Handbook of Philosophical Logic

Handbook of Philosophical Logic 2nd Edition Volume 6

edited by Dov M. Gabbay and F. Guenthner

CONTENTS

Editorial Preface

vii

Dov M. Gabbay

Relevance Logic

1

Mike Dunn and Greg Restall

Quantum Logics

129

Maria-Luisa Dalla Chiara and Roberto Giuntini

Combinators, Proofs and Implicational Logics

229

Martin Bunder

Paraconsistent Logic

287

Graham Priest

Index

395

PREFACE TO THE SECOND EDITION

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the rst edition and there have been great changes in the landscape of philosophical logic since then. The rst edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the rst edition as `the best starting point for exploring any of the topics in logic'. We are con dent that the second edition will prove to be just as good! The rst edition was the second handbook published for the logic community. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook of Philosophical Logic, published 1983{1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and arti cial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisation on the one hand and to provide the theoretical basis for the computer program constructs on the other. The result was that the Handbook of Philosophical Logic, which covered most of the areas needed from logic for these active communities, became their bible. The increased demand for philosophical logic from computer science and arti cial intelligence and computational linguistics accelerated the development of the subject directly and indirectly. It directly pushed research forward, stimulated by the needs of applications. New logic areas became established and old areas were enriched and expanded. At the same time, it socially provided employment for generations of logicians residing in computer science, linguistics and electrical engineering departments which of course helped keep the logic community thriving. In addition to that, it so happens (perhaps not by accident) that many of the Handbook contributors became active in these application areas and took their place as time passed on, among the most famous leading gures of applied philosophical logic of our times. Today we have a handbook with a most extraordinary collection of famous people as authors! The table below will give our readers an idea of the landscape of logic and its relation to computer science and formal language and arti cial intelligence. It shows that the rst edition is very close to the mark of what was needed. Two topics were not included in the rst edition, even though D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 6, vii{ix.

c 2002, Kluwer Academic Publishers. Printed in the Netherlands.

viii

they were extensively discussed by all authors in a 3-day Handbook meeting. These are:



a chapter on non-monotonic logic



a chapter on combinatory logic and -calculus

We felt at the time (1979) that non-monotonic logic was not ready for a chapter yet and that combinatory logic and -calculus was too far removed.1 Non-monotonic logic is now a very major area of philosophical logic, alongside default logics, labelled deductive systems, bring logics, multi-dimensional, multimodal and substructural logics. Intensive reexaminations of fragments of classical logic have produced fresh insights, including at time decision procedures and equivalence with non-classical systems. Perhaps the most impressive achievement of philosophical logic as arising in the past decade has been the eective negotiation of research partnerships with fallacy theory, informal logic and argumentation theory, attested to by the Amsterdam Conference in Logic and Argumentation in 1995, and the two Bonn Conferences in Practical Reasoning in 1996 and 1997. These subjects are becoming more and more useful in agent theory and intelligent and reactive databases. Finally, fteen years after the start of the Handbook project, I would like to take this opportunity to put forward my current views about logic in computer science, computational linguistics and arti cial intelligence. In the early 1980s the perception of the role of logic in computer science was that of a speci cation and reasoning tool and that of a basis for possibly neat computer languages. The computer scientist was manipulating data structures and the use of logic was one of his options. My own view at the time was that there was an opportunity for logic to play a key role in computer science and to exchange bene ts with this rich and important application area and thus enhance its own evolution. The relationship between logic and computer science was perceived as very much like the relationship of applied mathematics to physics and engineering. Applied mathematics evolves through its use as an essential tool, and so we hoped for logic. Today my view has changed. As computer science and arti cial intelligence deal more and more with distributed and interactive systems, processes, concurrency, agents, causes, transitions, communication and control (to name a few), the researcher in this area is having more and more in common with the traditional philosopher who has been analysing 1 I am really sorry, in hindsight, about the omission of the non-monotonic logic chapter. I wonder how the subject would have developed, if the AI research community had had a theoretical model, in the form of a chapter, to look at. Perhaps the area would have developed in a more streamlined way!

PREFACE TO THE SECOND EDITION

ix

such questions for centuries (unrestricted by the capabilities of any hardware). The principles governing the interaction of several processes, for example, are abstract an similar to principles governing the cooperation of two large organisation. A detailed rule based eective but rigid bureaucracy is very much similar to a complex computer program handling and manipulating data. My guess is that the principles underlying one are very much the same as those underlying the other. I believe the day is not far away in the future when the computer scientist will wake up one morning with the realisation that he is actually a kind of formal philosopher! The projected number of volumes for this Handbook is about 18. The subject has evolved and its areas have become interrelated to such an extent that it no longer makes sense to dedicate volumes to topics. However, the volumes do follow some natural groupings of chapters. I would like to thank our authors are readers for their contributions and their commitment in making this Handbook a success. Thanks also to our publication administrator Mrs J. Spurr for her usual dedication and excellence and to Kluwer Academic Publishers for their continuing support for the Handbook.

