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
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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
"!$#
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7 Operational rules
Definition 1.4 (principal, side, and auxiliary formulae). In all these rules, the formula occurrences in are called side formulae; the formula occurrence in the conclusion which is not a side formula is called principal, and the formula occurrences in the premisses which are not side formulae are called auxiliary. Definition 1.5 (proofs in LK). A proof in LK is a finite labelled tree whose nodes are labelled by sequents, in such a way that leaves are labelled by axioms and each sequent at a node is obtained from sequents at immediate predecessor(s) node(s) according to one of( the rules of LK. We shall denote !"# ! %$&$'$ ! ! ( 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
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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
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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
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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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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 <@BE > A ='' > A ='B ? 1 7#38 7#98 :38 :95 B ? B C DB&<&BE 1 243 ; 5 > A ='B ? CGF'DB&<@BE F
F
Let us pause for a while on the last equivalence. If we compare to each other the rules L and L', we readily see that they coincide with respect to the principal and the auxiliary formulae; on the other hand, they differ as regards the side formulae - or, as they are sometimes called by means of a collective noun, the context. The two premisses of L' share the same context, while the premisses of L do not. For this reason, such rules as L' are called sharing, whereas L and similar rules are called nonsharing. Rules of the former type are sometimes labelled also as contextdependent, for they can be applied only if the contexts of the premisses are the same; on the other hand, rules of the latter type are said to be context-free, since they do not have to abide by such a restriction7 . Curry (1960), among others, considered the following non-sharing versions of R and L:
F
F
H
I
F
F
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13
The reader can easily prove for himself the equivalence between these rules and . For each of the two and Gentzen's original two-premiss rules for connectives, thus, we can choose between two equivalent formulations of the left introduction rule and two equivalent formulations of the right introduction rule (the same fact happens with implication, since the rule
is easily seen to be interderivable with R in LK). But such equivalences heavily rely on the use of structural rules. What would happen, one might wonder, if some structural rules, say weakening and contraction, were left out of our calculus? If we adhere to Gentzen's methodological standpoint, discussed in § 1.1, according to which rules define logical connectives by giving them their operational meaning, we cannot help concluding that we should be in the presence of different connectives, since their defining schemata would no longer be equivalent. However, a word of caution is in order. We cannot take every combination of left/right introduction rules as defining a different connective in any theoretical framework whatsoever. For example, let us drop WL and WR. If we stipulate that our conjunction is defined by L and R' and our disjunction is defined by L' and R, we readily conclude that in such a calculus there can be no room for further conjunction and disjunction connectives. In fact, we can nearly recover both weakening rules as follows:
0324065725 < 2 = ! "$#&% 00 81 51< 2 = 0 1 5729085 0 1 51< 2 = ! .$' % < < 2 2 = 1 = 5;1 5 :0 57532:0356205241 0 0 < 2 = 1 51 0
! "$'() %
! .$#/) %
! *&+-, % ! *&+-, %
and these slightly restricted forms of weakening are enough to prove the equivalence of the alternative rules for conjunction and disjunction. Likewise, if we choose L' and R as rules for conjunction, L and R' as rules for
8
8
:
:
14
Substructural logics: a primer
disjunction, and drop CL and CR, we recover contraction rules as follows:
Therefore, pairing Ketonen-style one-premiss rules for and with Gentzen-style two-premiss rules leads to a recovery of the discarded structural rules in some substructural contexts; the same happens if we match Gentzenstyle one-premiss rules with Curry-style two-premiss rules. On the other hand, the pairs LR and L'R', as well as the pairs LR and L' R', would define different connectives if we dropped either weakening, or contraction, or both. It seems appropriate, then, to introduce different symbols for them right over. We choose to keep Gentzen's original symbols and for the connectives defined by Gentzen's original rules, respectively L- R and L- R. We do so not only on philological grounds, but also for a further reason, which will become clearer in Chapter 6: such symbols are reminiscent of the symbols for the lattice operations of meet and join, and the abovementioned rules are exactly what is needed to give our connectives the properties of those operations. For the same reason, following Casari (1997a), we call them lattice-theoretical connectives. Remark, however, that in the literature the terms "additive" and "extensional" are more frequently used in order to refer to this group of connectives. ! We adopt the symbol for the conjunction defined by L' and R' ! ! (and, consequently, we rename these rules L and R). Similarly, the " symbol will be employed for the disjunction defined by L' and R' (and " " these rules will be referred to by L and R). These connectives will be called group-theoretical, following once more Casari (1997a). Again, in the literature the terms "multiplicative" and "intensional" are more widespread8 . If we drop the exchange rules as well, negation splits up too. In fact, the pairs of rules
#$
%#$
Francesco Paoli
15
are no longer equivalent. These two negations are sometimes called, respectively, post-negation and retro-negation (see e.g. Abrusci 1991). Similarly, one has a post-implication and a retro-implication: the reader can devise for himself appropriate rules for these connectives by fiddling with Gentzen's L and R. We shall not introduce specific symbols for these connectives because, as we remarked in the Introduction, we shall not explicitly consider logics without the exchange rule in this volume9 . Summing up: if we relinquish some or all of the structural rules in Gentzen's LK (or, for that matter, in LJ), a plethora of new connectives emerges in place of the original four. Structural rules flatten this expressive wealth by reducing the defining rules of some connectives to the defining rules of other ones.
2. REASONS FOR DROPPING STRUCTURAL RULES But is that a problem?, a partisan of structural rules might interject. After all, he could rightly remark, such rules are essential to get all of the classical or intuitionistic logics, and these logics have proved very fruitful in the methodology of deductive sciences, while it is not at all certain, to say the least, whether substructural logics could turn out to be just as useful in this context. Moreover, he could continue, do all the above-mentioned connectives have an analogue either in natural language or in science, or else are they merely artificial constructions whose sole raison d'être is precisely that absence of structural rules which they should justify? If the latter alternative is true, why on earth should we dispense with structural rules? There are some reasons, indeed. In this section, we shall list and discuss a few of them.
2.1. Reasons for dropping structural rules altogether In § 1, we have already seen a reason why at least some of the structural rules should be dropped: if they are all present, the expressive power of our calculus is strongly restricted. Some linguistical distinctions are simply obliterated, since the use of structural rules makes the defining rules of different connectives collapse onto one another.
16
Substructural logics: a primer
Are, however, these distinctions desirable? Are there actually several kinds of "or", "and", "if... then" in natural language? The issue, as one can guess, is not easy to settle. Even if we confine ourselves to "if.. then", reams and reams were written to support either of the two possible answers (for a clear and concise survey, see Sainsbury 1991). We shall therefore limit our discussion to a couple of examples, regarding the interpretation of the natural language connective "and", which appears somewhat less disputed. To be as fair as possible in our presentation of these examples, we shall expound both an argumentative strategy for and a strategy against the thesis that there are several "ands" in ordinary English. Example 1.1 (the static "and" and the concurrent "and")10 . Consider the following sentence, adapted from Girard (1995): (1) For $1 you get a pack of Camel and a pack of Marlboro. Suppose that it is true - as it was back in 1995, when Girard's paper was written - that $1 is enough to buy just one pack of cigarettes of either brand. Is (1) true or false? Well, it depends on whether you interpret that "and" as a concurrent "and" or as a static "and". Both connectives are conjunction connectives, since they express the availability of two actions (getting a pack of Camel and getting a pack of Marlboro) given that a third action (paying $1) is performed. In the former case, however, both actions can be performed, while in the latter only one of them can, but we are in a position to choose which. If we interpret the "and" in (1) as a concurrent "and" - which is formally represented by our group-theoretical conjunction - then (1) is false, since one dollar is not enough to get both a pack of Camel and a pack of Marlboro; if, on the contrary, we interpret it as a static "and" - which corresponds to our lattice-theoretical conjunction - then (1) becomes true, for although the availability of one dollar is not enough to get two packs of cigarettes, it leaves you in a position to choose between the two brands11 . Those who maintain that natural language "and" has a substantially uniform meaning usually reply that in sentences like (1) there is a hidden possibility-like operator ("It is practically possible that...", "It is feasible that"); according to such an interpretation, the ambiguity of (1) is structural rather than lexical, and it depends on the fact that such an operator has wide scope on one reading, and narrow scope on the other. (1), thus, can be interpreted in either of the two following ways:
(2) It is feasible that (for $1 you get a pack of Camel and a pack of Marlboro). (3) It is feasible that (for $1 you get a pack of Camel) and it is feasible that (for $1 you get a pack of Marlboro).
Francesco Paoli
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If this line of reasoning were correct, clearly there would be no need to postulate two different meanings for the natural language conjunction "and", at least as far as the previous example is concerned. Example 1.2 (the sequential "and"). Now, let us focus on the next sentence: (4) John opened the door and stepped out of the room. Such uses of "and" are usually taken to have a sequential character: what they actually mean is something like "and then". If we were to permute the order of the two conjuncts, (4) would no longer be true. It follows that this occurrence of "and" cannot be properly formalized by the standard commutative conjunction, where the order of conjuncts is not relevant. Here, one could reply by appealing to Grice's notion of conversational implicature. According to Grice (see e.g. his 1975), a sentence may have, besides its literal content, also an "implicature". The term refers to what is implicitly conveyed by the sentence, although it is not literally said. For example (Sainsbury 1991), a hyperbolical expression like: (5) I can't tell a crane from a canary implicates that the utterer knows next to nothing about ornithology, although what it literally says is something definitely stronger (and probably false). This example also shows how the truth value of the implicatures of a sentence may differ from the truth value of the sentence itself. He who chooses this line of defence might claim that what (4) says is just that the indicated events both occurred, while the fact that they occurred in the given order is merely implicated. Likewise, the sentence (6) John stepped out of the room and opened the door is true, while only its implicature (that John went out before opening the door) is obviously false. After having examined a possible reason for discarding structural rules in general, let us go on to discuss some shortcomings of individual structural rules.
2.2. Reasons for dropping (or eliminating) the cut rule A negative attitude towards the cut rule could take two different shapes. On the one hand, someone could contend that such a rule should be rejected as incorrect, or at least severely restricted. Someone else, on the other hand,
18
Substructural logics: a primer
though accepting it as a valid inference pattern, could find it desirable to prove its redundancy, i.e. that it does not increase the stock of provable sequents of our calculus. The first view has noble and time-honoured roots, but at present is definitely a minority opinion. The second view, on the contrary, is quite widespread: all of today's logicians, except perhaps for a handful, prefer a cutfree calculus (that is to say, a calculus where cut is a redundant, or "admissible" rule) to a calculus without this property. Why is it hard to deny the correctness of the cut rule? Because such a schema embodies, in a very general form, the idea that derivability is transitive. Sometimes, indeed, the informal meaning of the cut rule is explained as follows, with reference to the methodology of deductive theories. Suppose that the lemma is derived from the axioms of a given theory, and that the theorem is proved by means of (plus, perhaps, other propositions, say ). It cannot be denied that, by doing so, we have shown that is derivable from the axioms . Anderson and Belnap (1975, p. 154) plead the cause of transitivity of deducibility with these words:
Any criterion according to which entailment is not transitive is ipso facto wrong. It seems in fact incredible that anyone should admit that follows from , and that follows from , but feel that some further argument was required to establish that entails . What better argument for could one want?
Surely, the great XIX century logician Bernard Bolzano would not have shared Anderson's and Belnap's confidence: in his Wissenschaftslehre (§§ 155 and 212), he investigated as many as two non-transitive entailment relations. More than a century later, Lewy (1958), Geach (1958), Smiley (1959) and Epstein (1979) have introduced systems of logic whose derivability relations are not unrestrictedly transitive12 . For to be deducible from , they argue, there has to be a meaning connexion between and . But the relation of meaning connexion is not transitive: it may be the case that although shares some common content with , and in turn with , fails to share any content whatsoever with . Still more recently, Tennant (1987) has challenged the validity of the cut rule on different grounds. Tennant denies that follows from and in case that either is inconsistent or is logically valid. Tennant's aim is twofold: first, he wants to achieve a notion of relevance of the premisses of an argument to its conclusion that - unlike in Anderson's and Belnap's mainstream relevance logic, of which more will be said presently - does not require the introduction of a new connective, but simply involves the analysis of the derivability relation; moreover, he aims at setting up a proof system which is adequate with respect to a natural semantic
Francesco Paoli
19
notion of entailment (an entailment, in this perspective, is any substitution instance of a valid sequent which has no valid proper subsequents). Apart from the previous exceptions, most logicians agree that the cut rule expresses a valid mode of inference, but believe that it is desirable to prove its redundancy in a sequent calculus, viz. to show that what can be proved with the help of cut can be proved just as well without it (cut elimination). This was, by the way, the opinion of Gentzen himself, who called such a result Hauptsatz ("main proposition") and considered it the principal motivation for carrying out an analysis of formal deducibility in the context of sequent calculi. The reason for such a belief can be summarized under the following headings. 1) Technical benefits of cut elimination. Cut elimination yields several advantages from a technical viewpoint; we shall now list but a few of them. As we shall see in Chapter 3, it sometimes allows to prove the decidability of a given calculus - in other words, it permits to show that there is a procedure which, given any sequent, checks in a finite number of steps whether it is provable or not in the calculus at issue. Moreover, by working on cut-free proofs we can show that intuitionistic logic is prime (if is provable, then so is either or ), and that both classical and intuitionistic logic have the interpolation property (if is provable, then either is provable, or is provable, or there is a formula , whose variables are among those in both and , such that both and are provable).
2) The concern for analyticity. In a more philosophical perspective, there is a simple reason to dislike a proof containing cuts. Such a proof may include formulae that disappear in the conclusion - in the words of Anderson and Belnap (1975, p. 53), "it might contain adventitious occurrences of formulas that have no connection whatsoever with what was to be proved" - whereas in a "normal" (i.e. cut-free) proof this is never the case, since all the remaining rules are such that the formulae in their conclusions contain all the formulae in their premisses as subformulae. Gentzen (1935, p. 69) explicitly acknowledges the importance of this property, called subformula property: Perhaps we may express the essential properties of such a normal proof by saying: it is not roundabout. No concepts enter into the proof other than those contained in its final result, and their use was therefore essential to the achievement of that result.
Why is the subformula property desirable? Technically speaking, it suffices to say that some of the benefits mentioned under the previous heading depend rather on the subformula property, than on cut elimination per se. On the
20
Substructural logics: a primer
methodological side, it is an optimal property for anyone who is interested in setting up an analytical calculus - taking the term "analytical" in what Jaakko Hintikka (1973) calls the directional sense. A calculus is called analytical in this sense if, given an arbitrary theorem, by analyzing its logical structure we have hope to succeed in a bottom-up search for its proof. "Bottom-up" means that in this process one is climbing up the proof-tree in a backward direction (from conclusions to premisses) until the basic ingredients (the axioms) are reached. This decomposition is not possible, of course, if the subformula property is not available: in our search for the premisses of a given inference, we should have to go through infinitely many candidates. In an analytical calculus the shape of proofs respects the intuitive methodical ordering which proceeds from simple axioms down to more complex theorems. And this concept has a highly respectable history. In fact, Descartes (in his Regulae ad directionem ingenii), Pascal and Arnauld ascribed to each explanatory process - especially to mathematical definitions and proofs - the countersign of complexity increase: in a deductive theory, we are usually led from simple ingredients (simple ideas as primitive concepts, simple propositions as axioms) down to more complex expressions; by analyzing a derived concept or a theorem, however, we can upset the procedure and track down its basic components. This tenet of the XVII century doctrine of method was perfected with admirable logical mastery by Bernard Bolzano, in the §§ 198-221 of his Wissenschaftslehre. In the light of the preceding discussion, it does not seem inappropriate - at least conceptually, if not historically - to connect Gentzen's concern for the subformula property to these philosophical antecedents. 3) The concern for conservativity. Cut elimination - more exactly, the subformula property - brings a further philosophical dowry. In § 1, we remarked that natural deduction and sequent calculi avoid any commitment to a holistic theory of the meaning of logical constants, as each connective is introduced by separate rules which do not depend on the definitions of other constants. Anyway, such a commitment cannot be completely excluded unless the subformula property has been established. In fact, take the fragment of LK containing just L, R, and the structural rules. Then extend it, say, by adding L and R. Can you rule out the possibility that some sequent containing no occurrence of conjunction and unprovable in the former fragment becomes provable in the latter? No, for you might prove it by resorting in an essential way to L and R and then cutting away the formulae introduced by these rules. Since it is possible to assume that whether a law involving a given connective holds or not is relevant at least to what we have called the
Francesco Paoli
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global meaning of the connective itself, in such a case the meaning of implication would be partly determined by the rules for conjunction. However, if our implication-conjunction calculus is known to have the subformula property, any formula occurring in the proof of our purely implicational sequent must occur as a subformula of the sequent itself - and this is enough to ensure that the rules L and R have never been used to establish it. In cases like this - namely, when a calculus S is obtained by adding to the calculus S one or more connectives and postulates concerning these connectives, in such a way that S proves no sequent containing just the old connectives which was not already provable in S - logicians say that S conservatively extends S. As we have just seen, conservativity is a condicio sine qua non for the soundness of a molecularistic semantics of logical constants in the framework of sequent calculi.
2.3. Reasons for dropping the weakening rules We shall consider three objections to the weakening rules: for reasons that will become clear in the following, we shall respectively label them the relevant objection, the paraconsistent objection and the nonmonotonic objection13 .
1) The relevant objection. Consider the following proof in LK (or, for that matter, in LJ)
is usually referred to as the law of a fortiori. The principle The reason why it is so called is that it can be given this intuitive reading: if holds even with no need for hypotheses, then a fortiori (all the more so) it holds under the hypothesis . This principle has been held to be rather counterintuitive for a long time. C.I. Lewis (1918), for instance, considered it the prototypical example of a paradox of material implication, signalling the deficiency of the classical rendering of "if... then" and its need to be replaced by a tighter notion (which he identified with the modal notion of strict implication). To convince ourselves that Lewis had a point, let us read the following passage by Anderson and Belnap (1975, p. 14):
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Substructural logics: a primer
as an entailment that It might be said in defense of at least it is "safe", in the sense that if is true, then it is always safe to infer from an arbitrary , since we run no risk of uttering a falsehood in doing so [...]. In reply we of course admit that if is true then it is "safe" to say so [...]. But saying that is true on the irrelevant assumption that , is not to deduce from , nor to establish that implies , in any sensible sense of "implies". Of course we can say "Assume that snow is puce. Seven is a prime number". But if we say "Assume that snow is puce. It follows that (or consequently, or therefore, or it may be validly inferred that) seven is a prime number", then we have simply spoken falsely.
According to Anderson and Belnap, thus, the classical derivability relation does not adequately mirror the usage of the words "follows from" either in ordinary language or in deductive sciences. When we say that follows from (or that implies ), we mean that actually depends on , that we are in a position to prove using the hypothesis . Viewed against such a background, weakening inferences are clearly not acceptable. Weakening says that if follows from the assumptions in , then it follows from any aggregate of assumptions which includes . But, generally speaking, some of the hypotheses in will not be used in deriving - worse than that, they may be totally irrelevant to whether holds or not. So it seems plainly incorrect to state that follows from the larger aggregate . Anderson, Belnap and their disciples devoted a lot of effort to characterizing relevant notions of implication and derivability. They assumed two criteria of relevance: a syntactic criterion (for to follow from it is necessary that the assumption be actually used in proving ) and a semantic one (for to follow from it is necessary that and share some common content). Both criteria, of course, can be made formally precise. This area of research is nowadays known as relevance logic14 (see e.g. Anderson and Belnap 1975; Routley et al. 1982; Dunn 1986; Read 1988; Restall 200+). In the following chapters, we shall have lots of opportunities to encounter various systems of relevance logic.
2) The paraconsistent objection. After having seen one of the drawbacks of admitting weakening on the left, let us verify that weakening on the right is not faultless either:
!#" $&%'" ()%+* ,#- . /0"
Francesco Paoli
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Also the principle is usually denoted by a Latin name: ex absurdo quodlibet. Literally, that means: from an absurdity anything follows. The gist of this law may be expressed by saying that since proving contradictions is obviously unacceptable, if I have done so I might as well have proved any old proposition. Is this reasonable? Here Lewis and relevance logicians part company. Lewis not only accepted the inference from to , but also devised an argument to justify it (the so-called "independent proof": Lewis and Langford 1932). Assume and . From I deduce . Then, by a disjunctive syllogism, from and I can conclude . This last step, according to Anderson and Belnap, is where Lewis' argument fails, for disjunctive syllogism cannot be accepted as a valid mode of reasoning (see e.g. Dunn 1986 for a discussion). In most systems of logic (but not all: see the already quoted Tennant 1987 for an exception), thus, the inference rules derived from the principles of disjunctive syllogism and ex absurdo quodlibet stand or fall together. But why should they be rejected? At least two reasons seem to recommend such a course of action. First and foremost, in everyday reasoning it is not true that any minor inconsistency permits us to draw any conclusion we like: most databases, for instance, are in a position to handle local inconsistencies without being driven to collapse (Routley and Routley 1972; Belnap 1977). To quote an example by Dunn (1986, p. 152): "One would not want trivially inconsistent information about the colour of your car that somehow got fed into the FBI's computer [...] to lead to the conclusion that you are Public Enemy Number One". In the second place, no logic which admits the inference from to can serve as the basis for a nontrivial inconsistent theory. But sometimes it may be worthwhile to study such theories: naive set theory (with Russell's paradox) and naive truth theory (with the liar paradox) are two cases in point (Priest and Routley 1989). A set theorist, for example, could be confronted with two alternatives: she might retain the simple and well-understood classical logic and go through all the complications of ZFC, or she may choose to keep the beauty and simplicity of the notion of set given by extensionality and comprehension, while dropping ex absurdo quodlibet in order to "localize" inconsistencies and render them harmless. If she does so, she somehow behaves like the relativity theorist who adopts a seemingly innatural and complicated geometrical system (elliptical geometry) to obtain great simplifications in the overall framework of his physical theory. These motivational stimuli originated a field of research now known as paraconsistent logic15 . A logic is said to be paraconsistent if it can be extended by nontrivial inconsistent theories; it is called dialethic16 if it contains
24
Substructural logics: a primer
explicit contradictions already within itself. Three trends of research in paraconsistent logics are usually distinguished (cp. Priest and Routley 1989): positive plus systems (Da Costa 1974), based on extensions of negationless intuitionistic logic; discussive logic (Jaskowski 1969), characterized by the rebuttal of adjunction; finally, depth relevance logics (Priest and Routley 1989). The pioneers of relevance logic, in fact, originally favoured two rather strong systems - called E and R - which were not suited for paraconsistent purposes. The concern for paraconsistency motivated then the study of weaker relevance logics with a simple and neat semantics, paving the way for a now thriving field of investigations (see e.g. Priest and Sylvan 1992; Brady 1996). Many important articles on paraconsistent logic are included in the miscellaneous volumes edited by Routley et al. (1989), Batens (2000) and Carnielli et al. (2001). 3) The nonmonotonic objection. We have seen above that the weakening rule can be interpreted as a "monotonicity principle": if follows from , then such a relationship continues to hold whatever additional information I may attach to . But everyday reasoning is often nonmonotonic, since our ordinary inferences are nearly always drawn under the implicit assumption that "normal" conditions obtain. For example (Stalnaker 1968), if I assume that a match is struck then I infer that it will light. This if normal conditions obtain; but if I assume that the match is struck and that it is wet, then I cannot draw the same inference as before. The concern for nonmonotonicity opened up a successful stream of research, which usually goes under the heading of nonmonotonic logics (Makinson 1993; Fischer Servi 1996) and has produced useful applications in artificial intelligence and computer science. It is worthwhile to remark one difference between nonmonotonic and relevance logics: the latter focus their attention on nonmonotonic implication connectives and syntactical deducibility relations, retaining at the same time a classical concept of logical consequence17 , whereas the former go well beyond that, focusing on nonmonotonic consequence relations. To better understand this point, consider the passage by Anderson and Belnap which we quoted earlier. The authors admit that if is true, it is "safe" to assert given the truth of , even though we cannot say to have deduced from . The nonmonotonic view is different: if holds under normal conditions, we cannot be sure that it will continue to hold if a modification of such conditions (namely, ) is considered.
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2.4. Reasons for dropping the contraction rules The contraction rules are liable to a number of criticisms, too. Let us examine some of them18 .
!
1) The intuitionistic objection. Consider the following proof in LK:
(CR)
By allowing contraction on the right, we are in a position to prove the law of the excluded middle, which is notoriously rejected by the intuitionists. All the well-known intuitionistic objections to such a principle, which we shall not repeat here, can thus be regarded as a first reason to restrict contraction. 2) The many-valued objection. The excluded middle may be viewed as a translation into the object language of propositional logic of the metalinguistical principle of bivalence, according to which any sentence is either (perfectly) true or (perfectly) false. Suppose, however, to have good reasons to believe that the truth values of some sentences are either undetermined, or somewhere in between the true and the false. Some pertinent examples could be e.g. sentences containing vague predicates ("John is a young person", if John is about 35) or sentences about contingent events in the future (like in Aristotle's famous example, "There will be a sea battle tomorrow"19 ). Then you will have qualms about accepting the excluded middle, and the contraction rules (or at least the right contraction rule) with it. The latter example was not chosen randomly. It was just reflecting on this passage by Aristotle that the Polish logician Jan Lukasiewicz (1930) came up with one of the earliest systems of many-valued logic, initiating what is now one of the most fruitful areas of contemporary logical research (see Urquhart 1986 for an introduction; Hajek 1998 and Cignoli et al. 1999 for much more advanced and selective presentations). Lukasiewicz investigated logics with finitely many as well as with infinitely many truth values. From our viewpoint, many-valued logical systems are important because they contain nonidempotent conjunction and disjunction connectives, and group-theoretical conjunctions and disjunctions are usually nonidempotent in substructural logics20 . Indeed, we shall see that several many-valued logics can be presented from a proof-theoretical point of view as logics where contraction is appropriately restricted.
26
Substructural logics: a primer
3) The linear objection. So much for contraction on the right. But also contraction on the left could be considered objectionable. Suppose, in fact, that formulae are interpreted as types of data, and formula occurrences as actual, concrete, bits of information. The rule CL does not seem to be supported by this interpretation: it would amount to saying that what can be deduced from two or more bits of information of type can also be deduced from just one. Suppose, for instance, that you are a judge and that you have inferred that someone is guilty of a given crime on the basis of three different pieces of evidence of the same type (take , for example, to be a witness' report saying "I have seen the defendant on the scene of crime"), plus possibly other relevant information. Should you have collected just one report of type , maybe you would not draw the same inference, for you might consider the available evidence insufficient to prove the defendant guilty. The idea that propositions and formulae are concrete resources whose multiplicity has to be taken into account is at the root of Girard's linear logic (see Girard 1987; Avron 1988; Gallier 1991; Troelstra 1992), which is now an extremely popular and rapidly growing subject, with an impressive range of connections with, and applications to, computer science and artificial intelligence. Linear logic rejects, on the grounds just given, all the weakening and contraction rules (retaining however, at least in its original version, the exchange rules), but is in a position to recover the full expressive power of classical logic by means of the introduction of special modalities (the exponentials) which permit to rescue the above-mentioned structural rules in a "controlled" way. We shall encounter linear logic nearly everywhere throughout this book; in particular, the role of exponentials will be clarified in the next chapter.
!" #$ % & ' is usually called the law of The principle
4) The Curry-Skolem objection. Consider the following proof in LK:
absorption. Even though few logicians have considered it objectionable in itself, it was discovered by Curry to have a role in the development of paradoxes in the context of naive set theory. Let us see why. It is well-known that Russell, at the turn of last century, showed that naive set theory, which includes an unrestricted comprehension principle, contains a formula which is provably equivalent to its own negation. This is not a
Francesco Paoli
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contradiction per se, but it becomes such if the underlying logic proves the law of excluded middle and other classical principles. Some mathematicians, including Brouwer, began therefore to nurture the belief that an intuitionistically correct set theory would have been free of Russell's paradox. Curry (1942), however, managed to obtain the following variant of Russell's paradox using nothing more than intuitionistically acceptable principles. Let ={ }, where is an arbitrary formula. Then:
) ) ) 3. 4 5 6
)
Def. of Def. of Law of absorption 1, 3, modus ponens 2, 4, modus ponens 4, 5, modus ponens
It follows that for a set theory, in order to be trivial, it is sufficient to contain, besides modus ponens, an unrestricted comprehension axiom and the (intuitionistically correct) law of absorption. From a substructural point of view, however, the excluded middle and the law of absorption are equally vicious: the former requires an use of contraction on the right, while the latter presupposes an application of contraction on the left. What would happen, then, if we were to reconstruct set theory dropping both contraction rules? The first logician who explored the possibility of building up set theory on a nonclassical logical basis was Thoralf Skolem, who devoted a series of papers to the subject in the late '50s and in the early '60s (see e.g. Skolem 1963; see also Chang 1965; White 1979; Brady 1983). Skolem added a restricted comprehension axiom to Lukasiewicz's infinitevalued logic (where the existence of formulae that are, like Russell's paradoxical sentence, equivalent to their own negations does not lead to absurdity: any formula whose truth value is 0.5 is such) and investigated the resulting system. As we shall see in Chapters 2 and 4, contraction does not hold unconditionally in Lukasiewicz's logics, and this property makes them suited to the present purpose. A result by Grishin (1982), however, indicated that Lukasiewicz's infinitevalued logic is still too strong to do naive set theory: if we add extensionality to plain contraction-free logic - which is weaker than Lukasiewicz's - contraction can be recovered, so that the system resulting by the addition of comprehension becomes trivial. It seems, therefore, that also weakening has a role in producing the paradoxes. The subsequent research on logical bases for naive set theory, therefore, has focused on systems in the vicinity of linear logic (see e.g. Shirahata 1996).
28
Substructural logics: a primer
It is instructive to compare the viewpoint of the paraconsistent set theories of § 2.3 with the viewpoint of these set theories based on contraction-free logics. Both kinds of theories aim at retaining the naive notion of set, fully specified by the axioms of comprehension and extensionality, and are ready to pay the price of abandoning the safe harbour of classical logic in order to achieve this goal. In particular, both have to reject absorption on pain of triviality. However, while paraconsistent set theorists are willing to accept an inconsistent theory so far as it is not trivial (thereby they regard Russell's paradox as a theorem about sets with "inconsistent properties": Priest 1979), Skolem and the other logicians who followed in his footsteps strive to obtain a consistent theory, even though this effort may lead - due to the lack of many classical inferential schemata - to considerably weak systems21 . 5) Other criticisms. From a technical point of view, the proof theory of logics containing contraction rules is far more intricate than the proof theory of contraction-free logics. In Chapter 3 we shall see two concrete examples, concerning cut elimination and the search for decision algorithms, which justify the previous assertion. One of the reasons of this lies in the fact that, even if contraction rules do not delete formulae from a proof like the cut rule, they may nonetheless delete from it some formula occurrences, and this complicates enormously the process of proof search.
2.5. Reasons for dropping the exchange rules Proceeding in our review of the weak spots of structural rules, there comes now the turn of exchange rules. There are two main sources of blame here; let us see what they are22 . 1) The linear objection revisited. As we have remarked in the preceding section, one of the main philosophical tenets of linear logic is the idea that formulae are concrete resources and that their multiplicity deserves consideration. A logic of concrete data, however, cannot disregard the problem of the accessibility of such resources (Abrusci 1992). Data often have spatiotemporal locations e.g. in the memories of humans or of computers, and sometimes remote data are less easily accessed than adjacent ones. In a logic of resources, therefore, not only the multiplicity of data, but also their order seems relevant. To access a resource we often have to overcome spatiotemporal obstacles; the exchange rules remove such obstacles and appear therefore inappropriate in the context of actual information processing situations, where the arrangement pattern of data is essential.
Francesco Paoli
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This concern is at the root of the thriving field of investigations about noncommutative linear logic and its ancestor, the Lambek calculus (Lambek 1958), whose original motivation will be examined presently. Noncommutative linear logic, once defined by Girard (1995) a "Far West" in virtue of its intrinsic difficulties, has been intensively investigated in the 1990s (see e.g. Yetter 1990; Abrusci 1991). The present trend of research into noncommutative logics seems to favour "mixed" logics, where both commutative and noncommutative connectives are present (Casari 1997a; Abrusci and Ruet 2000). 2) The categorial objection. Another motivation for the rejection of exchange rules comes from the field of linguistics, and especially of categorial grammar. The theoretical core of this discipline, foreshadowed by Husserl and Lesniewski and laid down in full detail by the Polish logician Ajdukiewicz, lies in the assignment of syntactic types to natural language expressions. Types describe the syntactic roles of the expressions they are attached to; in Ajdukiewicz's calculus, the basic types are (noun) and (sentence). More complex types can be subsequently constructed by means of suitable typeforming operations. For example, an expression of type is an expression which, if applied to an expression of type , yields an expression of type . In the original calculus by Ajdukiewicz, application of one expression to another (which corresponds to juxtaposition of expressions in natural language) is seen as the only operation by means of which complex expressions can be obtained out of simpler ones. How does noncommutativity enter into this picture? It is readily seen. Joachim Lambek (1958) remarks that most natural languages have precise rules for word ordering: for instance, John works is a grammatical sentence in English, whereas works John is not. The intransitive verb works has therefore type : when applied to the right of an expression of type (a noun), it yields an expression of type (a syntactically correct English sentence). On the other hand, the adjective poor has type : when applied to the left of an expression of type , it yields another expression of type (namely, a complex noun phrase). Lambek also considered rules for permissible transformations on syntactic types and devised a Gentzen-style sequent calculus for producing them. This calculus is like LJ except for: a) lacking all of the structural rules; b) having group-theoretical conjunction, retro-implication and post-implication as its sole connectives. It is generally known as Lambek calculus and has been intensively investigated by Došen, Buszkowski and others (see e.g. Buszkowski 1997)23 . In passing, we remark that Ajdukiewicz's and Lambek's type-theoretical grammars are only appropriate for very limited fragments of natural language.
30
Substructural logics: a primer
More powerful grammars - endowed with further expression-forming operations, such as -abstraction, and/or with further types, such as intensional types - have been subsequently produced, one of the most refined being surely Montague grammar (Montague 1974). Excellent introductions to this field of research are e.g. van Benthem (1991), Gamut (1991), Partee (1997).
2.6. Reasons for dropping the associativity of comma Seemingly, all the structural rules of sequent calculus have received due consideration in our discussion. However, there is a "covert" structural rule which we did not address so far. Remember that in Gentzen's view sequents are made up by two sequences of formulae separated by an arrow. In a sequence of items both multiplicity and order count, but the items are not grouped together in any particular way: is the same as . In other words, comma is taken to be associative. So, why not envisage the possibility of dropping this last structural rule? There are several grounds, indeed, for refraining from such a move. For example, the setting up of an algebraic semantics would become a rather desperate enterprise. However, there have been logicians who considered weak substructural logics where comma is not necessarily associative (see e.g. Došen 1988; see also Restall 2000). Even Lambek calculus has been given a nonassociative version (Lambek 1961).
3. WAYS OF READING A SEQUENT Summing up the preceding discussion, we have seen that there exist good reasons to drop one or more of the structural rules devised by Gentzen for his calculi LK and LJ. But what would be the intuitive interpretation of the resulting calculi? Classical and intuitionistic sequents, after all, reflect two analyses of deducibility which are by now well-understood; are we in a position to replace them by concepts which are just as sound? In the following, we shall analyze the classical and the intuitionistic readings of sequents and suggest two alternative readings, neither of which supports the plausibility of all the structural rules.
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3.1. The truth-based reading Both in his Investigations into Logical Deduction and in his The Consistency of Elementary Number Theory, Gentzen devotes a few lines to the explanation of the intuitive meaning of his sequents. According to him, a classical sequent of the form has the same meaning of the (classical) formula ("If the assumptions hold, then at least one of the propositions holds"). Anyway, since all classical connectives are truth-functional, classical sequents can be read in such a way that truth and falsity are the only concepts involved in the explanation. In other words, the following truth-based reading of sequents suggests itself:
(TB)
#holds ! 's is iff either at least one of the " false or at least one of the 's is true.
Commas, as we have just seen, are interpreted as "ands" on the left side of the arrow and as "ors" on its right side. What if either side is empty? Well, in that case two special fillers come into play. If the antecedent is empty, it must be read as if it were filled by the conjunction of all true formulae; dually, if the succedent is empty, we have to imagine that it is occupied by the disjunction of all false formulae, i.e. an unspecified false formula. As a consequence, the sequent holds iff implies something false, i.e. iff is false (in full accordance with TB); the sequent holds iff is implied by the conjunction of all truths, i.e. iff is true (again, in accordance with TB); finally, the sequent holds iff the conjunction of all truths implies something false, i.e. never. In the truth-based reading, then, the empty sequent has the same meaning as a contradiction. In fact, it is quite easy to show that adding the empty sequent to LK is equivalent to add both and for any formula :
$
%
%
% ) *,+ - '&( ) *,+ -
'&( % ) 2430'&( &($ ) .0/1 -
3.2. The proof-based reading In § 1 we have remarked that Gentzen's sequent calculi brought about a major shift of perspective in contemporary axiomatics, given by the fact that their focus is not on formulae, but on sequents, which are meant to represent
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Substructural logics: a primer
inferences. However, two aspects of the truth-based readings of sequents seem unsatisfactory from this point of view. In the first place, there is nothing in the explanans which refers to anything like a proof, a construction, or an inference: the talk is about truth values, not about deduction procedures. In the second place, an inference or a proof usually starts from a given number of assumptions and ends with a single conclusion, not with multiple conclusions. Intuitionists, therefore, suggest a different interpretation of sequents, which is only appropriate for their single-conclusion sequents and is coherent with their philosophy of logic, especially with the so-called "Brouwer-HeytingKolmogorov" interpretation of intuitionistic connectives (for which see e.g. Troelstra and van Dalen 1988). Such a reading is especially perspicuous if formulated with reference to the Curry-Howard isomorphism (also known as "formulae-as-types" isomorphism: Troelstra and Schwichtenberg 1996). The idea of this correspondence is the following: since what really counts about a formula is not whether it is provable or not, but how it can be actually proved, it is expedient to give proofs names, in order to distinguish between different proofs of a same formula. A formula, thus, can be identified with the set of its proofs, and inferences can be interpreted as "programmes" or instructions for manipulating proofs24 . According to this discussion, intuitionistic sequents admit the following proof-based reading: (PB)
proofs
holds iff there is a construction which transforms the , respectively of , into a proof of .
In this reading, commas have always a conjunctive role, for the succedent can contain no comma at all. The two fillers of classical logic have a different meaning here: an empty antecedent is read as if it were filled by the conjunction of all logically provable formulae, whereas an empty succedent is read as if it were filled by an unspecified absurdity, for which there can be no proof. The empty sequent, then, holds iff from no assumption at all (viz. by purely logical means) I can extract a proof of an absurdity, i.e. never. Also in intuitionistic logic, then, the empty sequent stands for a contradiction.
3.3. The informational reading According to the proof-based interpretation of sequents, inferences are read as instructions for constructively manipulating proofs. Could they be interpreted also as instructions for information processing? In other words, would it be permissible to replace the word "proof" by "datum" and the word "formula" by "type of data"? The answer is negative, and the reason is just the
Francesco Paoli
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presence of the structural rules of weakening and contraction, which would no longer be sound under such an interpretation. In § 2.4, introducing some of the philosophical motivations underlying linear logic, we remarked that contraction rules are not supported by an informational reading of sequents: what can be deduced from data of type is not necessarily derivable from a lesser amount of data of the same type. But the same thing might be repeated with respect to the weakening rules. Reconsider for a while the example of our judge, and suppose that he has decided to convict a defendant, inferring his guilt from the evidence . Now assume that a new piece of information is available, amounting to an alibi for the defendant; the judge may retract his previous inference on the basis of the new body of available knowledge . Linear logic, therefore, suggests the following informational reading of sequents:
holds iff there is a procedure which, if we , respectively of type , yields as an input data , respectively of type . output data
(I)
This interpretation, exactly like the preceding one, lays a special emphasis on the procedural or computational aspect of deductions. In the proof-based reading, however, manipulations were supposed to produce exactly one output, coherently with the belief that a construction cannot yield more than one conclusion. This is certainly plausible if we assume that the construction in point is carried out by a single agent; but it is no longer so if we conceive of constructions as effected in parallel by several agents who exchange data among one another in a cooperative setting (Abrusci 1992). The intuitionistic asimmetry between antecedents and succedents, corresponding to the asymmetry of the input-output relation (in any construction, many inputs produce a single output), readily vanishes in this perspective: suppose that the agent transmits a datum to the agent ; is simultaneously an output (from the perspective of ) and an input (from the perspective of ). Moving a datum from one side of the arrow to the other corresponds to dualizing it, i.e. transforming an input into an output and vice versa, and this operation is perfectly symmetrical. In the informational reading, comma is more difficult to interpret. All we can say is that it matches the connective (instead than ) on the left of the arrow and the connective (instead than ) on its right, and that it expresses a certain way of combining bits of information. Which way? To grasp such an idea, it can be helpful to read once again the remarks about the concurrent "and" in § 2.1. We could use the words "and-with" to denote the
34
Substructural logics: a primer
informal meaning of comma on the left, and "or-with" to denote its informal meaning on the right. The sequent means that inputting a datum of type leads to an inconsistency; means that by inputting no data at all (viz. by purely logical means) we can get as an output a datum of type . Finally, the empty sequent means that we can extract an inconsistency out of the empty input, which means out of pure logic.
3.4. The "Hobbesian" reading There is an interpretation of sequents which stresses even more profoundly the computational aspects of deduction. We call it "Hobbesian" because a parallel between computation and reasoning was first traced by Thomas Hobbes in his Elements of Philosophy (I, 1, 2): By ratiocination, I mean computation. Now to compute, is either to collect the sum of many things that are added together, or to know what remains when one thing is taken out of another. Ratiocination, therefore, is the same with addition and substraction [...] so that all ratiocination is comprehended in these two operations of the mind, addition and substraction.
We do not even attempt to provide an historically correct exegesis of Hobbes' remarks; rather, we content ourselves with an interpretation which is at least partly Hobbesian in spirit. Thus, it may be thought that information processing essentially takes place by means of two operations: combining two data (Hobbes' "addition") and erasing the informational content of a datum (Hobbes' "substraction"). The former operation is easier to grasp; as to the latter, consider once more our earlier forensic example. Suppose that we have recorded a witness' statement, whose content is that the defendant was on the scene of crime. Now, imagine that another witness maintains that the defendant was not on the scene of crime. Provided that the two witnesses are equally reliable, the latter statement erases the informational content of the former: if we combine them together, we are in a position to infer nothing about whether the defendant was or was not on the scene of crime. Summing up, the "Hobbesian" reading of sequents takes them to represent processes which restructure information by means of the above-mentioned operations: (H)
holds iff the informational content of the , respectively of type , can be restructured so data , respectively of type . as to yield data
Francesco Paoli
35
The operations of "erasing" the content of a datum and "combining" data are mirrored, respectively, by the connectives and . It is readily seen that the rules L, R, L, R and - if we take to mean the same as - also L and R make sense according to this reading. What about the structural rules? Apparently, they cannot hold unrestrictedly. Consider, however, the following two rules of "balanced" weakening and contraction:
They look all right: in fact, if the content of can be restructured so as to yield the content of can be restructured so as to yield , since the same informational content is being added on both sides. This accounts for BW. As
to BC, suppose that can be restructured so as to yield ; since BW is correct, we may add a datum of type on both sides. But a datum of type erases the content of a datum of type ; so can be taken as a
restructuring of the content of . As we shall see in Chapter 2, this interpretation of sequents is adequate for the group-theoretical fragment of Abelian logic, a logic whose models are Abelian lattice-ordered groups. Abelian logic, independently introduced by Meyer and Slaney (1989) and Casari (1989), is a dialethic logic, for it contains explicit contradictions. Unlike the logics examined so far, which are subsystems of classical logic, it licenses classically incorrect inferences (as it is apparent from the previous remark). The connectives and have the same rules and thus can be identified with each other; it follows that comma has the same meaning ("with") both on the left and on the right of the arrow. No fillers are postulated, for Abelian logic takes at face value the emptiness of antecedents and succedents: both an empty antecedent and an empty succedent correspond to the empty piece of information. So, both and mean that is uninformative: its informational content amounts, after some restructuring moves, to the empty datum. Finally, the empty sequent expresses a trivial truth: that the content of the empty piece of information is equal to itself (there is even no need for a restructuring).
Notes 1. See § 2 below. 2. We shall discuss in greater detail the informal meaning of sequents in § 3 below. 3. It is worth remarking that some authors (e.g. Girard et al. 1989; Wansing 1998) do not classify cut among structural rules, preferring to set it apart from this group of rules. 4. Cp. § 3.
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Substructural logics: a primer
5. All of this vague terminology will be made precise in Chapter 4, where display calculi will be dealt with in greater detail. 6. Visibility is one of the ideas underlying basic logic, a weak substructural logic to which we shall revert in Appendix B. 7. Good discussions of the distinction between sharing and non-sharing rules, as well as of different systems containing appropriate combinations of the former and the latter, can be found in Troelstra and Schwichtenberg (1996) and in Baaz, Ciabattoni et al. (1998). 8. The pair "additive/multiplicative" stems from Girard (1987) and is preferred by linear logicians, while the pair "extensional/intensional" is privileged in relevant circles. Our usage has therefore the advantage of granting us a position of neutrality in this terminological controversy. 9. Except for some hints in Appendix B. 10. This terminology is drawn from Casari (1997a), where an excellent discussion can be found of themes related to the present issue. 11. An amusing defence of the ambiguity of "and" and "or" in natural language, with examples drawn from a restaurant situation, can be found in Danos and Di Cosmo (200+). 12. For Epstein's relatedness logic this is not wholly correct: while relatedness implication is not generally transitive, the relation of derivability, which is defined classically in that logic, is such. But since this relation does not mirror the behaviour of provable implication (the usual deduction theorem, in fact, does not hold), one could just as well say that the real derivability relation of relatedness logic is not unrestrictedly transitive. 13. The merit of questioning the correctness of weakening inferences, however, should not be credited to modern logic. As Casari (1997a) remarks, Sextus Empiricus already acknowledged that an argument can be inconclusive by redundancy ( ) "when something external and superfluous is adjoined to the premises" (Against the Mathematicians, VIII, 431). 14. The term "relevance logic" is more fashionable in the United States, while "relevant logic" is more often used by the British and the Australasian logical communities. 15. The introduction of the expression "paraconsistent logic" is usually credited to the Peruvian philosopher Francisco Miró Quesada. 16. The adjective "dialethic" was first used in this context by Richard Routley. 17. In fact, the algebraic strong completeness theorem holds for relevance logics only if the deducibility relation at issue is not the "relevant" deducibility relation, but a rephrasing of the classical one. On this subject, see also Chapters 2 and 6. 18. More reasons for dropping contraction are mentioned and discussed in Restall (1994a): according to him, in fact, contraction-free logics can be applied - inter alia - to an analysis of vagueness, to issues arising in the logic of actions, and to model a notion of "information flow". 19. De interpretatione, 9, 19a. 20. See Chapter 2. There are however some exceptions: for instance the logic RM, on which we shall return, is a substructural logic, but its disjunctions and conjunctions (both the latticetheoretical and the group-theoretical ones) are idempotent. 21. For a philosophical defence of contraction-free systems of naive set theory, see Weir (1998).
Francesco Paoli
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22. Once again, some kind of distrust in the virtues of exchange can be traced back to the antiquity (Casari 1997a). Sextus Empiricus, in fact, came up with some counterexamples to premiss exchange in the context of his discussion of changing arguments ( ). 23. We shall return to these issues in Appendix B. The original Lambek calculus placed further restrictions on the R and R rules, which we shall disregard until then. 24. The expression "formulae-as-types" stems from the fact that in the Curry-Howard isomorphism proofs are named by terms of the -calculus, and formulae are viewed as the types of such terms: two terms with the same type are like two proofs of the same formula.
PART TWO THE PROOF THEORY OF SUBSTRUCTURAL LOGICS
Chapter 2 BASIC PROOF SYSTEMS FOR SUBSTRUCTURAL LOGICS
In this chapter, we shall introduce sequent calculi and Hilbert-style calculi for several substructural logics. A standard way to accomplish such tasks, in handbooks whose scope comprises various logical systems, is to focus on a basic system and then consider its extensions. Such extensions can be either axiomatic (the language remains the same as in the basic calculus, but more postulates are added) or linguistic (the language is enriched by new logical constants and, possibly, some postulates governing these new symbols are introduced). The choice of such a basic system must perforce be, to some extent, arbitrary. However, a delicate tradeoff is involved: this system must be neither too weak, for it would lack any intrinsic interest, nor too strong, since its extensions would be too limited in number. In view of these considerations, we choose to take as our starting point what is usually known as the "subexponential fragment of classical propositional linear logic without additive constants". This is a certainly interesting system and, as we shall see in this chapter and in the next, is general enough to admit a wide range of substructural logics as extensions. Before presenting our basic sequent calculus LL, however, we have to dispatch some tedious but unavoidable preliminaries.
42
Substructural logics: a primer
1. SOME BASIC DEFINITIONS AND NOTATIONAL CONVENTIONS Since our notation differs in part from other standard articles or textbooks on linear and substructural logics, we included a synoptic table of notations for the reader's convenience (Table 2.1): Table 2.1. A synopsis of notations
Our notation
Troelstra 1992
&
Girard 1987
?
?
Restall 2000
?
?
Definition 2.1 (some conventions about languages). Throughout this volume we shall be concerned with propositional languages, each containing a denumerable set of propositional variables and a given number of connectives, drawn from the set
"!#%$&%'&)(*+02+14, 36 5 .-/ ?
.
0214365
Of these, %$&%'&"!#)( are binary; .-/ ? are unary; +, are nullary. Nullary connectives are sometimes referred to as propositional 071 constants. The connectives " "!#+ are called group-theoretical; 6 3 5 the connectives $&%'&)(*, are called lattice-theoretical; finally, -/ ? are the exponentials. We adopt the convention according to which unary connectives bind stronger than either %$ or ' , which in turn bind stronger than either ! or ( . Hereafter, we list the languages on which our calculi will be based, together with their respective sets of logical constants:
Francesco Paoli £1 : £ : £ : £: £! : £"# : £:
43
; ; ? ; ; ; $&%$ ' $($)$+*,$-/$ .10 .
;
The letter "£" will refer to a generic language in the above list. By VAR(£) and FOR(£) we shall denote, respectively, the set of all the propositional variables and of all the well-formed formulae of the language £. Definition 2.2 (some conventions about calculi). Formal calculi - whether axiomatic or sequent calculi - will be referred to by boldface capital letters. The letter "S" will stand for a generic calculus; the letter "L", followed by a specific letter, will be employed to refer to sequent calculi (a convention which should be reminiscent of Gentzen's usage of the same letter in "LK" and "LJ"); likewise, the letter "H", followed by a specific letter, will designate Hilbertstyle calculi. If S is a calculus, then: 2 Si will denote its purely implicational fragment, based on the language £4 ; 2 Sg will denote its group-theoretical fragment, based on the language £3 ; 2 S+ will denote its positive fragment, based on the language £6 . If S is any sequent calculus, by writing 4 S 6 5 7 we shall mean that 6 5 7 is a theorem of S. Moreover, by writing 4 S 6 8 7 (or by saying that 6 8 7 is a theorem of S) we shall mean that both 4 S 6 5 7 and 4 S 7 5 6 . If S is any axiomatic calculus, by writing 4:S 9 we shall mean that 9 is a theorem of S. Moreover, by writing 4; (or by saying that S 9=<> 9=<> is a theorem of S) we shall mean that both 4:S 9,?> 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 # / * 4 0 2530 # 5 / * 0 6 / # 6 - * 7680 # * 0 - * 79;0 7980 # 9 / *
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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' & -$34 34 . # "#
'&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 #
)#'&1 7 .<$3;#. ) -84 $3 -34$ 8 ' & -84 : 34'&
6 34: 84
684 $34 &, -$3 84 &, -8 $ 3 -&9 $ 8 -& $3 = 8 7 '& $ 8 -3 >'& .& .?& #" & %) &
Proposition 2.1 (theorems of LL). The following sequents are provable in ; (ii) ; (iii) LL: (i) ; (iv) ; (v) ; (vi) ; (vii) ; (viii) ; (ix) ; (x) ; (xi) ; (xii) ; (xiii) ; (xiv) ; (xv) ; (xvi) ; (xvii) ; (xviii) ; (xix) ; (xx) ; (xxi) ; (xxii) ; (xxiii) ; (xxiv) ; (xxv) ; (xxvi) ; (xxvii) ; (xxviii) ; (xxix) ; (xxx) ; (xxxi) ; (xxxii) ; (xxxiii) ; (xxxiv) ; (xxxv) ; (xxxvi) ; (xxxvii) ; (xxxviii) ; (xxxix) ; (xl) ; (xli) . Proof. For its most part, this lemma will be left as an exercise for the reader. We only present a couple of examples: the left-to-right part of (viii) and the right-to-left part of (xxiv).
46
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Substructural logics: a primer
! " ' &' *&* $ # ! $ # ! % '&')(+* %&% *&')(,* "! "! %%& ' " %/'&&.%%/)/)00')')(,(+**1 1 %%-/* *&&%%/)/20')0')(+*(+1 *1 " #$ 0$%/'13(40$%/*15&%/)0')(+*1 )/ -87)98%:'
(
and are, in a Remark 2.1. By Proposition 2.1, the connectives way, redundant, since they can be reduced to other connectives of £1 . In fact, by (x) and (xii) is provably equivalent to ; by (x) and (xiii) is provably equivalent to ; finally, by (x) and (xxviii) 1 . By (iv), (v), (xxix) and is provably equivalent to (xxxii), moreover, we are allowed to write , , , and , disregarding parentheses.
%/' %=(+' H F K IJIBIK H G
%7'
H F L IBIBIL H G
9;0 98%7<95'1 9;0 98%<>?9'1 %@F /)ABABAC/D% G E % F 7)ABABAC7D% G
The next lemma clarifies the informal meaning of sequents in LL and justifies Definition 2.5 below. of sequents in ).SXW) (i) IBIBICWM S LL HOF NPProposition IJIBIBNHRG QSTF NC2.2IBIBIBNS Y (meaning Q E H ) U B I B I C I D U R H V F G F Z\[ N]_^<`ba ; (ii) M H_F NCiffIBIJIBNHRG Q M iff M Qdc Z HEF U2IBIBICUDHG Y a Z\[ ^<`eN]gf`bah (iii) M Q.STF NCIBIBIBNS Y iff M QSXF W)IBIBICWS Y Z$[ f`eN]_^<`ba . i Proof. (i) Left to right: H_NCIBIBIBNHRQSTNCIBIBIBNS HEF F UIJIBICUDG HRG QF STF NCIBIBIBY NS Y H@F U)IBIjIkUDHRG QSlF W)IBIBICWS Y mJrpq m sut q QHEF U)IBIjIkUDHRG VSlF W)IBIBICWS Ynm o pq
LL
LL
LL
LL
LL
LL
Francesco Paoli
47
, , and let be the + + , , $ %$ & %& + , + , '$ (%& $ (%&) *$ %& !+ """ #. %$ $ %& + """ #. %& Then we can graft onto the following proof tree: 14+ %1 + 14, %1 , 16+ 5714, %18+ 31 , + /"0"" #. %& & %12+ """ 31 9 + """ #. %18+ """ 31 9 (ii), (iii): left as an exercise. : of a sequent). If ;!@ %?8@ <2.5===<3? B (formula-translation is a sequent, its formula-translation CED ; @ <===< ;#A >%?8@ <==0=<3?4B F is defined as follows : G ;D @ ;H=H= = H ===;# A ;I%F ?@ J===J7? B DLDLKK < NPM OSNP< MUOQF T; OQF H G6?6 R @ A DLK TOS< M NPOQF ;; G @ J== = J7? B XLY[Z \U]^Q_ . GWV
Right to left. Let following proof:
2
LLg , the group-theoretical fragment of LL, has a nice syntactical property, for the statement of which we need a couple of definitions.
`
`
a X `cbed _
d f g ` a XX `hbi` _j_j]l]k^ a XX `hb3m _n_j]]^ ^ for `)om a X `hb } _j]ts X _ a `hb V a X `hbqprd _j] a X `ub d _zy X _ a XX `hb*dv%w _j_j] ] a X X `hb3w _|_ry s a X X `hb d _ _ a `hb*dx7w a `hb d a `hb3w a `hb d{7w a `hb d a `hb3w X ~ _ of a multiset of formulae ~ is obtained by defining The ` -count a `hb a X `ub*d Z/0Z d _ as
a X `hb d _ and by setting a X `ub ~ _j]^ if ~ is empty.
Definition 2.6 ( -count: van Benthem 1991). The -count of a formula FOR(£5 ) w.r.t. the variable is a function whose values are integers. It is inductively defined in the following way:
48
Substructural logics: a primer
Definition 2.7 (balanced calculi). The sequent calculus S is balanced iff, for any sequent which is provable in S, it is for every variable . Proposition 2.3. LLg is balanced.
Proof. Induction on the length of the proof of in LLg . Let us check e.g. the rules R and L. (Ad R). By induction hypothesis, . But then , i.e. . ! (Ad L). By induction hypothesis, and ) &'( * ) #"$ ! % ! % . It follows that * ) + #"$ , ! - ! % = , i.e. * + "$ !+ . . Proposition 2.4 (non-theorems of LLg ). The following sequents are /10 02 3 0 unprovable in LLg : (i) ; (ii) ; (iii) 40 5 #67 083 5 9 0 93 ; (iv) ; (v) . Proof. All these sequents contain some variable : such that the : -count of the antecedent does not equal the : -count of the succedent. Hence, by Proposition 2.3, they are not provable in LLg . . It might be the case, however, that some of these sequents be provable in the full system LL. So far we have no guarantee, in fact, that LL is a conservative extension of LLg . But the following lemma rules out such a possibility. Proposition 2.5 (non-theorems of LL: Avron 1994). The sequents of Proposition 2.4 are not provable in LL either. Proof. We construct a model of LL in the integers. Let ;< be any mapping from VAR(£1 ) to = (the set of the integers). A valuation > is inductively defined on FOR(£1 ) as follows:
>'?@!A BC> < ?(@!A ; >'?7D E A BGFIHJ>'?KE A ; ; >'?KE Z L1M$A BNFIHJ>2?KE A!OP>'?4M$AQ>'?KE+RM$A BC>'?KE A!OP>2?4M$A HF ; >'?JA BNF ; >'?KE [ SM$ABC>'?KE A!OP>'?4M$A >'?KEUTVM$A B min ?K>'?KE,A#W>'?4M$AA ; >'?JA BX ; >'?KEUYVM$A B max ?K>'?KE A#W>'?4M$AA .
is a sequent, is "$#&%('")+ **-, ! '." /0 * , while by definition '."213*5476 . of the proofs in LL) that, if *;:<6 . onButtheit length 8 It = is 1easy> ,tothencheck'."9? (by1 induction is just as easy to verify that all of > the sequents (i)-(v) of the previous lemma have values less than 0 for some valuation @ . Hence they cannot be provable in LL. A
Francesco Paoli
49
If defined as
LL
2.2 Adding the empty sequent: the dialethic route The system LL can be viewed as a crossroads whence several routes branch off. The first one we examine is the dialethic route, corresponding to that "Hobbesian" interpretation of sequents which we hinted at in Chapter 1. As the reader will recall, in such an interpretation the empty sequent is not read as a contradiction, but as a trivial truth (roughly amounting to the fact that "nothing follows from nothing"). It makes sense, thus, to add it as an axiom to LL.
