v CONTENTS
Preface ...

Author:
Paoli F.

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v CONTENTS

Preface .......................................................................................................ix Part I: The philosophy of substructural logics Chapter 1. The role of structural rules in sequent calculi...........................3 1. The "inferential approach" to logical calculus......................................... 3 1.1 Structural rules, operational rules, and meaning.......................... 5 1.2 Discovering the effects of structural rules..................................11 2. Reasons for dropping structural rules.................................................... 15 2.1 Reasons for dropping structural rules altogether........................15 2.2 Reasons for dropping (or eliminating) the cut rule..................... 17 2.3 Reasons for dropping the weakening rules.................................21 2.4 Reasons for dropping the contraction rules................................25 2.5 Reasons for dropping the exchange rules...................................28 2.6 Reasons for dropping the associativity of comma...................... 30 3. Ways of reading a sequent......................................................................30 3.1 The truth-based reading............................................................ 31 3.2 The proof-based reading........................................................... 31 3.3 The informational reading.........................................................32 3.4 The "Hobbesian" reading.......................................................... 34 Part II: The proof theory of substructural logics Chapter 2. Basic proof systems for substructural logics.......................... 41 1. Some basic definitions and notational conventions................................. 42 2. Sequent calculi....................................................................................... 44 2.1 The calculus LL.......................................................................44 2.2 Adding the empty sequent: the dialethic route............................ 49 2.3 Adding the lattice-theoretical constants: the bounded route........ 49 2.4 Adding contraction: the relevant route....................................... 50 2.5 Adding weakening: the affine route........................................... 55 2.6 Adding restricted structural rules.............................................. 57 2.7 Adding the exponentials............................................................65 3. Hilbert-style calculi................................................................................68 3.1 Presentation of the systems....................................................... 68 3.2 Derivability and theories...........................................................73 3.3 Lindenbaum-style constructions................................................ 81

vi 4. Equivalence of the two approaches.........................................................83 Chapter 3. Cut elimination and the decision problem...............................87 1. Cut elimination...................................................................................... 87 1.1 Cut elimination for LK.............................................................87 1.2 Cut elimination for calculi without the contraction rules............ 94 1.3 Cut elimination for calculi without the weakening rules............. 97 1.4 Cases where cut elimination fails.............................................. 99 2. The decision problem........................................................................... 100 2.1 Gentzen's method for establishing the decidability of LK.........101 2.2 A decision method for contraction-free systems....................... 105 2.3 A decision method for weakening-free systems........................ 106 2.4 Other decidability (and undecidability) results......................... 111 Chapter 4. Other formalisms.................................................................. 115 1. Generalizations of sequent calculi........................................................ 116 1.1 -sided sequents....................................................................116 1.2 Hypersequents........................................................................ 121 1.3 Dunn-Mints calculi.................................................................127 1.4 Display calculi....................................................................... 130 1.5 A comparison of these frameworks......................................... 136 2. Proofnets..............................................................................................137 3. Resolution calculi.................................................................................145 3.1 Classical resolution.................................................................146 3.2 Relevant resolution................................................................. 149 3.3 Resolution systems for other logics......................................... 153 Part III: The algebra of substructural logics Chapter 5. Algebraic structures..............................................................159 1. *-autonomous lattices...........................................................................161 1.1 Definitions and elementary properties......................................161 1.2 Notable *-autonomous lattices................................................ 165 1.3 Homomorphisms, -filters, -ideals, congruences.....................171 1.4 Principal, prime and regular -ideals....................................... 181 1.5 Representation theory............................................................. 186 2. Classical residuated lattices................................................................. 187 2.1 Maximal, prime, and primary -ideals..................................... 188 2.2 Subdirectly irreducible c.r. lattices.......................................... 190 2.3 Weakly simple, simple and semisimple c.r. lattices.................. 192

vii Part IV: The semantics of substructural logics Chapter 6. Algebraic semantics.............................................................. 201 1. Algebraic soundness and completeness theorems..................................202 1.1 Calculi without exponentials................................................... 203 1.2 Calculi with exponentials........................................................209 2. Totally ordered models and the single model property..........................213 3. Applications......................................................................................... 219 Chapter 7. Relational semantics..............................................................221 1. Semantics for distributive logics........................................................... 222 1.1 Routley-Meyer semantics: definitions and results.....................223 1.2 Applications........................................................................... 235 2. Semantics for nondistributive logics..................................................... 239 2.1 General phase structures.........................................................240 2.2 General phase semantics......................................................... 250 2.3 The exponentials.....................................................................252 2.4 Applications........................................................................... 254 Appendix A: Basic glossary of algebra and graph theory........................... 257 Appendix B: Other substructural logics..................................................... 271 1. Lambek calculus...................................................................... 271 2. Ono's subintuitionistic logics.....................................................277 3. Basic logic............................................................................... 281 Bibliography.............................................................................................289 Index of subjects.......................................................................................301

PREFACE

1. AN INTRIGUING CHALLENGE Whoever undertakes the task of compiling a textbook on a relatively new, but already vastly ramified and quickly growing area of logic - and substructural logics are such, at least to some extent - is faced with a baffling dilemma: he can either presuppose a high degree of logical and mathematical expertise on the reader's part, or else require no background at all except for a "working knowledge" of elementary logic. In our specific case, each one of these policies had its own allure. The former strategy promised to speed up the presentation of some advanced topics and to allow a more refined expository style; the latter one, on the other side, would have permitted to reach a wider audience, some members of which might have had the opportunity to study for the first time some elementary, but fundamental results - such as Gentzen's Hauptsatz - directly in the perspective of substructural logics. Teaching logic from this point of view to unexperienced, and presumably still unbiased, students seemed to us an irresistibly intriguing challenge - therefore, we opted for the second alternative. Thus, we assume that the reader of this book has attended an undergraduate course in logic and has a good mastery of the rudiments of propositional logic (Hilbert-style and natural deduction calculi, truth table semantics) and naive set theory. As for the rest, the volume is self-contained and gradually accompanies the reader up to some of the most recent and specialistic research developments in this area. Some prior acquaintance with either predicate logic or algebra is useful, but not indispensable; in particular, the algebraic notions used throughout the book are surveyed in a special glossary (Appendix A). Of course, this book is not meant only for students. The researcher in the field of substructural logics will find plenty of material she can directly exploit and draw from in her research practice.

x It is not easy, it must be confessed, to write a textbook on this subject short after such a wonderful volume as Restall's An Introduction to Substructural Logics (Restall 2000) has been sent to the press. Our intellectual debt towards this work is enormous, as the reader will notice. However, offering a different perspective on a same topic can be valuable, sometimes. Restall's book primarily focuses on natural deduction and display calculi, and on frame semantics. Our viewpoint is somewhat more traditional: we privilege ordinary sequent calculi on the proof-theoretical side, and algebraic models on the semantical side. We believe that readers who are scarcely at ease with the "punctuation mark" proof theory in the style of Dunn, Mints, Belnap, or with frame semantics - especially researchers belonging to substructural schools other than the relevant - could perhaps feel more comfortable in a setting like ours. Thus, we are confident that our book and the one by Restall can profitably integrate and supplement each other. We tried to arrange this book in such a way as to provide a (hopefully) useful tool for readers coming from any substructural tradition (linear logic, Lambek calculus, relevance logics, BCK-logic and contraction-free logics, comparative logic) and from a number of different backgrounds (philosophy, mathematics, computer science, linguistics). It is extremely important, in our opinion, that people from diverse provenances and academic environments, who often tackle the same problems using different jargons and being unacquainted with one another's results, can find a common ground for discussion and mutual interaction. Occasionally, some personal biases of the author - who is a philosophically oriented logician and a specialist of comparative logic - may show up. We hope that this won't happen too often, though.

2. OVERVIEW OF THE CHAPTERS Chapter 1 introduces the topic from both a historical and a philosophical perspective. After discussing the relationships between substructural logics and proof-theoretical semantics, we provide some reasons for dropping some or all of the structural rules in sequent calculi and, finally, we try to find plausible informal interpretations for substructural sequents. Chapter 2 contains a presentation of the main sequent and Hilbert-style calculi for substructural logics, and of their elementary syntactic properties. The cut elimination theorem for substructural sequent calculi is the heart of Chapter 3, where we also illustrate some decision procedures for these systems. Chapter 4 deals with more advanced formalisms, some of which have been introduced rather recently: we cover a few generalizations of sequent

xi systems ( -sided sequent calculi, hypersequent calculi, Dunn-Mints and display calculi) and of natural deduction (proofnets), as well as resolution calculi. Algebraic semantics will be in the foreground in Chapters 5 and 6, where we study the models of substructural logics at first in a purely algebraic perspective, and then linking them to the calculi of the preceding chapters by means of appropriate completeness results. Chapter 7 is concerned with a different kind of semantics, which generalizes Kripke-style semantics for modal and intuitionistic logics. We discuss models for both distributive logics (Routley-Meyer semantics) and logics without distribution (phase semantics). Appendix A provides a crib of elementary algebra, model theory and graph theory for those readers who are unfamiliar with even the most basic notions of these disciplines (we primarily thought of students in philosophy or linguistics, but also in computer science). Its main aim is letting the book be as selfcontained as possible. Appendix B surveys some logics which, regrettably enough, had not received adequate attention throughout the main body of the text.

3. WHAT HAS BEEN LEFT OUT Although we tried to cover as many topics as possible, due to obvious limitations of size we could not help making choices. In order to delimit the bounds of our enterprise, we imposed ourselves four constraints: The propositional constraint. Throughout this book, we shall remain within the boundaries of propositional logic. There exist interesting inquiries concerning quantified substructural logics, or even substructural arithmetic or set theory (see e.g. Meyer 1998), but in our opinion such a work will remain somehow foreign to the spirit of substructural logics so far as the difference between lattice-theoretical and group-theoretical quantifiers is not properly understood. We think that taking a firm grip on such a distinction is, at present, the most important task with which substructural logicians are confronted (a promising start is in O'Hearn and Pym 1999). The commutative constraint. We shall not consider logics without exchange rules, i.e. logics whose group-theoretical disjunction and conjunction connectives are not commutative. These logics pose tricky technical problems which by now, however, are beginning to find acceptable solutions. Some of the current work into noncommutative logics is reported in Appendix B; see also Abrusci and Ruet (2000), Bayu Surarso and Ono (1996), Ono (1999). The classical constraint. We shall focus on logics with an involutive negation, disregarding systems with minimal or intuitionistic negations.

xii Subintuitionistic logics are briefly surveyed in Appendix B, where the interested reader will find appropriate references to the literature. The -constraint. Although we shall generally consider logics with more than one pair of disjunction and conjunction connectives, in each case at least one such pair will exhibit lattice properties. Logics whose underlying algebraic structures are not lattice-ordered have recently emerged in the context of the "unsharp approach" to quantum logics (see e.g. Giuntini 1996), but the connection between these systems and substructural logics is still unclear. Besides abiding by these constraints, we had to leave out of this book other topics which would have surely deserved attention. For example, we neglected some items which have been exhaustively illustrated in the handbook by Restall - e.g. natural deduction, the Curry-Howard isomorphism for substructural logics, the semantics of proofs. Other important references for this constructive approach to our subject are Girard et al. (1989) and Wansing (1993). We shall spend nothing but a few words on Gabbay's approach to substructural logics in the framework of labelled deductive systems (Gabbay 1996), which represents one of the most innovative perspectives in contemporary logical research. Dunn's gaggle theory and Urquhart's inquiry into the feasibility of the decision problem for substructural calculi (Urquhart 1990) have been passed over as well, except for some occasional mentions.

4. ACKNOWLEDGEMENTS Our first heartfelt thanks obviously go to Ettore Casari, who first introduced us into logic in the mid-eighties, and into substructural logics, some years later. Studying and working under his guidance has been one of the luckiest opportunities we had throughout our scientific iter. His work on pregroups and comparative logic was, needless to say, a main source of inspiration for the general framework underlying the present book. We also thank Ettore Casari for consistently supporting in many ways the project of this volume. We are greatly indebted to Daniele Mundici for his encouragement and his invaluable suggestions, as well as for putting us in contact with his dynamic and stimulating research group. We gratefully acknowledge the friendly support and help provided by Roberto Giuntini and Maria Luisa Dalla Chiara. We feel extremely grateful to Heinrich Wansing, who supported this enterprise - from its very beginning - more than one could have asked for; to André Fuhrmann, who first led us into the territories of relevance logics; and to Pierluigi Minari, whose papers and oral remarks helped us to understand many things concerning these topics.

xiii Several people read portions of the manuscript and suggested precious improvements: among them, let us mention with immense gratitude Ettore Casari, Agata Ciabattoni, Enrico Moriconi, Hiroakira Ono and Heinrich Wansing. We also thank Matthias Baaz, Antonio Di Nola, Steve Giambrone, Sandor Jenei, Edwin Mares, Bob Meyer, Mario Piazza, Greg Restall, Giovanni Sambin, Harold Schellinx, John Slaney, Richard Zach, who answered questions, provided insights or discussed with us (orally or via e-mail) about relevant issues. Finally, we want to express our gratitude to an anonymous referee, for his/her precious remarks, and to Tamara Welschot and the editorial staff of the series Trends in Logic for their kind and competent assistance.

