice versa. In each case, R is supposed to be any two premise rule, and the usual assumptions apply to Ak and Dq. Some additional permutations are needed for D'T to alter the position of a thinning with respect to applications of the negation rules. The following will suffice for this purpose: (7)

II'

n

->AP ~>An Bk

reduces to

n At

Bk

IT -*An

and vice versa. If the step is read from left to right, the only occurrences

Reduction Procedures

125

of-.A n as a conclusion in

ir -»4» ->^n Bk are the ones displayed and t is supposed to be a new index; if the step is read from right to left, the only conclusions of

n Am At

Bk

having the form At are the ones displayed and n is a new index. (8)

nx n2 iii n Bk ~^Ap reduces to Bk Am n2 ~*An At ~~*Ap * * and vice versa, where similar assumptions apply to -*An and At as in the case of (7). (9) * * ->Ap Am reduces to -*An Aq -"An Bk Am Bk and vice versa, where q is any index if the step is read from left to right, and similarly for p if the step is read from right to left. IT Am

(io)

n Bk

* ~^Ap Am -*An

* reduces to

-iAn

n *

B

A

«

A

and vice versa—p and q as in the case of (9). Comment: These permutations are more than are needed for normalization. To reduce maximal segments to maximal formula occurrences, it would be sufficient to allow them to go only one way, either from left to right or from right to left, and apply only to certain rules. My motive in presenting them as I have done above is to leave open the possibility of equating derivations which differ from one another only by some permutations of thinnings. This concludes the list of possible reduction steps.

7

Correspondence Results I now want to consider briefly the question of a correspondence between the steps described in the preceding chapter and reduction in LK. I will adopt for a moment the terminology according to which the derivations of LK are mapped onto LKpT. Call this mapping (/> and let d, d ' , d i , . . . range over sequent derivations, d >i d' means that d reduces to d' by applying any one of the reduction steps listed in Appendix B, and > is the transitive closure of > i . It can reasonably be claimed that the reduction procedure characterized by > is in essence the familiar one which derives from Gentzen. It consists of three kinds of reduction step: the elimination of a cut one of whose premises is a logical axiom, the replacement of a cut by one of lower degree, and the permutation of a cut upwards past another inference. Unlike the traditional procedure, however, these steps are not applied in any systematic order (as determined, say, by the degree of the cuts or their position in a derivation); if a step can be applied, it may be applied. The other significant difference concerns the treatment of thinning. Ignoring distinctions which cannot be made in the usual formulations of the sequent calculus, it is fair to describe the reductions listed in the appendix as allowing any application of thinning to be permuted upwards. Although not part of the traditional procedure, these are perhaps not such a radical extension of it as they may appear to be, for if d! is obtained from d by such a permutation (or indeed by permuting any other inference upwards), we can find d1? d[ such that <j>(d) — (di), <j>{d!) — Bk as a conclusion in n Ap

IT

A-Bm A-+Bk iii.

Cn

are the ones shown, and q is supposed to be a new index. IIi IT Ux n II Cn A —• Bm reduces to Cn Ap W Ap A^Bk Aq A^B, Br Br where the only occurrences of A —• Bk as a conclusion in n;

iii ^n

A —» Bm

A^Bk are the ones shown, and q is supposed to be a new index. (3) If R is any other elimination, the step only applies from left to right. 1 1 (2ii) and (2iii) above can be included in X because, although they themselves are not members of (IHb), they can easily be derived from this group of reductions. ((2ii) is obtained by an application of (6a) from left to right, followed by an application of (6b) from right to left. Similarly, (2iii) is obtained by an application of (3b) from left to right, followed by an application of (3a) from right to left.)

128

Normalization, Cut-Elimination and the Theory of Proofs

(To establish a correspondence between the version of LK with rules for negation and LKJJT , the image of this calculus under , X must be augmented by (7) and (8) from left to right, and (9) and (10) from right to left.) It is not surprising that (IIIA3-6) are omitted from X. (IIIA3) is a part of the normalization procedure for N, rather than of cut-elimination for £, whereas (IIIA4-6) are best described as experimental steps. The restrictions on the members of (IIIB) are also to be expected. From one point of view thinning permutations are all alike in the sense that, if one is allowed, there is no reason to exclude any other; the members of (IIIB) were selected with this in mind. On the other hand, from the more limited perspective of a cut-elimination or normalization theorem, the criteria for including such reductions are stricter and take into account only what is necessary and convenient, given the format of the rules, for proving the theorem in question. Theorems 7.1 and 7.2 are established by lengthy and mostly routine inductive arguments. I shall merely outline these below, considering only one or two of the more interesting cases in any detail. Proof of Theorem 7. 1: If

ni . £ ^

n; S,

we show by induction on the length n of E that, for some d!', d such that (j){d) = ^

and cj){d') =

^

,d > x d'

Basis Step: n = 0. There are various subcases to consider according to which step transforms IIi into 11^. For reductions from group (I), it should be obvious that the result holds by virtue of the corresponding reductions in group (B). 2 Consider, by way of example, (II): If (j){di) = IIi, and (j){d2) = II2, then the image under / of

di

di AnhAn r\-A';Bm An; T'\-A;AABp

rt-A;An

BqVBq A A Bp r- Bq

Aw;r'hA,Bq

r,r'hA,A',£, is

ni

n2

AABP Bq. 2 See Appendix B below for a detailed description of the version of LK assumed here, together with a complete list of reduction steps for this calculus.

Correspondence Results

129

A single application of (Bib) to the former yields d2

rfx ri-AMn

r'hA';Bm An,r\-A',Bq BqhBq An;T'\-A',Bq r,r'hA,A',B,

er (j) whose image under <j>is is

ni

n2 Bq.

In group (II), (1) is taken care of by (C13), and (2) by (C16). Turning to (IIIA), consider first (la): Let <j)(di) = Hi (1 < i < n), {d!) = II', then the image under (j) of the derivation given below is the same as the figure on the left in the statement of (IIIAla) in Chapter 6 above. (A is supposed to stand

for {*£,...,££}.)

rt _.

d' d T h A'; Ct di A},;.. ;A£ \~A;Bk ft;r'hA',C, h A i ; A\ At;...; A? ; P h A, A', Cq 4;...;A?;r„ri-A1,A,A',C,

dn rnhAn;^.

."

r„,... 1 r l ,r'hA n ,...,A l ,A,A',c, An application of (E4) to the above gives d' d1 T'\-A';C, Tx h A i ; Al Aj;...; A? ; V \- A, A ' , C , . ^;...;i4?B;r1,r'l-A1,A,A',C,

Tn\-An;Al

r^.^ri.ri-An

A^A.A',^

The double horizontal line is meant to indicate a series of thinnings. 3 These can be assumed to have been ordered in such a way that the image under of the preceding derivation is just the figure on the right in the statement of (IIIAla). (lb) is just a special case of the above. As for (2), it is handled in the same way with (El) being used instead of (E4). It only remains to consider 3

A fuller explanation of this notation can be found in Appendix B.

130

Normalization, Cut-Elimination and the Theory of Proofs

the reductions in (IIIB), but these are routinely handled by their analogues in (F5-16) or in (G). For example, when R is rule (1), (IIIB3a) is treated as follows: Since rule (1) is an introduction, it is sufficient to find d and 6! such that

rii n 2

n2 n3

*4=£jy*. «'Q~i4rL£: BACm

and d>id

'

BACm

So, let Hi = (j)(di) (1 < i < 3) and take d to be d2

^3

r2hA2;^ r3hA3;Cn di r2,r3hA2,A3;gACq r t h Ai; ^ ^ ; T2, T3 h A2, A3, B A Cm r1,r2,r3hA1,A2,A3,5ACm An application of (F14bi) to the above gives d2 T2hA2;Bj

d3

di Ai;r2)r&2;Bk r3hA3;Cn TihAi;^ 4r2,r3hA2,A3,BACm r1,r2,r3hA1,A2,A3,SACm and this latter derivation can serve as d!. The remaining members of (IIIB) are dealt with in a similar manner. This establishes the basis of the induction. Induction Step: n = m + 1. There are only two cases to consider according to whether the last operation in E is substitution or an application of rule (5). Both follow trivially from the induction hypothesis, however. This concludes the proof of Theorem 7.1. d Proof of Theorem 7.2: The proof of Theorem 7.2 is similar, albeit a little more complicated. Given d and d' such that d >i d', we show by induction on the number n of inferences below the cut or thinning to which the reduction is applied that (d) h (d'). Basis Step: n = 0. There are innumerable cases to consider according to the nature of the step by which d' is obtained from d. (The cases below are numbered according to the classification of reduction steps for LK found in Appendix B.) A. It is obvious in this case that (j){d) = <j>{d'). B . The reductions in this group are handled rather straightforwardly by means of their analogues in (I) supplemented, in some cases, by the

Correspondence Results

131

use of conversions from (II) and (IIIA). As an example, I consider (B2a) below. The other cases are similar (and usually simpler): So, let d and d! be the figures on the left and right, respectively, in the statement of (B2a) in Appendix B; furthermore, let V be {A\x,..., Al} and A' be {/?/,,..., B£ }. Then

4>(d) =

4>(di) Bi Ay Bk Ap Bq

and (di) Bs

4>(d')

*L_

=

Bs Bs

B^

.Bh

Ba Bq

By £'

7"

where (for any r)

, is the result of substituting the conclusions Bq of 7-

r for the assumptions Bq of 4>{ds), and ^ is the result of substituting the conclusions Ap of r for the assumptions Ap of ^(cfe) and then substituting the conclusions Bq of the derivation thus obtained for the assumptions Bq of 4>{ds). (The usual provisions about subscripts apply to avoid unwanted clashes. This presents no problem since alike derivations are treated as indistinguishable from one another. In particular, I assume that Bq & T U T'.) Now,

(d) >i

Bl

B1 R

=

A q

y

P

Bq

Ap 4>{d2)

Xi <j>(d').

(The first reduction is justified by (12), the second by (IIIA2).) Notice that, in the case of (B2b), an application of (III) would be

132

Normalization, Cut-Elimination and the Theory of Proofs

required before (IIIA2) is applied, followed by a second application afterwards—the inverse of the first—to transform the resulting derivation into 0(d'). C l - 9 . In all these cases, it follows from the properties of substitution that (f)(d) and {d!) are the same.4 CIO—18. Here there are various possibilities, depending upon the last rule of the left-hand premise of the cut. (1) In case it is a one-premise left-rule, 0(d) and 0(d') are clearly identical. (2) In case it is —•-right, the image under 0 of the step by which d! is obtained from d is just (112). (3) In all the remaining cases except A-right, the image under 0 of the step by which d' is obtained from d is an instance of (III). This is obvious when the rule in question has only one premise or is cut. If it is V-left, take IIi in the statement of (III) to be BAAn &m Ap so that this step takes the following form (assuming Cr is among the conclusions of n 2 ) : BAAn Bm Ap

n; n 2

B\An reduces to

n

3 Similarly, if it —•-left, take fix to be

n; Bm

B *p>

so that (again assuming that Cr is among the conclusions of n 2 ) (III) becomes:

/nj |

£>TH

V

\ t>

n2

•

A

n; Of

n

)

CT

n3

reduces to

n3

Bm

B-*An Ap

n2 Or

n3

(4) It only remains to consider (CIO), the case in which the last rule See equations (5.4) and (5.5) above.

Correspondence Results

133

on the left is A-right. To simplify the writing I will omit subscripts and assume that the cut-formula occurs in both premises of the application of A-right. So, d and d! can be taken to be T\-A;C;A r'\-A';C;B T, Tf h A, A', AAB;C

d3 C; T" h A"

r,r',r"h

A,A',A",AA#

d% C;T"\-A"

d2 r'hA';C;B

and d\ FhA;C;A

r,r"FA,A";A

c/3 C',T" h A"

r',r"l-A',A";S A,A',A",AAB

r,r',r"h

respectively. Writing IIj for 4>{di) (i = 1,2,3), we must show that

n

c n3

t

n;

n2

A

B AAB

where

ru n = A

n2 iij B and Yl'} = C for j = l,2

n3

AAB Now EI can be written as

[njA] ( [ n 2 / B ] ^ _ | - ) hence by (III) (7.1) 11

C y,

n3

[n'JAjt( v

'^W^AA/0.

But, by (5.4), (7.1) can be written as

[ n i M ] [ n 2 / B ] ^ J - / c n3 which by (5.5) is the same as (7.2)

[n2/5][ni,'^-TAH n3

An application of (III) to (7.2) yields

[U'JBm/A}^^-

n3)

134

Normalization,

Cut-Elimination

and the Theory of Proofs

which is just

ni n'2 A. B_ AAB D l - 5 . In these cases, the fact that 4>(d) = (d') follows immediately from the following: Lemma 7.3 / / j occurs nowhere in . then Ai [TKA^/AjllU/A^W = [U/AiU^W This lemma is easily proved by induction on c(Hf); the basis step is trivial, and the induction hypothesis, together with the various clauses in the definition of substitution, takes care of the rest. D6—9. All these cases translate under (j) into the special case of (III) in which II2 is an assumption. They are all similar, so I will consider a single example, (D6): Let IIJ = 4>(di) (z = 1, 2,3), then it suffices to show that

n'

n

A KB,m

>-i

A A Bm

n3 -

and

n" =

ni AABm

AABS

n3

where n'3' = (A A Bsjm;)n'3,

n'

II

n'2

, Ai Bj , AABm

ni

AABt

n2 ,

AABt

AABS But this is just the instance of (III) in which IIi = II", Ii 2 = A A B^ and n 3 = n^. 5 E. The translation of (1) under (f> is an instance of (IIIA2). Similarly, (4) translates into an instance of (IIIAla) (or (IIIAlb) if T is empty). (3) too translates into an instance of (IIIAl), the only complication being that it is necessary to replace the subscript i whenever Ai G A— but this presents no problem. It only remains to consider (2). Let T = {Al,...,Al}, A' = {B}^...,BfJ and assume A% £ A', then we must show that Aj n Ai Bk y Ai {d2) 0(di) 5

The term on the left in each equation refers back to the formulation of (III) in Chapter 6. According to this same key, Ap is replaced by A A Bs and Cr by A A Bm.