Dov Gabbay King's College London

x

Logic

IT Natural language processing

Temporal logic

Expressive power of tense operators. Temporal indices. Separation of past from future

Modal logic. Multi-modal logics

generalised quanti ers

Action logic

Algorithmic proof

Discourse representation. Direct computation on linguistic input Resolving ambiguities. Machine translation. Document classi cation. Relevance theory logical analysis of language Quanti ers in logic

Montague semantics. Situation semantics

Nonmonotonic reasoning

Probabilistic and fuzzy logic Intuitionistic logic

Set theory, higher-order logic, calculus, types

Program control speci cation, veri cation, concurrency Expressive power for recurrent events. Speci cation of temporal control. Decision problems. Model checking.

Arti cial intelligence

Logic programming

Planning. Time dependent data. Event calculus. Persistence through time| the Frame Problem. Temporal query language. temporal transactions. Belief revision. Inferential databases

Extension of Horn clause with time capability. Event calculus. Temporal logic programming.

New logics. Generic theorem provers

General theory of reasoning. Non-monotonic systems

Procedural approach to logic

Loop checking. Non-monotonic decisions about loops. Faults in systems.

Intrinsic logical discipline for AI. Evolving and communicating databases

Negation by failure. Deductive databases

Real time systems

Semantics for logic programs

Constructive reasoning and proof theory about speci cation design

Expert systems. Machine learning Intuitionistic logic is a better logical basis than classical logic

Non-wellfounded sets

Hereditary nite predicates

-calculus extension to logic programs

Negation by failure and modality

Horn clause logic is really intuitionistic. Extension of logic programming languages

PREFACE TO THE SECOND EDITION

xi

Imperative vs. declarative languages

Database theory

Complexity theory

Agent theory

Special comments: A look to the future

Temporal logic as a declarative programming language. The changing past in databases. The imperative future

Temporal databases and temporal transactions

Complexity questions of decision procedures of the logics involved

An essential component

Temporal systems are becoming more and more sophisticated and extensively applied

Dynamic logic

Database updates and action logic

Ditto

Possible tions

Multimodal logics are on the rise. Quanti cation and context becoming very active

Types. Term rewrite systems. Abstract interpretation

Abduction, relevance

Ditto

Agent's implementation rely on proof theory.

Inferential databases. Non-monotonic coding of databases

Ditto

Agent's reasoning is non-monotonic

A major area now. Important for formalising practical reasoning

Fuzzy and probabilistic data Database transactions. Inductive learning

Ditto

Connection with decision theory Agents constructive reasoning

Major now

Semantics for programming languages. Martin-Lof theories Semantics for programming languages. Abstract interpretation. Domain recursion theory.

Ditto

Ditto

ac-

area

Still a major central alternative to classical logic More central than ever!

xii

Classical logic. Classical fragments

Basic ground guage

Labelled deductive systems

Extremely useful in modelling

A unifying framework. Context theory.

Resource and substructural logics Fibring and combining logics

Lambek calculus

Truth maintenance systems Logics of space and time

backlan-

Dynamic syntax

Program synthesis

Modules. Combining languages

A basic tool

Fallacy theory

Logical Dynamics Argumentation theory games

Widely applied here Game semantics gaining ground

Object level/ metalevel

Extensively used in AI

Mechanisms: Abduction, default relevance Connection with neural nets

ditto

Time-actionrevision models

ditto

Annotated logic programs

Combining features

PREFACE TO THE SECOND EDITION

Relational databases Labelling allows for context and control. Linear logic Linked databases. Reactive databases

Logical complexity classes

xiii

The workhorse of logic

The study of fragments is very active and promising.

Essential tool.

The new unifying framework for logics

Agents have limited resources Agents are built up of various bred mechanisms

The notion of self- bring allows for selfreference Fallacies are really valid modes of reasoning in the right context.

Potentially applicable

A dynamic view of logic On the rise in all areas of applied logic. Promises a great future

Important feature of agents

Always central in all areas

Very important for agents

Becoming part of the notion of a logic Of great importance to the future. Just starting