PART ONE THE PHILOSOPHY OF SUBSTRUCTURAL LOGICS

Chapter 1 THE ROLE OF STRUCTURAL RULES IN SEQUENT CALCULI

1. THE "INFERENTIAL APPROACH" TO LOGICAL CALCULUS Substructural logics owe their name to the fact that an especially immediate and intuitive way to introduce them is by means of sequent calculi à la Gentzen where one or more of the structural rules (weakening, contraction, exchange, cut) are suitably restricted or even left out. We do not assume the reader to be familiar with the terminology of the preceding sentence, which will be subsequently explained in full detail - but if only she has some acquaintance with the history of twentieth century logic, at least the name of Gerhard Gentzen should not be completely foreign to her. Gentzen, a German logician and mathematician who is justly celebrated as one of the most prominent figures of contemporary logic, introduced both natural deduction and sequent calculi in his doctoral thesis Untersuchungen über das logische Schliessen (translated into English as Investigations into Logical Deduction: Gentzen 1935). In a sense, as we shall see below, Gentzen can also be considered as the founding father of substructural logics (Došen 1993). Any investigation concerning this topic, therefore, cannot fail to take Gentzen's Untersuchungen as a starting point. And so shall we do. Gentzen describes as follows the philosophical motivation that led him to set up his calculus of natural deduction (p. 68): The formalization of logical deduction, especially as it has been developed by Frege, Russell, and Hilbert, is rather far removed

4

Substructural logics: a primer from the forms of deduction used in mathematical proofs [...]. In contrast, I intended first to set up a formal system which comes as close as possible to actual reasoning.

Natural deduction, according to Gentzen, has thus a decisive edge over Hilbert-style axiomatic calculi: its formal derivations reflect more closely some concrete structural features of informal mathematical proofs - most notably, the use of assumptions. But there is a further epistemological gain which can be achieved by resorting to a system of natural deduction. In the words of Haskell B. Curry (1960, pp. 119-121): In his doctoral thesis Gentzen presented a new approach to the logical calculus whose central characteristic was that it laid great emphasis on inferential rules which seemed to flow naturally from meanings as intuitively conceived. It is appropriate to call this mode of approach the inferential approach [...]. The essential content of the system is contained in the inferential (or deductive) rules. Except for a few rather trivial rules of special nature, these rules are associated with the separate operations; and those which are so associated with a particular operation express the meaning of that operation.

The outstanding novelty of Gentzen's standpoint, according to Curry, is thus a completely new approach to the issue of the meaning of logical constants. In axiomatic calculi, logical operations are implicitly defined by their mutual relationships as stated in the axioms of the system. No separate, operational meaning is ascribed to them. In the calculus of natural deduction, on the other hand, the emphasis is on laying down separate rules for each constant - rules which can be taken to express the operational content of logical symbols. In this way, any commitment to a holistic theory of the meaning of logical constants is avoided. It can be reasonably conjectured that this was the viewpoint of Gentzen himself, since he explicitly observed (p. 80): The introductions represent, as it were, the "definitions" of the symbols concerned, and the eliminations are no more, in the final analysis, than the consequences of these definitions.

We shall not dwell, for the time being, on this distinction between the respective roles of introduction and elimination inferences (but we shall return on this point). Suffice it to say that this fleeting remark by Gentzen was subsequently taken up and extensively developed by Dummett, Prawitz, Tennant, Schroeder-Heister and others, who started off a prolific trend of investigations into the relationships between natural deduction calculi and the

Francesco Paoli

5

meaning of logical constants (see Sundholm 1986 for detailed references on this topic). So much for the philosophical significance of natural deduction. What about sequent calculi? Gentzen seemed, prima facie, to award them a merely instrumental role, as these calculi appeared to him nothing more than an "especially suited" framework to the purpose of proving his Hauptsatz, a result whose importance we shall discuss at length1 . Looking in hindsight, however, we can legitimately say that Gentzen underestimated the philosophical status of his own creature, and that some issues concerning the meaning of logical operations can be framed and discussed in the context of sequent calculi just as well as (if not better than) in the context of natural deduction. Well: we believe that by now the curiosity of the reader should have been sufficiently aroused and that a presentation of the calculus can no longer be deferred.

1.1. Structural rules, operational rules, and meaning Gentzen's calculi LK (for classical logic) and LJ (for intuitionistic logic) are based on a first-order language; however, since the focus of this book is on propositional logic, we shall confine ourselves to their propositional fragments. Henceforth, then, by LK (LJ) we shall mean propositional LK (LJ). We shall now take on a slightly more formal tone for a short while, in order to state some definitions which will turn out useful throughout the rest of this volume.

Definition 1.1 (language of LK). Let £0 be a propositional language containing a denumerable stock of variables ( and the connectives , and . We shall use as metavariables for propositional variables. Formulae are constructed as usual; will be used as metavariables for generic formulae.

Definition 1.2 (sequents in LK). The basic expressions of the calculus are inferences of the form (read: "follows", or "is derivable from" ), where and are finite, possibly empty, sequences of formulae of £0 , separated by commas. Such inferences are called sequents. and are called, respectively, the antecedent and the succedent of the sequent.

! " #$ %& !

According to Gentzen, the sequent has the same informal meaning as the formula . This means that the comma must be read as a conjunction in the antecedent, and as a disjunction in the succedent, while the arrow corresponds to implication2 .

6

Substructural logics: a primer

Definition 1.3 (postulates of LK). The postulates of the calculus are its axioms and rules. Intuitively speaking, the rules encode ways of transforming inferences in an acceptable way, i.e. without perturbing the derivability relation between the antecedent and the succedent. More precisely, they are ordered pairs or triples of sequents, arranged in either of these two forms:

The sequents above the horizontal line are called the upper sequents, or the premisses, of the rule; the sequent below the line is called the lower sequent, or the conclusion, of the rule. Rules, moreover, can be either structural or operational3 . Here are the postulates of LK:

Axioms Structural rules Exchange

Weakening

Contraction

Cut

"!$#

Francesco Paoli

7 Operational rules

Definition 1.4 (principal, side, and auxiliary formulae). In all these rules, the formula occurrences in are called side formulae; the formula occurrence in the conclusion which is not a side formula is called principal, and the formula occurrences in the premisses which are not side formulae are called auxiliary. Definition 1.5 (proofs in LK). A proof in LK is a finite labelled tree whose nodes are labelled by sequents, in such a way that leaves are labelled by axioms and each sequent at a node is obtained from sequents at immediate predecessor(s) node(s) according to one of( the rules of LK. We shall denote !"# ! %$&$'$ ! ! ( proofs by means of the metavariables If is a proof, a subtree of ) ) which is itself a proof is called a subproof of . * A sequent is provable in LK (or LK-provable, or a theorem of LK) iff it labels the root of some proof in LK (i.e., as we shall sometimes say, iff it is the endsequent of such a proof). Definition 1.6 (sequents, postulates and proofs in LJ). The calculus LJ has the same language as LK, and all the concepts introduced in the Definitions 1.2-1.5 apply to it as well, with two sole exceptions. A sequent in LJ is an + , + , expression of the form , where and are finite, possibly empty, , sequences of formulae of £0 and can contain at most one formula. The rules given for LK, therefore, must be adapted accordingly. Definition 1.6 yields an immediate consequence as regards structural rules: the rules ER and CR have to be deleted from LJ, for they can only be applied to sequents with more than one formula in the succedent, while the rule WR

8

Substructural logics: a primer

must be restricted to the case where is empty. Keeping in mind the characterization of substructural logics that we suggested at the outset, the reader is now in a position to understand why we remarked that Gentzen can be reputed, broadly speaking, the first substructural logician. However, it must be noticed that, by suitably tinkering with the rules of the calculi, it is possible to build up multiple-conclusion versions of LJ (Curry 1939; Maehara 1954) and single-conclusion versions of LK (Curry 1952), although these variants are surely less elegant and more cumbersome than their counterparts. Is then the characterization of intuitionistic logic through the above-mentioned restriction on succedents a mere technicality, designed to the sole purpose of getting a manageable calculus and devoid of any philosophical significance? Not quite. We shall see how a profound epistemological meaning can be attached to it4 . Deferring until then any further reflection on the difference between LK and LJ, let us instead pause for a while on the distinction between structural and operational rules, a distinction which is common to both calculi. First, let us consider the latter group of rules. Like in the calculus of natural deduction, we have a pair of rules for each connective. However, while in that case we had an introduction rule and an elimination rule, here we are in the presence of two introductions - a rule for introducing the connective in the antecedent and a corresponding rule for introducing it in the succedent. This is because Gentzen intended to set up a calculus where nothing "was lost" in passing from the premisses down to the conclusion of each inferential step - and it is obviously hard to reconcile elimination rules with such a desideratum. Now, remember what Gentzen had to say about the role of introduction rules in a natural deduction setting: they give the operational meaning of the logical constant at issue. It can be supposed that Gentzen assigned a similar function to the introductions of his sequent calculi (see Hacking 1979 for an argument in defence of such a conjecture). However, a striking analogy and correspondence between introductions, respectively eliminations in natural deduction and right introductions, respectively left introductions in sequent calculi was soon noticed (see e.g. Sundholm 1983 for details). In the light of this, it is possible that Gentzen would have been reluctant to award his left introductions the status of meaning-giving rules. Be it as it may, we can safely assume that Gentzen viewed his operational rules (whether all of them, or the right introductions only) as means of specifying, entirely or in part, the "meaning" or "content" of logical symbols. The status of structural rules is less clear. They are so called since they do not introduce any logical symbol into discourse, but are concerned with the manipulation of the structure of sequents. In LK, they come in left/right pairs as well, with the exception of the cut rule. Gentzen characterizes them as

Francesco Paoli

9

follows (p. 82): Even now new inference figures are required that cannot be integrated into our system of introductions and eliminations; but we have the advantage of being able to reserve them special places within our system, since they no longer refer to logical symbols, but merely to the structure of the sequents.

After remarking this, however, he does not dwell any longer on this subject. As a consequence, if we want to understand better the role of structural rules in Gentzen-style calculi, we have to take a quick look at more recent papers on the philosophy of proof theory. In primis, we may wonder whether also structural rules have a meaninggiving role, i.e. whether they contribute to define the meanings of the constants introduced by the operational rules. Should we subscribe to the holistic viewpoint, there would be no doubt: if the meaning of the logical constants is implicitly given by the whole body of postulates of a system, then structural rules cannot be denied a meaning-giving function. As already remarked, however, such a viewpoint is irreconcilable with the very spirit of Gentzen's enterprise, whose aim is to provide each connective with a separate operational content - whereas on the holistic conception the meaning of each constant would also depend on the introduction rules for other constants. If we accept Gentzen's "inferential approach", then, two alternatives open up: either we assume that each connective has both an operational content, given by its introduction rules, and a global content, specified e.g. by what sequents containing that connective are provable in the system, or else we deny such a dichotomy. Partly depending on the answer given to such a question, we can distinguish at least four theories about the relationships between structural and operational rules in a sequent calculus. We shall list them according to the importance awarded to structural rules, in increasing order. 1) The nihilistic view (Negri and von Plato 2001). The sole meaning attached to a connective is its operational meaning, given by the operational rules. Structural rules correspond to rules concerning the discharge of assumptions in natural deduction; they are closely tied to the particular formalism chosen, and have therefore no meaning-giving role. 2) The ancillary view. It is not easy to credit such a view to any particular author, but Wansing (2000) quotes it as a widespread belief in current prooftheoretic semantics. According to it, connectives have both an operational and a global content, and operational rules are not sufficient to characterize the

10

Substructural logics: a primer

latter: the assistance of structural rules is needed. The global meaning of intuitionistic implication, for instance, depends both on its introduction rules and on the structural rules of the calculus for intuitionistic logic. 3) The dualistic view (Hacking 1979). In this perspective, the roles of operational and structural rules are kept quite separate. While operational rules give the meanings of connectives, structural rules "embody basic facts about deducibility and obtain even in a language with no logical constant at all" (Hacking 1979, p. 294). Structural rules, therefore, have to be postulated for atomic formulae and proved to hold for complex formulae containing logical symbols. A definition of a logical operation through introduction rules is a good definition only if it is not "creative", i.e. if it does not affect the facts about deducibility that obtain for the original "prelogical" language. 4) The relativistic view (Došen 1989a). The starting point of this approach is the idea that logical constants make explicit in a language of lower level some "structural features" of a language of higher level, formulated therein by appropriate "punctuation marks" (e.g. different ways of bunching the premisses together). For example, the formula reflects in the lower language the structural truth (" is deducible from "). Operational rules, in such a context, are simply translation rules from the higher language to the lower one. On the other hand, structural rules, which encode ways of manipulating the structure of sequents at the higher level, are what makes the real difference between the various systems of logic. Girard (1995, p. 11) supports an extreme version of such a view. He says that "the actual meaning of the words 'and', 'imply', 'or' is wholly in the structural group and it is not excessive to say that a logic is essentially a set of structural rules".