Correspondence Results where

A1

135

Aj

2±L-

A? U = B 3\

BV But Ai tfdi)

Bk 4>(d2)

^

yx

Ai

Bkl

Bk

<j>(d2)/Ai 4>{di)

by (III), from which

n Ai {di) is obtained by (IIIA2). In case Ai € A, the index i must first be replaced by one which does not occur on A in A. Again, this presents no problem. F l - 4 . In all these cases, with the exceptions of (F2c) and (F3c), it is an easy matter to verify that 0(d) = <j>(d'). As for (F2c) and (F3c), <j)(d) yi 0(d ; ) by an application of (III). To see this, consider by way of example the case of (F2c) in which R is V-left; the remaining ones are almost identical. We must show that EVFm En F0 EVFm 4>(d) (j>{df) y En F0 Ai n nr Ck where n = Ck

Ai_

and

n ' = Cfc

4>(d') Ai

Aj in other words, that (7.3)

£VFTO En F0

En

4>(d)/F0 4>{d')/Ai

Ck

A% Ai

136

Normalization, Cut-Elimination and the Theory of Proofs reduces to 1

EVFni \En F0

En

U/F0

n'

Ck

At

But, by (III), (7.3) reduces to E V F7 En

En 4>{d)IAi

A,

Fa

n'

and this is the same as (7.4), since by (5.4) E\/F„ En F.

En

m/Ai

Ck

A{

EVFm En F0

En

n

F 5 - 1 6 . Roughly speaking, the reductions in this group all translate into instances of (IIIB). There are, however, some minor exceptions and a few complications. Notice first that (j>(d) = <j)(d') in the case of (F9aii) or (F9bii). The remainder of (F5-10) all translate directly into members of (IIIB). When we consider (Fll-16), on the other hand, the situation is complicated by the fact that the active formula in the premise of the thinning being permuted may also occur in the premise(es) of the preceding inference. (Of course, this may happen in (F5-10) as well, but it causes no difficulty in these cases because they deal with left rules. As a result, given d and d' such that d >i d' by one of these reductions, we can find d" such that d" > i d! by the same reduction, 4>(d) = <j>{d") and the active formula in the premise of the thinning being permuted is introduced by the preceding inference.6) If it does, (III) must first be used to reduce cj){d) to a (d*) where d* is like d except that the final application of thinning has been split into two in such a way that (d*) y± (df) by an application of the appropriate member of (IIIB). This procedure takes care of all cases except for (F15aii) and (F15bii). These last two translate straightforwardly into instances of (III). (Cf. (F9aii) and (F9bii) above.) It is worthwhile to distinguish (F16) and (F13) from the other cases, however, because they require (III) to be applied even when the active formula of the thinning is introduced by the preceding inference. (In (F16) and (F13), (III) serves not only to split up the thinning, but also to permute one of the inferences which results.) 6

As an example, consider (F5a). Let d\ and d[ denote the derivations on the left and right, respectively, in the statement of this reduction step, then we can take d" to be {A3/i)d

Aa;(Bj);rh A As;(Bfe),(Cn),rhA R Bt;(gfc),(Cn),rhA

Bfc,cn,rhA It follows from the familiar properties of substitution that (p(d^) = (di). Furthermore, it is obvious that (d") y\ (d') by (5.4). F16a. We can take d and d' to be di

r;^hA;(Bro) r,Aj\-A;Bm r,Aj\-&,B *,. ~ ,k~,C*n, ~ „

and

r;^KA;(Bm) T;Ai\-A,(Bk),{Cn) r;At\-A,Bk,(Cn) r,^l-A,Bfc,CB

respectively. Also, let IT' be as above, then (j)(d) can be written as Aj m A{

Djn J

Ai

Bn

Hdi)/Bn

Bk

This reduces by (III) to Bm Bk Cnl

Ai

n'

Cn

138

Normalization, Cut-Elimination and the Theory of Proofs whether or not the conclusions of (di) include Bm. latter derivation reduces immediately to A, At _Ai

Bk

Now, the

I C n / A{ I

ir

by an application of (IIIB5a) from right to left, and this last is just cj){d') (again, by (5.4)). G. These reductions all translate into instances of members of (IIIB). More specifically, the image under 0 of (la) and (2d) is (IIIB5b) (in either direction); (lb) and (2b) translate into instances of (IIIB6a) from left to right, while (lc) and (2c) translate into the same reduction, but from right to left. Finally, (Id) and (2a) translate into instances of (IIIB3b) (in either direction). This completes the basis step of the induction.

Induction Step: n = m + 1. Suppose that the result holds when the reduction is applied to any inference with no more than m steps below it. Let d >i d', and assume that the number of steps below the inference I being reduced is m + 1. We must show that 0(d) y 0(d'). Let d\ be the immediate subderivation of d which contains J , and d[ be the result of applying the appropriate reduction to J , then (j)(d\) y (d[) by induction hypothesis. We distinguish two cases: (1) di is the derivation of the right hand premise of an application of —•left or cut, either premise of an application of A-right, or the premise of an application of right thinning (a), left thinning (b), —i-left, V-, —»-, V- or 3-right. In all these cases, 0(d) and 0(d') can be written as [(f>(di)/ Ai]U and [(^(d^/AilU, respectively (or as

0(di) Aj B->Ak

0(-right), for some Ai and II. Hence, it follows immediately from the induction hypothesis that 4>(d) y (d'). (2) In all other cases c/)(d) and (f)(df) can be written as [U/Ai](f)(di) and [II/i4i]^(d/1), respectively, for some Ai and II (or, if d\ is the derivation of the left-hand premise of an application of V-left, as [[Il/Aityid^/Bj]!!' and [[U/Ai]^^)/ Bj] II', respectively). Here, 0(d) y {df) follows immediately from the induction hypothesis and the following:

Correspondence Results

139

Lemma 7.4 If Hi >-\ H2, n is any member of LKr)T and Ai is any conclusion ofH, then [U/Ai]Ui >-i [ n / ^ ] r i 2 (by an application of the same reduction step). Proof of Lemma 7.4

ft ft Since II1 >-i II2, III can be written as * and II2 as 2 , for some E, where "III reduces to n' 2 " is an instance of one of the steps listed above. The lemma is proved by induction on /(E), the length of E. Basis step: /(E) = 0 In this case it is obvious that "[n/i4»]IIi reduces to [ n / ^ J I V is an instance of the same reduction step. (Strictly speaking, this has to be verified in the case of each reduction step with the aid of the definition of substitution, but it is a trivial matter to do so.) Induction Step: /(E) = n + 1 Suppose that the lemma holds for all E" such that /(E") = n, and let E' denote the first n terms of E. There are two cases to consider: i. The last term of E is an application of rule (5). Now, §

yx "'?, hence [U/A^]

^

[U/A^

by induction hypothesis. But it follows from the definition of substitution that, if II has no open assumptions of the form Cn or conclusions of the form Dm, for any II':

[n/Ai]W Dm C-+Dp

"' D \C^Dpm

VIM ( {n}

w,

So, in particular,

Dm C^DP

? 0-

[n/At] {n}

ii. The last term of E is an application of substitution—for the assumptions Bk of some II', say. Again, it follows from the induction hypothesis that [U/A^]

M

[Il/A^

Furthermore, by (5.4), if Ai is not among the assumptions of II' nor Bk among the conclusions of II,

[ n / ^ y ^ j i r = [n/^iQ^/sfcjn')

140

Normalization, Cut-Elimination and the Theory of Proofs IT (j = 1,2). But the term on the right is just [ I I / ^ ] ^ , hence [U/A^

yt

[U/A^

This completes the proof of the lemma and of Theorem 7.2.7

•

Whether all the reduction steps listed in the preceding chapter are considered, only that subset of them which corresponds to reduction in LK, or the steps for LK listed in Appendix B, it should be apparent that normal forms are not unique nor does every reduction sequence terminate. The latter feature can be attributed to two factors. The first is that some reduction steps are symmetrical. These comprise various instances of (III) as well as the thinning permutations in (IIIB); in the case of LK, they are basically those permutations which involve only cuts and thinnings, and do not result in the splitting up of an inference. The reduction sequences generated by steps of this kind can contain only a finite number of distinct terms—although some of them may be repeated infinitely many times. For this reason, they seem not to pose a serious problem. The same cannot be said of the second factor, which is that certain reductions, when applied to a derivation, may yield a more complicated one. Here again I am thinking of (III) or, in the case of LK, those steps which allow a cut to be split up, namely, (Cl-4), (C10-13), (Dl-5) and (D6-9). 8 It was this phenomenon which was exploited by Zucker to produce an infinite non-repeating reduc7

Using this lemma, it is easy to show (again by induction on / ( £ ) ) that, if n = 5A, II' = 5 ? and "III reduces to n 2 "

is an instance of one of the reduction steps, then n >-i II 7 —thus substantiating the claim made earlier that nothing is lost by formulating the reduction steps in terms of initial subderivations. 8 A11 I intend here is to draw attention to the fact that some reduction steps enable us to generate infinite non-repeating sequences. As a matter of fact, I am hard put to explain what it means for one derivation to be more complicated than another in the present context. A necessary condition seems to be that the former should contain more inferences or vertices than the latter, but this is not sufficient. For example, although the application of (I3a) may increase the size of a derivation, it seems inappropriate to assert that it also increases the complexity—because the strong normalization theorem holds (for NJ), if for no other reason. Yet it is notoriously difficult to specify what kind of simplification is accomplished by this step. (A way to do so, and hence to define a measure of complexity which decreased with each application, would yield a simple and direct proof of strong normalization.) In the case of LK, a thinning permutation such as ( F i d ) provides an example of a reduction which, although it may increase the number of inferences, seems to simplify rather than complicate a derivation. The situation here is further obscured by the fact that applications of at least two different reduction steps are needed to generate sequences of the kind described above. It is perhaps unreasonable, therefore, to claim that any particular step is by itself responsible for an increase in complexity.

Correspondence Results

141

tion sequence for LJ. His example can easily be adapted to the case of LKDT, as the following shows. To simplify the notation, I shall not write in subscripts; clearly, nothing is lost by this omission. Recall that by (5.5)

ni n 2 A B

n

may be used to denote ambiguously

n2

ri!

[Ui/A] B and [U2/B] A

n

n

(provided, of course, that II2 has no assumptions of the form A and II1 has none of the form B). Now, suppose that ITi and II2 are of the forms

ir

n"

CD

A

ni n2 A

diiu

F

ni' n2' B

A

respectively. Then,

B

n'

nx n2 A

E

C

B

n

D

A

n2 n2 B\

=

n; rr2

^1

A

B2

rr

n+ n+ (by (III) and (5.5)) where

n'

n2

C

n+ = B n

and

IT

=

D

ni ir2 A

Bx

B2

A

n

n

As is apparent,

rii n2 A

B

n2 n2 and

n

Bx

n*

B2

have the same form and II is a proper subderivation of II*. (III) can therefore be applied once more to obtain a figure of the form

n" n2

n2

#2i

B22

E where

11^ =

ni' B\

n2' B2i

n*

B\

B22

n*

142

Normalization, Cut-Elimination and the Theory of Proofs Clearly 11* properly includes II*. Furthermore, (III) can now be applied

to

n2 n 2 #2i

&22

n* and so on ad infinitum. With each application, the derivation which results is of increased size. (The use of the subscripts 1, 2, 2i and 2 2 on B is, strictly speaking, an abuse of notation. They are only intended as an informal device to make matters clearer by keeping track of various occurrences of B between figures.) As I remarked in Chapter 2 above, this sort of example cannot be carried out in the sequent calculus except under special circumstances. In particular, it does not apply to the version of LK presented here. The difference between LKDT and LK in this respect is accounted for by the fact that, if d is obtained from d' by a reduction step in (Cl-9) or (Dl-5) (i.e., by permuting a cut with the last inference in the derivation of its righthand premise, or by splitting the cut-formula in the right-hand premise of a cut), (j>(d) = <j>(df). As a result, the translation of a reduction sequence from LKDT back into LK may involve applying these steps from right to left, as well as from left to right. This is the case in the above example. It is easy to check that an infinite non-repeating reduction sequence analogous to the one described above can be generated in LK if the reduction procedure is augmented in this way. The conclusion to be drawn from all this, however, is not that there can be no infinite non-repeating reduction sequences in LK. I present an example of one below. Notice that it depends essentially upon allowing more than one formula to appear on the right-hand side of a sequent. In this respect, it differs significantly from the preceding example, which applies to LJDT no less than to LKDT. Again, I shall omit subscripts and all parts of the derivation (e.g., side-formulae) which do not affect the situation. Reading from top to bottom, each one of the figures below reduces to its successor by an application of (C4) or (C13). (7.5)