A new theory of logical agent

A new kind of model

J. MICHAEL DUNN AND GREG RESTALL

RELEVANCE LOGIC

1 INTRODUCTION

1.1 Delimiting the topic The title of this piece is not `A Survey of Relevance Logic'. Such a project was impossible in the mid 1980s when the rst version of this article was published, due to the development of the eld and even the space limitations of the Handbook. The situation is if anything, more diÆcult now. For example Anderson and Belnap and Dunn's two volume [1975; 1992] work Entailment: The Logic of Relevance and Necessity, runs to over 1200 pages, and is their summary of just some of the work done by them and their coworkers up to about the late 1980s. Further, the comprehensive bibliography (prepared by R. G. Wolf) contains over 3000 entries in work on relevance logic and related elds. So, we need some way of delimiting our topic. To be honest the fact that we are writing this is already a kind of delimitation. It is natural that you shall nd emphasised here the work that we happen to know best. But still rationality demands a less subjective rationale, and so we will proceed as follows. Anderson [1963] set forth some open problems for his and Belnap's system E that have given shape to much of the subsequent research in relevance logic (even much of the earlier work can be seen as related to these open problems, e.g. by giving rise to them). Anderson picks three of these problems as major: (1) the admissibility of Ackermann's rule (the reader should not worry that he is expected to already know what this means), (2) the decision problems, (3) the providing of a semantics. Anderson also lists additional problems which he calls `minor' because they have no `philosophical bite'. We will organise our remarks on relevance logic around three major problems of Anderson. The reader should be told in advance that each of these problems are closed (but of course `closed' does not mean ` nished'|closing one problem invariably opens another related problem). This gives then three of our sections. It is obvious that to these we must add an introduction setting forth at least some of the motivations of relevance logic and some syntactical speci cations. To the end we will add a section which situates work in relevance logic in the wider context of study of other logical systems, since in the recent years it has become clear that relevance logics t well among a wider class of `resource-conscious' or `substructural' logics [Schroeder-Heister and Dosen, 1993; Restall, 2000] [and cite the S{H article in this volume]. We thus have the following table of contents: D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 6, 1{128.

c 2002, Kluwer Academic Publishers. Printed in the Netherlands.

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J. MICHAEL DUNN AND GREG RESTALL

1. Introduction 2. The Admissibility of 3. Semantics 4. The Decision Problem 5. Looking About We should add a word about the delimitation of our topic. There are by now a host of formal systems that can be said with some justi cation to be `relevance logics'. Some of these antedate the Anderson{Belnap approach, some are more recent. Some have been studied somewhat extensively, whereas others have been discussed for only a few pages in some journal. It would be impossible to describe all of these, let alone to assess in each and every case how they compare with the Anderson{Belnap approach. It is clear that the Anderson{Belnap-style logics have been the most intensively studied. So we will concentrate on the research program of Anderson, Belnap and their co-workers, and shall mention other approaches only insofar as they bear on this program. By way of minor recompense we mention that Anderson and Belnap [1975] have been good about discussing related approaches, especially the older ones. Finally, we should say that our paradigm of a relevance logic throughout this essay will be the Anderson{Belnap system R or relevant implication ( rst devised by Belnap|see [Belnap, 1967a; Belnap, 1967b] for its history) and not so much the Anderson{Belnap favourite, their system E of entailment. There will be more about each of these systems below (they are explicitly formulated in Section 1.3), but let us simply say here that each of these is concerned to formalise a species of implication (or the conditional|see Section 1.2) in which the antecedent suÆces relevantly for the consequent. The system E diers from the system R primarily by adding necessity to this relationship, and in this E is a modal logic as well as a relevance logic. This by itself gives good reason to consider R and not E as the paradigm of a relevance logic.1

1.2 Implication and the Conditional Before turning to matters of logical substance, let us rst introduce a framework for grammar and nomenclature that is helpful in understanding the ways that writers on relevance logic often express themselves. We draw 1 It should be entered in the record that there are some workers in relevance logic who consider both R and E too strong for at least some purposes (see [Routley, 1977], [Routley et al., 1982], and more recently, [Brady, 1996]).

RELEVANCE LOGIC

3

heavily on the `Grammatical Propaedeutic' appendix of [Anderson and Belnap, 1975] and to a lesser extent on [Meyer, 1966], both of which are very much recommended to the reader for their wise heresy from logical tradition. Thus logical tradition (think of [Quine, 1953]) makes much of the grammatical distinction between `if, then' (a connective), and `implies' or its rough synonym `entails' (transitive verbs). This tradition opposes 1. If today is Tuesday, then this is Belgium to the pair of sentences 2. `Today is Tuesday' implies `This is Belgium', 3. That today is Tuesday implies that this is Belgium. And the tradition insists that (1) be called a conditional, and that (2) and (3) be called implications. Sometimes much philosophical weight is made to rest on this distinction. It is said that since `implies' is a verb demanding nouns to ank it, that implication must then be a relation between the objects stood for by those nouns, whereas it is said that `if, then' is instead a connective combining that implication (unlike `if, then') is really a metalinguistic notion, either overtly as in (2) where the nouns are names of sentences, or else covertly as in (3) where the nouns are naming propositions (the `ghosts' of linguistic entities). This last is then felt to be especially bad because it involves ontological commitment to propositions or some equally disreputable entities. The rst is at least free of such questionable ontological commitments, but does raise real complications about `nested implications', which would seem to take us into a meta-metalanguage, etc. The response of relevance logicians to this distinction has been largely one of `What, me worry?' Sometime sympathetic outsiders have tried to apologise for what might be quickly labelled a `use{mention confusion' on the part of relevance logicians [Scott, 1971]. But `hard-core' relevance logicians often seem to luxuriate in this `confusion'. As Anderson and Belnap [1975, p. 473] say of their `Grammatical Propaedeutic': \the principle aim of this piece is to convince the reader that it is philosophically respectable to `confuse' implication or entailment with the conditional, and indeed philosophically suspect to harp on the dangers of such a `confusion'. (The suspicion is that such harpists are plucking a metaphysical tune on merely grammatical strings.)" The gist of the Anderson{Belnap position is that there is a generic conditional-implication notion, which can be carried into English by a variety of grammatical constructions. Implication itself can be viewed as a implies that ', and as connective requiring prenominalisation: `that such it nests. It is an incidental feature of English that it favours sentences with main subjects and verbs, and `implies' conforms to this reference by