For the sake of completeness we quote two more viewpoints, indeed similar to each other, concerning the meaning-giving status of operational rules, though they do not directly bear on the issue of the role of structural rules.

5) The underdetermination view, first version (Belnap 1996). The operational rules of LK are not selective enough: a rule like R, for instance, says something not only about the meaning of conjunction, but also about the meaning of the comma and of . Therefore, one has to find systems where it is possible to "display" any part of a sequent, i.e. to make it the whole antecedent or the whole succedent of an equivalent sequent5 .

6) The underdetermination view, second version (Sambin et al. 2000). The meaning of a connective "is determined also by contexts in its rules, which can

Francesco Paoli

11

bring in latent information on the behaviour of the connective". It is then desirable that the rules of a system satisfy the requirement of visibility (similar to the above-mentioned property of display calculi): in such rules, there have to be no side formulae on the same side of either the principal, or the auxiliary formulae6 . The previous remarks about the nature of structural rules and their places within sequent calculi like Gentzen's LK or LJ can suffice for the moment. Now it is about time to see structural rules at work. The next section will be devoted exactly to this.

1.2. Discovering the effects of structural rules After Gentzen introduced his sequent calculi, it did not take long until some noteworthy effects of structural rules were discovered. In 1944, the Finnish logician Oiva Ketonen suggested a new version of LK where the rules L, R and L were respectively replaced by:

Ketonen devised these modifications in order to prove an "inversion theorem" for LK: in the new version, as Bernays (1945, p. 127) observes,

All the schemata by which the propositional connectives are introduced [...] can be inverted - i.e., the passage from the conclusion of each one of these schemata to its premiss or premisses can be accomplished by applying the other schema belonging to the same connective, together with the StrukturSchlussfiguren [structural rules].

A more refined version of Ketonen's result would have been proved some years later by Schütte (1950). It is nearly immediate to see that Ketonen's system is equivalent to LK, and that in proving such an equivalence an essential role is played precisely by the structural rules of weakening, contraction, and exchange. In fact, it is not difficult to see that the rule L' is derivable in LK:

12

Substructural logics: a primer

Conversely, the two halves of L are derivable given

L' and the rest of

LK:

# " "# ! ! $ L and $ L' is proved similarly. Finally, let us see The equivalence of % how L' can be derived in LK: 1 2435 % && ' % ( *,+ - *,./- 0,+ - 0,. ) 6

and how

L can be derived given

6

L' and the rest of LK:

< A = B C > A ? DBE 1 7#38 7#98 :38 :95 A ='' > A ='B ? 1 7#38 7#98 :38 :95 B ? B C DB& A ='B ? CGF'DB& and 4;S >@?A9 . Definition 2.3 (some conventions about sequents). Throughout this chapter, we shall adopt the same definitions and conventions about sequents that we stated in Definitions 1.1-1.6. With one notable exception, however: capital Greek letters will not stand for sequences of formulae of the language at issue, but for multisets of formulae of such language. Multisets can be rigorously defined (see e.g. Troelstra 1992, p. 2), but this is not necessary in our context: suffice it to say that multisets are aggregates where the ordering of the elements does not matter (whereas it matters for sequences), but their multiplicity does (while it does not for sets). So, for example, {9CBD> } is the same multiset as

44

Substructural logics: a primer

{ }, but { } is not the same multiset as . As a rule, outer brackets will be omitted: as it is customary to do, we shall write in place of the more correct .

2. SEQUENT CALCULI 2.1 The calculus LL It is now time to come to the heart of the matter, and present our basic sequent calculus. Definition 2.4 (postulates of LL). The calculus LL, based on the language £1 , has the following postulates: Axioms

Structural rules

! " #%$'&)(* Operational rules

, + # ,.- * 0+ #1 - * 2130+ 4! 0! 2530 ! 4+ 0+ 7680 + 7680 + 4! 0+ :980 + # 9

+ ! #5 - *

, # ,./ *

0 1 213 0 # / * 40 2530 # 5 / * 0 6 / # 6 - * 7680 # * 0 - * 79;0 7980 # 9 / *

Francesco Paoli

45

Notice that LL contains "covert" exchange rules: using multisets instead of sequences, we are allowed to perform arbitrary permutations either in the antecedent or in the succedent. Beside such rules, the only explicit structural rule of LL is the cut rule.

#) $)# '& ! %)#

( )# #" $"%# "#' & &*# $"#"+

("% , '&#"# # )# &*# $"#)+ / & 0 . ' . * .

* &

* ' . . #)#

&16'3 &1.' #.')## "$7.2' & -$3434 . # "#

'&1.'#$$)$3584 .2

:8 9 -:3484 :84

63 $84 63 '&$84 $ 34

68 $34 %!" $ -3 84 - 84' & %"#

68 ##"#) - 34 '& #)# 763 #)## " -34 %"#.

$63 34

#'"#&1 .' $8;.2 #)#.

6$8 84 #

)#'&17. 0 4 2 56.7,298@?A :"A 56.B*+,298;:=< , C * * , C *'+, -/.1 * , > 0 4 2 56.7*29856.7,298;:D< 56.7 * +,298;:=

;

Proposition 2.12. (i) LAB = LA ; (ii) LA is trivial.

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Substructural logics: a primer

; (F94) ; (F95) ; (F96) ; (F97) ; (F98) , if is any theorem of HA. Moreover, F1 becomes superfluous and R2 is derivable. ! Proposition 2.28 (theorems of#HC). The following theorems are provable in " # HC: ( )"*#&+# ;; (F101) $ (F99) % ; (F102) &;'(F100) ; (F103) (F104) , -, .# ; (F105) - , , &/# . Moreover, F1 becomes superfluous. ! Proposition 2.29 (theorems of HG). The following theorems in ( +are; provable F28, F87, (F106) (F107) HG: 0"1+# ; (F108) #2 0; (F109) 1 0; (F110) 0. ! . The following theorems are Proposition 2.30 (theorems of HRM ) ) * " 3 2 # ; (F112) provable F90, 4 )"*#in2HRM "*#: ;F89, )(F111) "*#2

"*# . ! (F113) Proposition 2.31 (theorems of HRM). The following theorems are provable , ' in HRM : F100, F102, -, (F114) ; (F116) - , ; 3(F115) . ! ND

ND

2 ! Proposition 2.33 (theorems of HLuk ). The following theorems are 0"*# provable 5"5"in HLuk : ; (F118) 0"*# (F119) "60"*' . ! Proposition 2.34 (theorems theorems 7 ). The # ; following are 2 7 provable 7 ; % 8 of,HL 7 7 7 (F121) in HL : ((F120) ; (F124) 7 7 = > ?7 @A92 7 > BC (F122) 7 %'9: ; ; (F125) 0 7 9 ? 7 < ;; (F123) @ >??0?@D ? => ?@A ; (F128) ? =E?> (F126) ?0?@A9 ? ? => ?@3AGF !? ? ; (F127) Proposition 2.32 (theorems of HLuk). The following theorems are provable in HLuk: F28, F114, F115, F116, (F117) . 3

3

E

E

Francesco Paoli

73

3.2 Derivability and theories In classical propositional logic, a (syntactic) theory is a set of formulae which contains the classical propositional axioms and is closed under modus ponens8 . In our substructural context, we need to draw some finer distinctions which the classical setting obliterates. Therefore, we set off with the following Definition 2.27 (S-theory). Let S be any of the previously introduced axiomatic calculi. An S-theory is a set of formulae of the appropriate language s.t. (i) if and S , then ; (ii) if , then .

An S-theory, therefore, need not contain any of the axioms of S: all that is required is that it be closed under adjunction and that it contain the consequent of an S-provable implication whenever it includes its antecedent. Next, we consider some "well-behaved" kinds of theories.

Definition 2.28 (some special kinds of S-theories). An S-theory

is said to

be:

regular, iff it contains all of the axioms of S;

detached , iff ; only if ; -consistent , iff -consistent for some ; simply consistent , iff it is -consistent, iff for no , both and ; -complete, iff for every , either or simply complete, iff it is -complete for every ; ; -complete , iff for every , either or

prime, iff

only if

or

;

.

To introduce suitable concepts of maximality for our S-theories, we need a preliminary definition.

"! # *,+.-0/1'2 '( 4

%& $ 3 % &

be a set of formulae. The SDefinition 2.29 (S-theory of sets). Let theory of (in symbols: ) is defined as is an S-theory and S . Moreover, by we mean the set is a regular, S detached S-theory and . From now on, we shall drop the subscript "S" wherever no danger of confusion is impending.

' ( 4 )

Definition 2.30 (maximal S-theories). An S-theory

is said to be:

74

Substructural logics: a primer

-maximal, iff it is -consistent but, for any

is not such; -maximal, iff it is -consistent but, for any is not such; weakly maximal, iff it is simply consistent but, for any

is not such; is maximal, iff it is simply consistent but, for any

weakly

not such. Classically, few of these distinctions make sense. As we shall see, indeed, any HK-theory is both regular and detached; hence, any weakly maximal HKtheory is maximal. Moreover, it is well-known that the two notions of simple consistency and -consistency, as well as the four notions of simple completeness, -completeness, primality and maximality, are classically equivalent to one another. The next few lemmata are devoted to establishing some of these relationships also for our substructural calculi.

Proposition 2.35. For any HL-theory : (i) if it is regular and detached, condition (i) of Definition 2.27 is redundant; (ii) if it is -consistent, it is simply consistent; (iii) if it is simply complete, it is -complete; (iv) if it is detached, -consistent and -complete, it is prime; (v) if it is maximal, it is weakly maximal.

! %+*, ( - " "$#&%'#&() %/.0"- (/.1"- 23%4.1 "657823(/.1"95:- %+*,( .0"- "- ; -theory : (i) if it is nonempty, it is detached; Proposition 2.36. For any HR < (ii) if it is regular and weakly -maximal, it is prime; (iii)< if it is regular and prime, it is = -complete; (iv) if it is regular and weakly -maximal, it is = complete. ?^ ad\OSM_ b;\OSMR bc>^^eb;\OS;_ fgSMR bc>^ h i jFk aAS;_ fgSMR bc l;mglX{ n"lop q r st . Finally, suppose that u-s(v wyxzl was and Q and |-}(~ by an application of I. Then, obtained from |}(~ and ~ M , for some by induction, Qbe U|VX " } . Let~ M zM yz ; using F74, we get: 3

5

w HL

w HL w HL

w HL

w HL

w HL

w HL

w HL

80

Substructural logics: a primer

!" #%$&#(4 '#*),+.-!/021 . and 3 57698 : then 8 ;=;2@BADC!E62F . We shall show how to replace ; I by 6HG , by inductionI on the 6HG ?@ ACBD ACEFHG .46 0 3 /24 1

Why, a reader could ask, did we introduce such a complicated and convoluted inference pattern as the mix rule, in place of the more natural and intuitively appealing cut rule? There is a reason, indeed, and it has to do precisely with the presence of contraction in LK. We shall explain our move in due course; thus, the curious reader is begged to wait patiently until § 1.2. What we shall do, for the time being, will be to prove a cut elimination theorem for LKM . To achieve this goal, we need a number of auxiliary notions. First of all, the concept of "mixproof" will permit us to focus on a quite small subset of the set of all proofs in LKM which contain one or more applications of mix1 .

I

Definition 3.2 (mixproofs and mix-free proofs). A proof in LKM is called a mixproof iff it contains just one application of mix, whose conclusion

Francesco Paoli

89

is the endsequent of the proof; it is called a mix-free proof iff it contains no application of mix at all.