B
B
Definition 2.8 (postulates of LL ). The calculus LL , based on the language £1 , has the same postulates as LL, plus the axiom
C D7F E
B M C L C L GIH L C L JKH L Proof. Exercise. A The system LLB is not very interesting in itself, but, as we shall see in § 2.6, can serve as a basis for the formulation of Abelian logic. C L
B
Proposition 2.6 (theorems of LL ). The following sequents are provable in ; (ii) ; (iii) ; (iv) . LL : (i)
2.3 Adding the lattice-theoretical constants: the bounded route
B
As we have just seen, the dialethic route diverges at once from the classical one: LL , in fact, contains explicit contradictions. The next path we inspect, on the contrary, comes much closer to the classical route. In classical logic there is an abundance of formulae which follow from everything - namely, all the true formulae. Likewise, there is an abundance of formulae from which everything follows - namely, all the false formulae. Now, we shall make a much more modest assumption: that there is at least one formula with the
50
Substructural logics: a primer
former property and at least one formula with the latter. That is, while LL has just a group-theoretical truth constant (1) and a group-theoretical falsity constant (0), now we want to add lattice-theoretical truth ( ) and falsity ( ) constants. The system thus obtained, which we name LLB , corresponds to subexponential linear logic (cp. e.g. Troelstra 1992). Definition 2.9 (postulates of LLB ). The calculus LLB , based on the language £2 , has the same postulates as LL, plus the rules
As we shall see below, if we had introduced lattice-theoretical constants right at the outset, the dialethic route would have turned into a blind alley quite soon: the interesting dialethic systems, such as Abelian logic, become trivial upon such an addition.
Proposition 2.7 (theorems of LLB ). The following sequents are provable in ; (ii) ; (iii) ; (iv) LLB : (i) ; (v) ; (vi) . Proof. Exercise.
2.4 Adding contraction: the relevant route We shall now consider what happens if we add structural rules to LL not in a body, but rather one at the time. We saw in Chapter 1 that relevance logicians share a strong dislike for the weakening rules, not so much however for the contraction rules. Indeed, the first relevance systems investigated by Anderson and Belnap (1975) - E and R - included explicit contraction rules; it was only much later, with the introduction of depth relevance logics, that contraction-free systems made their way into the relevant arena. It is not quite easy to write down a sequent calculus corresponding to Anderson and Belnap's R - we shall be in a position to do so only in Chapter 4, where some refinements of ordinary sequent calculi will be examined. In the meantime, we shall focus on a simpler system, due to Meyer (1966) and usually known as distributionless relevance logic (or R minus distribution: Dunn 1986).
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Definition 2.10 (postulates of LRND ). The calculus LRND , based on the language £1 , has the same postulates as LL, plus the rules
The name "distributionless relevance logic" stems from the fact that R contains the distributive laws and , but to get the corresponding sequents in a Gentzen-style calculus contraction rules are not enough - some amount of weakening is needed. The problem of picking out the right amount was successfully tackled by Dunn, Mints, and Belnap, as we shall see in Chapter 4.
! # $ %# ! " $ #)( %!)(* " #
Proposition 2.8 (theorems of LRND ). The following sequents are provable (ii) in (i) ; LRND : ; (iii) ; (iv) ; (v) ; (vi) ; (vii) ; (viii) ; (ix) ; (x) ; (xi) .
" " '& # )&*%#
Proof. We prove (iv) and (v).
:?7 : <"7 < 687#6 :?; <@=5:"7 < + ,.-0/ 6 9:"7#6 :"; <>=69:"7 6%;A:"; <B5=69:@=69:"7 6%;A:"; <B5=6 9:"7 < 6%;A:"; <B7!69:"; <
+ 12-0/ < + ,.-0/ <
+ 12-0/
+ 3-0/
6%7#6 :"7 : 6=6%; :"7 : + 12-0/ 6%; :"7DCE6=5: + H.45/ 68; :"7DCE6F:@=GC 6 F: + I.45J K L0M N OP/ + 345/ 6%; :"7DCE6F:
+ 1245/
Q
We now want to find a suitable analogue of Proposition 2.3. It turns out, in has a "mild" balanced character. To precisely state what fact, that also LRND g it amounts to, however, we need some definitions.
52
Substructural logics: a primer
2.11 (antecedent and consequent parts of a formula). If Definition FOR(£ ), the concepts of antecedent and consequent part of are
1
inductively defined as follows: is a consequent part of itself; if is a consequent (antecedent) part of , then is an antecedent (consequent) part of ; if is a consequent (antecedent) part of , then is an antecedent (consequent) part of and C is a consequent (antecedent) part of ; if is a consequent (antecedent) part of , then both and are consequent (antecedent) parts of ; if is a consequent (antecedent) part of , then both and are consequent (antecedent) parts of .
Proposition 2.9 relevance lemma: Anderson and Belnap 1975 . If the is provable in LR , then each variable in occurs in sequent "! # at least once as an antecedent part and at least once as a consequent ND g
part. member of VAR(£ ) a value in the set $ (& Proof. '*)+& ,-)/Assign .,-)/. to' % . each Then assign a value in the same set to all the other 1
members of FOR(£1 ), according to the following tables:
0
A
;2 3<873 8:6 2?658:6 8@6<276 8?3=243 C 24352@6D8:69873 2;3=243524352435243 2?6<2435276>2769873 8@6<243527698:69873 8?3=243>873>873>873
B
276
1
243527698:69873 42 3=24352435243>873 276<243527698:69873 8:6<243>8:698:69873 873<873>873>873>873 E 243527698:69873 243<873>873>873>873 276<243>8:698:69873 8:6<243527698:69873 873=24352435243>873
S NO-P/N Q T F G H I"JLR K H M f hi-j/h k g lLc mn[n[nZmol d IUWJ VXc PZY[Y[Y[P\V]d K ^_c PZY`Y[Y[Pa^be M p c q n[n[n q p e r s tUuL v x w
It can be checked, by induction on the length of proofs in LRND g , that if it is gets a value in (a the case that LRND g , then hint to save precious time: prove first that gets a value in iff gets a smaller, or at best equal, value than ). Now suppose that only occurs as antecedent or consequent part in . It suffices to show that, under such an
Francesco Paoli
53
assumption, gets a value in , for if we do so we can ND conclude that is not provable in LRg . We construct an assignment as follows. If occurs only as an antecedent part in , we set and for ; if occurs
only as consequent part in , we set and as before for . We shall show that, for every subformula of : !"$# % (i) if does not contain , then ; (ii) if contains and !" &' is a consequent (antecedent) part of , then . We prove this by induction on the construction of , checking here only a couple of cases3 . !)(+*,- , *, If does not contain , neither does , and by inductive , ,-$# % .*,-$# % hypothesis , whence as well. If / * , contains and is an antecedent (consequent) part of , contains ,- too and is a consequent (antecedent) part of . Hence if 0 1*,- 2' 3 , . !4(4,657- , 7 We distinguish four cases. (i) If neither nor contains , ,657 ,-8 7-$# 9 % neither does , and by inductive hypothesis , 6 , 5 7 $ #
% , 7 whence as well. (ii) If contains and does not, ,657 ,657 contains . Moreover is antecedent or consequent according to , , whether is antecedent or consequent. If is consequent, by induction 25;< = 25> < ,-:5 7- is either or , as desired; if , ,-:5 7-6? is antecedent, by induction in any case, as it should 7 , be. (iii) The case where contains and does not is treated symmetrically. , 7 ,657 ,657 (iv) If both and contain , then so does ; moreover, either , , 7 , and are all consequent parts of , or else they are all antecedent ,657 parts of the same formula. In both cases, keeps the value ( or ) assigned by induction to its components. Now, since is both a subformula and a consequent part of itself @ A ! and it contains 4 , , and the lemma is proved. The previous proposition does not hold for all of LRND , since e.g. although BCED"FHGJIKD BCLDMIKD"NHG G the sequents O andR 2 are provable in LRND , 6 B R 6 B PQ O PQ occurs only once in , 2 . However, the following generalization of Proposition 2.9 holds for such a calculus: Q
R
Proposition 2.10 weak relevance lemma: Maksimova 1967 . If the sequent S V T U V T Y WX is provable in LRND and contains neither any antecedent Z6[H\ Z6]H\ part of the form nor any consequent part of the form , then each U ^ T U V T Y WX variable in occurs in at least once as an antecedent part and at least once as a consequent part.
54
Substructural logics: a primer Proof. We add the following tables for the lattice-theoretical connectives:
( "!$#&%'! ) *+, /- .1 0 2 * , 3 4 -/.1 56.7*298;:=< > 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;:=
Then the proof of this lemma runs as the one of Proposition 2.9. We only check the case . (i) If neither nor contains , then ) as before. (ii) Now suppose that contains and does not. Since has to be a consequent part of , and are consequent parts of as well. Hence by induction , or , and . (iii) The case where contains and does not is treated symmetrically. (iv) If both and contain , so does , which is a consequent part of . Then so are and , whence by induction . It follows . Proposition 2.11 (non-theorems of LRND ). The sequents: (i) (ii) are not provable in LRND .
GNHPOQGIK@J
J
GIH@JLKMG ; R
S
Proof. The sequents (i) and (ii) contain no occurrence of either or ; on the other hand, the variable occurs just once in both sequents; hence, by Proposition 2.10, they are not provable in LRND .
E
Remark 2.2 ("mildly" and "fully" balanced calculi). We said earlier that
LRgND has a "mildly" balanced character, in contrast with the "fully" balanced character of LLg . This becomes even more evident if we translate Definition
G
G
2.7 and Proposition 2.3 into the language of Definition 2.11. Inspecting the definition of -count, in fact, it is apparent that the antecedent occurrences of in are counted negatively in , whereas consequent occurrences of in are counted positively in (by the way, this is the reason why antecedent and consequent parts of a formula are sometimes called, respectively, negative and positive parts: see e.g. Troelstra 1992). Now, let be the number of antecedent (consequent) occurrences of in . Proposition 2.3 says then that, if , then LLg = . It follows that =
T
T
UWV GQX TZY UWV G[X TZY
\ a V]TZY^V7Ua V_TZY`Y T G Uir V_T1p jegegefjkTlo Ykm b Tdn cfegegegc`T o H@h p cfegegegc h q sir t7uwp vxgxyxzv"u|q {&}=~kr t7uwp vxgxgxzv"u|q { sir t_1p xgxyxzko @uwp vxgxgxfv"uq {
G
V_T1p jegegefjkTlo Y \ r ~kr t_1p xgxgxfklo {
Francesco Paoli
55
= = " !# $ . Proposition 2.3, therefore, can be so % ' & is provable in LL , then each variable in ( ) & occurs translated: if ' & exactly the same number of times as antecedent part and as in * ( g
consequent part.
2.5 Adding weakening: the affine route
If we add weakening to LRND , we clearly get all of classical logic, for - as we know from Chapter 1 - the rules for the connectives and collapse onto the rules for and , due to the combined action of the structural rules. This is not the case, however, if we add weakening rules to LL.
+
,
Definition 2.12 (postulates of LA). The calculus LA, based on the language £1 , has the same postulates as LL, plus the rules
% / - 4 57698 20 1 . / -
. / - 457:28 . / - 130
=
Such a system, briefly discussed by Wang (1963), was explicitly investigated under the name of contraction-free logic by Grishin (1974; 1982) and studied from a semantical viewpoint by Ono (1985) and Wronski and Krzystek (200+) in the 1980s. Since then, Hiroakira Ono devoted a considerable effort to the analysis of subintuitionistic contraction-free logics4 . On the classical side, contraction-free logic was investigated by Casari (1989; 1997a) under the name of bounded comparative logic and by linear logicians as (the subexponential fragment of) affine linear logic (see e.g. Kopylov 1995). The first thing we want to show is that nothing is gained if the latticetheoretical constants are added to LA, whereas too much is gained if the empty sequent is added to it.
;
Definition 2.13 (the systems LAB and LA ). The calculus LAB is based on the language £2 and has the same postulates as LA, plus the rules L and R; the calculus LA is based on the language £1 and has the same postulates as LA, plus the axiom .
;
>
;
Proposition 2.12. (i) LAB = LA ; (ii) LA is trivial.
<
56
Substructural logics: a primer
Proof. (i) It suffices to prove that fact:
collapses onto 0 and
onto 1. In
& # " % ! ! $ $ ( )*+ % %! ! $ "$ ' ( ),+ # - ( . / 0 / 1 2 3 )*4 ),+ 5 8 6 : 7 < 9 = ; 6 > 6 @ 7 ? > 6 : ; 9 6B68AC;:9<9<;:7FD<EK6>7=;:6>DL;FIM;FEG9<EK;H6>;:DJI 9NACDLI 6>7:9<;=6BAC9 66P8OC;:9
6P7=OC69<7=6>6>;H7=9 6RQU6 (ii)
Proposition 2.13 (theorems of LA). The following sequents are provable in (i) ; ; (iii) (ii) ; (iv) ; (v) ; (vi) ; (vii) ; (viii) ; (ix) ; (x) ; (xi) .
LA:
VV WKXY Z[Y 6L6>d9<7=7=6 6 6J9ed9<7:7H9 9 V Z[Y 6L9;FEG9<;HDJI V \]X_^ `a b c Y 668>7=7=6J6 d9 V ZXY 9<7:9 ? 6Jdg6>?k7:6P9 OC9fdgV mK6>[Y 7:9f9 dg6>7:9 V Z[Y V nK[Y V \]XYo5 ? 6POC9j7l6>;:9 ph} pfqFqyrspfpfq=wytvp uxwypfpq=~ t } p pfqFrGz{qlt]u|pfq:z{wypfqlt Proof. We prove (iii) and (vii).
Proposition
2.14
;
; (iv)
(non-theorems of LA). (ii) are not provable in LA.
Proof. Consider the following tables:
The
sequents: (i) ; (iii)
Francesco Paoli
57
0 1/2
0 1/2
1 1/2 0
0 1/2 0 1/2 1 1/2 1/2 1
0
0 1/2
0 1/2
0 1/2 0 1/2 1/2 1
0 1/2 0 0 0 0 1/2 /2 0 /2
1
0 1/2
0 1/2
0 1/2 0 0 0 0 0 /2
0 /2
0 1/2 1 1 1 1/2 0 /2
It is easy, though tedious, to show by induction on the length of the proofs
in LA that, if , then gets 1 under any assignment of LA values in to the variables occurring therein. Now, assign both and the value 1/2, and the value 0. Under such an assignment the translations of (i)-(iv) all get the value 1/2. Hence these sequents are not provable in LA. ! Remark 2.3 (three-valued Lukasiewicz logic). The previous tables are characteristic matrices for three-valued Lukasiewicz logic, on which we shall return in this same chapter.
2.6 Adding restricted structural rules An alternative to the addition of full weakening or contraction to our basic system LL might consist in placing therein just a limited amount of the former and/or the latter rule. This is the option we are now going to explore. There are two ways to accomplish such a task - on the one side, we can expand some of the operational rules so as to build into them part of the deductive power of the missing structural rules; on the other side, we can explicitly introduce suitably restricted versions of weakening and contraction. The first choice works even for classical or intuitionistic logics: Dragalin (1988) formulated variants of LK and LJ which contained no structural rules at all, for they were absorbed into the axioms and rules of the system (cp. the systems G3i and G3c of Troelstra and Schwichtenberg 1996). Turning to substructural logics, Anderson and Belnap (1975) devised "merge formulations" of a wide array of relevance systems. Such formulations had some exchange built into the operational rules, so as to suit a modal system like E, for which the introduction of unrestricted exchange rules would have caused a collapse of modality.
58
Substructural logics: a primer
* ( ) !" & +' # ' $ ! & # & % , ,2 3 4 -0/+ . -151- 0 -3 . 2.- 4 6 7 8 99 8/6<5 ;: 852 8 3 3: :28 44 = 6<> ? 515 8: 3 2: 8 92 8 4 3 8 9 : 7<4; = 7<@A ?
In the following, we shall consider the next generalization of incorporates some weakening:
where: (i) if , then includes the rule as:
L, which
; (ii) if
, then , ; (iii) , . It is easy to see that L* L as a special case. Moreover, it licenses such inferences
Likewise, one can generalize above restrictions,
L and
R to obtain upon adapting the
The previous rules are useful to set up a sequent calculus LC for Casari's comparative logic (Casari 1989; 1997a; the Gentzen-style calculus is in Paoli 200+c), a logic that was originally introduced to model some features of natural language comparison.
6 7 B B 6 7 59 /8 5 :: 22 8 9 = ;DCE? 9 8 99 8//8 55 :: 22 88 99 8 9 = ;DFG?
Definition 2.14 (postulates of LC). The calculus LC, based on the language £1 , has the same postulates as LL, plus the rules L*, L* and R* (in place of L, L and R) and the rules of "balanced" weakening and contraction, already mentioned in Chapter 1:
From LC we easily get a calculus for Meyer's and Slaney's Abelian logic5 .
B 6 7
H
Definition 2.15 (postulates of LG). The calculus LG6 , based on the language £1 , results from the addition of the axiom to LC, or, equivalently, from the addition of L*, L*, R*, BW and BC to LL .
I
Let us now see some relationships among the previously introduced calculi and some variants thereof.
Francesco Paoli
59
Proposition 2.15 (generalized balanced rules). The rules: #
#
are derivable in both LC and LG.
in LC and LG. Proof. Easy induction on the length of the proof of Remark that cut is needed to dispatch the cases of the rules 0L and 1R. Proposition 2.16. In LA: (i) The rule rules BC and BW are derivable.
L is equivalent to
L*; (ii) The
L is a special instance of ( ) * +, ) ( * !"#%$'& becomes ) +, ) * ( *.- /1012301415361798 : 79;=< > . As far as BC is concerned, (ii) BW is trivially derivable in LA ?A@ ?A@ B H @ ?A@ ? ?C@ I B H @ ? DEFAG Proof. (i) We have already seen that Conversely,
becomes
L*.
?JBK? ?C@ I B H @ ? L M N O1N P Q1R SUTWV SUXZY
In LG: (i) The rule BC is superfluous; (ii) the postulates [ , Proposition \ L* can be2.17. replaced by the rule: ]_^ b ` c ^ a a \ ] ^ ` m dfehgjiki'l p r L o s R q is trivial. Moreover, (iii) LG n
60
Substructural logics: a primer Proof. (i) The proof:
becomes
! " # " $&% ' # " (ii) The rule ( L** is derivable in LG: ! ) ! # " ( ) ! # " Conversely, given ( L**, ( L* and * are derivable: # ! " - ! . +! ! 1 2 5 3 4 1 9 = 4 ) #/ . 0 - 0 ! ",, ! ( ( +! 1 9: ;<; 4 1 6&7 8 4 1 9 : < ; ; 4 ( ) #/ ! 0 - 0 ! ",. #
(iii)
>
> > > ?A@ > C ? > E FGH > @ > @ E FI J H D @ C ? B @ E K&L M HON D @ C B
P QPR,STQUS PWVYX STQ,SZ[QP V,PQP \ PQbS5cSTQPR,dTQbd ] V_^`XaPQP Z XeX STQ,dfZgQhXaPQ,dZZ[Q_XaPQ,SZ[R,ijQbi ^kXlP+QP Z[R,STQ,S ^`XaPQP Z[QhX SmQ,SZ[R,dTQ,d ^kXlP+Q,SZ[R,STQP STRPonp^qPcS ircsPutvX SWnwdZ[RhXaPutwSZxnvXaPotwdZcsi ircyXlPunwSZxtvXaPonwdZgRPonvX SWtwdZci
Proposition 2.18 (theorems of LC). The following sequents are provable in ; (ii) ; (iii) ; (iv) LC: (i) ; (v) ; (vi) ; (vii) ; (viii) ; (ix) ; (x) ; (xi) ; (xii) .
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61
"!$#% '& ,% 1 0 !(#% '& )#* & -/. !$#+ '& )#* & !$#% '& )#* &
Proof. We prove (v) and (viii).
#% & "!$#* 2'& !$#% '& 5 4)!(#% '& 6 4 2 3 54 6 728:9<;=8 8:>@?*AB9DCEF)?%8:>GAEH9@?I8:>GCE ?%8:9GAEH>@?%8:9GCEJF8:9@?*AB>GCE
Proposition 2.19 (theorems of LG). The following sequents are provable in (i) ; ; (iii) (ii) LG: ; (iv) ; (v) ; (vi) ; (vii) ; (viii) ; (ix) .
F 5 M R/STP 8F8 F M NOPQ 8 F8L 5 M NW P 8K8F 5 MUY OVP 8K8LF F 8K28L 5 5 F3;$?%8K8'E M X%W P ;$I? 8K8'EF 5 M X%OVP 0
Proof. We prove (iv).
Comparative logic and Abelian logic provide first examples of systems endowed with restricted forms of weakening and contraction. Now we shall see one more logic containing restricted weakening rules and a family of logics which instead contain restricted contraction rules. The availability of weakening allows to add an occurrence of any formula on either the left- or the right-hand side of a sequent. Suppose, however, that such a move is countenanced only whenever an occurrence of the same formula is already present on the same side of the arrow. The following rules would
62
Substructural logics: a primer
result:
In the literature, the rules ML and MR are variously labelled: they are known either as anticontraction rules (Avron 1991b), for they are like upside down contraction rules, or as duplication rules (Došen 1988), since they allow to "duplicate" formulae, or else as expansion rules (Avron 2000). With their help, it is easy to prove the sequent corresponding to the Mingle axiom , a distinctive postulate of Meyer's and McCall's "semirelevant" system RM (for a detailed analysis of RM, see § 29 of Anderson and Belnap 1975). However, due to the fact that lattice-theoretical connectives distribute over each other, it is not so immediate to find a sequent calculus for RM - to do so, we shall have to wait until Chapter 4, where some suitable proof-theoretical refinements of our calculi are discussed. For the time being, let us introduce two simpler systems.
Definition 2.16 (postulates of LRMND and LCM). The calculus LRMND , based on the language £1 , results from the addition of the rules ML and MR to LRND . The calculus LCM, based on the same language, results from the addition of the rules ML and MR to LC. The latter system is only seemingly a new calculus, as the next lemma shows.
Proposition 2.20. LCM
LA.
Proof. The inclusion from left to right is easy, given Proposition 2.16 and the fact that anticontraction is a special instance of weakening. From right to left: as to WL,
! " +,(-& " % # ' $ & " #%()*& . /1032 4
WR is derived similarly.
5
Somehow dual to anticontraction is bounded contraction. In the case of anticontraction, one is not permitted to create new formulae out of nothing, but
Francesco Paoli
63
only to duplicate already existing material. Bounded contraction, on the other hand, allows to cut down the number of occurrences of a given formula but only up to a point: more precisely, -contraction licenses arbitrary contractions on provided that astock of at least copies of is left untouched. If denotes the multiset ( times), the rules of -contraction are:
Bounded contraction was intensively investigated by Prijatelj (1996), who introduced the systems PLn (here called LPLn ) hereafter defined. Definition 2.17 (postulates of LPLn ). The calculi LPLn , based on the language £1 , are obtained for each ! by adding the rules CLn and CRn to LA. By adding suitable axioms to these calculi, it is possible to get sequent formulations of finite-valued Lukasiewicz logics (for which see e.g. Urquhart 1986 or Cignoli et al. 1999; cp. also Remark 2.3): each #" -valued Lukasiewicz logic contains rules of -contraction. Of course, if %$'& , then LPLn is included in LPLm (that is to say, any sequent which is provable in the former system is also provable in the latter), since LPLm permits more contractions than LPLn . Table 2.2 contains a visual summary of the relationships among the systems so far introduced.
64
Substructural logics: a primer Table 2.2. Sequent systems and their relationships. LL plus L, TR
plus CL, CR
LRND
plus WL, WR
LLB
plus BC
L*, L*, R*, BW,
plus ML, MR
LL
plus WL, WR
plus CLn, CRn
plus
plus
LPLn
LRMND plus WL, WR
LC
LA plus WL, WR
plus
L*,
L*, R*,BW
LG plus WL, WR, CL, CR
LPLn plus CL, CR
LK
Proposition 2.21 (theorems of LRMND ). The following sequents are provable in LRMND : (i) ; (ii) ; (iii) ; (iv) ; (v) ; (vi) . Proof. We prove just (iv).
$ %#&(' ! " # $ )+*-, .(/ 0 12' $ 3#*4' " 5 65 565 $ >?*4' 565 7#5 587#565 $ >?&(' $ @+*4' 5 65 7!95 $ %#*4'$ %#*4' $ @+&(A @+*4' 956:5 7!95 5 6;5<95 9565<95 $ )+&('CB 5=9565<95
(formula multiples and powers). We define: U if QD HN; GRV H GS EJETE-S G DFE GDefinition H GI EJEJE-I 2.18 GK(D times) if DMLNO ; DPE GH V X (D times) if DWLN ; GRH if DQHN .
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65
Proposition 2.22 (theorems of LPLn ). The following sequents are provable in LPLn : (i) ; (ii) . Proof. Immediate.
2.7 Adding the exponentials In the article where he introduces linear logic, Girard (1987) remarks that he is not interested in setting up a new logic which is weaker than classical or intuitionistic logic, but rather a logic which permits a better analysis of proofs through a stricter control of structural rules. With this aim in mind, he introduces two new operators, ! (read: of course!) and ? (read: why not?), especially designed to recapture the deductive power of weakening and contraction in a controlled way. ! and ? are called by Girard exponentials.
( ) "!$#&%' *, +-* .08 /111/2.19 * . ) (exponentiated 11/3. ) If+54 *6 Definition +2.08 /1sequences). + +74 then and ?
?
?
. If
, then
?
.
Definition 2.20 (postulates of LLE ). The calculus LLE is based on the language £3 ; its postulates are the same as in LLB , plus:
; 9 =?>A@-B * .3/ : ; 9
* .3/* .AC3B * .3/ : ; 9
.3/ : ; 9 = >ADEB * .3/ : ; 9
>: F , >, : F
9 G H =?>AJ3B G> H
: F 9 =K@-B : F 9 H ?
: F 9 HIG3H =KC3B ,? ? ? : F 9 H
9 =LDEB
: F 9 G H =KJ3B G H ? : F 9 3
? 9 ?
HIG>6 : F HIG>6 : F
?
? 9 ?
?
,?
,?
?
Some remarks concerning the previous rules are now in order. The addition of exponential connectives reintroduces weakening and contraction into our calculus for special classes of formulae; more precisely, a formula preceded by an exclamation mark can be the principal formula of an unrestricted left weakening and contraction inference, whereas a formula preceded by a question mark can be the principal formula of an unrestricted right weakening and contraction inference. Exponentiated formulae represent, intuitively speaking, "ideal" constraint-free resources, which can be duplicated or contracted at pleasure. Moreover, any reader who is acquainted with the proof
66
Substructural logics: a primer
theory of modal logic will have noticed that the exponentials obey S4-like introduction rules - the exclamation mark, in fact, behaves as an S4 box (necessity operator), while the question mark behaves as an S4 diamond (possibility operator). At first sight, it seems hard to reconcile with one another the rules listed above: why should a single operator abide at the same time by modal rules and structural rules? To convince yourselves that this is plausible, recall Definition 2.18, and try and read exponentiated formulae as infinite conjunctions and disjunctions, as follows:
? for any you like (even zero), while ? is yielded by So, yields for any as well (again, even zero). Finally, check out that the rules of Definition 2.20 make sense under this reading.
. !" $# &% '!()*%,+- $# ) /10(234*5768/ 23/94 /10"/ 23/94*5,:-/10(2;34*5 . :-/ < : = 0(2>4*5,: 2$?@4 2;>A4B6 0(2>4*5 0C2>A4*5,: 0(2>4*5ED
Proposition 2.23 (theorems of LLE ). The following sequents are provable in ; (ii) ; (iii) LLE : (i) ; (iv) ; (v) ; (vi) ? ; (vii) ? ? ? ; (viii) ? ? ? ; (ix) 0 ?? ? ?
/ 268/ 2 / 268/ 2 F P"NEI 6 2 Q 6 2 . F G HJI / 26 . / 2 6Q2 F G KLI F M(NEI S T*UVS TW8S T$XS T F G OEI / 26 . 3 2 / 268/ 2$R/ 2 F M(NEI \ S TW . YT YZ"S T$XS T&[
Proof. Consider for example (i).
By means of exponential connectives, it is possible to embed both of LK and LJ into LLE . This is what we meant when we observed that ! and ? are "designed to recapture the deductive power of weakening and contraction". To be sure, in order to embed LK into LLE exponentials are not needed: Ono (1990), in fact, developing an idea by Grishin (1974), came up with a translation from (propositional) LK to (propositional) LLE which makes no use of them. ! and ?, however, are still needed to deal with quantifiers and cannot be dispensed with in the embedding of LJ (even at the propositional level). Here, we shall focus on the embedding of LK into LLE by means of exponentials, which is somewhat more perspicuous.
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67
As Girard (1987) remarks, this translation preserves not only provability, but also the structure of proofs.
). The translations and from Definition 2.2 (the translations FOR(£0 ) to FOR(£3 ) are simultaneously defined in an inductive way, as follows: + + ! " # $# ? ? + + % %$& # % (' %)' # + *,+" - %$ # .,+" /'0- %)' #
;
.
<
Definition 2.22; (translations of multisets). ; 2143030< 3015 6879: is < = 1430303015 ; similarly, 2143030301> is defined as defined as ; < ?@ 1A3B303015 . Moreover, #C: ?C: ,C . E H
Proposition 2.24 (an embedding of LK into LLE : Girard 1987). F F K P M= I J O M G KL iff LLE N ?N . D
LK
Proof. From left to right, the theorem is proved by induction on the length G L F of the derivation of in LK. To appreciate how the procedure works, consider how the translations of the group-theoretical versions of the rules for conjunction in LK express correct inferences in LLE . We shall not explicitly mention which rule is being applied at each step, but the reader can Q easily reconstruct the missing glosses. (In the following tree, let be O MRKS O M#T K N N .) U) c VBWYX e ] e
^] ^]
e e
U) c V0WYX
U) c V Z[X e
U) c V0WYX
]
U) c V0WYX
^]
^ U) c V0WYX
^] ` ^] e
^] ^ U) c V0WYXba
U) c V Z\X U) c V Z[X
e e e
U) c V Z[X
^ U) c V Z[X ^ U) c V Z\X
^ ] ` ^ ] ` ^ U) f X e c V8 ^ ] ` ^ U) f X e c V8 ^ U) c V0Wih
_ ^ U) f X e U c V8 c V0WYX#` ^ U) c V Z[X#` ^ ) ^ U) ^ ^ f X e U c V Z\X#` ) U c V8 c V0WYXba )
U V,X ? d g U d V,X ? g f X e Z[X#` ^ U) U V,X c V8 ? d g
?
U d V,X g U V,X ? d g
68
Substructural logics: a primer
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? " ? ! ? ? From right to left, we rely on the fact that LLE is cut-free (Girard 1987; see also Chapter 3), a result which has as an immediate corollary the subformula property for such a calculus. Now, a cut-free proof of the translation of a classical sequent in LLE involves nothing more than translations of classical sequents. Hence, it suffices to erase the exponentials and the 's occurring in it to get a proof of the desired sequent in LK.
$
#
Exponentials can be added not only to LLB , but to other substructural calculi as well. However, we shall not consider such an option (the interested reader is referred to Došen 1992b).
3. HILBERT-STYLE CALCULI 3.1 Presentation of the systems Even though the most expedient and intuitive way to introduce the proof theory of substructural logics is by means of sequent calculi, many of such logics were first packaged in a different syntactical form, viz. by means of Hilbert-style axiomatic calculi. This is certainly true at least for Lukasiewicz logics, comparative and Abelian logics, and most relevance logics. Hilbert-style calculi are less than optimal tools for theorem proving and efficient proof search, whereas sequent calculi fare much better in this respect. However, they are sometimes more manageable than their Gentzen-style counterparts when it comes to proving completeness theorems. Furthermore, some logics which are difficult to formulate in a sequential setting are very easily presented as axiomatic calculi. It seems therefore appropriate to introduce and develop both methods, employing each time whichever seems better suited for the task at hand. To begin with, we shall set up an axiomatic calculus corresponding to LL; then, we shall progressively enrich it by adding new postulates, in such a way as to get Hilbert-style versions of the calculi of § 2, as well as new calculi
Francesco Paoli
69
which do not correspond to any of the sequent systems defined therein. In the next section, we shall prove the mutual equivalence between these calculi and their sequential counterparts. Definition 2.23 (postulates of HL). The calculus HL is based on the language £1 and has the following postulates:
(F5) (F6'') (F9) (F10'') (F11'') ! ! ! !"!! "! ! (F12'') # infer (modus(F13'') ponens) #$ infer (adjunction) % & #$ will be used as shorthand for In the following, Definition 2.24. ')(*(+-,'/. *.+ (F1) (F2) (F ) (F4) (F6') (F7) (F8) (F10') (F11') (F12') (F13') (R1) from (R2) from
.
Now consider the following formulae and rules, formulated in the appropriate languages (containing, if necessary, also the lattice-theoretical constants and the exponentials !, ?):
0 31 2 ')(*4')(*. + + *5')(*. + (*4'/.*(+ (F14) (*4')(*(+ (F15) ')(*(+ *4'/. *.+ (F16) ' ')(*. + *. + *5' '/. *(+ *(+ (F17) (F18) ')(*(+ *4'/. *. + (F19) 6' ')(*(+ *4'/. *. + + *5'/7 *7 + (F20) 6 '8(*(+ (F21) 6'9')(* (+ *(+ *( :* ; (F22) ')(*(6 + * (F23) (* 0 : (F24) (F25) * * 1 0 0 1 6(* ( (F26') (F26'') 6 (F27) <> (=BC?A'/@ .D7 <+ @ *4')(B. +DC')(B7 + (F28') ')(D. +BC')(D7 +E*(DC'/.B7 + (F28'').*4' F (*. + (F29)
70
Substructural logics: a primer
(F30) (F31) (F32) (F33) (F34') ? % ! (F34'') ? #"$ (R3) from % infer &"' #($ )&"$ *(')+"$ (R4) from , infer , (R5) from infer Definition 2.25 (other axiomatic calculi). The axiomatic calculi HL- , HRND , HRW, HR, HRMND , HRMI, HRM, HA, HC, HG, HPLn , HLuk, HLuk3 , HK (on the language £1 ), HLB (on the language £2 ), and HLE (on the language £3 ) are defined as follows: HL - = HL + F23; HRW = HL + F28; HRMND = HRND + F16; HRM = HR + F16; HC = HL + F17, F19, F20, F24; HPLn = HA + F27; HLuk3 = HLuk + F22; HLB = HL + F25, F26;
HRND = HL + F14; HR = HRND + F28; HRMI = (HL-R2) + F14, F16, R3-4; HA = HL + F15; HG = HC + F21; HLuk = HA + F18; HK = HA + F14; HLE = HLB + F29-F34, R5.
As we shall prove in the next section and as these names suggest, the systems HL - , HRND , HRMND , HA, HC, HG, HPLn , HK, HLB and HLE are nothing more than the Hilbert-style versions of the previously introduced Gentzen systems. HR and HRM correspond to the relevance logics R and RM, which we already had several opportunities to hint at. HRW corresponds to distributive linear logic, also called contractionless relevance logic (see e.g. Brady 1990; 1991 or Slaney 1992). HRMI corresponds to a variant of RM, investigated at length by Avron (1990; 1991b; 1992) and motivated as a logic of relevance which is devoid of the philosophical and technical inconveniences of the mainstream relevance systems7 . Finally, HLuk and HLuk3 are axiomatic systems, respectively, for the infinite-valued and the three-valued Lukasiewicz logics. Of these systems, HRMI is the only one which is not an extension of HL. Throughout the rest of this chapter, it will not be further considered. .0/2111/./ 3
5 are formulae of3 any Definition 2.26 (iterated implications). If 4 .76 5 of the previously defined languages, the iterated implication 8 is
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71
inductively defined as follows: ;
We record without a proof the following propositions.
# $ ! " $ %
"
) & ' *% 1, 0- ,/ (
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
Proposition 2.25 (theorems of HL). The following theorems are provable in ; (F36) ; (F37) HL: (F35) ; (F38) ; (F39) ; (F40) ; (F41) ; (F42) ; (F43) ; (F44) ; (F45) ; (F46) ; (F47) ; (F48) ; (F49) ; (F50) ; (F51) ; (F52) ; (F53) ; (F54) ; (F55) ; (F56) 0 ; (F57) ; (F58) ; (F59) ; (F60) ; (F61) ; (F62) ; (F63) ; (F64) ; (F65) ; (F66) ; (F67) ; (F68) ; (F69) ; (F70) ; (F71) ; (F72) ; (F73) ; (F74) ; (F75') ; (F75'') ; (F76) . Proposition 2.26 (theorems of HRND ). The following theorems are provable in HRND : (F77) ; (F78) ; (F79) ; (F80) ; (F81) ; (F82) ; (F83) ; (F84) ; (F85) ; (F86) ; (F87) . Proposition 2.27 (theorems of HA). The following theorems are provable in ; (F89) ; (F90) ; (F91) HA: (F88) ; (F92) ; (F93)
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
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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.
! %+*, ( - " "$#&%'#&() %/.0"- (/.1"- 23%4.1 "65 7823(/.1"95:- %+*,( .0"- "- ; -theory : (i) if it is nonempty, it is detached; Proposition 2.36. For any HR < (ii) if it is regular and weakly -maximal, it is prime; (iii)< if it is regular and prime, it is = -complete; (iv) if it is regular and weakly -maximal, it is = complete. <$>? K F QRassume . It follows that <CNMO P and but and Proof. (i) Let and HL . Since is regular, ; since it is detached, . (ii) If any formula whatsoever belongs to , trivially for some both and . (iii) If is simply complete, then in particular either or , and so by F44 . (iv) Let and . Since is -complete, in particular and , whence . By F76, then, . Since is detached, , contrary to our hypothesis. (v) Trivial.
Francesco Paoli
75
, i.e. . Now we , which implies shall show that , against the hypothesis of the weak -maximality of . The right-toleft inclusion is obvious. To prove the converse, we need a result which we , shall pages: namely, that for $% any -theory HR prove in' a few " ! # & ( iff 9 . Suppose that 8 )%*+,- . /01 . Thus 243 ( 5")%* and 063 ( 5")7* , and :<;6= ( >"?A@=B:"?A@ED whence F
~
~
Proposition 2.38. If is a set of regular, detached S-theories, then is a regular, detached S-theory.
Proof. Left to the reader.
c
We shall now turn to the issue of defining a suitable concept of derivability in our axiomatic calculi. Classically, as it is well-known, a derivation of in S from the assumptions is a finite sequence of formulae of the language of S such that every is either (i) an axiom of S or (ii) one of
76
Substructural logics: a primer
the 's or else (iii) is obtained from preceding formulae of the sequence by one of the rules of S. However, it turns out that such a notion is too lax for most systems that we are dealing with. A concept of derivability works if we are in a position to prove an appropriate version of a deduction theorem; but for this purpose we need a tighter notion, which we are now going to examine. Definition 2.31 (S-derivation and derivability in S: Troelstra 1992). Let S be an axiomatic calculus. An S-derivation tree whose leaves are is a, labelled labelled by expressions of theform where is a formula of the language of S, or of the form , where is an axiom of S. As regards the labels of the other nodes of , they are obtained by means of the following are finite, possibly empty multisets of formulae of the rules (where language of S):
A formula (in symbols, labelled by
is said to be derivable in S from the assumptions ! ! "S ) iff there is an S-derivation # whose root is .
Remark 2.4 (comparison with the classical definition). It is easy to see that this definition rules out the case, permitted by the classical notion, where a formula is derived from without depending on itself. Moreover, $inthe classical case, if a formula in the sequence is obtained from % and by modus ponens and the latter both depend on the formula , then also depends % - ones on the formula and that's all. Here, however, we are dealing with multisets and&so formula occurrences take into 'have to, and % isit isobtained andthat we from if account: if
occurs ( times in and ) times in , it will occur (+*,) times in . From this definition it is apparent that is derivable in S from no assumption just in case it is provable in S. Such a definition gives us exactly what is needed to prove an appropriate deduction theorem for our substructural logics. Let S Proposition 2.39 (substructural deduction Troelstra iff . $ 1992). -theorem: be HL or one of its extensions. Then . S S Proof.
From right
to left,
consider the following S-derivation:
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77
of ! from ! . Then there is an S-derivation let " # Conversely, by induction on the construction of , we shall associate to itself. Proceeding " an S-derivation $ of % ! from . (Base) has a single leaf, which has to be of the form (otherwise there is nothing to prove). Now, F1 is a theorem (or even an axiom) of any extension of HL. Then $ will be any proof of F1 in S, rewritten in tree form. " Suppose that ! is obtained by % E from the premisses " & (Step) " " from the premisses & ' and " ''(% and! (" ') (" & % * " ! ,). orIn theelseformer " & + % ' and thus, using the S-axiom F2: case, by induction hypothesis , 7 ; -+./ 0 -1 ./ 032/1140(/ 52/61 ./ 522 ; -8140(/ 52/1 . / 52 < -0(/ 5 :=>9 < -./ 5 -./140(/ 52 and thus, In the latter case, by induction hypothesis using the S-axiom F3: 7 -./140(/ 52?-81 ./140(/ 5@22A/140(/1 ./ 5B22 7 < -0(/1 ./ 52 = -C0 + . / 5 D=E9 < .-5GF0 is obtained by F I from 9 .-5 and suppose that 9 -0 . By 9 .Finally, -./ 5 and -. / 0 , and thus, induction hypothesis using the axiom F7: 7 7 -+./ 5 -./ 0 -81 ./ 52HF1 ./02-1 ./ 52HFI1 ./ 02/1 ./5GF02 -./ 5GF0 J N , then KLM Proposition 2.40. (i) If S contains F14 and K#LKLM N ; (ii) if S contains F15 and MON , then KLMON .
S
2
2
2
2
2
S
S
S
S
78
Substructural logics: a primer Proof. (i)
!
%&' %&'())*'+%&'()
, (
!
,
(ii)
"$#!
(/' %0&1'() -
"$#!32
&'( -
.
& -
(
!
We now wish to try to relate to each other the concepts of S4 -theory and Sderivation. Classically, the relationship is simple: a formula is derivable 4 5 from the set of 5 assumptions iff belongs to the intersection of all the theories containing . Here, however, we need to be more careful, since our notion of theory is set-theoretical, while our concept of derivability is multisettheoretical. Anyway, there is a weaker notion of derivability which lends itself perfectly to a generalization of the above result. We obtain such a notion by closing an S-derivation under suitable rules of weakening and contraction. Definition 2.32 (weak S-derivation and weak derivability in S). A weak Sderivation differs from an S-derivation exactly in that its nodes can be labelled also by formulae obtained through the following rules: 6 7
8 4
6 7
90:<;
8 4
8 4 8 4
6 7 6 7
9>=;
4
A formula is said4 to be 4 weakly derivable in S from the assumptions 4 8@?A? ? 8 4 8B? ? ? 8 6 w 7 C D (in symbols, E D ) iff there is a weak S-derivation F GIHBJ J J HGLKSG E D whose root is labelled by . There exists a version of a deduction theorem also for the relation of weak derivability, first obtained by Meyer et al. (1974) for HR. There it is called "enthymematic deduction theorem" since the kind of entailment relation at issue in the statement of the theorem (where the antecedent implies the consequent when conjoined with some truth) is peculiar of Anderson and Belnap's (1961)
Francesco Paoli
79
account of enthymemes. Its formulation requires the introduction of a new notion:
FOR(£ ). The FOR(£ . ) which
Definition 2.33 (conjunction set of a formula). Let ( ) is the smallest set conjunction set of contains and and is closed w.r.t. the connectives and
1
1
2.41 ("enthymematic" deduction theorem: Restall 1994a). (i) Proposition , for some !" #$ % & ' ; (ii) iff ( iff *)"+, ; (iii) -( iff .2 , for some $0/*1 . w HL w HRND
w HL w HRND
w HA
w HA
Proof. We only take care of the left-to-right arrows, leaving the converse implications to the reader. (i) The proof proceeds by induction on the length of the weak HLderivation of from , as in Proposition 2.39. We just consider three cases of the inductive step. w w Suppose first that was obtained from by an HL HL application of W. Using F3 and F9, we easily prove in HL the theorem . Thus:
4 -6 7 4 67(8 5 4 8 5 59;:< 9 5>= ? 4 8@5 8A 5 9;:< 9 5>= 4 8 9 5 B C DFE J was obtained J by an Now, suppose that GH(I from G-H>G-H(I I M K ; L O N M K L > J P application of C. Then, by induction, Q R , for some SU_ TVSXR W"SYZ [ \ ]^ . Using F46, we get: ` a@SM_ b;\OSMR b c>^ ad\OSM_ b;\OSMR b c>^^eb;\OS;_ fgSMR b c>^ h i jFk aAS;_ fgSMR b c l;mglX{ n"lop q r st . Finally, suppose that u-s(v wyxzl was and Q and |-}(~ by an application of I. Then, obtained from |}(~ and ~ M , for some by induction, Qbe U|VX "
} . Let~ M zM yz ; using F74, we get: 3
5
w HL
w HL w HL
w HL
w HL
w HL
w HL
w HL
80
Substructural logics: a primer
!" #%$&#(4 '#*),+.-!/ 021 . and 3 57698 : then 8 ;=<: , for some (ii) By (i), if I ;?>;2@BADC!E 62F . We shall show how to replace ; I by 6HG , by inductionI on the 6HG <: construction of ;2@,A.C!E 62F . If ; is either 6 or , then 8 I I I 6H G <JOH6 P IandP OH8 P I Q6HR G < . If ; is ;K3 GL; M , Nthen by since 8 N induction , whence by F62 SUT I VW . If X is XZ3 Y[X b , then by induction \ ` SHT^a ]_b VW , _c dHe I fg . whence by F79 d9c g then c i=fg , for some ikji*lBmDn!o d2p . (iii) By (i), if h i by a suitable power of d , once again by We shall show how to replace 2 i l,m.n!o d2p . If i is either d or I q_d r , we are induction on the construction of s K s L t s { , then by induction u vw| t v } xy , and by F94 home. If is z u v |"} xy . If s is sZz ~[s { , then by induction u v |"} xy . w HRND
w HRND
w HRND
w HRND
w HRND
w HRND
w HRND
w HRND
w HRND
w HA
w HA
w HA
w HA
w HA
Now we are ready to establish the desired relationship between the notions of weak S-derivability and of S-theory.
k
is a finite multiset of Definition 2.34 (contraction of a multiset). If formulae, its contraction is the set which contains exactly all the formulae occurring in .
k
Proposition 2.42 (a characterization of weak S-derivability). Let S be HL w or one of its extensions. Then iff . S S
K
Proof. From left to right, we proceed by induction on the construction of a designated weak S-derivation of from . If , then of course belongs to any S-theory which contains it. If is an axiom of S, it will belong to any regular S-theory whatsoever. Now, suppose that is obtained in by E from and ( . By inductive hypothesis, belongs to any regular, detached S-theory containing and belongs to any regular, detached S-theory containing . Let be any regular, detached Stheory containing . , then, contains and . Being detached, it contains as well. The case of I is left to the reader.
& =¡J¢ £ ¤ ¥ ¦©§ ¬ ¢ ¥ ¨
3 ¡J¢ = ¦§ ¥ ¦§ ¥ ¬ ¥«ª ¨ ¬ ¥«ª« ¡J¢ = ¦ § ¦§
Francesco Paoli
81
Suppose is obtained by W from . By inductive hypothesis, belongs to any regular, detached S-theory containing and so, in particular, to any regular, detached S-theory containing . Finally, the fact that takes care of the case involving the rule C. . Now, From right to left, the induction is on the construction of the mapping of Definition 2.34 is by no means injective; thus, it is necessary w for every which contracts to . If is an axiom of to show that S w w and, by successive applications of W, belongs to S, then S S . If w w , then and, by successive applications of W, S S ; since w
contracts to , and so by C is derived in S . If ! from and #"$ , then by inductive hypothesis w #"$ S and wS for every which contracts to , whence by " E we have % w w & ' and by C S S . The case of adjunction is left to the reader. (*) +
(
w Proposition 2.42 justifies the notation (where is a set of S +-,./0213( 4 formulae), meaning that ( . For weak derivability, in fact, the S multiplicity of formulae in does not matter, as we have just seen. In such a + ( case, we shall say that is weakly derivable in S from .
3.3 Lindenbaum-style constructions When proving completeness theorems, it is often useful to be in a position to extend a given consistent theory to a consistent and prime (or complete) theory. In classical logic, this goal is achieved+ by means of Lindenbaum's Lemma, which shows us how to construct an -maximal theory + out of an + + arbitrary -consistent theory. Classically, -maximality (whenever is e.g. a contradiction) suffices to obtain 5 -completeness, simple completeness and primality. In substructural logics, however, as we know from the preceding subsection, this is not necessarily the case. Now, we shall prove an appropriate version of Lindenbaum's Lemma and examine what we can get out of it. For a start, we need the following Proposition 2.43 (chain lemma). If H , G 6?9<9<9<6 of £ £1 £I s.t. \ M NPOQR S Y[Z T ] ]_^=` U , then for every a-bcde fgih j . Proof. As usual.
'
(7 @ 68(7 A 6:9;9<9<6=(> B 6?9<9<9 V
(
J
CD(
in £,
V
K
are sets of formulae C9<9<9 and L X iff there exists an s.t.
W
C9<9<9ECF( M
82
Substructural logics: a primer We are now ready to prove:
Proposition 2.44 (a version of Lindenbaum's Lemma). Let S be HL or one of its extensions. If is an -consistent S-theory, there is a weakly maximal S-theory s.t. .
!#" %$'&- () 2 - * & -/.0 +,& 1 43 798!:#;%< @BA = 5 CE56 D > ? GF IH just constructed? Not very much, What elseJ can we say about the theory indeed. It is -consistent and thus simply consistent, but we have no guarantee that it be either K -consistent, or prime, or K -complete. However, it is regular
Proof. We proceed as usual. We enumerate all the formulae of the language of S; let be such an enumeration. Then we construct a sequence s.t. and either (if ) or else . By construction and by Proposition 2.43, is a weakly -maximal S-theory including .
and detached; by Propositions 2.36 and 2.37, therefore, we can legitimately infer:
H
J
L HF
is an -consistent HR-theory, there is an Proposition 2.45. If consistent and prime HR-theory s.t. .
J
-
J K J H L HMF IH in Proposition 2.45 might be K -inconsistent, The problem is that the IH while the in Proposition 2.46 might be K -incomplete and/or not prime.
Proposition 2.46. If is an -consistent HA-theory, there is a -consistent and weakly -maximal HA-theory s.t. .
Nonetheless, Meyer (1976) managed to prove, using the technique of metavaluations, that:
K
4H
L IHNF
Proposition 2.47 (Meyer 1976). If is a regular prime HR-theory, there is a regular, -consistent and prime HR-theory s.t. . The full power of Lindenbaum's Lemma (if not for arbitrary HR-theories, at least for HR) is then achieved in two steps: a "way up" lemma, by means of which HR is pumped up to a prime, but not necessarily -consistent HRtheory ; and a "way down" lemma, where is suitably pruned so as to obtain a -consistent and prime containing all of HR. Moreover, a more refined completion technique due to Belnap (for which see e.g. Restall 2000) permits to ensure that:
H
K
H
K
Francesco Paoli
83
is an -consistent HRW-theory, Proposition 2.48 (Belnap 200+). (i) If there is an -consistent and prime HRW-theory s.t. ; (ii) if is a set of formulae closed under disjunction (i.e. imply ), is an HRW-theory and , there is a prime HRW-theory s.t. and .
! # #
For systems S which do not contain F28, an arbitrary -consistent S-theory cannot in general be extended to an -consistent and prime S-theory . In fact, might contain without containing . In such a case and, since is supposed to be prime, either or . It would follow that either or , whence , against our assumption.
$#
" # # ! %#
4. EQUIVALENCE OF THE TWO APPROACHES The aim of this section is proving that the two proof-theoretical approaches so far developed - namely, Gentzen-style sequent calculi and Hilbert-style axiomatic calculi - are equivalent to each other, in an appropriate sense of this term. This will be done by means of the next Proposition 2.49 (equivalence of sequent and axiomatic calculi). If LS is any of the previously introduced sequent calculi and HS is the corresponding axiomatic calculus (where "corresponding" means that the final letters of their denominations are the same), then LS iff HS
&
* ' +
&
( * ' + )
& & ' ,.D - /0/0/1-2,4E 3658F 7 /0/0/1795 G :<;=?>A@CB= , 5 H65 HI,>JHK,365 HL5 HK,>JH65
Proof. From right to left, we shall prove a more general result: for any formula of the appropriate language, if , then . In HS LS particular, if has the form ( ), our result follows from Proposition 2.2. We proceed by induction on the length of the proof of in HS. First, we prove that for any axiom of HS, is provable in LS; then we check that the rules of HS preserve such a property (i.e. if are provable in LS, then so is ; if are provable in LS, then so is )10 . The truth of the former claim can be easily inferred from Propositions 2.1-2.23; as to the latter, it is enough to consider the following
HK, M5
,
84 proofs in LL:
Substructural logics: a primer
!
From left to right, we proceed once again by induction, this time on the length of the proof of in LS. Here, we shall be somewhat less sketchy we shall check in detail a number of cases, leaving the remainder of the task up to the reader. It goes without saying that each of the cases listed below applies only to the Gentzen systems containing that specific axiom or rule: for example, the translation of the rule WL is proved to hold only in HA and its extensions. In what follows, F50-F55, as well as F4 and F40, will be applied without special notice. Furthermore, we shall resort to the following convention: if is a sub-multiset of the antecedent of , and is a sub-multiset of its succedent, we shall use the lowercase Greek letters to denote , and to denote . (Ad Ax). is F1, an axiom of HL and a theorem of all its extensions. (Ad ). is F110, a theorem of HG. (Ad R). Suppose is provable in HS; then by F47 is provable in HS as well. (Ad R). Suppose is provable in , whence by F12 HS. By F46, we readily deduce therefrom and F38 . In virtue of F39, we get , whence again by F12 and F37 it results . (Ad L). Suppose that and are both provable in HS. From the former, by F47, we get , whence by F35 and F37 , and by F46 . On the other hand, in virtue of this last principle, entails , which together with gives by F2. F46 yields then , which becomes through F12 and F37. F47 is now enough for .
" '% &). (+*,**( & / # - $ % 012 (+*,**( 0 4 3 5 0=2 <8***+<>0 3 &?A@ -&CBD%8&7&2 68E**9*:& 6;& / -IBD% J W FG H ? @ LAZ M NPQ ORW Q OHK^NCSDS T8N>U X V Y X V K Y [K\N T8LHM] V LAM NPORZ Q W O _PSDX T8V`N>M UNX V V Y [>Y [>_P_ S XX V`YMKbY VaM XNV`VM _PNScSVaZ MKbY W V_PScS V [8MdN V V_PSDT8LHM]Q O N V_PS LHM]Z Q W O NCSDT X V Y [eN LAMf_gOhi Q j SDT_U Xk UV Kbl Y VN X UmKbY VaM M NV_PSDV_PS X Uk KnY U8l M NV_PSDV_ k l _mK=U Y U8V M NV_PSDV_ _ VaMoV S X U XX UK=Y U8M NV_PSDYVaMok k V l S l _P_PSDSDVa MKb[KR V`k Mo[ l V S S X U8U8M M NNV Y V V M NV_PSpU X U k V Y [ l T8LHM NV_gOoZoOhi Q WOj S
Francesco Paoli (Ad
85
L). It suffices to remark that = = . (Ad R). Suppose that and " # $ ! are both provable in HS. By F47, we get %& ' # (%) $ * and ; on the other hand, by F37 and F45, first we %& +,-'. obtain and then, by F3 and F38, %& +/ # (%) $ ' . In virtue of F37 and F46-48, we conclude # $ 1 2/ 03 ! ** " ( .
2 (Ad 4 L). Suppose that is provable in HS. 5+6 Then by F46 we obtain , which can be chained with F6 to get -+6 4 . Another application of F46 yields 2 7 , 4 4 . With we proceed analogously. 8
Sources of the chapter. Our presentation of the embedding of LK into LLE (§ 2.7) closely follows Gallier (1991). Troelstra (1992), Dunn (1986) and Restall (1994a) were especially useful in drawing up §§ 3.2-3.3.
Notes 9 : 1. By saying that is provably equivalent to in S we mean, of course, that the sequent 9<;=: is provable in S. 2. Recall that the antecedent and the succedent of a sequent are multisets. Hence, strictly speaking, a sequent may have several different formula-translations, differing for the association pattern and/or the order of its group-theoretical disjuncts and/or conjuncts. Such differences, however, may be disregarded as irrelevant in most contexts. Hereafter, thus, we >,?A@7B shall assume that is fully specified, and we shall do the same whenever we introduce a notion of formula-translation throughout this volume. 3. Throughout the rest of this proof, we shall use the same symbols for the connectives of £1 and their matrix interpretations, since no confusion can arise. 4. See Appendix B. 5. Cp. Chapter 1 once again. 6. We already used an "A" for the calculus LA. The "G" in LG should be reminiscent, in our intentions, of the word "group", since LG is a logic whose models are Abelian C -groups. 7. The third paper focuses on the philosophical aspects of the issue, while the first two articles contain a technical presentation of the system's proof theory and semantics. 8. Henceforth in this section, systems with exponentials will be excluded from our discourse. 9. Except, of course, for HRMI, which we agreed not to reconsider until Chapter 4. D 9 D=E 9 is provable in LLE , then so is . 10. For LLE , we also have to add: if
Chapter 3 CUT ELIMINATION AND THE DECISION PROBLEM
In Chapter 1, we discussed at some length the importance of cut elimination, both from a philosophical and from a technical viewpoint. Hitherto, however, we did not prove the cut elimination theorem for any of the systems so far introduced. This will be exactly the task of the present chapter. For a start, we shall present Gentzen's proof of the Haupsatz for LK; coming to know how such a proof works is essential also from our perspective, for it allows to appreciate the role that structural rules play in it. Subsequently, we shall assess how Gentzen's strategy should be modified in order to obtain the elimination of cuts for systems lacking some of the structural rules. We shall also show, with the aid of appropriate counterexamples, that not all of our sequent systems are cut-free. In the second part of this chapter, we shall examine one of the main applications of cut elimination, seeing how to extract out of it a decision procedure for many of our sequent calculi. Again, such methods do not work invariably in all cases: there are, indeed, systems which are known to be undecidable (and we shall briefly discuss them).