4

J. MICHAEL DUNN AND GREG RESTALL

the trick of disguising sentences as nouns by prenominalisation. But such grammatical prejudices need not be taken as enshrining ontological presuppositions. Let us use the label `Correspondence Thesis' for the claim that Anderson and Belnap come close to making (but do not actually make), namely, that in general there is nothing other than a purely grammatical distinction between sentences of the forms 4. If A, then B , and 5. That A implies that B . Now undoubtedly the Correspondence Thesis overstates matters. Thus, to bring in just one consideration, [Casta~neda, 1975, pp. 66 .] distinguishes `if A then B ' from `A only if B ' by virtue of an essentially pragmatic distinction (frozen into grammar) of `thematic' emphases, which cuts across the logical distinction of antecedent and consequent. Putting things quickly, `if' introduces a suÆcient condition for something happening, something being done, etc. whereas `only if' introduces a necessary condition. Thus `if' (by itself or pre xed with `only') always introduces the state of aairs thought of as a condition for something else, then something else being thus the focus of attention. Since `that A implies that B ' is devoid of such thematic indicators, it is not equivalent at every level of analysis to either `if A then B ' or `A only if B '. It is worth remarking that since the formal logician's A ! B is equally devoid of thematic indicators, `that A implies that B ' would seem to make a better reading of it than either `if A then B ' or `A only if B '. And yet it is almost universally rejected by writers of elementary logic texts as even an acceptable reading. And, of course, another consideration against the Correspondence Thesis is produced by notorious examples like Austin's 6. There are biscuits on the sideboard if you want some, which sounds very odd indeed when phrased as an implication. Indeed, (6) poses perplexities of one kind or another for any theory of the conditional, and so should perhaps best be ignored as posing any special threat tot he Anderson and Belnap account of conditionals. Perhaps it was Austin-type examples that led Anderson and Belnap [1975, pp. 491{492] to say \we think every use of `implies' or `entails' as a connective can be replaced by a suitable `if-then'; however, the converse may not be true". They go on to say \But with reference to the uses in which we are primarily interested, we feel free to move back and forth between `if-then' and `entails' in a free-wheeling manner". Associated with the Correspondence Thesis is the idea that just as there can be contingent conditionals (e.g. (1)), so then the corresponding implications (e.g. (3)) must also be contingent. This goes against certain Quinean

RELEVANCE LOGIC

5

tendencies to `regiment' the English word `implies' so that it stands only for logical implication. Although there is no objection to thus giving a technical usage to an ordinary English word (even requiring in this technical usage that `implication' be a metalinguistic relation between sentences), the point is that relevance logicians by and large believe we are using `implies' in the ordinary non-technical sense, in which a sentence like (3) might be true without there being any logical (or even necessary) implication from `Today is Tuesday' to `This is Belgium'. Relevance logicians are not themselves free of similar regimenting tendencies. Thus we tend to dierentiate `entails' from `implies' on precisely the ground that `entails', unlike `implies', stands only for necessary implication [Meyer, 1966]. Some writings of Anderson and Belnap even suggest a more restricted usage for just logical implication, but we do not take this seriously. There does not seem to be any more linguistic evidence for thus restricting `entails' than there would be for `implies', though there may be at least more excuse given the apparently more technical history of `entails' (in its logical sense|cf. The oed). This has been an explanation of, if not an apology for, the ways in which relevance logicians often express themselves. but it should be stressed that the reader need not accept all, or any, of this background in order to make sense of the basic aims of the relevance logic enterprise. Thus, e.g. the reader may feel that, despite protestations to the contrary, Anderson, Belnap and Co. are hopelessly confused about the relationships among `entails', `implies', and `if-then', but still think that their system R provides a good formalisation of the properties of `if-then' (or at least `if-then relevantly'), and that they system E does the same for some strict variant produced by the modi er `necessarily'. One of the reasons the recent logical tradition has been motivated to insist on the erce distinction between implications and conditionals has to do with the awkwardness of reading the so-called `material conditional' A ! B as corresponding to any kind of implication (cf. [Quine, 1953]). The material conditional A ! B can of course be de ned as :A _ B , and it certainly does seem odd, modifying an example that comes by oral tradition from Anderson, to say that: 7. Picking a guinea pig up by its tail implies that its eyes will fall out. just on the grounds that its antecedent is false (since guinea pigs have no tails). But then it seems equally false to say that: 8. If one picks up a guinea pig by its tail, then its eyes will fall out. And also both of the following appear to be equally false: 9. Scaring a pregnant guinea pig implies that all of her babies will be born tailless.