Proposition 3.2 (circumscription of cut elimination). In LKM , if any mixproof of can be transformed into a mix-free proof of the same sequent, then any arbitrary proof of can be transformed into a mixfree proof of the same sequent.

in LK . Take the leftmost Proof (sketch). Let be any proof of be its conclusion. The and uppermost application of mix in , and let subproof of whose endsequent is is a mixproof which can thus

be turned into a mix-free proof of . Now consider the result of replacing in by , call it , and take the leftmost and uppermost application of mix in . By repeating this procedure as many times as there are applications of mix in , we get the required transformation. The details are left to the reader. lemma, it will suffice to show that any mixproof ofintheLKpreceding ofIn virtue can be turned into a mix-free proof of the same sequent M

M

M

in LK . To do so, we shall argue by induction on a special parameter, to be specified presently. Definition 3.3 (rank of a sequent in a mixproof). Let whose final inference is:

be a mixproof

and is# so defined: The rank of the sequent in is denoted by " If belongs to the subproof ! of ! whose endsequent is % $ , is the of an upward of sequents &(> '*)+maximal ),),'-& ? s.t. length &/> .0& (diminished &132547by684:one)9 J ? L @E E@& K ? M B@ N OPQ R H ? I @A J1@ K ? L)@ M B@C N S5TR J1@& K ? I!@) L)@ M AGF!B@#H#@& U

If was contained in the left premiss of the L inference, we argue analogously. 3.3) Transformations T10 and T11. This subcase of the inductive step, however, needs no longer to be treated separately. Now we can simply push the cut upwards, cutting away one of the side occurrences of . For example:

V

W W VYX[ Z X \ ` a V ! U ] X ] N S5dR V ! ` Z a X\V U!] ^X V ! b Z c U ] U ] X N OPQ Rfehj gi `#X_ b Z a)X! c X\V U!] > > kYl[ r m s l^kon!plp kon!pql t m u v w xy z kYl#r0l t m s!l! u lp v {5|z r#l t m s)l! u l^kon!p

The previous discussion provides a hint for the proof of the next proposition, whose details are left to the interested reader:

}

~

Proposition 3.4 (cut elimination theorem for LA: Grishin 1974). Any cutproof of in LA can be turned into a cut-free proof of the same sequent in the same calculus.

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1.3 Cut elimination for calculi without the weakening rules

When proving Proposition 3.3, we had to introduce the mix rule in order to cope with cases involving contraction. Mix, however, is a rather "brutal" inference rule: if the premisses of the relevant application of mix are and , it makes a clean sweep of any occurrence of the mixformula in . In LKM this is not a problem: if we need to reintroduce some of the deleted occurrences, we can do so by means of weakening moves. Yet in systems like LRND , where the weakening rule is not available, this is not possible. Hence, we need a more selective rule than mix - an "intelligent" mix, so to speak.

Definition 3.8 (postulates of LRNDM ). The system LRNDM is exactly like LRND , except for the fact that the cut rule is replaced by the following rule (called intelligent mix, or intmix):

Here, both and contain , and ( ) is obtained from ( ) by deleting at least one occurrence of in it. Again, when there is no danger of ambiguity, we shall drop the subscript " ". The formula is called intmixformula.

Proposition 3.5 (equivalence of LRNDM and LRND ). . LRNDM

LRND

iff

Proof. We only need to show that intmix is equivalent to cut. The derivability of intmix in LRND is shown as in Proposition 3.1. The derivability of cut in LRNDM is trivial, as cut is nothing but a special instance of intmix.

Now, to obtain a cut elimination theorem for LRNDM , it is sufficient to adapt the definitions and proofs of § 1.1 as follows. 1) All the definitions containing the word "mix" must be pruned in the obvious way. Thus, for example, instead of "mixproof" and "mix-free proof" we shall have to speak of "intmixproof" and "intmix-free proof". 2) The proof of Proposition 3.3 must be adjusted in an appropriate way. In particular, we have to show that all the uses of weakening in such a proof can be dispensed with. Weakening has been employed in four subcases, exemplified by the following transformations: 2.1) Transformation T6, which now becomes:

98

Substructural logics: a primer

5

2.2) Transformation T8, which now becomes:

"! 6 # (

$

$ % &

'

)

2. ) Transformation T , which now becomes:

/

/ 0 1

314 6 2 7 13 8 2 9 ? @A ; 01 6:14 8 2 71 9 6:15 8=1 ; 2 7>1 J%M A R HF @ LQ K6R MOJ?VJ' @ ? I > JKSR M T I > JU

Francesco Paoli

! " # $& %$& %$& $& ! '($& / 0 1 0 +&45 2 )! *(+&,- 1 . 0 3 2 1 . 0 - * 2 1 . 0 -, +&65 2 1 . 0 7 45 2 1 . 0 - *%+&, 2 1 . 0 - *%+&, 3 2 7 - 1 . 0 3 < 8 =B< A@ 8 7 =?7 >@ 9 D / : C C < =B< >@ 9 : ;

123

Remark 4.4 (on the rules of LLuk3 '). In LLuk3 ', structural rules are split into two groups: external rules, by which whole components are added or deleted in a hypersequent, and internal rules, acting on formulae within each component. Notice the lack of internal contraction rules, which justifies the inclusion of three-valued Lukasiewicz logic among substructural logics. The operational rules of LLuk3 ' are the same as in LL, with side sequents added. Remark 4.5 (An alternative axiomatization of LLuk3 '). Ciabattoni et al. (1998) suggested an alternative axiomatization of LLuk3 ', where the rule Mx is replaced by the following, simpler rule:

E D"FH C G KJF L I MNFH C G OF L E !I DPFJM G KJF O C G L QSRUTW V

124

Substructural logics: a primer

The informal meaning of hypersequents in LLuk3 ' is emphasized by the next definition and lemma.

Definition let 4.6 (formula-translation be a hypersequent,ofandafor hypersequent). Let , as defined in Definition be the formula-translation of the component 2.5. The formula-translation ! of the hypersequent is the formula . # , . &%('')'*%+ - / # , / . "$ 0 iff 4.3 (meaning of hypersequents in LLuk '). 0 Proposition #1! . Proof. To avoid notational we consider the simple example of 2 redundancies, hypersequents of the form #43657#48 ; the general case is left up to the reader. From left to right: 9;:4 :@?;A4 M NPORQ S TVU W XZY :EDF?;A4IGJ :KDF?;A4IG M [FORQ S TVU W XZY M \"]RY :KDF?;A4IG ^7_ ?;A4< ; 2) ` _ =7AC> ; 3) a _;^ HI` . From right to left, let 1) b Let moreover be the following proof: f d fKg d g h ikjml g d g f d g6 g d f h nFoRl c f d f e7d g6 g d f h nFoRl c d4e e7d g e7d f h pVqr l dCe ed g d f h pVqr l d g d f 3

LLuk3'

LLuk3'

b w t4wKx6t4x CsB t4u +s vRw7t4x } ~PV } V y t y{z t z } ~PV u|v y t z tCu6 w7t4x } V y t z w7t4x Proposition 4.4 (completeness of LLuk ': Avron 1991a). " ! . The result we are after can be obtained thus:

3

HLuk3

LLuk3'

iff

Francesco Paoli

125

Proof. We proceed as in the proof of Proposition 2.49. From right to left, then . This is we prove that for any formula , if HLuk3 LLuk3' HLuk done, of course, by induction on the length of the proof of in 3 . The desired conclusion follows then from Proposition 4.3, upon considering formulae of the form

! " ! # $ $ % $ $ & ' ( )+* ( ) ( ) ( (-,8 (.(/8 ,( 9 :;!< ,0(1,(28 (435(/, 9 =>?< ,0(2(48 35(/, * 9 @BA?< 0 , 2 ( / ( , ( 6 ( )* ( ) (-,(28 (/, * ( 9 CDE< 9 FGA?< ( ))+* *( ) ) (-,(18 ,8 ( )+)+* * ( ) (-,( 9 FGDE< ( ( (-),* (7( ) ( (/,( 9 =?>?< ( ) ( * (-) ,( ) 9 FGA?< , ( ( ( ( H I As an example, we prove the hypersequent

((

)

)

.

)

(

)

(

)

( )

(

((

)

)

)

In the opposite direction, the proof proceeds by induction on the length of the proof of in LLuk3 ', and is omitted. LLuk3 ' is a cut-free calculus. Indeed, by using a rather complicated method (the "history" method, necessary to deal with the case where one of the premisses of the relevant cut is obtained by external contraction)2 , it is possible to prove:

I

Proposition 4.5 (cut elimination for LLuk3 ': Avron 1991a). LLuk3 ' is cutfree. As we hinted earlier, also RM and its "cousin" RMI have been given by Avron cut-free hypersequential formulations. Here they are: Definition 4.7 (postulates of LRMI: Avron 1991b). The hypersequent calculus LRMI, based on the language £1 , has the same postulates as LLuk3 ', except that: The internal weakening rules, WL and WR, are replaced by the following internal contraction rules:

J

126

Substructural logics: a primer

(so that, The rules L and R must abideby the restriction ! and #)" for example, we cannot conclude from The mixing rule Mx is replaced by two relevant Mingle rules: - $&% . / $' 354687:9 $0 - %( $) . / 346) . ? * - >/ 2 0 ? -@ . ? / ABDCFEHGJILK

Definition 4.8 (postulates of LRM: Avron 1987). LRM is exactly the same as LRMI, except for the fact that relevant Mingle is replaced by the following combining rule:

M N O N M !O Q N R S T,P R * Q P S AUV:WXK and no restriction is imposed on Y L, Z R.

Remark that the hypersequential version of the anticontraction rule MR, a distinctive postulate of LRMND , is derivable in LRM:

M N[ M N[ M8T 8M Q P\ N T Q P\ ] ^_ `:a M T,P T N Q*P*Q P\P\ ] bL^La T,M P T N Q*P*Q P\P\ ] ^ced ^fLa T Q P\P\

Likewise, ML is also derivable. The cut elimination theorem for both LRMI and LRM was proved by Avron with the help of the history method3 .

Francesco Paoli

127

1.3 Dunn-Mints calculi Hypersequents are a powerful tool for setting up proof systems for several substructural logics. However, they seem of little avail in the case of relevance logics like R or RW. A first step towards giving a proper Gentzen-style formulation of such logics was made by Dunn (1973) - and independently by Mints (1972) - who found a calculus for positive (i.e. negation-free) R. As we remarked back in Chapter 2, the disturbing axiom of HR and HRW is the distribution axiom (F28), whose proof requires, in ordinary sequent calculi, the use of both weakening and contraction. Dunn and Mints overcame this hurdle by dropping Gentzen's tenet according to which the antecedent and the succedent of a sequent are sequences of formulae separated by commas. In their calculi, the formulae occurring in the antecedent of a sequent can be bunched together in two different ways: by means of commas (to be interpreted as lattice-theoretical conjunctions) and by means of semicolons (to be read as group-theoretical conjunctions). The behaviours of these punctuation marks are governed by different structural postulates: weakening, in particular, is available for comma but not for semicolon. This is what makes distribution provable in the system, while still hindering the proof of relevantly 4 unacceptable sequents such as . Let us now present in some detail Dunn's version of the calculus, hereafter labelled LR+ .

Definition 4.9 (£6 -structure). An £6 -structure (henceforth in this subsection, a structure) is inductively defined as follows: Any formula of £6 is a structure; The empty set is a structure; If and are structures, then is a structure; If and are structures, then is a structure.

Definition 4.10 (substructure and substitution). The concept of substructure of a structure is inductively defined as follows: is a substructure of ; Any substructure of and of is a substructure of and of . By , or simply by whenever no confusion can arise, we mean the result of replacing in the indicated occurrence of its substructure by an occurrence of .

Definition 4.11 (sequents in LR+ ). A sequent in LR+ (henceforth in this subsection, a sequent) is an expression of the form , where is a structure and is a formula of £6 .