CY- B

A h C\ B

B hE

A\-B

B-E\~D BY- D

Ah D (7.6) B\-E

AV-C\B

B;E\-D B\- D A\-C;D

C\- B C\~ D AhD

BhE B;EhD BY- D

Correspondence Results

143

(7.7) AhC;B B\- E AhC;E

A\-C;B B;EVD A;E\-C;D A\-C,D Ah D

BhE CVB C\- D

B,E\-D BhD

(7.8) A\-C,B BhE A\-E\C

A\-C;B B \-E C h B B \- E C h B B\-E A-S\-C\D CVe C;£\-V A\-C-,D C\-V AV-V Now, it is clear that any reduction step which applies to (7.5) can be applied with the same result to the part of (7.8) which is written in calligraphic characters. So, this sequence of steps can be repeated, beginning this time with (7.8), to obtain a larger derivation, and so on ad infinitum. Other similar examples of infinite non-repeating sequences can be constructed, but the above is as simple as any. I turn now to the issue of uniqueness of normal forms. The above considerations are already sufficient to rule out the possibility of each derivation having a unique normal or cut-free form. (For the purposes of the present discussion, I will regard as normal any derivation of LKDT which has the subformula property, and lacks maximal segments and formula occurrences.) In the first place, some of the symmetrical reduction steps, most notably the thinning permutations, apply to derivations which may already be normal or cut-free. By itself, this is perhaps not so disturbing since it implies only that a derivation may reduce to a finite number of normal ones and that these are all reducible to one another by means of such permutations. In fact, however, a derivation may have infinitely many distinct normal forms. This is obvious in the case of LKr>T, since each derivation in the above example of an infinite reduction sequence may be normal, and it is also true for LK. (To see this, suppose that AY- C\B, C h B, B h E and B\ E h D all have cut-free derivations and that C, B and E are atomic. Then, it is an easy matter to specify a reduction procedure, namely always eliminate the left-most cut with no cuts above it, which will yield a distinct cut-free form for each term of (7.5)-(7.8).) In addition to the problems caused by the failure of strong normalization, there is a further difficulty which stands in the way of uniqueness. In LK it has to do with the manner in which cuts are to be eliminated, and in LKDT it concerns the pruning of redundant parts of a derivation {i.e., those attached by thinning). Notice first that the proper reduction steps for LKDT a r e conservative: each application of any one of them removes exactly two inferences from a derivation (an introduction together with the elimination following it). Those which remove maximal formula occur-

144

Normalization, Cut-Elimination and the Theory of Proofs

rences whose principal connective is A, V or —• differ from the conventional reductions in this respect. (This holds for —• only when the occurrences are introduced by an application of rule (5) which discharges no assumptions.) The usual procedure is to remove some or all of the subderivation which culminates in [or derives from] the redundant premise [or conclusion] of the inferences eliminated by the reduction—whenever such redundancies occur. How much of this subderivation is to be removed depends upon considerations which vary from calculus to calculus. Now, as an illustration of the difficulty which presents itself, consider a derivation of the form (7.9)

n' A CV A C A

n n" A B AAB A

where II, n ' and n " are assumed to be normal, Applying (II) and (12) in any order yields (7.10) ir

C

A

n n" A

B A The problem is that, although this latter derivation contains no maximal formula occurrences, it may not be normal, i.e., it may lack the subformula property. A solution is provided by the pruning reductions in (IIIA). Unfortunately, however, when used without restriction, they lead to distinct normal forms. In particular, they can be used to convert (7.10) into . or IT into .. Before discussing the possibility of restricting them in some way, it is worthwhile to consider how this matter is handled by the reduction procedures for NJ and LK. In NJ (7.9) cannot be represented by a derivation whose last inference is A-elimination, rather it corresponds to one of the form [C]

rr

A CvA

n n" A B AAB A

[A]

Correspondence Results and this reduces to

145

. no matter which maximal formula occurrence is

operated upon first. In LK, (7.9) corresponds to a number of different sequent derivations, among them the following: (7.11)

d \-A

d' \- A \-CVA

d" C\- B A\- A C\JA\-B;A

\-B;A \- AAB;A

A\- A AAB\-A \-A

and (7.12)

df \- A \-CVA

d d" \- A CY- B A\- A C\-AAB AAB\-A C\-A CV AY- A h A

A\- A

(d, d' and d" are supposed to be such that (j)(d) = II, (j){d') = 11' and cj){d") = II". Also, I have not bothered to write in any side-formulae.) No matter how the cuts are eliminated from (7.11) and (7.12), the result is that the former reduces to d and the latter to d!. This would be perfectly satisfactory were it not for the fact that the following derivation can easily be seen to reduce to both (7.11) and (7.12): d' \- A \-CvA

d" C\-B AY- A CVA\-B;A h B\A

d \- A BY- B A\- A B \-A A B A A B Y-A BY- A \- A

The problem is that when a cut is eliminated or reduced in complexity, its location in the derivation affects the result of reducing it. Unfortunately, however, this location is not in general uniquely determined—except by adding ad hoc restrictions. (Another illustration of the difficulty this causes is provided by the procedure for eliminating a cut, one of whose premises is introduced by thinning.) The uniqueness of normal forms in natural deduction can be explained, as far as the negative fragment is concerned, by two features of this calculus. The first is that there is a natural ordering of the inferences which constitute an N derivation, and this ordering is not affected by any of the reduction steps. The second is the fact that any branches made redundant by the application of a reduction are composed entirely of inferences

146

Normalization, Cut-Elimination and the Theory of Proofs

subordinate (in the sense of this ordering) to the ones being removed. In view of this, they may be pruned in their entirety without spoiling uniqueness. When we turn to the full calculus, it is no longer obvious that the ordering of inferences is entirely natural. In addition, it becomes necessary to allow reductions which alter this ordering. To preserve uniqueness, ad hoc restrictions must be placed on these. By itself, however, this is not sufficient because the branches made redundant in the process of normalization may contain inferences which are not subordinate. So, to ensure that the calculus possesses the second feature mentioned above, the meaning of "redundant" is altered by treating inessential applications of V- and 3-elimination as though they were essential.9 In NJpT and NKQT, this last translates into some ad hoc conventions concerning when the rule (LT) can be applied and how much of a redundant subderivation is to be pruned. In light of the preceding considerations, we can understand better why normal forms are not unique in LK and LKDT. TO begin with, there is in general no satisfactory way to order the inferences of a multiple-conclusion derivation. This can be seen from the example of (7.9), which can be interpreted as having been constructed either from II and IT A CV A C A

n" by applications of A-introduction and elimination, or from C

IT A CV A

n n"

B_ AAB A and A by V-elimination. Although there are a number of other similar examples, I will present only one more: 9

A

T h e branches of a derivation made redundant by a reduction are those which, if they are to be retained after the removal of some maximal formula occurrences, must be reattached by means of thinning. I argued earlier that applications of V-elimination which do not discharge assumptions in both minor premises and applications of 3-elimination which discharge no assumptions both involve the tacit use of thinning. If this interpretation is kept in mind and reduction in the N calculi is taken to consist of removing maximal formula occurrences together with the branches made redundant by their removal, then it can fairly be claimed that what it means for a branch to be made redundant in the negative fragment is not the same as in the full calculus.

Correspondence Results

147

Given ~ ^, let

rii n

n n4 n2 = E

F G

and

n 3 = j4 B_ C

Then iii _^4

n2 n3 n4 B_ = ]Z F_ C G

(provided that E is not among the conclusions of Iii, nor B among those of n 4 ). Of course, the inferences of an LK derivation can be ordered. The problem is that the ordering is rather artificial and can be changed radically— most notably by the upward permutation of cuts—in the course of cutelimination. In LKDT too, even when the order of inferences is determined by the structure of the derivation, it can often be reversed by applying (III). As a result, any reduction procedure (for LK or LKoT) which allows entire branches to be pruned as they become redundant will not yield unique cut-free or normal forms. It is however hard to envision natural reduction steps which prune enough to ensure the subformula property, but not so much as to destroy uniqueness. Certainly those reduction steps for LK which involve pruning, namely (Bl-3), do not fit this description. They correspond in LKpT to (11-3), respectively, followed by an application of (IIIAl) (in the cases of (II) and (I3b)) or (IIIA2) (in the case of (12)). In effect, they allow a segment of variable length to be removed from each redundant branch—the only constraint being that no open assumptions or conclusions are to be lost. It is obvious that this is too drastic for uniqueness. Furthermore, because the extent to which a branch is pruned depends upon the position of the inferences used to construct it relative to the cut whose complexity has been reduced (those above it being removed, while all others are retained), relatively insignificant permutations of inferences will alter significantly the effect of these reductions, and it is by no means clear that they have any claim to be called natural. If we contemplate replacing (IIIA1-2) by pruning reductions which are more systematic and will preserve uniqueness, there seems to be only one reasonable possibility: restrict (IIIAla) to the case in which II consists of a single introduction and, dually, (IIIB2) to the case in which II' consists of a single elimination. The drawback to this idea is that, if the connective being introduced is —•, II will not fit into the format of (IIIAla). To take account of this, (IIIAlb) must be replaced by (IIIA5). Now, if the somewhat trivial step (IIIA6). is also included, we have a group of pruning reductions which will ensure that normal derivations possess the subfor-

148

Normalization, Cut-Elimination and the Theory of Proofs

mula property. (In fact, _L may occur in a normal derivation even though it is not a subformula of any assumption or conclusion. For all practical purposes, however, the claim is true.) Furthermore, because no maximal formula occurrences are removed by these reductions, they do not by themselves threaten uniqueness and leave open the possibility of proving that the normal forms of a derivation are all equivalent in some suitable sense of the word. What vitiates this approach, of course, is the fact that (IIIA5) may introduce new open assumptions into the derivation on which it operates, and hence is unsuited to be a reduction step. The disappointing conclusion, therefore, appears to be that the only natural way to prune a normal form of an LKDT derivation II in such a way that it will have the subformula property and bear some structural resemblance to the other normal forms of II leads to an insuperable difficulty. The preceding remarks notwithstanding, by choosing an appropriate notion of normal form and placing sufficient restrictions on the reduction steps, it is clearly possible to prove for any of the calculi under consideration not only the uniqueness (up to some equivalence) of normal forms but also the termination of each reduction sequence in a normal derivation with the subformula property. The problem is that this will be an ad hoc procedure designed expressly for the purpose of obtaining these results. On the other hand, unless we are prepared to interpret 'natural' as 'natural relative to the rules of a particular system', it is not clear that there is any such thing as a natural reduction procedure. Despite a fundamental similarity between reduction in all the systems discussed above, we are faced with a bewildering number of choices about matters of detail which are decided for each particular calculus in what appears to be a reasonable way only by respecting its combinatorial peculiarities. As a result, these decisions often seem pointless and arbitrary when translated from one calculus into another. Furthermore, it is upon these apparently trivial decisions that the possibility of proving strong normalization and Church-Rosser type theorems depends. When we do come across a calculus like NJ for which such theorems hold with respect to a relatively straightforward set of reduction steps, it seems to be more a matter of combinatorial accident than a reflection of some profound truth about normalization. For this reason, it seems unwise to use (as Zucker appears to do) the normalization procedure for NJ as a kind of benchmark by which to judge other reduction procedures. Once we go beyond the negative fragment, no method of reduction stands out as privileged; they all appear to be more or less satisfactory compromises between competing requirements. An investigation of their formal properties does not provide sufficient grounds for choosing between them or assessing their wider significance. So, rather than pursuing such an investigation further in the hope of discovering some clue as to how the relationship between a derivation and its normal form(s) is to be interpreted, it might prove more fruitful to

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consider directly various interpretations which have been suggested for the derivations of a formal system with a view to drawing up a set of criteria, independent of the rules of any particular formalism, by which to judge reduction procedures and proposals regarding their significance. Before turning to this task in the next chapter, I would like to conclude the present one with a brief discussion of a topic which, although peripheral to my main theme, is nevertheless of some interest—namely, the advantages of presenting classical logic in a multiple-conclusion framework. As I observed earlier, the rules of DT are all classically valid so that, for example, the sequent T h A is derivable in LK iff there is a derivation II G DT of A from T. Some of them, however, are not intuitionistically valid. In particular, both (5) and (10 v ) need to be restricted to obtain intuitionistic logic. It is rather a complicated matter to formulate such restrictions if the rules are to be applied downwards in a straightforward manner, although there are quite naturally generated subsets of DT which are adequate for intuitionistic logic—the most conspicuous example of one being LJDT> Nevertheless, it seems fair to say that these multiple-conclusion rules express more naturally the classical interpretation of the logical connectives than its intuitionistic counterpart. The situation is reversed in the single-conclusion case. It has often been remarked, for example, that NJ is extended to classical logic at the cost of a certain artificiality, or that NK is perhaps not "the proper way of analyzing classical inferences."10 Furthermore, NKDT can only be described as an arbitrary subset of DT- The claim I wish to defend here is that D? (or some variant of it) is the proper generalization of NJ to classical logic and that it is superior to NK as a natural system of classical deduction. In the first place, the rules of classical multiple-conclusion logic can be formulated in a completely explicit way without restrictions. (This was observed by Kneale and seems to have motivated, in part at least, his interest in multiple-conclusion derivations.) The only rule which does not conform to this description is (5) and, following Kneale, it could be replaced by:

< 5 '*

A-^TB

«"«

A

(All occurrences of the assumption A are discharged by the final application of

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As pointed out earlier, we cannot afford to be too dogmatic about what constitutes the correct definition of interreducibility, but so unacceptable a consequence makes it tempting to reject Martin-Lof's candidate out of hand; whatever applications his notion of convertibility may possess, it is surely not the relation we are interested in analyzing. At the very least, we should be cautious about accepting it and subject the arguments offered in its favor to careful scrutiny. There are only two such arguments, and one of them is not really relevant to the purpose at hand. The first is that, by weakening the definition of convertibility in the manner suggested above, certain technical advantages are obtained. 34 The weaker relation is certainly more manageable (cf. the case of weak vs. strong reduction in combinatory logic), although it is not without its drawbacks too. Even if we grant the point, however, it establishes only that the relation defined by Martin-Lof can be useful—and this is not in dispute. As he himself writes (on page 96): "we are free to define many different relations between terms and call them convertibility relations." He then adds: "but my claim is that only one of these correctly formalizes the [intended interpretation of convertibility]." I am interested in his contention that he has supplied the correct formalization. This brings me to the second argument: (a) The intended interpretation of convertibility is a relation between linguistic expressions called definitional equality. (b) Definitional equality is the least equivalence containing various defining equations and closed under (f) above. (c) It follows that the correct definition of convertibility simply formalizes the properties mentioned in (6) (and that is exactly what Martin-Lof's does). Once (a) and (b) are accepted, there is obviously no denying (c). It would appear therefore that, no matter which interpretation of the conjecture is favored, MartinLof's remarks about it depend ultimately on his conception of definitional equality. As was hinted earlier, however, he seems to think that this relation is a familiar one whose role needs no explanation and whose properties, once stated, are easily recognizable as such. Consequently, he does not embark on a systematic defense of (a) or (b). In introduction.) This example depends upon the fact that its conclusion is introduced by an application of —•-introduction. It is the only rule which will produce such a result in DT although similar examples can be constructed in NJ and NK using V- or 3-elimination. (A notion of normal form which is sensitive to differences of this kind is perhaps not entirely satisfactory.) Notice, however, that the subformula property will hold for certain subclasses of derivations—for example, derivations of atomic formulae in the negative fragment—so that many applications of normalization arguments will not be much affected by this weakening of the reduction relation. 34 Some of these are listed in Section 2.1 of Martin-Lof's paper, op. cit

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view of the importance of definitional equality to the discussion, it is worth considering briefly whether he is justified in treating it in this way. Martin-Lof claims that by definitional equality he means "the relation which is used on almost every page of an informal mathematical text." He also claims that it is to be found in the writings of such authors as Frege and Godel. None of these claims will bear much scrutiny, however. It seems obvious that definitions are used informally in more than one way, and it may plausibly be argued that in one of these usages they express a relationship between signs. Nonetheless, whatever relation R holds between definiens and definiendum, R is not an equivalence. (It is probably not reflexive, symmetric or transitive, let alone all three.) In the second place, it is not closed under (f) above. (This is not to deny that the inference c(a) = c(b) is often drawn from a =df. b, but this is simply to justify the conclusion that c(a) and c(b) are the same by reference to the definition of a. It certainly does not suggest that there is some special relationship between "c{af and "c(6).") The evidence for these assertions is contained in any "informal mathematical text," where the reader will be hard put to find statements like "b =df. a because a =df. b" or "since a =df. b, it follows that c(a) =df, c(6)." This sort of nitpicking about usage does not really get us very far. It certainly provides no argument against the possibility, or even the desirability, of introducing a relation like definitional equality. On the other hand, it is surely sufficient to establish that, far from playing the central role attributed to it by Martin-Lof, this relation is not to be found in mathematics as it is currently written. Turning now to Frege, we find according to Martin-Lof that definitional equality may be identified with the relation of equality of content found in the Begriffschrift (and symbolized there by "=") "provided one disregards the geometrical example" with the aid of which it is introduced (in Section 8). He goes on to say that Frege's axiomatization of = (in Sections 20 and 21) is not "compatible with the analysis of the relation given earlier" (presumably, earlier in the Begriffschrift). The reason is that Frege gives the familiar identity axioms for =, and Martin-Lof contends that, because "a" and "6" stand for themselves in some occurrences and for their contents in others, statements like a = b —• (A(a) —* A(b)) are meaningless. He concludes that this led Frege to replace = with the more familiar equality relation analyzed in "Uber Sinn und Bedeutung" and the Grundgesetze. The implication is that Frege gave the wrong axioms for the notion he was trying to capture, and that the consequences of this mistake led him to abandon it altogether. Unfortunately, none of this is in the least plausible. It is obvious from the outset that Frege's identity of content resembles Martin-Lof's definitional equality in only one respect: it too is a relation between expressions. As the example we are asked to disregard makes clear, it is the relation

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which holds between two terms when they denote the same thing. The axioms presented in Sections 20 and 21 are, of course, valid under this interpretation. Furthermore, Frege seems not to have been unduly disturbed by the need to use names in a systematically ambiguous way. (Granted that this usage offends the ears of formal language speakers nowadays, it surely does not reduce the axiom quoted in the preceding paragraph to meaninglessness.) It is unlikely, therefore, that this impelled him to revise his treatment of identity. A more plausible suggestion is that, finding himself unable to distinguish between more than two judgable contents, he abandoned the notion of content altogether in favor of the doctrine of sense and reference. He then found it convenient to reformulate his treatment in terms of this distinction. The point to stress is that no great discontinuity exists between the earlier and the later accounts of identity. In fact, identity judgements are analyzed in remarkably similar terms in Begriffschrift and "Uber Sinn und Bedeutung." 35 The geometrical example of a single point determined in two distinct ways which appears in both works, albeit in slightly different guises, makes these similarities particularly evident. Frege takes pains to stress the connection between names and ways of determining. In Begriffschrift he writes that the different "ways of determination" correspond to the "different names of the thing thus determined," while in "Uber Sinn und Bedeutung" he speaks of "different designations of the same point" and states that these names "indicate the mode of presentation." 36 It is this connection which saves judgements of identity from triviality. The difference between the two works is that in the later one he grants these modes of presentation an existence apart from the names to which they correspond. Of course, he also reformulates his account of identity statements so that the relation asserted to hold is between the individuals named rather than the names themselves, but this seems to be of less interest. I am not denying the obvious formal differences between these two sorts of relation, but from most perspectives—including Frege's, I suspect—it matters little whether "a = 6" is taken to mean that "a" and "6" name the same individual, or that the individual named by "a" is the same one named by "6." Frege nowhere suggests that its way of determining is part of the content of a name. On the contrary, his analysis precludes such a possibility since it is only when the same content is determined in different ways that a non-trivial identity judgement can be made. For singular terms, at least, content can be identified with what Frege later called reference. This much seems uncontroversial and is sufficient to refute Martin-Lof's interpretation 35 For this observation, and for much else in this paragraph, I am indebted to the interesting discussion of Frege's views in Chapter 4 of An Essay on Facts by K.R. Olson (Stanford, 1987). 36 T h e phrases in quotation marks are those used by Geach and Black in their Translations from the Philosophical Writings of Gottlob Frege (3rd edition, Oxford, 1980).

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of identity of content, even if one prefers his account of why Frege eventually revised the Begriffschrift theory. When Frege turns to definitions in Section 24 of Begriffschrift, he distinguishes them from identity judgements in the previous sense not because they are concerned with a different kind of relation—they are not—but because, being prescriptive, they are not to be regarded as judgements at all. Baker and Hacker comment on this section: "If a symbol is introduced by a formal definition, the fact that it designates an entity in a particular way . . . seems to be an altogether objective feature of it, and hence there seems pressure towards adopting the principle that in this special case the way of regarding (or the mode of determining) an entity is part of its content. Frege did not draw this conclusion."37 So, there is no comfort for MartinLof here, either. Somewhat ironically, his ideas are more easily reconciled with Frege's later views. It is only after the mode of determining has been separated from the name that it makes sense to ask whether two names determine an object in the same way, and I take this to be the question underlying his conception of definitional equality. In the final analysis, however, nothing approximating to this relation is to be found in Frege's writings. It only remains to consider what Godel has to say about definitional equality and, in particular, whether it plays any role in his Dialectica paper cited earlier. The paper describes a translation of intuitionistic arithmetic into a system T of computable functional of finite type. Roughly speaking, T comprises: (i) certain equational axioms for defining these functions, (ii) the principle of proof by induction (with respect to a numerical variable), (Hi) the usual axioms for identity, and (iv) the propositional calculus—including axioms of the form (s = £) V (s ^ t) for all terms s and t. Martin-Lof contends that 'equality' in (i) is not the relation which is described in (Hi) or appears in an equation obtained with the aid of (ii). His reason is that he takes the former to be definitional equality, whereas "we cannot convince ourselves [of the validity of (ii) or (Hi)] unless, when reading the formulae, we associate with the terms not themselves but the abstract objects which they denote." (page 106). In fact, he upbraids Godel for remarking in a footnote to the identity axioms that this relation is to be understood as "intensional definitional equality." 38 Rather sur37

Frege: Logical Excavations by G.P. Baker and P. Hacker, New York, 1984, page 160. T h e reason for the remark is that from a constructive viewpoint (extensional) equality between functions of higher type is not a decidable relation. The axioms mentioned in iv) are essential for the translation, however, so " = " must be given some other interpretation if T is to be constructively acceptable. 38

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prisingly, this remark probably gave rise to the idea of definitional equality as a relation between terms which is determined by their conversion rules. Tait, while discussing the interpretation of equality in (certain extensions of) T, observes that according to Godel's own interpretation: "s = t means that s and t denote definitionally equal reckonable terms." (The term he translates as "reckonable" is rendered by "computable" above.) He goes on to say: "Lacking a general conception of the kinds of definitions by which an operation may be introduced, the notion of definitional equality is not very clear to me. But if . . . we can regard the operations . . . as being introduced by conversion rules . . . then definitional equality has a clear meaning: s and t are definitionally equal if they reduce to a common term by means of a sequence of applications of the conversion rules." 39 This is essentially Martin-L6f 's view, except that he is less diffident about the general notion of definitional equality.40 There is a difference, however. For the reasons alluded to earlier, Martin-Lof regards definitional equality, not as a possible interpretation for identity in T (or any other theory), but as a relation satisfying different laws. Despite what Tait writes, it is doubtful whether Godel really intended s = t to express a relationship between terms. If he had, it is difficult to understand why he regarded T as constituting an extension of finitary mathematics. According to his own account: Bernays' observations . . . teach us to distinguish two component parts in the concept of finitary mathematics, namely: first, the constructivistic element, which consists in admitting reference to mathematical objects or facts only in the sense that they can be exhibited or obtained by construction or proof; second, the specifically finitistic element, which requires in addition that the objects and facts considered should be given in concrete mathematical intuition. This, as far as the objects are concerned, means that they must be finite space-time configurations of elements whose nature is irrelevant except for equality or difference. . . . 39

Tait, op. cit, page 198. He is no doubt encouraged in this by Kreisel who makes light of Tait's misgivings in a review of the latter's paper (in Zentralblatt fur Mathematik, Vol. 174, 1969, pages 1213). Kreisel attributes them to the view—mistaken, in his opinion—that to make sense of definitional equality for constructive operations it is necessary to have a listing of their possible arguments. (He makes the same point in "A Survey of Proof Theory II," page 156: "Tait expressed doubts in [1967] about the sense of definitional equality t = t' unless all possible arguments o f t and t' are listed.") This criticism seems misplaced, however. The passage quoted above points out only that the notion of definitional equality depends upon what constitutes a definition, and that we lack a general answer to this question. It is hard to find fault with this observation. (Presumably, even definitional equality between number-theoretic functions is not very clear to Tait, although there can be no doubt about their possible arguments.) 40

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Normalization, Cut-Elimination and the Theory of Proofs It is the second requirement which must be dropped [in the face of negative results about the provability of consistency].41

He goes on to say that his theory T is one result of doing so. Now, the terms of T conform to the finitistic description of objects; the conversion rules which are supposed to settle questions of their identity and difference are finite combinatorial operations, and the theorems of T are just (propositional combinations of) equations of the form s = t. The intended interpretation of T, i.e., what its theorems are supposed to be about, must therefore lie outside this domain. In fact, it is clearly stated that T is a theory of certain abstract objects, the computable functionals of finite type; this granted, equations in T must surely express a relation between these abstract objects. It should be apparent by now that there is little in Godel's paper to support Martin-Lof's analysis of definitional equality. In fact, I have tried to argue that there is little reason to accept it at all. Few arguments, and no compelling ones, have been advanced in favor of the thesis that the intended interpretation of convertibility is a relation between linguistic expressions, and the same can be said of his claim that, because its intended interpretation does not satisfy the laws of identity, the definition of convertibility must be weakened in the manner indicated above. Furthermore, even if one accepts definitional equality (in the sense of Martin-Lof) as the correct interpretation of convertibility, it seems to me that this is quite simply a case of explaining obscururn per obscurius: Martin-Lof succeeds neither in establishing that the former is a familiar relation, nor in explaining its significance. These strictures notwithstanding, Martin-Lof's discussion of identity between proofs does focus on the central questions: where proofs are to be located in the scheme of things, and what sort of equivalence relations might hold between them. That these issues should be contentious at all is due, in part at least, to the view that proofs are intensional and that a relation other than the familiar one of extensional equality holds between them. The notion of intensionality is a notoriously problematic one. It appears under a variety of different names, and the term "intensional" has been used to express a variety of different distinctions. 42 This is not the place 41 T h e quotation is from a revised English version of Godel's Diaectica paper (see Vol. II of his Collected Works, page 274), but a similar passage occurs in the original (ibid., page 244, or Dialectica, Vol. 12, page 282). It is unclear why Tait interprets s = t to mean that s and t denote definitionally equal terms. Three or four lines earlier he suggests that s and t denote functionals. These statements are hard to reconcile unless terms are supposed to be used ambiguously in the manner recommended by Martin-Lof. There is no mention of this, however, which makes it seem an unlikely possibility. Perhaps it is simply a slip on his part, occasioned by the existence of term models for T or his reservations about the notion of definitional equality. 42 This is well illustrated by the appendix to Fr. Frisch's essay Extension and Com-

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to survey or evaluate these usages, but I do think it worth distinguishing between two of them which, although not independent of each other, can be separated. In so doing, I do not mean to suggest that these two are the only ways to understand the term or, least of all, that they are the only correct ones. My claim is simply that elements of both are present in discussions about proofs and that some advantage is to be gained from distinguishing between them. The first sense is easy to explain: intensional means not satisfying an extensionality principle of the appropriate kind. The second is harder to make precise, so I propose to give only a rough idea of what I have in mind. A distinction can be drawn between some domain of elements and a system which describes or represents it. The domain can be anything from the universe in which we find ourselves to a mathematical structure. As for the representational system, it is usually thought of nowadays in linguistic terms, and I shall follow this practice below, but it need not be. Ideas, for example, could serve just as well, and would have done so in another age. I am less concerned with the exact nature of these two items than with the relationship between them. To explain this, i.e., how language can refer to the elements of the domain or, alternatively, how these can be grasped in linguistic terms, an intermediate realm is sometimes postulated which consists of 'ways of determination' or 'modes of presentation'—to borrow a pair of phrases from Frege. In its second sense the word "intensional" characterizes the denizens of this realm. I prefer this usage and propose to adopt it below, reserving "non-extensional" for the first sense. 43 Kreisel respects none of the distinctions drawn above. His conception of intensionality is patterned after the paradigm of a predicate or formula which represents a property and its extension. Here, formal systems play the role of predicates. The proofs expressed by such a system can be compared to the property, while the conclusions established by these proofs are analogous to its extension. In case the system generates arguments from assumptions, comparison with a functional term is more appropriate: prehension in Logic (New York, 1969) in which he lists one hundred and seventy eight modern (i.e., since 1662) logicians, the terms they use to express the distinction between intension and extension, and what they understand it to be. The remarkable feature of this list is not that there are similarities between its entries, but that scarcely any two of them are exactly alike. 43 I am quite comfortable with the division into objective (i.e., pertaining to the objects of our interest—not, as opposed to subjective), intensional and linguistic spheres. I realize, however, that there are those who deny the independent existence of one or more of them and others who deny their existence outright. Although this presents a potential source of problems, I hope that the discussion which follows will be acceptable even to those who reject my metaphysical prejudices. All it depends upon really is the claim that we can draw some kind of distinction between these three elements of our experience. Whether it is a distinction in re or merely in intellectu need not be decided here.