1. CUT ELIMINATION 1.1 Cut elimination for LK The next definition introduces a calculus which is equivalent to LK, as we defined it back in Chapter 1.
88
Substructural logics: a primer
Definition 3.1 (postulates of LKM ). The system LKM is exactly like LK, except for the fact that: 1) its basic expressions are inferences of the form , where and are finite, possibly empty, multisets of formulae of £0 ; 2) it contains no exchange rule; 3) the cut rule is replaced by the following rule (called mix):
and contain , and ( ) is the same multiset as Here, both ( ), except for containing no occurrence of . As far as there is no danger of ambiguity, we shall drop the subscript " ". The formula is called mixformula.
Proposition 3.1 (equivalence of LKM and LK). .
LK
iff
LK M
Proof. We confine ourselves to showing that the cut rule is equivalent to the mix rule. In fact:
!
% " - (& ')&+* * * ' ',&+* * * ',& # - % $ ! % " - &' # -% $ , ' & !& # - % " - & $ . 3 / 45 5,46 0 3 1 .4 0 - 3 / - 4 1 7 89 : ; < =< > ?@ ACB D 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<; contains = in itspath and @ D succedent; E G F , B 1 & ; A If & belongs to the subproof C of C whose endsequent is H L I " by "J L K " is defined in the same way, except for replacing " MOand NQ P HZ"succedent" RTSUN P H L I RWVXN P J L K R-Y [ J \ L I \ by[ K "antecedent"; ] Definition 3.4 (rank of a subproof in a mixproof) . Let be a ^ mixproof ^ ] ] ] in ] is and be any of^ its subproofs (possibly itself). The rank of ^ ] ] `ba or simply] by^ _c`ba and coincides by definition with _ ` d a , denoted by _Q d where is the endsequent of . 2
90
Substructural logics: a primer
Definition 3.5 (grade of a subproof in a mixproof). Let be a mixproof and be any of its subproofs (possibly itself). The grade of in is denoted by or simply by and is the number of logical symbols contained in the mixformula .
!!#"$ %&
Definition 3.6 (index of a subproof in a mixproof). Let be a mixproof and be any of its subproofs (possibly itself). The index of in is denoted by or simply by and is the ordered pair . Indexes are ordered lexicographically: that is, iff either or else ( and ).
'#"
)("
*%
Before proving our cut elimination theorem, we settle some notational matters. When drawing the proof tree of a mixproof , we shall sometimes write , meaning thereby that is . We shall also write to denote the maximum of the ranks and . Now, we are ready to start.
35476
+ - .,
1/ 0 + 3- , 2 6
8 ; 9
:
Proposition 3.3 (cut elimination theorem for LKM : Gentzen 1935). Any mixproof of in LKM can be turned into a mix-free proof of the same sequent in the same calculus.
:
8 ; 9 < ; = @BADC 8>? < E ; 9 E ? = FGH We . I FGproceed HKJLMObyNMQinduction PR . SinceonSGHK,JTtheM, index U is aofvariable, V WXGHKJTM , say . As Y W G Z H T J M ] \ . Hence, either [ ] \ is an axiom, or else V is in particular [ ] \ , which must the principal formula of the inference whose conclusion is [ perforce be WR. We have the following transformations: _ _ b c ^ ] ^ b E c ] g hji df e b b ]E c c g k l mnl o pnq rsti ^a` ] ^a` ] u u yy v z z g Oi { u | y v z g k l mnl o pnq Ost Oi v y}wtx { z~v | X g j h i } y { w E v z~w | w Ev w Proof. Let
be a mixproof whose final inference is:
Francesco Paoli
91
" or *,+ - &
!
#
.
If , then either $ $ . We distinguish the two subcases. % ( ) ! ! $ $ ' . Thus is the conclusion of an inference where can be either a principal, or an auxiliary, or a side formula. If it is a side formula, our strategy consists 0 in "pushing cuts upwards" in such a way as to ." ! 0 # $ / construct new proofs of / containing mixproofs of grade 0 and of lower rank (hence of lower index) as subproofs. This entitles us to exploit our inductive hypothesis. Some examples:
1 6 6 2
1 1
; 7 G H IJ 8 ;>=@? 7 < 9 3 4 2 6: 8 7 9 A 2 ;B=
M T
T N T. O V
3
o
X v
p u q w :
EF
M T N Y U OQ OP j kdlm Z [B\ YD]_^@` U OPSRQ V a W N Z [ e \ Z U W b N ^ac^@`d] YD^ac^@`d] b O5OPSRQ
o u
G KLJ
8 ; 7 < 9 2 2 6. 7 A 3 9 ; =
w
qr |B} y v
{ p
j nLm
y x p
{ 5 q x qr u: q w { p
~ L v
u
p u q w .
o
z v
w
qs z } y v
{ p
y x p
§ ¨
X °
¯µ© ± ¸ ¨
~ d
x qrSts { q5
£ ¡ ¥ ¢ ¤ B : D_@ 5 5
.
e .
c@d Dc@d
5
.
¯
~ L
x qs { q5
X
G H IJ
i g fh
o
G KLJ
M
X U N
6
1
§ ±
©ª ¹>º ¶ °
¨
§
¦L¤
¶ ²
¼ ½L¾
³ «© ¨
h
§ · ´
± ¨
¶ ²
² ©ª ³© ± ¸ ¨ ¸ ©¬ · º ¶ ´ ¸ © ² «© » ¸ ¸ ¸ ¸ : ¯ ³ ® ± ± ° ´ ¬ ² ª5«©.© ©®© ¨ © © ¬ © ² ¼ ¿À¥Á ¿Â¾ ¯: ³© ± ¸ ¨ ° ¸ © ´ ¸ © ² ª5«©. ©
¼ ½L¾ ¼ ãÀ¥¾
If the mixformula Ä is auxiliary, the strategy is basically identical, except that we possibly need to perform some adjustments by means of structural
92
Substructural logics: a primer
rules. For instance:
"! $ # *( ) & % #
+
%&# '!$# "! $ #
Finally, if , is principal, it must have been obtained either by a weakening or by a contraction move. We proceed as follows:
2 . 6 3 /$01/$0 F G J H I 2 . 86 7:9 3 /$0 4 . ; 5 BDC 2 . < 3 $/ 01/$0 4 . ; 5 F KLI E F KLI 2/ 4 = . <>?7 ;@7A9 3 = / 5 2/ 4 = . <>7?; 3 = / 5
M PRXZQ Y N [ SUTWV O . This subcase is dealt with in a symmetric manner. \ ] ^` r _badcWe fJgihjf"kmlWen . Since o?^p r _bajg , the mixformula q must be s u t and Y u [ ; principal in the inferences whose conclusions are t moreover, such inferencesq cannot be contractions, and and Y cannot contain s u t or Y u [ is further occurrences of . The case where either obtained by weakening is left to the reader. If both sequents are obtained by means of operational rules, our target will be to perform our mixes on the immediate subformulae mixformula. In such a way we shall obtain new s f Y v u t v f of[ the proofs of containing mixproofs with lower grade (hence with lower index) as subproofs, and we shall be able to apply our inductive hypothesis. Some examples:
w
w w x yJz x }{y| x z{y|
* J
*$ x yJz ~} x z ~}{y|
*J y x y
J {| v v J $ J
Francesco Paoli
93
! #"%& $ , ' ( / ) 0 (* *( 1 ) 2 < = > '(+/3( 1 87 ) 0 87 ( 2 4 ) 5 ( ' < D> / 9 7 (4( 1 9;7 : 8 ) 0 87 ( 5 9 7 ( 2 < A ? @ B A ? C > /(4(. 1 ) 06( 5 ( 2 E
EGF
Transformation T9 must be interpreted in the following way. Let be the proof which precedes the transformation and the proof which results from it. The subproof of which ends with the mix inference whose principal formula is (call it ) has a lower grade, hence a lower index, than . Thus, by inductive hypothesis, there is in LKM a mix-free proof of . It follows that, if we replace in by , the subproof of such a proof ending with the mix whose principal formula is is a mixproof of lower grade (hence of lower index) than , and can be replaced by a mix-free proof by inductive hypothesis. . We have to distinguish once again the cases and . . If the mixformula is a side or an auxiliary formula in the inference whose conclusion is , we resort once more to the strategy of pushing cuts upwards. These easy cases are left to the reader. We are now left with the case where is principal. We distinguish three subcases. 1) If occurs in , then is:
H
I
EGF
J
K L+NPL O US M Q US L R
I
J T
E
J TT
J FF
V
W X Y[E Z]\^`_;a bdceagfdhiba_Aj Y k o p t q Zlc`m k o Y r t s Znc`m u xzo y p t q {l|`} v w p t q
w
w
p
~dF
A A +
2) If occurs in , we argue symmetrically. 3) Now, let occur neither in nor in . We illustrate this subcase by e means of a simple example where the mixformula has the form :
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! " #! $ contains at least one occurrence of % & ' Under such circumstances, /10 3 2 45 6 7 +-, . ) whereas, by our hypothesis, (otherwise, it would be (* 8 does not contain 5:6 7 ) . We transform the 9 previous proof, call it , in this < 9 ; =
5 6 7 ): way, getting the following proof (where is short for N N OQP W R C X PS T P Y R D Z V OPUWP Y E R FHGIK X J P Z PS [ \] [ ^ _ ] T P WUP Y J R LM X J P Z PT Y R I Z >@?A? [ \] P P R P P J J K I J U W Y Y X Z Z B [ \`ba \_] WP Y J R X J P Z c d ; which ends with the first application of mix. c Let be the subproof of d . We can has the same grade and a lower rank (hence a lowerd index) than ; c thus apply our inductive ;gfQhiUhypothesis, h j o m k o and h l hn replace in the subproof by a e d ; mix-free proof of . The result iUh j of o such h j o am replacement k o h l h l p , ending is a proof which contains a mixproof with p has the same grade and a lower rank (hence a lower index) than d : in fact. qs|~r }{ E E { {x y z tvu w , since x y z does not occur in E { . This leaves us in a position to apply once again our inductive hypothesis. The other cases (some of which slightly more complicated) are treated similarly. . This subcase is dealt with in a symmetric manner.
q
/r t1w
Remark 3.1. In virtue of Proposition 3.1, the cut elimination theorem for LKM has as an obvious consequence a cut elimination result also for LK.
1.2 Cut elimination for calculi without the contraction rules If you try to prove the Hauptsatz for LK without resorting to the mix rule, everything goes all right for a while; you soon get stuck, however, when you consider the case where the cutformula is obtained by a contraction move:
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You cannot push this cut upwards, since you would be left with an extra occurrence of which you could remove only by a further cut, getting a proof whose index is not necessarily lower than the original one. Hence you cannot apply your induction hypothesis to this case. That is why Gentzen introduced LKM and the mix rule3 . If our sequent calculus does not contain CL and CR, however, we do not need to worry about all this. In such a case, it is enough to show that ordinary cut can be eliminated from the proofs of the system at issue. To obtain a cut elimination theorem for LA, for example, it is sufficient to adapt the previous definitions and proofs 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 "cutproof" and "cut-free proof". 2) The definition of rank of a sequent can be simplified. Let us see how. Definition 3.7 (rank of a sequent in a cutproof). Let final inference is:
be a cutproof whose
!#"%$& ' & and is so defined: * ) The, rank of the sequent ' in ( -/is.!,1denoted by + If is not8#9 26 3 7 4 ,6 5 , * 0 is the total number of nodes of the J F K =?with +;subproof :/* < HF I L ending >@:/* < H; L J F D =BAC:/* <ED F I L K =G
It is easy to see that this simpler definition could not have worked for LK (even upon replacing cuts by mixes in it). Take, for instance, Transformation T11 in Proposition 3.3. It is apparent that such a transformation would not have brought about a reduction of , if such a concept had been defined as in Definition 3.7. 3) The proof of Proposition 3.3 must be adjusted in an appropriate way. In particular, we have to show that all the uses of contraction in such a proof can be dispensed with. Contraction has been employed in three subcases, exemplified by the following transformations:
:/* < HF I M L J M F K =
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3.1) Transformation T1. But this application of contraction depended strictly on the presence of mix. Now we can go a much easier way:
3.2) Transformation T5. But at present contraction is no longer indispensable, since there is just one occurrence of the mixformula to be cut:
' * + , - ! "#*#" , $ +!"! - "% 2 3546 %"& . $ / 2 789 6;:=< 13 ! "#*0"& ,1"( . $ +)") -)" / D D > 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!p lp 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:
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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 <1 9
*+-, . C
2. ) Transformation T
DFE
. Hence, let
G
qsrPtu
< 10
? BA
be the proof:
H IKJ
L Q U R JM Q L UWVX R NJ IPOM ^ _ `a IPOMJ% S L Z [ Z J L \ V ] Y V X J QS R T p
2
Y T
^ bN_ ca
rPtu
gihkj and n contains lke m n . If o does not where df P r t u contain , we can still resort to T11 (try to work out for yourself, as an rPtu is cut away from n ). If o exercise, the case where no occurrence of P r t u does contain (a case which we can no longer rule out), we adopt the transformation ~
v"w 8 x
~ rKq
rPtuq% m y n qu z o N
{ { } y \ | z rq mq n q o qu mq { n { q o qNrPtu N m q { n {qo
The previous discussion can be seen as a sketch of the proof of the next proposition, whose details are left once again to the reader: Proposition 3.6 (cut elimination theorem for LRNDM : Meyer 1966). Any
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intmixproof of in LRNDM can be turned into an intmix-free proof of the same sequent in the same calculus. By appropriately combining the modifications to Proposition 3.3 of § 1.2 with those discussed in the present section, one can rather easily achieve the next Proposition 3.7 (cut elimination theorem for LL: Girard 1987). Any in LL can be turned into a cut-free proof of the same cutproof of sequent in the same calculus. In his 1987 paper, Girard actually proved more than this: he showed that LLB and LLE are cut-free (and this last result was a little bit harder to obtain than the previous ones).
1.4 Cases where cut elimination fails Gentzen's procedure for showing that the cut rule is redundant in a given system works perfectly for a variety of calculi, but fails under some circumstances. Indeed, it can be shown - by exhibiting appropriate counterexamples - that there are calculi where the cut rule actually increases our stock of provable sequents. For example (Prijatelj 1996): Proposition 3.8. For each , LPLn is not cut-free.
Proof. formula , and be the%' formula Let be . It the ! " $# & is easy to see that the sequent is not provable in LPL2 without using cut. On the other hand, it is provable in LPL2 with the help of cut:
%'./%' ', %' 4 : 698 % +*, -" 0%' 4 : 698 %' (% 4 : 698>= 4 7 5 9 6 8 + , 1 " $# & ' % ! $)% 4 : $ ; 8 4 < 608 (%' +*, -"$#&%' 4 <9?A@ 8 ! "2#3%'
Similar counterexamples to cut elimination can be found for each LPLn . Quite analogously (Paoli 200+a):
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Proposition 3.9. LG is not cut-free. Proof. Let the following proofs be called, respectively, be the formula :
and
, and let
"!# "!# $ %& '
Combining them in an appropriate manner, we get a proof in LG of the sequent ( )*+) . In fact:
C CED 1 - 31 2 415 , 1 0 1 - 132 415 , 7 = <; ,.-/, ,.-/, 56132 41 - 7 8:<> ; 7 : 8 : 9 ; . , / 0 . , -/,.0/, 132 41 7 ?6@BA ; - 132 41 (GFIHJF provable in LG without using cut, a hypothetical proof of Were it could contain nothing but applications of the negation and disjunction rules K and of BW, because the axiom can play no role here. Hence the sequent ( LMNL would be provable in LA, where such rules are either primitive or derivable. But it is not (cp. e.g. Ono 200+a), whence our conclusion. O
By slightly generalizing the previous proof it is possible to show (Paoli 200+c P that LC is not cut free either. It is however interesting to remark that both LGg and LCg , if formulated without constants, are cut-free, and that the former result (unlike the latter) can be proved in an extremely simple and direct way.
2. THE DECISION PROBLEM When faced with a logical calculus (whether axiomatic or sequential), it is of great importance to know if there is an effective way to ascertain, given an arbitrary formula or sequent, whether it is provable or not. By an "effective" method (or algorithm), we mean a procedure which takes place in a finite number of steps and thus can - at least in principle - be carried out by a human or a machine. This demand, which first arose in the context of Hilbert-style
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calculi, led to the following Definition 3.9 (decidability of an axiomatic calculus). An axiomatic calculus S is decidable iff, given any formula of its language, there is an algorithm which determines whether is provable in S or not. It is well-known that classical propositional logic (unlike classical firstorder logic) is decidable. In fact, there is a very elementary strategy (the method of truth tables) to check whether a given formula is provable in HK. But, although it is the simplest and most popular, this decision method is not the only one available for classical logic. Suppose to possess an algorithm which tells you, for any sequent, whether it is provable in LK (or, for that matter, in any of the sequent calculi so far examined). In virtue of Proposition 2.49, then, you will get for free an algorithm which establishes the decidability of the corresponding axiomatic calculus. This is exactly the route which Gentzen took in his doctoral dissertation, by now quite familiar to the reader. There, he devised a procedure to find out, given any sequent in LK, whether it was provable or not in such a system. A sequent calculus with this property is called decidable as well:
Definition 3.10 (decidability of a sequent calculus). A sequent calculus S is decidable iff, given any sequent in S, there is an algorithm which determines whether is provable in S or not.
Let us now take a closer look at this method.
2.1 Gentzen's method for establishing the decidability of LK
Throughout the rest of this chapter, by "LK" we shall mean the equivalent be finite, system obtained by deleting the exchange rule and by letting possibly empty multisets of formulae of £0 . Suppose to be shown an arbitrary sequent in LK, and to be confronted with the task of finding out in a finite number of steps whether it has a proof. A promising strategy (known as the proof search algorithm) to accomplish this task might consist in generating one by one all the possible proofs of it - i.e. all the "candidates" for being a proof of in LK. To do so, we pick and consider all the possible inference rules that could have such a sequent as a conclusion. In this way, we start to construct a number of labelled trees, one for each distinct possibility, writing down the premisses of such rules above . Then we repeat our procedure for the
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premisses of these rules, and so forth until we reach terminal sequents which can no longer be the conclusion of any inference rule. At this point we scan one by one the labelled trees thus constructed; given any one of them, if all such terminal sequents are axioms, we have found the desired proof, otherwise we need to continue in our search. The crucial question is now the following: does this procedure terminate in a finite number of steps? In other words: can we restrict our proof search to finitely many candidates? In general, the answer is negative. In fact, three sources of potential infinity lie hidden in the very notion of proof, and need to be kept under control. 1) There is no upper bound on the number of formulae that an arbitrary proof of in LK could contain. This, however, is only seemingly a problem, for by Proposition 3.3 we can restrict our attention to cut-free proofs of , where such a number is bounded upwards by the number of subformulae of formulae in . 2) In spite of this, there is no upper bound on the number of occurrences of a given formula that an arbitrary cut-free proof of in LK could contain. Consider for instance the following candidate for a proof of :
is obtained by CL from , which in turn is Here,
obtained by CL from , which in turn... It is obvious that such
a chain cannot go on ad infinitum; but when is it going to stop? It is easy to see that, for each natural number , there is a distinct candidate for a proof of containing applications of contraction whose principal formula is . Hence, there are infinitely many candidates for a proof of . 3) Finally, there is no upper bound on the length of branches in arbitrary cut-free proofs of in LK, for a branch might contain arbitrarily many "detours" such as
Francesco Paoli Consider for example the following candidate for a proof of
103 :
is obtained by CL from , which in turn is Here, , which in turn is obtained by CL from obtained by WL from , which in turn... Again, it is easy to see that, for each natural containing number , there is a distinct candidate for a proof of
W
"cycles" made up by an application of WL immediately followed by an application of CL, where in both cases the principal formula is . Hence, there are infinitely many candidates for a proof of . It is therefore vital, to be in a position to narrow down our proof search to candidates which are "well-behaved" with respect to the above-mentioned parameters. In order to do this, we need some definitions.
is called a Definition 3.11 (contraction of a sequent). A sequent contraction of a sequent iff it is obtained from it by successive applications of CL and CR.
is called 3-reduced Definition 3.12 (reduced sequents). A sequent iff every formula in it occurs at most three times in the antecedent and at most three times in the succedent. is called 1-reduced iff every formula in ( ) occurs exactly once in ( ). Remark that any sequent whatsoever has a unique 1-reduced contraction.
, its 1-reduced contraction 3.10. Given any sequent is provable Proposition is such. in LK iff provable. Then, by successive applications of CL Proof. Suppose . Conversely, if is provable, a and CR, we get a proof of can be obtained by applying repeatedly WL, WR. proof of
is its 1
is provable in LK and Proposition 3.11. If reduced contraction, there exists in LK a cut-free proof of containing nothing but 3-reduced sequents. Proof. Let
be provable in LK. By Proposition 3.3, without loss of
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generality we can pick a cut-free proof of it. Let be the maximum among the lengths of the branches in . We shall prove this lemma by induction on . The base of the induction is trivial. As to the inductive step, let us discuss an example. Suppose that has the form and that it is obtained by an application of L:
Let be the 1-reduced contraction of . By induction hypothesis, there exists a cut-free proof of it containing nothing but 3reduced sequents. Now consider the following proof:
!
!
"
is 3-reduced, since ! occurs at most once in . Now, if ! does not occur in , then the desired 1-reduced contraction of ! is ! itself. If it does, it suffices to apply CL to obtain it. The proof thus obtained, moreover, contains only 3-reduced sequents. The other cases are treated analogously4 . # Now we are ready to define a class of "well-behaved" proofs to which we can restrict our proof search algorithm.
$
% '
Definition 3.13 (concise proofs). A cut-free proof of called concise iff 1) no sequent occurs twice in a branch of sequents occurring in ' are 3-reduced.
%
&
in LK is and 2) all the
&
is an LKProposition 3.12 (properties of concise proofs). (i) If provable 1-reduced sequent, there exists in LK a concise proof of it. (ii) Given % & a 1-reduced sequent , the number of possible concise proofs of it is finite.
%
&
Proof. (i) Let % be an& LK-provable 1-reduced sequent. Of, course, its 1-reduced contraction is % & itself. By Proposition 3.11, there exists in LK a cut-free proof ' of containing ( % nothing & but 3-reduced sequents. ' can be turned into a concise proof ' of by deleting as follows any ) * repeated occurrence of a sequent in the same branch of ' :
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(ii) Since concise proofs are cut-free proofs, the number of formulae which
may occur in a possible concise proof of in LK is bounded. Moreover, such formulae can appear at most three times both in the antecedent and in the succedent of any sequent contained in these proofs. It follows that also the
number of sequents which may occur in a possible concise proof of in LK is bounded. Finally, each one of these sequents appears just once in any branch of such proofs; hence, the length of each branch is bounded upwards by the number of sequents occurring in the proof, which, by the way, can
contain no more than branches. Consequently, the number of nodes of any
possible concise proof of in LK is bounded upwards, and therefore the number of such proofs must be finite. Proposition 3.13 (decidability of LK: Gentzen 1935). LK is decidable.
Proof. Suppose to be shown the sequent . Is it provable in LK or not? By Proposition 3.10, its 1-reduced contraction is LK-provable
iff is such. It is therefore sufficient to identify an algorithm which establishes whether is provable in LK or not. Such an algorithm is the proof search algorithm. By Proposition 3.12.(i), in fact, we can restrict our search to possible concise proofs of , and by Proposition 3.12.(ii) there are ony finitely many of these. This means that, if we fail to find a correct proof of among them, we are in a position to conclude that such a
sequent (hence also ) is not provable in LK.
2.2 A decision method for contraction-free systems It is quite immediate to see that the decision method suggested by Gentzen does not work for calculi which lack either the weakening or the contraction rules: our lemmata on reduced sequents, in fact, rely rather heavily on the presence of such rules. For contraction-free systems, however, we do not really need such a machinery. Indeed, reduced sequents were introduced with the sole
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purpose of picking out a finite class of proofs to which we could narrow down our search; more precisely, they were employed to prove that only finitely many sequents could occur in a possible concise proof of a given sequent. Anyway, let us examine all the rules of LK different from cut and contraction. The premisses of such rules contain at most the same number of symbols as their conclusions. Thus, we can infer rather easily that, in contraction-free calculi, only finitely many sequents may occur in any possible cut-free proof of a given sequent. In these systems, a concise proof of can be defined simply as a cut-free proof of it where no sequent occurs twice in the same branch. According to the above discussion, for every sequent there exist only finitely many such possible proofs. Thus:
Proposition 3.14 (decidability of LL, LLB , LA: Wang 1963). LL, LLB , and LA are decidable. But there is more than that. Surprisingly enough, it is possible to show that even the first-order versions of LL, LLB , and LA are decidable (Komori 1986; Kiriyama and Ono 1991), a fact which makes a striking contrast with the well-known undecidability of first-order LK.
2.3 A decision method for weakening-free systems If contraction is available in a sequent calculus, but weakening is not, our job gets tougher. The situation is somewhat parallel to the case of cut elimination: the contraction rules caused trouble in LK, but given the presence of weakening we were in a position to overcome such difficulties by introducing the mix rule. In weakening-free systems, as the reader will remember, we were forced to modify slightly our procedure to work things out. Here, we have solved the problems brought on by contraction by introducing the apparatus of reduced sequents, yet our lemmata concerning them relied once again on the availability of weakening. As a consequence, we are once more compelled to come up with some new ideas. To illustrate the situation, we shall discuss a simple example: the implicational fragment of LRND .
Definition 3.14 (postulates of LRi ). The calculus LRi contains the following postulates (as usual, are multisets of formulae of £4 notice that here sequents are single-conclusioned): (Ax)
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!
For reasons that will become clear in the following, however, it is expedient to work on an equivalent version of LRi , where contraction is "absorbed" into the operational rules. Definition 3.15 (the calculus LRi *). The calculus LRi *, formulated in the language £4 , contains the same postulates as LRi , except for the fact that the rule CLi is deleted and the rule Li is replaced by:
" * # , $ # is any contraction of
where occurs in it %&(' or ) times fewer than in such that $
, whereas occurs in it % or ' any other formula in
$ time fewer than in . Anderson and Belnap (1975, p. 125) explain the purpose of such a contrived rule in this way:
*,+.-
* ] amount only to The multiplicity restrictions [in the rule the requirement that [it] not be used to perform any contraction in its conclusion which could instead have been carried out in the premisses. They have no point other than to render finite the number of possible premisses from which a conclusion might follow, a feature essential, as we shall see, to our decision method.
Resorting to the strategies of § 1, it is not difficult to prove that LRi * admits cut elimination: Proposition 3.15 (cut elimination for LR*i ). LRi * is cut-free.
/
At the beginning of this subsection, we claimed that LRi * is an equivalent version of LRi . To prove that this is actually the case, we need the following, decisive lemma (named after Curry since a similar lemma is proved, in the context of LK and LJ, in Curry 1950):
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Lemma). Let be a contraction of . If 3.16 is (Curry's Proposition provable in LRi * by means of a proof of length , then is provable in LR * by means of a proof of length . i Proof. Induction on , left to the reader.
Proposition 3.17 (equivalence of LRi and LRi ). . *
LRi
iff
LRi
Proof. That LRi * is included in LRi is obvious, since any application of Li* can be simulated by successive applications of Li and CLi . Conversely, just remark that Li can be subsumed under Li* as a special case and that, by Curry's Lemma, CLi is a derivable rule in LRi *. We shall now present a decision procedure for LRi *, due to Kripke (1959b). Our strategy will be slightly different to the one we adopted in § 2.1. , whose provability we want to assess, we shall now Given a sequent build a complete proof search tree for it. The method works as follows. We at the root of the tree we want to construct, and then set above it place all the premisses from which it can follow in virtue of one of the rules, except cut (remark that in § 2.1 we had constructed separate trees for each different has a contraction below itself, we fail to alternative). Moreover, if include it in the tree as well. Notice that disregarding cuts is legitimate in virtue of Proposition 3.15; the latter restriction, on the other hand, will be justified presently. Then we repeat the procedure for each of the premisses thus obtained, and so on. is a provable sequent, then its proof is a If we can show that: i) if subtree of the complete proof search tree for it; ii) such a tree is finite, then we have attained decidability. In fact, it suffices to check one by one the finitely many subtrees of such a tree and see whether at least one of them is a proof of . However, neither i) nor ii) is so easy to show. As to the first sentence, has been proved by applying the rule Li* in it might happen that appearing in the proof tree is a such a way that some sequent contraction of a sequent above it. But such a proof tree would not be a subtree , given the way it has been of the complete proof search tree for constructed. To rule out such cases, we need a suitable analogue of the notion of concise proof.
Definition 3.16 (succinct proofs). A cut-free proof of i * is andin LR called succinct iff none of its branches contain sequents such that the latter occurs below the former and is a contraction of it.
Francesco Paoli Proposition 3.18 (properties of succinct proofs). If LRi *, there exists in LRi * a succinct proof of it.
109 is provable in
Proof. If is provable in LRi *, by Proposition 3.15 there exists, in the same system, a cut-free proof of it. Now we shall prove our lemma by induction on the length of . The base of the induction is obvious. Suppose now that contains the following branch:
ending with has length , where has length , its subproof length . By Proposition 3.16, and its subproof with has a proofending of length has. Hence we may replace by the following proof :
! " #%$ . By induction hypothesis, then, & is a succinct proof whose length is ' "). ( of * is provable By the previous proposition, we have made sure that if in LRi *, then its proof is a subtree of the complete proof search tree for it. This yields the first of the above-listed desiderata. What about the second? It is about time to recall what is perhaps the best-known result in the theory of trees, known as König's Lemma, according to which a finitary tree is infinite iff it contains at least an infinite branch. Thus, we only need to show that our complete proof search tree is finitary and contains no infinite branch. The first part is easy to settle: since we are in a position to disregard the cut rule, it is clear that any sequent in the tree can only have finitely many sequents as immediate predecessors. The second part is a little harder. To establish it, we need a couple of definitions.
Definition 3.17 (sequences of cognate sequents). A sequence of sequents is called a sequence of cognate sequents iff all of its members have the form
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Substructural logics: a primer
, where
are positive integers.
Definition 3.18 (irredundant sequences). A sequence cognate sequents is said to be irredundant iff, whenever contraction of '& .
!#"
,
%$
of is not a
To attain our goal, it is enough to prove that any irredundant sequence of cognate sequents is finite. In fact: (i) we have constructed our complete proof search tree in such a way that every sequence of cognate sequents occurring in any of its branches is irredundant; (ii) the subformula property for LRi * implies that only finitely many such sequences can occur in any branch of this tree. Hence, we shall now prove the following Proposition 3.19 (Kripke's Lemma: Kripke 1959b). Any irredundant sequence *) ( 2 +-,. ( +,/ 10 of cognate sequents is finite. Proof. We proceed by induction on the number 3 of formulae occurring in . For 3465 , the result is obvious. Let now contain 7685 formulae; we 9 9 choose one of them, call it 9 . We say that (%$ +-, is -critical (in the : ; 0 sequence ) iff for any occurs in ( < +-, at least as many times as it (%$%+=, does in . Now we define a new sequence > @) ? ? 9 2 +-,. +-,. 10 as follows. Let ( & +-, be the first critical sequent in (our previous definitions ensure A that such a sequent ? +=, ( & A +-, 2 always exists), and put , where ( & is the same as ( & , 9 ? < +-, except for containing no occurrences of . If has been defined as ( $ A I-J B K@L ( CED1F M-J.N ( CEDHG M-J.N O O O1P , consider the sequence (if Q CED1F M=J R SD1F M-J does not exist, we are home), and take as the sequent T W M=J T U M-J X V , where V is the first -critical sequent in . Y Z Y [ It is easy to see that is finite iff is finite. Moreover, contains formulae, hence by inductive hypothesis it is finite if irredundant. But it is Z \ such, given the way it was constructed. Therefore, is finite.
From the discussion above it follows: Proposition 3.20 (decidability of LR*i : Kripke 1959b). LRi * is decidable.
\
Such a result was extended, with minor modifications, first to the grouptheoretical fragment of LRND (Belnap and Wallace 1965) and then to all of LRND :
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Proposition 3.21 (decidability of LRND : Meyer 1966). LRND is decidable.
2.4 Other decidability (and undecidability) results As we have just seen, the decidability of an axiomatic calculus S can be established, so to speak, "parasitically", by showing the decidability of the corresponding sequent calculus and then referring to the theorem which asserts the equivalence of the two systems. But there are also more direct ways to obtain the same result. One of the most popular has a semantical character and consists in proving that S has the finite model property, a countersign which can be so defined:
Definition 3.19 (finite model property). An axiomatic calculus S has the finite model property iff any S-unprovable formula can be falsified in a finite model.
If S has such a property, it is easy to see that it is decidable. All the proofs of S, in fact, can be enumerated ( ), and so can all its finite models ( ). Given an arbitrary formula , we first check if is a proof of , a task we can perform in finitely many steps. If it is, all the better; if it is not, we check if falsifies , and once again we shall be finished after finitely many steps. If it does not, we pass to , and so on, leaping back and forth from the list of proofs to the list of finite models and vice versa. We shall not have to wait forever, since sooner or later either the right proof or else the right falsifying model will turn up. In the literature on substructural logics, a number of decidability results of this kind can be found. One of the earliest is due to Meyer (200+), who made use of a version of Kripke's Lemma to show the finite model property for HRi . Adopting a variant of the same method, Meyer and Ono (1994) proved that also HAi has the property. Buszkowski (1996) obtained an analogous result for HLi by resorting to a different technique (the method of barriers). From the previous lines, it would seem as though the only calculi whose decidability can be proved in this way were purely implicational calculi. But this is not quite true. For example, Lafont (1997) established the finite model property for all of HL and HA. A very good survey concerning the abovementioned results can be found in Ono (1998a). For the sake of completeness, we also mention a further semantical technique which is often used to show that a calculus has the finite model property: the method of filtrations. Such a strategy, quite widespread among
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modal logicians, is not very suitable for axiomatic calculi containing the axiom F2. This is the reason why it has been employed only for very weak substructural logics (cp. e.g. Fine 1974). Before we close this subsection, we cannot help mentioning two areas of research whose in-depth analysis would require on the reader's part an amount of logical expertise which we do not want to presuppose. The first focuses on proofs of undecidability for various logical calculi, the second is about the computational complexity of decision algorithms. As to the former domain of investigations, we only mention the two most important results. For many years since its explicit formulation, the problem concerning the decidability of the relevant system HR had remained open. Finally, Alasdair Urquhart (1984), by means of an extremely ingenious and sophisticated argument, succeeded in proving that HR is actually undecidable. Some years later, Lincoln et al. (1992) showed that HLE and LLE are undecidable as well. Moreover, Lincoln and his colleagues pointed out that even the exponential-free calculi HLB and LLB are in general unable to decide effectively whether a formula (a sequent) is derivable from a multiset of formulae (of sequents). It is interesting, anyway, to compare these limitative results with a theorem by Kopylov (1995), stating that the sequent system obtained by adding to LA the rules for exponentials (known as full propositional affine linear logic) is decidable. To end off, let us spend just a couple of words on the research about computational complexity. Some decision algorithms for either axiomatic or sequent calculi are not very efficient. If given as an input a formula (or a sequent) of the appropriate language, they do indeed yield as an output the required verdict on its provability in the calculus at issue, and they do so after finitely many steps, but sometimes - if the input formula is rather complex such a number of steps is so high that even the fastest available computers could not perform the job in a reasonable period of time. Discovering feasible decision algorithms, i.e. procedures that can be actually implemented on real machines, is therefore a task of great practical importance. Over the last few decades, a copious literature on the subject of computational complexity has been rapidly piling up (two readable introductions are e.g. Garey and Johnson 1979; Floyd and Beigel 1994). These investigations now profitably interact with the research on systems of propositional logic (Krajicek 1995; Urquhart 1995). As to the computational complexity of decision algorithms for substructural logics, the reader is referred to three very useful survey articles: Urquhart (1990), Lincoln (1995), and Ono (1998a).
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Sources of the chapter. The whole of this chapter owes much to Ono's (1998b) tutorial article on the proof theory of nonclassical logics. The present proof of the Hauptsatz is substantially the one to be found in Casari (1997b). Other texts on which we drew heavily in this chapter are Dunn (1986) and Anderson and Belnap (1975).
Notes 1. From now on, the terms "inference" and "application of a rule" will be used interchangeably. 2. Remark that our definition of rank is not the standard one to be found in most textbooks. As a rule, the notion of rank is undefined for proper subproofs of a mixproof; we believe, on the other hand, that our definition may have some didactical advantages, since it allows to "visualize" the gain in rank obtained with each reduction step. 3. To be sure, such a recourse to the mix rule can be avoided. As von Plato (2001) recently suggested, it is possible to prove the Hauptsatz even for LK and LJ by eliminating plain cuts: one has to distinguish several cases according to how the auxiliary formulae of contraction inferences were introduced into the proof at issue. 4. It is left up to the reader to verify that, in the subcases of the inductive step involving rules with two premisses, one may bump into sequents whose antecedents, or succedents, contain three occurrences of a formula; this is the reason why we need to consider 3-reduced sequents.
Chapter 4 OTHER FORMALISMS
In Chapter 2, we examined two different kinds of formalisms whose role is undoubtedly central in the proof theory of substructural logics: sequent calculi, on the one hand, and Hilbert-style systems, on the other. On that occasion, we noticed that there are at least six well-motivated axiomatic calculi - HRW, HR, HRMI, HRM, HLuk, and HLuk3 - which do not have any sequential counterpart, in that they seem scarcely amenable to a treatment by means of traditional sequents. As we already remarked, Hilbert-style calculi are definitely not the best one could hope for when it comes to engaging in proof search and theorem proving tasks. As a consequence, it seems desirable to find efficient and manageable formalisms also for the above-mentioned logics. In the present chapter, we shall briefly illustrate three types of formalisms. In the opening section, we shall delve into some generalizations of ordinary sequent calculi. Each one of them arises when we abstract from some peculiar features of traditional sequents as conceived of by Gentzen, and allows to deal with one or more of the logics we mentioned in the previous paragraph. Subsequently, we shall try to get the hang of proofnets, which according to Girard (1987) constitute a sort of natural deduction calculus for linear logic. Finally, we shall turn to resolution calculi, taking a rapid look at resolution systems for some substructural logics. The content of this chapter, as the reader will guess, is of a more advanced level than the material so far presented. The topics we shall discuss are at the very centre of contemporary research into the proof theory of substructural logics. This does not imply that we request, on the reader's part, any previous knowledge of the subject or technical prerequisites of any kind; rather, this means that we shall often be forced to skip a number of details and to present
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many results in a "dogmatic" manner, omitting their proofs or replacing them by examples which can give the reader some clue about how these proofs go through, or about how the calculi at issue actually work.
1. GENERALIZATIONS OF SEQUENT CALCULI As we repeatedly pointed out, Gentzen's sequents are meant to represent inferences in a fully general and abstract form: we have an antecedent, which contains the premisses of the inference; a succedent, which contains its conclusions; and the arrow, which stands for the relation of consequence between the former and the latter. If we examine with care the structure of sequents, however, we can spot at least three somewhat accidental features, three expressive limitations due to which our calculi do not seem to stand up to the required level of abstraction and generality: The arrow denotes a binary relation of consequence between an antecedent and a succedent. Why should it be so? In other words, why should the number of "cedents" be just two, and not three or more? " " means, informally, " follows from ". What if I want to say that "either follows from , or follows from "? There seems to be no way to express such a disjunction by means of ordinary sequents. Should not we enrich the expressive power of our calculi to allow also for such patterns? Comma is the only way of bunching formulae together inside sequents. Cannot we think of other ways of providing them with an internal structure? Should not we try to refine, when circumstances seem to require it, our analysis of the logical role of comma? All of the above questions have been successfully addressed by logicians working in the proof theory of nonclassical logics, who came up with such flexible formalisms as -sided sequent calculi, hypersequent calculi, DunnMints calculi and display calculi. Let us examine them one by one.
1.1
-sided sequents
Finite-valued Lukasiewicz logics, like many other logics whose semantics is smooth and attractive, have always proved very resistant to any prooftheoretical analysis, especially by means of sequent calculi. In 1967, however, Rousseau (foreshadowed to some extent by Schröter 1955) thought out a promising new idea. His strategy can be summarized with a slogan: if two-
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sided sequents are good for two-valued logic, multiple-valued logics need multiple-sided sequents. Let us now try to be more precise. The classical sequent holds, informally speaking, iff at least one of the 's is false or at least one of the 's is true. In other words, it holds iff at least one of the 's assumes the value 0 ("false") or at least one of the 's assumes the value 1 ("true"). Two values, two multisets of formulae. Now, suppose to be faced with an -valued logic; what you will need, then, is an ordered -tuple of multisets of formulae which holds true iff there is a such that at least one of the 's assumes the value . This seems a fair generalization of the two-valued case; and such, in fact, is the intuitive idea behind Rousseau's investigations. After Rousseau introduced (or rediscovered, if we take into account Schröter's contribution) -sided sequents, many authors followed in his footsteps. Similar techniques are employed in several interesting papers and monographs on the proof theory of finite-valued logics, such as e.g. Carnielli (1991), Zach (1993), Baaz et al. (1994), Baaz, Fermueller et al. (1998), Gil et al. (1997, 1999). To give a significant example of this area of investigations, we present hereby a calculus for three-valued Lukasiewicz logic which is substantially equivalent to the one to be found in Baaz et al. (1994) - and not very distant from the original system by Rousseau. A 3-sided sequent is an expression of the Definition 4.1 (3-sided sequent). form , where each is a finite, possibly empty multiset of formulae of £1 . Intuitively, ! " says that at least one of the 's is false, or at least one of the ! 's is intermediate, or at least one of the " 's is true. Definition 4.2 (postulates of LLuk3 ). LLuk3 is a calculus whose basic expressions are 3-sided sequents. Its postulates are: Axioms
#$ % #% #
& ' ( -/.%021 #*) &' ( 8*9:8,9 B @ A C -<;=021 8*9 @A C
Structural rules
& ' ( -/.4351 & +,) ' (
8 9:8,9 ? @ , A C -/;>351 @ 8,9 ? A C
& ' ( -/.%671 +,) &' ( A 8,9:8,9 C @ -/;=621 A 8,9 C @
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( * + , * + * + ,
!#" %$'& , ) ! %$'& 0
Operational rules
21 - - . 1 / 4 - 21 . / 4 . / 4 . / 63 5'7 - 3 21 .0 / 3987 . 3 21 / 36:'7 3 21 - - ; < > = B> ; < = @'E F ?A@BC> ; D < = G JLKCJ H O I KJ G H MJ I NPOQR G M KCJ H I MJ U X V YCX W ZX U YCX V KCJ W NSOT'R U V Z\[YCX W ] XS^_X6`
a b
c
Remark 4.1 (definability of connectives in LLuk3 ). The rules for the connectives , , and can be easily derived from the above rules, taking into account the mutual relationships among different connectives that hold in all our substructural logics and recalling that in all of Lukasiewicz logics is equivalent to and is equivalent to .
ZdaeY
fgZA[Yh6[Y
b
ZA[iZ
Remark 4.2 (on the rules of LLuk3 ). Notice that in LLuk3 operational, weakening and contraction rules are not divided into left and right rules, as usual, but rather into 0-, 1- and 2-rules. So, for example, there are as many weakening rules as there are sides in a 3-sequent, i.e. three. We also have three cut rules, but for a different reason - in fact, there are as many cut rules as there are pairs of different truth values in our logic. Operational rules are directly obtained from the three-valued truth tables for the corresponding connectives. For example, negation shifts formulae from the "false" side on to
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the "true" side and vice versa, while leaving where they stand formulae on the "intermediate" side. Remark 4.3 (on the substructural character of LLuk3 ). The reader will have noticed that LLuk3 contains both weakening and contraction rules; therefore it is not, strictly speaking, a substructural calculus. In the next section, however, three-valued Lukasiewicz logic will be given formulations in terms of hypersequents where (internal) contraction rules do not hold. So, how does the matter stand? This seeming contradiction can be explained away by remarking that contraction rules, as we shall presently see, express here nothing more than the idempotency of lattice-theoretical disjunction. Therefore, they are similar to external, not internal, contraction rules in hypersequent calculi1 , and external structural rules always hold in such calculi. Now, let us prove that LLuk3 actually corresponds to HLuk3 . Definition
4.3
(formula-translation
of
a
3-sided
sequent).
Let
, and . Let moreover
! # $ " % ! '( '( '( ' ( & '( be the 1* formula 1 1+1. The formula ++ ) *+ ,.-0/ translation of the three-sided sequent , -0/ is defined as 24365 78365 9:365
follows if either
or
or
:
A < ; < F G =4>>>?=@;!< F H B ! = A>>?=B
If L4MN8MPO:M6Q , then RTS
X
1* 1V V+ V+ F. G0 J U
GI C#=:< JG =4>>>?=:< J K
W
is .
Proposition 4.1 (equivalence 1+ of 1 LLuk3 and HLuk3 ). (i) 1_1 1+If 1 X HLuk3 Y , then ++ ++ . ` 0 a b ` 0 a b ^ . , then X HLuk3 \T] LLuk3 Z[Z Y ; (ii) if X LLuk3
Proof. (i) Induction on the length of the df proof of e g6h example, let us consider the axiom F22. Let the following proofs:
kpo
gf1y gf1 g
gf1f gf1 g
gq1 d:jgrjsgt1 iug[jg
gf1 g[jTi!g[jvdt1 i!g[jg
w 1 d:jgq1 d:jxg dyh&gq1 d:jgf1 g
c
in HLuk3 . Byjway of kmlPj n , and let be
i!g
gf1f gf1 g gq1 d:jgf1 g
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)) )) )) ) ) )) )) )) ) ) )) ! ) ) #)*"$) % & )") '( )*") 2)2) 2 .).) . /%0 123) /0 12) 2 23) /%0 12) 425 2 2) 42*5 2) 425 2 6 )2 ) 75 2 2)2) 2 +-, 78(2)2) 2 6*)*6 ) /78(291:8(2 can be proved as follows: Then ; < = ) ) ) ) 78(2 75 22 25>78(2 6 75 2 78(2)2) 2 78(2 ) 78(2) 2 6)*6 ) /78(291:8(2 ?@)A ?A) ? (ii) Induction on the length of the proof of BDCFE in LLuk . The proof is rather tedious and convoluted, and we omit it. G 3
By resorting to semantical methods (which is often an expedient shortcut, when one is working with many-valued logics) it is possible to prove that Cut01, Cut02, Cut12 prove exactly the same LLuk3 and LLuk3 sequents:
HJI
K
? LN)A A ?AMF) ? O ?ALN)A ?AMF) ? O in
Proposition 4.2 (redundancy of cut in LLuk3 : Rousseau 1967). If is provable in LLuk3 , then there exists a proof of Cut01, Cut02, Cut12 . LLuk3
HQP
R G
As Baaz et al. (1994) correctly point out, the previous proposition does not deserve the name of "cut elimination theorem", for it simply guarantees the existence of a cut-free proof for any provable -sided sequent of LLuk3 , without yielding an effective method to produce it. In the same paper where such a remark can be found, Baaz and his colleagues prove a proper cut elimination theorem for LLuk3 (and other logics) by a direct constructive
S
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argument, i.e. by a stepwise cut removal procedure similar to the ones already encountered in the preceding chapter. The sample derivation contained in the proof of Proposition 4.1 makes it quite clear that LLuk3 is a rather cumbersome and inefficient calculus from a computational viewpoint. A more elegant calculus along the same lines was recently suggested by Aguzzoli, Ciabattoni and Di Nola (2000), who took their cue from a well-known decomposition of formulae of three-valued Lukasiewicz logic into ordered pairs of classical, two-valued formulae. Loosely speaking, their "sequents" are ordered pairs of classical sequents, where formulae in (resp. ) correspond to the first (respectively the second) projection of such a representation. This idea was also generalized to obtain a cut-free calculus for infinite-valued Lukasiewicz logic (Aguzzoli and Ciabattoni 2000) which avoids, unlike other cut-free calculi to be found in the literature, any deductive apparatus involving nonlogical elements external to the language.
1.2 Hypersequents The fact that three-valued Lukasiewicz logic is a substructural logic is less than evident in the calculus we have just seen. On the other hand, such a property becomes crystal clear if three-valued Lukasiewicz logic is formulated in terms of hypersequents. Hypersequents were introduced independently by Pottinger (1983) and Avron (1987), who managed to provide cut-free Gentzen-style formulations of systems that had been frustrating such attempts for decades - viz. the modal system S5 and the relevance logic RM, on which we shall return presently. Since then, hypersequents have proved a fruitful and extremely flexible tool for the study of nonclassical logics, giving rise to neat and thorough prooftheoretical analyses of many intermediate, modal, many-valued and substructural logics (for a survey, see Avron 1996). The next definition explains what we mean by a hypersequent.
! #" "
Definition 4.4 (hypersequent: Avron 1987). A hypersequent is a finite multiset of ordinary sequents. It is customary to write hypersequents in the form:
Each sequent is called a component of the hypersequent. We use as metavariables for (possibly empty) hypersequents.
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It is evident that ordinary sequents may be regarded as hypersequents with a single component. It follows, in particular, that traditional sequent calculi are special instances of hypersequent calculi. The intuitive interpretation of the vertical bar is disjunctive: may be read as " follows from or follows from ". As a first example of hypersequent calculus, we present Avron's system LLuk3 ' for three-valued Lukasiewicz logic.
Definition 4.5 (postulates of LLuk3 ': Avron 1991a). The hypersequent calculus LLuk3 ', based on the language £1 , has the following postulates:
Axioms External structural rules Internal structural rules "!$#% Mixing + &&,& . +' , . + , . '+/' ( ',' . )+ &, , ,0 *,1 &. . .' . 3 46587 2 (+ 9 +' Operational rules 2 : > ;< ACBEDF 2 <;&? : > AGBEHF @ = <;&? : > @ ? : > ; = < @ ? I > JK LN MOJ' T I U A R DF @P KJ&? I > J%M A R HF @ LQ K6R MOJ ?VJ' @ ? I > JKSR M T I > JU
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! " # $& %$& %$& $& ! '($& / 0 1 0 +&4 5 2 )! *(+&,- 1 . 0 3 2 1 . 0 - * 2 1 . 0 -, +&65 2 1 . 0 7 4 5 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 O F L E !I DPFJM G KJF O C G L QSRUTW V
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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<6 =7:4> :@?;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
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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
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Substructural logics: a primer
(so that, The rules L and R must abide by the restriction ! and #)" for example, we cannot conclude from The mixing rule Mx is replaced by two relevant Mingle rules: - $ &% . / $' 354687:9 $0 - %( $) . / 346<;=9 2 % 01$) ! 2 %+ $,0,$) . -*$*/ $ . -*$ / and a splitting rule:
2 01>) . ? * - >/ 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 .
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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 .
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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) # $ ) # :<5 ) :<6 2$9 :<) ;:<) # ; # 4) # =<5 ) ;=<) # ;=<) =<) =<6 4) ?A> @CB D > ?A> EB ) 1 >
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Remark 4.6 (associativity rules). LR+ has, beside the usual structural rules of weakening (for comma), contraction, cut, and exchange, also associativity rules for both comma and semicolon. In fact, Definition 4.9 offers no guarantee that either comma or semicolon is associative. Associativity rules can be dropped, but this simplification has a price: we must either replace them by equivalent structural rules (as in Anderson and Belnap 1975), or else muddle up somewhat the definition of structure (as in Dunn 1973). Remark 4.7 (restrictions on the rules). The reasons for restricting (W,) and cut become evident upon considering the following "proofs", where unwarranted inferences are countersigned with exclamation and question marks:
)
!?
!?
It can be shown (see e.g. Anderson and Belnap 1975, pp. 372 ff.) that LR+ is actually equivalent to HR+ and that it is cut-free. Instead of proving these results, we shall present an example of a proof in LR+ in order to illustrate the functioning of such a calculus. In what follows, let and ; moreover, let be the following proof:
"! $# %& ' 47636 838 ( ./ , ( -+* , 2 4:35 ( )+* , 4:9 9 663534<= ; 6 449 89 833 8 = ( )+* , ( )+0, ?@3 ? 4:9 6 > 83= ': ( 10, 4:9BAC?ED FGH3= AI?@J54KG&; F3?@J= can now be proved as follows: ' 49BAC?ED FGH3= ?L3? ( 10, AI?ED ?@J54KG 9BAC?EDMFGH3= AC?EDBAC?@J549 FGNGM9BAC?EDBAC?@J549 F:GNGH3= ( ./ , ( S / , ?ODBAC?@J54P9MFGH3= ( -+0, ?ODBAC?@J54:GQ; F3= ( 1* , AI?@J54KG&; F3?RJ =
These ideas have been developed in several directions. Using calculi in the style of Dunn and Mints where formulae are prefixed either by a positive or by
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Substructural logics: a primer
a negative sign, and building on previous work by Giambrone (1985), Brady (1990; 1991; 1992) managed to provide cut-free calculi also for RW and some weaker relevance logics, showing their decidability as well. More recently, O'Hearn and Pym (1999) took their cue from Dunn-Mints proof theory and suggested a new logic of bunched implications, with interesting computational applications. However, the most successful development of Dunn-Mints calculi over the last two decades has certainly been Belnap's display logic, to which we now turn our attention.
1.4 Display calculi Dunn-Mints calculi marked a significant advance in the history of prooftheoretical research into relevance logics. Nonetheless, they seem to be plagued by four manifest defects: It is unclear how to accommodate negation into them, unless we resort to signed formulae à la Brady (which, on the other hand, would make our logical apparatus quite cumbersome). It is often necessary to perform substitutions inside structures, a feature which renders their rules a little opaque. The restrictions placed on some rules seem somewhat artificial. Dunn proved the admissibility in LR+ of the rule called "cut", which is however a restricted cut rule - even though it is all that is needed in order to prove completeness with respect to HR+ . Full cut (from and infer ) is not even a sound rule, because - as we have seen - it permits to prove some HR+ - invalid paradoxes of material implication.
Belnap's (1982) display logic overcomes all of the above inconveniences. Here are its core features: Like in Dunn-Mints calculi, formulae can be bunched together in several different ways. Here, however, we do not have just comma and semicolon, but also other structural connectives. In full accordance with the "underdetermination view" of operational rules we sketched in Chapter 1 - according to which the meaning-giving role of such rules, in ordinary sequent calculi, is blurred by the presence of comma and of side formulae - the principal and auxiliary formulae of operational inferences are required to be visible, i.e. to constitute the whole of the structures where they occur. More than that, if we are given a formula and a sequent containing it, it is always possible to transform that sequent into an equivalent one where the
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given formula is either the entire antecedent or the entire succedent (such an operation is called displaying the given formula, whence the name of display logic). As a consequence, the only cut rule we really need is simple cut:
All other forms of cut (including "nested" cuts, used to delete formulae inside wider structures) are reducible to simple cut. Therefore, to achieve cut elimination for a given calculus it is sufficient to show that simple cut does not lead to the proof of new theorems therein. Belnap (1982) introduced display logic as a proof-theoretical tool for the investigation of a wide range of logics, not only of R. The most astonishing result of his paper was the proof of a general cut elimination theorem which holds for all display calculi whose rules obey a small number of easily checked formal conditions. This result certainly recommends display logic as a powerful framework for a unified treatment of the proof theory of several logics; as regards R, however, Belnap's presentation rested on the addition of a classical negation connective which, given the philosophical motivations underlying this system, seemed rather unacceptable. The amended version by Restall (1998), which we are going to illustrate, solves such a problem (another display calculus for R can be found in Goré 1998). Definition 4.13 (£1 -structure and substructure). £1 -structures (henceforth in this subsection, structures) are either antecedent or consequent structures: Any formula of £1 is both an antecedent and a consequent structure. is both an antecedent and a consequent structure. If and are antecedent (consequent) structures, so is . If and are antecedent structures, so is . If is an antecedent and a consequent structure, then is a consequent structure. If is an antecedent (consequent) structure, then is a consequent (antecedent) structure. The concept of substructure of a structure is defined as in Definition 4.10, taking of course into account the presence of the new structural connectives.
Definition 4.14 (sequents in LR). A sequent in LR (henceforth in this subsection, a sequent) is an expression of the form , where is an antecedent structure and is a consequent structure.
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Definition 4.15 (antecedent and consequent parts of sequents). If is a sequent, then is an antecedent part and is a consequent part; If is an antecedent (consequent) part of a sequent, so are and ; If ( ) is an antecedent (consequent) part of a sequent, is an antecedent and is an antecedent (consequent) part; If is an antecedent (consequent) part of a sequent, is a consequent (antecedent) part.
Definition 4.16 (display equivalence). Below, sequents in the same row are said to be immediately display equivalent:
are called Two there exist sequents / "!#!#!#sequents 0 s.t. / and $ %, &0 $ 2 anddisplay '(*equivalent ),+.-. 1 iffis immediately for display equivalent to 14365 . Definition 4.17 (postulates of LR ). The system LR has the following postulates:
97 Axioms 87 Display rules
8 ? ;<= ; 8 >"= < ?
:
; 8 "> = < ? 8 ? ;<=
The display equivalences of Definition 4.1 , read as two-way rules. E.g.:
Structural rules ? B E 8 D ;<"@A=< ;<"@C=< ?B 8D @ 7 2 < B @ ;<= B < ? 8D G@ FH< B 7 B @ < @ ;<= B < ? 8ED @ =<; B < ? 8D @ ;< ?B <= 8D @ ;<= B <= 8? @AIH< B ; "@C= ?B 8D @ 7 B ; 8 = @ D B ;<= 8? @ ; = B ? 8D ; ? 8 =
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"!$#&% The rules
029 143 + -.
'
Operational rules R,
(
R,
)
R and
*
R of LR+ , plus:
/ 9 069 573 , / 08, 143 . / : 02573 . 0 > 143 . /=;$< 0 > 573 $ ; < / , . / , >< / . . / >< < ?A@8/ . . 0 B 143 </ . @8/ . . 0 E 143
< J W @KEXU K
Remark 4.8 (the "intuitionistic" structural connective). Suppose you want to display in . You have two alternatives: either you introduce a "Boolean" negation " " with the property that iff , as Belnap (1982) did, or you follow Restall (1998) in resorting to a structural connective , not corresponding to any object language connective of £1 . can be read as " intuitionistically implies ", since the display equivalences in the fourth line of Definition 4.16 can be interpreted as versions of the deduction theorem for intuitionistic implication ( is deducible from iff is deducible from , iff is deducible from ).