128

Substructural logics: a primer

In the following presentation of LR+ , the symbol "*" will ambiguously denote both commas and semicolons. For instance, the rule E* actually embodies two different rules, one where stars are replaced by commas and one where they are replaced by semicolons. Definition 4.12 (postulates of LR+ ). LR+ has the following postulates:

Axioms

rules !" Structural " #!" *($+ ) #,.-& %$& '( ) must be nonempty; in the cut rule, '/ denotes the In the rule (W,), result of replacing the indicated occurrence of by if the latter is nonempty, by 1 otherwise. Operational rules 4 ) # 35 21 ) 36 43) 0 21 3) # 1 ) # 785 ) 786 219 878) 878) # $ ) # : 143 . /=;$< 0 > 573 $ ; < / , . / , >< / . . / >< <?A@8/ . . 0 B 143

Preface .......................................................................................................ix Part I: The philosophy of substructural logics Chapter 1. The role of structural rules in sequent calculi...........................3 1. The "inferential approach" to logical calculus......................................... 3 1.1 Structural rules, operational rules, and meaning.......................... 5 1.2 Discovering the effects of structural rules..................................11 2. Reasons for dropping structural rules.................................................... 15 2.1 Reasons for dropping structural rules altogether........................15 2.2 Reasons for dropping (or eliminating) the cut rule..................... 17 2.3 Reasons for dropping the weakening rules.................................21 2.4 Reasons for dropping the contraction rules................................25 2.5 Reasons for dropping the exchange rules...................................28 2.6 Reasons for dropping the associativity of comma...................... 30 3. Ways of reading a sequent......................................................................30 3.1 The truth-based reading............................................................ 31 3.2 The proof-based reading........................................................... 31 3.3 The informational reading.........................................................32 3.4 The "Hobbesian" reading.......................................................... 34 Part II: The proof theory of substructural logics Chapter 2. Basic proof systems for substructural logics.......................... 41 1. Some basic definitions and notational conventions................................. 42 2. Sequent calculi....................................................................................... 44 2.1 The calculus LL.......................................................................44 2.2 Adding the empty sequent: the dialethic route............................ 49 2.3 Adding the lattice-theoretical constants: the bounded route........ 49 2.4 Adding contraction: the relevant route....................................... 50 2.5 Adding weakening: the affine route........................................... 55 2.6 Adding restricted structural rules.............................................. 57 2.7 Adding the exponentials............................................................65 3. Hilbert-style calculi................................................................................68 3.1 Presentation of the systems....................................................... 68 3.2 Derivability and theories...........................................................73 3.3 Lindenbaum-style constructions................................................ 81

vi 4. Equivalence of the two approaches.........................................................83 Chapter 3. Cut elimination and the decision problem...............................87 1. Cut elimination...................................................................................... 87 1.1 Cut elimination for LK.............................................................87 1.2 Cut elimination for calculi without the contraction rules............ 94 1.3 Cut elimination for calculi without the weakening rules............. 97 1.4 Cases where cut elimination fails.............................................. 99 2. The decision problem........................................................................... 100 2.1 Gentzen's method for establishing the decidability of LK.........101 2.2 A decision method for contraction-free systems....................... 105 2.3 A decision method for weakening-free systems........................ 106 2.4 Other decidability (and undecidability) results......................... 111 Chapter 4. Other formalisms.................................................................. 115 1. Generalizations of sequent calculi........................................................ 116 1.1 -sided sequents....................................................................116 1.2 Hypersequents........................................................................ 121 1.3 Dunn-Mints calculi.................................................................127 1.4 Display calculi....................................................................... 130 1.5 A comparison of these frameworks......................................... 136 2. Proofnets..............................................................................................137 3. Resolution calculi.................................................................................145 3.1 Classical resolution.................................................................146 3.2 Relevant resolution................................................................. 149 3.3 Resolution systems for other logics......................................... 153 Part III: The algebra of substructural logics Chapter 5. Algebraic structures..............................................................159 1. *-autonomous lattices...........................................................................161 1.1 Definitions and elementary properties......................................161 1.2 Notable *-autonomous lattices................................................ 165 1.3 Homomorphisms, -filters, -ideals, congruences.....................171 1.4 Principal, prime and regular -ideals....................................... 181 1.5 Representation theory............................................................. 186 2. Classical residuated lattices................................................................. 187 2.1 Maximal, prime, and primary -ideals..................................... 188 2.2 Subdirectly irreducible c.r. lattices.......................................... 190 2.3 Weakly simple, simple and semisimple c.r. lattices.................. 192

vii Part IV: The semantics of substructural logics Chapter 6. Algebraic semantics.............................................................. 201 1. Algebraic soundness and completeness theorems..................................202 1.1 Calculi without exponentials................................................... 203 1.2 Calculi with exponentials........................................................209 2. Totally ordered models and the single model property..........................213 3. Applications......................................................................................... 219 Chapter 7. Relational semantics..............................................................221 1. Semantics for distributive logics........................................................... 222 1.1 Routley-Meyer semantics: definitions and results.....................223 1.2 Applications........................................................................... 235 2. Semantics for nondistributive logics..................................................... 239 2.1 General phase structures.........................................................240 2.2 General phase semantics......................................................... 250 2.3 The exponentials.....................................................................252 2.4 Applications........................................................................... 254 Appendix A: Basic glossary of algebra and graph theory........................... 257 Appendix B: Other substructural logics..................................................... 271 1. Lambek calculus...................................................................... 271 2. Ono's subintuitionistic logics.....................................................277 3. Basic logic............................................................................... 281 Bibliography.............................................................................................289 Index of subjects.......................................................................................301

PREFACE

1. AN INTRIGUING CHALLENGE Whoever undertakes the task of compiling a textbook on a relatively new, but already vastly ramified and quickly growing area of logic - and substructural logics are such, at least to some extent - is faced with a baffling dilemma: he can either presuppose a high degree of logical and mathematical expertise on the reader's part, or else require no background at all except for a "working knowledge" of elementary logic. In our specific case, each one of these policies had its own allure. The former strategy promised to speed up the presentation of some advanced topics and to allow a more refined expository style; the latter one, on the other side, would have permitted to reach a wider audience, some members of which might have had the opportunity to study for the first time some elementary, but fundamental results - such as Gentzen's Hauptsatz - directly in the perspective of substructural logics. Teaching logic from this point of view to unexperienced, and presumably still unbiased, students seemed to us an irresistibly intriguing challenge - therefore, we opted for the second alternative. Thus, we assume that the reader of this book has attended an undergraduate course in logic and has a good mastery of the rudiments of propositional logic (Hilbert-style and natural deduction calculi, truth table semantics) and naive set theory. As for the rest, the volume is self-contained and gradually accompanies the reader up to some of the most recent and specialistic research developments in this area. Some prior acquaintance with either predicate logic or algebra is useful, but not indispensable; in particular, the algebraic notions used throughout the book are surveyed in a special glossary (Appendix A). Of course, this book is not meant only for students. The researcher in the field of substructural logics will find plenty of material she can directly exploit and draw from in her research practice.

x It is not easy, it must be confessed, to write a textbook on this subject short after such a wonderful volume as Restall's An Introduction to Substructural Logics (Restall 2000) has been sent to the press. Our intellectual debt towards this work is enormous, as the reader will notice. However, offering a different perspective on a same topic can be valuable, sometimes. Restall's book primarily focuses on natural deduction and display calculi, and on frame semantics. Our viewpoint is somewhat more traditional: we privilege ordinary sequent calculi on the proof-theoretical side, and algebraic models on the semantical side. We believe that readers who are scarcely at ease with the "punctuation mark" proof theory in the style of Dunn, Mints, Belnap, or with frame semantics - especially researchers belonging to substructural schools other than the relevant - could perhaps feel more comfortable in a setting like ours. Thus, we are confident that our book and the one by Restall can profitably integrate and supplement each other. We tried to arrange this book in such a way as to provide a (hopefully) useful tool for readers coming from any substructural tradition (linear logic, Lambek calculus, relevance logics, BCK-logic and contraction-free logics, comparative logic) and from a number of different backgrounds (philosophy, mathematics, computer science, linguistics). It is extremely important, in our opinion, that people from diverse provenances and academic environments, who often tackle the same problems using different jargons and being unacquainted with one another's results, can find a common ground for discussion and mutual interaction. Occasionally, some personal biases of the author - who is a philosophically oriented logician and a specialist of comparative logic - may show up. We hope that this won't happen too often, though.

2. OVERVIEW OF THE CHAPTERS Chapter 1 introduces the topic from both a historical and a philosophical perspective. After discussing the relationships between substructural logics and proof-theoretical semantics, we provide some reasons for dropping some or all of the structural rules in sequent calculi and, finally, we try to find plausible informal interpretations for substructural sequents. Chapter 2 contains a presentation of the main sequent and Hilbert-style calculi for substructural logics, and of their elementary syntactic properties. The cut elimination theorem for substructural sequent calculi is the heart of Chapter 3, where we also illustrate some decision procedures for these systems. Chapter 4 deals with more advanced formalisms, some of which have been introduced rather recently: we cover a few generalizations of sequent

xi systems ( -sided sequent calculi, hypersequent calculi, Dunn-Mints and display calculi) and of natural deduction (proofnets), as well as resolution calculi. Algebraic semantics will be in the foreground in Chapters 5 and 6, where we study the models of substructural logics at first in a purely algebraic perspective, and then linking them to the calculi of the preceding chapters by means of appropriate completeness results. Chapter 7 is concerned with a different kind of semantics, which generalizes Kripke-style semantics for modal and intuitionistic logics. We discuss models for both distributive logics (Routley-Meyer semantics) and logics without distribution (phase semantics). Appendix A provides a crib of elementary algebra, model theory and graph theory for those readers who are unfamiliar with even the most basic notions of these disciplines (we primarily thought of students in philosophy or linguistics, but also in computer science). Its main aim is letting the book be as selfcontained as possible. Appendix B surveys some logics which, regrettably enough, had not received adequate attention throughout the main body of the text.

3. WHAT HAS BEEN LEFT OUT Although we tried to cover as many topics as possible, due to obvious limitations of size we could not help making choices. In order to delimit the bounds of our enterprise, we imposed ourselves four constraints: The propositional constraint. Throughout this book, we shall remain within the boundaries of propositional logic. There exist interesting inquiries concerning quantified substructural logics, or even substructural arithmetic or set theory (see e.g. Meyer 1998), but in our opinion such a work will remain somehow foreign to the spirit of substructural logics so far as the difference between lattice-theoretical and group-theoretical quantifiers is not properly understood. We think that taking a firm grip on such a distinction is, at present, the most important task with which substructural logicians are confronted (a promising start is in O'Hearn and Pym 1999). The commutative constraint. We shall not consider logics without exchange rules, i.e. logics whose group-theoretical disjunction and conjunction connectives are not commutative. These logics pose tricky technical problems which by now, however, are beginning to find acceptable solutions. Some of the current work into noncommutative logics is reported in Appendix B; see also Abrusci and Ruet (2000), Bayu Surarso and Ono (1996), Ono (1999). The classical constraint. We shall focus on logics with an involutive negation, disregarding systems with minimal or intuitionistic negations.

xii Subintuitionistic logics are briefly surveyed in Appendix B, where the interested reader will find appropriate references to the literature. The -constraint. Although we shall generally consider logics with more than one pair of disjunction and conjunction connectives, in each case at least one such pair will exhibit lattice properties. Logics whose underlying algebraic structures are not lattice-ordered have recently emerged in the context of the "unsharp approach" to quantum logics (see e.g. Giuntini 1996), but the connection between these systems and substructural logics is still unclear. Besides abiding by these constraints, we had to leave out of this book other topics which would have surely deserved attention. For example, we neglected some items which have been exhaustively illustrated in the handbook by Restall - e.g. natural deduction, the Curry-Howard isomorphism for substructural logics, the semantics of proofs. Other important references for this constructive approach to our subject are Girard et al. (1989) and Wansing (1993). We shall spend nothing but a few words on Gabbay's approach to substructural logics in the framework of labelled deductive systems (Gabbay 1996), which represents one of the most innovative perspectives in contemporary logical research. Dunn's gaggle theory and Urquhart's inquiry into the feasibility of the decision problem for substructural calculi (Urquhart 1990) have been passed over as well, except for some occasional mentions.

4. ACKNOWLEDGEMENTS Our first heartfelt thanks obviously go to Ettore Casari, who first introduced us into logic in the mid-eighties, and into substructural logics, some years later. Studying and working under his guidance has been one of the luckiest opportunities we had throughout our scientific iter. His work on pregroups and comparative logic was, needless to say, a main source of inspiration for the general framework underlying the present book. We also thank Ettore Casari for consistently supporting in many ways the project of this volume. We are greatly indebted to Daniele Mundici for his encouragement and his invaluable suggestions, as well as for putting us in contact with his dynamic and stimulating research group. We gratefully acknowledge the friendly support and help provided by Roberto Giuntini and Maria Luisa Dalla Chiara. We feel extremely grateful to Heinrich Wansing, who supported this enterprise - from its very beginning - more than one could have asked for; to André Fuhrmann, who first led us into the territories of relevance logics; and to Pierluigi Minari, whose papers and oral remarks helped us to understand many things concerning these topics.

xiii Several people read portions of the manuscript and suggested precious improvements: among them, let us mention with immense gratitude Ettore Casari, Agata Ciabattoni, Enrico Moriconi, Hiroakira Ono and Heinrich Wansing. We also thank Matthias Baaz, Antonio Di Nola, Steve Giambrone, Sandor Jenei, Edwin Mares, Bob Meyer, Mario Piazza, Greg Restall, Giovanni Sambin, Harold Schellinx, John Slaney, Richard Zach, who answered questions, provided insights or discussed with us (orally or via e-mail) about relevant issues. Finally, we want to express our gratitude to an anonymous referee, for his/her precious remarks, and to Tamara Welschot and the editorial staff of the series Trends in Logic for their kind and competent assistance.