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its consequence relation is analogous to a function in extension, while its proofs correspond to the procedure by which values are obtained from arguments. For example, Prawitz, who shares this general outlook, comments on Frege's rules for the predicate calculus that they "may be understood as an extensional characterization of logical proofs (i.e., a characterization with respect to the set of theorems) within certain languages . . . but the characterization is only extensional since the formal derivation[s] may use quite different methods of proof and have a structure different from the intuitive proof[s]."44 It is not just the system as a whole which may have an intensional feature however. Individual proofs may also be described as intensional objects. Kreisel, for example, at the beginning of his review of The Collected Papers of Gerhard Gentzen refers to them as such when emphasizing that "the distinction between different formal systems with the same set of theorems, in terms of the proofs expressed by their derivations, is meaningful," and he goes on to note: "I use 'express' for the relation between a formal expression E and the intensional object meant by E, and 'denote' for the case when we suppress the intensional features of the object, for example, in model theory." 45 It seems then that there are intensional objects but these are continuous with extensional ones—the latter being simply the former minus some of their characteristics. (This is a sort of opposite to the view, espoused by Quine amongst others, that a property is a set with something added.) In the present case, presumably, what the object asserts is an extensional feature, how this assertion is established an intensional one. This view may reasonably be described therefore, in the terms used earlier, as one which associates intensions with how objects are presented and, at the same time, insists that objects cannot be separated from their mode of presentation. As for the criterion of what constitutes an intensional feature, this is provided by a principle of extensionality. So, for Kreisel, there is no difference between intensional and non-extensional. What form such a principle should take is assumed to be uncontroversial. This assumption—although almost universal—is, I think, unfortunate. Consider, for example, proofs as conceived above. Once their intensional features are suppressed, we are left with the conclusions they assert (on the basis of their assumptions). The extensionality principle underlying Kreisel's analysis seems, therefore, to be one derived from regarding the consequence relation as the analogue of a function in extension, namely: two proofs are extensionally equal if they have the same assumptions and conclusion. Now, this is certainly a possible criterion, but it is not the only one. In fact, it even conflicts "Ideas and Results in Proof Theory," page 238. Journal of Philosophy, Vol. 68, 1971, page 243.

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with the obvious extensionality principle for proofs, when each one of them is regarded as a function along the lines suggested by the analogy with A-terms.46 Intensionality appears in another guise too, associated now with the way in which an expression determines (rather than with the mode of presentation of an object). Somewhat confusingly, therefore, intensional objects seem to be continuous not only with extensional ones, but also with linguistic representations. For example, after an analysis of Godel's Second Theorem, Kreisel remarks that "not only deductions treated as extensional objects are relevant here . . . but even additional information or 'structure', namely the sequence of operations involved in building up the deductions." 47 It is true that the features of a derivation which, by implication, do not count as extensional are here associated with how it is constructed (i.e., with its mode of presentation) but, thought of in this way, the distinction is purely conventional and arbitrary. I raised this same point earlier in connection with the elimination rules for V and 3, and when comparing L with N rules. The fact is that formal derivations always bear traces of their construction; the extent to which they do so varies from calculus to calculus, and even from rule to rule within a calculus. It only makes sense to distinguish between extensional and intensional features in this context if derivations are thought of not merely as formal objects, but as expressions. Their extensional features are sufficient to determine what they express, their intensional ones indicate how. This does indeed seem to be the point of the remark quoted above, namely, that for the theorem to hold it matters not only what proof a derivation expresses, but also how (as "the proof X," for example, or as "the proof X than which there is no earlier proof of the negation of its conclusion in some fixed listing"). The intensional features described in the preceding paragraph differ from those discussed earlier. Proofs were originally said to be intensional because they mediate between derivations and assertions, now there are intensional elements which mediate between derivations and proofs. Perhaps these too deserve to be called proofs. Prawitz seems to encourage this usage when he implies that derivations which represent the same proof are synonymous. 48 Certainly, it underlies the distinction drawn by Martin-L6f between (linguistic) proofs and abstract ones. His point, apparently, is that a proof can be a certain kind of abstract object or a way of determining (or defining) such an object. He has little to say about what constitutes 46

T h e division of an object's characteristics into extensional and intensional ones is oddly reminiscent of the traditional distinction between primary and secondary qualities. The former inhere in the object itself, the latter have to do with how it appears to us. The problem here is that it is not altogether obvious where the line is to be drawn. 47 "A Survey of Proof Theory II," page 179. 48 "Ideas and Results in Proof Theory," page 237.

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identity between proofs in the first sense. As for the second, identity does not mean determining the same object, but determining it in the same way; hence it is a matter of definitional equality. In the terms I introduced earlier, proofs in the second sense are intensional objects. Unfortunately, Martin-Lof, for reasons which he never makes explicit, treats the intensional as a subdivision of the linguistic. As a result, he refers to them as expressions of an informal language. What emerges from the above is, I think, not only that "intensional" and "extensional" can each be used in different ways, but also that they are so used—sometimes by the same author in a single work. This is relevant to our inquiry because, as I suggested earlier, the language of extensions and intensions provides the framework within which the most basic and general questions about the nature of proofs are to be settled. I do not wish to imply that any of the authors whose views I have been discussing are confused in their use of these terms, but they are perhaps a little confusing. I have tried to indicate above, in a reasonably unambiguous manner, what I understand by "intensional" and "extensional." It remains for me now to classify proofs according to the concepts so introduced. My proposal is that we should not regard them as intensional objects at all, but simply as the denotations of derivations. This does not rule out a study of the relationship between proofs and what they establish, but it does affect how the relation is to be described. It is also not intended as a comment upon the possibility or the interest of a study of how derivations denote proofs. Contrary to what Martin-Lof suggests, however, I believe that, if such a study is to be successfully undertaken, it can only be after we have gained a better understanding of proofs themselves (including their identity criteria). We would be ill-advised, therefore, to concentrate upon it at the outset. As for the question of whether or not proofs are extensional, and even what constitutes an appropriate criterion of extensionality for proofs, these are matters for investigation. Certainly, we cannot simply assume that they are non-extensional.49 49 Those who are made uncomfortable by talk of intensional objects can interpret my proposal simply as a methodological principle (or even, less kindly, as a terminological one). It amounts to no more than the claim that the objects of interest in a particular field of inquiry should be separated from the way in which they are presented. If it should happen that we are interested in a domain O of objects which comprises the members of another domain O' together with their modes of presentation 7£, we should be especially careful not to confuse the members of 71 with the manner in which we represent the members of O. I realize that, even interpreted in this way, the principle is not philosophically neutral. In fact, it contradicts a famous view of the foundations of mathematics. Its rejection, however, seems to lead almost inevitably to obscurity and confusion. For what it is worth, I think that even writers from the intuitionistic standpoint adopt it de facto. Because it violates their principles, however, they do so neither very explicitly nor always consistently.

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There are immediate benefits to be obtained from adopting this proposal. (i) There is a gain in clarity. Viewed in the way suggested above, the formal study of proofs takes place within a well-developed conceptual framework, that of model theory. We are interested in models of certain formal systems whose terms are derivations. Our ideas about their intended interpretations are, perhaps, not as precise as we would like. But, as I argued in the introduction, we may hope to clarify them by investigating what interpretations they will in fact admit. (2) I remarked at the beginning of this chapter on the possibility that the analogy between derivations and A-terms might be incompatible with that between derivations representing the same proof and sentences having the same meaning. It clearly is if meanings are supposed to be intensional. From the perspective of the A-calculus, the appropriate comparison is not with sentences and their meanings, but with terms and their denotations. In other words, the view of the relationship between derivations and proofs taken above is forced on us if we want to take seriously the analogy between derivations and terms. This analogy, however, has been the mainstay of the subject as developed by Prawitz et a/., and there is no reason to suppose that its usefulness—as a source of results, for example—has been exhausted yet. (3) I think it is regrettable that the interest of a general theory of proofs for classical mathematics has seldom been emphasized. The fact that proofs play a special role in mathematics conceived intuitionistically, as part of the subject matter of ordinary mathematical assertions, should not blind us to their importance for mathematics on any conception of the subject. As I remarked earlier, the claim that classical mathematicians are interested only in consequence, and not in the proofs themselves, does not do justice to the facts. Questions concerning the identity of proofs, for example, or their constructive content are no less interesting from a classical, than from any other point of view. On the other hand, intensions have no role in classical mathematics. It seems to me, therefore, that there ought to be a classical theory of proofs, and that the distinction drawn above between a derivation's denotation and its manner of denoting provides a convenient way of differentiating classical from intuitionistic approaches to the subject. (4) The distinction is also useful for resolving, or at least clarifying, the nature of the disagreement between Prawitz and Feferman alluded to earlier.

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On the one hand, the view that TV derivations adequately represent proofs, their similarity to the terms of certain calculi and the interpretation of the convertibility relation in these calculi combine to make it almost impossible to deny that interreducible derivations represent the same proof. On the other, such derivations are clearly not just linguistic variants of one another; there can be no doubt that information is lost in the process of reduction. The only apparent way to reconcile these facts is to classify the information lost as relevant not to the proof itself but to its mode of presentation. In other words, although interreducible derivations describe the same proof, they may do so in different ways (just as "the author of Waverly" and "the author of Waverly and Kenilworth" describe the same individual, although information is lost when the first replaces the second). This is not to deny the possibility of other conceptions of the subject, according to which distinctions between proofs are expressed by differences between such derivations; they may well be necessary for some purposes. Yet, I want to claim more than that a coherent and interesting notion is also arrived at by identifying the denotations of interreducible derivations. It seems to me that the ideas and methods currently employed in the general theory of proofs—the emphasis on strong normalization, the comparison of proofs with functions, etc.,—presuppose a notion of this kind. (This seems to be Prawitz's point too, when he remarks that the real issue is whether a proof can indeed be identified with "the result of applying certain operations to obtain a certain end result.") If it should turn out to be inadequate, new insights and techniques will be needed to study its replacement. (5) Finally, I think the distinction between a derivation's denotation and its way of denoting helps to clarify the status of permutative reductions. The idea that any reduction step preserves the way of denoting is, in my opinion, wholly implausible. Recall that, given any derivation in N or L, it is easy to construct another one with the same normal form which is arbitrarily complex (on any measure of complexity). Although I do not subscribe to the view that meanings are psychological, it does seem to me that psychological criteria can play a role in evaluating a theory of meaning. For example, if two expressions present an object in the same way, anyone familiar with the conventions governing their use should be able to recognize this fact immediately. By the above, however, it is far from obvious in general when two derivations reduce to the same normal form. (In my opinion, this argument counts also against Martin-L6f's conception of definitional equality.) As for permutative reductions in particular, they alter those features of a derivation which, according to Kreisel at least, are paradigms of intensional ones. Once the idea in question is abandoned, it is possible to look at permutative reductions in a new light. Viewed in the abstract, there is little to

Interpretations of Derivations

179

distinguish one permutation of inference from another. As I observed earlier, it is difficult even to imagine what kind of evidence would legitimate some of these whilst ruling out others. The obvious conclusion to be drawn is that, given any property of proofs, it is either preserved by all such permutations or by none. The circumstances described in (4) above, as well as the interpretations of derivations discussed below, would seem to favor the first of these alternatives. Unfortunately however arbitrary permutations can alter radically the structure of a derivation, which implies that much of this structure cannot correspond to features of the proof being described. Its importance, on the other hand, is undeniable and not explicable simply as a matter of syntax. The position being advocated, then, is that permuting any pair of inferences in a derivation leaves its denotation unchanged, but alters the manner in which this denotation is presented. This does justice to the importance of the structure of a derivation, whilst removing the need to perform the seemingly impossible task of judging between permutative reductions. I do not mean to imply that there are no grounds for distinguishing between different sets of permutative reductions. For certain applications it may obviously be necessary to restrict attention to a group of them which is adequate (in the sense that it allows every derivation to be reduced to a normal form), yet has certain desirable properties; it ensures that normal forms are unique, for example, or that every reduction sequence must terminate. But these distinctions are based on considerations of technical expediency. It is not surprising therefore that they should be made differently in different formal contexts, and unlikely that any profound consequences can be drawn from their being so. One of the virtues of the multiple-conclusion approach to these matters is that it reveals very clearly both the arbitrariness of the restrictions placed upon permutative reductions in the conventional reduction procedures (in particular, how they are motivated by syntactic features of the calculus concerned) and the wide variety of choices available when it comes to selecting such restrictions. The preceding provides, I think, a reasonable explanation of this situation.