<
G
U
@
.ZY G
."Y G
@8J U
.
<
The next proposition - which holds not only for LR, but for any calculus with the same display and operational rules - yields the distinctive feature of display logic:
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Substructural logics: a primer
Proposition 4.6 (display theorem: Belnap 1982). Each antecedent (consequent) part of a sequent can be displayed as the entire antecedent (succedent) of a display equivalent sequent ( ), where is determined only by the position of in , not by its structure.
Proof. Induction on the construction of the structure containing . If , we are finished. By way of illustration, let us check one subcase of the inductive step. Let and let be, without loss of generality, a substructure of and an antecedent part of . This sequent is display equivalent to . By induction hypothesis, there is a sequent , display equivalent to , whose antecedent is . By transitivity of display equivalence, is equivalent to . If is a consequent part of , we argue analogously.
We are now ready to prove a completeness result for LR. Henceforth in this subsection, by "LR" we shall mean its linguistic extension based on the language £7 , which contains the new binary connective of "intuitionistic" implication (it is not an axiomatic extension, since no new postulates are added).
Definition 4.18 (formula-translation of a structure and of a sequent). If is an antecedent (consequent) structure, its antecedent (consequent) formulatranslation ( ) is a formula of £7 , inductively defined as follows:
#! ' " %$ !&( " %$ !&' "*) $,+ !&( "-) $,+ ) !#' "/. $ + C ; !&( "/. $,+ D !#' " 10324$5+ !&' " %$76 !&' " 28$ !&( " 10328$,+ !&( " 9$7: !&( " 28$ !#' " 1;324$5+ !&' " %$=< ! ' " 28$ !&( " ?>@28$,+ !&' " 9$ !&( " 28$ !#' "3A %$,+B !&( " %$ !#( "3A %$,+ EGF&K H-I%J If ILM is a sequent, its formula-translation FNH-ILM8J is defined as F#K H-IOJ5PF&Q H-M8J . S iff R 4.7 (completeness of LR: Restall 1998). R T L Proposition S. HR
R ILM
LR
Proof. From right to left, the proposition can be proved by semantic methods: it is possible to prove, more generally, that if , then LR is valid in all the structures of a certain kind which satisfy a number of semantic conditions. It can likewise be shown that any formula of £1 which is valid in such structures can be proved in HR. Upon remarking that
FNH-ILM8J
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does not contain , and that HR iff HR , we get the desired conclusion. From left to right, we proceed as usual by induction on the length of the proof of in HR. We provide a couple of examples:
* +-,/.
! !"
* +-,/. * 021 3 465 7 86.
#$ "%
* 92: .
#$ & "$
* +-;!< =/1 > ?@.
' ($
* A@BC.
"$ ' ($ ) DE F(G$H F IHJ&K"K H F IH F(G$H'J K"K
In the proof below, let GEGWIEI G&YIE'GQPRI G&YIE
L I E'G[ L
LNM
FOGQPRI KTSUF(GVPRJ K G$EXGWJEJ
* _(;.
G YJEGVPZJ
* `(;. * 0!a .
G YJE
L JEG\[ L
I]SRJE'G[ L G&YI^SRJE GQPUF I^SRJ K E
* +-;.
: * _(;. * `(;. * 0!a . * `(,/. * 0!a .
L
* _(,/. L
DEGVPUF I^SRJ K H L
* A@BCb +-;.
c
As we hinted above, Belnap (1982) singled out eight structural properties of rules that are necessary and sufficient for a calculus in order to admit the elimination of simple cuts (hence, in virtue of Proposition 4.6, of any cut whatsoever). Since LR obeys such rules, we are in a position to conclude: Proposition 4.8 (cut elimination for LR). Simcut is eliminable in LR.
c
As Restall (1998) acknowledges, however, this calculus does not solve the problem of providing a cut-free sequent calculus for R in its original vocabulary. Although the intuitionistic conditional can be conservatively added to R, in fact, it is not definable in terms of the connectives of £1 .
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1.5 A comparison of these frameworks In a survey paper on the role of hypersequents in the proof theory of nonclassical logics, Avron (1996) lists six desiderata by which any generalization of ordinary sequent calculi should abide5 :
1. It should be able to handle a great diversity of logics of different types ... On the other hand, the construction of the framework might suggest new logics that should be important. 2. ... The framework should be independent of any particular semantics ... . 3. The structures used in the framework should be built from the formulae of the logic and should be not too complicated ... . Most important - the subformula property they allow should be a real one. 4. ... The applicability of a rule should depend only on the structure of the premisses and not on the way they have been obtained. 5. ... The rules for the connectives should be as standard as possible. The difference between logics should be due to some other rules, which are independent of any particular connective. Such rules are usually called "structural rules". 6. The proof systems constructed within the framework should give us a better understanding of the corresponding logics and the difference between them.
As Avron points out, the second requirement (as well as the first, we dare to add) is violated by formalisms like -sided sequents. According to Avron, Dunn-Mints calculi and display calculi fail to meet the third desideratum. For instance, go back for a while to the proof of distribution in the calculus LR: what formulae do "really" appear in that proof? One should perhaps count also in that number, although the intuitionistic conditional is disguised as a structural connective. Such a formula, however, is not a subformula of formulae in the endsequent. In Avron's opinion, hypersequential calculi meet all the above standards, and should therefore be awarded a distinguished role among proof systems for nonclassical logics. On the other side, Wansing (1998) manages to translate hypersequents into display logic in at least three sample cases. For example, building upon Avron's hypersequential calculus LLuk3 ', he provides a cut-free display calculus for three-valued Lukasiewicz logic. Wansing argues that display logic has a greater expressiveness and generality than hypersequential calculi, because - in virtue of its diversified range of structural connectives - it allows
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to tackle logics with more than two "families" of connectives, and not with just the group-theoretical and the lattice-theoretical ones. According to Wansing, in particular, the display calculi for modal and tense logics have a clear edge over the corresponding calculi of hypersequents. In fact, he lists a number of properties that the operational rules of a calculus must satisfy if they are to be interpreted as assignments of meaning to the logical constants they introduce; for example, the rules for the constant are called weakly explicit if they exhibit in their conclusions but not in their premisses. Some rules of hypersequential calculi for modal and tense logic, however, fail to satisfy such a property.
2. PROOFNETS Cut-free sequent calculi, at a first glance, might seem formalisms which one could hardly hope to improve upon. If we examine them more carefully, however, we can spot at least three flaws that it could be desirable to amend: Consider the following proofs in LL:
(
(
! and ! correspond to the same informal proof of It is evident that "#$% &%'" ; formally speaking, however, they count as distinct
proofs thereof because the order of application of the rules is different. As a consequence, we might want to find a calculus where such proofs are identified with each other, as it would seem appropriate. Sequent proofs are extremely redundant, because side formulae keep being copied again and again with each inference of the proof itself. The cut elimination procedure is highly nondeterministic. For example, consider the proof:
) ) 3 #"*&+*&% 5 678 % &,-&. # 4 5 9:;8 3 # "0/+*&% % & ,1'. # 4 5 <;=?> 8 ,'.*&23 # 4 & "0/+ The indicated cut can be pushed upwards and replaced by a cut whose @ B C*DE*DF and F DG-DH I A . But what comes next? We premisses are J L inference and then apply the K R rule, or can either go on with the proceed the other way around. Both choices are equally legitimate.
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Substructural logics: a primer
Girard (1987) devised a simple calculus which is free of these shortcomings and works perfectly well at least for the constant-free group-theoretical fragment of linear logic: the calculus of proofnets. Formulating such a system is not difficult, but before doing that it is expedient to have a look at a "onesided" variant of LLg , hereafter named LLg
Definition 4.19 (the language £8 ). Let £8 be a propositional language containing a set LIT of literals, such that belongs to LIT iff it is a propositional variable or the negation of it. Every literal is, by definition, a formula of £8 . £8 contains also the connectives and , so that both and are formulae if are formulae. Negation can now be defined for any formula of £8 by stipulating that , and .
Definition 4.20 (sequents in LLg ). A (one-sided) sequent in LLg is a finite, possibly empty multiset of formulae of £8 , separated by commas. Definition 4.21 (postulates of LLg ). The system LLg has the following postulates:
ules ! "! #%$'& Axioms
Proofs of LLg , just like all sequent proofs, are not free from the defects we mentioned at the outset. In order to avoid them, we now consider a representation of such proofs by means of particular labelled graphs.
( /) .0 (
Definition 4.22 (proof structure and conclusion of a p.s.). A proof structure (p.s.) is a graph whose nodes are labelled by formulae of £8 or by the symbol . The multiset of the conclusions of is a distinguished submultiset of the multiset of such labels. Both notions are inductively defined as follows: (Hypothesis clause) A graph with a single node, labelled by , is a p.s. whose conclusion is . (Axiom clause) A graph with two nodes, respectively labelled by and and connected by an edge (axiom link), is a p.s. with conclusions , .
1
)+*-,
1
2
(
2
2 342 2 342 5
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) If clause
.
(Join
are
p.s.,
then
so
is
139 ,
and
# " # # # & # #
(Cut clause) If is a p.s. (with , , then so is the graph obtained by connecting the nodes labelled by and to a new node labelled by the symbol (cut link); . ( -clause) If is a p.s. (with , , then so is the graph obtained by connecting the nodes labelled by and to a new node labelled by ( -link); . ( -clause) If is a p.s. (with , , then so is the graph obtained by connecting the nodes labelled by and to a new node labelled by ( -link); .
! " " # " & & # &
$#% $#%
Hereafter, with an abuse of notation, we shall often use the labels of the nodes to refer to the nodes themselves. Within the larger class of proof structures, we now single out a subclass of structures which properly correspond to proofs in LLg .
'
Definition 4.23 (inductive proof structure and conclusion of an i.p.s.). Inductive proof structures (i.p.s.) are generated by the following clauses: A graph with a single node, labelled by , is an i.p.s. whose conclusion is . A graph with two nodes, respectively labelled by and and connected by an edge, is an i.p.s. with conclusions , . If and are i.p.s. (with and , then so is the graph obtained by connecting and to the symbol ; . If and are i.p.s. (with and , then so is the graph obtained by connecting and to ; . If is an i.p.s. (with , then so is the graph obtained by connecting and to ; .
(
)
)
) *) )*) ) /
. 2 0 1 *)-/. 03+ 11 ( + , ) *) . 56 4 . 074 180 . 02, 19:. 0;+ 19?C )A@B*) D HI/J K2G L M%I/J K;F LNL E F H M H=P=M O J K7O LQK J K2G LR:J K3F L!RSC H=P=MTD LU?C HAVWM D HAVWM%I/J K2LL O E G H M H=X=MYJ K7G O LQK J KZG LRSC H=X=MTD LU?C HAVWM D (
J
H
For the sake of perspicuity, we shall represent proof structures by means of a "quasi-deductive" notation; for example, if is connected by edges to both and , we shall write either or
M
HAU U[J U U[M H M J
instead of the more commonplace:
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Substructural logics: a primer
Axiom links will be often represented by arcs. Example 4.1. The following proof structure is inductive:
_________________________ ____________ ____ _______
On the other side, the proof structure ___________ ___ _______
is not inductive, for according to Definition 4.23 you cannot paste together two different i.p.s. by means of a -link. It is nearly immediate to prove:
Proposition 4.9 (equating sequent proofs and i.p.s.). is provable in LLg iff it is the multiset of conclusions of an i.p.s. without hypotheses.
Proof. From left to right, we proceed by induction on the length of the proof of in LLg . From right to left, the induction is on the construction of . The result is immediate given the similarity between the clauses of Definition 4.23 and the clauses for the construction of a sequent proof in LLg .
Inductive proof structures have been picked out among general proof structures since they seemed to follow the shape of sequent derivations. Yet we might ask ourselves: if viewed as labelled graphs, what do they look like? Is it possible to give an abstract, independent characterization of them? Can they be
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identified through an intrinsic property? Luckily, the last two questions can be answered in an affirmative way.
Definition 4.24 (switching). Let be a p.s. A switching of obtained by omitting one of the two edges for each -link in .
is a graph
Example 4.2. Here is a switching for each of the proof structures of Example 4.1:
While form:
______________________________________ ____________________ ____ __________ _
______________ ________ ____ ___
is a tree,
is not, as is apparent by rewriting them in the following
Definition 4.25 (proofnet). A p.s. is a proofnet just in case every switching of is a tree.
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Remark 4.9 (omission of cut links). Whenever our sole concern is determining whether a given p.s. is a proofnet, we can disregard cut links: in fact, the result of replacing, in a p.s. , a cut link by a link is a p.s. which is a proofnet iff was a proofnet.
Our aim is showing that the concrete, proof-theoretical notion of inductive proof structure and the abstract, graph-theoretical notion of proofnet coincide with each other. To do so, we first need to state a result which is nothing else than the application to proof structures of a purely graph-theoretical theorem (whose proof is omitted here, but can be retrieved from Troelstra and Schwichtenberg 1996). be a p.s. containing a -link of a p.s.). Let Definition 4.26 . (section is called a section of iff each path in from to or toSuch link passes through one of the edges , . first p.s. of Example 4.1. The link Example 4.3. Consider the is a section; the link is not, for you can go from to via ! , ", #$ , # %$ , # . Proposition 4.10. Any proofnet containing at least one -link contains at least one section. & We are now almost ready to prove that the notions of i.p.s. and of proofnet amount to one and the same thing. But first, we need just one more definition.
,-/.1032
'
-
0
(*)' +
is Definition 4.27 (index of a proofnet). Let be a proofnet. Its index an ordered pair of natural numbers, where and respectively correspond to the number of -links and to the number of -links in . Indexes are ordered lexicographically.
4
5
'
'
Proposition 4.11 (equating i.p.s. and proofnets: Danos and Reignier 1989). A p.s. is inductive iff it is a proofnet.
'
Proof. From left to right, we proceed by induction on the construction of . Graphs obtained by either the hypothesis or the axiom clause are obviously acyclic and connected. Joining two proof structures whose switchings are trees by means of a -link results in a proof structure whose switchings are trees. If all the switchings of (with are trees, and results from adding the link to it, just select an arbitrary switching of
5
8 .:9<;>= )?6 1+ + 6 8@@8 4 9 @@ 9
7
A
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, which by inductive hypothesis is a tree. The graphs:
are trees, hence any switching of is such. As to the opposite direction, we argue by induction on . . If contains neither -links nor -links, then it consists either of a hypothesis, or of an axiom link: in fact, the join clause cannot have been used in the construction of , which is a connected graph. Hence is inductive. . In this subcase, contains no -links and at least a -link. What is the last clause applied in the construction of ? Were it the join clause, would not be connected, against the hypothesis. Hence, it was the -clause; consequently, has the form:
/
!"#$!" % ('
/
.
+ , )' & ( * )
Consider the subgraphs 1 and . We lay the following claims: They are connected: if they were not, in fact, would not be connected either. They are disjoint: if they were not, in fact, would contain a cycle. Linking them by means of the link , one gets all of : were it otherwise, would not be connected. By induction hypothesis, then, 1 and are inductive proof structures; hence, so is . . Since contains -links, by Proposition 4.10 it contains a section. Hence has the form:
/
8:9 ; <= >@?BAC"D#A?2DE G
disjoint. with Q RTSV UXW6 Y , Z . . Thus 1
1
0 1213 0 4 5612175 . F
O H KMJ L I K N L
P J By induction hypothesis, [ \ _`S b ] _ a ^ S
1
is an i.p.s. with
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Substructural logics: a primer
obtained by replacing in the hypothesis coincides with .
. By induction hypothesis, is an i.p.s. with i.p.s. with as a hypothesis. Hence:
2
2
is also an
by , is an i.p.s. which
Remark 4.10 (cut elimination for proofnets). As remarked above, one of the main advantages of the introduction of proofnets is the fact that cut elimination can be made more deterministic. What does cut elimination mean, in the case of proofnets? As one can easily guess, it means just finding an effective method to do away with cut links. We have two sole cases to consider: if either or is the conclusion of an axiom link, we use the following transformation:
If both
! ! "$&# %
! (&* ) ' !
are conclusions of logical links, we proceed as follows: !+! ! ! ! ! ! ! " ,- " /. " , // " ( * -"$#& ,% " "$#& % " "$#&%
and
2
The exact details of the procedure are left to the reader. Girard (1995), moreover, lists further pleasant features of this simple cut elimination procedure. Remark 4.11 (proofnets for lattice-theoretical connectives). Is it possible to include in a calculus of proofnets also the lattice-theoretical connectives of LL? It is; however, the resulting system does not meet analogous standards of beauty and simplicity. Initially, Girard (1987) coped with his "additives" (as he calls them) by introducing special "boxes", which however, as he himself acknowledged, were nothing else than "sequents in disguise". A more satisfactory solution was advanced in Girard (1996). Calculi of proofnets were devised for still other fragments of linear logic and for other logics as well (Abrusci 1995; Bellin 1997). Remark 4.12 (on the original definition of proofnet). Definition 4.25, in its
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present formulation, is due to Danos and Reignier (1989). Girard's (1987) original definition was based on a different correctness criterion for inductive proof structures (the longtrip condition). Since the appearance of the paper by Danos and Reignier, however, the modified definition has become customary in the literature on proofnets. It is generally acknowledged that natural deduction, as a formalism, is better suited for intuitionistic logic and its neighbours, whereas sequent calculi work better for logics whose negation has classical properties. As we remarked at the beginning of this section, proofnets can be regarded as an attempt to set up an intuitively satisfactory system of "natural deduction" for logics whose negation is similar to the classical one. Two more attempts in this direction deserve at least a mention: calculi based on Gabbay's labelled deductive systems (Gabbay 1996; Basin et al. 2000; Viganò 2000) and Orlowska's relational proof systems (Orlowska 1992; 1994). A presentation of these systems, however, lies beyond the scope of this book.
3. RESOLUTION CALCULI Informally speaking, both Hilbert-style and sequent calculi are meant to determine whether a formula follows from other formulae, or whether it is logically valid (according to a given concept of logical validity, which depends on the logic at issue). Sometimes, however, it is convenient to possess a method to decide whether a given formula is unsatisfiable (again, according to a predetermined notion of satisfiability). If the syntactic counterpart of validity is provability, the syntactic counterpart of unsatisfiability is refutability. Resolution calculi, indeed, were originally devised as an efficient method to establish whether a classical formula in conjunctive normal form is refutable (Robinson 1965). Nevertheless, this "refutation-oriented" feature of such calculi (called negative in Eisinger and Ohlbach 1993) is rather inessential - resolution systems can be presented so as to work equally well as "proof-oriented", or positive, tools for automated theorem proving. Resolution calculi are thus, above all, computationally efficient devices for all kinds of automated deduction, even though at present their applications extend far beyond this original aim. As Eisinger and Ohlbach (1993) point out: Even though the original motivation for developing these systems was to automatically prove mathematical theorems, their applications now go far beyond. Logic programming languages
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Substructural logics: a primer such as PROLOG have been developed from deduction systems like resolution calculi and these systems are used within natural language systems and expert systems as well as in intelligent robot control. In addition, this field's logic-oriented methods have influenced the basic research of almost all areas of artificial intelligence.
In this section, we shall concisely present the main features of classical resolution; subsequently, we shall show how to generalize this calculus in such a way as to cope with some substructural logics.
3.1 Classical resolution In Definition 4.19 we have seen what a literal is. Henceforth, we shall denote literals by means of the letters Definition 4.28 (complementary literals). If is a literal, its complementary literal is:
, if , i.e. if is a variable;
, if , i.e. if is the negation of a variable. Definition 4.29 (clause). A finite, possibly empty, set of literals is called a clause. We shall denote clauses by the letters and the empty clause by instead of and the symbol . Moreover, we shall often write . in place of Throughout this section, we shall repeatedly encounter sets (or multisets) of clauses; in order to avoid notational redundancies, and unless the risk of ambiguity is impending, we shall denote a set (or multiset) of clauses of the !# ()( "%$ $ $ "! (+* . "%$ $ $ "& !' ,%( "%$ $ $ "! ,-* / by the notation form
02 8)8 1433 3 10 8+9 ; 563 3 36570' :%8 1%3 3 3 10 :-9 /
using commas to separate literals of a same clause, and colons to separate clauses from one another. Definition 4.30 (postulates of RK). The calculus RK of classical resolution has no axiom and a single rule, called resolution rule:
<=10
<>D 1 0 E
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In such an application of R, the clauses and are the parent clauses; the clause is the resolvent; and are the literals resolved upon.
be a set of clauses. A proof in RK of Definition 4.31 (proof in RK). Let the clause from the set of assumptions is a finite sequence of clauses s.t. and, for , either or is obtained from ( ) by an application of R. If there exists a proof in RK of from , we say that is provable in RK from the assumption set , and write . RK
% & & (-/ . *) +, # # #10
2 3 46587 3 58794 3 465 3 :7946587 3 46587 3 :794 ;*<>=@?ACB DEDF< G8Q MRK8? O
"' !$#
'
2
Example 4.4. Here are two examples of proofs in RK. The first is a proof of from and the second is another proof of from .
*; <>=@?A B DEDF< G8<>=@?:HJA B DEDF< K8<>= M ;*?NG O Q MRK8?S
With the help of the calculus RK, it is possible to single out in a computationally efficient way all and only the classically unsatisfiable formulae. Before doing that, we introduce two further definitions.
BUVBX[ WY
Definition 4.32 (clause set of a formula). If of £0 in conjunctive normal form, where for its clause set is the set
j kmlRn
clauses in z|Definition {(}tt ~Z~} 4.33 `(formula-translation `F} ~~} w:v y is ofa asetsetof ofclauses, m t v w t
its z : translation is the formula in conjunctive normal form } tut $E,} tmv
@$E } wt $E,} w:v y
z
j m: z9{ z
RK). If formula-
It is immediate to see that, if is a set of clauses, . By the conjunctive normal form theorem, every formula of £0 is classically equivalent to a formula in conjunctive normal form. Let
R
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Substructural logics: a primer
henceforth be a designated conjunctive normal form of (since clause sets are sets of sets, it is irrelevant which one). The clause set is the assumption set of a proof of the empty clause in RK iff is classically unsatisfiable, as the next theorem shows. Proposition 4.12 (completeness of RK). Let classically unsatisfiable iff RK .
be a formula of £0 .
is
Proof. From right to left, we prove the contrapositive. there exists a If
. Now, for valuation s.t. , then also ! #"$ let * be %' * &(&)% *,+.- . It follows that, for every #"$ there / ! 0 1 2 . 3 4 8 s.t. 8 9 5 67 . Now, consider a generic proof : in RK from exists a ; 3<= 5 . It is easy to prove, by induction on the construction the assumption ? set ? > > 4 2 3.4@5 67 (therefore of , that if occurs in then contains a literal s.t. A cannot occur in : , whence our conclusion). In fact, if BDC EFHGJ I , we have P PR KMN Q just verified that this is the case. If BLKMB was obtained from B KMN and BOPS Q by an application of R, then by inductive hypothesis bothP B KMN and BOKMN contain a literal true in T ; but such a literal must be in B KMB , for T F N I UV iff Q T F N I UXW . From left to right, we argue by induction on the number Y of variables in G . J [\ Y If UV , then, G being unsatisfiable, E FGI must contain the clauses Z and [ ]_^ \ (where ^ is the unique variable contained in ` ). As a consequence, we d have a proof of a in RK from b `c :
ge fih jlkkmf nf oph'jlkkmf qf t r egsMn
If uvwx
e
, fix a literal
y
and let:
y
~D z jd { v X | } | z jd { y
~ |D z j d{ z jd { v X | }
y |
! |
y and and
.
We claim that, for every valuation there is a if is unsatisfiable, then in , .# X , and that the same clause in s.t. for every literal happens for H . In fact, suppose that there is a s.t. for any clause in H there exists a literal in s.t. .# . Let be so defined: ¡ ¡ , if does not occur in ; ¡ ¢¤£g¥§ if ¦¨¢!©pª «¬² ® ©¡¯ ¢X° , if ¦¨¢O±p© . ³ ®´ ² ¯ , then it either belongs to ®´ ²¯ µ , or contains Now, if belongs to S · ³ ¦ , or else has the form ¥M¦ ¸ for ³· ¶ ®´ ²¯ µ . In the first case, by hypothesis
- "!$#&%' %()- ) ( - !*#&%' %+(),- ) / 027 13 /08 15: 7 ; 4 : 9 /067 1 . 3 /0<=28 1 4 <=6@ 9 > @ 7 >? A C B <=D > FE C G EIHKJMP OLN N Q ASR Z J PVPUO TXT W LN NYQ [U\^]\`_acb dfegd hi lkmoqn p j
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and by definition of there is a s.t. . In the second case, this last property holds since . The third case is settled just as easily, since and contains an s.t. . For we argue analogously. As a consequence, is satisfiable and then so is . Thus, our claim is proved. Let us now return to our inductive step. If is unsatisfiable, by our previous claim the formulae and are unsatisfiable as well, and contain at most variables. By inductive hypothesis, RK and RK . But then, if we reintroduce the literal ( ) into the clauses of ( ) whence it had been deleted, we easily obtain a proof of in RK from and a proof of in RK from . Combining such proofs as follows:
R
we get the desired proof of in RK from
.
3.2 Relevant resolution
s
r
r
t
If we reflect for a while on the calculus RK, we soon notice two distinctive features which render it unsuitable for an application to substructural logics: When proving a clause from the set of assumptions , we do not necessarily use all the clauses in ; thus, the very definition of proof in RK fails to satisfy what one could consider a minimal criterion of relevance. Moreover, the use of sets of clauses - instead of multisets makes the calculus insensitive to multiplicity distinctions. The completeness theorem for classical resolution rests on a peculiar property of classical logic (the conjunctive normal form theorem) which does not hold for most substructural logics. Some years ago, Garson (1989) devised an appropriate resolution system for LRND g (or, which is the same, for LRg ). Its main feature is a device which allows to "keep track" of the clauses that have actually been used in a given proof. The basic statements of the calculus, in fact, are no longer plain clauses,
t
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Substructural logics: a primer
but "sequents" whose "succedent" is the derived clause , and whose "antecedent" is the multiset of the clauses actually used to derive it. Definition 4.34 (the calculus RR). The calculus RR differs from RK in that: its clauses are defined as finite, possibly empty multisets (not sets) of literals; it contains axioms of the form ; the resolution rule is modified as follows:
where are multisets of clauses; the following contraction rule is added:
Definition 4.35 (proof in RR). A proof in RR is a finite labelled tree whose leaves are labelled by axioms, and such that each sequent at a node is obtained from sequents at immediate predecessor nodes according to R. The sequent is said to be provable in RR (or a theorem of RR: in ) iff it labels the root of some proof in RR. symbols, RR
! "$# % "&# %('& " '$" 0 1 2 "$# %*)+"&'&# "%*)+'&"-% )+',% /. ',% ',% 0 1 2 '$"'&"',% '&"$#3'&"& #3',% " " 0 1 2 '$"'&'&"&"',#3'&%*"&)4#3" ',%*)4'$"-"&)4#3" ',% ',% " " 0 1 2 ',% ',% 0 1 2 '&"&'"&'"&'"&',#3%*',%*)4"-)4"-)4"-)+',)+%',% / /. . 0 5 2 6C 87 < #:9;9;9;# 7 = @ > B ? ; A ; A 4 A ? > >E< DFA;AGAHDI> = < = Example 4.5. Here are two examples of proofs in RR.
In the calculus RK, the intuitive meaning of the clause was the formula . Here, the intended meaning of is the formula , where literals are linked to one another by means of grouptheoretical disjunction, as the next definition shows:
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multiset of clauses in Definition 4.36 (formula-translation is a clause, itsof aformula-translation RRis ).theIf !"$#% %# formula of £ of clauses, ' ( is defined as the multiset . Ifof formulae ) * & +is ,-a& multiset . Although it is an injection, the mapping from the set of clauses to FOR(£ ) is by no means a surjection. make no sense to talk of !1 /.0 for . Hence, it may The 1 an arbitrary formula of such language. inverse mapping * " 6 * 6 & 253 is well-defined only for formulae of the form 243 (identified with one another up to association of disjuncts), 8.9 * and +. permutation 7 which we agree to call clausal . If is a multiset :1 )7 ; !1 .< =#% %#>!1 .? is also well-defined.of clausal formulae, then 5
5
Unfortunately, it is not the case that any formula of £5 is provably equivalent in LRg (or in HRg ) to a clausal formula or to any combination thereof. Given the clausal formula , there is always a way to derive in RR the empty clause from and a particular multiset of clauses, which we are now going to define.
:1 .0
.
4.37 6"(opposite of a clausal formula). C .0If .@ Definition * * B 6 & 253 2A3 #% %# 3 is a# 2 clausal 6D #% %formula, # 2 6 & . Ifits7 opposite 8.9* +. is the multiset of clauses 3 E is a multiset of clausal formulae, then C )7 is the multiset of clauses CF .< =#% %#C . , with C )7 ;G whenever H" .
1 .0;IJ!1 .0 3 I 3 R S T : !1 .0=# 3 I 2A3 O * 243 P 6K * 6 & 3 O I 3 O R S T !1 .0=# 3 # 3 I 2A3 Q * 2A3 P 6K * 6 & I 3 Q R S T Q 3 L R S T !1 .0=#C .M;ING
Such a derivation can be carried out as follows:
2
Now we have exactly what is needed to prove a completeness theorem for RR:
U 4.13 (completeness of RR). Let VX[ Proposition W%Y Y Y%WV \ . Then ] U!^ Z iff ] _'`U(a;^ RR
LRg
be the multiset of clauses .
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Substructural logics: a primer
Proof. From left to right, we prove a more general statement: if RR , then LRg . The proof is by induction on the length of in RR. Hereafter we check the cases of the inductive the derivation of step corresponding to the rule R, leaving the rest up to the reader. Thus, suppose by inductive hypothesis that and LRg , and let be the variable . Let moreover be the LRg proof:
! ! (#$ )# , -+.0/ ! " # ! '&'"# #$ % '&'# , 30465 / #*7 &+ , 12.0/ 8 E < = 9 :>E 9:;< = 9 :>E <?@ 7 9 :>E <E ?@A7 = 9 :>BE <'C'@ 7 F J0K6L I 9 :;< = 9 :><'C'@ 7 MNODPRQS T M NUWV'UBX S 8 Y Z[\] ^ Z [_X ]'`'a f abf `+Z [c ] ^ g Z [_] h i0j6k l Z[c ]'`+Z [\] ^ Z [d_]'`+Z [_g ] h m+n'l Z[cDeR\] ^ Z [_W`'_B] oqp*rtuws v y z x{ v z |
}u ~RR} ~
2 RR2
¡
¢ R
R
2
RR £'+ ¡¤RRRR and
be the proof:
can be proved as follows:
The symmetric case is left to the reader. From right to left, we claim that if every formula in is clausal and LRg , then RR . Our theorem follows from such a claim by considering the case where is empty. The proof can be carried out by induction on the length of a cut-free proof of in LRg . Such a proof always exists, for LRg (like the whole calculus LRND ) admits the elimination of cuts. The base of the induction follows from the fact, already remarked above, . As to that the empty clause can always be derived in RR from the inductive step, we check the case corresponding to the rule L. So, let the last rule in be the following application of L:
By inductive hypothesis, .
RR
We
have
and RR to show RR . Thus, let us pick a
! " # !" $ # %
Francesco Paoli proof of labelled
153
in RR. Let us replace therein a leaf by a leaf labelled by . Since this replacement has no repercussion whatsoever on the feasibility of resolutions and contractions, what we get is a proof of . By grafting such a proof onto a proof of , we get the desired proof of . by
Remark once again that the above completeness theorem is much less general than Proposition 4.12. Thanks to the conjunctive normal form theorem of classical logic, the calculus RK singles out exactly the classically unsatisfiable formulae; on the other hand, RR only refutes multisets of clausal formulae of £5 .
3.3 Resolution systems for other logics If Garson's system is a rather straightforward generalization of classical resolution - for it simply adds to it the opportunity of keeping track of the clauses used to derive a given clause - other resolution systems for substructural logics involve more profound modifications of the classical notions. We shall briefly hint at two such systems, without going into details. The main idea of a resolution system for linear logic devised by Mints (1993), henceforth called RL, is that literals need not necessarily be atoms or negated atoms; they can be more complex formulae as well. Disregarding exponentials (but recall that this system accommodates them successfully), the literals of RL are either classical literals or formulae of the form , where is one of . Clauses are, in turn, group-theoretical disjunctions of literals of RL, treated up to permutation - which means that, for any permutation of , the clauses and (1) are identified with each other. A consequence of this modified notion of literal is that the resolution rule now splits into four different rules, according to the literal resolved upon:
'
+-,/.0,21 3 4 687:9;9;9;7 < 5
(& ')&$* =?B >@9;9;9:>A= C = G >@9;9;9:>A= G D CFE
HJIAKLHM* ION/K PWVOIOR OH I@PK-QAKS* RTHM* ION/KUION/K * PVOQOR HJIAH * HOIAH * HOI@PKYXZKS* RTHM* ION/K OH I@PKYXZKS* R[HM* ION/K * PV@X\R HOIAH * HOIAH *
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Substructural logics: a primer
Mints proves an appropriate version of a completeness theorem for his calculus with exponentials with respect to LLE . He does so by means of a "general method for transforming a cut-free sequent system into a resolution type system, preserving the structure of derivations" (p. 59). The basic concepts underlying the resolution calculus RLuk for infinitevalued Lukasiewicz logic introduced by Mundici and Olivetti (1998) are not too different. Here, too, literals can be more complex than variables or negated variables. More precisely, a positive literal (in ) is defined as a negationless formula of £1 in which no variable occurs other than , while a by repeated negative literal (in ) is built from the negated variable applications of the connectives . A clause is a finite set of literals such that for each variable , contains at most one positive literal in and at most one negative literal in . Now the classical satisfiability notion can be generalized in the following way. Let be a number in the semiopen real unit interval ; the set of clauses is -satisfiable iff there is an assignment VAR(£1 ) such that if , then for any in it is the case that (where is calculated by uniformly assigning the value to and interpreting the connectives by means of the standard Lukasiewicz operations on : see Cignoli et al. 1999). The -support of a literal is the set of all assignments which can be extended to valuations where has a value i.e. . The notion of -support paves the way for a nice generalization of the classical notion of complementary literals. In RK, in fact, two complementary literals are nothing else than literals which cannot be true together, for any assignment whatsoever of truth values to the variable occurring therein. In our many-valued context, two -complementary literals are literals whose supports have an empty intersection. If you set , you can appreciate how close this notion comes to the classical one. Now it is possible to formulate our resolution rule. As a preliminary step, Mundici and Olivetti describe an effective method to transform any finite set of literals into a clause . Suppose this has been done, and let and be clauses, such that: occurs in the positive literal and in no other literal of ; and in no other literal of ; occurs in the negative literal and are -complementary.
!
""#) %$&$'$' ( * + 3 + :<;!=
?A@>@GFHBIB 8 K/L1M J PRQ/STS _ UVCWXZ` Y + 7 [\Y/UV]WE^ + a
>
+
e j fgeih kls b"m#s n%o&o'o'n(m t p k u bqrs n1o'o'o'n(q v mz wyx q { wyxm q | w z {
, -/.1012 6 8 /- .10 9 ?A@>CBED + ? 7 @GFHB F N O +
4 7 5
:
+
+
+cb#d
ks ku
Francesco Paoli Then rule is:
and
155
are said to be -resolvable w.r.t. . The binary resolution
The calculus of binary resolution, however, is not complete w.r.t. the unsatisfiability problem. On the other hand, a suitable extension of it - called by Mundici and Olivetti calculus of multiple resolution - is such. This means that the empty clause is derivable from some finite , where is a set of clauses, through finitely many applications of binary and multiple resolution w.r.t. if and only if is not -satisfiable.
Sources of the chapter. The content of § 2 is largely based on the monographs by Troelstra (1992) and Troelstra and Schwichtenberg (1996). Our presentation of classical resolution owes a great deal to Lolli (1991). For the rest of the chapter, we used the research papers referred to in the text.
Notes 1. See the next section for a thorough explanation of this terminology. 2. There is, however, a simpler cut elimination method for hypersequent calculi (see e.g. Ciabattoni and Ferrari 2001); its key feature is that the number of applications of EC is considered as an independent parameter in the induction. 3. For details, see Avron (1987; 1991b). 4. Slaney (1990) motivates very well these two different forms of bunching. is viewed as "the result of taking [the body of information] as the determinant of available inference ! and applying it to ", whereas is taken as "the result of pooling information with information ". 5. More precisely, Avron maintains that the first three properties are indispensable, whereas the last three are simply desirable.
PART THREE THE ALGEBRA OF SUBSTRUCTURAL LOGICS
Chapter 5 ALGEBRAIC STRUCTURES
When studying a logical calculus S of any kind, it is extremely important to be in a position to find a class of adequate models for it - i.e. a class of algebraic structures which verify exactly the provable formulae of S. Thus, for example, it turns out that the algebraic counterpart of classical propositional logic are Boolean algebras, while intuitionistic propositional logic corresponds to Heyting algebras. As a rule, these correspondences pave the way for a profitable interaction: the investigation of models may yield several fruitful insights on the structure of the given calculus, and, conversely, it may even happen that proof-theoretical techniques be of some avail in proving purely algebraic results (Grishin 1982; Kowalski and Ono 2000). Ono (200+a) compares proof-theoretical and algebraic investigations into the field of substructural logics with these remarks: Proof-theoretic methods have shown their effectiveness for particular substructural logics, e.g. logics formalized in cut-free sequent systems [...]. On the other hand, the semantical study up to now is quite unsatisfactory, and therefore the general study of substructural logics is far behind e.g. that in modal logic in recent years.
In our opinion, the reasons of the difficulties experienced in the algebraic study of substructural logics are essentially two: 1) It is not difficult to identify suitable classes of algebraic models for the different logics encountered in the previous chapters. Such structures, however, seem to have a restricted mathematical interest (with some notable exceptions, nevertheless). On the one hand, they do not arise often in areas of mathematics
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Substructural logics: a primer
other than logic; on the other hand, they are usually rather weak structures, with few interesting properties, and thus their theories of ideals, congruences and representation are either scarcely developed or even lacking. 2) A second ground of such a disrepair has to do with the idiosyncratic research styles of the various substructural schools. Linear logicians, for one, seem to assign a limited value to model-theoretic semantics in general (to which they prefer, far and away, the proof-theoretic semantics developed within the logical calculi themselves) and to traditional algebraic semantics in particular (usually replaced by phase semantics, of which more in Chapter 7, and by a denotational semantics for proofs). Relevance logicians, in turn, tend to focus on the lattice-theoretical properties of the different -semigroups they study, often viewed as lattices with additional operators (Urquhart 1996; Hartonas 1997). For example, the notion of filter one can often find in the literature of relevance logic (see e.g. Dunn 1986; Restall 2000) coincides with the concept of filter of the underlying lattice. Such a tendency, however, may sometimes obscure some fundamental algebraic properties more directly related to the behaviour of the semigroup operation1 . Beside researchers working either in the linear or in the relevance tradition, over the last few years there have been authors who have undertaken a serious and in-depth investigation of the algebra of substructural logics in a general framework, i.e. without committing themselves to the partial perspective of a given logic. In particular, we are greatly indebted to the contributions of Avron (1988; 1990; 1998), Ono (1985; 1993; 200+a; 200+b) and Kowalski and Ono (2001), both for the results they proved and for the ésprit de système which underlies their works on the subject. In the present chapter, we shall introduce the algebraic structures for substructural logics according to a rather traditional expository pattern. To begin with, we shall define such structures and study their elementary arithmetical properties. Subsequently, we shall introduce and investigate suitable concepts of homomorphism and ideal. Finally, we shall lay down the fundamentals of a representation theory. In the next chapter, we shall see how to match the logical calculi of Part II with the structures of the present chapter. With regard to the definitions of the concepts needed for our treatment, we chose - as already pointed out in the Preface - to adopt a particular strategy. If we had started from scratch, defining and illustrating even such notions as "lattice", "congruence", "subdirect product", the chapter would have reached exorbitant lengths and would have become virtually unusable for readers with some algebraic background. As a consequence, we took a few basic notions for granted. However, we did not want to scare away the other readers; therefore, the unexperienced student will find all that is needed for a thorough comprehension of the chapter in the glossary of Appendix A.
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1. *-AUTONOMOUS LATTICES 1.1 Definitions and elementary properties In Chapter 2, we remarked that whenever one wants to investigate various logical systems in the lump, it is often expedient to focus on a suitable basic system and then consider its extensions. And so we did, appointing LL as the starting point for our syntactical investigations. Here we shall do the same - we shall pick a basic class of structures and see what happens by adding further axioms to its defining conditions. In the next chapter, we shall verify that the resulting atlas of algebras presents an exact correspondence with the atlas of logics of Table 2.2.
Definition 5.1 (*-autonomous lattice). A *-autonomous lattice (or, briefly, *-lattice) is an algebra of type , such that:
a
is an Abelian monoid; is an involutive lattice; !" $#% '& iff #!& . ! . In C3, " " denotes the induced lattice ordering of Henceforth, we() $ agree to denote the element " " by the symbol " " and to % # * $ & + # . , & abbreviate by . (C1) (C2) (C3)
Remark 5.1 (denominations of *-lattices). The name "*-autonomous lattice" is not standard in the literature. Rosenthal (1990) uses the expression "*autonomous poset" for partially ordered structures satisfying C1, C3 and (C2')
/!0
is an involutive poset.
He chooses such a label in view of the connection existing between these structures and *-autonomous categories (Barr 1979; Seely 1989). Avron (1988; 1994), on the other hand, employs the denomination "additive relevant disjunction monoids" in order to refer to our *-lattices.
9;:= 3
1 2435 678%
be an Abelian Example 5.1 (Girard 1987). Let monoid, and let . We define the following operations on subsets of :
3
! ! "! 4 5 6 ) ' + ( * . , 1 / 0 2 , 3 * , % # $ & =?> < 79:@8 ; 79:)8 ; M H N C MPO N AMPCEI N D C B :M =?>3Q =3N =F;S 6
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Substructural logics: a primer
for every , and
only if
Let moreover and let be a subset of which contains and is closed w.r.t. the operations and settheoretical intersection ( itself will do, for example). Then is a *-lattice where and This example will play a key role in Chapter 7, in the context of the relational semantics for nondistributive logics. is a *Example 5.2. Any Abelian -group lattice where . Any Boolean algebra can be presented as a *lattice where and . We list below some elementary arithmetical properties of *-lattices, mainly discovered by Avron (1988; 1990) and Casari (1989).
bdbdcc XaXa:e:eXX [[ WW3^:e;fX Wg^ WVXVik:h;lX ;fW3^ WV:eX ij;f^ WgW3X :eX [ [WVWai ij;l; bdbdcc XX [[ WWV^i bmbmc c ^ X [ W [ bmc X [ [ W c ^ XV:eX bm[ c W ^X ;fbm^ W c WV^ [ i [[]Wg\ b XVC :e[ X ^ W ^ ; [ c c [ ^ WVi [ \ : ^mX \ ^ij; c C X : [ [ \_WV^ ; i \ i [ \_^ [ C c ^m^ \_[ [[ WVW [i n c c ^ ^ WVW i ^ n [ \ i c ^m\ i [ \_[[ n c c ^ ^m\_^ n [ n c ^ n XaXa:: [[ WW ^b ;;fWgc XVX :[ ^ WoW bX ; ^ WgXV: b W [ b ; \_^ c i [ c XaX :^ [ WVW i ^ W b ; c [: [ W3W : ^^ ;eHIij:; [ C WV: ij[ ; W c ^ [ ;eHW3: : [ ^ WVIijij;; [ \ [ : ^ \ Hp: ^ ijIp; ijc ; :C [ :\_[ ^ ;e\_H^ ;e: I[ :\ [ ik; \ ij; X [ W3:eX ^ WVij; c X ^ W3:eX [ WVij; Proposition
5.1
(arithmetical
, then ; (v)
properties ; ;
; (iv) if
;
of
(iii) and
*-lattices).
if
,
(vi)
; (viii)
; (xi)
; (ix) ; (xii) if ; (xiii) if and and ; ; (xv) (xvi) iff ; (xviii) ; (xx)
;
iff
and
;
, ,
;
(i) (ii) and then (vii) ; (x) then then (xiv)
(xvii) ; (xix)
.
Proof. We prove (ii), (xi), the first half of (xii), (xiv), (xv), (xvii), leaving the rest as an exercise. (ii) By C1, C2 , whence our conclusion follows by C3. (xi) Immediate, applying C3 twice.
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whence by C2 and C3 (xii) By (v), . Now, suppose ,, which of C2 implies ; we immediately get in ,virtue whence by C2 and . (xi) and , by (xiii) (xiv) Since , whence by (xi) we get our conclusion. and thus, via (xii), By (xv) (vi), , whence by (xi) the desired conclusion ! similarly ! . Hence, by (xii), !" # $ and . Conversely, # $ ! , by C3 % # $ % & , since $% # $ ' % $ ! whence $% # $ ' % $ !( , which together . with the Similarly $% # $ ' % $ ! ) . Twoprevious inequality yields more %
# $
%
applications of C3 give first and # $ ! +. * then follows. (xvii) From
Proposition 5.2 (*-lattices form a variety: Minari 200+). The class of *,.-/0/21#/4365 , lattices can be presented as a variety in the signature defined by the equations:
798: ; < - 7= -> ;4?@7< - #= ; - > 798 C ; < -1 ?< 798HG ; 0+13 7 0 < - <$;4? 0I1 798 K ; < 3 = ?= 3 < 798HM ; < ?< 3N0 7 0 < 3 =#;
7A8 B ; < - = ?= - < 7A8ED#; 0F0 < ?< 7A8 J ; < 3 7= 3)> ;4?@7< 3 =#; 3 > 7A8 L ;#< - 7= 3 > ;4?@7< - #= ; 3 7< - >; 7A8: 1 ; < ?< 3 7 0 7 0 < - =#; - =#;
Proof. The previous equations are directly implied either by C1-C3 or else by Proposition 5.1. Hence, they hold in any *-lattice. Conversely, it is P < O = = O 0 0 < sufficient to :E prove by means ofOE1-E10 that (A) if , then O < = < = 0 and that (B) < iff3 = ?< . . It follows 0 < ? 0 7'< 3 =#; , i.e. (= Ad A). Suppose 0 3N0 < ? 0 = 3N0 7< 3 =#;4? 0 = by E9, E7 and E4. :Q?R: 3 7 0 < - =#; . It follows that 1 ? 0 7%: 3 7 0 < - =#;%; (Ad B). Let 1Q3N0 7 0 < - =#;4? 0 7%: 3 7 0 < - =#;%; 3R0 7 0 < - =#; ? and7 thus = 0 0 < - =#; ?T= 3 7 0 7 0 to< - both =#; - =#sides, ; , by 0 7 0 < - =#; ,- = by?S7 1E4,3N0 E77 0 < and - =#;%; E9. - =NAdding < 7 7 < # = ; # = N ; ? 3 0 0 - E2, E3 and E8. As a consequence, using E6, < 3 = 3 7 0 7 0 < - =U; - =#; , whence by E10 < ?< 3 = . For the other
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Substructural logics: a primer
sides, suppose . Adding ,toby both E2 and E8. Hence, in virtue of E2, E5 and E6, . is called arithmetical Definition 5.2 (arithmetical variety). A variety whenever, in any one of its"!members, the lattice of congruences is distributive $# %&!$# ' and the relational % product # ' of any two congruences (defined by ( and ( ) is commutative. iff there is a ( s.t. direction,
recall that, according to Pixley (1963), in order to prove that isWearithmetical, ) * %,+-'.a+ variety (0/ and it is sufficient to exhibit two polynomials 1 * %,+-'.+ (0/ s.t. the equations 1* %,+-'.+-' / 2 % 1* %,+3%,+-' / 2 ' , % + , % + ' 4 % + . ' + % . ' + , % 3 + % % ) * / 2) * / 2) * / 2 are satisfied in any member of . *-lattices form an arithmetical variety, as the next proposition shows. Proposition 5.3. The variety of *-lattices is arithmetical.
5 ) , and 1 be defined as follows: 5 * %,+6'7+ (8/92:*;<*; % =' / = (0/>( )* %,+-'.+ (0/ 2 5 * %,+3'.+-' /? 5 * '.+ ( + (0/? 5 *@( +-%,+-% / 1* %,+-'.+ (0/ 2 5 * %,+-'.+ (0/? 5 *@( +-'.+-% / ) * %,+-%,+-' / 2 Then, by C2 and Proposition 5.1.(vi)-(vii), 5 * %,+-%,+-% /? 5 * %,+-'.+-' /? 5 * '.+-%,+-% / 2 *-*;<*; % =% / =% /> % / % = ' A = ' ' ' = % = % ?*-*;<*; / /> /?*-*;<*; / /> % / 2 % ?*-*;<*; % =' / =' /> ' /?*-*;<*; '=% / =% /> % / 2 % ?*-*;<*; '=% / =% /> % / 2 % . Moreover, ) * %,+-'.+-% / 2 5 * %,+-'.+-' /? 5 * '.+-%4+-% /? 5 * %,+-%,+-% / 2 % and ) * '.+-%,+-% / 2 5 * '.+-%,+-% /? 5 * %,+-%4+-% /? 5 * %,+-'.+-' /2 % . 1 , we have 1* %,+-'.+-' /B2 5 * %,+-'7+-' /? 5 * '.+-'.+-% /B2 As regards % = ' = *-*;<*; / ' /> ' /?*-*;<*; '=A' / =% /> % / 2 % = ' % . Finally, 1* %4+-%,+-' / 2 *-*;<*; / =' /> ' /? % 2 5 * %,+-%,+-' /? 5 * '.+-%,+-% / 2 *-*;<*; % =% / =' /> ' / ?*-*;<*; '=% / =A% /> % /2 ' ?*-*;C*; '=% / =% /> % /2 ' . Proof. Let ,
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1.2 Notable *-autonomous lattices In Example 5.2, we have already recorded two extremely significant subclasses (indeed, two subvarieties) of the variety of *-lattices: Abelian groups and Boolean algebras. But these varieties are not the sole mathematically and logically interesting classes of *-lattices. Hereafter, we mention some more. Definition 5.3 (commutative Girard quantale: Avron 1988, Yetter 1990). A commutative Girard quantale is a *-lattice such that:
(C4) # %'&()$ %* ++
is a complete lattice2 .
Example 5.3 (Girard 1987). In Example 5.1, let . Under such an assumption, the structure is a commutative Girard quantale where and .
! %'& " %
,-/.0 1 2 1431516178 .0 16178 is a distributive lattice; (C5) 2 9;:9 . (C6) 9 2 9 , then it is called an idempotent De If it moreover satisfies (C7) 9;:9
Definition 5.4 (De Morgan monoid: Dunn 1966). A De Morgan monoid is a *-lattice such that:
Morgan monoid (Dunn 1975). Example 5.4. Interesting examples of De Morgan monoids can be found e.g. in Slaney (1989). A well-known example of idempotent De Morgan monoid is the infinite-valued Sugihara matrix (Sugihara 1955; Meyer 1975). is the structure , where: is the set of the integers; retain their usual arithmetical meaning; is , if ; is , if ; is , if . Likewise, one can define finite-valued Sugihara matrices , where .
.@?A1 B 14315C1EDGF 9 1HDG<>= IKJ 8 < > = L ? 3151HDGF 1HDGIKJ L BNM 9 RGR 8OR M R M RGR .OR M R 7PM RGR -QR M R 9 9 9 9 9 L 9 S>TVUXWZY [\r ] ^ _ ^`^a^HbGced ^HbGfKg h s ]i[ `jg4^lkmkmkm^4`no^a^ p no^lkmkmkm^qpNg t -pregroup: Casari 1989). A lattice-ordered pregroup (or, Definition 5.5 ( t for short, an -pregroup) is a *-lattice uwvxy z { z|z}z~z such that:
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Substructural logics: a primer
; . If it moreover satisfies (C10) , then it is called a strongly distributive pregroup (for short, an m-pregroup: Casari 1991). Remark that Abelian -groups are exactly those -pregroups where (C11) . (C8) (C9)
!#"$%, "&'," , "(), "+, * be an . of its subgroups, and / 0 !1#"$05 " 256 / " 5 "(35 "45 * beanya Boolean The -splitting of w.r.t. . is the structure 7 0 89"$ "&:" "(;algebra. " < * , where: =?>A@ BDCFEHNP G% Q O IJBDK3EML9I ; R ST$UWVYX[Z-S\]n UW^_Xa`0STbZce \]UWVZHf ^_XAd e T$U gaf VX if Tijkd Rhg)ST$UWVXa` S)S)gaga e T$UmlFf X otherwise; Rpo_`0S!oe' UWoqf X ; R ST$UWVYXpr2S\]UW^_X iff sTS4e \ or sT` \ and Vrtf ^vuwu . x is an z -pregroup. The condition that y be a Boolean algebra, indeed, may y be weakened: in order to achieve this result, it is sufficient that be a classical z residuated lattice (see below). Moreover, members of a remarkable class of { | pregroups are representable, for some{ and | as above and for some classical } } residuated lattice , as -splittings of w.r.t. . Example 5.5 (Casari 1989). Let Abelian -group, be
Definition 5.6 (classical residuated lattice: Ward and Dilworth 1939; Grishin 1974). A classical residuated lattice (for short, a c.r. lattice) is a *lattice such that:
x ~ 9$ &:W
; ~ . (C12)
Example 5.6 (Paoli 1998-1999; Minari 200+). In Example 5.1, let be an idempotent element of - which always contains at least one such element, viz. the identity . Let moreover be . Then is a c.r. lattice. As a further example, let be an involutive lattice with top element . Let , where: , with ;
© p¡+¢A¡£¥¤¦m§ ª
~ !
; x ~0!- $< W
¤¦m§_¨¡
x
!" # ( $ !% ) &' !" * ) +-,". / 01/32"/54/768/79;: 2"<=2 >0?@A0?+$2"<=2 ?B0>=@A0> CED3F G H IKJLNMPO QRTSTU'R VWRYXZR5QR [ \^] R [ _'`8a [ \b] Rc[ _'` dd X O Q]RTSTLeU Sgf ] d ]oVip3hF G H LWIKJ [ _'`kj SlR ]mhn op3F G H IKJ q
Francesco Paoli
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if if if
if if
if either
Then
iff or is a c.r. lattice.
or
or
or ( otherwise
and
)
.
Definition 5.7 (MV-algebra: Chang 1958). An MV-algebra can be defined as an -pregroup such that: (C13)
Example 5.7 (Chang 1958). The standard example of MV-algebra is the structure , where retain their usual meaning and: is the closed real unit interval; ; . is of primary importance in the theory of MV-algebras: it can be shown (Chang 1958; 1959) that an equation holds in all MV-algebras iff it holds in . Remark that the c.r. lattices of the second part of Example 5.6 are not MValgebras if contains a chain of at least three elements. Remark 5.2 (on the logical significance of these structures). We shall see in the next chapter that the structures introduced in Definitions 5.2-5.7 form classes of adequate models for many of the logical calculi defined in Chapters 2 and 4.
r
Remark 5.3 (on the definitions of c.r. lattice, MV-algebra, and Boolean algebra). The denomination "classical residuated lattice" is due to Ono (200+a). Remark that c.r. lattices are sometimes called -zeroids (Casari 1991; 1997a), l-L0 -algebras (Grishin 1974; 1982), or integral, commutative, Girard monoids (Höhle 1995)3 . The definition of MV-algebra here given, due to Casari (1991), is different from, but equivalent to, the original one by Chang. A simplified version of Chang's definition was provided by Mangani (1973)4 .
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Substructural logics: a primer
Boolean algebras can be equivalently defined as either -pregroups satisfying both C6 and C7, or as De Morgan monoids satisfying C12, or else as c.r. lattices (or MV-algebras) satisfying C6. Let us now list some elementary arithmetical properties of the abovementioned structures. Most of the following lemmata can be found in Casari (1991), Minari (200+), Rosenthal (1990), Avron (1988; 1990; 1998), Ono (200+a), Höhle (1995). Proposition 5.4 (properties of De Morgan monoids). In any De Morgan monoid: (i) ; (ii) . In any idempotent De Morgan monoid: (iii) . 5.1.(vi), Proof. !We only " ; proveon (ii). theBy C2otherand Proposition side, by (i), ! # $ , whence by transitivity " . Now our claim follows from C2, C3. % & & ' Proposition 5.5 (properties of -pregroups). In any -pregroup: (i) ; () ; #* ! ; (ii) (iii) (iv) + ) , !) . - . 0/1 -" -0/1 ; (vi) In .-0any /1 m-pregroup: "- 0/1 (v) (vii)
2 - 3 4-; 3 . Proof. We prove (i)-(iv) and (vii). A proof of (v)-(vi) 'can be found in Casari (1991). (i) By C2 and C9, 5) , whence by C3 76 (ii) By (i) ' and Proposition 5.1.(v), ! ; thus, in virtue of Proposition 5.1.(xi), (8) . (iii) By C9 and Proposition 5.1.(v)-(xii), 5) *! . An application of C3 yields our conclusion. (iv) By C8 and C9, 9 5) )!! ; adding on both sides of the equality gives the desired result in virtue of C8. (vii) By C8, C10, and involutive lattice properties,
!
! 4 !
:( - ;( - . In virtue of Proposition 5.1.(xvii)
!C8, - this <latter !-expression
!
- 3 2 !- 3=6 %
and
amounts
to
Francesco Paoli
169
In any c.r. lattice: (i) Proposition ; (ii) 5.6 (properties ; (iii) of c.r. lattices). ; (iv) conditions C8 and C9 are iff . satisfied; (v)
Proof. (i) For a start, it is easy to become convinced that items (i) and (ii) of Proposition 5.5 hold for c.r. lattices as well. Thus, by C12, . From it follows then, by C2, that . (ii) By C1, (i) and Proposition 5.1.(xii), . (iii) Immediate from (ii) and lattice arithmetic. (iv) C9 is trivially satisfied in c.r. lattices. As to C8, it is enough to show that , since the converse is granted by Proposition 5.1.(v). But this follows easily from (i). (v) Trivial.
!
An extensive list of arithmetical properties of MV-algebras may be found in the first chapter of Cignoli et al. (1999). It is interesting to remark that conditions C10 and C12 are always satisfied by MV-algebras (hence MValgebras are both c.r. lattices and m-pregroups), and that coincides with .
#"$ &%'
"$ &%'
Proposition 5.7 (properties of MV-algebras). In any MV-algebra: (i) condition C12 is satisfied; (ii) ; (iii) ; (iv) condition C10 is satisfied.