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J. MICHAEL DUNN AND GREG RESTALL

10. If one scares a pregnant guinea pig, then all of her babies will be born tailless. It should be noted that there are other ways to react to the oddity of sentences like the ones above other than calling them simply false. Thus there is the reaction stemming from the work of Grice [1975] that says that at least the conditional sentences (8) and (10) above are true though nonetheless pragmatically odd in that they violate some rule based on conversational co-operation to the eect that one should normally say the strongest thing relevant, i.e. in the cases above, that guinea pigs have no tails (cf. [Fogelin, 1978, p. 136 .] for a textbook presentation of this strategy). Also it should be noted that the theory of the `counterfactual' conditional due to Stalnaker{Thomason, D. K. Lewis and others (cf. Chapter [[??]] of this Handbook ), while it agrees with relevance logic in nding sentences like (8) (not (10) false, disagrees with relevance logic in the formal account it gives of the conditional. It would help matters if there were an extended discussion of these competing theories (Anderson{Belnap, Grice, Stalnaker-Thomason-Lewis), which seem to pass like ships in the night (can three ships do this without strain to the image?) but there is not the space here. Such a discussion might include an attempt to construct a theory of a relevant counterfactual conditional (if A were to be the case, then as a result B would be the case). The rough idea would be to use say The Routley{Meyer semantics for relevance logic (cf. Section 3.7) in place of the Kripke semantics for modal logic, which plays a key role in the Stalnaker{Thomason{Lewis semantical account of the conditional (put the 3-placed alternativeness relation in the role of the usual 2-placed one). Work in this area is just starting. See the works of [Mares and Fuhrmann, 1995] and [Akama, 1997] which both attempt to give semantics for relevant counterfactuals. Also any discussion relating to Grice's work would surely make much of the fact that the theory of Grice makes much use of a basically unanalysed notion of relevance. One of Grice's chief conversational rules is `be relevant', but he does not say much about just what this means. One could look at relevance logic as trying to say something about this, at least in the case of the conditional. Incidentally, as Meyer has been at great pains to emphasise, relevance logic gives, on its face anyway, no separate account of relevance. It is not as if there is a unary relevance operator (`relevantly'). One last point, and then we shall turn to more substantive issues. Orthodox relevance logic diers from classical logic not just in having an additional logical connective (!) for the conditional. If that was the only dierence relevance logic would just be an `extension' of classical logic, using the notion of Haack [1974], in much the same way as say modal logic is an extension of classical logic by the addition of a logical connective 

RELEVANCE LOGIC

7

for necessity. The fact is (cf. Section 1.6) that although relevance logic contains all the same theorems as classical logic in the classical vocabulary say, ^; _; : (and the quanti ers), it nonetheless does not validate the same inferences. Thus, most notoriously, the disjunctive syllogism (cf. Section 2) is counted as invalid. Thus, as Wolf [1978] discusses, relevance logic does not t neatly into the classi cation system of [Haack, 1974], and might best be called `quasi-extension' of classical logic, and hence `quasi-deviant'. Incidentally, all of this applies only to `orthodox' relevance logic, and not to the `classical relevance logics' of Meyer and Routley (cf. Section 3.11).

1.3 Hilbert-style Formulations We shall discuss rst the pure implicational fragments, since it is primarily in the choice of these axioms that the relevance logics dier one from the other. We shall follow the conventions of Anderson and Belnap [Anderson and Belnap, 1975], denoting by `R! ' what might be called the `putative implicational fragment of R'. Thus R! will have as axioms all the axioms of R that only involve the implication connective. That R! is in fact the implicational fragment of R is much less than obvious since the possibility exists that the proof of a pure implicational formula could detour in an essential way through formulas involving connectives other than implication. In fact Meyer has shown that this does not happen (cf. his Section 28.3.2 of [Anderson and Belnap, 1975]), and indeed Meyer has settled in almost every interesting case that the putative fragments of the well-known relevance logics (at least R and E) are the same as the real fragments. (Meyer also showed that this does not happen in one interesting case, RM, which we shall discuss below.) For R! we take the rule modus ponens (A; A ! B ` B ) and the following axiom schemes.