The views expressed above, especially in (5), have an apparently disturbing implication. It is that in some respects proofs might be better represented by derivations whose inferences cannot be permuted. A single such derivation could then be associated with each group of N or L derivations whose members differed from one another only in the order of their inferences. As I argued at the end of Chapter 4, however, the possibilities for a representation of this kind seem rather limited. The most promising appears to be a calculus of the sort mentioned there in which the conclusions of a rule need not be connected to its premises; but it re-

180

Normalization, Cut-Elimination and the Theory of Proofs

mains to be seen whether this possibility can be realized.50 One objection raised against this sort of calculus in Chapter 4—that the representation it provides is not uniform because the relationship between the premise and conclusions of rule (4) is treated differently from that between the premises and conclusions of the other rules—could perhaps be overcome by making more widespread changes in the structural effect of applying these rules. Such an expedient would only serve however to reinforce the other objection, namely, that a calculus of this kind is less than ideal for representing the actual process of reasoning. That this should be so is, in my opinion, simply a reflection of the difference between a proof as conceived above and a particular piece of reasoning. The latter is properly regarded as a mode of presenting the former. My proposal now is that this difference can be regarded as a special case of the distinction between the grounds for an assertion and the procedure by which they are established. Expressed in these terms the distinction is a familiar one—albeit one which, because it corresponds to the distinction between truth and proof, is usually held not to apply to the present situation. This simple dichotomy is, however, misleading. It seems to me that the ambiguity inherent in the notion of justification is more accurately conveyed by a whole spectrum of possible meanings than by a pair of clear-cut alternatives. At one end of this spectrum is the view, espoused by Frege amongst others, that all justifiable assertions have the same justification, namely, they denote the True. At the other is Brouwer's conception of a justification as a singular process that takes place at a particular point in time; of course, an extrapolation from or description of such a process may also be called a justification, but this is clearly a secondary meaning which applies to something only insofar as it can serve to produce a justification in the primary sense. Justification in Frege's sense is certainly general enough, but this generality is purchased at the expense of informativeness; exactly the reverse could be said of Brouwer's conception. Our general philosophical perspective may determine how wide this spectrum appears to be, and may blind us to certain parts of its range. I am however more concerned with the other factors which influence where along this spectrum an acceptable justification is thought to be located. One such factor—perhaps the decisive one—is the nature of the assertion to be justified. As a rule of thumb, the distance from the Fregean end seems to be inversely proportional to the degree of obviousness and accessibility of the procedure by which an assertion of the kind in question is established. So, for example, a claim to the effect that its grounds can 50 T h e project of characterizing the equivalence relation generated by permutations of inference is of interest even if one rejects the thesis that interreducible derivations represent the same proof. Whatever view is taken of the proper reduction steps, if order of inference has some real significance for proofs, I do not see how either N or L derivations can be said to represent them adequately.

I n t e r p r e t a t i o n s of Derivations

181

be established is usually sufficient to justify an assertion about observable events close at hand. If there are obstacles in the way of observing an event, the justification of assertions about it may require additional information indicating how their grounds were established. To describe a justification in this way, however, is also to interpret it. For example, the grounds for asserting that it is 70° and sunny in Athens are certain meteorological conditions at a particular time and place, and the weather report in The Times provides a means of ascertaining whether these conditions do in fact obtain; but, if for some reason I am not particularly interested in Athens itself and have made the assertion on the basis of the weather report, it might be more natural to describe the report itself as the grounds for my assertion. Of more interest in the present context are mathematical assertions. Here, the procedures by which they can be established are well known, but relatively inaccessible. As a result, they normally require a more elaborate justification. (A simple "factual" claim may still suffice, however, when these procedures are accessible—e.g., in the case of a particularly obvious assertion, or one intended to convince only specialists.) Here too there is a question as to how such a justification is to be characterized. The difference is only that in the present case the question is taken more seriously. (After all, the view that Athens exists only insofar as it is reported on in newspapers is currently out of favor even amongst philosophers; its mathematical analogue, however, is actively discussed.) It may be helpful to compare a mathematical assertion to an empirical one pertaining to a state of affairs which cannot be investigated directly. For example, on a foggy night the location of a sandbank may be inferred from the sound of a bell or a siren, the sight of a warning buoy or light, from depth soundings or from any combination of these. According to some mariners, the sandbank itself, a particular configuration of matter, is distinct from all of the above; they only constitute ways by which we may come to know about it, albeit indirectly. Under other circumstances, we might hope to see the bank directly and comprehend all its properties including its location. 51 According to others, there is no such configuration of matter. To assert the existence of a sandbank at some location is simply to make a statement about the possibilities of undergoing certain kinds of experiences (seeing buoys and hearing sirens, for example, in a particular place). The sandbank itself is one of those "noxious ornaments, beautiful in 51 This somewhat heavy handed extension of the analogy is intended to provide a metaphorical account of Plato's conception of mathematics. Mathematical objects enjoy an independent existence and under the right circumstances, with the appropriate effort and philosophical training, we may hope to know them directly (i.e., when the fog lifts and the sun comes up we may hope to see them). In the meantime, we must learn about them with the aid of objects we do know (can see) which reflect their properties. See, for example, Republic, 510e.

182

Normalization, Cut-Elimination and the Theory of Proofs

form, but hollow in substance/' which must be excised from our ontology; any risks incurred in so doing will be "at least partly compensated for by the charm of subtle distinctions and witty methods" by which it enriches our thought. 52 My purpose in mentioning all this is not to adjudicate between different views, but to illustrate how many different distinctions can be drawn when it comes to a matter of justification. There are at least the following: I.

the particular configuration of sand and its location [mathematical objects and facts about them] II. evidence for I (for example, sirens and lights, or bells and buoys) [proofs] III. particular presentations of II (for example, the sighting of a buoy followed by the sound of a bell) [modes of presenting proofs] IV. linguistic expressions which describe II in terms of III [derivations] V. a particular experience (for example, seeing the buoy and hearing the bell) [the grasping of a proof]. As I have already indicated, there are those who would dispute the significance of some of the above. Furthermore, the way in which I have chosen to describe them is not neutral. (For example, to characterize the activity V as the grasping of a proof is to give II a priority or, at least, an independence which will offend those for whom II is derivative upon V.) Nevertheless, this list does serve to indicate where proofs, conceived of as the denotations of formal derivations, can be located in the scheme of things. Of course, I have not provided an analysis of this notion of proof, nor can I do so here, but I hope that I have done enough to make the idea of distinguishing between proofs and reasoning at least seem coherent. 53 The ideas discussed in (l)-(5) above are of a rather general and speculative kind, although some do contribute to the solution of quite specific problems. As a result, their value depends to some extent on their ability 52

A. Heyting, Intuitionism, 3rd edition, Amsterdam, 1971, page 11. I realize that the view I have espoused is an unconventional one, violating both linguistic and philosophical orthodoxy, but I do think it has much to recommend it. For example, I remarked earlier that mathematicians are said to be interested in questions about the identity of proofs. It is doubtful, however, whether they are as interested in such questions in the abstract as they are in new proofs, even of known results. Whatever "new proof" means in this context, it surely is something more than "obtained from the old one by the kind of structural operations on derivations considered above." (Anyone who attempted to publish a proof which was "new" in this sense would invite ridicule.) On the other hand, a determinate structure is an integral part of an argument or a piece of reasoning. There seems to be no other way of reconciling these facts than by distinguishing proofs from reasoning. 53

Interpretations of Derivations

183

to interact fruitfully with more formal and specialized work in logic and mathematics. By a fruitful interaction, I mean both that there should exist formal results which support them and which acquire significance when interpreted in terms of them, and that they should suggest new lines of inquiry. Whether such interaction is in fact possible remains to be seen. Even at this stage, however, it seems to me that there are interesting connections between these ideas and some formal work. I would like to conclude by mentioning two examples which, in my opinion, illustrate this fact. (i) The analogy between derivations and terms, coupled with the idea of looking for models which the former will admit, leads naturally to a consideration of the full type-structure over some domain. Harvey Friedman has considered structures of this kind and characterized the identity relation between their elements in terms of convertibility (or provable equality) in a version of the typed A-calculus.54 The version in question includes— in addition to axioms and rules for identity, an axiom for A-conversion and an axiom which allows changes of bound variables—an extensionality principle: ((Xx)(sx)) = s if x is not among the free variables of s. After defining a notion of structure appropriate to the typed A-calculus, he prove its soundness—showing by induction on the length of the derivation of s = t that , if h s = t, then M f= s = t for all structures M. He then constructs a particular Mo whose elements are terms of the calculus factored out by the equivalence relation of being provably equal and shows that it is a structure in his sense. Once this has been established, it follows almost immediately that, if M 0 |= s = t, h s = t. (Friedman refers to this as a completeness theorem for the typed A-calculus because, when taken together with the soundness result quoted earlier, it establishes the equivalence of the following three conditions: (i) h s = t; (ii) for all structures M, M (= s = t, and (Hi) Mo \= s = t.) Friedman goes on to establish what he calls an extended completeness theorem for the typed A-calculus. He first defines a notion of partial homomorphism between structures which has the property that, if there exists a partial homomorphism from M onto M and M (= s = t, then Af (= s = t. Given any set B, T#—the full type-structure over B—is the paradigm of a structure for the typed A-calculus. Friedman proves that, for any infinite 54 "Equality Between Functional," Springer Lecture Notes in Mathematics, Vol. 453, 1975, pages 22-37. Actually this paper deals with a wider range of topics than the above description indicates. Friedman was interested in classifying the relations of equality and being everywhere different between functionals of various classes according to their complexity. The completeness theorem quoted in the text is an intermediary towards showing that equality between simple functionals (i.e., those in the full type-structure over to which are defined by a closed term of the typed A-calculus) is recursive.

184

Normalization, Cut-Elimination and the Theory of Proofs

set B, there is a partial homomorphism from TB onto Mo* This allows him to conclude that TB f= s = £ is equivalent to any of (i)-(iii) above. Equality in the structure TB is of course set-theoretical equality between functions. Translated into the language of derivations, the extended completeness theorem asserts that, when closed derivations in the pure implicational fragment of NJ are interpreted as denoting functionals in the set-theoretic sense over some (infinite) domain of atomic proofs, II and IT are interreducible (using Prawitz's reductions steps augmented by expansions) iff they denote the same functional. I think this is an interesting statement; it provides an illuminating characterization of interreducibility, albeit for a restricted class of derivations, and it demonstrates that proofs need not disappear into the consequence relation when they are interpreted as extensional objects. In addition, it suggests some interesting questions for further investigation. For example: (a) Can this result be extended to all of propositional NJ? It seems to be a routine matter to extend the theorem to the negative fragment of NJ by adding product types and modifying the notion of structure accordingly. If the type-structure is then extended to take account of disjunction, it seems clear that all permutations of inference will preserve equality. A more problematic issue is whether the converse holds, z.e., whether it will still be true that derivations which denote the same functional are interreducible. The difficulty lies not so much with disjunction itself as with the thinning that accompanies it. It is unclear how best to treat this latter operation in a functional context. (b) Can it be extended to NIC! Multiple-conclusion logic suggests a natural generalization of the notion of function and of the type-structure which may provide an appropriate framework within which to tackle this question. (c) What is the significance of the expansions? In view of Prawitz's remark quoted earlier, it may be that expansions are of no particular interest, and the need for them here is neither significant nor disturbing. On the other hand, it may be worth considering the possibility of obtaining the same result for the interreducibility relation generated by the reduction steps alone. This could perhaps be accomplished by interpreting derivations not as particular proofs, but as proof patterns, z.e., by regarding their minimal formulae as variables and identifying derivations which differed only with respect to these. Derivations might then be interpreted as denoting (untyped) partial functions—the difference between n and IT, when the latter is obtainable from the former by expan-

Interpretations of Derivations

185

sion, being reflected in a difference in their domains of definition ( Dom(W) C Dom(U) ) . (2) Another possible interpretation for derivations is as the morphisms of a category. Again this might provide evidence for the choice of reduction steps—in particular, for whether there is any reason beyond expediency to restrict permutations. The first obstacle to be overcome on this approach is the need to find a suitable generalization of the notion of category which will accommodate morphisms with a series of domains and, if LK derivations are to be interpreted, a series of ranges as well. In Appendix C below, I have sketched very briefly some work in this direction. I have also outlined what I think is a natural generalization of categories to morphisms with more than one domain and range, suggested by the multiple-conclusion calculus considered earlier. It then turns out that the structural axioms for these generalized categories force us to identify derivations which permute to one another—thus providing some additional support for (5) above. Zucker's non-terminating reduction sequence appears in this context as an innocuous example of an infinite series of terms all of which refer to the same morphism.

The conclusions I have reached may seem surprising at first sight but there is, I believe, much to recommend them. On quite general grounds, the conception of proofs as rather loosely structured objects is a plausible one. Once it is accepted and the denotations of formal derivations are viewed in terms of it, a unified treatment of cut-elimination and normalization becomes a possibility and the general theory of proofs is freed to some extent from the shackles of syntax. Although it is impossible to predict with any certainty how the subject will develop in the future, the direction I have indicated does show some promise of being a fruitful one—or so, at least, I have tried to argue.