#"$"( &%') &%* +
,
"$ -%'."/0 $%' 12 13 # "$ &%'
45#"$ &%'
76
476 8 6 8 6 9#"$ 6:%*8"$ 6:% 8"$ 6:%' ";<60 &%'
8"$<60 &%'
8"$<60 &%' "$ "$ &%=% "$<60 &%' "$#"$ &%' &%'> "$"$ &%'> &%' "$ 6:%'#6 "$"$ &%'> &%'6 "$ &%? 76 "$ &%$+1"$ $%* 6@8"$ &% A78#"$ $% 6+A 6CBD"$<60 AC% BD"$ AC%EBD"="(4B0AC%' &%, A#BD"$A< $%EBD"="(B0AC%' &% BD"( AC%EBF4B0ABD"$"(B0AC%? &% , for
Proof. (i) By C1, C2, C8, C13 and Proposition 5.5.(iv), . (ii) holds by Proposition 5.1.(iv); on the other hand, holds by (i) and Proposition 5.6.(ii). Now, suppose and ; then and , whence . Recall that, in virtue of Proposition 5.1.(i) and Proposition 5.6.(i), , whence . By Proposition 5.6.(i)-(ii), , whence by C13, applied twice, . It follows , i.e. . (iii) Similar. (iv) First, we prove that (*) in any MValgebra. Let and . By resorting repeatedly to C13, we get:
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Substructural logics: a primer
and . Now, as we saw in the proof of Proposition 5.5.(vii), ! " # $ !% ! & $ ' () ) ! & ' () . By (*), (i) and Proposition 5.6.(i), then, ' (*+ $ ' !, ' ) $ '&, ! ' () ) ! & $ ' ()%- ! ) $ ! . By C3 and Proposition 5.1.(i), . / 01'23 / 01!4 5 ' , whence, applying Proposition 5.1.(xii) and transitivity, $ / 01 26 / 01!4 ! 5 ' ) 01! 0 7 / 01 28 / 01!4 ! 5 ' ) 01! ,. Similarly, $ / 01! ) 90 / 01! 2 which / 01!: ! yields ' ) 01! . But we established that $; $ 7 / 01! ) 90 7 / 01! , whence $ / 01!4 ! 5 ' ) 01! . An application of Proposition 5.6.(v) gives now / 01<2 ) 01 . The converse inequality follows from Proposition 5.1.(xix). = In the following, we shall use without a special mention the fact that MValgebras are c.r. lattices. Table 5.1 summarizes the mutual relationships among the classes of structures introduced thus far. Table 5.1. Algebraic structures and their relationships. *-autonomous lattices
plus distribution, lower semiidempotency
plus lattice completeness
plus x+ 1=1
plus 1+ 1=1, x+-x=1
Classical residuated lattices De Morgan monoids
Comm. Girard quantales
plus upper semiidempotency
plus x+ 1=1
l-pregro ups plus 0=1
plus Lukasiewicz axiom
Idempotent De Morgan monoids
Abelian l-gro ups
MV-algebras plus x+ 1=1 plus idempotency
Boolean algebras
plus idempotency
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171
1.3 Homomorphisms, -filters, -ideals, congruences Whenever we want to build up the fundamentals of the algebraic theory for a given class of structures, the indispensable ingredients are a notion of homomorphism and a notion of ideal (and, dually, of filter). It is therefore desirable to endow our theory of *-lattices with such basic concepts. As regards homomorphisms, this task is not at all difficult. In fact:
$ $%!$ "$ #$ &(' *) + -, . /103254670$ . /8254 $ . /92;:=<46*. /82541:>$ $ . /?<4 .@/92BAC<4$ 6*. /82545AD. /?<4E. /82 FC<46*. /82145FD. /8<4 . /9GH4 6*G O P . Q 3 I L J M K N / 5 . 4 R SUTWVDXZY [8S5\]*^ $ S(_` Y [8S5\a_b$ Y [8`\
Definition 5.8 (homomorphism of *-lattices). A homomorphism of *-lattices from to is simply a homomorphism of algebras as defined in Appendix A, i.e. a mapping which satisfies the following conditions for every in :
If is a homomorphism of *-lattices from defined as .
to
, its kernel
is
It is easy to see that a homomorphism of *-lattices preserves the induced lattice ordering: if , then . The notions of ideal and filter are less immediate. A good concept of ideal (but similar reflections hold, in a dual way, for filters) should satisfy at least the following requirements:
c
(R1) It should be consistent with the definitions of ideal already existing in the literature on special *-lattices, e.g. Abelian -groups, MV-algebras, Boolean algebras. (R2) For any *-lattice , there should be a 1-1 correspondence between the ideals of and the congruences on . (R3) The ideals of should coincide with the kernels of homomorphisms with domain .
d d d
d
d
The solution we are going to suggest, first advanced in Paoli (2000; 200+b), meets - as we shall presently see - all of the above adequacy standards.
g
e
Definition 5.9 (complement operations and We define on two more complement operations:
f
). Let
d
be a *-lattice5 .
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Substructural logics: a primer
(intuitionistic complement) (anti-intuitionistic complement)
Remark 5.4 (on the denominations of and ). The operations and have been labelled, respectively, "intuitionistic" and "anti-intuitionistic" complements because, in -pregroups, the former behaves much like the pseudocomplement operation in Heyting lattices, whereas the latter has a dual behaviour w.r.t. the other. In fact, the following holds (Paoli 1999):
then then !"#!$% !&' !$("
(i) (ii) if (iii) if (iv) (v) (vi)
if then if then !$"# ! !") !$
The properties of lines (i), (iii), (iv) can be summarized by saying that and induce, respectively, a preclosure and a preinterior operation on the poset underlying an -pregroup.
+-,
. .
*
Definition 5.10 (absolute value and dual absolute value). Let be a *lattice and let . The absolute value of the element is defined as ; the dual absolute value of the element is defined as .
/ !0 /$&!$
-filter). Let * be a *-lattice and let ( -ideal and 1325Definition 476 , . 4 5.11 * is said to be an -ideal of iff the following three conditions are satisfied: 4 , then ;+ 4 ; (I1) if 98:(+ 4 , then + 4 ; (I2) if + 4 and ./<. .- . , then + 4 . (I3) if + 4 * On the other hand, is called an -filter of iff the following conditions are satisfied: 4 , then ?@+ 4 ; (F1) if =8>+ 4 , then + 4 ; (F2) if + 4 and , then + 4 . (F3) if +
Francesco Paoli
The sets of all -ideals and of all -filters of respectively, by the symbols and .
173 will be sometimes denoted,
5.5 (special cases of the previous definitions). Notice . Hence that: , and
Remark In Abelian -groups, . It readily follows that (i) conditions I2 and !#" , then ##" , whereas (ii) conditions I3 and F3 F2 become: !if#" $%$'&( ) , then $'#" . Moreover, recall become: if and that in such structures the operations of sum and product coincide with each other. In Abelian -groups, therefore, the two notions of -ideal and -filter coincide with each other; furthermore, they coincide with the standard notion of -ideal (Darnel 1995). In c.r. lattices (hence, in particular, in MV-algebras and Boolean algebras), and . Hence . It readily follows that: (i) condition I2 becomes: ; (ii) condition F2 becomes: 1 ; (iii) condition I3 becomes: if and , then ; (iv) condition F3 becomes: if and , then . In c.r. lattices, therefore, our definition of -ideal ( -filter) coincides with the standard definition of ideal (filter) (Höhle 1995; Ono 200+a).
+*
-,
##"
. , #" # #" $'&( # &($ $'#"
$'#"
#"
According to Remark 5.5, our definitions of -ideal and -filter abide by the desideratum previously referred to as R1. In what follows, we shall exclusively focus on -ideals, disregarding filters and their properties. This choice is more convenient for us, given the way we have defined *-lattices - nothing, however, would prevent us from adopting a dual perspective and reformulating the results of the present chapter with reference to -filters. Let us now see some basic properties of the notion of absolute value.
, & / ; / / ,&( #&( 10 $ .-, , ; (v) 2/ 52 6/ 107$ &/ 3 / 08$ '$ & 94 $ &/ 3 08$ . ,&( , by Proposition 5.1.(xii) *:&( 10* , whence Proof. (i) Trivial. (ii) If 10* &;,&( . It follows that / . (iii) From (i) and by C2 Proposition 5.1.(xii)./(iv) From right to left, the implication follows from (ii). . , 1 0 * ; & , Conversely, let . Then *:&< 10* . It follows *:&<= , i.e. #&;, , andand, *:&( 1by0* , C2, that i.e. ,&( by C3. Proposition 5.8 (properties of absolute value). In any *-lattice: (i) implies ; (iii) (ii) ; (iv) iff ; (vi) ; (vii)
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Substructural logics: a primer
and . By Proposition (v) From (i) and (ii). (vi) Obviously 5.1.(xiii), . On the other side, by Proposition
. 5.1.(xv), by (i) and Proposition 5.1.(xiii). It followsFinally, that
. (vii) implies , whence * $ ! "# . %Consequently, % ! + * "& ' ' ( ) . On the other hand, , by (iii) and lattice properties. The notion of sigma term, hereafter introduced, will play an important role in the algebraic theory of *-lattices.
. is a *-lattice and 021!345 9 6 9 0 : 3"8';=< .
Definition 5.12 (sigma term). If we mean the element
, by
/ 67021!3"8
/ 670>3?1!0'8 / / / 6A0 >3?1!3"8 67021!0;B3"8 673?1!0;3"8 < (ii) / @ 6C021!3"8 ; / 67021EDF8 @ / @ 67021G3 >=D 8 ; (iii) / 6702@ 1EDF8 ; / 673?1EDF8 @ / 6 0 @ ;3 1HD 8 ;; (iv) / 6C021!0>3"8 @ / 6C021!3"8 . Proof. (i) 0 >3I0 implies that J I 9 670 >3"8: 0 , which in turn entails / 6 7 6 0 > " 3 8
: ' 0 K 8 L I < 7 6 0 >3?1!0'8 @ < . The other equations are proved 9 9 , / i.e. / 6C021NDO8 @ 9 6 9 0 : 3"8'; 9 6 9 0 :DF8';=< @ similarly. (ii) 6C021M3 8 ; ! 6 6 0
: " 3 P 8 > 6 0 : F D 8!8';=< @ 9 6 9 0:673%>=DF8!8';=< @ / 6C021!3 >=D 8 . 9 9 / 9/ 6C021EDF8 ; 6C3?1NDO8 @ 9 6 9 0 :DF8'; 9 6 9 3:DF8';=< @ (iii) 9 6!6 9 0 :DF8P> 6 9 3Q:DF8!8';=< / @ 9 6!6 9 0> 9 /3"8:DF8';=< @ 6 7 6 0 ; " 3 8 : F D ' 8 R ; < 6 0 ;
3 H 1 D 8 . (iv) 67021!0>3"8 @ @ 9 9 @ 9 6 9 0 :670>3"8!8S;=< 9 6!6 9 0 : 0S8P> 6 9 0 : 3"8!8';=< @ 9 6 9 0 : 0'8'; 9 6 9 0 : 3"8';=< @ / 67021G3"8 by C2 and Proposition 5.1.(v). Proposition 5.9 (properties of sigma terms). In any *-lattice: (i)
.
T
The next lemma concerns homomorphisms and their kernels. In particular, (iii) establishes that kernels of homomorphisms with domain are -ideals of . This settles the first half of our desideratum R3. Later on, we shall see that every -ideal of is the kernel of a homomorphism with domain .
.
T
.
.
U homomorphisms). Let . ofW V 512:1 9 1E<"1XU >1X;ZY^ U [ @WVL\ 12:]^ 1 9 ^Z1E< ^ 1X>(^ 1X;*^ Y . @ Then 6C0'8#I 76 3"8 : (i)
Proposition 5.10 (properties homomorphism from
be a to iff
Francesco Paoli
175
; (ii)
ideal of
iff
is injective; (iii)
is an -
.
!"# $%1 !'& # ()12 $+13 *,1 -!'"#/.01 !'& # Proof. (i)1 iff , iff 1 1 D 1 1 465-48795:;/<075'= ;; >%? , i.e. 5 795:;E@ 75'= ; ;AB75C5 :@= ;;AB? 1 , which means 51 :@= ; FGHI57; . (ii) From left to right, suppose that for any J , 7-5JK;9AB? implies JLAB? , and that 75':;AB795= ; . Then 75':; >%1 75'= ; , whence 1 by (i) 75C5 :-@= ;;AB? . Then 5 :@= ;AB? , whence ML>N4:O<)= and :>C= . Similarly, =P>C: . From right to left, suppose that 75':;AB75'= ; implies :QAR= ; for =OAB? , we have our conclusion. 795:;AB75'= ;AB? 1 . Then (iii) As to 1 I1, suppose that 7-5S:O%suppose 1 1 75_` 75:;AB? ] = Y > : 7 ' 5
= ^ ; W A 9 7 9 5 :_ ; and . Then 1 _ aBg 1 1 ac_ bd'efT_ ac_B gh . By Proposition 5.8.(iv), it follows that bdi faBg . j k
Next, we record some properties of -ideals.
t q u lCmo rBn] s p . Then: (i) wE{P|P}~E}CwEP|
k
Proposition 5.11 (properties of -ideals). Let v wyxz w0 xz the smallest -ideal of ; (ii) iff ; (iii) w-|
xz ~Exz imply .
t u
v
is and
Proof. (i) First, we show that I2 are satisfied B is an -ideal of . I1 and E |] } by C1 and C2. As to I3, since , it suffices to show that implies | B w v , which follows from Proposition 5.8.(i)-(iv). Now, let belong to the z w O }Y` xz ideal . Since , by I3 it follows that . (ii) By I3 and Proposition w5.8.(v). |
xz wE{P|P})~E}CwP| (iii) Let and . By C2 and Proposition 5.1.(xii), _ Y ^[RZTC'P /RZ . Consequently, _c D\^[RZQ
P\C'EP RZQ EP\^'RZ\C'DZQ _ _ ]_E _ . On the other hand, by Proposition 5.8.(vii), _ _ ]_E]_ Y_` \_¡ ¢_E]_ _'_` \_¡ £_E _'_ . In virtue of I1 and (ii), however, _` \_¡ ¢_E _ ¤ belongs to , whence by I3 so does . j ¥ ¤ ¦ § Now, we associate to and prove it any -ideal of a binary relation
to be a congruence on . Such a relation will be of great importance in laying down a representation theory for *-lattices.
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Substructural logics: a primer
iff
Definition 5.13 (congruence associated with an -ideal). Let we associate a binary relation on , defined by:
and
. To
Remark 5.6 (distances in special *-lattices). In many classes of *-lattices, given an -ideal , it is possible to define a congruence in such a way that iff a given term built out of and (the distance between and ) belongs to . For example, in Abelian -groups (the distance between and is the absolute value of their difference); in c.r. lattices, on the other side, . However, it is not difficult to show that our Definition 5.13 leads exactly to this result in both cases: in fact, both in Abelian -groups and in c.r. lattices iff .
! ! " #%$&(' )+*-,.#/,$102' )304,#5,'602$5) 7 8$&'9;:=< $?> ' ABEGCF D , then @ is a Proposition F 5.12 ( @ is a congruence). If
congruence on
.
C K IL DJI H C LMIJN DJI H C LMI K @ DJI H C N IL DBA H H C K IL D3O C H C LMIJN O2DPL BDA U C IL D H C LMC K IJIN L D D3O H C LMIJN D QSR C V L O W N D U H T T C K O2L D3O H K C LVONWT DUT R K T T C T T K T ONWDU T C T K O2L D3O T C T L6ONWD T C ONWDUR XYUR ! T Z [ eT fJg K \^! ] Z [ efJg \_`a]b!S` ! [Z efJg \cd Z [ ghfe \Pcd i Since l j hasm been shown to be a congruence on k , whenever we are given an -ideal of k we are entitled to construct the quotient algebra j G
Y k W n q o ps rtunu vx wyz w|{}
wJ~u w w . The elements modulo kYnWo are denoted by w1 w , with the obvious meaning that is the of equivalence class of in u u . In such a structure, we have ¥ Z ¡ ¢¤£ . By iff iff Z M J
¦ Z^ § ¢£ . In Proposition 5.9, however, this¨Y© last condition amounts to
¨ © ± ¬ iff Z2 ²³´ ®¯° . virtue of Proposition 5.11.(i), ª «
Proof. We check transitivity of , leaving the remainder to the reader. Suppose . Then by I1 . For the sake of brevity, let . Since and , by Proposition 5.1.(xii) . But Proposition 5.1.(i), together with C3, implies that . Hence, . On the other hand, it is immediate that . Therefore, . Consequently, . Quite similarly, we prove that .
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canonical . The defined by:
Definition 5.14 (canonical homomorphism). Let homomorphism of is given by the mapping
is actually a homomorphism from to to show that !#It" , isasimmediate $ " the name suggests. Moreover, we now prove that the -ideal is the $ kernel of . Since kernels of homomorphism with domain are -ideals of $ in any *-lattice, we conclude that our concept of -ideal fully satisfies our previous requirement R3.
"&%(* '+ ) . Then ,.-0Proposition /1'2 7 )4365 . 5.13. Let be a *-lattice and let ,.-0/1'8 7 )3:H 9<; 9 =?> %@5 and A&'B9DCFEG)=?> %@5 I . If 9<%@5 , it Proof. JKBLDMONGPRQ@S . By Proposition 5.11.(i)-(iii), follows by I2 LUT?VXWYJ&KBLDMONGPT?VUQ@S . Conversely, if L T?V Q@S and^_J&^ KBLZMONGPT?V Q@S , L T[J&KBLDMONGPT?V]\ L Q@S , whence by by Proposition 5.11.(iii), Proposition 5.11.(ii) L`Q@Sb .a c is a homomorphism with domain d , if d egBy f@h0i1Proposition jlkm is perfectly5.10.(iii), well-defined. Furthermore, we can prove: k`n d o p then q egProposition f@h0i1jlkmrts5.14. uev If4
B wyxz wY{|isw~} awsurjective w4 homomorphism, is isomorphic to prswyxOwY{&w}Xw<w . 4
B jBmr: n1kyj& j y m~m4r } and Proof. First remark that y&© ª ¡ « . Then the mapping ¢_£ ¤¥¦ ¬4B®¯ °± §z¨ defined by ²³ ¸ ¬yB®¯ °± ³B´µµ¶ ·y³B´µ is injective by Proposition 5.10.(i), and surjective as · is such. It is easy to verify that the of the structure are preserved by ² . Furthermore, remark that if ÀU´ Á operations ÃyÎÐÄBÅÏ Æ ÇÈ ¹ ÉÎyÀÂºÏ Á ÃyÄBÅÆ ÇÈ , i.e. ©Ë³ ÊyÌÍ µR»<¼@½¿¾#³l·µ , ÍXÑ . a we have by Proposition 5.10.(i) that ÊÑRÒ Our next move consists in introducing some useful relations of orthogonality between elements of a *-lattice.
Õ Ó ÖØÏ × Ñ Ù Õ
d be a *-lattice, ÊyÌÍÓ_Ô Ý_ÊÜÝÛÝ Í ÝßÞ.à ;
Definition 5.15 (orthogonality relations). Let . We define: ( -meet-orthogonality) iff
Ê?Ú Í
and
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Substructural logics: a primer
iff ; iff for every , if , then %&' %& (* ) ' . +-,/.0 will be short Definition 5.16 (multiple of an element). In the following 2 0 4 1 5 3 5 3 6 3 7 1 8 0 , for ( times).
"! # $
(meet-orthogonality) ( -orthogonality) ; (orthogonality) iff
In any *-lattice: 0:9 Proposition ; and <25.15 9 @ ; (properties 0217of<2orthogonality 9 @ ; ; (ii) forrelations). ,>=? (i), @ imply any positive integers 0:9 @ < implies +-,/.09 @ +-?A.< ; (iii) BDCE0F=< implies (0G @ < iff 09 @ < ).
MPHDOHJQSHKRHI MPHDO8H T#U . By I1, H"LNHI V H"LNProof. MHI PHDO H WYX4V HK(i)RHI MPHDO H WSuppose T#U . Distributing, V H\L8HI MPHDO H WZX4V HDRHI MPHDO H W [ V H"L H_ X HDR`H W] V X ] W V X ] W V X W ^ MaH\L H_HDObH MaHDR H_HDObH McHDO H_HDO H . Let abbreviate the last X ReHI X HKR` MPHDO` ObH f ^ , H f V H\L Hd HfhW MP D H expression. By Proposition 5.8.(vi), H L X MPHDO H H L ^kj ReHI in virtue of Proposition 5.8.(iii) and lattice properties. Since g ^ X [ X ^ i [ f , we conclude HlH L ReHI MPHDO H-H H L RHI MPHDO H and H4H Thus, I3 X X L R P M O T U L 2 R m O HIHDH , i.e. n . ol(ii) implies that H pDqsNearly rutpv4immediate w o-pDqsrufrom txrv4(i). w (iii) Leftpto right. By Proposition 5.9.(i), | r o D p s q r ~ v } 4 w D p s q 2 r w 8 p q r yDz#{ . As n , it follows that y K F
x
; H\3)HI HDHY #. Right ; 2) to left. Let us suppose: 1) ; 4)
K: # . We have to show that # . By 2), 3), Proposition 5.11.(iii)
]
Ds # . In virtue of 4) and Proposition 5.9.(ii), K
s Z
Ds . On the other hand, by Proposition and I1, we get -
KsZS l
Ds -
DsZ
Ds . 5.1.(vi), K-
DsY
Ds . Hence H H Moreover, by 1), we have that -
KsZ
Ds H -
DsZ
D H . By I3, then, " # .
The following theorem shows that there is a one-one correspondence between -ideals and congruences in *-lattices, whereby our desideratum R2 concerning the notion of -ideal is satisfied. Indeed, we shall prove something more, namely that:
¡£ ¢
¤ ¨©¡ ¢¥¦§
Proposition 5.16 (correspondence between -ideals and congruences). Let be a *-lattice and let be the set of all congruences on . Then there is an order isomorphism between the posets and .
¤ #¡ ¢¥¦§
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and define two mappings: Proof.
We ,shall and show that: 1) they are order homomorphisms , andthen2) they are inverse to each other. If
! ; if " # , then is the congruence $ % associated with , as defined in Definition 5.13. Now, we must check that and & have the required properties. 1) ' ( 2 is a congruence 3 . This has been proved in Proposition )+*,.-05.12. /1*, imply is an -ideal. I1 is satisfied since 2) )546/1*,748,:9, . I2 holds because )+*, implies )14<;=*,74<;=9>; and )14<;CB *, . Lastly, suppose )+*, and DE/D GHDF) D . Now, we have thus ?A@ )+*, implies ?A@ )14<;CB *, , whence DF) D *, . Consequently seen that DE/D GI, and /19, , by Proposition 5.8.(iv). This implies /1*, . JLK<M K JLK<M and O+NQR P , i.e. UT OW3)V0P.XYV T U PZV0O[X \ implies J . Then T N U OWR V0P.XYV T N U PZS V0.O[X Let M \ h , i.e. O+N]S P . i K M < K M ^ _ ` ` N j implies a j bi . If c+dfe j implies c+m. ¯ we conclude Now, we rapidly investigate two important subvarieties of the variety of *lattices. In the subvariety of weakly contractive *-lattices, the definition of absolute value can be simplified: Definition 5.17 (weakly contractive *-lattice). A *-lattice contractive iff it satisfies: (C14)
°
is called weakly
DF±³D ²<±E´¶µA·±1¸<¹Cº
Even more significant - in that, as we shall see, they can be given subdirect representations with totally ordered factors - are representable *-lattices, which can be defined as follows:
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Substructural logics: a primer
Definition 5.18 (representable *-lattice). A *-lattice representable iff it satisfies the strong algebraic De Morgan law: (C15)
is called
Not every *-lattice is weakly contractive, as the next example shows. Example 5.8 (rectangular quantales: Piazza and Castellan 1996). Let be positive integers with . A rectangular quantale is a Girard ! "# $&% (*),+ ' -/.0-21 -43 - 6 -7 5 , where: quantale ' M 8 +9( );:-=@:A(B3 -DCECFC GHI<J(B3 -KCFCFCF-L N ; R 5Q( ),:-4< 5O.P),:ARS -4< g h _UaWZY g []\ [, i i \ ^` i \ YXbUc_Y2_4^d\ ^ i h TVUXWZY[]\ ^J_U_4[,
8
eBf
jlknm,op4qartsum vwkxopyXkxq rBz j|{]sum vp4{Jr j
;
i r m,op4qar|}?m,oAiS p4q
i
i
iff ~oX}o and q }q . Now, consider the rectangular quantale
and let sum, p4Jr . It turns out that J k ~ 0 sum p4Jr m,{ p4JrQs*m, p4Jr ; but {sum, p4{Jr and it is not the case that m, p4{r?}?m, p4Jr . On the other hand, the variety of weakly contractive *-lattices includes as subvarieties the varieties of -pregroups (hence also of c.r. lattices, of MValgebras, of Boolean algebras, and of Abelian -groups) and of De Morgan monoids. It also turns out that m-pregroups are representable. More generally, a *-lattice is weakly contractive iff its order is 2-semiclosed (Darnel 1995), i.e. iff {} implies {J} . Such results are summarized in the next:
is weakly contractive iff, for every Proposition 5.17. (i) A *-lattice X , {J} implies {J} ; (ii) De Morgan monoids and -pregroups (hence in particular c.r. lattices) are weakly contractive; (iii) m-pregroups (hence in particular MV-algebras, Boolean algebras and Abelian -groups) are both representable and weakly contractive.
Proof. (i) Suppose that is weakly contractive and that {J} . Then k } 0 k ~ ; } , i.e. . But this means s J kV~ 0 ; since {} is weakly contractive, . Conversely, by Proposition 5.1.(vi), k }nk ~ 0 ; and, in virtue of Proposition 5.1.(xi), {} k ~ 0 . Now, p2k ~ 0 } J k ~ ; . By isotony, {} k ~ 0 }~ k ~ 0 ~ J k ~ 0 . In force of our hypothesis, {} J k ~ 0 .
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Proposition 5.4.(ii), in any De Morgan monoid (ii) By , whence by C3 . As regards implies . Suppose pregroups, by (i) it is enough to show that
. Then "! $# . then , or equivalently
! "! the other ##! side, %! in## virtue of Proposition (5.1.(xvii) &') %! and &C9, # "! &#( "! ) &# %! #( "! *# "! # , since %! # . Therefore, by C8, C9 and Proposition ! #( "+ ,! -#( "!&#. ,!/ "! ##0 5.1.(xvii), ! "! ##! %! ##21 ! %! ## ! "!3 ## 1 ! "! ##" 4 . So , whence by C3 .
On
(iii) By (ii) and the fact that C15 is satisfied in any m-pregroup by Proposition 5.5.(vii). 5 Before closing this we remark that the quotient of a *-lattice 6 subsection, 7 is again modulo a generic -ideal a *-lattice, and that passages to the quotient preserve representability.
8 is a *-lattice and (i) If preserve structure). 7:9 Proposition <>!+ = # , then5.18 = ; 7 (quotients = a *-lattice; (ii) if is a representable *-lattice and 7:9 <>!+ = # , then = ; 7 isisrepresentable as well. = ;7
=
Proof. Since is a homomorphic image of , both (i) and (ii) are trivial, for the classes of *-lattices and of representable *-lattices are Evarieties. % ? implies @ A D B@ ?CA D and Nonetheless, it may be instructive to see how F'GIH JLK are preserved in how the leastE bounds of FMG NOPT H JQK upper R"J S then . As regards our first statement, if E X R Y H [ S W Z \ ] ` ^ _ a R b c J a C S b UWV T T . As to the second claim, , whence a Rb T JcE a RdISCb T iff UWV RXH Rd SfZ ^e_ ; but the latter condition surely since JcE a RdISCholds b T . Lastly, UWV RXH Rd SfZW\] by Proposition 5.9.(i). Analogously, a SCb T a RbIT Ha SCb T JcE a ghb T , i.e. U V RXHigCZiH U V SjHigCZ ^`_ . Then, by suppose that RXHigCZ(d U V SjHigCZW\ U V RdISjHigCZ ^`_ , Propositions 5.9.(iii) and 5.11.(iii), U V E a R I d C S b c J a h g b T T .5 which means
k
1.4 Principal, prime and regular -ideals In this l subsection, we shall investigate the properties of three l remarkable classes of -ideals in *-lattices: principal, prime, and regular -ideals. All of these classes are of primary importance in the development of the representation theorems of § 1.5. We set off with the following
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Definition 5.19 (principal -ideal) a *-lattice, .byLet - inbesymbols, and
- is the smallest. The principal -ideal generated in -ideal of containing . instead of when With a slight abuse of notation, we shall write is a singleton.
Proposition 5.19 (properties of principal -ideals). Let / be a0 *-lattice. (i) "! $ # % If , then there exist #'&)(*(*(*&+#-,. s.t. 12791 8:13 7< 7 N 165 ; (ii) if @A8 7CB+D , then E7FHGEDHF"IJE7AKLDHF . M 1 ;>=*=*=);?14
O
V
I 7.P there exist 7QM BR=S=*=*B+7N TU Proof. (i)13 First, have 1WZ91 [: 1` \>]*we 14 1a YXto show that _ < Z * ] R ] ^ \ Z s.t. is an 1 -ideal 13of1 . As to I1, if 12c91 d:13 c< c k 1 and 1Ag1 dh1b gi j 1 e>f*f*f)e?14 j1 e>f*1 f*f)e?g k , then 1Wby1lPropositions monqp r msn?1Ap 1 r 5.1.(xiii) and 5.8.(vi) 1 1 4 1 l 1 b 1 1 3 1 1 1 mt ~ nvuSu*u)n?mw n?pi ~ n>u*u*u)n?p . As regards I2, we have that x ymzn>{}| 1 Qo
>}s z
}
>} ; in virtue 1 of Proposition 5.1.(vi) and z
>}
>} , whence z
>} 1 1W 1 C2, however, 1 t 1 , which yields the desired conclusion. The easy verification of 1 vS*)?14 an -ideal of which clearly contains . I3 is left to the reader. Hence is L ¡ , then . If , then What remains to show is that if 121 3 1 1 ' ) ¢ * £ * £ * £ + ¢ ¥ : ¤ < ¦ ¨ 1 ¤ 113 < ¨ 11 . ¨ § £*£*£)¦?14 § 1 ¦>£*£*£)¦?14 there exist § s.t. ©Q ª)«*«*«*ª+© ® belong to ¬ as well, whence by I1 and Proposition 5.11.(ii) However, 1 1 ¯t ¯ ® 1 ²2³ . By I3, then, ¯²W³ . °>±*±*±)°?14 ´Aµ ¯C¶+· ; 2) º ² ¸¯¹ ; 3) º ² ¸·H¹ . By (i), (ii) From left to right, suppose: 1) 1A¼1 ¾¿»ÁÀÃ1W 1 »½ items 2) and 3) are respectively equivalent to: 2') for some , and 1A¼1 ¾¿ÄÀÃ1AÅ 1 Ä 3') 1Ô1 for 1Wsome , , whence we easily1A1 conclude Æ ¿Ç¼2È)¿6»ÇÀÇÂ1 ÀÉÈvÆ ¿1A¼Ê1 ÈR¿6ÄËÀÇ1AÅ21 ÀÍÌÎAÏпÂAÑLÅHÒ . Now, by 1) 1WÂ1Õ ÑÅÓÌ>ÂAÑLÅ , A  L Ñ L Å Ð Ï ¿ A  L Ñ H Å Ò Â Ö Å × ØÚÙÜÛ ] , whence by Proposition and by definition. So 6 Ý Á Þ ß à á Ý â ß ã ÝÞÁßäàå × ØÚÙÜÛ Ýâß ã . Then × Ú Ø Ü Ù Û 5.15.(ii)-(iii) and ] ] 1Ôæv1 çèêéë1Aæ1 ìÐíîAïLðHñ and thus æAìÐíîAïLðHñ . Conversely, since îÔïLðLò îCó+ð , we have îAïLðLì:íîôñöõíðHñ , which is easily shown to be an ÷ -ideal of . By definition, then, íîAïLðHñøíîñHõíðöñÊ .ù It can be proved without particular difficulties that:
ÿ ú Proposition í ûÚóüø4ý
. ù
5.20 (the lattice of ÷ -ideals). In a *-lattice, the poset þ and is a complete lattice under the operations þé
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Remark 5.7 (distributivity of the lattice of -ideals). Proposition 5.16 can indeed be strengthened: there is a lattice isomorphism between the lattices and . Combining this result with Proposition 5.3, we conclude that is a distributive lattice.
In the algebraic theory of particular *-lattices, say e.g. MV-algebras or Abelian -groups, prime -ideals play a significant role. Among such ideals, moreover, a special importance is attached to regular -ideals, i.e. those -
ideals which are maximal for the property of not containing a given element. Such notions can be suitably reformulated within our framework. Furthermore, since the strong algebraic De Morgan law is not necessarily satisfied in arbitrary *-lattices, we can even distinguish two different notions of prime ideal (cp. Casari 200+):
(prime and weakly prime -ideal). Let be a *-lattice and Definition . 5.20 is called: weakly , if !#" and %$ , then either & orprime , iff; for any prime, iff for any , either * ' & or * ( ) . Remark 5.8 (an alternative definition of weak primality). It is easy to see that is weakly prime iff the following condition holds: ,+% For any , if .- / , then either & or 10 / ; since !%" 23 2412% 2 , we conclude In fact, let be weakly prime and .3 2 2 % 2 2 78 or 6 78 , whence either 5 78 or 6 78 . Conversely, let that either 5 78 . Then (C) hold true and suppose that 9#: 5<; 6 and 5%=>6 2 C 2 % 2 2 5#=>6@?A5B=6 78 and thus either 5 7&8 or 6 78 . Hereafter, we shall resort to both characterizations of weak primality, using whichever will be more convenient according to circumstances.
D D FHG 5<; 6'I &7 8 / %9 : 5<; O 6 5%=J6@P?AQ 25B2C=2%6 2 78 5LKM6 G 5<; 5NIH? 9 R#STVUWR PQ 5.21 (linear and weakly linear *-lattice). E A *-lattice X is called: ] ^ Y Definition weakly linear, iff Z\[ is a weakly prime -ideal of ; E ] ^ _ linear, iff `a[ is a prime -ideal of . E
Remark 5.9 (prime -ideals are weakly prime). Remark that any prime ideal of a *-lattice is weakly prime. In fact, suppose without loss of , that generality6 that and . By Proposition 5.15.(iii), . Since , we infer that .
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Substructural logics: a primer
By our previous remark, all linear *-lattices are weakly linear.
What do weakly linear and linear *-lattices look like? The following definitions pave the way, as we shall soon see, for an answer. Definition 5.22 (orthogonally indecomposable and totally ordered *-lattice). A *-lattice is called: orthogonally indecomposable iff for any , if then either or ; totally ordered, iff for any , either or .
The main properties of weakly prime and prime -ideals are summarized in
the next
Proposition 5.21 (properties of weakly prime and prime -ideals). (i) Let be a *-lattice and . Then the following are equivalent: (A) is weakly prime; (B) is orthogonally indecomposable; (C) is a finitely meet-irreducible element of the lattice . Also the following are equivalent: (D) is prime; (E) is totally ordered. (ii) If is a representable *-lattice, then all of the previous items and (F) is totally ordered by set-theoretical inclusion are equivalent to one another.
' )*+ # ) (
, 3 6 7/8 2 3 < > 3 6 918 2 3 = 6 :;8 2 3? 35 N ILK0M A 3 @ 3 BCEDGF HJ O P/Q 2 T O R;Q 2 O S1Q 2 2 T O R;Q 2 P P]\ [^ N T UEVXW1YJ2 Z a b [ _
$#&% ! "
- /.02 - 1. 2 61347 35 < 3 93 8 2 = 6 :;8 2 S N2Z UEVXW1YJ [ 34P 35 ` 3 S 3 \ [ 2 fgc h1e 2 c / d 0 e N c d/e 2 j c i;e 2 dklEmXn1oJ2 p j0q r sutwvyxz s^{v s^{ z | } v~ys& } z$~ys ] |3 3 | { v ~ys
} v~ys 3 3 4y 0 ]
Proof. First, let us prove (i). (A B). Suppose weakly prime and , i.e. , which amounts to , i.e. , by Proposition 5.13. We must prove that either or , i.e. that either or belongs to . Since is weakly prime, w.l.g. , whence our conclusion. (B A). Assume orthogonally indecomposable and . But the latter condition holds, as we saw before, iff . W.l.g., then, we may safely assume that , i.e. , by Proposition 5.13. (A C). Suppose weakly prime, and . Let , . The assumption entails no loss of generality, since given an in , , and similarly for . As , it follows that . But is weakly prime, whence either or , a contradiction.
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! #" ! $" % &" ( ) '" * + 7 4 ,.-0/ 13254 6 21 7 4;9 7 < 8 8'9#,-0/ 1: =>4 = > 1 7 ? I 1A@B CD45EFI 1GCHB @54 L M L L L 9 N 1A@B CD4E N 1GCHB @54 9 J%K8 8 ON APQ RDSUTV N OGRHQ P5SUT.V Z Y V W X Y V ][_ ^ ` a `%a i chgje \ bdcfeHgh b%*kl bqprs i bqp t i a z u!v imcfeHnoh w { b x] y ^;w ~ i x y` { ] bn|i ^ c v} h bn b ch `~ bch % b v} eHnoh v ~ e
eqh ~ v e } h
(C A). Suppose finitely meet-irreducible, and . By Proposition 5.19.(ii) and in virtue of the distributivity of (cp. Remark 5.7), we have that , for . As is finitely meet-irreducible in , w.l.g. , i.e. . (D E). Let be a homomorphism of *-lattices from to ; by Proposition 5.10.(i), is a prime -ideal iff is totally ordered. In particular, by Proposition 5.13, is prime iff is totally ordered. (ii) If is representable, the circle of equivalences is closed as follows: (A D). By C2 and C15, in any representable *-lattice . Since is weakly prime, either or . (E F). Let and be mutually incomparable members of . W.l.g. pick and . Arguing as above, . By Proposition 5.15.(iii), then, and . Now, since was assumed to be totally ordered, w.l.g. , i.e. . Moreover, by C2 and Proposition 5.1.(v), . It follows that , a contradiction. (F C). If , then and w.l.g. , whence .
*
Corollary 5.1 (a characterization of weakly linear and linear *-lattices). A *-lattice (i) is weakly linear iff it is orthogonally indecomposable; (ii) is linear iff it is totally ordered.
'
* *
Proof. In Proposition 5.21, let . Then is orthogonally indecomposable iff is weakly prime and is totally ordered iff is prime.
We now turn to the notions of value and of regular -ideal, which can be defined as follows.
*
f # q Two useful properties of regular -ideals are collected in the next
q#_ .
Definition 5.23 (value; regular -ideal). Let be a *-lattice and is called: a value of , iff , and for any -ideal , ; regular, iff it is a value of some ( ).
$
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Substructural logics: a primer
Proposition 5.22 (properties of regular -ideals). In any *-lattice : (i) regular -ideals are meet-irreducible elements of ; (ii) regular -ideals are weakly prime.
, then if . But then of and (i)If is a value
Proof.
, whence is meet-irreducible in . (ii) and Immediate from (i) and from Proposition 5.21, because meet-irreducibility implies finite meet-irreducibility. 1.5 Representation theory We conclude this section with two subdirect representation theorems, respectively for *-lattices and representable *-lattices. From such results one can easily derive as corollaries some well-known representation theorems for special subvarieties of these varieties. For a start, we prove the next
is a *"!$#%& # 59; )8 6 + and #34/ 8 . 5 )*#,+ #'! ( 9:and8 let 5 )*#,+.- 7 /01/%2 Proof. Suppose (<%=5 )>#,+ . Now,7;letO / ? ?A@CB be a subchain of is nonempty since, e.g., D M E>F,GH I J . By Proposition 5.20, E N K PRQ SUT L is its least upper bound; N]\;Z P]^ _ `Ua [ , for suppose otherwise. Then, by Proposition moreover, VWYX Nq;i p rs tUu s.t. v]b v x vwbkn v jlegegefjmv b o v . 5.19.(i), there exist bdn cfegegegchb o in | }~ s.t. d
fgggh ~4| . By However, as y{P z C is a chain, there must be an ~4| , which is impossible as Proposition 5.11.(ii), I1 and I3, it follows that | ~=M >, . We have thus shown that the least upper bound of every subchain M > , M * , M * of is in , whence by Zorn's Lemma contains a maximal element. Proposition 5.23 (existence of values of nonzero elements). If lattice and , then has at least a value.
Proposition 5.24 (subdirect representation of *-lattices). (i) Any *-lattice is a subdirect product of weakly linear *-lattices; (ii) any representable *-lattice is a subdirect product of linear *-lattices.
¤¦¥ ª «§ ¬ ¨
.7 R¡ ¢ £ ¬ ¤ © ¯ ± ® ° ² ³µ´· ¶¸² ¹±´
Proof. (i) Let be a *-lattice, and let . If , then by Proposition 5.23 contains a value of , call it . By Propositions 5.18, 5.21, 5.22 and Corollary 5.1, is a weakly linear *lattice. Let us agree to denote its elements by etc. Now it suffices to
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. Let show that is a subdirect product of the family be ! " % # $ & ' ) + ( *
,.-%/ . Is one-one? Suppose ; then w.l.g. it is not 6 4 078395> , -%4% / 5 . As the case that 021 3 . Thus 6 4 078395;:=< , whence L E C D D D 6;4 078395? F GIH J8K 7 it is not the case that @ 0 A F GIH J8K 1B@ 3!A F GIH J8K , whereby a fortiori MN O P;Q=MN R9P ST N%P YZM U V%W , X [ . Lastly, for every is nothing but the \ ] ^ canonical homomorphism _ from W onto W `!a .
(ii) Immediate from (i), Proposition 5.21.(ii) and Corollary 5.1. b
Corollary 5.2 (Paoli 200+b). Any De Morgan monoid is a subdirect product of weakly linear De Morgan monoids. Corollary 5.3 (Dunn 1966). Any idempotent De Morgan monoid is a subdirect product of Sugihara chains (cp. Example 5.4). Corollary 5.4 (Minari 200+). Any c.r. lattice is a subdirect product of weakly linear c.r. lattices. Corollary 5.5 (Paoli 2000). Any m-pregroup is a subdirect product of opregroups (linear m-pregroups). Corollary 5.6 (Chang 1958). Any MV-algebra is a subdirect product of linear MV-algebras. Corollary 5.7 (Stone 1936). Any Boolean algebra is a subdirect product of Boolean chains.
c Corollary 5.8 (Clifford 1940). Any Abelian -group is a subdirect product of Abelian o-groups (totally ordered Abelian groups). Proofs. It is not difficult to check out that the quotient structures `a in Proposition 5.24 satisfy the remaining conditions that characterize the structures mentioned in the above corollaries. Then, these results are an easy consequence of Proposition 5.24. b
2. CLASSICAL RESIDUATED LATTICES Among the various kinds of *-autonomous lattices, classical residuated lattices deserve a special mention, in view both of the substantial amount of literature hitherto devoted to this subject, and of the important role that
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Substructural logics: a primer
particular classes of c.r. lattices (especially MV-algebras) play in the algebraic analysis of nonclassical logics. Since c.r. lattices are *-lattices, all the theorems of § 1 concerning *-lattices hold for them. Furthermore, the algebraic theory of c.r. lattices is especially smooth and pleasant. This is, above all, a consequence of the fact that most notions of § 1 are dramatically simplified once they are applied to the more restricted domain of c.r. lattices. The next lemma records, for the reader's convenience, some of these simplifications, most of which were already mentioned in the preceding section. of c.r. lattices). Given any c.r. lattice : (i) Proposition and 5.25 (basic
for features any ; (ii) for any ; (iii) is an ideal of iff (I1) implies and (I3) and imply ; (iv) is an -filter of iff (F1) implies !" and (F3) #$% and #& ' imply '$% ; (v) 5 ()# *',+.-0/ (1/#2',+ ; (vi) #346 ' iff 5 (8# *',+92 5 ()':*#1+;$< ; (vii) 7 (=/#2',+1>(1/'?2@#1+.-A holds for any # *' . is representable iff Proof. We only prove (vi), all the rest being immediate. If #346 ' , then obviously B (8# *',+92 B ()':*#1+;$< by I1. If B ()# *',+92 B ()':*#1+;$ < , then, being B (8# *',+;& B ()# *',+92 B (8':*#1+ and B ()':*#1+;& B ()# *C',+92 B ()':*#1+ by Proposition ' .D 5.6.(ii), in virtue of I3 we conclude that #346E The complement operation of representable c.r. lattices has a nice property:
FHG
FI0JF FKI0JF LI0JL . Let and M IOProof. JFPL FS L M ION=N=JJFFPPL,L,Q1Q1RRN1N1JJL?L?P@[email protected] I0JL?PF ,, whencewhence LS F . Thus FI ,L . D
Proposition 5.26 (fixpoints of the complement operation). If representable c.r. lattice, there is at most an s.t. .
T
2.1 Maximal, prime, and primary -ideals
U
U
is a
Then and
Besides having the properties of -ideals of general *-lattices, -ideals of c.r. lattices are countersigned by additional properties of their own. Thus, the present subsection will be devoted to an illustration of the main relationships among different classes of -ideals in c.r. lattices. We begin with a characterization of maximal -ideals.
U
U
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Proposition 5.27 (characterization of maximal -ideals). is a maximal ideal of iff, for any , either or, for some positive integer , .
is maximal and that . Then left to right, suppose that Proof. From , whence . This means, by Proposition 5.19.(i), that there exist and a positive integer ! s.t. "$#% $&%'(! )+* , whence , '(! )+*- of I3, , '.! )+* / . Conversely, let 10324 58:'9 by)6, *C3 2 and,, *in7 virtue . Then there exists a positive integer ! s.t. , '(! )+* <;<2 I1 implies that '.! )+* 2 ; it follows that '(! )+*=& , '.! )>*?#@. " But2 , i.e. 2A#B . C D D In arbitrary *-lattices, the notions of maximal -ideal and prime -ideal are not known to be linked D by any special relationship. In c.r. lattices it is quite otherwise: any maximal -ideal is weakly prime, as we immediately check. D E of a c.r. lattice is weakly Proposition 5.28. Any maximal -ideal prime.
HF G?W I J E , FLXYKMIOW N E . If E is maximal, then as in Proposition P JRQ ES FTUKVQ ES I@T . Applying once more Proposition 5.19.(i), Z\[^]%_ ` positive integers ab[Mc s.t. [email protected] hUi=j Z and d$ekg(ch+lfjm] . By and d e g.a h>inj Zojm] e Proposition 5.6.(ii), g(ch+lfj Zojm] , whence dpeqdsrdAetgMg.a h+i=j Zoju]vhwrgMg.ch+lfj Zxjm]vh eygzg.a h{g.iwhwrg.ch{g.l\hMh|j Zojm] , by Proposition 5.1.(xvii). Since i}rOlO_/` , however, 5.15.(ii) and I1 it is gzg.a h{g.iwhwrg.ch{byg.l\hMh|j ZoProposition j ]<e@d_ ` , which ` is impossible as is proper. C ~ We now introduce a new class of -ideals, which will prove rather useful in
Proof. Let 5.27 there exist
the following.
Definition 5.24 (power of an element). In the following, abbreviation for ( times).
i
will be an
of is called -primary LM OO . It is called primary A 1-primary -ideal is a sort of "complete" -ideal, as the next lemma
Definition 5.25 (primary -ideal). The -ideal iff, for any , if , then either or iff it is -primary for some positive integer .
shows.
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Substructural logics: a primer
Proposition 5.29 (properties of 1- and 2-primary -ideals). (i) If is a 1primary -ideal of , then it is prime; (ii) if it is prime, then it is 2-primary; (iii) it is 1-primary iff, for every , either or .
(i) Remark that Proof. !" , for any -ideal " . In particular or
# $ , which is 1-primary. It follows that either #$
$ . (ii) If is prime, w.l.g. #% . Let &' #% . . In virtue of Propositions 5.1.(i) and 5.6.(i), By I1, #( #$ ) , whence by C3
# (
# ( however, *- $+ #( # and by I3 ,- . (iii) Left to /0!" for any -ideal "1, & , which is 1right. Since . 2 , then or . Conversely, primary. It follows that either if is trivially #$ but 1-primary. If is proper, let .' 3 .4 . Then our hypothesis implies that & , whence by I1 & and #5 , which contradicts our assumption. 6 2.2 Subdirectly irreducible c.r. lattices
7
7
Given any class of algebras, its subdirectly irreducible members play a special role in the algebraic theory of . In virtue of Birkhoff's subdirect representation theorem, in fact, any element of is representable as a subdirect product of subdirectly irreducible members of . As a consequence, it is extremely interesting to obtain some kind of additional information on the structure of such members. In virtue of Proposition 5.2, c.r. lattices form a variety in the signature (simply add to E1-E10 the equation ). We shall now characterize the subdirectly irreducible members of by means of the following
8
7
7
9;:2<>= <@?#<5ACB
D :2EFGE 8
H
Proposition 5.30 (characterization of subdirectly irreducible c.r. lattices). Given a nontrivial c.r. lattice , the following are equivalent: (A) is subdirectly irreducible; (B) has a single atom; (C) There is an s.t. for any there is a positive integer s.t. .
H
9 N IKH J@<LMB P&BO? Q D BO? DR ISQJTP U B). A theorem of universal algebra (cp. e.g. Grätzer 1979, p. Proof. (A H is subdirectly irreducible iff the lattice 124) states that a nontrivial algebra of its congruences has exactly one atom, contained in every congruence other
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than the identity. The equivalence of (A) and (B) follows now from Proposition 5.16 and Remark 5.7. (B C). Suppose that has a single atom, call it . Now, there must be an s.t. . Let , and consider the -ideal . Since , there must be an s.t. . Conversely, let be an element of with the property indicated in (C), and let . Being a principal -ideal with a single generator, is atomic in ; we shall show that , whence is the single atom of . Pick in ; it follows that for any positive integer . If , it follows that for some . In virtue of our hypothesis, however, for some , and thus , whence . But then, by I3, .
#% $ " ! & '(*) + 20 '1 ,-. / '(*)3.54 ' (*) + 7'1 ,-.6/ ' :, 8; 4 : 8 ) / 9 4 ' * ( ' A @ , ( @ =< ; ' @A, ( > ' @ B , 8 B 4D( C > ' B , 8 ?< > ' @ B , 8
The next theorem, together with Birkhoff's subdirect representation theorem, yields an independent proof of Corollary 5.4.
E
E
Proposition 5.31 (properties of subdirectly irreducible c.r. lattices). For any subdirectly irreducible c.r. lattice : (i) is orthogonally indecomposable; (ii) is either atomless or has a single atom.
E
E ?F MIKL N=KL O2KL P N?QRPAS O N?Q2R PAS F N=QRTS I PUGJT ]_^2` V TXWFYR TZGJPAS N?Q2R V[S FHG\R V[S I Vacb p R ] SR F?MIdS ^ kmj lon R efS Fhg5R ] a"efS I . But it is easy to see that either eUicV or qsrutwvcx . In the former case, y?z2{ x[| }z{~f| }z2{ ~f| }5{ qArt| ; in the y?z2{ x[| z2{ qArct| z{ ~f| }5{ qArct| . It follows that latter case, ?
y?z2{ q|{ }?d| , whence ?
}? . E y? is an atom , either y?zc or (ii) Suppose of ; then if E 2 By (i), however, is orthogonally whence 2 .C implies 2 or 2 . Hence, if E containsindecomposable, an atom, such an atom FHGJIKL
Proof. (i) Let ; we have got to show that . Since is subdirectly irreducible, by Proposition 5.30 there is a s.t. for any there is a positive integer s.t. . In particular, and for some . Let be : it follows that . Let moreover . By Proposition 5.1.(xvii),
is unique.
The next lemma provides a neat criterion to single out c.r. lattices with a single atom.
Proposition 5.32 (characterization of c.r. lattices with a single atom: Kowalski 1995). A c.r. lattice has a single atom iff there exist and a positive integer s.t. for any , .
u2 A
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Substructural logics: a primer
Proof. From left to right, let be the single atom of . Then for any , , which is enough for us. From right to left, let and let be the smallest positive integer s.t. for any . By Proposition 5.30, is subdirectly irreducible, hence it has either no atom or a single atom. We shall show that the latter alternative is the correct one, arguing by induction on . If , we are done. Let now . By our assumption, there is a s.t. for any , but not . Consequently, , whence . We shall show that is the single atom of . In fact, let and . Then and by Proposition 5.31.(i). Since , by C3 (applied twice), . Hence . As , by our assumption . It follows that , whence . We conclude that is the single atom of .
!" #%$& '( ) #%$& *+ , ) #%$& - ." ) #%$& - )/ . 0 ) #%$1 + 32 . 4 ) #%$& - 5 . ' . . ) #%$& - 5 ) #%$& .6 5 . ) #%$& - &. 87 ) #%$& -
2.3 Weakly simple, simple and semisimple c.r. lattices The aim of this subsection is investigating the mutual relationships among some important classes of c.r. lattices. In § 1 we defined weakly linear *-lattices as those *-lattices where is a weakly prime -ideal, and linear *-lattices as those *-lattices where is a prime -ideal. Although we did not do so, we might have called semilinear (weakly semilinear) those *-lattices which are representable as subdirect products of linear (weakly linear) *-lattices. In such a case, we might have rephrased Proposition 5.24 by stating that every *-lattice is weakly semilinear, and every representable *-lattice is semilinear. The following definition makes a nice contrast with the picture just sketched:
=
9<: ; ?@> :
=
Definition 5.26 (weakly simple, simple, and semisimple c.r. lattice). A c.r. lattice is called: weakly simple, iff is a primary -ideal of ; simple, iff is a maximal -ideal of ; semisimple, iff it is representable as a subdirect product of simple c.r. lattices.
A
A
A
?@> :
?@> :
=
=
Throughout this subsection, we shall try to find independent characterizations of these concepts. The task is easy as regards semisimple c.r. lattices. In fact, if we define:
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Definition 5.27 (radical of a c.r. lattice). Let be a c.r. lattice. By its radical (in symbols, ), we mean the intersection of all its maximal ideals. We immediately get:
Proposition 5.33 (characterization of semisimple c.r. lattices). A c.r. lattice is semisimple iff . Proof. A theorem of universal algebra (cp. McKenzie et al. 1987, p. 179) states that any algebra is semisimple iff the intersection of all its maximal congruences is the identity. Our conclusion follows then from Proposition 5.16 and Remark 5.7.
The next lemma gives us an idea of what the radical of a c.r. lattice looks like.
! "$#
be a c.r. lattice. Proposition 5.34 (characterization of the radical). Let Then iff for every positive integer there is a positive integer s.t. .
& !'&
%
%'&
) !*'&
(
Proof. Suppose first that ; then , for some maximal ideal of . By Proposition 5.27, there is a positive integer s.t. . In virtue of I1, for every it holds that , which is a proper -ideal, whence . Conversely, suppose that there is an s.t., for every , . Consider the principal ideal . By our assumption and by Proposition 5.19.(i), such an -ideal is proper. In virtue of Zorn's Lemma, then, there is a maximal -ideal s.t. . Suppose : then , whence , a contradiction since was supposed to be proper. Therefore and a fortiori .
(
) / !10 ) / !1023& !56 !7"$#8'& 9%'&
! +$# , !*-.# '&
%
&
4!'&
(
(
(
&
Any simple algebra is subdirectly irreducible, but not conversely. Therefore, even in the light of Birkhoff's subdirect representation theorem, the task of finding out which members of a class of algebras are semisimple is not at all trivial. Grishin (1982) managed to show that every free c.r. lattice is semisimple; interestingly enough, he proved such a result by means of purely proof-theoretical tools. Grishin's results have been recently generalized by Kowalski and Ono (2000).
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Substructural logics: a primer
Now that we are done with semisimple c.r. lattices, it remains to describe the structure of weakly simple and simple c.r. lattices. To do so, we need some more definitions.
Definition 5.28 (order of an element in a c.r. lattice). Let be a c.r. lattice and . The order of (in symbols, ) is defined as the smallest positive integer s.t. , if such an exists; otherwise, . If , we say that has finite order.
Definition 5.29 (locally finite c.r. lattice). A c.r. lattice finite iff every has finite order. Definition 5.30 (local c.r. lattice). A c.r. lattice , either or !" has finite order.
is called locally
is called local if, for any
The connection between these notions and Definition 5.26 becomes apparent upon remarking that
$ %)+ &*
$
* '($
#
$
Proposition 5.35 (quotients modulo primary and maximal -ideals). Let be a c.r. lattice and . Then: (i) is primary iff is local; (ii) is maximal iff is locally finite.
*,'-$
$ $ for any $ , it is either (i) Let be primary. Since /.0!", $ :1Proof. : : ; 324(65 ; 7289 !" : ; 8 !" : 24(65 or : : ;<>=7V ?@< A"B C/C9D"E , whence eitherV F BHGJN V IKF =LG N or FM< A"B C,: G3N IKF =G N , i.e. either AO< PCQF BRG3N IKF =LG N AS< PTCRAUF BRG3N IKF =LG N . Consequently, either <PCQF BRG3N IKFMW@G N or < PorCRAXV F BRG3N IKFMW@G N , which means Y(E is local. that ,Z ( [ / \ Q ] ` ^ b _ d a c " a ` \ U e f a 7 ^ 9 g h"[ , which Conversely, suppose local and a o aUo i \Rj ekop aUo i ^lj,q g_mi nLj q , whence aUo i \Rj q ekop aUo i ^lj amounts _Ki r@j q . toBy C3,( i \RqjKq stup aUu i ^lj q , and by Proposition 5.1.(xiii)q cvLgQi \RjKq swu c vLgRaUu i ^lj q for every v . Now suppose that [ is not primary, i.e. for ~ ~ every x it is both y{z"| and }wz"| . From the former assumption, we infer x d xyz"| , whence that U xR"fory6
3 bany S xU y6
3 whatsoever K L
and xRX y6
3 K M@
. Since (| is U }(
3 KM@
, as we KM@
, whence local, there is an s.t. Q y6
3
Q R y K
w
R X }l
It follows that previously S RU }(established
3 Kd Rthat "}(
3 K L
, which . implies }p R"}/"| , a contradiction. K L
, i.e. yz"| . (ii) From left to right, suppose | maximal and let yR
3 x d xy"| , which In virtue of Proposition 5.27, for some positive integer Kd xy6
3 bO xQ y6
. But then M@
3 xTQ y6
. From right means L
3
Francesco Paoli
! " #$ $ # $ #$ $ +', $ .- /
195
to left, suppose locally finite and . It follows that is nonempty; let thus and . From the latter assumption we get and, in virtue of our hypothesis, for some positive integer . As , it also holds that . But , whence and . Consequently , i.e. , which yields .
"% & (') * &
Corollary 5.9 (characterization of weakly simple and simple c.r. lattices). A c.r. lattice (i) is weakly simple iff it is local; (ii) is simple iff it is locally finite. Proof. We argue as in Corollary 5.1.