A!A (A ! B ) ! [(C ! A) ! (C ! B )] [A ! (A ! B )] ! (A ! B ) [A ! (B ! C )] ! [B ! (A ! C )]

Self-Implication Pre xing Contraction Permutation:

(1) (2) (3) (4) A few comments are in order. This formulation is due to Church [1951b] who called it `The weak implication calculus'. He remarks that the axioms are the same as those of Hilbert's for the positive implicational calculus (the implicational fragment of the intuitionistic propositional calculus H) except that (1) is replaced with A ! (B ! A) Positive Paradox: (10 ) (Recent historical investigation by Dosen [1992] has shown that Orlov constructed an axiomatisation of the implication and negation fragment of R

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J. MICHAEL DUNN AND GREG RESTALL

in the mid 1920s, predating other known work in the area. Church and Moh, however, provided a Deduction Theorem (see Section 1.4) which is absent from Orlov's treatment.) The choice of the implicational axioms can be varied in a number of informative ways. Thus putting things quickly, (2) Pre xing may be replaced by (A ! B ) ! [(B ! C ) ! (A ! C )] SuÆxing. (20 ) (3) Contraction may be replaced by [A ! (B ! C )] ! [(A ! B ) ! (A ! C )] Self- Distribution,

(30 )

and (4) Permutation may be replaced by

A ! [(A ! B ) ! B ] Assertion:

(40 )

These choices of implicational axioms are `isolated' in the sense that one choice does not aect another. Thus THEOREM 1. R! may be axiomatised with modus ponens, (1) Self-Implication and any selection of one from each pair f(2); (20 )g; f(3); (30)g, and f(4); (40)g.

Proof. By consulting [Anderson and Belnap, 1975, pp. 79{80], and ddling.



There is at least one additional variant of R! that merits discussion. It turns out that it suÆces to have SuÆxing, Contraction, and the pair of axiom schemes [(A ! A) ! B ] ! B Specialised Assertion, A ![(A ! A) ! A] Demodaliser.

(4a) (4b)

Thus (4b) is just an instance of Assertion, and (4a) follows from Assertion by substitution A ! A for A and using Self-Implication to detach. That (4a) and (4b) together with SuÆxing and Contraction yield Assertion (and, less interestingly, Self-Implication) can be shown using the fact proven in [Anderson and Belnap, 1975, Section 8.3.3], that these yield (letting A~ abbreviate A1 ! A2 ) A~ ! [(A~ ! B ) ! B ] Restricted-Assertion. (400 ) The point is that (4a) and (4b) in conjunction say that A is equivalent to (A ! A) ! A, and so every formula A has an equivalent form A~ and so `Restricted Assertion' reduces to ordinary Assertion.2

2 There are some subtleties here. Detailed analysis shows that both SuÆxing and Pre xing are needed to replace A~ with A (cf. Section 1.3). Pre xing can be derived from the above set of axioms (cf. [Anderson and Belnap, 1975, pp. 77{78 and p. 26].

RELEVANCE LOGIC

9

Incidentally, no claim is made that this last variant of R! has the same isolation in its axioms as did the previous axiomatisations. Thus, e.g. that SuÆxing (and not Pre xing) is an axiom is important (a matrix of J. R. Chidgey's (cf. [Anderson and Belnap, 1975, Section 8.6]) can be used to show this. The system E of entailment diers primarily from the system R in that it is a system of relevant strict implication. Thus E is both a relevance logic and a modal logic. Indeed, de ning A =df (A ! A) ! A one nds E has something like the modality structure of S4 (cf. [Anderson and Belnap, 1975, Sections 4.3 and 10]). This suggests that E! can be axiomatised by dropping Demodaliser from the axiomatisation of R! , and indeed this is right (cf. [Anderson and Belnap, 1975, Section 8.3.3], for this and all other claims about axiomatisations of E! ).3 The axiomatisation above is a ` xed menu' in that Pre xing cannot be replaced with SuÆxing. There are other `a la carte' axiomatisations in the style of Theorem 1. THEOREM 2. E! may be axiomatised with modus ponens, Self-Implication and any selection from each of the pairs fPre xing, SuÆxingg, fContraction, Self-Distributiong and fRestricted-Permutation, Restricted-Assertiong (one from each pair). Another implicational system of less central interest is that of `ticket entailment' T! . It is motivated by Anderson and Belnap [1975, Section 6] as deriving from some ideas of Ryle's about `inference tickets'. It was motivated in [Anderson, 1960] as `entailment shorn of modality'. The thought behind this last is that there are two ways to remove the modal sting from the characteristic axiom of alethic modal logic, A ! A. One way is to add Demodaliser A ! A so as to destroy all modal distinctions. The other is to drop the axiom A ! A. Thus the essential way one gets T! from E! is to drop Specialised Assertion (or alternatively to drop Restricted Assertion or Restricted Permutation, depending on which axiomatisation of E! one has). But before doing so one must also add whichever one of Pre xing and SuÆxing was lacking, since it will no longer be a theorem otherwise (this is easiest to visualise if one thinks of dropping Restricted permutation, since this is the key to getting Pre xing from SuÆxing and vice versa ). Also (and this is a strange technicality) one must replace Self-Distribution with its permuted form: (A ! B ) ! [[A ! (B ! C )] ! (A ! C )] This is summarised in

Permuted Self-Distribution.

(300 )

3 The actual history is backwards to this, in that the system R was rst axiomatised by [Belnap, 1967a] by adding Demodaliser to E.