Appendix A

A Strong Cut-Elimination Theorem for LJ This appendix discusses the possibility of avoiding Zucker's negative result about strong cut-elimination for LJ by altering his conventions for the indexing of formulae. As I remarked in Chapter 2 above, his particular counterexample to strong cut-elimination depends upon a special, and perhaps rather unnatural, feature of these conventions. The question that remains is whether an alternative indexing system could avoid such counterexamples altogether. The version of LJ presented below is essentially the one adopted by Zucker in his paper. 1 A sequent has the form T H A, where T is a set of indexed formulae; negation is defined in terms of a constant _L for falsity, and there is no thinning rule. I have however altered the conventions which govern the indexing of formulae and, as a result, contraction becomes a derived rather than a primitive rule. My desire to follow Zucker's treatment as closely as possible explains the unusual formulations of V- and 3-left. As before, F, A , . . . are supposed to range over sets of indexed formulae, and z, j , k,... over indices. I will write T, A for TUA and T, Ai for r u { ^ i } ; this notation is not intended to imply either that m A = 0 or that Ai £ T. When I do want to indicate this, I will use T; A and T;Aj, respectively. Finally, T;(Ai) will be used to denote ambiguously T;Ai and T. (In the latter case, it is assumed that Ai & T.) Similarly, T,(Ai) will be used to denote T, A\ and T, when it is left open whether or not A% G T. The notations T; (A) and T, (A) are explained in an analogous way. I take the Calculus LJ to consist of the following. Axioms: Ai h A

±i\- P (P atomic and different from _L)

lu O n the Correspondence between Cut-Elimination and Normalization," Annals of Mathematical Logic, Vol. 7, 1974, pages 1-156.

186

A Strong Cut-Elimination Theorem for LJ Logical Rules: Right

187

Left

ri- A A h B r,AI-^AJ5

T;(Aj)\-C r,(AABj)\-C

T;(Bi)\-C Y,{AABj)\-C

T\-A T\-B TbAVB ri-AVB T; (A) h B Th A^B

T;{Ai)\-C A;(g,)hC T,A,AvBk\-C Tt- A A; (Bj) \- C (r),A,(A->Bj)hC

T h A{a) * T h VxA(x)

r;(^)Qi-B r.O/a;^!^)!-^

r i- A{t)

r;(^(q)8)hg f T, 3xA{x)j h S

T I- 3xA(x)

* where a does not occur in Y \ where a does not occur in T, B. Cut Rule: r\-A

(At);A\-B (T),A\-B

The application of cut, or of a left rule other than V- or 3-left, to a premise which contains no formula occurrence involved in the inference is empty. This means in the case of cut that, if A; (Ai) = A, then d T\-A

d' A;{AZ)\-B (r),AhB

is just another notation for the derivation d!. Empty applications of A-, —*-, V- and 3-left are treated similarly. Formulating these rules so as to take account of empty applications is just a convenient way of introducing some notation which will be useful later on. The use of sets in place of sequences of formulae makes a rule of interchange redundant. As for contraction, in this calculus it takes the form r,AjY- B and is not included among the basic rules because of the following:

188

Normalization, Cut-Elimination and the Theory of Proofs

Lemma A . l If d is a derivation ofT\Ai h B, then there is a derivation d(Ajfi) o/T, Aj h B which differs from d only in that some formula occurrences are assigned different indices. (In particular, there are no cuts in d(Aj/i) which are not already in d.) This lemma is proved by a straightforward induction on the construction ofd. Reduction steps are of three kinds: A. Elimination of trivial cuts B. Permuting cuts upwards C. Reducing the complexity of cuts. The following reduction steps are to be read from left to right. A. Elimination of trivial cuts: a.

d r hA

Ai\-A

d T\- A

r\-A b.

d Ai\- A T; Aj 1- B r,At\-B

diAi/j) T,Ai\-B

B. Permuting cuts upwards: These reductions divide into two groups according to whether the cutformula is passive on the right or on the left. (1)

Cut-formula a.

d

ei-,4

passive on the right d\ d

r;(4)KBA;(4)hC F,A;Ak\BAC e,r,Ah5AC d d2 Q\- A A;(Ak) VC (6) , A I - C (e),ri-B e.r.AhfiAC

d 6 h i

b. d T\-A

d\ T]{Ak)hB

d' A;Ak\-B A; Ak\-B'

A.rhs' where R is V- or 3-right.

d T\-A A,T\-

d' A;Aky-B B

A,ri-s'

A Strong Cut-Elimination Theorem for LJ c d F\-A

d' A; Ap; (Bq) V- C A;AP\-B^C r.AKB^C

189

d d'(Bm/q) T \-A A;ip;(Bm)hC r,A;(5m)hC r.AhB^C

where m does not occur as an index in d or d'. d. d D-B

d'(a) A; Bfc h A(a) A;BkhVxA(x) T, A I- Va;A(x)

d d'{b) T \-B A;Bk\-A{b) r , A h 4(6) r , A h V«i4(i)

where 6 is a parameter not occurring in d or d'(a). e. d Y\-D

di d-i A;(Dn);(Ap)hC 8;(D„);(B q ) h C A,9,iVBt;PBhC T,A,e,AVBk\-C d di(AT/p) r h - D A;(Dn);(Ar)\-C A,(r);WhC

d ThD

d2(Bs/q) 9;(D„);(B,)hC e,(r);(B,)hC

r,A,e,i4vBfchc where r, s do not occur in d, d\ or ^2. f.

d' d A;Ap;Bq\-C F\-B A,A'k;Bq\-CR A,r,4HC

d Th B

d'(Am/p) A;Am;BqhC A,T-Am\-C A,T,A'khCR

where m occurs nowhere in d or d', and R is A- or V-left. g. d e\-C

di d2 T;{Cm)\-A A;(Cm);Bk\-D T,A,A~>Bp;Cm^D r,A,e,A-+Bp\D d S\-C

dx d d2(Bq/k) r ; ( C m ) h ^ @\-C A;(Cm);Bq\-D r,(8)hi4 A,(@);Bq\-D T, A, 9 , A - • Bp h D

where q occurs nowhere in d or d2.

Normalization, Cut-Elimination and the Theory of Proofs h.

d'(a) d A;Ap;(B(a)q)\-C Tt-A A,3xB(x)r;Ap\-C A,3xB(x)T,r\-C

d FhA

d'(b)(B(b)s/r) A;Ap;(B(b)s)i-C A,T;(B(b)s)\-C A,3xB{x)r,V\-C

where 5 occurs nowhere in d or d'(b), and b is a parameter not occurring in d or d'(a). Cut-formula passive on the left a.

d\ d2 T;(AP)\-C A;(Bq)\-C T,A,AVBkl-C r,A,AVBk,Q\-D

d @;Cm\-D

d d2{Bs/q) d di(AT/p) r;(Ar)hC e;Cm\-D A;(BS)\-C 9 ; Cm \- D Q,T;(AT)\-D e, A; (Bs) r- D T,A,A\/ Bk,@\- D where r, s do not occur in d, d\ or d2. b.

d(Ak/p) A;Ak\-B A,T;Ak\-C

d A;AP\-B d' A,A'a^BR Bn;ThC A,A'q,T\-C

d! Bn;T\-C

A,A'q,r\-cR

where k occurs nowhere in d or d' and R is A- or V-left. c.

d\ d2 T\- A A;Bk\-C T,A,A^ BpbC r,A,A^Bp,

di

d Q;Cm\-D

e\-D d2(Bq/k) A-Bq\-C

d " ; Cm \-D

T\-A

r,A,A^Bp,et-D where q occurs nowhere in d or d2. d.

d(a) r ; (A(a)p) \- B d' r,3xA(x)q\-B A;Bm\-C A,3xA{x)q,T hC

d(b)(A(b)T/p) d' r ; (i(i) r )hg A;Bm\-C A,T; (A(b)r) h- C A,3xA(x)q,T \- C

where r occurs nowhere in d(b) or d', and b is a parameter not occurring in d' or d(a).

A Strong Cut-Elimination Theorem for LJ

191

C. Reducing the complexity of cuts: (1)

a.

di di YV A A\-B r,A\-AAB

d e;{AABk);Ap\-C 9;AABk\-C

e,r,A\-c dt T\- A

di d2 r\~A A\-B d(Aq/p) T,A\-AAB Q;(AABk);Ag\-C Q,(T,A);Aq\-C

e,r,(A)i-c where q does not occur in d, d\ or di. b. Like (la) with Bv instead of Av and 0*2 playing the role of d\. (2)

a.

d &^A Oh AVB

d\ d2 r;(AVBk);(Ap)\-C A;(AV Bk);(Bq)\-C r , A; A V Bk H C

r,A,ei-c d d

Q\-AVB

e\-A

r;(AVBk);(Aq)\-C

r,(ey,(Aq)\-c r, (e)»- c

where q occurs nowhere in d or d\. b. Like (2a) with B instead of A, and d2 playing the role of d\. (3)

d r ; (Ap) h B r\-A- ->B d T-AAP) \-B r\-A- -*B

d\ {A^Bk);A\-A

0Z2 (A -> Bk);G;Br

hC

A^Bk;A,Q\-C r,A,0(-C7 di

(A->Bk);A\-A ,A\-A T-{AP)\-B (r) T,{A)\-B / / T,(A)hB

T;(AP)\-B d2(Bq/r) T\-A-> B (A^Bk);@;Bq (V).,e;B,xvc r,(A),6hC

where q occur s nowhere in d, di or aV

\-c

192 (4)

Normalization, Cut-Elimination and the Theory of Proofs d(a) T h A(a) n-VxA(g)

d' A;(Vx(Ax)p);A(t)q\&;VxA{x)phB &,T\-B d(a) r I- A(a) T h VJA(X)

d(t)

r i- ACQ

B

d'(A(t)r/q) A; (VaA(g)p); i4(-left, left thinning (a) or (b)—provided that if R is left thinning (a), Ai is not the active formula in its premise. (4) di r\-A;Az

d2 Ak;T'\-A' 4„i,;rhA'LT(a)

r, r h A, A' di{As/i) rhA;i8

di T\~A;Ai

r, r (5) di r h A; 3xA(x)t

d^xAix)^) T h A; 3xA(x)s

d2 Ak;F' h A' LT(a) Ai\As;T ' h A ' i s ; r , r ' h A ,A' h A, A'

d2(a) A(a)n;3xA(x)i;T' \- A' 3xA(x)j, 3XA{X)J;T' \- A' r,r'h A, A' d2(a) A{a)n;3xA{x)i;T'h A' rfl r h A; 33^4(3:), 3 ^ ( 3 : ) , ; a ^ Q c ^ ; T' h A' 3xA(x)s; T, T' h A, A' T,T'\- A, A'

A Formulation of the Classical Sequent Calculus

211

Cut-formula active on the left (6)

di d2 T\-A;(AABm);Aj V h A'; (A A Bm); B, r,r'hA,A';AABm,iABm

d3 ,4 A flm;r" I-A"

r,r',r"h A,A',A" d\

d>2

rhA;(AABm);Aj r'hA';(AABmy,Bj r , r ' h A,A';AABm;AABs

d3 AABm\T"\-

A"

r,r',r"i-a,A',A";iAB 8 r,r',r"i-A,A',A" (7)

dx r'hA'^j „ r,lhA";Ai,ii

(A A Ba/m)d3 AABS-,T"\- A"

d2 Ai;rhA

r",ri- A", A di T'\-A';Ai

d2

r"hA";^;A8R ^fhA T",r\-A",A;AS r",ri-A",A

(J4s/i)d2 A3;r\-A

where R is V-, —•-, 3- , -i-right, right thinning (a) or (b)—provided that, if R is right thinning (a), Ai is not the active formula in its premise.

(8)

^ T\-A;Aj

d2

r\-A;Ak,AkKTW Ak;T'\-A' r,ri-A,A'

F h A

^

.RT(a)

d2

rhA;i,;4 W 4r'HA' iTFKATATA:

(^s/fc)d2 A,;fhA'

212

Normalization, Cut-Elimination and the Theory of Proofs

(9)

di(o) r \-A;VxA(x)f, A(a)n r h A; V x ^ x ) ^ V x ^ x ^

d2 V x ^ x ) ^ T h A'

r, r \- A, A' di(a) T\-A^xA(x)i;A(a)n r h A; Vx^(x) 8 ; Vx^(x)j Vx^(x) i ; T' I- A' r,ri-^.A^Vs-Apc), r , r i - A, A'

(Vx>l(x) s/i )d 2 ViA(i)8;r'hA'

£ . Elimination of Cuts with a Thinned Premise (1)

di Tl-A;^

d2

d'(Aj/s) A,A',A,

rhA,^;gfc fit;rhA' r,r'hA,AUj

r,r h where d' =

T h A; ^

r,r'HA,A';^ s (2) is like (1) except with Ai, ^ and As on the left. (3) dl

d2 Tj^hA'

(^/s)d'

ri-A;g fc B t ; r',^i-A' r,r,^hA,A'

r, r , ^ \- A, A' where d'=

r';^hA r,r';ishA,A'

(4) is like (3) except with .4,, Aj and As on the right. F. Thinning Permutations Premise of the thinning passive in the preceding inference Left Thinning (a) (1)

a. b.

Aj r- Aj Ak,Bnh Aj d

4rhA ii;r"HA" ^,c*fc,r"i-A"

Ak\-As Ak,Bn\-Aj d

A-, —>-right, A-, 3-, V-left, right thinning (b) or left thinning (a), provided that, if R is 3-left and the proper parameter of R occurs in C, it is replaced by a parameter which does not occur in the figure on the right. (Bn is supposed to be a premise of R in d, and Bs in (Bs/n)d.)