PART ONE THE PHILOSOPHY OF SUBSTRUCTURAL LOGICS

Chapter 1 THE ROLE OF STRUCTURAL RULES IN SEQUENT CALCULI

1. THE "INFERENTIAL APPROACH" TO LOGICAL CALCULUS Substructural logics owe their name to the fact that an especially immediate and intuitive way to introduce them is by means of sequent calculi à la Gentzen where one or more of the structural rules (weakening, contraction, exchange, cut) are suitably restricted or even left out. We do not assume the reader to be familiar with the terminology of the preceding sentence, which will be subsequently explained in full detail - but if only she has some acquaintance with the history of twentieth century logic, at least the name of Gerhard Gentzen should not be completely foreign to her. Gentzen, a German logician and mathematician who is justly celebrated as one of the most prominent figures of contemporary logic, introduced both natural deduction and sequent calculi in his doctoral thesis Untersuchungen über das logische Schliessen (translated into English as Investigations into Logical Deduction: Gentzen 1935). In a sense, as we shall see below, Gentzen can also be considered as the founding father of substructural logics (Došen 1993). Any investigation concerning this topic, therefore, cannot fail to take Gentzen's Untersuchungen as a starting point. And so shall we do. Gentzen describes as follows the philosophical motivation that led him to set up his calculus of natural deduction (p. 68): The formalization of logical deduction, especially as it has been developed by Frege, Russell, and Hilbert, is rather far removed

4

Substructural logics: a primer from the forms of deduction used in mathematical proofs [...]. In contrast, I intended first to set up a formal system which comes as close as possible to actual reasoning.

Natural deduction, according to Gentzen, has thus a decisive edge over Hilbert-style axiomatic calculi: its formal derivations reflect more closely some concrete structural features of informal mathematical proofs - most notably, the use of assumptions. But there is a further epistemological gain which can be achieved by resorting to a system of natural deduction. In the words of Haskell B. Curry (1960, pp. 119-121): In his doctoral thesis Gentzen presented a new approach to the logical calculus whose central characteristic was that it laid great emphasis on inferential rules which seemed to flow naturally from meanings as intuitively conceived. It is appropriate to call this mode of approach the inferential approach [...]. The essential content of the system is contained in the inferential (or deductive) rules. Except for a few rather trivial rules of special nature, these rules are associated with the separate operations; and those which are so associated with a particular operation express the meaning of that operation.

The outstanding novelty of Gentzen's standpoint, according to Curry, is thus a completely new approach to the issue of the meaning of logical constants. In axiomatic calculi, logical operations are implicitly defined by their mutual relationships as stated in the axioms of the system. No separate, operational meaning is ascribed to them. In the calculus of natural deduction, on the other hand, the emphasis is on laying down separate rules for each constant - rules which can be taken to express the operational content of logical symbols. In this way, any commitment to a holistic theory of the meaning of logical constants is avoided. It can be reasonably conjectured that this was the viewpoint of Gentzen himself, since he explicitly observed (p. 80): The introductions represent, as it were, the "definitions" of the symbols concerned, and the eliminations are no more, in the final analysis, than the consequences of these definitions.

We shall not dwell, for the time being, on this distinction between the respective roles of introduction and elimination inferences (but we shall return on this point). Suffice it to say that this fleeting remark by Gentzen was subsequently taken up and extensively developed by Dummett, Prawitz, Tennant, Schroeder-Heister and others, who started off a prolific trend of investigations into the relationships between natural deduction calculi and the

Francesco Paoli

5

meaning of logical constants (see Sundholm 1986 for detailed references on this topic). So much for the philosophical significance of natural deduction. What about sequent calculi? Gentzen seemed, prima facie, to award them a merely instrumental role, as these calculi appeared to him nothing more than an "especially suited" framework to the purpose of proving his Hauptsatz, a result whose importance we shall discuss at length1 . Looking in hindsight, however, we can legitimately say that Gentzen underestimated the philosophical status of his own creature, and that some issues concerning the meaning of logical operations can be framed and discussed in the context of sequent calculi just as well as (if not better than) in the context of natural deduction. Well: we believe that by now the curiosity of the reader should have been sufficiently aroused and that a presentation of the calculus can no longer be deferred.

1.1. Structural rules, operational rules, and meaning Gentzen's calculi LK (for classical logic) and LJ (for intuitionistic logic) are based on a first-order language; however, since the focus of this book is on propositional logic, we shall confine ourselves to their propositional fragments. Henceforth, then, by LK (LJ) we shall mean propositional LK (LJ). We shall now take on a slightly more formal tone for a short while, in order to state some definitions which will turn out useful throughout the rest of this volume.

Definition 1.1 (language of LK). Let £0 be a propositional language containing a denumerable stock of variables ( and the connectives , and . We shall use as metavariables for propositional variables. Formulae are constructed as usual; will be used as metavariables for generic formulae.

Definition 1.2 (sequents in LK). The basic expressions of the calculus are inferences of the form (read: "follows", or "is derivable from" ), where and are finite, possibly empty, sequences of formulae of £0 , separated by commas. Such inferences are called sequents. and are called, respectively, the antecedent and the succedent of the sequent.

! " #$ %& !

According to Gentzen, the sequent has the same informal meaning as the formula . This means that the comma must be read as a conjunction in the antecedent, and as a disjunction in the succedent, while the arrow corresponds to implication2 .

6

Substructural logics: a primer

Definition 1.3 (postulates of LK). The postulates of the calculus are its axioms and rules. Intuitively speaking, the rules encode ways of transforming inferences in an acceptable way, i.e. without perturbing the derivability relation between the antecedent and the succedent. More precisely, they are ordered pairs or triples of sequents, arranged in either of these two forms:

The sequents above the horizontal line are called the upper sequents, or the premisses, of the rule; the sequent below the line is called the lower sequent, or the conclusion, of the rule. Rules, moreover, can be either structural or operational3 . Here are the postulates of LK:

Axioms Structural rules Exchange

Weakening

Contraction

Cut

"!$#

Francesco Paoli

7 Operational rules

Definition 1.4 (principal, side, and auxiliary formulae). In all these rules, the formula occurrences in are called side formulae; the formula occurrence in the conclusion which is not a side formula is called principal, and the formula occurrences in the premisses which are not side formulae are called auxiliary. Definition 1.5 (proofs in LK). A proof in LK is a finite labelled tree whose nodes are labelled by sequents, in such a way that leaves are labelled by axioms and each sequent at a node is obtained from sequents at immediate predecessor(s) node(s) according to one of( the rules of LK. We shall denote !"# ! %$&$'$ ! ! ( proofs by means of the metavariables If is a proof, a subtree of ) ) which is itself a proof is called a subproof of . * A sequent is provable in LK (or LK-provable, or a theorem of LK) iff it labels the root of some proof in LK (i.e., as we shall sometimes say, iff it is the endsequent of such a proof). Definition 1.6 (sequents, postulates and proofs in LJ). The calculus LJ has the same language as LK, and all the concepts introduced in the Definitions 1.2-1.5 apply to it as well, with two sole exceptions. A sequent in LJ is an + , + , expression of the form , where and are finite, possibly empty, , sequences of formulae of £0 and can contain at most one formula. The rules given for LK, therefore, must be adapted accordingly. Definition 1.6 yields an immediate consequence as regards structural rules: the rules ER and CR have to be deleted from LJ, for they can only be applied to sequents with more than one formula in the succedent, while the rule WR

8

Substructural logics: a primer

must be restricted to the case where is empty. Keeping in mind the characterization of substructural logics that we suggested at the outset, the reader is now in a position to understand why we remarked that Gentzen can be reputed, broadly speaking, the first substructural logician. However, it must be noticed that, by suitably tinkering with the rules of the calculi, it is possible to build up multiple-conclusion versions of LJ (Curry 1939; Maehara 1954) and single-conclusion versions of LK (Curry 1952), although these variants are surely less elegant and more cumbersome than their counterparts. Is then the characterization of intuitionistic logic through the above-mentioned restriction on succedents a mere technicality, designed to the sole purpose of getting a manageable calculus and devoid of any philosophical significance? Not quite. We shall see how a profound epistemological meaning can be attached to it4 . Deferring until then any further reflection on the difference between LK and LJ, let us instead pause for a while on the distinction between structural and operational rules, a distinction which is common to both calculi. First, let us consider the latter group of rules. Like in the calculus of natural deduction, we have a pair of rules for each connective. However, while in that case we had an introduction rule and an elimination rule, here we are in the presence of two introductions - a rule for introducing the connective in the antecedent and a corresponding rule for introducing it in the succedent. This is because Gentzen intended to set up a calculus where nothing "was lost" in passing from the premisses down to the conclusion of each inferential step - and it is obviously hard to reconcile elimination rules with such a desideratum. Now, remember what Gentzen had to say about the role of introduction rules in a natural deduction setting: they give the operational meaning of the logical constant at issue. It can be supposed that Gentzen assigned a similar function to the introductions of his sequent calculi (see Hacking 1979 for an argument in defence of such a conjecture). However, a striking analogy and correspondence between introductions, respectively eliminations in natural deduction and right introductions, respectively left introductions in sequent calculi was soon noticed (see e.g. Sundholm 1983 for details). In the light of this, it is possible that Gentzen would have been reluctant to award his left introductions the status of meaning-giving rules. Be it as it may, we can safely assume that Gentzen viewed his operational rules (whether all of them, or the right introductions only) as means of specifying, entirely or in part, the "meaning" or "content" of logical symbols. The status of structural rules is less clear. They are so called since they do not introduce any logical symbol into discourse, but are concerned with the manipulation of the structure of sequents. In LK, they come in left/right pairs as well, with the exception of the cut rule. Gentzen characterizes them as

Francesco Paoli

9

follows (p. 82): Even now new inference figures are required that cannot be integrated into our system of introductions and eliminations; but we have the advantage of being able to reserve them special places within our system, since they no longer refer to logical symbols, but merely to the structure of the sequents.

After remarking this, however, he does not dwell any longer on this subject. As a consequence, if we want to understand better the role of structural rules in Gentzen-style calculi, we have to take a quick look at more recent papers on the philosophy of proof theory. In primis, we may wonder whether also structural rules have a meaninggiving role, i.e. whether they contribute to define the meanings of the constants introduced by the operational rules. Should we subscribe to the holistic viewpoint, there would be no doubt: if the meaning of the logical constants is implicitly given by the whole body of postulates of a system, then structural rules cannot be denied a meaning-giving function. As already remarked, however, such a viewpoint is irreconcilable with the very spirit of Gentzen's enterprise, whose aim is to provide each connective with a separate operational content - whereas on the holistic conception the meaning of each constant would also depend on the introduction rules for other constants. If we accept Gentzen's "inferential approach", then, two alternatives open up: either we assume that each connective has both an operational content, given by its introduction rules, and a global content, specified e.g. by what sequents containing that connective are provable in the system, or else we deny such a dichotomy. Partly depending on the answer given to such a question, we can distinguish at least four theories about the relationships between structural and operational rules in a sequent calculus. We shall list them according to the importance awarded to structural rules, in increasing order. 1) The nihilistic view (Negri and von Plato 2001). The sole meaning attached to a connective is its operational meaning, given by the operational rules. Structural rules correspond to rules concerning the discharge of assumptions in natural deduction; they are closely tied to the particular formalism chosen, and have therefore no meaning-giving role. 2) The ancillary view. It is not easy to credit such a view to any particular author, but Wansing (2000) quotes it as a widespread belief in current prooftheoretic semantics. According to it, connectives have both an operational and a global content, and operational rules are not sufficient to characterize the

10

Substructural logics: a primer

latter: the assistance of structural rules is needed. The global meaning of intuitionistic implication, for instance, depends both on its introduction rules and on the structural rules of the calculus for intuitionistic logic. 3) The dualistic view (Hacking 1979). In this perspective, the roles of operational and structural rules are kept quite separate. While operational rules give the meanings of connectives, structural rules "embody basic facts about deducibility and obtain even in a language with no logical constant at all" (Hacking 1979, p. 294). Structural rules, therefore, have to be postulated for atomic formulae and proved to hold for complex formulae containing logical symbols. A definition of a logical operation through introduction rules is a good definition only if it is not "creative", i.e. if it does not affect the facts about deducibility that obtain for the original "prelogical" language. 4) The relativistic view (Došen 1989a). The starting point of this approach is the idea that logical constants make explicit in a language of lower level some "structural features" of a language of higher level, formulated therein by appropriate "punctuation marks" (e.g. different ways of bunching the premisses together). For example, the formula reflects in the lower language the structural truth (" is deducible from "). Operational rules, in such a context, are simply translation rules from the higher language to the lower one. On the other hand, structural rules, which encode ways of manipulating the structure of sequents at the higher level, are what makes the real difference between the various systems of logic. Girard (1995, p. 11) supports an extreme version of such a view. He says that "the actual meaning of the words 'and', 'imply', 'or' is wholly in the structural group and it is not excessive to say that a logic is essentially a set of structural rules".