/
Remark 5.10. Items (i) and (ii) in Proposition 5.35 were first proved for MV-algebras, respectively by Belluce at al. (1993) and Chang (1958). The generalization to arbitrary c.r. lattices is due to Casari (200+). Remark 5.11 (simple c.r. lattices are weakly simple). Since locally finite c.r. lattices are obviously local, it follows from Corollary 5.9 that simple c.r. lattices are weakly simple. This justifies the denomination we have chosen for them.
0
Remark 5.12 (simple representable c.r. lattices are totally ordered). In representable c.r. lattices, any weakly prime -ideal is prime. Hence, by Proposition 5.28, any maximal -ideal of theirs is prime. It follows that, if is simple, then is prime, whence is totally ordered. Summing up, every simple representable c.r. lattice is totally ordered. The converse does not hold: there are totally ordered representable c.r. lattices (in fact, MVchains: see e.g. Chang 1958) which are not simple.
14 2
0
53682 7
By means of Corollary 5.9, we have thus reduced the ideal-theoretical concepts of simplicity and weak simplicity to the purely "arithmetical" notions of locality and local finiteness. The notion of weak simplicity can be characterized in further ways as well: Proposition 5.36 (another characterization of weakly simple c.r. lattices: Belluce et al. 1993). Let be a c.r. lattice. Then the following are equivalent: (A) is weakly simple; (B) is local; (C) has a unique maximal -ideal; (D) is maximal;
:<;(=?>A@
9
196
(E)
Substructural logics: a primer .
"!# $%&'() *+ '(-, . /102'3 ) ( ( 4 . (5 4 $%&'() *+ 3 $%&' 6 '7 3 ) () *+ 6 '73 ) (8 4 $%&'9'3 ) ();:<+ $%&='();:<+ 4 . ) *A4 > . 4 ? @&' ?@&') > . );BCE (D$%&'() *+ F ?@&' ) ) ?@&' $%&'() *+ (5?@&' 3 6 '3 ) (8?@&') G 'G)H' 6 '3 ) 6 '73 ) ( ) *"'9'7G)H'9'3 ) ()IK ) *"/ 'J'3 ) () K LJLM N ONQK PLM N ORLMSN O O Z[ W LON X\ N X E OYZ[ W LON X\ ] T UVW L e e ^`_abcd f ^gahbcd ^hijf ^lk"m#_abcde abncde o fp^`g o fqdr esf ^_ o o d7r esf ^g drteufp^`gabcde vxwcudJdr esf ^e;y
Proof. (A B) This is Corollary 5.9.(i). (B C). First remark that an element of has infinite order iff it belongs to some proper ideal : in fact, if , then , since otherwise for some it would be , and would not be proper. On the other hand, if , then is a proper -ideal, since for any . Now suppose that is local, that is a maximal -ideal of , and that , with . By Proposition 5.27, there is an s.t. , whence by our previous remark . Since is local, , which implies , a contradiction. Hence any element of infinite order belongs to , which is therefore - once again by our previous remark - a unique maximal -ideal of . (C D). If has a unique maximal -ideal , then , and thus is maximal. (D E). As we already recalled, the elements of any proper -ideal have infinite order, which entails that . Now, suppose maximal, and . Then, by Proposition 5.27, there is an s.t. , whence by Proposition . So 5.34 there is an s.t. has finite order, and since , has finite order as well. This implies that has finite order, a contradiction. (E B). Suppose and . Then . This implies that , for suppose otherwise: then by I1 , and would not be proper. So there is a maximal s.t. . By Proposition 5.27, then, , whence and thus . This means and thus .
Corollary 5.10. Every local semisimple c.r. lattice is simple.
Proof. Let be local and semisimple: then i.e. is simple.
|
E2 }h~
is maximal,
Sources of the chapter. The main results of § 1 are due to the present author; sources for § 2 are essentially Höhle (1995), Ono (200+a), Casari (200+), Minari (200+).
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Notes 1. To avoid possible misunderstandings of the previous remark, we point out that we do not intend by any means to underestimate the extremely important contributions of great logicians working in the relevant tradition, like Belnap, Dunn, Meyer and the other people mentioned above, to the algebraic knowledge of substructural logics. 2. For the sake of precision, the structures just defined are the duals of commutative Girard quantales as usually defined in the literature. Here and in the following, however, we shall feel free to disregard such distinctions. 3. To be precise once again, these notions coincide with one another up to dualities and up to minor differences in the presentation (w.r.t. e.g. the choice of primitives). 4. A systematic and thorough treatment of the theory of MV-algebras is contained in the volume by Cignoli et al. (1999); see also Hajek (1998). 5. Hereafter, when no confusion can arise, we shall fail to mention explicitly the carrier and the operations of the indicated structure. Thus, for example, in this case it is tacitly assumed that is the carrier of . 6. Henceforth, the expression "without loss of generality" will be abbreviated by "w.l.g.".
PART FOUR THE SEMANTICS OF SUBSTRUCTURAL LOGICS
Chapter 6 ALGEBRAIC SEMANTICS
In the preceding chapter, we introduced and investigated at some length several classes of algebraic structures, claiming that there exists a correspondence between the diagram of logics in Table 2.2 and the diagram of algebras in Table 5.1. Our present task will be to show that our claim was sound. In fact, we shall prove completeness theorems for most of the Hilbertstyle calculi of Chapter 2 using the algebraic structures of Chapter 5. Subsequently, we shall see that - at least in some cases - such classes of structures are even too large for our purposes: due to the representation results of Chapter 5, in fact, the theorems of the logics at issue coincide with the formulae which are valid in a smaller (and usually much easier to tinker with) class of structures. In a few lucky cases, it will be sufficient to consider a single manageable structure, just as it happens for classical logic (even though this structure may not be just as simple and wieldy). Finally, we shall quickly browse through some applications of algebraic semantics to the solution of purely syntactical problems concerning our substructural calculi. Before we start, a couple of remarks are in order. In the first place, it seems worth emphasizing that we shall prove strong completeness theorems. This will be done, chiefly, for the sake of greater generality; nonetheless, we do not place a high value on such a formulation. The consequence relations we are going to define, in fact, do not correspond to the substructural derivability relation w of Chapter 2, but to its "classical" counterpart . For this reason, after proving our theorems, we shall quickly forget about consequence relations and concern ourselves, in the remainder of this chapter and in the next, only with sheer validity of formulae.
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Substructural logics: a primer
Secondly, in the light of the emphasis hitherto laid on sequent calculi, it might appear somehow bizarre that we use their Hilbert-style counterparts in order to get the adequacy theorems contained in the next pages. Such a choice depends, however, on the fact that Hilbert-style formalisms - as already remarked - are definitely not the best friends of the theorem prover, but are quite comfortable to work with when your goal is just attaining soundness and completeness results. Moreover, our option is after all rather inessential in the light of Proposition 2.49, which entails that (weak) completeness of the Hcalculi is automatically inherited by their L-counterparts.
1. ALGEBRAIC THEOREMS
SOUNDNESS
AND
COMPLETENESS
The formulae of a formal language are, stricto sensu, mere strings of symbols - they mean nothing. However, it is possible and sometimes desirable to assign them some sort of meaning - in other words, to provide the language at issue with an interpretation. How can such an aim be achieved? We might wish to take as a starting point Gottlob Frege's answer, which is based on two very simple and appealing key ideas: 1) A sentence, in his opinion, was nothing else than a name for a truth value. This tenet, at least prima facie, seems to have little to do with algebra; however, take the set , the carrier of the two-element Boolean algebra given a sentence to be interpreted, we might agree to consider it a name for the element 1 just in case it is true, and for the element 0 just in case it is false. 2) Another cornerstone of Frege's semantics was a rigid principle of compositionality for meanings, according to which the meaning (the truth value) of a compound sentence uniquely depends on the meanings (the truth values) of its component sentences. Once again, this non-mathematical principle can be given an algebraic reading: the connectives of the language are mirrored by the operations of the two-element Boolean algebra, in such a way that conjunction correspond to lattice meet, disjunction to lattice join, and so on. It is easily seen that such an interpretation of logical operations guarantees compositionality. Frege's idea, as you can guess, works perfectly also in the presence of larger sets of truth values. An increase of the number of truth values can be intuitively buttressed by thinking of sentences as names for truth degrees, which are true just in case the degrees they name belong to a preferred subset (the set of designated values, which in the previous example was simply ) of the set of truth degrees. If the latter set is ordered, and the designated values
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stay on top of the ordering, it is tempting to read such an ordering as leading from "falser" degrees on to "truer" degrees. This is not always straightforward, nonetheless - under some circumstances, it could be difficult to specify an informal interpretation for each and every element of the set of truth values, or to interpret the ordering as suggested above. Sometimes, therefore, it could even be appropriate to abandon the idea that the meanings of our sentences are truth values or truth degrees, and think of them simply as "senses", "states of affairs" or whatever you may like. This is, very roughly, the intuitive idea underlying algebraic semantics. In what follows, we shall treat the formulae of our formal languages as names for elements of algebraic structures, and the connectives as names for operations on such structures. In § 1.1, the letter "S" will range over the set containing the axiomatic calculi listed in Table 6.1, whereas "£" will range over the set £1 ,£2 . It will be tacitly assumed that each of the Hilbert-style systems referred to in the sequel is formulated in the appropriate language.
1.1 Calculi without exponentials To implement the chief idea of algebraic semantics, we need to specify what it means to "name" an element of an algebraic structure. In other words, we have to replace this informal relationship by an appropriately rigorous counterpart, and we shall do so by means of the concept of valuation. Definition 6.1 (absolutely free algebra of formulaeof a language). The
absolutely free algebra of formulae of £ is the structure £ , whose carrier is FOR(£) and whose operations are the connectives of £. Definition valuation). Let 6.2 !"$#%(algebraic $&(' be a *-autonomous lattice, where " !*() ,+ is -)./+ ) have been included in the the defined operations is a signature. An algebraic valuation of the language £ with values in
$ 3 021 homomorphism £ which extends the arbitrary mapping , 054.1 VAR(£) to the effect that: 0 768 $ 094 768 :5;=<>?@$A: ;=<.@CBD: ;E?@ F : ;=<G?2@$A: ;=<.@HI: ;E?@ F :5;KJ<.@$A L-: ;=<.@MF : ;=<*NO?@$A: ;=
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Substructural logics: a primer
If the lattice-theoretical constants are present, then element and a bottom element , in which case
must contain a top
Definition model). An algebraic model for £ is an ordered 6.3 (algebraic , where is a *-lattice and is an algebraic valuation of pair £ with values in .
the axiomatic calculus Definition 6.4 (matching). The *-lattice matches S (or, as we shall often say, is S-matching) iff belongs to the class and the names of S and appear in the same row of Table 6.1 below. is S-matching whenever We say that the algebraic model is such. Table 6.1. Matching relations between algebras and calculi
HL HL HLB HRW HRND HR HRMND HRM HA HC HG HLuk HLuk3 HK
(
(
(
(
(
(
(
(
(
(
= *-autonomous lattices = *-lattices where $ = *-lattices with top and bottom " %'& # ! = *-lattices where is distributive + ) , * ) ) = *-lattices where = De Morgan monoids = *-lattices where )+*,)/.0) = idempotent De Morgan monoids = classical residuated lattices 1 = -pregroups 1 = Abelian -groups = MV-algebras 24365 )/. 287 5 ) = MV-algebras where = Boolean algebras
We now introduce appropriate notions of truth, validity, and consequence.
9;:
FOR(£) is said to be true in the algebraic model Definition 6.5 (truth). >? = @BADC (in symbols, < EG AF ) iff HJIKAMLNF/O ; it is said to be true in = = EP < AF the *-lattice (in symbols, ) iff it isQS true = R in every algebraic model < whose first projection < E A Q is . Finally, the set Q FOR(£) is< called true in (in symbols, ) iff every formula F;T is true in .
<
Definition 6.6 (validity). FUT FOR(£) is said to be (algebraically) valid
Francesco Paoli
with respect to S (in symbols, matches S.
A S
205
) iff it is true in every *-lattice which
Definition 6.7 (consequence). FOR(£) is said to be a consequence of A FOR(£) w.r.t. S (in symbols, ) iff it is true in every Sthe set S matching algebraic model where is true. We are now ready to prove a soundness theorem: if is weakly derivable in S from the assumptions in , then is a consequence of w.r.t. S. Recall that, in the light of Proposition 2.42 and the remark immediately following it, w means that ; then: S S Proposition 6.1 (algebraic soundness theorem). If w A FOR(£), then implies S S .
FOR(£) and
Proof. Since by hypothesis belongs to every regular and detached Stheory containing , it is sufficient to prove that, given an S-matching algebraic model , the set of all formulae true in is a A regular and detached S-theory. In such a case, if , it follows A automatically that . In the light of Proposition 2.35.(i), it suffices to show that the axioms of S are true, and that its rules of inference are truthpreserving, in any S-matching algebraic model . The correctness of the former claim can be verified by checking all of the axioms of S one by one. We shall consider just a few cases, leaving the remainder up to the reader. (Ad F2). By Proposition 5.1.(i), in any *-lattice = . (Ad F5). By Proposition 5.1.(v), in any *-lattice = . (Ad F7). By Proposition 5.1.(v), in any *-lattice . According to Proposition 5.1.(xvii), however, , whence
! # "%&.$ -/ ) $ -0& ( *' )1 -/*) , (' -0$ &. $ -/ +, ) (' *, (' $ & (' *, 2 &.-435) !#-6" 7)8$ -6 $ 39 & (' $ *) (' $ *) (' $ & !#"%$ & (' *) : *, (' $ $ $& ($ ' & *) : **) , *, $$ < $$ && (' **)) $ : &$ & ;*', *, $ &( ' +) : ! * , " &.-/) (= ' +> -/ , : -0&.-/()' = , 2(' ? ' ? " (' ? : $ & $ (' & $ & $ "@ & $ & *) "/$ & *) (' (' ('
(Ad F14). If monoid), then and C3,
. is a *-lattice which satisfies (e.g. a De Morgan , whence by Proposition 5.1.(xii) and
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! . " is a c.r. lattice, by Propositions 5.1.(v) and 5.6.(ii), (Ad F15). If #$% &' () .
It remains to show that* modus ponens adjunction are truth-preserving *0/ A andin ,+ " -. and any algebraic model . Thus, suppose *0/ A 1 i.e. 2 and 34 5 ( ; by C3 64 , ,whence 3 , viz. * / A . Finally, in virtue of C2 * / A and * / A , i.e. 2 and 34 ; we assume that 4 78 9: <;= , viz. * / A <;= . > conclude by C2 that Proposition 6.1 established that the Hilbert-style calculi in Table 6.1 are sound with respect to our algebraic semantics. We are now going to show that they are also complete. Definition (an equivalence ?5@BAD C FOR(£).6.8We stipulate ? E relation A iff FGbetween ? H A andformulae). F$A)H? Let. that S
S
E
S
Proposition 6.2 ? ( IKJ S is a congruence). If S is based on £, the relation is a congruence on (£).
E
S
E
Proof. In virtue of F1 and F2, S is an equivalence relation on FOR(£); in virtue of F2, F6-F7, F12-F13, F37, F41, F75-F76, it preserves the operations ? K I J > of (£). The last result legitimates the following, important Definition 6.9 (Lindenbaum algebra of a system). The Lindenbaum algebra language is the structure @UT0@WonVX@Bthe Y@ Z[ @ \^] ,£,where: L6M=Nthe O PRcalculus Q FOR(£)S,S E based S S S S S S S
of
_ ?3`T _ A` P _ 4 ? a4Ab` S Y P S _ f S` S _ ?bS ` \ _SA` P _ < ? e=A` S
S
S
V _3 ? ` P _c 3 ? ` S _ ?b` Z S_ A` P S_ ?
S
S
S
S
g hbikjGg li iff m'h n l ; in fact, if g hi#opg li S l S g h3S i , S thenS m'h1Snh
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now on, whenever no danger of confusion is likely to arise, we shall omit the indexes S which tag the equivalence classes and the operations of the structure S . Proposition 6.3.
S is an S-matching *-lattice.
Proof. First, let us show that S is a *-lattice. C1 holds by F51, F53, F55. As to C2, first remark that iff S , which holds true because of F1. By definition, iff and if and , which ; moreover, S and S , by F2 S , i.e. . amounts to Thus, the relation S is a partial ordering on FOR(£) S . Theorems F6-F7 and F75-F76 express nothing but the fact that S is a lattice ordering where ! "$#&% $'( !) *,+.- $'( . Finally, and /0/ 121& by F4 and F40, and if (i.e. S ), then by F41 S 1341& , that is to say / / . This takes care of C2. : : As regards C3, / / 65 iff 17 1&989 ; but < = 17 < by F43, and 13;89= by F2, F40. It follows I F12, = @ >
? A B ! > ?
C 6 A D ? E A ? 9 A B ? C F E A K F J C FEHG . This that iff I CFE , i.e. S last condition, however, holds iffI , in view of F3, F8, F9, F35. S < Hence, >@? AB>!? C A6D ? ELA iff S CFE , which means ? C AB? EA . It remains to show that MON J S G matches S; this can be verified by checking that, in each and every case, the additional conditions characterizing the class of structures whose members match S hold in MON JP S G . For example, suppose you want to show that MON J HC G is an Q -pregroup; you have to check that C8 and C9 hold in MN J HC G . But this is rather easily done. In fact, C8 holds iff >@? < ARS? C A6D>!? C A iff ? = AR ? C9T!U&C A . But ? C9T@U&C AR ? C FC A by F2, = F12, F40, while ? AR ? C FC A by F9 and F24. In virtue of F9, F20, F24, !? < AR ? = AR ? = T = ARK>!? < A6D>@? < A , which accounts for C9. The remaining cases are left to the reader. W Remark that the preceding construction can be carried out not only for the Hilbert-style systems SX of Table 6.1, but also, any] regular YZ more [ generally,Y for \[ X ^ and holds iff and [\_detached Y`]aX S-theory , stipulating that X . Therefore, every regular and detached S-theory can be provided with a Lindenbaum algebra j ^ X e gih bc df ^ v ^ ^ ^ ^ l&m l2nolqpfl rsl t u , where the defining FOR(£) k
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clauses for the operations of this structure are obtained by replacing "S" by " " in the above definition. Proposition 6.3 can be accordingly adapted as follows: Proposition 6.4. If matching *-lattice.
is a regular and detached S-theory,
is an S-
A notion of canonical model is now needed. But it is easy to get it, in the light of the preceding remarks: Definition 6.10 (canonical algebraic model). To each regular and detached associate a canonical algebraic model S-theory we
, where for every formula , "!$# &% . With these ingredients at our disposal, we can now prove: Proposition 6.5 (algebraic strong completeness theorem). If )+* )-, A )-. w and FOR(£), then implies S S .
('
FOR(£)
/ Remark that, if is a regular and detached S-theory, 0 (Proof. 1&2 #&5 %4 3 6 7 8 94 = : / , since by F3, F8, F9 and F35 8<; / iff > ?A@
theory containing but not . Let be the canonical algebraic model associated to it; by Proposition 6.4 it is S-matching [ A\ [OA ] and by our initial remark we have that Z yet not Z . \-[OA ] Consequently, it is not the case that S . Remark 6.1 (the algebraic completeness theorem: historical overview). The techniques just used for proving our algebraic completeness result are rather standard. Therefore, it is difficult to credit Proposition 6.5 to any particular author. To the best of our knowledge, the algebraic completeness theorem for HL and HLB is due to Avron (1988); the corresponding results for HR and other relevance logics should be ascribed to Dunn (1966). Grishin (1982) established the completeness of HA, while Casari (1989) did the same for HC and HG1 , and Chang (1958) for HLuk and HLuk3 .
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1.2 Calculi with exponentials Up to now, we did not consider the calculus HLE for propositional linear logic with exponentials. It is not hard to extend the results of § 1.1 in such a way as to deal with this system, but such a generalization requires some refinements in our algebraic equipment. Thus, let us introduce a few useful concepts. Definition 6.11 (closure *-lattice). of Atypeclosure*-lattice is an, algebra where !" # is a *-lattice and is an operation (called closure) which satisfies the following conditions: %$&%'( %$&%$*)+','!%$*)+' (C16) (C18) )-%$*)+' %$*)+'. %$*/'(%$*)0/' (C17) (C19) )12 , the element 34$5 6)7' will be referred to by 8,$*)+' and will For every ) be called the interior of .
9-Proposition %$*)+' ; (ii)6.6 (properties $*) :of 6)+closure ';-%$**-lattices). /' ; (iii) In any)<-closure ) %$**-lattice: /' ; (iv)(i) %$=)+'(%$*)+'. %$*)5' ; (v) if )-/ , then %$*)+';-%$*/4' ; (vi) if )- , then %$=)+'( if )-%$*/' , then >$*) ' -%$*/' ; (viii) if )-/?D @A@B@ C/ E , then >$*)+';-%;$*/6(vii) '. @A@A@ %$*/E ' ; (ix) 8,$, F' ; (x) 8,$*)5';-) ; (xi) 8,$&8,$*)+','(8,$=)+' ; D 8G$*)+' H 8,$*/'(8,$*)I/' 8,$*)+';-( 8,$*)+';-/J : 6/ (xv) (xii) )IHK8G$*/';-) ; (xvi) 8,$*)+'(8,$=;)+' (xiii) H 8,$*)+' ; (xvii) if;)(xiv) -/ , then 8,$*)5';-8,$*/' ;;(xviii) L-) , then 8,$*)+' ; (xix) if 8,$*)+';-/ , then 8,$*)+';-8,$*/' ; (xx) if if )MD H @A@A@ HN)OE -/ , then 8,$*)D ' H @A@A@ H 8,$*)(E ';-8,$*/' . 4$=)+'P %$*)+'. Q Proof. We confine ourselves to (i)-(viii). (i) %$=)+'. %$&%'6R%$S)9T%'3U:)TVUP , by C16, C17, C19. (ii) From (i) and Proposition 5.1.(xi). (iv) >By C19, %$=)+'. %$*)+'5.1.(v). (%$*)I(iii))+'From (%$*)+(ii), ' . (v)C2If and )-Proposition / , then by (iii) * $ + ) ' and C19 )- %$=)+'. %$*/'CW%$*)I/'CW%$*/' . (vi) By (i), -%$*)5' ; moreover, %$=)+';-%$&4'X( . (vii) If )-%$*/' , then byif(v) and, in virtue of (v) and C16 %$=)+';-%$&%$*/','(%$=/' . (viii) Since /?Y -%$*/Y ' by C17, Proposition C18 /Z @A@[@ C/\-%$*/6D '. @A@A@ %$*/E ' and thus 5.1.(xiii) )<-%$*/6D '. !implies @B@A@ %$*/E that'(%$*/]D D <@A@A@F0/EE ' . In virtue of (vii), we get our conclusion. ^
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Substructural logics: a primer
The set of all closed elements of a closure *-lattice - i.e. the set of all elements such that - has interesting properties. Let us see some of them.
"# ! $ %& ')(+*-,.'0/ 1(+*-,.12/ '3!14(5*-,.'0/63!*7,812/9( *B:,.'Proof. ; 12/<(i)( If*-,=*-,.';>12/?/@and( B *2,A*-,.'0/63!, then *-,.12/?/@( *:,.'312/ B. (ii) Since CED ' G for every ' G in C , *7, C / D *-,.'G /(F' G , whence *-, C / D B C B B J . (iii)-(iv) Immediate from Proposition 6.6.(i)-(iv). (v) and thus H:IKJLM SupposeY NOM+H-I.N2L and PQ!N ; thenZ H-I.PRLSQTH-I.N2LMFN . Moreover, ]I^ L and PQ!N ; therefore, H-I.P0L M \ [ N X N byUW]C17-C18, ^ L and H-I8P0LVUW > N X N W U I _`!a Z . b ). (i) is closed under addition; (ii) Proposition 6.7 (properties of if is complete, then is closed under arbitrary meets; (iii) if , then ; (iv) if , then ; (v) for every , and .
It is worth noting that a commutative Girard quantale can always be viewed as a closure *-lattice, as the next lemmata show: Proposition Girard quantales: Girard 1987). Let cedg f!hikjFim6.8lnip(closure o2iqir9operations s be a incommutative Girard quantale, and let tvu h be a set with the following properties: (i) it is closed under addition and d j o y t w x w w w w under arbitrary (ii) if , then c
nkand !p2.Moreover, p s letis z-{8w0| d B} > meets; and ! ~ . Then a closure *-lattice.
- 0 -A
} > ~
} >
! ~ - = - 0
Proof. By definition, is a lower bound of and , whence . This settles condition C17 and implies, in particular, that . Our assumption (ii), moreover, entails that for every in , whence and, by (i), . This is enough to conclude that C16 holds. Once more in virtue of our assumption (i), , being a greatest lower bound of elements of , belongs to . It follows that and , which entails condition C18. Finally, recall (cp. Rosenthal 1990) that in a Girard quantale addition distributes over arbitrary meets. Hence and and and and . Thus, we have to show that and . From left to right, let , and . By assumption (i), as well. By assumption (ii), furthermore,
!
T-A :- 0?
} > - 0S!~ - 0 -.R6!-.2 B¤ ¡- ¢£ ª ª « ¤ « ¥T ª ¡- ¢£ ¥¦ ¤ §¨+¡- 2©?¨+¢£ ¥T « ¤ >¥!¨ ª ¬¡ ¬¢£ ®)>¥!¬¯+-.06!-.7 2©?¨+¢£ ¥T ¥n¨ Vn¨¢£
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211
and . Hence and & , whereby . Thus and and so !" $#% . Since and were completely arbitrary, it follows that !" $#' !"(#) !*$# . Conversely, if and +, , then a fortiori .-/& . Thus 01 2 ! $-/3 and , whence !"(#)4 !"$#' and so and !"(#) !"$#' !" $# , since was arbitrary. C19 is thus satisfied. 5 Proposition 6.9 (building closure *-lattices out of Girard quantales: Avron 1988). Every commutative Girard quantale can be turned into a closure *lattice.
6
7
Proof. It suffices to show that it is always possible to define, in any commutative Girard quantale , a set with the properties of Proposition 6.8. Thus, we put:
and so
S
798 @ :; :08:=< :
and
> ?: A
BDC*E(FHG9IKJ LM L=GLON L
and
E P0Q R4L A .
S U E V T L W S Y ZS E1N L=GE=N E=N LONXL [ + Q R 1 E
N L = E
N L W \ Y N \ Y G \ J E=N4LM E.T/LW Y A G \ Y Q R \ Y Q RE EW Y 5
Therefore, satisfies property (ii) simply by definition. Closure w.r.t. addition follows from the fact that, if , then and , whence . Finally, if , then by distribution and , as for every . We are now in a position to devise an appropriate semantics for HLE . To this purpose, we have to adapt our definitions of § 1.1. This task is obvious for its most part, and the reader can carry it out for himself. Let us stress just the modifications required to adapt the concepts of valuation and algebraic model: Let Definition valuation). 6^]` _a b.cb%debgfb'6.12 hibkj$bml bHnbH(algebraic obkp!bkqgr be a closure *-lattice, where d b'hibslebkq have been included in the signature. An the defined operations 6 algebraic valuation of the language £ with values in is a homomorphism a m v ! w z x H y h 6 h|a to t+u £ which extends the arbitrary mapping t${=u VAR(£ ) 3
3
the effect that:
3
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Substructural logics: a primer
is defined as in Definition 6.2, if FOR(£ ); 2 ? . Definition 6.13 (HLE -matching algebraic model). An HLE -matching algebraic model for £3 is an ordered pair , where is a complete closure *-lattice and is an algebraic valuation of £3 with values in . We can now prove: Proposition 6.10 (algebraic soundness theorem for HLE ). If FOR(£3 ) w ! A and FOR(£3 ), then implies HLE HLE . Proof. Let us check F29-F34 and R5. (Ad F29). By Proposition 6.6.(iii), "$#&%')(*"#&%',+.-#0/1"#2' '43 "$#&%',+4/65 #"#2''738"# 9 2:;%' . Thus <=( /1"#&%',+>"# 9 2?:;%'3 "$#&%7:)# 9 2?:;%' ' . 3 (Ad F30). By Proposition 6.6.(iv), "$# 9 2:)# 9 2?:;%' '
/ 5#"#2' ',+4/65 #"#@2' ',+>"#&%';3 6 -#0/1"#2' ',+.-#A/1"$#2' ',+>"#&%'43 -#0/B"$#2' ',+>"#&%' 3 "$# 9 2:;%' . Hence "#&%' 'I( "$#&%' , whence by Proposition 6.6.(xx) 5 #"#2' 'JFK5 #0/1"#2',+>"#&%' ')( 5#"#&%' ' , i.e. 5 #A/1"#2L',+>"#&%' '( /65 #"$#2' ',+.5 #"#&%' ' , which means < (4"$# 9M#2:;%N'O:)# 9 2:)9P%' ' . (Ad F32 and F33). Immediate from C17 and C18. (Ad F34). Immediate from the definitions.
HLE
6.6.(xviii)
in the obvious way, we get:
Definition 6.14 (Lindenbaum algebra of HLE ). The Lindenbaum algebra of HLE is the structure TVUW# HLE ' 3
X
Y
FOR(£3 ) S
HLE Z
+
HLE Z
/
HLE Z\[ HLE Z ]
HLE Z ^
HLE Z
where the operations of the structure are as in Definition 6.9 and
-
HLE #\` 2cb HLE '
3a`J?2cb HLE
-
HLE
_
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As a consequence, we obtain: Proposition 6.11.
HLE is a closure *-lattice.
Proof. C17 and C18 hold by F32-F34 and C19 follows from As to F126. C16, is a consequence of F32 and F34, while holds iff in turn amounts to ! #"$ 1. By F8 and R5, however, 1 ,! which E E , and by F3 and F9 1 HL ! HL %" "$ ! $ " 1 . Consequently, HLE 1. & HLE 1 However, the canonical model constructed upon ')(+* HLE , need not be a complete closure *-lattice. To ensure that it is such, we need the following result, whose proof is omitted: Proposition 6.12 (Embedding theorem: Avron 1988). Every closure *-lattice can be embedded into a complete closure *-lattice in such a way that existing infs and sups of subsets of are preserved. &
(
The reason why we require of our structures to be complete as lattices may not be too perspicuous at present, but will become clear in Chapter 7, when we show the equivalence between the algebraic semantics and the phase semantics 2 for HLE . Once we are guaranteed that our canonical models are based upon complete lattices, we can extend the above results to arbitrary regular and detached HLE -theories. Arguing as above, then, it is possible to prove: 6.13 (algebraic strong completeness theorem for HLE ). If 021 043 A 065 w & FOR(£3 ) and FOR(£3 ), then implies HLE HLE .
-/. Proposition
2. TOTALLY ORDERED MODELS AND THE SINGLE MODEL PROPERTY
04798
In Proposition 6.5, let . Under this assumption, we get a special case of the algebraic strong completeness theorem, i.e. what is usually called a weak completeness theorem: Proposition 6.14 (algebraic weak completeness theorem). If 3;A 5<& then implies S S .
-:.
FOR(£),
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Substructural logics: a primer
Likewise, if in Proposition 6.1, we conclude that only if S A . What does this mean for each one of our calculi - for example, for S HL? It means that a formula is HL-provable just in case it is true in every *lattice. In other words, if we want to make sure that A is a theorem of HL, we have to check its truth in every member of the indicated class of structures. Not a very simple task, indeed. For classical propositional logic, the job would be easier: it would be enough to focus on a single structure where it is quite easy to carry out computations - the two-element Boolean algebra . If has been proved to hold in , we are assured that it holds in every Boolean algebra and so - via the algebraic weak completeness theorem - that it is provable in HK. This pleasant situation is a direct consequence of a nice property of classical logic, a property that we are now going to define in a general form.
Definition 6.15 (single model property). An axiomatic calculus S, based on £, has the single model property iff there is a "manageable" S-matching *A A iff . lattice such that, for every FOR(£), S The single model property is, of course, a vague property3 . What does it mean for a structure to be "manageable"? We take it to mean that: (i) it has strong additional properties besides those which define S-matching structures, and that (ii) the semantic values of our formulae are easy to compute therein, , so that focusing on the given structure gives you as it happens e.g. with a clear advantage as regards the problem of checking whether a formula is valid with respect to S or not. The class of manageable structures, therefore, has intrinsically unsharp boundaries, but should include at least , (the standard MV-algebra) and , the -group of the integers. Why does classical logic have the single model property? To understand why it is so, recall the definitions of term and equation (see the corresponding items in the algebraic glossary of Appendix A). Applying very general algebraic principles, it is possible to show that:
&
!" #%$
'
+ ' - , - .0/ >= ?
(*1 )
Proposition 6.15 (validity of equations in subdirect factors). If the *-lattice is a subdirect product of the family of *-lattices, and are terms of type , then the equation holds in iff it holds in every .
@-A
243 1657198:1 3<;
BB
@
B
Now, Birkhoff's subdirect representation theorem says that, if is a variety of algebras, every member of is a subdirect product of a family of subdirectly irreducible members of ; but it can be proved that the two-element
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215
Boolean algebra is the only subdirectly irreducible Boolean algebra. It follows that every Boolean algebra is representable as a subdirect product of , whence by Proposition 6.15 an equation of type copies of holds in every Boolean algebra iff it holds in . However, the problem of determining whether a formula is true in a *-lattice can always be reduced to the problem which consists in finding out whether a ! certain equation of the above type holds in : in fact, is true in iff for #%$&"' !)( holds in / *,+ 0 - ". , iff the every valuation " !)the relation (7* 182 0 equation 13254 "6' holds in . It follows that classical logic has the single model property, as previously claimed. But we can draw further consequences from Proposition 6.15. In the first place, we know that several subvarieties of the variety of *-lattices are such that their members are subdirectly representable with totally ordered factors. 9 at This fact implies that an equation holds in all members of the subvariety 9 issue just in case it holds in all the totally ordered elements of . Translated into logical jargon, this property amounts to the fact that a formula is valid with respect to the calculus S just in case it is true in all the totally ordered Smatching *-lattices. In other words, if I want to disprove a given formula by exhibiting a counterexample, i.e. by finding a model where it does not hold, I can restrict my search to the models on totally ordered structures. The availability of such an option, sometimes, turns out to be quite useful and convenient. Among logics with the single model property, there is the infinite-valued Lukasiewicz logic. Showing that this is the case requires a huge amount of work, and all of the existing proofs presuppose for their comprehension some degree of mathematical expertise (Chang 1958; 1959; Rose and Rosser 1958; Cignoli 1993; Panti 1995; Cignoli and Mundici 1997). It is definitely easier, on the other side, to show that the single model property is possessed by Abelian logic. We sketch below a nice proof of this fact, given by Meyer and Slaney (1989). We begin with a lemma:
!
Proposition 6.16 (normal -form ;<;<;<- : = theorem for HG). Every formula of £1 containing the variables : can be equivalently expressed in the conjunctive normal form
I H L M ? ACB P O T MRQE * K , # D 2 V Y [ Z is an integer and V Y [ Z W is either the formula V Y [ Z X W or where Y Z \ V [ X ]^W or L according to whether V Y [ Z is, respectively, positive, negative,
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Substructural logics: a primer
or null. That HG .
means that
can be "equivalently expressed" as
HG
iff
Proof. It is sufficient to apply normal form principles available in HG. First, remark that F100 and F106 imply, together with F4, F12-13 and F40, that and . Now, you HG HG can get rid of group-theoretical connectives other than disjunction, negation and by F4, F12-13, F40, F101, F109. Then, use F4-F5, F40 and to drive negation signs inwards in each latticetheoretical disjunct (simplifying double negations if necessary) and F56-F59, F68-F73 (plus the equivalence of and ) to push group-theoretical connectives into the various lattice-theoretical disjuncts. Finally, use F28, F70F71 to put everything into conjunctive normal form.
Once equipped with this lemma, we are ready to prove our:
Let
Proposition 6.17 (single model property for HG: Meyer and Slaney 1989). FOR(£1 ). The following are equivalent: (A) HG ; A (B) HG ; A (C) for every totally ordered Abelian group ; A (D) , where is the -group of the integers.
!
Proof. The equivalence between (A) and (B) is nothing else than the weak completeness theorem for HG. (B) and (C) are equivalent by Proposition 6.15 and Corollary 5.8. Since (C) trivially implies (D), it remains to show that (D) implies (A). This is done by induction on the number of variables contained in . ( ). Suppose it is not the case that HG , where contains just the variable . Then it is also not the case that HG , where is a conjunctive normal form of provided by Proposition 6.16. It follows, in particular, that is not a generalized conjunction all of whose members either have as a disjunct or have disjuncts of the form and , where and , for conjunctions of this form are rather easily seen to be HG-provable. As a consequence, some conjunct of has the form
(
"$#&% '
(
)+*
,-*
"
(
)/.0
2, 10
*( 356 4 ) 7 8:9<;=;=;>9 ) , 8 ? where, for every , ) 6 @A0 , and all the ) 6 's have the same sign (positive or E BDC. and 1 E BDCA( . , where negative). Now consider the models 1 FHGJILKM5NO and FP( GJILKMRQSO . The former model falsifies T V (hence also U W
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217
$& 9 3% % $ 9 3 ' !#" (*),+,- 3 ; . 8 4 /04 .1112. 8 57963 4 / 5764 8 : <>= ? A&B @ CEDGF @ AHe BJIGCLKCNMOCQPRCTSUCTVWF DYXZIj[\k ; B,P j k p ]_^a` bc^,d iDYX l f i m `0fggg2f i njok m ` q7rp [\BhP ; w yHz shx {Et v u |~} JGQ
U * \ c G E 0 2 E 7 0 2 7 In the pairs ¡ ¢H ¡ what £,¤Wvirtue ¢&£,¤ we¥; ¦ allhavebelongjustto © ª said, ¥; ¦~¥2of§§§¥¨ . Now we transform ; such pairs in the following way. If «&®7¬ ¯ °c±h² , then ¡Y· ³&±,´· µ¶ becomes ¡ ·¸ ¼&½ Å Æ É Ç ¾0¹ ¿ÁÀÀÀ2¿ Å ÆÈÊ Ç ¾º Â_Ã Å Ê ÆÈËÇ É ¾ º7»¹ Ä . ; If Å&ͬ ÎÌ Ï ÄhÐ , then ¡ Ì ¼&½,ÑÌ ÂÄ becomes ¡RÒÓ ×&Ø á äâ åã æ Ù Ô7ÕÖ Ú_Û á âÈæ ã Ù0Ö ÜÁÝÝÝ2ÜÞÛ á âÈä ã ÙàÔ ß . ì If á&è7¬ éç ê ×Áë , then ¡ ç is left untouched (i.e. ¡ ç × ¡ ç ). It is easily seen that, í on totally ordered Abelian groups, if î ç belongs to ï , for every model ñ ñ õ ö ò7óô appears in then so does ¡ ð . Thus, the ¡ ð 's still belong to ÷ ; furthermore, at most one member of each pair, and always with a positiveø coefficient. At this point, we proceed with our transformations. Let ñ be the product of ò ó ô ñ ö ¡ ð all the positive coefficients of in the 's. If ¡ ð has the form ùOúGû_ü ¬ ý þ , it becomes ¡ ÿ&ù úGû ý þ ; likewise, if and ) if the 's are positive, the latter falsifies it if the 's are negative. In either case, is not true in . ( ). Suppose it is not the case that HG , where contains the variables . Then, by Proposition 6.16, there is a formula s.t.: 1) for every , has the form ; 2) for every , has the form ; 3) is unprovable in HG as well. The item 3) means that there is a model , where is a totally ordered Abelian group, s.t. and, consequently, that there is an s.t. for every , . By F55, however, group-theoretical disjuncts of the form can be safely omitted. Now, let be an algebraic model whatsoever. We define the relation (FOR(£1 ))2 in such a way that just in case in . Thus, consider the disjuncts:
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has the form , it becomes , ! ; finally, if does not occur in #" , then the latter is left untouched ( and so "$ " % "$ ). & ' Once again, for every model on totally ordered Abelian groups, if $" ( ( belongs to ) , then so does $" " . Thus, the $" " 's still belong to * . In other
words, our transformation process did not bring about any change as regards + whether our inequalities hold in , or, for that matter, in any algebraic model of the specified sort. 4 Summing up, our , $" " 's have one of the following forms: 1) -/.1032 , < where 5 9:; does not occur in . ; 2) -/67085 9:; 2 , where 5 9:; does not < 5 9?:; 0!=>2 occur in 6 ; 3) , where 5 9:; does not occur in = . Suppose that all of these three types are actually inhabited (otherwise, the falsifying model on the integers is rather easily found). Thus, we have the pairs: 4 2B0?CDCDCD0E-/.803 4 2 -/.@; 0A P < < -/6F; 0G5 9?:; 2H0ICJCDCD0E-/6K Q 085 9:; 2 < 5 - < 5 9?:;R 0L=M 0!=O ; 2N0ICJCDCD0E9:;T S 2 U
V
We claim that, given a totally ordered Abelian group , there exists a model XY W Z [N\ s.t. these pairs simultaneously belong to ] ) iff there exists a V ^ XbY W Z [3 ^ \ _?`a model s.t. the following pairs, not containing c : 4 gBf?hDhDhDfEd/e8u f3 4 g d/e@t fA d/iNv f wyx g jlknmofqprkns
, for all
z
simultaneously belong to ) " . This can be seen as follows. It is evident that any model verifying the old inequalities in {7|~} variables also verifies the new inequalities with one N
variable less. Conversely, suppose that satisfies the new ^ 3 ^ inequalities, and consider the model , where: ^> 7¦ Nrr7¦ , for l ; ª DD « ¡ ¢£¥ ª ¤DD¤£K ¬ §> ¨?©ª ~r M
. ¦ 4
We have to show, for any n®q¯rn°o!±n , that: (i) ²³3µ´¶ ) " ; Å º ÀÁ »µ¼½ ) ·/¸$ ¹8 · Å º ÀÁÂÄ ¹L¾¿ à »µ¼½ ) . In the following, (ii) " ; (iii) " Æ ÇÉÌ ÈËÊDÊDÊDÈqÇ Í we shall use without explicit mention the fact that, if contains just , Ö Õ ÎÏ ÆÑÐÒNÓÔ ÎrÏ ÆÑÐ thenÐ , a consequence of the fact that, in all Abelian -groups, Ï × ÏÙØÑÚÉÛ ÐÒ ÏÙ× Ð ØÑÚ~ÏÙ× Ð Û . Ö ÐÒNÓÔ ÎrÏÜ â Ð Ðàß/á Î/Ö ÏÜ â Ðàßá (i) ÎÏÝÜ â Ðàß .ÊDÊDÊ But ÎÞÐ ÏÜ â , whence . ÎÞÏã é ÎrÏä¥ å åä ê ÎrÏã é Ðàæ ÎrÏçãM è ÊDÊDÊ èã ë Ð Ì Ì (ii) , while . Then Ó ÎrÏã é Ð
ß
ÎrÏãM Ì è
ÊDÊDÊ èã ë Ð ÚÉÎrÏä¥ Ì å
ÊJÊDÊ åä ê Ð
Ò
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. This implies ! & "$#% , i.e. '& ! ()+*,! -*.& "3#% /(0& *1 "3#% 2
(iii) Similar. Our claim in particular, that there exists a 6 ? s.t.Thisthemeans, 4 6 :)settled. @ 7$8%9 , model new pairs, not containing ACBED . Looking at our new inequalities, it is evident simultaneously belong to S S TO ' F ) G H K I J L BED M.R NG0Q OPFM.R NK Q T that iff 6UVWYe X[Z\U.6 VK B,iffD f g X^]0U.6 VW`e _a f g X-bc . Hence, the inequalities belong to d iff
h.6x ijlw k m mnmok jy k i { z | ip`z qr } | ssts-uv ~ Let us take~ stock. We assumed that , and thus any of its conjunctive w have
m m m normal forms , isz unprovable in HG . We have transformed z one of its falsifiable conjuncts into a formula , containing just variables, way that ordered there exists a model awhere Q forisanynottotally < in such true iff there a model < where is not true. But this means exists that is HGunprovable. By inductive hypothesis, therefore, it is falsified in a model of the , whence also , and are falsified in a model whose form first projection is . 3. APPLICATIONS
The algebraic study of substructural logics is interesting in its own right; often, however, it also provides useful syntactical information concerning the substructural calculi themselves. For example, it was through the use of algebraic techniques that Meyer (1973) proved that HR (as well as other calculi) is a conservative extension of its own negation-free fragment. Indeed, algebraic machinery is often used in order to prove conservative extension results (see e.g. Avron 1988; Restall 1994b). In any case, this is not the only possible application for algebraic semantics. We confine ourselves to a couple of examples, relative to the problem of the admissibility of rules in both Hilbert-style and sequent calculi. Meyer and Dunn (1969) proved with the help of algebraic tools that disjunctive syllogism , is an admissible rule in HR - in other words, if and HR HR then we are in a position to conclude that HR . On the other side, Piazza and Castellan (1996) investigated concrete Girard quantales in order to give a characterization of formulae in LLg for which the rules of weakening and
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contraction are admissible, viz. those formulae ( ) is LLg -provable, then also
such that, if is such.
Sources of the chapter. The sources of § 1.2 are essentially Avron (1988) and Gallier (1991). The proof of Proposition 6.17 is directly borrowed from Meyer and Slaney (1989).
Notes 1. Yet Meyer and Slaney, who published an independent proof of the latter result in the same year, should be awarded a chronological priority, since a typescript of their proof was already circulating in the early 1980's. 2. Ursini (1995) develops both an algebraic and a phase semantics for linear logic with exponentials using structures which are not necessarily complete as lattices. 3. A patent abuse of language is being committed here. By saying that S has the single model property, of course, we do not mean that it is enough to consider a single model of a certain kind in order to check whether a formula is S-provable, but that it suffices to focus on all models on a single structure.
Chapter 7 RELATIONAL SEMANTICS
We have traced back the conceptual roots of algebraic semantics to Frege's idea according to which sentences are names for truth values. But there is another standpoint one can assume about the semantic value of sentences. In fact, a true sentence like "Brutus killed Caesar" could have been false if Brutus had not killed Caesar; it is true in the light of what actually happened, but could have been false if human history had been different - in another "possible world", so to speak. In this perspective, it seems natural to view sentences as names not for truth values, but rather for sets of possible worlds. According to this approach, in fact, the meaning of a sentence is given by specifying which states of affairs, courses of events etc. render true. Although this way of thinking originated with Leibniz as early as three centuries ago, the first logician who tried to give it a formal clothing was Carnap (1947). Subsequently, Kanger, Hintikka, and above all Saul Kripke (1959a; 1963) devised a semantics for modal logic based exactly on this idea. Carnap, following a suggestion by Leibniz, had identified necessary (respectively, possible) sentences with sentences true in every possible world (respectively, in at least one possible world); his theory, however, could not properly account for iterated modalities, i.e. for sentences like "It is necessarily possible that ". Kripke overcame this obstacle through the introduction of an accessibility relation between possible worlds. Intuitively, one can interpret such a relation as follows: a world is accessible from if whoever is in is acquainted with the events in ; roughly speaking, if can "see" . Whether a modal formula is true or false at a world depends on the semantic status of its components not only at that world, but also at worlds accessible from it. The
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evaluation of connectives and of modal operators does not change across different logics - what distinguishes modal logics from one another are the conditions imposed upon the accessibility relation. Later, Kripke (1965) extended his possible world semantics - also called relational semantics, because of the role played by accessibility relations - to intuitionistic logic. There, the informal interpretation of possible worlds and accessibility becomes different (see e.g. van Dalen 1986), but the core ideas of the semantics are not altered. For example, if we read " " as "The formula is true at the world ", and " " as "the world is accessible from ", the evaluation clauses for intuitionistic implication, disjunction and conjunction are:
, then not
iff for every , if iff and iff or
or
The accessibility relation , here, is a reflexive and transitive relation. One may wonder, then, whether a variation of the conditions imposed on - in analogy with the story we told about modal logics - could yield a suitable semantics also for our substructural logics. This is quite unlikely, however. On the one side, in fact, the evaluation clauses for disjunction and conjunction satisfy in a natural way the axioms of distributivity, which fail instead in many of our logics; on the other side, the clause for implication satisfies just as naturally such paradoxes of material (and strict) implication as , which fail in relevance logics. These facts hold true independently of any condition imposed on . It seems, then, that in order to get an adequate relational semantics for substructural logics we ought to undergo a deeper reorganization of the general framework of Kripke semantics. We shall examine these problems in reverse order. First, we shall see how to interpret substructural logics where distribution holds; subsequently, we shall face the issue of failure of distribution.
1. SEMANTICS FOR DISTRIBUTIVE LOGICS Not surprisingly, the problem of finding an appropriate Kripke-style semantics for logics without the paradoxes of material implication was first tackled by relevance logicians. Urquhart (1972) was the first to devise an operational semantics for the calculus of relevant implication, using a set of "situations" with the algebraic structure of a semilattice. It soon emerged, however, that Urquhart's models could not be straightforwardly extended to the whole of HR.
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Reasoning along the same lines, but taking as a starting point Kripke's relational approach, Routley and Meyer (1973) found the proper key to access the universe: one has to replace the binary accessibility relation substructural . of Kripke semantics for intuitionistic logic by aternary relation Dunn (1986, p. 200) explains the intuitive meaning of with these words:
In interpreting perhaps the best reading is to say that the combination of the pieces of information and (not necessarily the union) is a piece of information in [...]. Routley himself called the etc., "set-ups" and conceived of them as being something like possible worlds except that they were allowed to be inconsistent and incomplete [...]. On this reading can be regarded as saying that and are compatible according to , or some such thing.
As Dunn suggests, therefore, the "worlds" of Routley-Meyer semantics can be given a traditional "modal" interpretation (see e.g. Routley et al. 1982), according are conceived courses of events where andto which (orthey norof as) bizarre both else neither might hold1 ; still, they are also can be seen, in fact, as amenable to an "informational" reading - the pieces of information which, possibly in combination with other relevant bits of information, may or may not support the truth of a sentence. (This is hardly surprising, if the reader recalls our remarks in Chapter 1 about the informational sequents.) In what follows, we shall reading usingofthesubstructural refer to the rather neutral term of situations. Situations, we said, differ from possible worlds in that they may be inconsistent and/or incomplete. The need for inconsistent situations becomes apparent once you realize that most substructural logics lack the principle of ex absurdo quodlibet, that implication in a "! , and " ! anfor . holds given model just in case implies every situation Thus, we hold) yet fails. need situations where holds (i.e. where $# % , one needs incomplete Since most substructural logics also lack situations as well.
1.1 Routley-Meyer semantics: definitions and results After having introduced Routley-Meyer semantics by means of some informal explanation, let us see how it actually works for a number of distributive substructural calculi, viz. HRW and its extensions HR and HRM. Throughout this section, the letter "S" will range over the set containing these axiomatic calculi, except where otherwise specified.
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Definition 7.1 (some notational conventions). Let be a nonempty set , and containing the designated element 1; moreover, let
. We introduce the following abbreviations:
means that there is an s.t. and ; means that "! . # $ % In the following, we shall sometimes use the symbols " ", " ", " ", "&", & " as metatheoretical abbreviations for "for every", "for at least one", "v", " "not", "and", "or", "implies" (in their classical, truth-functional sense). The sole purpose of such shorthands is to facilitate reading whenever the use of plain English could be detrimental to clarity. Definition 7.2 (frame). A frame is a structure ')(+*-,).0/.012.3)4 type *5*-6 .7/4 .8*-94:4 , where: ; , is a set (the set of situations) containing 1 (the base situation); ; ' satisfies the following conditions regarding the relation 3 : [AR1] < = G [AR3] 3A
;
< < > B?C &
G 3A <@?DBE>C
[AR2] 3 [AR4] 3
< >2? & 3 ><@? <>2? & C = < &
of
3 CF>2?
' satisfies the following conditions regarding the operation 1 : & 3 <@? 1 > 1 [*1] < 121H( < [*2] 3 <>2?
Definition 7.3 (Routley-Meyer valuation). Let 'I(+*-,).0/.012.3)4 be a frame, and let J 1 be a mapping from VAR(£1 ) KL, to M T,F N which respects the constraint OQPQRTSUWVXZY T & V\[ ]_^`O P2RaSbUW] XcY T. A Routley-Meyer valuation O of the language £1 with values in ' maps FOR(£1 ) dLe to M T,F N according to the following clauses:
OQRaSbUVbXcYfO P2RaSbUVgX , for S a variable; OQRih jlk"UVbXcY T iff m@]Un REopV]2n & O RihU]QXcY OQRih q k"UVbXcY T iff m@]Un REopV]2n & OQRihU ]PXcY OQRih s k"UVbXcY T iff t2]Un REop]2nV & O RihU] XZY uQviw-xzy"{|b}c~ T iff u viw{|b}c~ T v u vEy"{|b}c~ uQviw-zy"{|b}c~ T iff u viw{|b}c~ T & u vEy"{|b}c~ uQvw{|b}c~ T iff uQviw{|}c~ F; uQv {W|b}c~ T iff
| ; uQv 0 {W|b}c~ T iff
| .
T ^`O REk"Un2XcY T X ; F ^rO REk"Un2XcY T X ; T & O REk"Un2XcY T X ; T; T;
Throughout this section, we shall use the notation "|zw " as a shorthand for uQviw{|b}c~ T.
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Definition 7.4 (Routley-Meyer model). A Routley-Meyer model for £1 is an ordered pair , where is a frame and is a Routley-Meyer valuation of £1 with values in . Remark 7.1 (on the ternary accessibility relation). Now the reader has some clue about the reason why the introduction of a ternary accessibility relation can block the above-mentioned paradox of material implication . In Kripke semantics for intuitionistic logic, in fact, holds in every possible world, since holds iff, for every s.t. , either or , which is clearly true independently of . But iff, for every s.t. , either or , which has to be, as we have seen. Hence, is true everywhere simply because is true everywhere. On the other side, this is no longer the case in Routley-Meyer semantics: here, it is not at all granted that for an arbitrary , as you can easily infer from the clause for implication. In other words, the principle of identity may fail at some situations, although it always holds at the base situation 1. As we shall see below, however, what really matters for the validity of a formula is how it is evaluated at 1. Thus, though it may fail at some situations in some frames, counts as a valid formula in all of the logics we shall consider.
Remark 7.2 (on the Routley star). In Kripke semantics, the accessibility relation is all you need in order to lay down the evaluation clauses for any connective. Routley-Meyer semantics needs instead an additional tool, especially designed to take care of negation: the operation , usually referred to as the Routley star. Its informal content is clarified by Routley and Meyer (1973, p. 202) in these terms:
Negation, on the other hand, requires [...] the admission of theories that are inconsistent, incomplete, or both [...]. We save nevertheless something like the familiar recursive treatment of negation by distinguishing a strong and a weak way of affirming a sentence in a given set-up2 . The strong way is to assert ; the weak way is to omit the assertion of . This yields for each setup the complementary set-up , where what is strongly affirmed in is weakly affirmed in and vice versa.
"
! !
Dunn (1993b) and Restall (1999) try to buttress the plausibility of the Routley star by introducing a more intuitive evaluation clause for negation. Let be a binary accessibility relation between situations, whose informal meaning is that holds whenever and are compatible. Then is said to hold at whenever fails at all 's compatible with . Remark that it is not
# $" #!% '
%# %
#
&('
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Substructural logics: a primer
sensible to require reflexivity of : an inconsistent situation is evidently incompatible with itself. On the other hand, assume that is symmetric and directed3 ; moreover, suppose that for every there is a greatest (under ) situation compatible with , and call it . Then the clause for negation in Definition 7.3 becomes equivalent to the new clause involving compatibility.
Definition 7.5 (matching). Let S be a Hilbert-style calculus and be a frame. matches S (or, as we shall sometimes say, is S-matching) iff it satisfies the condition which, in Table 7.1 below, appears in the same row as the name of S. We say is S-matching that the Routley-Meyer model whenever is such. Table 7.1. Matching relations between frames and calculi
HRW HR HRM
No additional condition besides AR1-AR4 and *1-*2 [AR5] [AR6] v
Remark 7.3 (an alternative condition for HRM-matching frames). According to Table 7.1, a frame matches HRM iff it satisfies AR6. This condition was suggested in Dunn (1979); the original article by Routley and Meyer (1973) has instead the constraint [*3]
!#"$ !#"% v
&('
FOR(£1 ) is said to be true in the Routley-Meyer Definition 7.6 (truth). + ,
. / 0 model ) * ) 13R 2 ), (in symbols, where = 7 > 1 2 + 5 , 6 4 8 . 9 7 8 . : < . 6 ; 0 * , iff ; it is said to be true in the frame * (in R2 symbols, * 1? ) iff it is true in every Routley-Meyer model ) whose first projection is * . Definition 7.7 (validity). 2A@ FOR(£1 ) is said to be (Routley-Meyer) valid with respect to S (in symbols, 1 RS 2 ) iff it is true in every frame which matches S. Our task is now quite standard: we have to show that the notions of provability in our axiomatic calculi and validity with respect to the same calculi coincide in each case. As usual, we start with the easy half, proving a soundness theorem. This requires two lemmata.
Francesco Paoli
Proposition
227 7.1 (hereditary and for every
.
condition lemma). FOR(£1 ), if
In any and
model , then
Proof. Induction on the complexity of . If is a variable, the thesis of the lemma is contained in Definition 7.3, and this much is enough for the basis. Let us now check a couple of cases of the induction step: ( ). By inductive hypothesis, if and , then . Suppose and , whence it is not the case that . Since , by *2 we obtain . Hence, by induction, it is not the case that , i.e. . ( ). Suppose and , which means & . Assume further that and ; but AR4 entails that , whence . This proves .
! - !"2540%#%* $ $ 1 4 $ & - 325#%#%4 $ $7 26 '()+*),.- /*
: ; "#% 8+9 ; "#% ' , "0 81
< 7
= >
Proposition 7.2 (verification lemma). In any model for every FOR(£1 ), if implies for every . Proof. and 7.1 .
means . By our hypothesis,
&
and , then
. Then, assume , whence in virtue of Proposition
Proposition 7.3 (Routley-Meyer soundness theorem). If R S , then S .
FOR(£1 ) and
Proof. We follow the customary inductive procedure, where the induction is, of course, on the length of the proof of in S. We shall check the axioms F2, F5, F7 and F28' of HRW; the axiom F14 of HR; the axiom F16 of HRM; and the rules R1, R2. The rest is left as an exercise. In what follows, we shall apply Proposition 7.2 with no specific mention. (Ad F2). Suppose that ; we have to show that . Suppose then that:
& ,.#%$ 1#?,@ #%$ 1 & "#% ' +*),.- * "0%* 1 B B 2 2
5 2 2 2 6 A #%$ BCED&F"G%H B C6DJIGKH CEDF"G%H L M/NOMQB BP.R C M+B?M B B M DJIS M B DJB H T LVU(N+U P.R C U!U U DF"S U DJH T (1)
&
We have to prove that, for every . In turn, and (2) (3)
& &
.