(At);r!-A;EP WirhA'jf, 4r,r'hA,A',£AFm A j ,C fc ,r,r'l-A,AM5AF ra (^) ; rhA ; ff p (^r'l-A'if, (^),(C fc ),r h A;EP (^),(C fc ),F h A';F, A i ,C f c ,r,r / r-A,A',£AF r o ^ i ; r,r,£vF m i-A,A' ^,Cfc,r,r',£VFro(-A,A' (Es/P)di (4,);r;£8HA (Aj), (Ck), F;ES\-A

(Ft/q)d2 (^);r';F(hA' (Aj), (Ck), P ; Ft \~ A'

i j ,ft,r,r',£vF m hA,A' (Aj);r\-A;EP

( ^ r ' j f ^ A ^

4r"hA,A' ^,C f e ,r"hA,A' di(Es/p) (Ai);T\-A;E, (^),(Ct),rhA;£8

H

(Ft/q)d2 (4i);r';F«hA' {As),(Ck),r;Ft ^A'

^,C f c ,r"hAA'

R

where R is —> -left or cut, with premises Ep and Fq in the figure on the left, and Es and Ft in the one on the right. If R is cut, I assume that s = t. (Notice that the possibility of Fq being equal to Ai has not been excluded.)

214

Normalization, Cut-Elimination and the Theory of Proofs

(lb-f) above simply state that an application of left thinning (a) the active formula of whose premise is passive in the conclusion of the preceding inference can be permuted upwards past that inference. Ignoring complications which arise from the need to keep the active formulae in the premises of the inference distinct from the active formulae in the conclusion of the thinning (and, in case the inference is an application of V-right or 3-left, from the need to ensure that the proper parameter does not occur in the conclusion of the thinning), there are really only two cases to consider according to whether the inference has one or two premises. Having shown how these complications are handled in the case of left thinning (a), there is no reason to give a similarly detailed treatment of the remaining thinning rules since no new problems arise. d So, given T h A where / is an application of any one premise rule, T'hA'7 let the derivation dj be obtained from d by applying whatever contractions may be necessary to preserve the distinctions mentioned above and, in case J is an application of V-right or 3-left, by possibly replacing its proper parameter. (It is convenient to write the conclusion of dj as T h A, even though it may differ from the conclusion of d in minor respects.) If J is an application of a two premise rule which operates on the conclusions of d and d', the derivations dj and d'j are explained similarly. With the help of this notation the remaining cases can be presented in a simple and uniform way. Left Thinning (b) (2)

a.

As h Ak Bn,Ai\- Ak

Aj h Aj Bn,Ai\-Ak d T\-A;Ai T'l-A';^

b.

di r l

rh A Ck,T\-A,Aj -5=

c.

d

T-.

n

:— 1

d'

r\-A;{Aj) r'hA';(ii) F'hA";^

Cfe,r"hA",A

•J

dj

d'j

r\-A;(Aj) r'HA';^) ( Bj, r r A b. Like (2a) with Cn on the right. c. Like (2a) with Bj on the right. d. Like (2c) with Cn on the right.

d Ai-,T\- A As;Cn,Th-A Am,Gn,Bj,l r A

219

Appendix C

Proofs and Categories This appendix outlines one way in which the derivations of a formal system may be regarded as representing the morphisms of a category with some additional structure. The possibility of such a representation arises from the similarity between the rules of Gentzen's N and L systems, on the one hand, and the definitions of product, exponent, etc. in category theory, on the other. Because these morphisms must satisfy certain identities, they are not in general represented by a unique derivation. This naturally suggests the question of what logical sense can be made of the notion of two derivations representing the same morphism and whether this relationship can be characterized solely in terms of structural properties of the derivations themselves, without reference to their categorial interpretation. These topics were first investigated by Lambek in a series of papers on deductive systems and categories.1 Subsequently, Szabo gave an account of the negative fragment of LJ in terms of a relation between derivations which he called 'equi-generality'.2 Mann reproduced Szabo's results for the negative fragment of NJ and showed that equi-generality was equivalent to the relation 'being reducible to the same expanded normal form' (in the sense of Prawitz). 3 When Szabo extended his treatment to the whole of intuitionistic (first-order) predicate logic, he abandoned his original equivalence relation between derivations in favor of one which, like Mann's, is defined in terms of unique normal forms.4 (The word "unique" needs some qualification, but I shall ignore that complication here.) Derivations most naturally represent morphisms having a sequence of 1 "Deductive systems and categories I," Mathematical Systems Theory, Vol. 2, pages 287-318, "Deductive systems and categories II" in Springer Lecture Notes in Mathematics, Vol. 86, pages 76-122, and "Deductive systems and categories III" in Springer Lecture Notes in Mathematics, Vol. 274, pages 57-82. 2 "A categorical equivalence of proofs," Notre Dame Journal of Formal Logic, Vol. 15, 1974, pages 177-191. See also the addendum to this paper in the same journal, Vol. 17, 1976, page 78. 3 "The connection between equivalence of proofs and cartesian closed categories," Proceedings of the London Mathematical Society (3), Vol. 31, 1975, pages 289-310. 4 Algebra of Proofs, Amsterdam, 1978.

220

Proofs and Categories

221

objects, rather than a single object, as their domain. There is, however, no standard way of treating such morphisms within category theory. Of course, in the presence of pairing (or products) an n-place function can always be regarded as a function of one argument, but this approach is not well suited to the present purpose. On the other hand, the multicategories of Lambek and sequential categories of Szabo are cumbersome and difficult to work with. I propose to adopt here an alternative notion of 'multicategory'. Rather than defining it at the outset, however, I shall try to show how it arises from the attempt to introduce additional structure into an ordinary category. The idea is to give a categorial interpretation of propositional logic (or, more properly, of its derivations). So, let Lp be a language comprising a set P of propositional variables, the propositional constants T and _L, and the binary connectives A, V and —>. Let Cp be the discrete category whose objects are the members of P. I want to extend Cp to a cartesian category CA which will serve to interpret logic based on the A- fragment of Lp. The objects of C A will be the formulae built up from the members of P by means of conjunction. The morphisms of CA will include an identity morphism for each object. In addition, for each object of the form A A B, there will be projection maps ir^AB: A A B »-• A and n^3'- A /\ B *-+ B. (I will suppress reference to their domains when these are obvious from the context.) Finally, if / : C*-> A and g: £)»-• B, there will be a unique pairing map (f,g):

C, £)•-• A A B which makes the following diagram commute:

CJD

f/ /

/

A, hf: D'^ and C=D,D',

B

then:

a. if E, then fog

= (fohJoh'):

{C

/E)DJ)'^AAB

By convention, / o g is just # when the range of / does not appear among the domains of g. (C IA) D is supposed to be the result of replacing each occurrence of A in D by the sequence C(1) follows from the properties of identity morphisms in a category, (2) from the commutativity of the product diagram, and (3) from the uniqueness requirement on morphisms of the form (x, y). To see this last notice that f oh: (C /E) D>-> A and / o h'\ (C /E) D'*-> B, so there is a unique (fohjo

h'): {C IE) D, D'^

A/\B

such that

(foh,hof')oiri

= f oh

and (foh,foh')on2

=

foh'

But (fog)oTTi

= fo(goTTi)

=

foh

and ( / og)o7T2

=

foh'

Hence f°g = ifohjoh') Suppose we have closed the class of projection maps under composition. (Since these always have a single domain, n o n' for example will be a map from the domain of n to the range of 7r'.) We can now define the class of multimaps of C A to be the closure under pairing of the identity, projection and compositions of projection maps. It is then easy to infer from conditions (l)-(3) above, using the associativity of composition, what might be called the cut-elimination theorem for the A-fragment of LJ, namely:

Proofs and Categories

223

The morphisms of C A are closed under composition. In other words, CA is a cartesian category.5 I now want to extend this treatment to include implication. The idea is to construct a cartesian closed category, C A -M whose objects will be the formulae built up from P by conjunction and implication. The morphisms of CA— will include identity, projection and pairing maps for this enlarged set of objects together with an evaluation map eA~*B: A,{A-* B) *-+ B for each object of the form A —• B. In addition, for each morphism / : A, ( 7 ^ B, there will be a unique exponent map e ( / ) : C^ (A —• B) such that the following diagram commutes:

The notation ^ * • for arrows is used to make it easier to read the domains of the various maps from the diagram. Here, e(f)oe and / have the same domains and the commutativity of the diagram allows us to identify these two morphisms. In addition, to (1), (2) and (3) above, composition must satisfy: (4) If g is for the form e(h): C*-> {A —• B) and / : D*-* J5, where E appears among the terms of C, then / o g: (£) jE) C*-* (A —> B) is just

e{foh). (5) If / : D»-> {A —> B) is of the form e(/i), where h: A,D*-+ B, and g is eA^B, then / o g = h. (Again, (4) follows from the uniqueness of e(f o h) and (5) from the commutativity of the exponent diagram.) It might seem, by analogy with C A , that we need only close the evaluation and projection maps (taken together) under composition, add the resulting class to the identity morphisms of CA— and then close everything under pairing and exponentiation in order to obtain a cartesian closed category. Even with these additions, however, and conditions (l)-(5), there 5 C/. the discussion of logical calculi at the beginning of Chapter 4 above, where it was pointed out that the meaning of 'application of a rule' determines the notion of substitution for derivations and that the class of derivations is closed under the latter operation.

224

Normalization, Cut-Elimination and the Theory of Proofs

is still one possibility left uncovered, namely the case in which / : D*~+ A is composed with eA^B'. This makes it necessary to introduce morphisms / o eA~*B: D,{A-+ B) \-+ B for each such / if the cut-elimination theorem is to hold for CA—• An alternative (more in the spirit of Gentzen's —»-left rule) is to replace the maps eA~~*B by ef~*B: D, (A-> B) *-> B, for each / : D*-+ A, and require that, for any h: A,C>-> B, there is a unique map e(h) such that the following diagram commutes:6 / A,C

C D

i

V

e(h)\

i A-+B -

_ _ ^ >

* > B

If this alternative is adopted, condition (4) remains unchanged, condition (5) generalizes to: (5') If / : D*-> (A —> B) is of the form e(ft), where h: A, D>-> B, and g is of the form g: C,(A-> B) »-• B is of the form eArB, then fog — h' oh.

where h': C^

A,

and the last remaining case is dealt with by the following: (6) If / : E^> F (where F # A -+ B) and p = e*: 5 , (4 - • B) *-> B, then / o g: (E /F) D,(A->

B)^B

is efoh.

The cut-elimination theorem for CA_>—i.e., the statement that its morphisms are closed under composition, or that it is a closed cartesian category—can now be proved with the help of conditions (l)-(6) by a slightly more complicated inductive argument than was the case for C A . That the maps eA~*B do not suffice for cut-elimination is reflected in the sequent calculus by the fact that cut-elimination does not hold when the —•-left rule is replaced by T;Bh A Y,A,A->B\A 6

Notice that e(h) does not depend upon / .

Proofs and Categories

225

In natural deduction calculi, it is reflected in the structure of normal derivations—more precisely, in the fact that a branch of such a derivation cannot pass through the minor premise of an application of —^-elimination if it is to consist of a series of eliminations followed only by introductions. Despite the added complications associated with implication the above seems to provide rather a satisfactory interpretation of a fragment of the sequent (or natural deduction) calculus and its associated normal form theorem. Furthermore, it is easily extended to the full negative fragment (of intuitionistic propositional logic) by introducing an initial object _L into CA— together with a unique morphism JL^: _!_»-*• A for each object A. Unfortunately, the picture is spoiled somewhat by a number of complications having to do with the 'structural' properties of morphisms which I have chosen to disregard here. For example, we really need some principle corresponding to Gentzen's interchange rule which, given / : A,B »-• C, say, will ensure the existence of / ' : B,A*-+C and allow us to treat / and / ' as equivalent in some sense. For reasons such as this, the approach sketched above does not work out very well in detail. It can, however, be modified in such a way as to avoid these difficulties. The modified approach is connected to the present one as the formulation of sequent calculi in terms of sets of indexed formulae is connected to the usual formulation in terms of sequences. Basically, the idea is to operate with sets of indexed formulae rather than sequences. The objects of the category should still be formulae, however, rather than indexed ones. One way of accomplishing this is to think of multimaps as having arbitrarily long sequences of sets of domains, not excluding the empty set. The occurrence of A as a member of the mth set of a sequence will be associated with the indexed formula Am. Within this framework, the interchange principle mentioned above is no longer needed, and structural operations corresponding to contraction and thinning can be conveniently introduced if desired. My aim in this appendix has just been to give a preliminary exposition. For that purpose, the approach taken above seems the most perspicuous and is easiest to motivate. Rather than modifying it now and trying to spell out in detail the interpretation of derivations from the negative fragment, I want to conclude with a brief sketch of how disjunction might be incorporated into this framework. Let us begin by reconsidering Cp to see what is involved in extending it to the dual of C A , the co-cartesian category C v - The objects of C v are just the formulae built up from P using V, and its morphisms include identity maps (one for each object), injection maps z^ VB : A »-> A V B and %2yB' B H-> A V B for each object of the form A V B, and, given / : A *-+C and g: B

H-ȣ),

a unique map [/, g]: Ay B i-+C, D, such that the following

226

Normalization, Cut-Elimination and the Theory of Proofs

diagram commutes: AVB

Dually to the case of C A , the commutativity of the coproduct diagram means that / = ii°[f>9] and g = i2[/,0] so that the ranges of ij o [/, g] (j = 1, 2) will be a subsequence of the ranges of [/,