For the sake of completeness we quote two more viewpoints, indeed similar to each other, concerning the meaning-giving status of operational rules, though they do not directly bear on the issue of the role of structural rules.

5) The underdetermination view, first version (Belnap 1996). The operational rules of LK are not selective enough: a rule like R, for instance, says something not only about the meaning of conjunction, but also about the meaning of the comma and of . Therefore, one has to find systems where it is possible to "display" any part of a sequent, i.e. to make it the whole antecedent or the whole succedent of an equivalent sequent5 .

6) The underdetermination view, second version (Sambin et al. 2000). The meaning of a connective "is determined also by contexts in its rules, which can

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11

bring in latent information on the behaviour of the connective". It is then desirable that the rules of a system satisfy the requirement of visibility (similar to the above-mentioned property of display calculi): in such rules, there have to be no side formulae on the same side of either the principal, or the auxiliary formulae6 . The previous remarks about the nature of structural rules and their places within sequent calculi like Gentzen's LK or LJ can suffice for the moment. Now it is about time to see structural rules at work. The next section will be devoted exactly to this.

1.2. Discovering the effects of structural rules After Gentzen introduced his sequent calculi, it did not take long until some noteworthy effects of structural rules were discovered. In 1944, the Finnish logician Oiva Ketonen suggested a new version of LK where the rules L, R and L were respectively replaced by:

Ketonen devised these modifications in order to prove an "inversion theorem" for LK: in the new version, as Bernays (1945, p. 127) observes,

All the schemata by which the propositional connectives are introduced [...] can be inverted - i.e., the passage from the conclusion of each one of these schemata to its premiss or premisses can be accomplished by applying the other schema belonging to the same connective, together with the StrukturSchlussfiguren [structural rules].

A more refined version of Ketonen's result would have been proved some years later by Schütte (1950). It is nearly immediate to see that Ketonen's system is equivalent to LK, and that in proving such an equivalence an essential role is played precisely by the structural rules of weakening, contraction, and exchange. In fact, it is not difficult to see that the rule L' is derivable in LK:

12

Substructural logics: a primer

Conversely, the two halves of L are derivable given

L' and the rest of

LK:

# " "# ! ! $ L and $ L' is proved similarly. Finally, let us see The equivalence of % how L' can be derived in LK: 1 2435 % && ' % ( *,+ - *,./- 0,+ - 0,. ) 6

and how

L can be derived given

6

L' and the rest of LK:

< A = B C > A ? DBE 1 7#38 7#98 :38 :95 A ='' > A ='B ? 1 7#38 7#98 :38 :95 B ? B C DB& A ='B ? CGF'DB& and 4;S >@?A9 . Definition 2.3 (some conventions about sequents). Throughout this chapter, we shall adopt the same definitions and conventions about sequents that we stated in Definitions 1.1-1.6. With one notable exception, however: capital Greek letters will not stand for sequences of formulae of the language at issue, but for multisets of formulae of such language. Multisets can be rigorously defined (see e.g. Troelstra 1992, p. 2), but this is not necessary in our context: suffice it to say that multisets are aggregates where the ordering of the elements does not matter (whereas it matters for sequences), but their multiplicity does (while it does not for sets). So, for example, {9CBD> } is the same multiset as

44

Substructural logics: a primer

{ }, but { } is not the same multiset as . As a rule, outer brackets will be omitted: as it is customary to do, we shall write in place of the more correct .

2. SEQUENT CALCULI 2.1 The calculus LL It is now time to come to the heart of the matter, and present our basic sequent calculus. Definition 2.4 (postulates of LL). The calculus LL, based on the language £1 , has the following postulates: Axioms

Structural rules

! " #%$'&)(* Operational rules

, + # ,.- * 0+ #1 - * 2130+ 4! 0! 2530 ! 4+ 0+ 7680 + 7680 + 4! 0+ :980 + # 9

+ ! #5 - *

, # ,./ *

0 1 213 0 # / * 40 2530 # 5 / * 0 6 / # 6 - * 7680 # * 0 - * 79;0 7980 # 9 / *

Francesco Paoli

45

Notice that LL contains "covert" exchange rules: using multisets instead of sequences, we are allowed to perform arbitrary permutations either in the antecedent or in the succedent. Beside such rules, the only explicit structural rule of LL is the cut rule.

#) $)# '& ! %)#

( )# #" $"%# "#' & &*# $"#"+

("% , '&#"# # )# &*# $"#)+ / & 0 . ' . * .

* &

* ' . . #)#

&16'3 &1.' #.')## "$7.2' & -$3434 . # "#

'&1.'#$$)$3584 .2

:8 9 -:3484 :84

63 $84 63 '&$84 $ 34

68 $34 %!" $ -3 84 - 84' & %"#

68 ##"#) - 34 '& #)# 763 #)## " -34 %"#.

$63 34

#'"#&1 .' $8;.2 #)#.

6$8 84 #

)#'&17. 0 4 2 56.7,298@?A :"A 56.B*+,298;:=< , C * * , C *'+, -/.1 * , > 0 4 2 56.7*29856.7,298;:D< 56.7 * +,298;:=

;

Proposition 2.12. (i) LAB = LA ; (ii) LA is trivial.

72

Substructural logics: a primer

; (F94) ; (F95) ; (F96) ; (F97) ; (F98) , if is any theorem of HA. Moreover, F1 becomes superfluous and R2 is derivable. ! Proposition 2.28 (theorems of#HC). The following theorems are provable in " # HC: ( )"*#&+# ;; (F101) $ (F99) % ; (F102) &;'(F100) ; (F103) (F104) , -, .# ; (F105) - , , &/# . Moreover, F1 becomes superfluous. ! Proposition 2.29 (theorems of HG). The following theorems in ( +are; provable F28, F87, (F106) (F107) HG: 0"1+# ; (F108) #2 0; (F109) 1 0; (F110) 0. ! . The following theorems are Proposition 2.30 (theorems of HRM ) ) * " 3 2 # ; (F112) provable F90, 4 )"*#in2HRM "*#: ;F89, )(F111) "*#2

"*# . ! (F113) Proposition 2.31 (theorems of HRM). The following theorems are provable , ' in HRM : F100, F102, -, (F114) ; (F116) - , ; 3(F115) . ! ND

ND

2 ! Proposition 2.33 (theorems of HLuk ). The following theorems are 0"*# provable 5"5"in HLuk : ; (F118) 0"*# (F119) "60"*' . ! Proposition 2.34 (theorems theorems 7 ). The # ; following are 2 7 provable 7 ; % 8 of,HL 7 7 7 (F121) in HL : ((F120) ; (F124) 7 7 = > ?7 @A92 7 > BC (F122) 7 %'9: ; ; (F125) 0 7 9 ? 7 < ;; (F123) @ >??0?@D ? => ?@A ; (F128) ? =E?> (F126) ?0?@A9 ? ? => ?@3AGF !? ? ; (F127) Proposition 2.32 (theorems of HLuk). The following theorems are provable in HLuk: F28, F114, F115, F116, (F117) . 3

3

E

E

Francesco Paoli

73

3.2 Derivability and theories In classical propositional logic, a (syntactic) theory is a set of formulae which contains the classical propositional axioms and is closed under modus ponens8 . In our substructural context, we need to draw some finer distinctions which the classical setting obliterates. Therefore, we set off with the following Definition 2.27 (S-theory). Let S be any of the previously introduced axiomatic calculi. An S-theory is a set of formulae of the appropriate language s.t. (i) if and S , then ; (ii) if , then .

An S-theory, therefore, need not contain any of the axioms of S: all that is required is that it be closed under adjunction and that it contain the consequent of an S-provable implication whenever it includes its antecedent. Next, we consider some "well-behaved" kinds of theories.

Definition 2.28 (some special kinds of S-theories). An S-theory

is said to

be:

regular, iff it contains all of the axioms of S;

detached , iff ; only if ; -consistent , iff -consistent for some ; simply consistent , iff it is -consistent, iff for no , both and ; -complete, iff for every , either or simply complete, iff it is -complete for every ; ; -complete , iff for every , either or

prime, iff

only if

or

;

.

To introduce suitable concepts of maximality for our S-theories, we need a preliminary definition.

"! # *,+.-0/1'2 '( 4

%& $ 3 % &

be a set of formulae. The SDefinition 2.29 (S-theory of sets). Let theory of (in symbols: ) is defined as is an S-theory and S . Moreover, by we mean the set is a regular, S detached S-theory and . From now on, we shall drop the subscript "S" wherever no danger of confusion is impending.

' ( 4 )

Definition 2.30 (maximal S-theories). An S-theory

is said to be:

74

Substructural logics: a primer

-maximal, iff it is -consistent but, for any

is not such; -maximal, iff it is -consistent but, for any is not such; weakly maximal, iff it is simply consistent but, for any

is not such; is maximal, iff it is simply consistent but, for any

weakly

not such. Classically, few of these distinctions make sense. As we shall see, indeed, any HK-theory is both regular and detached; hence, any weakly maximal HKtheory is maximal. Moreover, it is well-known that the two notions of simple consistency and -consistency, as well as the four notions of simple completeness, -completeness, primality and maximality, are classically equivalent to one another. The next few lemmata are devoted to establishing some of these relationships also for our substructural calculi.

Proposition 2.35. For any HL-theory : (i) if it is regular and detached, condition (i) of Definition 2.27 is redundant; (ii) if it is -consistent, it is simply consistent; (iii) if it is simply complete, it is -complete; (iv) if it is detached, -consistent and -complete, it is prime; (v) if it is maximal, it is weakly maximal.

! %+*, ( - " "$#&%'#&() %/.0"- (/.1"- 23%4.1 "657823(/.1"95:- %+*,( .0"- "- ; -theory : (i) if it is nonempty, it is detached; Proposition 2.36. For any HR < (ii) if it is regular and weakly -maximal, it is prime; (iii)< if it is regular and prime, it is = -complete; (iv) if it is regular and weakly -maximal, it is = complete. ?^ ad\OSM_ b;\OSMR bc>^^eb;\OS;_ fgSMR bc>^ h i jFk aAS;_ fgSMR bc l;mglX{ n"lop q r st . Finally, suppose that u-s(v wyxzl was and Q and |-}(~ by an application of I. Then, obtained from |}(~ and ~ M , for some by induction, Qbe U|VX " } . Let~ M zM yz ; using F74, we get: 3

5

w HL

w HL w HL

w HL

w HL

w HL

w HL

w HL

80

Substructural logics: a primer

!" #%$&#(4 '#*),+.-!/021 . and 3 57698 : then 8 ;=;2@BADC!E62F . We shall show how to replace ; I by 6HG , by inductionI on the 6HG ?@ ACBD ACEFHG .46 0 3 /24 1

Why, a reader could ask, did we introduce such a complicated and convoluted inference pattern as the mix rule, in place of the more natural and intuitively appealing cut rule? There is a reason, indeed, and it has to do precisely with the presence of contraction in LK. We shall explain our move in due course; thus, the curious reader is begged to wait patiently until § 1.2. What we shall do, for the time being, will be to prove a cut elimination theorem for LKM . To achieve this goal, we need a number of auxiliary notions. First of all, the concept of "mixproof" will permit us to focus on a quite small subset of the set of all proofs in LKM which contain one or more applications of mix1 .

I

Definition 3.2 (mixproofs and mix-free proofs). A proof in LKM is called a mixproof iff it contains just one application of mix, whose conclusion

Francesco Paoli

89

is the endsequent of the proof; it is called a mix-free proof iff it contains no application of mix at all.