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J. MICHAEL DUNN AND GREG RESTALL

THEOREM 3 (Anderson and Belnap [Section 8.3.2, 1975]). T! is axiomatised using Self-Implication, Pre xing, SuÆxing, and either of fContraction, Permuted Self-Distributiong, with modus ponens. There is a subsystem of E! called TW! (and P W, and T W in earlier nomenclature) axiomatised by dropping Contraction (which corresponds to the combinator W) from T! . This has obtained some interest because of an early conjecture of Belnap's (cf. [Anderson and Belnap, 1975, Section 8.11]) that A ! B and B ! A are both theorems of TW! only when A is the same formula as B . That Belnap's Conjecture is now Belnap's Theorem is due to the highly ingenious (and complicated) work of E. P. Martin and R. K. Meyer [1982] (based on the earlier work of L. Powers and R. Dwyer). Martin and Meyer's work also highlights a system S! (for Syllogism) in which Self-Implication is dropped from TW! . Moving on now to adding the positive extensional connectives ^ and _, in order to obtain R!;^;_ (denoted more simply as R+) one adds to R! the axiom schemes

A ^ B ! A; A ^ B ! B [(A ! B ) ^ (A ! C )] ! (A ! B ^ C ) A ! A _ B; B ! A _ B [(A ! C ) ^ (B ! C )] ! (A _ B ! C ) A ^ (B _ C ) ! (A ^ B ) _ C

Conjunction Elimination Conjunction Introduction Disjunction Introduction Disjunction Elimination Distribution

(5) (6) (7) (8) (9)

plus the rule of adjunction (A; B ` A ^ B ). One can similarly get the positive intuitionistic logic by adding these all to H! . Axioms (5){(8) can readily be seen to be encoding the usual elimination and introduction rules for conjunction and disjunction into axioms, giving ^ and _ what might be called `the lattice properties' (cf. Section 3.3). It might be thought that A ! (B ! A ^ B ) might be a better encoding of conjunction introduction than (6), having the virtue that it allows for the dropping of adjunction. This is a familiar axiom for intuitionistic (and classical) logic, but as was seen by Church [1951b], it is only a hair's breadth away from Positive Paradox (A ! (B ! A)), and indeed yields it given (5) and Pre xing. For some mysterious reason, this observation seemed to prevent Church from adding extensional conjunction/disjunction to what we now call R! (and yet the need for adjunction in the Lewis formulations of modal logic where the axioms are al strict implications was well-known). Perhaps more surprising than the need for adjunction is the need for axiom (9). It would follow from the other axioms if only we had Positive Paradox among them. The place of Distribution in R is continually problematic. It causes inelegancies in the natural deduction systems (cf. Section 1.5) and is an obstacle to nding decision procedures (cf. Section 4.8). Incidentally, all of the usual distributive laws follow from the somewhat `clipped' version

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(9). The rough idea of axiomatising E+ and T+ is to add axiom schemes (5){(9) to E! and T! . This is in fact precisely right for T+ , but for E+ one needs also the axiom scheme (remember A =df (A ! A) ! A):

A ^ B ! (A ^ B )

(10)

This is frankly an inelegance (and one that strangely enough disappears in the natural deduction context of Section 1.5). It is needed for the inductive proof that necessitation (` C ) ` C ) holds, handling he case where C just came by adjunction (cf. [Anderson and Belnap, 1975, Sections 21.2.2 and 23.4]). There are several ways of trying to conceal this inelegance, but they are all a little ad hoc. Thus, e.g. one could just postulate the rule of necessitation as primitive, or one could strengthen the axiom of Restricted Permutation (or Restricted Assertion) to allow that A~ be a conjunction (A1 ! A1 ) ^ (A2 ! A2 ). As Anderson and Belnap [1975, Section 21.2.2] remark, if propositional quanti cation is available, A could be given the equivalent de nition 8p(p ! p) ! A, and then the oending (10) becomes just a special case of Conjunction Introduction and becomes redundant. It is a good time to advertise that the usual zero-order and rst-order relevance logics can be out tted with a couple of optional convenience features that come with the higher-priced versions with propositional quanti ers. Thus, e.g. the propositional constant t can be added to E+ to play the role of 8p(p ! p), governed by the axioms. (t ! A) ! A t ! (A ! A);

(11) (12)

and again (10) becomes redundant (since one can easily show (t ! A) $ [(A ! A) ! A]). Further, this addition of t is conservative in the sense that it leads to no new t-free theorems (since in any given proof t can always be replaced by (p1 ! p1 ) ^


^ (pn ! pn ) where p1 ; : : : ; pn are all the propositional variables appearing in the proof | cf. [Anderson and Belnap, 1975]). Axiom scheme (11) is too strong for T+ and must be weakened to

t:

(11T)

In the context of R+ , (11) and (11T) are interchangeable. and in R+ , (12) may of course be permuted, letting us characterise t in a single axiom as `the conjunction of all truths':

A $ (t ! A)

(13)

(in E, t may be thought of as `the conjunction of all necessary truths').