, if
and , then mean, respectively: ;
.
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Substructural logics: a primer
, (5) and (6) (1), (2), and in addition also (4) Thus, . Ourassume , final aim isshowing . Now, (4) and (5) entail and whence by AR3 ; that is to say, there is an s.t. & . Since and ! " (by (6)), according to (1) # %$ . On the & is the same as ! . In the light of (2), it other side, by AR2 ' % follows that , i.e. what mattered to us. " " ' * ( + ) $ (Ad F5). Suppose that , i.e. , 0 .
/ / " '132'/ 4 %$ . (1) & and (3) 5 %$ . We must get to 267 4 " . By (2) and assume (2) 7 8 4 4 7 4 " , by (1) we would obtain 294 4 %$ , and *2, . Were it 4 4:6 , whence a contradiction follows. Therefore while us that 297 4*1 "tells . " ;'(<$!>=?@(<! , then (Ad F7). Our aim is proving that, if A "'(<$6=" . However, if " B'(<$!>=?'(C! , then " "'(<$ A "'(< , i.e.: and ,D-.E/F GH- / & - "'1P?7="! . and A "'(Q'(<$! , viz. (Ad F14). Suppose ,D-.0/ - / & - "'1X HR-matching U Y Z T U W Z Y T U W Z AR3 i.e. there is a s.t. and . But Z"\ ^'_ is the TWHUZ , in the, light ] ; same as of AR2. Then, byT(1) and (3), we get now, ZUY Y\%] . use (3) again together with the fact that WA\ ^ to deduce W"\ ^' _N^ on the (Ad F16). We must show that implies S T W` a and Therefore, let assumption a HRM-matching ` \ ^ . Then,thatby AR6,is either W?b7a or `cb7model. a . In both cases, by Proposition a \ ^ . 7.1, \ ^ and d \ ^'_ ] , which means that for every WfeK` , (Ad^ R1). Suppose d A W \ ? W g b ` ` \%] . As d b d by AR1, we immediately get d \%] . if and , then
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(Ad R2). The verification of adjunction is immediate. The converse result is a bit more laborious, since it involves, as usual, the construction of a canonical model. Hereafter, we present first in full detail the completeness proof for HR and HRM, and only afterwards briefly discuss how it should be amended to cope with HRW. Throughout the path leading to the completeness result, therefore, "S" will range over the set HR, HRM . The first concept we need is a generalization of the notion of S-theory:
be a regular S-theory. A (not necessarily -S-theory iff, for FOR(£ ): if 1 . was assumed to be Remark that any -S-theory is an S-theory, since
Definition 7.8 ( -S-theory). Let regular) S-theory is called a and , then
regular, but the converse does not necessarily hold. Another important thing to observe is that Proposition 2.48 can be modified in such a way as to hold for -S-theories. Now we are ready to introduce our notion of canonical frame:
be a formula of £1 which is not Definition 7.9 (canonical frame). Let provable in S. The canonical frame for S w.r.t. is the structure , where: is a designated prime, regular, -consistent S-theory; is the set of all prime -S-theories; holds iff for any FOR(£1 ), and imply ; .
# # # #
"! $ &%(' ), -' $. 4 1/ 02 3+$ 5
)
6
* ) +$
+%
6
Definition 7.10 (canonical Routley-Meyer model). The canonical RoutleyMeyer model for S w.r.t. ( being a nontheorem of S) is , where is the canonical frame for S w.r.t. and, for every formula and every in , T iff .
7 9 :<; 8 9 ?= >A9 @ C
D
8 9
E J 9 + > 9 F C = HD G :
C I+D
B
K
From now on, we shall omit most of the time the superscript " " when referring to items of our canonical model. The canonical accessibility relation has a useful property:
NOD L M D
D ?= L LR= M P =?>
M PQ D NOD L M NTD P >UL V P
be S-theories and be a prime Proposition 7.4 (squeeze lemma). (i) Let holds, then there is a prime S-theory such that S-theory. If . (ii) Let be a prime S-theory and and be S-theories. If , then there are prime S-theories such that , and
N P LM C S M
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Substructural logics: a primer
. (iii) The items (i) and (ii) continue to hold true if "S-theory" is replaced everywhere by " -S-theory", whenever is a regular S-theory. # & & "! $ . We want% to make sure that this set of formulae meets the conditions imposed on in Proposition 2.48.(ii). First, we show that it -. is 'closed under ( & ) ' + & * , / * 0 3 1 2 3 1 2 disjunction. then9 are9 4=< and ' 462 57 Suppose 9 ;-6 4 1? 8 &: 8 &>there 1 that;-61 9 4=1 < and, whence s.t. . By F2, F75-F76, 8 &A1 @&>9 ;-61 9 4=1 < @;-69 4=< , Proof. (i) Let
S
S
S
S
2
2
2
2
2
2
& 1 @&>9 ;-C1 BD-69 4 1 @ 4=< . 8 A 0 7 As is an S-theory and is a prime S-theory, however, we conclude that -E1 BD-/*0 and 4 1 @ 465F7 , whence &A1 @&+*, , as desired. Next, we prove that GIHJK L . Should M NGIHJ , there would be O=NP and Q=RS T s.t. Z MVU WO U QIX ; but M NYG , whereby O U Q is in G too. However, since G[P\S , it would be Q=NS , a contradiction. Summing up, J is closed under disjunction and does not overlap with G ; by Proposition 2.48.(ii), then, there a prime S-theory ]V^ G such that ] HJK L . It remains to show that Z ] P\S is holds. M NP . Since ] HJKVL , Thus, suppose M U O=N ] and T M U O=RJ , whence for arbitrary Q_` , if WaM U OIXbU WQ Uc`DX and QNP , then `NS . For Q KdM and `=K O , we get O=NS , as desired. (ii) By Proposition 2.48.(i), if O=RS , there is a prime, O -consistent Stheory e with SIfge . Now, let h K l Q=ij`W`Re & Q U `NGkX m . h Arguing much in the same way as h above, it is possible to prove that is K L . Thus, by Proposition 2.48.(ii), closed w.r.t. disjunction and that h PnH P H K L there is a prime h ] ^ s.t. ] . Now, let o Ucp=NG and o=N ] . K L , for every formula q either o U qrRG or qr Ne ; Then, since ] H Z s q K p = p N e G ] e. choosing , we get , whence we conclude that e Consequently, ] and are the required prime S-theories. (iii) In (i), let JK l M ijO_QWONP & QtRFS & M U WO U QIXuN m . whereby in virtue of F2, F6-F7, F37, F75-F76, R2, S
2
2
2
2
2
2
S
S
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231
Since the prime S-theory provided by Proposition 2.48.(ii) is a -S theory provided that is such, the proof runs through and we get the desired conclusion. Likewise, under the indicated hypotheses, the and in (ii) are prime -S-theories as well. The first task we must carry through is verifying that the canonical frame is actually a frame, and that the canonical model is actually a Routley-Meyer model. Proposition 7.5. For S-matching frame.
whatsoever, the canonical frame for S w.r.t.
is an
Proof. First and foremost, we must show that is well-defined. Well, is a set and is a ternary relation among -S-theories; what remains to prove, then, is that 1 exists and belongs to and that is a prime -S-theory if so is . However, by Proposition 2.48.(i), any -consistent HRW-theory can be extended to an -consistent, prime HRW-theory; so, in particular, can HR and HRM, and the resulting S-theory will be obviously regular. Is it a 1-Stheory? To be such, it has to be detached - but in virtue of Proposition 2.36.(i), every nonempty HR-theory is, in fact, detached. Now, we want to make sure that is a prime -S-theory if so is .
, and that Suppose that , i.e. that . We must show
. Indeed, assume the contrary: then from that , i.e. that we should conclude (by F41) and thus ,
. against our assumption. Secondly, suppose , i.e. , then by F58 ! , and, by primality of , Were it
!
either or " , a contradiction. Finally, let and, for . Our first assumption entails # the sake of reductio, and thus, by F59, . Our second assumption entails instead and then , a contradiction again. Thus if is a prime -S-theory, $ is also such. The next step consists in proving that satisfies the conditions AR1-AR5 (for HR) and AR1-AR6 (for HRM).
(Ad AR1). We have to show that, if % and % , then . But this is obvious, since is a -S-theory.
)(*+ -,. % 02 30 & % (Ad0 AR2). Suppose , and let /
)(1
that &'% , whence 0 -, . . By F35, 8 7 4 (5 6, (Ad AR3). Suppose , which means that there is an s.t.
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Substructural logics: a primer
(1) & ; (2) & . We must find an (3) (4) So, let
s.t.
& ; &
" !
.
&
#
.
% &' $
$ is, Is it a -S-theory? To prove ( that it (
% , whence there is an s.t. )* & % &+, -.( s.t. . But (
% , /( &01 $ F37, whereby ) (
suppose first that ) and % . We have to find a % , /( &0. (* & is an instance ( of and . Since , we $ can choose . So is closed w.r.t. -provable implications. To show 23 5 s.t. that it 8 is 9 closed w.r.t. adjunction, suppose that there are 46 7 9 ; : % % 4 4<5 5 4 7 5 . . We have to> find a :? s.t. @ !9 ' = : 9
, / : 7 $ 4 4 5 5 4 5 4 5 Since (being an !9
: 9
1 4 4 5 5 instance of F74), and since , it follows that >4 :;9 5 4 :;9 5 . Hence it suffices to take % A4 :; 5 . Does satisfy (3) and (4)? As to (3), suppose and . % % B We to find a s.t. it is
enough clearly 01to choose % ought % , and % / . As to (4), suppose that there is a s.t. and C / % 01 % that . C By F3, , and by (1) . According to (2), then, , and we are home. $ Now, by Proposition 7.4 DE5 FGHIcan be extended to a prime -S-theory still satisfying (3) and (4). So , as desired. (Ad AR4). Suppose that:
(1) (2)
$
H
& ; & .
% & % and & . By F1, however, % Let 0 & ,moreover / % &01 $ & % , whence and thus , DJKH establishing . (Ad AR5). Let and . Since every nonempty HR-theory is detached (by Proposition 2.36.(i)), we have that . L H (Ad AR6). Suppose that & , and that it is not the case either that MNPO or that QNPO . So there are RTU SR V s.t. RX S] V s.t. U W R V Y[ZISR U YM and R V \O . Likewise, there are ]^U
"!# $% $% '&(&*) 3 +-,.+- /&04 ) 3 +-,.1+- 4 )2 F =G-? @ F 5769?8 D.:9H ;<6>F =I :-FK? J @ D 6-? ACIB G>=:-I ?F D(E ? J @ I G ? A I ? D I G A G-? ALHNM O O PRQ Q S [^\W T T-Q Q ULV(WYX>Z[-\ U X-VL_ \]W V `-a b c7d9`pe f hgd-ha fKi aji dka]bmlfKan(o d-an p ` h d q i r bse n `-a bteud-qn b2{{ v || fKw(`>hxf ar }} nyv z-w(`>hxz^a i `^`pa h b { ~ ar d-qn { ~g ~ hh-a ohCi g<`>h~ oaa br { { p ` h a i a n b{
i/e en bm{r {e n { r { a b ~ ~ h-ar ~ an { `>h ~ a i `>h~ -h`>a-h r a~ r a-r a b { { < g > ` h o C h < g > ` h o a r p ` h a i a n ~ b{ n{ r
Francesco Paoli
233
and . Thus, and . By F35, , whence by F2, F40, F41 and thus by our hypothesis. In virtue of F12-F13, then and, applying F16, F46, F75, F76, . Since is prime, however, it should be , a contradiction. Finally, we check the conditions *1 and *2 concerning negation. (Ad *1). , by F4, F40. (Ad *2). Suppose that & , and assume and , i.e. . By F41, hence, should , it would be , a contradiction. Thereby , i.e. . Proposition 7.6. For S-matching model.
whatsoever, the canonical model for S w.r.t.
is an
Proof. Since has already been shown to be an S-matching frame, it suffices to prove that satisfies the appropriate valuation clauses, in compliance with Definition 7.3. We confine ourselves to the clause for implication, leaving the rest as an exercise. The easy part is showing that if , and , then - simply recall the definition of . The tricky part is showing the converse. Suppose that for every , if both (1) and (2) & hold, then . We must prove that . Actually, we prove the contrapositive of such an implication: we assume that and find prime -S-theories such that (2) above holds and . Let ; .
According to these definitions, and . Moreover, let and . Then and thus, by F2, , whence , which means . Thus, (2) holds for . We show that are -S-theories. First, let us prove that they are closed w.r.t. implications in . Suppose first that , i.e. that , and that . By F2, applied twice, it follows , i.e. . Now assume that , i.e. that , and that . By F37, it follows and , i.e. . Finally, and are closed w.r.t. adjunction in virtue of F7, R2. So, we have three -
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Substructural logics: a primer
prime and -consistent. An S-theories - - with , application of Proposition 7.4.(ii)-(iii) yields prime -S-theories s.t. , ! (as and . The completeness of HR and HRW is now an immediate consequence of Proposition 7.6. Proposition 7.7 (Routley-Meyer completeness theorem: Routley and Meyer " $ " # R" , then S . 1973). If FOR(£1 ) and S
"
is not provable in S. Consider the canonical model 0 S w.r.t. , which0 is an S-matching 1 Routley-Meyer ' model in virtue of Proposition 7.6. Clearly does not hold at ,! whence it is & ' not true in the canonical frame and thus it is not valid w.r.t. S. Suppose that % ' Proof. ' . ) * & '( + ,-/ for
Remark 7.4 (on the completeness theorem for various fragments and subsystems of HR). The proof of Proposition 7.7 has been somewhat intricated. Such an intricacy can be considerably lessened when we consider some fragments of HR. For example, in the completeness proof for HRg we need not, of course, worry about primeness, as lattice-theoretical disjunction is simply not in the vocabulary - it suffices, 1 then, to take the set of all theorems of the fragment as our base situation , and HRg -theories (where, of course, closure under adjunction is no longer required) as canonical situations. Therefore, no appeal to the Lindenbaum-style results of Chapter 2 is necessary. On the other side, in the completeness proof for HR+ we must keep prime theories, but a result by Meyer (1976) guarantees that HR+ is indeed such, whereby we are once again enabled to choose the fragment itself as our base situation. When it comes to the subsystems of HR, on the contrary, we are forced to introduce further complications. First, consider HRW. It is easy to see where the proof of Proposition 7.7 breaks down - at the very beginning, indeed. In fact, since 1 HRW does not 1 contain F85, an HRW-theory need not be detached, whence might not be a -S-theory. However, Slaney (1987) showed that in spite of all this it is possible to choose a detached prime theory as the base situation of the canonical frame in the completeness proof for HRW and other systems4 . In the semantics for very weak logics which do not contain even F2 and F41, the truth of a 1 formula in a model does not depend on how it is evaluated at a single situation 1 , but, more generally, on how it is evaluated at a whole set 2 , containing , of regular situations. Canonical models for such logics are 3 constructed by taking S-theories, and not -S-theories, as situations. Routley
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235
et al. (1982) label reduced the models we have presented, because the set of regular situations reduces to the singleton . Not all relevance logics can be characterized through reduced models, although many can (Slaney 1987; Priest and Sylvan 1992; Restall 1993)5 . Remark 7.5 (relational semantics for other distributive logics). Throughout this subsection, we did not discuss some of the axiomatic systems of Chapter 2 that contain the distribution axiom. For example, we did not say a single word about HLuk and HG. There is, however, a semantics for HLuk which is not relational, but is nonetheless akin to Routley-Meyer's, in that it can be seen as a development of the basic ideas of Urquhart's (1972) paper; it can be found in Scott (1974) and Urquhart (1986). There, the set of situations has the algebraic structure of a totally ordered Abelian group. No Routley-Meyer semantics for HG has hitherto been worked out, at least to the best of our knowledge. However, Paoli (200+a) contains a Routley-Meyer semantics for its grouptheoretical fragment. Remark 7.6 (some objections to Routley-Meyer semantics). Routley-Meyer semantics has been variously attacked over the years. Some authors have maintained that it ought to be considered as a purely formal semantics, rather than as a genuinely "philosophical" one. The clause for negation, in particular, has been repeatedly under crossfire. Copeland (1979) believes that the very fact that there are situations where and are both true speaks against counting relevant negation as a negation, and that the meaning of the Routley star is utterly unclear. Read (1988), on the other hand, claims that the adoption of Routley-Meyer semantics commits the relevantist to embracing dialethic ideas (still in virtue of the presence of inconsistent situations), while motivations for relevance and dialetheism are quite independent of each other (in Read's opinion, moreover, relevance is well-motivated, while dialetheism is not). This hurdle can be removed, according to Read, only by abandoning Routley-Meyer semantics with its classical metatheory in favour of a "homophonic semantics" based on a relevant metatheory. In that semantics, is a consequence of iff it is impossible for to be true and false, where that "and" must be read as a group-theoretical conjunction, not as a latticetheoretical one.
1.2 Applications In their 1973 paper on the semantics of entailment, Routley and Meyer point out that their models have at least two immediate applications: they allow
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Substructural logics: a primer
to prove several conservative extensions results concerning HR and its fragments - for virtually every combination of connectives, all theorems of HR in those connectives are derivable from the axioms and rules of HR containing only those connectives - and to obtain a new proof of the admissibility of disjunctive syllogism in HR. The latter result, originally proved - as we hinted in Chapter 6 - by Meyer and Dunn (1969) in an algebraic fashion, has a distinctive philosophical importance in the context of relevance logics. Anderson and Belnap's claim that the implication of R is an adequate formal representation of "real" logical implication has been harshly criticized on various grounds, including the observation that R lacks intuitively plausible modes of inference like the disjunctive syllogism, or "Gamma", as the relevantists use to call it (see e.g. Bennett 1969). Showing that this inference is admissible is therefore important in that it shows that disjunctive syllogism is a perfectly sound inference at least in the "logical" situations. The proof by Routley and Meyer, which we now present in some detail, yields the further advantage of simplifying tremendously the rather convoluted algebraic proof by Meyer and Dunn. Definition 7.11
(normal frame). A normal frame is where the following postulate is satisfied: [AR7]
.
a
frame
&( ' "!# $ % +*, .-/ 01% ) *2$ .-/ 0134 $5*/- 0 6% .*7% $98 8:* ;*, 6% *2 6% 9$ 8:*,8 ! .*, .-9% + *, .-9% $98:*/- +*, .-/ 6% 5$ *<-/8 ! ) !>=! *@?( !A=( . The next lemma explains the name we have given to
Definition 7.12 (normalization of a frame). Let be a frame, and let . The normalization of is the structure , where: holds exactly between the following triples: (i) , if and ; (ii) ; (iii) , iff ; (iv) and , iff ; (v) and , iff ; (vi) iff . for , and .
Proposition 7.8 (normalizations of frames are normal). If matching frame, then is a normal HR-matching frame.
B C<E F D C2K.E/KF1L4M
is an HR-
BG CHFJI E/I
Proof. We devote ourselves to a single passage of the proof, viz. the verification of *2. Suppose ; we have to show that . We argue case by case. If we are done, as is an HR-matching
Francesco Paoli
237
that , whence and , we are entitled to assume ; and similarly for . If , we infer that ,
, i.e. . Other cases are handled in a similar way.
frame. If thus whence
Using the notion of normal frame, we are in a position to narrow down our range of "significant" models. In fact, if we want to disprove a formula in HR, it is sufficient to focus on models whose first projection is a normal frame, as the next theorem shows.
Proposition 7.9 ("normal" completeness theorem). If following are equivalent: (A) HR ; R (B) HR ; (C) is true in every normal HR-matching frame.
"
#
!
"
FOR(£1 ), the
"
$ "% ) ' ( & ,* +.8 # " ;
$
$
Proof. We established the implication (A) (B) in Proposition 7.3, while (B) (C) is obvious. Thus, it remains to prove (C) (A). Suppose then that is not provable in HR. By Proposition 7.7, there is an HR-matching model , with , such that it is not the case that R , , where is the HR . Consider the model normalization of and is so defined on variables:
& / ' ( 01*32* 4*657- 8:9 '/( ; <>9 = 9< ?9 @BADC,EDFHG T iff ? @BADC,EDFHG T, for EJIJK ?9 @BADC6LFHG T iff ? @BADC MNFHG T.
;
;
?9 EPOQ9 A
As far as non-atomic formulae are concerned, the behaviour of is determined by the inductive clauses of Definition 7.3. To count as a valuation, anyway, must satisfy the hereditary condition according to which, if and , then . We distinguish four cases. (i) If , there is nothing to prove. (ii) If , we have , whence as the hereditary condition holds for . Therefore, . (iii) If , then and mean, respectively, and , whence again and . (iv) If , then means , i.e. . Since holds in all models (by AR5 and *2), and thus also in , by heredity , i.e. . What remains to show is that, given FOR(£1 ): (i) iff , for ; (ii) if , then . (i) and (ii) are proved by simultaneous induction on the length of . The most interesting case, and the sole we are going to examine, concerns implication. (i) Suppose , and assume first that . Then there are in s.t. , yet not . It follows ; moreover, by
9 R L ?E S S OQ9 A EGTS GL R MES EULVUTS S OWA % 9 XZYQ[ \P]^_X `^ ^VYa[ bc ^N^ X dfehg iddj j ehg jZeQg9 i m km ihd k dp d pVehg dp qd klmn jZo m % rtshu vVsQu9 wyx zPshw zPs79 w zJxJ{ vVs79 w rfshw | zP}v ~TP z shQ { z h s h s z
6
,
238
Substructural logics: a primer
. induction and not . Thereby, it is not the case that s.t.
Conversely, . Then there are in ) !#"$ &%assume
' , yet not % ( . Four subcases must be considered. If * +-,/.10 , it suffices to reverse the previous argument to get 243 576 89 . If *:<; and ,=.>0 , then ?@576 by the inductive hypothesis of (ii); moreover A ?B3C, , whence 243 576 89 . If *D.>0 and ,:<; , the fact that A ) ?E?EFG; , A ) 3 *!; implies jointly with Proposition 7.1, implies that 2H?EFI5J9 , whereas A 3 *K?EF , which in turn yields 243 576 89 , given the inductive hypothesis. A A ) 3C;G; , whereas by induction Finally, if *&:<,:<; , we get 3L?E?EF from ?M576 and 2H?EFI579 , which is enough for 243N576 89 . This much O establishes (i) for any formula . (ii) is proved similarly. P P does not hold in Q , by what we Now, return to R our nontheorem . Since
Q by have just proved does not hold in either, which is HR-matching R is not Proposition 7.8; thus, there is a normal HR-matching frame where true. S To prove the admissibility of Gamma in HR, we need only one more ingredient, provided by the next lemma.
Q
UW] V Proposition 7.10. Let [ HR-matching frame, and let e -consistent, prime HR-theory.
T X$Y=Z be a model such that T ^`b _Ha \ d R _ c . Then [ ]
is a normal is a regular,
f ]
Proof. By Proposition 7.2, is closed under modus ponens, and in virtue f ] of the definitions it is closed w.r.t. adjunction. By Proposition 7.9, contains all the axioms of HR. Since every HR-theory ] is detached, by f Proposition 2.35.(i) this suffices to establish that is a g regular HRf ] much theory. By definition, is g prime; it remains tog show that it is -consistent. h@i j and h@i j ; by AR7 hEkIi j , i.e. it is not the case that Suppose that hEklknmpoHhGqri j , a contradiction. S Proposition t 7.11 (admissibility of Gamma x in HR: Routley and Meyer L u w t v x 1973). If s HR and s HR , then s HR .
t
uLtwv x
t
s HR and s HR , then by Proposition 7.9 both and uLtwProof. v x areIf true Q in all the normal ] HR-matching frames, and in all models y {BuLtwisv defined based thereupon. That is, if t= Proposition 7.10,] for every x| y ]as. inSince u z such model upt we have that all of these y 's are consistent, belongs to none of them; as they are prime, however, each one
Francesco Paoli
of them must include either or . Therefore, , whence HR by Proposition 7.9.
239 must belong to every such
2. SEMANTICS FOR NONDISTRIBUTIVE LOGICS As we have just seen, the evaluation clauses for disjunction, conjunction and implication in Routley-Meyer semantics - but also in most generalizations of Urquhart's operational semantics for relevant implication (cp. Bull 1987) naturally force the distribution axiom. Routley-Meyer semantics, therefore, is inadequate to model nondistributive logics; yet, many substructural logics are such. As regards the negation-free fragments of these logics, however, as well as logics whose negation has intuitionistic-like properties, the problem can be bypassed: it is enough to introduce a nonstandard clause for disjunction, involving an intersection operation on situations. Along this way, an Urquhartstyle semantics for many substructural logics can be built up (Ono and Komori 1985; Došen 1989b). Once an involutive negation is added, however, the problem of warranting an appropriate interaction between intersection and the Routley star, or other devices necessary to handle such a negation, becomes very thorny. A deeper change of perspective seems thus needed. This shift was brought about in the field of quantum logic, a branch of logic which is concerned with the foundation of physical theories, and where the issue of failure of distribution arose for the first time. The algebraic models of quantum logics, in fact, are possibly nondistributive lattices. Relational semantics for systems of quantum logics were devised in the 1970s by Dishkant (1972), Goldblatt (1974), Dalla Chiara (1977). The conceptual kernel of this approach can be synthesized as follows: Some sets of situations (though not all) are intensions of formulae. The meaning of , i.e. the intension corresponding to it, is not the settheoretical complement of the intension of , but the "orthogonal" of such a set. The appropriate notion of "orthogonality" is formulated with reference to a suitable accessibility relation, and the operation of double orthogonal, mapping the set of situations to , has the algebraic properties of a preclosure. Only the closed sets of possible worlds, i.e. the worlds which coincide with their double orthogonal, may count as intensions of formulae. While the intension of a conjunction is the intersection of the intensions of and the intension of a disjunction is the double orthogonal of the union of the intensions of and - an essential feature for invalidating distribution.
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Substructural logics: a primer
Nothing in this semantical analysis seems, after all, specific to quantum logic. So, why not carry over such a perspective - one might wonder - to the substructural field? In fact, this has been done. The most successful development of the approach at issue is Girard's (1987) phase semantics for linear logic, which can be extended to other substructural logics as well. The next subsections will contain a rapid excursus on the concepts and methods of phase semantics.
2.1 General phase structures In this subsection, we shall discuss one more representation of (some of) the algebraic structures of Chapter 5. This time, we shall examine a general method to build up such structures, and then we shall prove that all the abovementioned structures can be obtained, up to isomorphism, by this construction. Our starting point will be Example 5.1, which we are going to rephrase to better suit our new context. Throughout this subsection and the next, the letter "S" will range over the set containing the Hilbert-style systems listed in Table 7.2 below, while the letter "£" will range over £1 , £2 . Once again, it will be tacitly assumed that each of the Hilbert-style systems referred to in the sequel is formulated in the appropriate language.
[PS1] is an Abelian monoid; . [PS2] is called the set of phases, the set of antiphases . Henceforth, the symbol for the monoid operation " " will be often omitted in favour of simple juxtaposition, while ! " $#%#&# will denote generic phases. Definition 7.13 (phase structure: Girard 1987). A phase structure is a structure of type , where:
6
< ' = )! (+* -" ,."/ '0 ! "/132 > ; (orthogonal) '34 = ! " (+! / ' & "/ 4 > ; (multiplication) '6574 8, '34 2 << ; (sum) '89:4 8, ';< 4 <72 < ; (product) ?A@CBDE? < F7B ; (implication) ?HG BD8IJ?LKMB N << . (join)
Definition 7.14 (operations on a phase structure). We define the following operations on subsets of :
Francesco Paoli Definition 7.15
241
set: ) iff ! " #$ % & ) iff '* iff (-/ . +,+ . 021 is a preorder, i.e. a reflexive and transitive It is easily seen that 3 relation, on . (an inclusion relation in phase structures). Let be a phase structure, and let and . We
453 4 6,6 4 6H6H6H6 CD4 6H6
Proposition 7.12 (the double orthogonal operation is a preclosure). The , is a preclosure on double orthogonal operation, mapping to , i.e.: (i) ; (ii) ; (iii) implies .
3=@5BA 4E5 4 6H6 4E5GF 4 796H8;6 : 5GF 6H6 4 6H6 CJ- I K"LNM : MO 4 6 PQIRMOST< . . Now, let UVO 4 , and (i) MWO Proof. 4 6 . Then U"MOS , establishing the conclusion. (ii) 4 6,6 5G4 6H6H6H6 I O 4 6,6H6H6 and follows from (i). As for the converse inclusion, suppose MWO 4 6 MO 4 6,6H6 hence IRMOS . But this means nothing but that I&O 4 6H.6 .By(iii)(i),We show first; that 45GF F 6 5G4 6 . Let 45GF and I&O F 6 . Then IRMOS , for every M=O&F . Inimplies IHUVOS for particular, therefore, UXO 4 I O 4 6 every 4 6H6 5GF 6H6 . ifSo4$5GF . Y . Arguing in the same way, it is easy to conclude that By a principle of universal algebra, an immediate consequence of g h , i i _ ] ^ ` a d b e c ` D f ` \ Z [ Proposition 7.12 is that, upon defining the tvkdu m n@oBp is a complete latticeas where w qsy r x q y , and l j poset z |sy }~{ |y; H . The sets in \~d are called closed sets of phases, or else facts. There is a further result of the general theory of structures which will turn out useful in the following. We state it below, omitting its proof.
| and let 2
D R , it is V iff X| } = ¢e¡!£T¤ ¥e¦ ¥ N ~ | § e£ R § ¤ Y be a groupoid, let Proposition 7.13. Let be defined as follows: is a symmetric relation on s.t., for every in ; & . Then iff for every , .
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Substructural logics: a primer
Phase structures are all we need to build up our semantics. Here, however, a parallel with modal logic can be enlightening. In modal logic, too, you can do semantics with frames. Still, the correspondence between modal algebras and frames is best explained by resorting to general frames, where we specify in advance which sets of possible worlds are to be viewed as intensions of formulae (see e.g. Fitting 1993). In a like manner, we might wish to work with general phase structures, where we specify in advance which facts can function as intensions of formulae. This discussion leads straight to the following
, closed w .r.t. and containing [P S 3 ] is a designated sub set of
Definition 7.16 (general phase structure). A general phase structure is an ordered quintuple , where is a phase structure and .
We are now ready to enter some correspondences between general phase structures and axiomatic calculi. Definition 7.17 (matching). Let S be a Hilbert-style calculus and be a general phase structure. We say that matches S (or, as we shall sometimes say, is S-matching) iff it satisfies the condition which, in Table 7.2 below, appears in the same row as the name of S.
Table 7.2. Matching relations between general phase structures and calculi
"
HL HL HLB HA HRND HRMND HC HG
#%$& '& $ 9 & '( )+*-,/.102*3.4. $&65 *3$. $& &87 $&65 $& )+*-,/.:,<;:02*3. *1; *3.=; RSR6TUR >+?:>3@BAC@ED 9 FGA2?IHKV JMLN?IHOJ4PQP
No additional condition besides PS1, PS2, PS3 [PS4] [PS5] [PS6] [PS7] [PS7] + [PS8] & [PS9] [PS10] [PS4]+[PS9]+[PS10]
!
)
General phase structures are closely related to *-autonomous lattices. In fact, there is a uniform method to convert S-matching general phase structures into S-matching *-lattices. Let us see what it amounts to.
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Proposition 7.14 (turning general phase structures into *-lattices). Let be an S-matching general phase structure. Then is an S-matching *-lattice. is a *-lattice. Verifying that Proof. First, let us show that & ! "!#%$ is an Abelian monoid is tedious but easy, and such a task is ('-/*. )!+!, 0 $ is a safely left to the reader. As we already remarked, complete lattice with intersection as meet and the double orthogonal of union as & , together with the fact that join, and the closure properties of 132*456 78192 :5 >6 , ensure that ; ? <:<4 = is a sublattice of is an involution since 2@72 holds by definition of ? , such.2BMoreover, C A 5 5 AC2 by the proof of Proposition 7.12.(iii). and implies 2BAC5 is equivalent to Thus, it remains to show that C3 holds, i.e. that D AC2 E>5 . Assume first that D AC2 E>5F78192 J5G >6H 78192I5G >6 , 2 belongs to ? . In Proposition 7.13, let ;K< M = be ;L< M = , and as D . Then our assumption becomes: identify N with and OQPSR with TVUXW Y[Z \]YV^_\]^`a*bH^0ZX`acdbHYQef\9e`g%bheiZX`jckc . ZSlm . Since the antecedent of the resulting conditional In particular, choose gonCj . Conversely, let is trivially satisfied, the consequent is also true, and gonCj ; we must show that a n\9gIjG >c , or else \9gIjG >c na - which is a equivalent ^p`q\9gIjG >c to the, i.e.former, for is a fact and is an involution. Let thus Y_Z \rZX`q\9gIjG scH bH^0ZX`aIc .
Y_Z \]YQef\9e`g%bheiZX`jctbH^VZX`aIc . Zulm ; then, since gonCj , we get ^`a , as required. Let Once again by Proposition 7.13, this becomes
v w xw
Let us now check the various correspondences between the additional constraints on on the one side, and the additional conditions on on the other. For a start, we suppose that PS4 holds and show that . Let ; since , then . the bottom element of If PS5 holds, we must prove that is the top and . While the former claim is trivial, the latter holds for, as it is easy to verify, is included in every fact . Moreover, PS6 implies , i.e. . In fact, in virtue of PS6, and thus . Conversely, our hypothesis readily implies that ; hence for every , and in particular .
{z w ypz w y|y } { ~ z w ~ ? ~ 98C
3 >H 89 H o 9 9
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Substructural logics: a primer
, or equivalently that If PS7 holds, wemust . Now,prove that ' !" $ #$% !& ( . and !) . Then *% !& , whence % !& by PS7. But Let ) . this is enough to conclude that that both PS7 andPS8 we have to show that + ,Assume now , hold; . We already have since we know that - ./0 1PS7 . holds; what we 2 must show is the converse inclusion, i.e. that Thus, 5 3 6 4 % 7 3 8 4 & suppose . In virtue of PS8, 48& , i.e. ) and 90 1 . , which implies & :& (in fact, let 3; !& ; since & is If PS9 holds, it follows that !& ), and that & & :& , i.e. closed operation, & =& w.r.t. < the:& monoid (in fact, & :&>& implies &&< :& , which is the ? & = & < : & same as ). 0@< :& , the converse being It remains to prove that, if PS10 holds, ! #$% ! , easy. A By Proposition 7.13, it is enough to show that, if * 4
8 4 & B C # 8 4 & then . But the means nothing but 2D F E , while the latter amounts D:former E . Thecondition to implication is thus warranted by PS10. G There is also a method for converting S-matching *-lattices into S-matching general phase structures. This construction will serve as the basis for the announced representation of *-lattices. Before examining it, however, we need a further definition. Let Definition sets in XY*-lattices). H IK JL!MON,[email protected] M;R MTSMTUW(downward V be a *-lattice, L and let . The downward X (in symbols, Z[X ) is the set a \^]_\>`X b . The family of all the set of c deH . downward sets of elements of will be denoted by Proposition into general phase structures). Let H rtfK gsKuh!v!iOwyj,x_iQw{7.15 kz|@w~i;}l i (turning nw~} iTo m *-lattices be an S-matching *-lattice. Then p q is an S-matching general phase structure. r is a general phase}structure. Proof. We u@ first show that By Proposition ! v y w _ x w z is a nonempty 5.1.(viii)-(x), is an Abelian monoid. subset of v , because it contains at least . In this structure,
s >|AO
>x>:* ,
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# " . ()# *+-, $ %'& ! # /102,3/54,64879%;:4<7=0&>:?07=*5& /102, %<7=@0A:07=*5& A 0 C B = % = < % 7 = * D * ) 7 % " *D+ $ % $$GF=E *H$ BI$J* $ *LKM0NBO$ *L$ PQ" $ 0 $ *<$TR=0L*HB B $ V *<X8SYZX$ [0 \] W $ *U B V XY ^5_`6_ [)]Ha _ [\Xb W X[\ _ _ [_ [)\]X ]LfMX \e ]V [_ Y X\ _ c _ [)d ]L] fM X W X8_;[e \] ] `k d ] d Xhd Mb inmok e _ ` d W \] d \X_ b e i V _ Y ^1ijlk;d ` ]Li k g[d\X ] e [\XLa [p\ b iFmok [G[]A\fM] X m \[X \] [\qX _ [G]AfMX e \ _ ` c \] m]L\qg XbrX[\ ` inmok b i;e \] kTe \inmoX k [\ _ _ cd d ]LfMd X2s tv u w;xTy w;z {ny | ~ } { | u u { L w w ol
2x) z {ny;8o
z {n ly'
2 LxTMy{nyx)C o
{nxTy y' x)o ooxTy ox ooxTy o
; oo
; xGyny y o o
xTy xGy i.e., by Proposition 5.1.(xvi),
.
Now, let is a fact, i.e. that
, i.e. let
, for some . Thus, let
in . We must show that , which means .
But the last condition is easily seen to be equivalent to
(see below). Choosing , we get and thus , i.e. . What remains to show is that is closed under the operations of orthogonal, sum and intersection ( is obviously a member of ). It clearly suffices to prove that: (i) ; (ii) ; (iii) . (i) , and . If , then ; but if , by transitivity , i.e. . Conversely, if , for we have the required conclusion. (ii) and, using (i), & . Thus, let , and . By Proposition 5.1.(xiii), and thus . It follows that . Conversely, let . We choose and and, in a few moves, we get (iii) By standard lattice arithmetic. We are left with a single task - showing that matches S whenever does. (Ad PS4). Suppose that . Then , which is all that we had to prove. (Ad PS5). Suppose that and are, respectively, the top and the bottom element of . Then of course belongs to . (Ad PS6). Suppose that . Then, by Proposition 5.6.(ii) and C2, , whence . But this means , i.e. . (Ad PS7). Suppose that , whence , and that . Then, by Proposition 5.1.(xiii), . (Ad PS8). Suppose , and . Then .
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and . Since and PS10). Suppose that (Ad PS9 and ; by isotony, , it follows that . Now, let . Moreover, since , it follows that ; we & $ & ! #"% must show that . However, & ,/H .1032547698:';4<* 8<* = 4 >8?<%@ .A+ I ')(*(*#+& ,/H .1032547';4<*=C4<%&@ .A+I ,/H .10D.<EGI F (*B .JK'!(*(*#+ & 4,8 ,L* assume . It follows that *?<Thus, *>*<% @ . that .188<*#we + have to show that 2A8N'O8888
P#+ , we8, have that . Conversely, let . P .>RP?*<* . Multiplying both sides by , and @Q'A@ .SET+U<* E applying C2 and C8, we get , i.e. . Adding on @V'A@ .SE3+5SE
? In other words, are and isomorphic to, respectively, and ? The answer to the first question is affirmative. In fact, we have the following representation theorem for *-lattices: Proposition 7.16 (representation of *-lattices: _ _ `#a . essentially, Girard 1987). If is a *-lattice, then is isomorphic to
_
_ dfeQghjihlkVh9mNh!n1h!oqp . Proof. Let Then ] # W t w u Y v z x j y { y | y x N } y & [ [ y! &N& ~ . Let r Dx s be such that
A!BX . Then
is clearly a bijection, and
O#!B . That
preserves Q! and & w & & is preserved as has Proposition &N& ; 7.15, been proved and ;in .Z Answering the second question is a somewhat tougher issue. After all, we do not even know as yet what it means, for two general phase structures, to be isomorphic to each other. To fill this gap, we agree on the following
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! "$%'# & "$%(# & %*%8),7+.&9-/6&0/7 %*)1&1+2 %3-& :<%345; &0/64 %=:>&?; :<; B )@;@: %*)1&?;2 %=:A& B B # );>7 %*)1&D;E 7C%37; &9B /67 # # # F # GIH 6JKLNMPO5MRQ5MTWS'XYMUW1WSX ML X V L GZH\[]^1^a_ bcd_ b eYbcfl bU gZga'h bc h1h i j^1c<^a k _1_Yg1b g^1a^1h ^`ba c _1_ h b mG
Definition 7.19 (homomorphism of general phase structures). Let and be general phase structures. A homomorphism from to is given by two mappings, , and , such that: ;
if
; , then
iff
; iff
;
.
A homomorphism of general phase structures is called an isomorphism iff both and are bijective. Remark that, since in every general phase structure , it follows that iff whenever and yield a homomorphism from to . Once this definition is granted, it is not hard to devise a counterexample to : Example 7.1. Let , where is the set of natural numbers and the other symbols retain their usual arithmetical meaning. Then , and of course there is no bijection . Remark that is the two-element Boolean algebra. Therefore, the task of specifying which general phase structures are descriptive (to borrow a term from Chagrov and Zakharyashev 1997), i.e. which general phase structures are such that , is not a trivial one. In the general frame semantics for modal logics, there is a nice answer to this question: a general frame is descriptive iff it is differentiated, tight, and compact. In our case, it turns out that we can go an even better way. Now for some preparatory definitions which will help us to carry out this characterization.
m n m GIH
wyz x }~} @z m6{>o| pq r s rtruv
Definition 7.20 (principal polar). Let structure. A fact of the form , for .
@
T Definition
7.21
be a phase , is called a principal polar of
(MacNeille space). An ordered is called a MacNeille space iff: is a phase structure; is a lower semilattice;
quintuple
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for every in there is a greatest (under ) s.t. ; is the set of all the principal polars of . Remark 7.7 (on the terminology of the$ preceding definitions). The !#" term "polar" is borrowed from -group theory. If is %'& % an7 -group and , then the polar of is the -ideal % )8 ( *,+.-/0- %2139: (49; 29 -59 6 < . It is possible to prove that the set of = polars of an -group, partially ordered by inclusion, forms a complete Boolean 84< algebra, and that the principal polars of the form >2? ? form a sublattice of such. On the other hand, nothing in this construction is specifically grouptheoretical, and in fact it can be mimicked for a much wider class of *-lattices. In particular, for Boolean algebras, this implies that every Boolean algebra can be embedded into the complete Boolean algebra of its polars, which is nothing but a different statement of MacNeille's theorem. This justifies the denomination we gave to MacNeille spaces. Our first aim, of course, is showing that all MacNeille spaces are general phase structures. And, in fact, they are: Proposition 7.17. Every MacNeille space is a general phase structure.
@
Proof. It clearly suffices to show that is closed w.r.t. ?ABA C , because G6H ? ?JI @ F and K5L6K ? ? . We principal polars are trivially facts, D ? E @ shall show that is closed w.r.t. ?MONM P , whence closure under sum follows easily. SUT S [ Y Z\Y^] QR T ; on the other hand, (Ad N ). On the one hand, QR ?V?XW S_T ? ? S`T SaT S`T Q N3R ? ? Wab Q ? ? R ? ? c ?V? is
S Y[Zedgfhbid.jkrtjs# b0ja r ] Q & ja s ] Rml jkrtjns f^o pqc l Y,f[o pqc T . }0{ r vxw & { s vyn~ {krt{ns ^ q . Let Thus, suppose u^vxwy and z.{|r{s# r w , {6 s y ; then wy ^ and thus u ^ . Conversely, suppose that {
zgh}iz.{krt{s# }0{ r vxw & { s vym~ {krt{ns ^ q ~u [ q and let wVy\ . For k
, we obtain zg{|rt{s# }0{ r vxw & { s vyn~ {krt{ s q ~u, . It is clearly enough to prove that wy\ implies the antecedent of this conditional. Therefore, assume wy\ , { r vxw and { s vy . Since { r vxw ,
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implies ; since , implies . Thus, and is closed under product. (*),+&$% $% '& ! #" (Ad ). We must show that for every . 589 / 3;= . . / 1#. / 436. 587589 0 / 0 2 2 : * 0 , @ 2 But A3 & 5B9 . Now, let . Then .: / . 5B7589 : / 3C. 587589 0 / 2 & 'D , which is enough for us. . 0 / (Ad ). We must show that for every 0 S in E . But 4 6 I 8 J M K * L P N Q O 8 N T R V S W U M J 8 N ' X 4 Y Z F G [ \ H . However, for every thereN8RT is S a S XY J8R [ \ IC[ J8KJ8R ] \ ] ] ] ] greatestS X s.t.Y , andJ^S'X'Y and , JMN8XY . Now, if ] since weY4Z get and suppose that L_NPOQN8RTSAU`JM,N8X NIa S Sb.J8Conversely, XY J8R ] as ] is . Choosing , we get , whence the greatest such element. c But there is more than this. MacNeille spaces are exactly the descriptive general phase structures.
d
Proposition 7.18 (characterization of descriptive general phase structures). d d is a MacNeille space iff is isomorphic to ef .
d
g
kVlnml,oPl,pql r be a MacNeille space. Proof. Left to right. Let hjia d s s e ftvu wny x w,z{w^|~}Qzwa|~}'~
. Then Let ~= , and . Then and are bijections: in fact, is injective in MacNeille spaces. We must show that: (i) as is antisymmetric H¡ £ ; (ii) ¤ ¥ ¦ ; (iii) ¨~©Qª4«§¨~©Qª4« ; (iv) § ¢ ¬® ¨~© ¬ « ¯ ±³²=´µ~¶´¸· ° ; (v) if ´· , then ¹q·q´ iff º¼iff» ¹ ° ¯ . However, (ii), (iii) and (iv) are trivial, while (i) has already been proved in Proposition 7.17. »¾½ À ¿ Á » ¿ Å Âqà ÂqÄ . Then, if ÄT , it As to (v), Å suppose and , i.e. ¿ º À ¿ Á À  Á Âqà À  Á follows that Ä . Conversely, let ; since (as ¿ ÂÄT ), it follows that Âqà ÀÆÁ . Right to left. To prove our conclusion, it is enough to show that every d general ÇÈjÉËphase ÊÌnÍy̳structure μÌ,Ï ÌÐÌofÑÓthe Ò form e is a MacNeille space. So, let be a *-lattice, and ÈjÉËÊÌÖÕÌØ×ÌÙÏ ÌÙ Ç Ò Ô e be its dual general phase structure. Then: Ü iff ÚÝËÜ . In fact, Ú'ÛËÜ iff Þ*ß àQßMÜVá ÙÏ â ßMÚá ÙÏPã iff - ÚÛË Þ_ßàßäÝ Î Ü â ÚÝ Î ß ã iff ÚÝËÜ . Thus, ÉËÊÌ Û Ò is a lower semilattice Ð Ü å=ænç è Ú Ì Ü é where is simply Ú .
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Substructural logics: a primer
- For every , the greatest (under ) s.t. is . In fact, it is always the case that , i.e. . Moreover, suppose that . Then . . By the previous items, in fact, iff iff - . It follows that is the set of all the principal polars of "!$#%!'&%! ()"* . +
2.2 General phase semantics Up to now there has been a lot of talk about algebra, but stricto sensu not a single word has been spent about logic. Moreover, we carefully avoided to hint at any accessibility relation at all, so much so that the reader might rightly suspect that our inclusion of phase semantics under the heading of relational semantics was simply a mistake. However, it is possible to exploit the results of the preceding subsection to provide a relational semantics for nondistributive substructural logics - even though accessibility relations play no special role in it, being so to speak simply an extra. Definition 7.22 (phase model). A phase model for £ is an ordered pair . - !0/"* , where - . 2!43!657!98:! 1 * is a general phase / ;=< . structure is an (algebraic) valuation of £ with values in > @Band ? ? A C0D"E is said to be S-matching whenever is such.
,
Following Dunn (1993b) and Restall (2000), we are now ready to introduce the accessibility relations we need (cp. also Ursini 1995): 7.23 (accessibility relations? in a phase model). If G E B @ A H 4 C I 6 C 7 J 9 C : K C is a phase model, , we LNM HPwhere O (combination of information) and define the following relations QNM HPR (compatibility):
F
Definition ?
@BA
C0D"E
SUTVXW
holds iff
TVPYZW
;
[TV
holds iff
T\V"]^
.
Recall our Remark 7.2: in Dunn's compatibility semantics for distributive [ substructural logics, the meaning of the accessibility relation is [`_a _ intuitive a b4c _ that holds whenever and are compatible, and is said to hold at just c a _ in case fails at all 's compatible with . Then, the evaluation clause for negation involving the Routley star becomes equivalent to such a clause [ _ a whenever is symmetric and directed and for every there is a greatest _ compatible with . Anyway, it is rather immediate to see that, according to
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Definition 7.23 (and recalling Proposition 7.13),
iff ; ! iff " #$%&'# & ()# .
Therefore, phase semantics and Routley-Meyer semantics handle negation and implication (and, more generally, group-theoretical connectives) essentially in the same way. On the other side, they differ profoundly in their treatments of lattice-theoretical connectives - and unsurprisingly so, for phase semantics is meant to invalidate distribution, whereas Routley-Meyer semantics is not.
* FOR(£) is called true in the phase model + Definition 1 3 ), where , -/.54 " 6"87$"89" 2 0 , 0 (in(truth). - . , " 7.24 / symbols, , iff 7: ; it is called true in the general phase structure , 3 ) iff it is true in every phase model 1 whose first projection(inissymbols, , . P
P
*
3
Definition 7.25 (validity). FOR(£) is said to be (phase-semantically) P valid with respect to S (in symbols, S ) iff it is true in every general phase structure which matches S.
*
1 < - . ; " 0 >@? A A= 1 =3 1 = -/. , = 0 A = CBD(EGF A C BD HJI L RTS LWV RUS Y V
Proposition 7.19 (correspondence between algebraic models and phase models). (i) Let FOR(£), and let be an algebraic model A P iff , where and is such for £. Then that, for every in FOR(£), . (ii) Let FOR(£), and let P A be a phase model for £. Then iff , where . (iii) The correspondences and map (both ways) Smatching models to S-matching models.
1 3 B + M
L M
Proof. (i) Let be an S-matching algebraic model for £. Then, by Proposition 7.15, is an S-matching general phase structure. Moreover, exploiting what we proved on that occasion, , while , and so forth for the other connectives. Hence, is a perfectly well-defined algebraic A valuation with values in . Furthermore, iff iff P iff . (ii) Let be an S-matching phase model for £. Then, by Proposition 7.14, is an S-matching *-lattice. By definition, is trivially a well-defined algebraic valuation with values in . Remark, in passing, that if , then iff . In fact, if and , then ; conversely, if , then and thus . Hence
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Substructural logics: a primer
P
A iff iff iff . (iii) We already got it from the proof of (i) and (ii).
Proposition 7.20 (phase-semantical P FOR(£). Then S iff S .
completeness
Proof. Immediate from Propositions 6.1, 6.5 and 7.19.
theorem).
Let
Remark 7.8 (the phase-semantical competeness theorem: historical overview). The phase-semantical completeness theorem for HLB was proved by Girard (1987). The relationships between phase semantics, the theory of Galois connections and MacNeille's completion theorem for partially ordered sets are emphasized in Avron (1988), Gallier (1991) and Ono (1993)7 . In the last paper it is shown how to extend phase semantics to other substructural calculi. However, neither Ono's contribution nor most of the literature on the subject contain any correspondence theory between conditions on phase structures and conditions on algebraic structures. Exceptions are Lafont (1997), where the condition PS6 is first mentioned with regard to phase structures for affine linear logic8 , and Girard (200+), where the condition PS7 is quickly hinted at. It must be remarked, too, that phase semantics for HA is foreshadowed in Ono (1985), where the evaluation conditions for some connectives are, however, a little bit more contrived.
2.3 The exponentials Can the exponentials be accommodated into phase semantics? Of course they can, as Girard (1987) limpidly shows. Yet, to do so we have to touch up a little our notion of phase structure. The next definition does the appropriate job.
/ . " $ ! % # ' ! ( & ' ! * ) + ! ! , 1 00 1 "5!$24#%683 !'&(7 !')*! , 0 .91 : .) :;<: 5 :
Definition 7.26 (topolinear space: Girard 1987). A topolinear space9 is an ordered sextuple , where: is a general phase structure; ; contains and , is closed under arbitrary intersections and finite sums; moreover, if , then: (i)
It is now possible to prove:
;
(ii)
)9:
.
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Proposition 7.21 (turning topolinear spaces into closure *-lattices). Let a topolinear space. Then & ' ! " '(' #%be$ , where )+*-,!. / 8:9 0213024 < and ,6570 ; is a closure *-lattice.
D ' (' ' & HJEGILF K M R = N O!N > P ? (' > ' >Q @> A!> B C M R N S N ' NTUNVO!N P W%'( ' Q X
Proof. By Proposition 7.14, is a *-lattice. Moreover, since is a complete lattice and thus , is a commutative Girard quantale. By Proposition 6.8, then, is a closure *-lattice. Proposition into topolinear spaces). Let Y2Z []\_^ ` ^b7.22 a7^c(^(turning d!^ e!^fhclosure g be*-lattices a [7\_^ dG^Veig is complete. Then x y Zj[]\_closure ^lk3^nm3^poqc(*-lattice ^porz ^stvz where uwg , where s{t:z uVZ o}|!~3|!\ and |Zft-|u is a topolinear space.
Y Y G y 7
_l3n3p (p vY }hh}h} X
Proof. By Proposition 7.15, is a general phase structure, and by the hypothesis of completeness. Moreover, by Propositions 6.7 and 7.15, satisfies the restrictions imposed on in Definition 7.26 (recall, in fact, that , and so forth). Thus, is a topolinear space.
y
Our topolinear spaces are thus well-behaved structures, and are to complete closure *-lattices as general phase structures are to *-lattices. Consequently, they are exactly what we need for the semantics of exponentials.
£j¤ ¥§¦_¨
£j¤©¥ ª«¥¬+¥®¥j¡j¥¯ ¢ ¨ °
Definition 7.27 (topolinear model). A topolinear model for £3 is an ordered pair , s.t. is a topolinear space and is an (algebraic) valuation of £3 with values in .
¦
It is not difficult, at this point, to rephrase our notions of truth in a model (or in a structure) and validity in such a way as to take into account the new concepts just introduced. In particular, a formula of £3 will be valid w.r.t. HLE iff it is true in every topolinear model (or in every topolinear space). Thus, we get an equivalent version of Proposition 7.19 above (which we omit)
±
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and, finally, the sought completeness theorem for linear logic with exponentials: 7.23 (phase-semantical theorem for HL ). Let Proposition iff is truecompleteness FOR(£ ). Then in every topolinear space. E
HLE
3
2.4. Applications As we hinted in Chapter 3, one of the principal techniques for showing the decidability of a calculus is proving that it has the finite model property. We also recalled that Lafont (1997) established the finite model property, and thus the decidability, of LL and LA, from which the decidability of their Hilbertstyle counterparts easily follows. What we did not say is that he used phase semantics in order to achieve the result. In fact, he constructed a finite phase structure whose phases are sequences of formulae made out just of the subformulae of the formula to be disproved. The resulting canonical model can be employed to prove the decidability theorem. Lafont's theorem can be extended also to the intuitionistic versions of subexponential linear logic and subexponential affine linear logic (Okada and Terui 1999).
Sources of the chapter. The main sources for § 1 are Routley and Meyer (1973), Routley et al. (1982), Dunn (1986), Read (1988), Fuhrmann (200+). For § 2, we borrowed something from Gallier (1991); Restall (2000) was also useful. However, the concept of general phase structure and the overall setting are due to the present author.
Notes 1. Restall (1994a; 1999) suggests an intuitive interpretation for inconsistent and incomplete worlds: he describes the former worlds as ways the world couldn't be (e.g. states of affairs with square circles), and the latter ones as ways that parts of our world could be (e.g. limited portions of either the actual, or of some other possible world). 2. Recall that "set-up" is Routley's and Meyer's wording for what we called "situation". As for the rest, the notation in this quote has been adapted to comply with our standards. 3. I.e. that for every there is a such that . 4. Slaney also showed that HRW is, after all, prime, so that it could even be possible to take = HRW - however, he provides compelling philosophical reasons for not doing so.
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5. For details about non-reduced models, see e.g. Routley et al. (1982), Read (1988), Fuhrmann (200+). 6. The term "phase" is borrowed from physics - not surprisingly, given the fact that this approach originated within the field of quantum logic. In particle mechanics, in fact, a phase is a pure state that a given physical system may assume, whereas a set of states represents a possible "property" of a pure state and then, by extension, a "proposition" which may or may not hold true of such a state. Hence, roughly speaking, a phase represents a "situation" which may or may not satisfy (i.e. belong to) a proposition as so conceived of. 7. Dunn's gaggle theory (see e.g. Dunn 1993a), which is a powerful generalization of relational semantics, underscores the relationships between the theory of Galois connections and relational semantics in general. 8. See Paoli (1998-1999) for a different approach to phase semantics for subexponential affine linear logic. 9. The etymology of the word "topolinear" is simple: "linear" stems from linear logic, while "topo" refers to the fact that the exponentials ? and can be viewed as topological operators, respectively of closure and of interior.
Appendix A BASIC GLOSSARY OF ALGEBRA AND GRAPH THEORY
In this Appendix we collect a number of extremely basic definitions of algebra, general theory of structures and graph theory, which may help the unexperienced student over some technical passages throughout the book. As we remarked in the Preface, we assume that the reader of this volume has attended an introductory course in logic; therefore, he/she should be acquainted with at least the most rudimentary notions of naive set theory, such as the Boolean operations, cartesian product, the concepts of relation, function, and so on. To this last notion, however, we devoted a specific item in the glossary in view of its primary importance for any kind of mathematical investigation. Of course, this glossary is by no means intended as a complete introduction to the subject. It is only meant to be a "first aid" for whomever ignores the meanings of some technical terms used in the book but not explicitly defined therein, and thus it contains nothing that exceeds this set of notions. For a far more systematic treatment of the topics at issue, we refer the reader to the appropriate handbooks of universal algebra (e.g. McKenzie et al. 1987), lattice theory (e.g. Birkhoff 1940), and graph theory (e.g. Diestel 1997). Due to an obvious lack of space, we had to skip each and every kind of exemplification, but the mentioned textbooks contain plenty of examples which will undoubtedly help the reader to understand and assimilate more thoroughly these notions. In this glossary, an arrow of the form Item after a given term or in the body of a definition means that such a concept is either defined under the entry of the glossary referred to by the arrow, or involves notions and/or notations that are explicitly introduced under the entry pointed by the arrow.