Proposition 3.2 (circumscription of cut elimination). In LKM , if any mixproof of can be transformed into a mix-free proof of the same sequent, then any arbitrary proof of can be transformed into a mixfree proof of the same sequent.

in LK . Take the leftmost Proof (sketch). Let be any proof of be its conclusion. The and uppermost application of mix in , and let subproof of whose endsequent is is a mixproof which can thus

be turned into a mix-free proof of . Now consider the result of replacing in by , call it , and take the leftmost and uppermost application of mix in . By repeating this procedure as many times as there are applications of mix in , we get the required transformation. The details are left to the reader. lemma, it will suffice to show that any mixproof ofintheLKpreceding ofIn virtue can be turned into a mix-free proof of the same sequent M

M

M

in LK . To do so, we shall argue by induction on a special parameter, to be specified presently. Definition 3.3 (rank of a sequent in a mixproof). Let whose final inference is:

be a mixproof

and is# so defined: The rank of the sequent in is denoted by " If belongs to the subproof ! of ! whose endsequent is % $ , is the of an upward of sequents &(> '*)+maximal ),),'-& ? s.t. length &/> .0& (diminished &132547by684:one)9 J ? L @E E@& K ? M B@ N OPQ R H ? I @A J1@ K ? L)@ M B@C N S5TR J1@& K ? I!@) L)@ M AGF!B@#H#@& U

If was contained in the left premiss of the L inference, we argue analogously. 3.3) Transformations T10 and T11. This subcase of the inductive step, however, needs no longer to be treated separately. Now we can simply push the cut upwards, cutting away one of the side occurrences of . For example:

V

W W VYX[ Z X \ ` a V ! U ] X ] N S5dR V ! ` Z a X\V U!] ^X V ! b Z c U ] U ] X N OPQ Rfehj gi `#X_ b Z a)X! c X\V U!] > > kYl[ r m s l^kon!plp kon!pql t m u v w xy z kYl#r0l t m s!l! u lp v {5|z r#l t m s)l! u l^kon!p

The previous discussion provides a hint for the proof of the next proposition, whose details are left to the interested reader:

}

~

Proposition 3.4 (cut elimination theorem for LA: Grishin 1974). Any cutproof of in LA can be turned into a cut-free proof of the same sequent in the same calculus.

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97

1.3 Cut elimination for calculi without the weakening rules

When proving Proposition 3.3, we had to introduce the mix rule in order to cope with cases involving contraction. Mix, however, is a rather "brutal" inference rule: if the premisses of the relevant application of mix are and , it makes a clean sweep of any occurrence of the mixformula in . In LKM this is not a problem: if we need to reintroduce some of the deleted occurrences, we can do so by means of weakening moves. Yet in systems like LRND , where the weakening rule is not available, this is not possible. Hence, we need a more selective rule than mix - an "intelligent" mix, so to speak.

Definition 3.8 (postulates of LRNDM ). The system LRNDM is exactly like LRND , except for the fact that the cut rule is replaced by the following rule (called intelligent mix, or intmix):

Here, both and contain , and ( ) is obtained from ( ) by deleting at least one occurrence of in it. Again, when there is no danger of ambiguity, we shall drop the subscript " ". The formula is called intmixformula.

Proposition 3.5 (equivalence of LRNDM and LRND ). . LRNDM

LRND

iff

Proof. We only need to show that intmix is equivalent to cut. The derivability of intmix in LRND is shown as in Proposition 3.1. The derivability of cut in LRNDM is trivial, as cut is nothing but a special instance of intmix.

Now, to obtain a cut elimination theorem for LRNDM , it is sufficient to adapt the definitions and proofs of § 1.1 as follows. 1) All the definitions containing the word "mix" must be pruned in the obvious way. Thus, for example, instead of "mixproof" and "mix-free proof" we shall have to speak of "intmixproof" and "intmix-free proof". 2) The proof of Proposition 3.3 must be adjusted in an appropriate way. In particular, we have to show that all the uses of weakening in such a proof can be dispensed with. Weakening has been employed in four subcases, exemplified by the following transformations: 2.1) Transformation T6, which now becomes:

98

Substructural logics: a primer

5

2.2) Transformation T8, which now becomes:

"! 6 # (

$

$ % &

'

)

2. ) Transformation T , which now becomes:

/

/ 0 1

314 6 2 7 13 8 2 9 ? @A ; 01 6:14 8 2 71 9 6:15 8=1 ; 2 7>1 J%M A R HF @ LQ K6R MOJ?VJ' @ ? I > JKSR M T I > JU

Francesco Paoli

! " # $& %$& %$& $& ! '($& / 0 1 0 +&45 2 )! *(+&,- 1 . 0 3 2 1 . 0 - * 2 1 . 0 -, +&65 2 1 . 0 7 45 2 1 . 0 - *%+&, 2 1 . 0 - *%+&, 3 2 7 - 1 . 0 3 < 8 =B< A@ 8 7 =?7 >@ 9 D / : C C < =B< >@ 9 : ;

123

Remark 4.4 (on the rules of LLuk3 '). In LLuk3 ', structural rules are split into two groups: external rules, by which whole components are added or deleted in a hypersequent, and internal rules, acting on formulae within each component. Notice the lack of internal contraction rules, which justifies the inclusion of three-valued Lukasiewicz logic among substructural logics. The operational rules of LLuk3 ' are the same as in LL, with side sequents added. Remark 4.5 (An alternative axiomatization of LLuk3 '). Ciabattoni et al. (1998) suggested an alternative axiomatization of LLuk3 ', where the rule Mx is replaced by the following, simpler rule:

E D"FH C G KJF L I MNFH C G OF L E !I DPFJM G KJF O C G L QSRUTW V

124

Substructural logics: a primer

The informal meaning of hypersequents in LLuk3 ' is emphasized by the next definition and lemma.

Definition let 4.6 (formula-translation be a hypersequent,ofandafor hypersequent). Let , as defined in Definition be the formula-translation of the component 2.5. The formula-translation ! of the hypersequent is the formula . # , . &%('')'*%+ - / # , / . "$ 0 iff 4.3 (meaning of hypersequents in LLuk '). 0 Proposition #1! . Proof. To avoid notational we consider the simple example of 2 redundancies, hypersequents of the form #43657#48 ; the general case is left up to the reader. From left to right: 9;:4 :@?;A4 M NPORQ S TVU W XZY :EDF?;A4IGJ :KDF?;A4IG M [FORQ S TVU W XZY M \"]RY :KDF?;A4IG ^7_ ?;A4< ; 2) ` _ =7AC> ; 3) a _;^ HI` . From right to left, let 1) b Let moreover be the following proof: f d fKg d g h ikjml g d g f d g6 g d f h nFoRl c f d f e7d g6 g d f h nFoRl c d4e e7d g e7d f h pVqr l dCe ed g d f h pVqr l d g d f 3

LLuk3'

LLuk3'

b w t4wKx6t4x CsB t4u +s vRw7t4x } ~PV } V y t y{z t z } ~PV u|v y t z tCu6 w7t4x } V y t z w7t4x Proposition 4.4 (completeness of LLuk ': Avron 1991a). " ! . The result we are after can be obtained thus:

3

HLuk3

LLuk3'

iff

Francesco Paoli

125

Proof. We proceed as in the proof of Proposition 2.49. From right to left, then . This is we prove that for any formula , if HLuk3 LLuk3' HLuk done, of course, by induction on the length of the proof of in 3 . The desired conclusion follows then from Proposition 4.3, upon considering formulae of the form

! " ! # $ $ % $ $ & ' ( )+* ( ) ( ) ( (-,8 (.(/8 ,( 9 :;!< ,0(1,(28 (435(/, 9 =>?< ,0(2(48 35(/, * 9 @BA?< 0 , 2 ( / ( , ( 6 ( )* ( ) (-,(28 (/, * ( 9 CDE< 9 FGA?< ( ))+* *( ) ) (-,(18 ,8 ( )+)+* * ( ) (-,( 9 FGDE< ( ( (-),* (7( ) ( (/,( 9 =?>?< ( ) ( * (-) ,( ) 9 FGA?< , ( ( ( ( H I As an example, we prove the hypersequent

((

)

)

.

)

(

)

(

)

( )

(

((

)

)

)

In the opposite direction, the proof proceeds by induction on the length of the proof of in LLuk3 ', and is omitted. LLuk3 ' is a cut-free calculus. Indeed, by using a rather complicated method (the "history" method, necessary to deal with the case where one of the premisses of the relevant cut is obtained by external contraction)2 , it is possible to prove:

I

Proposition 4.5 (cut elimination for LLuk3 ': Avron 1991a). LLuk3 ' is cutfree. As we hinted earlier, also RM and its "cousin" RMI have been given by Avron cut-free hypersequential formulations. Here they are: Definition 4.7 (postulates of LRMI: Avron 1991b). The hypersequent calculus LRMI, based on the language £1 , has the same postulates as LLuk3 ', except that: The internal weakening rules, WL and WR, are replaced by the following internal contraction rules:

J

126

Substructural logics: a primer

(so that, The rules L and R must abideby the restriction ! and #)" for example, we cannot conclude from The mixing rule Mx is replaced by two relevant Mingle rules: - $&% . / $' 354687:9 $0 - %( $) . / 346) . ? * - >/ 2 0 ? -@ . ? / ABDCFEHGJILK

Definition 4.8 (postulates of LRM: Avron 1987). LRM is exactly the same as LRMI, except for the fact that relevant Mingle is replaced by the following combining rule:

M N O N M !O Q N R S T,P R * Q P S AUV:WXK and no restriction is imposed on Y L, Z R.

Remark that the hypersequential version of the anticontraction rule MR, a distinctive postulate of LRMND , is derivable in LRM:

M N[ M N[ M8T 8M Q P\ N T Q P\ ] ^_ `:a M T,P T N Q*P*Q P\P\ ] bL^La T,M P T N Q*P*Q P\P\ ] ^ced ^fLa T Q P\P\

Likewise, ML is also derivable. The cut elimination theorem for both LRMI and LRM was proved by Avron with the help of the history method3 .

Francesco Paoli

127

1.3 Dunn-Mints calculi Hypersequents are a powerful tool for setting up proof systems for several substructural logics. However, they seem of little avail in the case of relevance logics like R or RW. A first step towards giving a proper Gentzen-style formulation of such logics was made by Dunn (1973) - and independently by Mints (1972) - who found a calculus for positive (i.e. negation-free) R. As we remarked back in Chapter 2, the disturbing axiom of HR and HRW is the distribution axiom (F28), whose proof requires, in ordinary sequent calculi, the use of both weakening and contraction. Dunn and Mints overcame this hurdle by dropping Gentzen's tenet according to which the antecedent and the succedent of a sequent are sequences of formulae separated by commas. In their calculi, the formulae occurring in the antecedent of a sequent can be bunched together in two different ways: by means of commas (to be interpreted as lattice-theoretical conjunctions) and by means of semicolons (to be read as group-theoretical conjunctions). The behaviours of these punctuation marks are governed by different structural postulates: weakening, in particular, is available for comma but not for semicolon. This is what makes distribution provable in the system, while still hindering the proof of relevantly 4 unacceptable sequents such as . Let us now present in some detail Dunn's version of the calculus, hereafter labelled LR+ .

Definition 4.9 (£6 -structure). An £6 -structure (henceforth in this subsection, a structure) is inductively defined as follows: Any formula of £6 is a structure; The empty set is a structure; If and are structures, then is a structure; If and are structures, then is a structure.

Definition 4.10 (substructure and substitution). The concept of substructure of a structure is inductively defined as follows: is a substructure of ; Any substructure of and of is a substructure of and of . By , or simply by whenever no confusion can arise, we mean the result of replacing in the indicated occurrence of its substructure by an occurrence of .

Definition 4.11 (sequents in LR+ ). A sequent in LR+ (henceforth in this subsection, a sequent) is an expression of the form , where is a structure and is a formula of £6 .

128

Substructural logics: a primer

In the following presentation of LR+ , the symbol "*" will ambiguously denote both commas and semicolons. For instance, the rule E* actually embodies two different rules, one where stars are replaced by commas and one where they are replaced by semicolons. Definition 4.12 (postulates of LR+ ). LR+ has the following postulates:

Axioms

rules !" Structural " #!" *($+ ) #,.-& %$& '( ) must be nonempty; in the cut rule, '/ denotes the In the rule (W,), result of replacing the indicated occurrence of by if the latter is nonempty, by 1 otherwise. Operational rules 4 ) # 35 21 ) 36 43) 0 21 3) # 1 ) # 785 ) 786 219 878) 878) # $ ) # : 143 . /=;$< 0 > 573 $ ; < / , . / , >< / . . / >< <?A@8/ . . 0 B 143

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