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Substructural logics: a primer
Absolutely freealgebra Free algebra Acyclic graph Graph First order structure Algebra Atom Partially ordered set First order structure Basic operation Basic relation First order structure Bijection Function Bijective function Function Boolean algebra Lattice Bottom element Partially ordered set Branch Tree Carrier First order structure Chain set Partially ordered Codomain Function Comparable elements Partially ordered set Complement Lattice Complete poset Partially ordered set Congruence Equivalence Connected Graph graph Cycle Graph
DIRECT ( PRODUCT. Let !" % ) ###" % * ( $ & ' be similar algebras. The direct product of + is the structure 7 7 < 7 7 < , - + . / 0-21!34 : -24 : 355534 ; -24 ; 6 , 8 8 9 9
and and
=?> 9 where for every , [ ] [ ]^ [ ] [ ]^ @ \ AB@D F C 2 E ! G J H L I M K O H N N P H 2 E G J H Q I M K T R U S 2 E V @ C W G X H N N N J H L G Y R J H D @ ` _ _ _ a a d RFK . bc d . \ \ \ CIZ_ HXN N N HJI dcb d e f e g The algebras and are the factors of . The previous definition can be generalized so as to cover the case of an arbitrary number of factors. In h f i k k lXm be a nonempty indexed family of algebras of type j . The fact, let n o k k lXm is the algebra: direct product over | | kt p k lXmr q k uwv p k lXms x p k lXm y } xzzzx p k lXm y ~ { , where:
Xs
X ; is the cartesian product over
Francesco Paoli for every
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,
!". #$ !&% / (')$ *. + , / %- ! 0132 , the 0 -th projection 6) 4 7 : <>=? 9; If 8 : 5 of 7 : <>=? 8 : is simply 8 9 . Disjoint graphs @ A Graph B Distributive lattice @ A LatticeB Domain @ A Function B Edge @ A Graph B Embedding @ A Homomorphism B Equation @ A LanguageB EQUIVALENCE. A relation C abides by the following constraints: [E1, reflexivity] E [E2, symmetry] E [E3, transitivity] E
on a set
D
is an equivalence on
C E ; C FHGIF C E ; C F &F C K J GLE C J
D
iff it
.
Y \ R VWVWVWR T Y ] X MONQP S D R TU Z [ be an algebra. An equivalencem on D is lm bia kn Mm ^ _ ` , if acj bdafk j e gWgWgWe ah iff for every , then noqpsr$j t uWuWuWt rwn m v z ox n psr$ m y y k j t uWuWuWt rwkn v .
M M|{q}f~ U WWW and is a congruence on If is the algebra to construct the quotient algebra modulo , i.e. MS, it is possible Q « O O WWW Q « , where: ¬ ¬ ?KK ¢¡¤ ¥¦ , where ¢¡, £ 3 ¨§©£ §©Ã ¥ª & § i ; à ® for every ¯°±,, ²Ä È ³K´Oµ ¶ ·¹ÅǸ3Æ º »W»W»Wº¼¶ ·½Ã È ¸¦Æ ¾À¿Á¶ ²qÄ È µs·$Å º »W»W»Wº ·wÃ È ¾Â¸ Æ . ¶ÊÉ Direct product ¸ Factor Now, let m a congruence on
M|ËqÌÍSÎÀÏ3ÎÀÐ¤Ñ (of a lattice). ÒFILTER, ÓiÔ½Õ ÍIDEAL Ô M Let let . is a filter of iff: ÎØ×ÚÙ ÔÜÛ Ö Ï©×©Ù Ô [F1] Ö Ù Ô Î Ö3Ý × Û ×©Ù Ô ; Ö [F2] . Ô M Dually, is an ideal of iff: ÎØ×ÚÙ ÔÜÛ Ö Ð©×©Ù Ô [I1] Ö Ù Ô Î × ÝÖ Û ×©Ù Ô ; Ö [I2] .
be a lattice, and
.
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Substructural logics: a primer
A filter (ideal) of is proper iff . A proper filter (ideal) of is maximal iff there is no proper filter (ideal) s.t. . If , the smallest filter (ideal) containing is the principal filter (ideal) generated by , and is denoted by (respectively ). If is the singleton , braces are usually omitted. In lattices, it is possible to prove that, for every ,
&
!" ; #$% !" .
& Lattice FIRST ORDER')STRUCTURE. a (+*,*.P -> /1020202/3A-5P ? first 46/7Q order *98:> /102structure 02Q 02/;8<@ 4=4 - or,is ansimply, structure - of type ordered ACB9DEBF -tuple (G*"HI/3JK> R /1020202/3JL? S /3M > R /1020202/3M @ T 4 (A / DN.O ), where: UWV is a nonempty set, called the carrier of the structure; U=X^ Y[Z\Z2Z2Y3X ac_ b"arednonegative ]:^ Y1Z2Z2Z2Y;] ` are positive integers; U forj any , egf h i is integers and k l l prq to l , if k)o m.n a o -ary operation on , i.e. a function from l a designated element of , otherwise. s for any tcu"vx, wzy { q is a | { -ary relation on } , i.e. a subset of } y q . In other words, a structure is a set with finitely many (possibly zero) operations and finitely many (possibly zero) relations defined on it. A structure ~=~,~ | [\22 |
, , is a relative; with no operation, which has type ~,~. 122235 67~,, , is a structure with no relation, which has type an algebra. However, when no danger of confusion is impending, we commit an abuse of notation and denote the above types, respectively, by ~ | 1\22 |
and ~. 122235 . Remark that a type of a structure is sometimes called a similarity type. Structures of the same type are said to be q q { w y { similar. The 's are the basic operations of , the 's the basic relations of . ª ª "I3K 12223L 3 12223 and If ¡ ¥ § [¢\¢2¢23 ¬ ¥ ¨ 3£ )¦ § ®"ª 1¢2¯ ¢2¢23£ ¦ © ª ¤ are similar structures, « is a substructure µ ²q ²q´ ° each¸ ¹ q ± ² ³ q ´ ofis theiffrestrictionand: to of ± ³ (i.e., ± ³ yields the same output as ¶ when applied to elements¹ of · ¸ q ); Ã Ä Äq º whenever ÅÇÊ Ë q Æ.Ì È1É2É2É2ÈÀÆ theÊ q iffordered Å Ê Ë q Í Æ.Ì È1» É2É2-tuple É2ÈÀÆ Ê q . ¼"½.¾1¿2¿2¿2¾À½Âq Á belongs to · , it is Finitary tree Tree Finitely meet-irreducible element
Francesco Paoli
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In particular, a substructure of an algebra is a subalgebra of such.
! + , . %)))% ' + - / * " $ # & " % ( ' 4 2 3 0 57698;:<0 2 =&6>? : ? 2 1 5 @BADCFEHGJIKADCFE CMLON P FREE
ALGEBRA. Let be a class of algebras of type , and let be an algebra of the same type. Moreover, let . is free for over the set X of generators iff: (i) the smallest subset of containing s.t. is a and for every function subalgebra of is itself; (ii) for every , there is a homomorphism that extends (i.e. such that for ). An algebra which is free for the class of all algebras of a given type is called absolutely free.
IMS QJTUR C Q CC I Q IWA Q EYX R I Q I R Q R R I
Q
R
FUNCTION. Although the concept can be given a precise set-theoretical definition, a (total) function (or mapping) from the set to the set (in symbols, ) can be intuitively described as a correspondence which assigns to every element of a unique element of . If is the element assigned to by , it is customary to write , and is called the image of . The set is the domain, and the set the codomain, of . The set of all the -images of members of is the range of . A function from to is an injection (or an injective function, or a into ) iff whenever function from it is also . On the other side, it is a surjection (or a surjective function, or a function from onto ) iff the codomain of equals its range. Finally, is a bijection (or a bijective function, or a 1-1 correspondence between and ) iff it is both an injection and a surjection.
T\ \ T
V V R VV IKADCFEHR G I Q I I IKADCZEWGJIKA V E C[G V Q QI R
] ] ^ G`_$acb dfe _ g e ^ d ^h ^ ^ ^ Ga _$^ acb dfe ^ G`_$acb dfe ^ h G_$aihkb djh e aml[anh GJo dpC s l[dqh CGrt o ^ G_$acb dfe z y z y aiG C{ buuubvC | dwG _C${nbvC} erbx_C$}`bvC~ eJbuuubx_C |}jbvCj| e
$c
fK v& MM
Kv& M $ Function symbol Language Generator Free algebra
GRAPH. A graph is a relative of type . The members of are the nodes of , while the members of are called the edges of . If is a graph, then any substructure of is said to be a subgraph of . Two graphs and are disjoint iff and . A path from to is a nonempty graph , where and . The length of a path is the number of elements in , and the path passes through (through ) iff (iff ). Let , and let
262
Substructural logics: a primer
a path from ! to ! . Then the graph " obtained by adding #be$&(* % $ ) ' to the edges of is a cycle. An acyclic graph is a graph which contains no cycle as a subgraph. A connected graph is a 6 nonempty graph + #,-%/.0' s.t., if $1% 2435, , there exists a subgraph of which is a $ 2 path from to . An acyclic connected graph is a tree. # %/7%/85' where graph is an ordered triple #,-%/.0a ' labelled 7 8 + Finally, is a graph, is a nonempty set (the set of labels) and is a , 7 7 function from to . Throughout this volume, the set of labels is usually a set either of formulae, or of sequents. Greatest lower bound Groupoid Group
9;:
<
9;:
Partially ordered set
<
= + #>?%A@'
#B?'
GROUPOID. A groupoid is an algebra of type . In an arbitrary groupoid, we do not make any assumption about the basic operation. However, consider the following equations:
$C@EDF2@HGJI + DF$C@K2 IL@HG ; $M@K2 + 2@K$ ; $M@N$ + $ .
[G1, associativity] [G2, commutativity] [G3, idempotency]
A groupoid satisfying G1 is a semigroup. If it satisfies G2, it is Abelian. An Abelian semigroup which satisfies G3 is a semilattice. An algebra of type s.t. is a semigroup and
= + #>?%A@E%/OC'
#BP%RQ?'
#>?%A@'
9 G4, existence of a neutral element< $M@HO + ON@K$ + $ = #>?%A@E%TS%/OC' is satisfied is a monoid. Finally, an algebra + #B % UV%RQ?' s.t. #>?%A@E%/OC' is a monoid and 9 G5, existence of inverses< $C@S$ + SW$C@K$ + O
of type
holds is a group. The classes of (Abelian) semigroups, semilattices, (Abelian) monoids and (Abelian) groups are varieties in the appropriate signatures Variety , and as such they are closed under subalgebras. A subalgebra of a monoid is called a submonoid of such; moreover, if is a monoid and is closed under the basic operation , is called a subsemigroup of .
=
= + #>?%A@E%/OC' @ #Y4%A@'
9X:
<
Y
Francesco Paoli
Language Holding Homomorphic image
263
Homomorphism
HOMOMORPHISM. Let and ! + / . """ ! + 0 . # $&%')(* , be similar algebras. A function is a 1 2 354 6 homomorphism just in case for every , P T >A@CBEDFDDBG@IH :LKM>N@ toHBDDDBOKM>N@I 798 =? P T ; =J P T HGH . R R Q Q 1 V U U iff V is a function W is a homomorphism of X into, respectively onto,Y from into, respectively into is also called Y V onto, . A homomorphism of Y V an embedding Y V of into . An isomorphism between and is Y an embedding V of onto . Whenever there exists an isomorphism between and , the Y Z mentioned algebras are said to be isomorphic . If is a homomorphism from V onto , the latter algebra is a homomorphic image of the former. Ideal []\ Filter, ideal ^ Image []\ Function ^ Immediate predecessor [\ Tree^ Immediate successor [\ Tree^ Incomparable elements [\ Partially ordered set ^ Inf [\ Partially ordered set ^ Injection [\ Function ^ Injective function [\ Function ^ Interpretation [\ Language^ Involutive lattice [\ Lattice^ Isomorphic algebras [\ Homomorphism ^ Isomorphism [\ Homomorphism ^ Join [\ Lattice^ _ _ -group [\ -groupoid ^
Y `a W bdceb
n p s c t r
n u p v c r o o o n rvcwp ; l for every o in , implies and G b q p b r W e c N x e p y h { r z x w c | p M z } h x v c { r z xNpehyr{z~c o ` ` o o o l for every o in , and xNpuc o zMhgxLrvc o z Y . `a W bdceb
264
Substructural logics: a primer
is an ! -groupoid and of "type# is a group. ! -monoid $ % ! -groupoid& ! -semigroup $ % ! -groupoid& $ % & Tree Label $ % Tree& Labelled tree ' (). +*,*,*,(-/ is a similarity type of some algebra If 0 , LANGUAGE. we associate to it a language, containing a nonempty set 1 (whose 34257 3+6,6,6 ) and9 B a function members are called variables and are denoted by 2 3425.8 > 0 A 9 < : ; = 9 >@ @ ? symbol for each . If has type , the interpretation ? of ? in C is the operation D E F @ . GIHKJL)R MONPN,N,ML-S Q be a type with its associated language. The Now, let G T UWVXTY s.t.: set of terms of type over is the smallest set Z []\I^_WZ `X\b a ; cedgf h i @ j _WZ `X\a ; [ for every s.t.n k oIthepfunction symbol l k q r + r , s , s , s r tvuWZ wXxy , then the string } m for~ z | @ every s.t. if } { @ } +P,,
} @ vWZ X . OP, denote arbitrary terms; the notation 5 +,,,4 The letters }g emphasizes the fact that } contains at most the variables 5 + , , ,4 . An ¡ ¤v¦Z §©¨ª , is called an equation of expression of« the form } £ ¡ , where ¢¥ G type ¯over °5±+²,²,²,±4.°· ³´vµWZ ¯ « ³ and is an algebra of type ® , the term function If ¬ ¶ 0 ¸ ¸¹"º¼»¾· ½W» defined as follows: on ¸ corresponding toÀ¼Á is the¸mapping ¹  à + Ä , Å , Å , Å 4 Ä ¿ if is the variable , then ÐÉ @ Â¥ÂÃgÎ5Ê ÆÄOÅPÇÈÅ,Å,Ä4à ÎË ;ÆÄ+Å,Å,Å,Ä ÂÎ5Ä+Å,Å,Å,Ä4ÎÏÆ4Æ Í Ñ Ñ Ì ifB ÓÕÔ} Ö+×,×,×,has BÝ ÓÔÝ Ö×,×,×,Ö4Ë ÔgÞ ØÉWÖ+×,×,×,É Ö Ñ ßB @ ÓÔÊ Ý Ö+×,×,×,ÖÜÔgÞ Ð Ø4@ Ø É Ê , then Ò Ý Ö4ÔgÞ ØÙgtheÚß à @ ÓÛ Ñform áãâgäåB ìîáæ ñ íB áçí ò è+é,é,é,è4çgï êè+é,é,é,è ñ ðB @ áçí è+é,é,é,è4çëï ê4ê4êó . ÷,÷, ö4ÿ õû ,ÿø4þ ù ñ ôõ5 ú ö+÷,÷,÷,öÜõû ø iff satisfies equation ôÕõ5ú ö+÷,þ þ ÿ say,ÿ4þ that ü B ýþ ÿ Pÿ4þ thefor every B} ýÕWe choice of 0 0 in . Instead of " satisfies " we sometimes say that holds in . group is an algebra s.t.
The notion of satisfaction can be extended also to more complex formulae, built out of equational atoms by means of classical connectives and quantifiers. We do not spell out in detail the clauses for these formulae, since their behaviour is as expected. Throughout the text, we permit ourselves a slight notational abuse: we use the letters to refer ambiguously both to the variables of the language associated to a given type and to generic elements denoted by them.
Francesco Paoli
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of type ; [L1, associativity] !" #" ; [L2, commutativity] ; [L3, idempotency] ; $ . [L4, absorption] A lattice algebra, but presented as a poset , ) can - ( also % '&(is thuswherean each ) *+be) has pair both a sup and an 0 . / 1 inf Partially ordered presentations are equivalent can be 23%4set56 .7 That 6 8these 9 satisfies seen as follows. If L1-L4, then the relations 3 7 : ;=? < iff : 3 : 8 < : ;>@ < iff < : < A is any of A ? , A @ , it is areV partial orderings and,V where W W BDCFEH F G ) I J " K G L J O M N P F G ) I J " K G Q J for any GFI)J RTS . Conversely, U U and 2 % X Y Z ' [ ( \ ] in a poset o and infs the operations o fFg)` p sups ^_ `#acbDdFeH iOk l n binary n fFg)` p and fh `!ajwith satisfy L1-L4 and f m` holds ^ q # r " s t t u ! r " s r iff iff . 2 svwx'y(z is a lattice In the light of the above discussion, if o p { F | } (presented as a poset), o tFx)r p is often written tq r and is called the meet of t and r ; similarly, ~tFx)r is often written tu r and is called the join of ^ and . More generally, if
, FF - whenever it exists - is often written and called the meet of ; O -2 again, whenever it exists - is often written and called the join of . If is presented 2 as an algebra , is called the induced ordering of . We freely use w.r.t. such an ordering all the terminology of partially ordered sets Partially ordered sets . LATTICE. A lattice is an algebra which satisfies the following equations:
¡¢£¤¥ ¦¡ §¢¥ £ ¡ ¤¥ £ ¡¢ ¤¥ ¦¡£ ¢ ¥ ¡£¤¥ .; 2 A distributive lattice ¦¨©ª ª £« with top element ¬ and bottom is a Boolean algebra iff for every ®+© there is a ¢ s.t. element ¢!¦ and £ ¢!¦"¬ . Such an ¢ can be proved to be unique, and is ¯° ). Boolean algebras termed the complement of (in symbols, can be 2 % ¦ ¨ ©
ª
ª £ ± ª ¯ ) ª ² ¬ ª « equivalently presented as algebras type ¨´³ª³ªOµ0ª¶ª¶« , (redundantly) defined by the conditions L1-L5of plus: A lattice is distributive iff it satisfies either one of the equivalent conditions: [L5, distributivity]
266
Substructural logics: a primer
; ; ; ; . "!#$%$ $ '& of type !)(*$,+ $,+& s.t. !An #$ algebra $ '& is a lattice and satisfies L8 and L9 is an involutive lattice. [L6] [L7] [L8] [L9]
L9 can be equivalently replaced by the condition [L9']
-/.0 -1
.
Throughout the text, we generally take up L9' instead of L9 in the definition of involutive lattice. Finally, an element of a lattice is meet-irreducible iff, for every , whenever it is for some ; it is finitely meet-irreducible if the same happens for every finite .
"!#$ $ '& 5 2
23 #
9;:
<
Leaf Tree Least upper bound Partially ordered set Length Graph, Tree Lower bound Partially ordered set Lower semilattice Semilattice Mapping Function Maximal element Partially ordered set Maximal filter, ideal Filter, ideal Meet Lattice Meet-irreducible Lattice Minimal element Partially ordered set Monoid Groupoid Node Graph 1-1 correspondence Function
9;:
9;:
9*:
9*:
9*:
9*:
9*:
<
*9 : < 9*: 9*: < 9*: 9*: < < 9*:
4 2
<
<
<
<
<
<
678
<
<
="> 8?A@B
PARTIALLY ORDERED SET. A graph is a partially ordered set, or briefly a poset, if its basic relation is a partial ordering, i.e. if satisfies the following formulae:
C @ C; C @D & D @ C EFC = D ; @ D D
@ G C EFC @HG . & D 8 @D Two members C and of are comparable if either C [P1, reflexivity] [P2, antisymmetry] [P3, transitivity]
incomparable otherwise. If
D @ C
or , and contains no incomparable members, it is a
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totally ordered set, or a chain. A totally ordered substructure of a poset is called a subchain of . A poset can contain at most one element such that, for any , it is . If such a exists, it is called the bottom element of . Likewise, is the top element of iff for any (again, such an element is unique whenever it exists). On the other side, is minimal iff, for no , it is ; dually, is maximal iff, for no , it is (where means, of course, that and ). If contains a top (respectively, a bottom) element, such an element is maximal (respectively, minimal), but the converse statements need not hold true. If contains a bottom element , then is an atom of iff and there is no s.t. . Let be a poset and let . Then is an upper bound of iff for every . Dually, is a lower bound of iff for every . For every , there is at most one upper bound of s.t. for every upper bound of ; if such an exists, it is said to be the least upper bound, or the sup, of , and is denoted by Dually, for every , there is at most one lower bound of s.t. for every lower bound of ; if such an exists, it is said to be the greatest lower bound, or the inf, of , and is denoted by (The subscript " ", however, is omitted as often as possible.) is complete iff every has both a sup and an inf. A fundamental result in the theory of partially ordered sets where most of the previously defined notions occur is Zorn's Lemma: if is a poset and every subchain of has an upper bound in , then contains a maximal element.
!#"%$& ')( + ,. + */0. / + . + 1&2435&+ 6)7 0 8 9* -: 0 ; + ,89-: 0 ; < = - = >@?
>@? >@?
A
A B> ?
Passing through Graph Path Graph, Tree Poset Partially ordered set Principal filter, ideal Filter, ideal Projection Direct product Proper filter, ideal Filter, ideal Quotient algebra Equivalence Range Function Relative First order structure Root Tree Satisfaction Language Semigroup Groupoid
>B?
>B?
>B?
>B?
>B? >B?
A
>B?
>B?
A
A
A
A
A
A
A
A
A
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Substructural logics: a primer
SEMILATTICE. We have seen Groupoid that a semilattice is an algebra of type which satisfies the following equations:
[G1, associativity] ;
! [G2, commutativity] ;
[G3, idempotency] . Like lattices, however, semilattices can be presented relatives, too. An " as #$ upper% (lower) semilattice, in fact, is a poset where each & pair '(*) has a sup (has an inf). That these two presentations are equivalent +,"- . (/0 can be seen as follows. If is a semilattice, then the relations 1 2 1 3 and defined by: '547= 6 iff 8:98!; 6 854<3 6 iff 6!98!; 6
8>?6A@CB , are partial orderings and, for every S T S T 8D; 6!9FEGH P Q 8>?6F 9 IKJ L P R 8>?6 . Thus, M B >4O = N is a lower UWV XY [ Z semilattice and is an upper semilattice. S X T Conversely, in an upper semilattice the operation \D] ^!_F`Ka b c \ ^ Y ^ holds satisfies G1, G2, G3, and \ S X iff T ^!_\!] ^ . Dually, in a lower Y c ^ semilattice the operation \D] ^!_edgfhi\ ^ satisfies G1, G2, G3, and \ holds iff \j_\!] ^ . S T S T c \ X ^ (`Ka b c \ X ^ ) is often called the In lower (upper) semilattices, dfhi meet (join) of \ and ^ and is denoted by \ kA^l(\ mA^ ). Signature npo Varietyq Similar algebras npo First order structureq Similarity type npo First order structureq Structure npo First order structureq Subalgebra no First order structureq Subchain npo Partially ordered set q Subdirectly irreducible algebra no Subdirect product q Subdirectly representable algebra no Subdirect product q
~ vK{|{|{|vyx ~ } +r"sutwvyxz SUBDIRECT PRODUCT. An algebra is a +!
subdirect product of an indexed family of algebras iff there exists an pF injection s.t. (the composition of and the -th C projection of a homomorphism from onto for every . +!) is ¡¢ The elements of are the factors of the subdirect product, and the
Francesco Paoli
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function is sometimes called a subdirect embedding of into . An algebra is subdirectly representable iff it is isomorphic to a subdirect product of an appropriate family of algebras. On the other side, it is subdirectly irreducible iff, for any subdirect embedding , there is a s.t. is an isomorphism between and . Intuitively, therefore, an algebra is subdirectly irreducible whenever it cannot be subdirectly decomposed into smaller algebras. An important result by Birkhoff (the Subdirect Representation Theorem) says that every algebra is subdirectly representable with subdirectly irreducible factors (and that whenever belongs to a variety , so do the factors as well).
Subgraph Graph Submonoid Groupoid Subsemigroup Groupoid Substructure First order structure Subtree Tree Subvariety Variety Sup Partially ordered set Surjection Function Function Surjective function Term Language Term function Language Top element Partially ordered set Partially ordered set Totally ordered set
!#"%$'&)(+*-, /0* . 2
TREE. We have seen [ Graph] that a tree is an acyclic and connected graph. Equivalently, a tree is a poset such that, for every node , the set & is linearly ordered. If has a bottom element, such an element is called the root of the tree. In this book, we uniquely consider trees with a root. Likewise, any maximal element of is a leaf of . A substructure of a tree is a subtree of such. In this new context, a path in (from to ) can be defined as a linearly ordered subset (where for every in ). A branch of is a maximal path in ; the length of is the number of nodes in its longest branch. The node is an immediate predecessor of (and is an immediate successor of ) iff and, for every node , if then or . A tree is finitary iff every node has at most finitely many immediate successors. The most important result in the theory of trees - König's Lemma - says that any finitary tree contains infinitely many elements iff it contains at least an infinite branch. Finally, a labelled tree is a labelled graph where is a tree.
1/ / &
. & 3
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3
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3
3#4%5'6)7+8-9 : 6 ? : ' 8 A @ B 8 ; <>= 3 3 : : :?8B;
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Substructural logics: a primer
First Type order structure Upper bound Partially ordered set Semilattice Upper semilattice Variable Language VARIETY. A class of algebras of type is a variety iff there is a set of equations of type over some s.t. contains exactly the algebras satisfying each member of . Sometimes, if is a variety of algebras of type , we say that is a variety in the ! . However, when talking about signatures, we are signature usually quite loose as regards the distinction between function symbols and their allowing ourselves some abuses of language. " # interpretations, " " A subclass of a variety which is itself a variety is called a subvariety of . A celebrated theorem by Birkhoff (affectionately called the HSP Theorem) says that a class of similar algebras is a variety just in case it is closed under subalgebras, homomorphic images, and direct products.
Appendix B OTHER SUBSTRUCTURAL LOGICS
Although the range of substructural logics that we have encountered throughout this book is in itself quite broad, we did rarely venture outside the boundaries of subexponential linear logic (without additive constants) and its extensions. In this Appendix, we wish to take care of some logics which do not fall within this scope. To begin with, we survey in an extremely succinct way the main results concerning Lambek calculus and its applications to linguistics. Next, we consider a family of substructural logics which has been intensively investigated over the last few decades: Ono's subintuitionistic logics. Finally, we examine a recently discovered logic which, in our opinion, is likely to play an increasingly important role in the substructural panorama: Sambin's basic logic.
1. LAMBEK CALCULUS As we hinted in Chapter 1, one of the main grounds for rejecting exchange (as well as weakening and contraction) rules in sequent calculi is given by the availability of interesting applications for the resulting calculi in the field of linguistics. To understand how such a connection can arise, we need a couple of preliminaries about the concept of grammar in formal linguistics. A formal grammar is made up by an initial part and a deductive part. The initial part operates on lexical atoms (for example: Joan, smiles, charmingly) by assigning to each one of them a grammatical category, or type (e.g. NP = noun phrase; V = verb; Adv = adverb; VP = verb phrase; S =
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Substructural logics: a primer
sentence). For example: Joan: NP; smiles: V; charmingly: Adv. Statements like the above are called lexical assumptions. The deductive part is given by a consequence relation , which is the smallest relation containing the identity axioms of the form and a set of axiomatic patterns , and being closed under the following cut rule:
, are types and are finite, possibly empty sequences of where
types. How can the machinery of a formal grammar assign types to the compound expressions of natural language? Simple: the concatenation of lexical atoms is assigned type just in case there are lexical assumptions in the initial part s.t. the pattern is validated by the deductive part of the grammar. A grammar is context-free whenever formulae are nonterminal symbols (Chomsky 1963) and the consequence relation is defined by a finite collection of axiomatic patterns of the form . For example:
NP, VP
S,
V
VP,
V, Adv
VP.
The stock of basic grammatical categories can be substantially reduced if we employ suitable type-forming operators, as in Ajdukiewicz's categorial grammar. There, we have two operators and , whose intuitive meaning is easy to grasp: an expression has type (respectively ) iff, whenever the expression has type , the expression (respectively ) has type . The deductive part of categorial grammar is defined by the modus ponens patterns:
.
In Ajdukiewicz-style grammars it is possible to do away with many basic types: thus, for example, it is not necessary to assume the category V. For instance, if the initial part contains the lexical assumption John: NP, the verb runs can be assigned type NP S, getting type S for the expression John runs. Actually, Ajdukiewicz built up his system out of just two basic types: (noun) and (sentence). Lambek (1958) considerably extended the deductive power of categorial grammar by setting up a Gentzen-style calculus for permissible transformations on types. More precisely, he introduced a new type-forming operator , to the effect that has type whenever , with of
!
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type and of type . Furthermore, he empowered the deductive part of the grammar by admitting patterns of hypothetical reasoning (corresponding to introduction rules on the right) aside modus ponens. Now, let be metavariables for types constructed out of suitable grammatical categories by means of the type-forming operators , and let stand for finite, possibly empty sequences of types. The postulates of the Lambek calculus LLk are:
Axioms
Structural rules
!#"%$'&( Operational rules
) * " -, + ) * " ! ( * / " , + ( 1 01 " ! " , + ( 3 2 " ! . and . . with nonempty in 4
! -. ( ! . ( ! . (
Significant variants of the calculus are: LLkN, i.e. the nonassociative Lambek calculus (Lambek 1961), where are not sequences, but structured databases (in the terminology of Gabbay 1996); this means that comma is taken as a nonassociative binary operation on types. LLkP, i.e. the commutative Lambek calculus (van Benthem 1986), where are treated as multisets of types. Due to the covert exchange rule thereby introduced, the connectives and collapse onto each other. LLk1, LLkN1, LLkP1, corresponding respectively to LLk, LLkN, LLkP, but with no restriction on the rules and . It is not difficult to prove cut elimination theorems for all of these calculi. Due to the absence of contraction, such results immediately lead to decidability proofs. Lambek-style calculi also admit natural deduction versions, as well as
56 +
4
4
+
.
.
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formulations in the framework of Gabbay's labelled deductive systems (Kurtonina 1995). Which of the various Lambek systems proves best suited for an application to linguistics? There seems to be no definite answer to this question, as Moortgat (1997, pp. 112-113) remarks: [In LLkN] grammatical inference is fully sensitive to both the horizontal and the vertical dimensions of linguistic structure: linear ordering and hierarchical grouping. As in classical Ajdukiewicz-style categorial grammar, application is the basic reduction law for this system. But the capacity for hypothetical reasoning already greatly increases the inferential strength [of LLkN] in comparison with the pure application fragment. The principles of Argument Lowering [...] and Value Raising [...] turn out to be generally valid type change schemata. [...] An essential limitation of [LLkN] is its rigid concept of constituency.
On the other side (ibidem, pp. 113-114): [In LLk] associative resource management makes the grammar logic insensitive to hierarchical constituent structure. [...] The free availability of restructuring makes it possible to give alternative constituent analysis for expressions that would count as structurally unambiguous under rigid constituent assumptions [...]. Unfortunately, the strength of [LLk] is at the same time its weakness. Associative resource management globally destroys discrimination for constituency, not just where one would like to see a relaxation of structure sensitivity [...]. At the same time, the order sensitivity of this logic makes it too weak to handle discontinuous dependencies.
In other words, Lambek-style calculi with fewer structural rules provide a more faithful representation of natural language in that they are sensitive to a greater number of syntactical dimensions, but a price is paid in terms of flexibility in type assignment, and the availability of fewer inferential mechanisms ultimately cuts down the range of linguistic phenomena the calculus can properly account for. In stronger calculi the situation is reversed; thus, there seems to be a sort of tradeoff between the factors of sensitivity and flexibility. As regards the semantical interpretation of Lambek calculus, most researchers incline towards the use of a proof-theoretical semantics. However, several kinds of model-theoretical semantics have been devised as well. Let us survey some of them.
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Algebraic semantics. It is quite straightforward to design appropriate algebraic models for our calculi, along the lines of the semantics developed in Chapter 6. Let us define a residuated groupoid as a structure of type , where: is a groupoid; is a poset; iff iff . A residuated groupoid is a residuated semigroup just in case is a semigroup. On the other hand, a structure of type is a residuated groupoid with unit (a residuated monoid) iff is a residuated groupoid (a residuated semigroup) and . Moreover, any of the previously defined structures is Abelian iff so is . It is not difficult to show that, upon defining adequate notions of valuation, model, truth, and validity, a sequent turns out to be provable in LLkN just in case it is valid in all residuated groupoids, i.e. just in case in every model , where is a residuated groupoid. Adding associativity (resp. exchange) to the calculus yields completeness with respect to residuated semigroups (resp. Abelian groupoids), while lifting the restrictions on the rules R and R corresponds model-theoretically to the addition of the neutral element.
"
$ # % ! #&#' (*),+ ( / -. + / 0 . 1 " ! - ! 2 3 1 5 4 -
4
Routley-Meyer semantics. The relational semantics of Chapter 7, § 1, was extended to Lambek calculus by Došen (1992a) and Dunn (1993a). Models for, say, LLkN are ordered pairs , where is a drastically simplified frame made out of just a set of situations and of a ternary accessibility relation , on which no special condition is imposed. The valuation clause for is what one expects, while the clauses for and are:
67
-8-8.. +9?2:3 9+ // ;<;< = = .. 66 @ --.. ++ >> // -8.>ABHJABAB. I,+ DFK + E / ABABA + G / -.>ABABAB. + D 7H L NO M K P>QR L N"O M P>QR Q8T P>SZWYN T iff T iff
If sequent
& &
2
)) -. 9 / 9 / . + E / 7 ABK ABAC/ 7 + G / S HJM I,K QTUH PVSXWYN
T T
3
T); T).
is equated to , then a is called LLkN-valid just in case entails in every model , i.e. for every and in , T implies T. We can thus show that the sequent is LLkNvalid iff it is LLkN-provable. By adding suitable conditions to the accessibility relation, we recover adequate models for LLk and LLkP, while the introduction of a base situation 1, to which truth in a model can be relativized, accounts for the systems with no restriction on the right introduction rules for both implications.
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Operational semantics. An operational semantics for Lambek calculus was introduced by Buszkowski (1986), who drew his inspiration from Urquhart's operational semantics for relevant implication. If is a groupoid, we can define for :
/ & 0 ;
!"/ $#% &' )( * + *, 0 ; - "/ $#% &' )( . + *, 0 . 132547698;: < :5=>:5?@:.ACB is a residuated groupoid, The structure 6 called the powerset residuated groupoid over . The powerset residuated 6 groupoid over is a residuated semigroup, residuated Abelian semigroup, 6 itself is a semigroup, an residuated monoid etc. according to whether Abelian semigroup, a monoid etc. Conversely, one can give representations of arbitrary residuated groupoids in terms of powerset residuated groupoids of the same kind. In such a way, it is possible to exploit the results of algebraic semantics in order to prove completeness theorems for any Lambek system we may like.
D
Free semantics. A fourth sort of semantics for LLk and LLk1 can be obtained by considering models on free semigroups over an alphabet . Elements of such semigroups are strings of elements from , and the semigroup operation is simply concatenation. Completeness w.r.t. these models is not as easy to prove as in the context of the other kinds of semantics. However, Pentus (1995) managed to prove completeness for LLk w.r.t. models on free semigroups, and later extended such a proof to cover also the case of LLk1 (Pentus 1998).
D
To round off our survey, let us spend a couple of words about some linguistic extensions of Lambek calculus. The logical vocabulary of the Lambek systems can be enriched by further connectives, although the linguistic content of such additions is not always so crystal clear. Negation can be used to make sense of a notion of negative information (Buszkowski 1995); it is also possible to add modalities (Moortgat 1996) and even quantifiers. Yet, the best motivated move is perhaps the introduction of lattice-theoretical disjunction and conjunction connectives (Kanazawa 1994). These connectives are useful in reasoning about multiple type assignment - in fact, there seem to be expressions in natural language which must be assigned different types according to circumstances. It is natural, then, to assign type to expressions which are both of type and of type , and to assign type to expressions which can be either of type or of type . However, if we add
E
E
H
H
EGFIH EGJIH
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the obvious Gentzen rules for conjunction and disjunction:
we do not capture properly the intended meaning of such intersection and union types, which should distribute over each other, while - as the reader by now knows - the lattice-theoretical rules for and are not sufficient, by themselves, to grant lattice distributivity. Two routes can be taken: you can either modify the semantics for disjunction as suggested in § 2 below, in order to block distribution, or else resort to a Dunn-Mints proof system where distribution is provable (Restall 1995).
2. ONO'S SUBINTUITIONISTIC LOGICS Logics weaker than the intuitionistic did not receive a special attention in this book, where our primary concern was to focus on systems with an involutive negation. Anyway, subintuitionistic logics were extensively studied over the last 20 years by Hiroakira Ono and his collaborators (see e.g. Ono and Komori 1985, Ono 1993, Ono 1998a, Ono 200+a). In this section, we shall try to fill this gap, at least in part, by presenting a concise survey of some of the work done so far in this area. Ono's starting point is the system FL, which bears a precise relationship to Lambek calculus - as the name itself suggests, for FL is nothing but a shorthand for "full Lambek (logic)". FL closely resembles LLk1 of the previous section, extended with rules for the lattice-theoretical connectives. For the sake of precision, we lay down in full detail the postulates of FL. stand for finite, possibly empty, sequences of formulae of a language ! " containing the connectives , , , # , % , $ , &( , ' , ) . * is a sequence of formulae of the same language containing at most one formula. Axioms +(,+
;
, $
;
&
,
;
.
, '
;
.
)
- / , *
.
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Substructural logics: a primer
"$ ! "$ #"$ %" & '& % & (& ) *. +* , -*. $ *. /. +*. +/. $ +/. +/. /.
0 1 2 2 1 - Structural rules
Operational rules
Now, consider the following structural rules of weakening, exchange, contraction:
4356 :;
= B@ ADC A8E
43879 <; :>3<
?
and let be any combination of the letters , , . Then the calculus FL is defined as the system obtained by adding to FL the rules corresponding to the letters in ( and stand or fall together, and are added just in case is in
A
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). Of course FL is plain intuitionistic logic, while FL corresponds to Girard's intuitionistic linear logic without exponentials1 . What are the main properties of the FL systems? Let us examine first their main proof-theoretical features. First, and foremost, the cut elimination theorem holds for FL when , but not for FL . In fact, the cut elimination procedure for weakening-free systems described in Chapter 3, whose key feature is the introduction of an intmix rule, essentially relies on the presence of exchange. With this sole exception, the FL systems are cut-free. And cut elimination has, as usual, some pleasant consequences: Primality. A prime theory, as the reader will recall from Chapter 2, cannot contain a disjunction unless it contains either or . Similarly, a Gentzen system is prime if it proves either or whenever it S proves . LJ, for example, is prime, whereas LK, of course, is not !" think of , which is LK-provable for an arbitrary , even though " neither * nor are generally such. It turns out that FL is prime + #%$ &'&(&'(&') when , as a consequence of the cut elimination theorem and of the fact that the succedents of sequents in these systems contain at most one formula. Decidability. As we remarked in Chapter 3, it is not hard to prove decidability for sequent systems lacking3 contraction 4 rules. And, in fact, FL is easily shown to be decidable when , - ./0.21./1 . Remarkably enough, the first-order counterparts of these calculi are decidable as well, even if the language contains function symbols and individual constants (Kiriyama and Ono 1991). Propositional FL576 , on the other hand, can be shown to be decidable by resorting to the techniques expounded in Chapter 3 under the heading "A decision method for weakening-free systems". The predicate logic corresponding to FL 56 , however, is undecidable. Interpolation. As Craig proved in 1957, if an implicational formula is a theorem of classical propositional logic, formulated with the = > ? @:A
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Substructural logics: a primer
theorem: we simply claim that such a exists if and share at least a variable, while otherwise either or must be provable. This property is called interpolation property. A well-known method due to the Japanese logician Maehara allows to derive the interpolation property for classical (and intuitionistic) logic from the cut elimination theorem for LK. Such a pleasant method, unlike the semantical techniques used by Craig, contains instructions for actually constructing the interpolant . And, applying this procedure, it is possible to prove that FL has the interpolation property when .
Variable sharing and variable separation. A logic has the variable sharing property iff, whenever a constant-free implication is provable in it, and share at least a variable. As we remarked in Chapter 2, the variable-sharing property for relevance logics was thoroughly investigated by Anderson and Belnap and by Maksimova. Adapting Maksimova's proofs, it can be shown that most weakening-free logics in the FL family - i.e. FL, FL , FL - have the variable-sharing property. As one can expect, logics with weakening lack this typically "relevant" property. On the other side, consider the following principle of variable separation (whose formulation is due to Maksimova): if a logic proves the implication , where are such that and share no variable, then the logic 2 . FL has the variable separation at issue proves either or property when (Naruse et al. 1998).
+* ",* &,* )
$& "%$) -$& "%$) 5 /. 0 1 2131 231 24 6
!#"%$&('#)
This much can suffice for the proof theory of Ono's logics. An in-depth investigation of the models for the FL family is contained in Ono (1993), where each logic in the family is matched with an appropriate class of algebraic structures, and where phase semantics for intuitionistic linear logic is adapted to suit FL and its extensions. Finally, a Kripke-style semantics is introduced as in Ono and Komori (1985) and Došen (1989b). It may be interesting to recap here, too, the leading ideas of this kind of semantics. A semilattice-ordered monoid is a structure of type , where: is a lower semilattice with top element ; is a monoid; for every ; for every . A model is an ordered pair , where is a semilatticeordered monoid and the valuation obeys the following clauses3 :
9 9!H <JI< I+F-<E9!H+FGF K 9!G K 9!;=;=@DG<AF F KML >NB!8(B LPO ; KGQ >@R L+S,T U >NV!8WR Q > L >NV UXS Y R Q > T >NV U 8:9 7
7%8:9!;=@<A@
Francesco Paoli If T and , then implies
If T and , then implies
$ !" # T for every ! ; &%('() !" # T iff *+, - &+/.0-1(! & &% +" # T & 2) -" # T " ; &%435) !" # T iff 67+, - &+/.8!1- & &% 9+" # T :; 2) -" # T " 2)=<;% !" # T iff 67+, - &!.8+1- & &% 9+" # T :; 2) -" # T " &%>?) !" # T iff *+, - +CB?-1(! & ( &% +" # T v 2)@ +" # T " & ( &% -" # T v 2) -"A# T " " ; &%D?) !" # T iff &%E !" # T & 2) !" # T; F 9!" # T iff !J# K ; G 9!" # T iff HI1(! .
281 T; T;
; ;
Remark the nonstandard clause for disjunction, whose aim is preventing lattice distribution from being valid. By induction, one can prove that if !L +MN &%O" and -EMQP , then !EBJ+@1- implies -EMR &%O" for every formula % . The formula % is true in a model S #VU T 9 @W iff &% !" # T for every !RX4H , and is true in a semilattice-ordered monoid T/Y just in case it is true in Z ^ Y [\] all models of the form . Employing familiar techniques, it is _;` is FL-provable iff ` is true in all possible to prove that the sequent semilattice-ordered monoids. By adding suitable conditions to the above models, one can recover adequate soundness and completeness theorems for other logics of the FL a family.
3. BASIC LOGIC Basic logic4 (Battilotti and Sambin 1999; Sambin et al. 2000) has been introduced in the attempt to provide a common framework for a wide variety of logics, including linear logic, intuitionistic logic, and quantum logic. From a philosophical and methodological viewpoint, it is not hard to discern behind its leading intuitions the influences of Martin-Löf's approach to logic and type theory (see e.g. Martin-Löf 1984) and of the investigations by Dummett, Prawitz, Došen and others into the meaning of logical constants. Three general assumptions lay the groundwork for the development of basic logic: the principles of reflection, symmetry, and visibility. Let us examine them one by one, beginning with the first. The principle of reflection amounts, roughly speaking, to the assumption that in the framework of sequent calculi each propositional connective reflects at the level of object language a link between assertions in an appropriate metalanguage. Which assertions, and which metalanguage? Like Martin-Löf,
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Substructural logics: a primer
Sambin carefully distinguishes between formulae and assertions concerning formulae. A sequent calculus, in fact, can be conceived of as making implicit assertions about the formulae of its language. The basic assertions can be thought of as expressions of the form , where is a formula and the interpretation of is deliberately left in the vague (it could mean "is true", "is available", etc., according to the desired interpretation of the calculus). Basic assertions may enter into more complex assertions, built up by means of the links and . These links are sufficient to express all the assertions occurring in sequent calculi: in fact, abbreviates the conjunction of atomic assertions , while abbreviates the assertion . On the other side,
" # $#
+ % ,
! + &' ()* , "
+ % , - % . / % 0 1 / % 0 + % , !2 - % . 3&4' ()*5 assertion
abbreviates the , and similarly for inferences with a single premiss. The truth (or availability) conditions for a connective are given by its definitional equation, a metalinguistical biconditional relating two assertions: a definiendum, containing the connective, and a definiens, not containing it. Definitional equations provide the justification for the inference rules of the calculus, which are obtained by "solving" such equations according to a general method. (Below, we shall see suitable examples of definitional equations and of the method used to solve them.) More precisely, every connective has two introduction rules: a formation rule, derived from the direction of the definitional equation which gives sufficient conditions for asserting a formula containing the connective; and a reflection rule, stemming from the converse direction of the equation, which licenses deductions from an already available formula containing the connective. Basic logic is formulated in a language containing the connectives , , , , , , , , , . Here are the definitional equations for some of the connectives; remark that symbolizes the empty assertion and that stand for finite, possibly empty, multisets of formulae:
6 7
8 ; 9 : <>=@? A B CEFD EGGG HJILKNMNOPHQRMRSUTJV OWTX Y)Z*V _ iff HQTV\[4] Z^SUTV OWTX Y)Z*V _ ; HJILKN`NO5_ WTX Y)Z*VaHbQR`RSUTV O iff _ WTX Y)Z*VcHQTV\[] Z^SUTV O ; HJILKedfO5_ WTX Y)Z*VaHQgdhSUTV O iff Hi_ WTX Y)Z*VcHQTV OO [] Z!Hi_ WTXY)Z*VaHSUTV OO ; HJILKejfOPHQNjhSUTVkOWTX Y)Z*V _ iff HHQTV O4WTX Y)Z*Vl_ O [] Z!HHSmTV OWTX Y)Z*Vl_ O ; HJILK2p OPHnp TV OWTX Y)Z*V _ iff o\WTX Y)Z*V _ ; qJrLs2| t5_ uvw x)y*zaqn| vzkt iff _ uvw x)y*z\{ ; qJrLs t}qi_ uvw x)y*zaq~vz tt y!qi_ uvw x)yzaq vz tt iff _ uvw x)y*zaq~vz t ;
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283
!
"
iff
#
. !
As you can see, , , , all reflect
links. Yet, and reflect links within the scope of a principal link, respectively onthe " # left and on the right of it; on the contrary, and reflect principal links. Incidentally, this dichotomy sheds new light on the distinction between lattice-theoretical and group-theoretical connectives. Solving these definitional equations one gets the following rules, which, if %$& added to axioms of the form , form the calculus B0 for basic logic without implication and negation: ')(+*-,
'101*-,
,&')(+* @
@
@
,&'171*
,&' @
@ '=,
@
3 9;4)5
,&':9;* *-, @
@
':>;*-, ,
@ ,
B
, ,
@ ,
3 E
*-, @
'171*-,
'%,
3 016)5
,/* @
,&':>;*
,&':>;* @
@
3 9;6)5
3 >;6)5
3 B 6)5 BC ,
3 D 6)5 ,
D
4)5
@
'<9;*-, @
,&' @
3 716)5
*8,
':9;*-,
@
?
( ? @A
@
3 D 4)5 D
E
'%,
3 B 4)5 @
@
@
3 >;4)5
@
,/* ?
(2 ,&'101* @A ?
3 714)5
,/*
,.'
3 014)5
, F
@
3 F
4)5 0
To show how to solve0 a definitional equation, we justify the rules F and R on the basis of DE . The right-to-left side of the equation immediately gives, by replacing GHI and J K with comma and LJ MNIK with the inference sign, 0
')(+*-, 'O01*-, 0
0
@ @
which is F. To'1get R, we first trivialize the')other direction of the 01* (+*-,&'10O* @=P equation, letting . The result is the "axiom" , which displays the connective sign on the proper side of the arrow. Then we assume ' * @ ? that and are produced from and respectively, and apply composition
284
Substructural logics: a primer
- a form of cut which is later shown to be admissible in basic logic - to obtain R in this way:
The full system B of propositional basic logic results from B by adding two connectives of implication ( ) and dual implication, or exclusion ( ). 0
The rules for such connectives are grounded on appropriate definitional equations as well, but in a more mediate way (cp. Sambin et al. 2000). Furthermore, two order rules need to be postulated in addition to the ordinary formation and reflection rules. Thus, we get:
% % % % #$ #$ ! " Two negations can be defined within B, according to whether we take & ' ( as equivalent to ("implying the false") or to ("excluding the
true"). By now the principle of reflection should be sufficiently clear; therefore, let us move on to symmetry. If you take a look at the rules of B, you soon notice that its connectives can be divided into left connectives - whose formation and reflection rules operate respectively on the antecedents and on the succedents of sequents - and right connectives, where the roles of antecedents and succedents are reversed. Every left connective is accompanied by its symmetric right connective, according to the following schema:
( , - '). ' , - ( / *, + + , * ,0 -12.$1 , - 0 / 34, - -!56. . 54, -7- 36/ / 8 ,- 9 . 9 ,- 8 . : More than that, : for every formula it is possible to define inductively its , symmetric formula : ,, - < , if, < is a variable; ;! < ;> = B = A = @ ? , if EG D : F C E , , , where : is a connective. ; ? A - , isCHa: constant;
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285
It is then possible to prove that coincides with , for any formula . But we can go even a couple of steps further, defining the notion of "symmetric" also for multisets of formulae, sequents, and rules:
;
;
.
The importance of these notions will become apparent in a short while, when we discuss the extensions of basic logic. But first, a couple of remarks are in order concerning the third fundamental tenet of basic logic - the principle of visibility. In Chapter 1, when we examined the "underdetermination view" of operational rules in sequent calculi, we fleetingly hinted at Sambin's idea according to which the meaning of a constant is "determined also by contexts in its rules, which can bring in latent information on the behaviour of the connective". Hence, the presence of side formulae on the same side of either the principal or the auxiliary formulae in the introduction rules for a connective makes a difference as to the meaning of the connective itself. As the reader may have remarked, all the rules of basic logic satisfy visibility: in such rules, there are no side formulae on the same side of either the principal, or the auxiliary formulae - these formulae are "visible", in Sambin's terminology. Basic logic, in other words, adds to the control of structural rules typical of linear logic a control of side formulae, both on the left and on the right of the arrow. What gain can we expect from the three principles just expounded? The main advantages are a common framework where linear, intuitionistic, and quantum logics can be accommodated, and a general procedure for proving the admissibility of cut - or, to be more precise, of composition - which has eventually led, among other things, to the formulation of a cut-free sequent calculus for orthologic5 (Faggian and Sambin 1997). We can extend basic logic in at least three ways: relaxing the visibility constraints, either on the left (L) or on the right (R), and restoring the deleted structural rules of weakening and contraction (S). Conceive of L, R, S as of three "actions" to be performed on B. The following combinations are possible as a result of the performance of one or more of such actions: and )6 ; BL, subexponential intuitionistic linear logic (with BR, subexponential "dual intuitionistic" linear logic; BS, basic orthologic7 ; BLR, subexponential linear logic; ); BLS, intuitionistic logic (with
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Substructural logics: a primer
BRS, "dual intuitionistic" logic; BLRS, classical logic. Adding further structural rules to BS, moreover, we can get a sequent calculus for orthologic. The previous notion of symmetry can now be extended to the calculi S') iff it contains the themselves. A calculus to S (S S' is symmetric of symmetric of every axiom of S, and the symmetric every rule of S. According to this definition, BR = BL and BRS = BLS , while B, BS, BLR, BLRS coincide with their own symmetric and are thus termed self-symmetric. More than that, every proof in the calculus S is matched by a symmetric proof replacing every in S , obtained from by assumption of an axiom with an assumption of and every application of a rule with an application of . It is possible to prove that,
for any of the above-listed calculi and for arbitrary , is a proof of S in S iff is a proof of in S . If S is a self-symmetric calculus, of course, the symmetric derivation can be carried out within the calculus itself. Substituting a proof with its symmetric is sometimes quite useful in cut elimination proofs.
Sources of the Appendix. § 1 is essentially based on Moortgat (1997) and Buszkowski (1997). The main sources for § 2 were Ono (1993) and Ono (1998b), while for § 3 we used primarily Sambin et al. (2000).
Notes
1. As regards FL , the exchange rule is admissible in it, but the cut elimination theorem does not hold therein. In the following, we shall disregard this system altogether. 2. The logical significance of the variable separation property is highlighted in Chagrov and Zakharyashev (1993). 3. For a less rushed definition of valuation in Kripke-style semantics, see Chapter 7. 4. The reader should not confuse Sambin's basic logic, which is here at issue, with Hajek's basic logic (Hajek 1998), an important member of the family of fuzzy logics. 5. Orthologic is one of the mainstream quantum logics. As we recalled in Chapter 7, its models are possibly nondistributive lattices. On the proof-theoretical side, however, it is not easy to block distribution in the presence of structural rules; to do so, one has to place appropriate visibility restrictions upon the rules needed to prove it. Yet, such restrictions make it extremely difficult to eliminate cuts (cp. Dalla Chiara and Giuntini 200+). Basic logic extends the visibility restrictions to each rule of the calculus, restoring symmetry and turning what was a "constraint" into a "strong point" of the system. Along this way, it is possible to regain a limpid cut elimination proof for B and its extensions, including orthologic.
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6. Remark that the group-theoretical disjunction of this linguistical extension of intuitionistic linear logic is nonassociative. 7. This logic was introduced by Dalla Chiara and Giuntini (1989) under the name of paraconsistent quantum logic.
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INDEX
Abelian logic 35, 58ff, 100, 162, 215ff, 235 Absolute value 172 Affine linear logic 55ff, 94ff, 105106, 111-112, 187ff, 252 Antecedent part 52, 54 Antiphase 240 Arithmetical variety 164 Balanced calculus 48, 54 Barriers 111 Basic logic 36, 281ff Bounded contraction 62ff, 99 Bunched implications 130 Canonical model 208, 209 Categorial grammar 29-30, 272 Classical residuated lattice 166, 187ff free 193 local 194 locally finite 194 semisimple 192 simple 192 subdirectly irreducible 190 weakly simple 192 with a single atom 191 Clausal formula 151
Clause 146 parent 147 resolvent 147 Cognate sequents 109 Comparative logic 58ff, 100, 165 Concise proof 104 Conjunction set 78 Connective 14 group-theoretical 14, 42 lattice-theoretical 14, 42 Consequence 205 Consequent part 52, 54 Curry-Howard isomorphism 32 Curry's Lemma 108 Cut elimination 17ff, 87ff, 144 Decision problem 100ff Deduction theorems 76ff De Morgan monoid 165 idempotent 165 Derivability 75ff Dialetheism 23-24, 49, 235 Disjunctive syllogism 219-220, 236ff Display equivalence 132 Display logic 130ff Distance 176 Downward set 244
302 Dual absolute value 172 Dunn-Mints calculi 127
Substructural logics: a primer
Exclusion 284 Exponentials 65ff, 209ff, 252ff External rule 122-123 Filtration 111 Finite model property 111, 254 Formula auxiliary 5 clause set of a - 147 principal 5 side 5 Frame 224 normal 236 normalization of a - 236 Gaggle theory 255 Grade 90 Grammar 271 context-free 272 History method 125 HL and its extensions 68ff, 203ff, 223ff Hypersequent 121ff
LC 58ff, 100 Lexical assumption 272 LG 58ff, 100 Lindenbaum algebra 206 Lindenbaum's Lemma 81 Linear logic 26, 28-29, 42ff, 65ff, 68ff, 99, 111-112, 137ff, 153154, 161ff, 203ff, 240ff, 252ff, 298 Literal 138, 146, 153-154 resolved upon 147 LJ 5, 7, 8 LK 5ff, 87ff, 101ff LL 44ff, 99 LLB 50 LLE 65ff LL 49 LLuk3 117 LLuk3 ' 117 LR 131 LR+ 127 LRND 51ff, 97ff, 106ff LRM 125-126 LRMI 125-126 LRMND 62 Lukasiewicz logics 25, 57, 63ff, 117ff, 154-155, 167, 215, 235 -filter 172 -group 162 -ideal 172 canonical homomorphism of a 177 congruence associated with a 176 maximal 189 primary 189 prime 183 principal 182 regular 185 weakly prime 183 -pregroup 165
Index of a proofnet 142 of a sequent proof 90 Internal rule 122-123 Intmix 97 Kernel 177 Kripke's Lemma 110
LA 55ff, 94ff, 105-106 Labelled deductive systems 145, 274 Lambek calculus 29-30, 271ff
Francesco Paoli MacNeille space 247 Many-valued logics 25 Matching 204, 226, 242, 250 Meaning of logical constants 4ff Mingle 62 Mix 88 Mixproof 88 Model algebraic 204 phase 250 Routley-Meyer 225 topolinear 253 Multiple resolution 155 MV-algebra 167 Naive set theory 23, 26-28 Nonmonotonic logics 24 -sided sequent 116ff
Operational semantics 222, 276 Order 194 Orthogonality relations 177 Paraconsistency 22ff -count 47 Phase structure 240 descriptive 247ff general 242 ff Post-implication 15 Post-negation 15 Principal polar 247 Proofnet 137ff Proof search algorithm 101 Proof structure 138 inductive 139 section of a - 142 switching of a - 141 Quantale Girard 165, 210-211 rectangular 180
303 Quantum logic 239, 281ff Radical 193 Rank 89, 95 Reduced sequent 103 Relatedness logic 36 Relational proof systems 145 Relevance logic 21-22, 50ff, 97ff, 106ff, 127ff, 149ff, 165, 223ff, 235ff Residuated groupoid 275 powerset 276 Residuated semigroup 275 Resolution calculi 145ff Resolution rule 146 Retro-implication 15 Retro-negation 15 RK 146 RL 153 Routley star 225 RR 149 Rule 6ff context-dependent 12 context-free 12 S-derivation 75 weak 78 Semilattice-ordered monoid 180 Sequent 5 Sigma term 174 Simple cut 135 Single model property 214 Splitting 166 Squeeze lemma 229 *-autonomous lattice 161ff, 243ff closure 209 homomorphism of a - 171 linear 183 orthogonally indecomposable 184 representable 180
304 totally ordered 184 weakly contractive 179 weakly linear 183 S-theory 72 -complete 73 -consistent 73 -maximal 73 detached 73 -complete 73 -consistent 73 maximal 73 prime 73 regular 73 simply complete 73 simply consistent 73 -S-theory 229 weakly maximal 73 Strong algebraic De Morgan law 180 Structural connective 130 Structure 127 Subdirect representation 186 Subintuitionistic logics 277
Substructural logics: a primer Succinct proof 108 Sugihara matrix 165 Topolinear space 252 Truth algebraic 204 phase-semantical 251 Routley-Meyer 226 2-semiclosed order 180 Type 271 Validity algebraic 205 phase-semantical 251 Routley-Meyer 226 Valuation algebraic 203, 211 Routley-Meyer 224 Value 185 Variable separation 280 Variable sharing 51ff, 280 Visibility 11, 281ff