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`Little t' is distinguished from `big T ', which can be conservatively added with the axiom scheme

A!T

(14)

(in intuitionistic or classical logic t and T are equivalent). Additionally useful is a binary connective Æ, labelled variously `intensional conjunction', `fusion', `consistency' and `cotenability'. these last two labels are appropriate only in the context of R, where one can de ne A Æ B =df :(A ! :B ). One can add Æ to R+ with the axiom scheme: [(A Æ B ) ! C ] $ [A ! (B ! C )] Residuation (axiom):

(15)

This axiom scheme is too strong for other standard relevance logics, but Meyer and Routley [1972] discovered that one can always add conservatively the two way rule (A Æ B ) ! C a

` A ! (B ! C )

Residutation (rule)

(16)

(in R+ (16) yields (15)). Before adding negation, we mention the positive fragment B+ of a kind of minimal (Basic) relevance logic due to Routley and Meyer (cf. Section 3.9). B+ is just like TW+ except for nding the axioms of Pre xing and SuÆxing too strong and replacing them by rules:

A ! B ` (C ! A) ! (C ! B ) Pre xing (rule) A ! B ` (B ! C ) ! (A ! C ) SuÆxing (rule)

(17) (18)

As for negation, the full systems R, E, etc. may be formed adding to the axiom schemes for R+ , E+ , etc. the following 4 (A ! :A) ! :A Reductio (A ! :B ) ! (B ! :A) Contraposition ::A ! A Double Negation.

(19) (20) (21)

Axiom schemes (19) and (20) are intuitionistically acceptable negation principles, but using (21) one can derive forms of reductio and contraposition that are intuitionistically rejectable. Note that (19){(21) if added to H+ would give the full intuitionistic propositional calculus H. In R, negation can alternatively be de ned in the style of Johansson, with :A =df (A ! f ), where f is a false propositional constant, cf. [Meyer, 1966]. Informally, f is the disjunction of all false propositions (the `negation' of t). De ning negation thus, axiom schemes (19) and (20) become theorems 4 Reversing what is customary in the literature, we use : for the standard negation of relevance logic, reserving  for the `Boolean negation' discussed in Section 3.11. We do this so as to follow the notational policies of the Handbook.

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(being instances of Contraction and Permutation, respectively). But scheme (21) must still be taken as an axiom. Before going on to discuss quanti cation, we brie y mention a couple of other systems of interest in the literature. Given that E has a theory of necessity riding piggyback on it in the de nition A =df (A ! A) ! A, the idea occurred to Meyer of adding to R a primitive symbol for necessity  governed by the S4 axioms.

A ! A (A ! B ) ! (A ! B ) A ^ B ! (A ^ B ) A ! A; and the rule of Necessitation (` A ) ` A).

(1) (2) (3) (4)

His thought was that E could be exactly translated into this system R with entailment de ned as strict implication. That this is subtly not the case was shown by Maksimova [1973] and Meyer [1979b] has shown how to modify R so as to allow for an exact translation. Yet one more system of interest is RM (cf. Section 3.10) obtained by adding to R the axiom scheme

A ! (A ! A) Mingle:

(22)

Meyer has shown somewhat surprisingly that the pure implicational system obtained by adding Mingle to R is not the implicational fragment of RM, and he and Parks have shown how to axiomatise this fragment using a quite unintelligible formula (cf. [Anderson and Belnap, 1975, Section 8.18]). Mingle may be replaced equivalently with the converse of Contraction: (A ! B ) ! (A ! (A ! B )) Expansion:

(23)

Of course one can consider `mingled' versions of E, and indeed it was in this context that McCall rst introduced mingle, albeit in the strict form (remember A~ = A1 ! A2 ),

~ A~ ! (A~ ! A~ ) Mingle

(24)

(cf. [Dunn, 1976c]). We nish our discussion of axiomatics with a brief discussion of rstorder relevance logics, which we shall denote by RQ, EQ, etc. We shall presuppose a standard de nition of rst-order formula (with connectives :; ^; _; ! and quanti ers 8; 9). For convenience we shall suppose that we have two denumerable stocks of variables: the bound variables x; y, etc.

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J. MICHAEL DUNN AND GREG RESTALL

and the free variables (sometimes called parameters) a; b, etc. The bound variables are never allowed to have unbound occurrences. The quanti er laws were set down by Anderson and Belnap in accord with the analogy of the universal quanti er with a conjunction (or its instances), and the existential quanti er as a disjunction. In view of the validity of quanti er interchange principles, we shall for brevity take only the universal quanti er 8 as primitive, de ning 9xA =df :8x:A. We thus need

8xA ! A(a=x) 8-elimination 8x(A ! B ) ! (A ! 8xB ) 8- introduction 8x(A _ B ) ! A _ 8xB Con nement:

(25) (26) (27)

If there are function letters or other term forming operators, then (25) should be generalised to 8xA ! A(t=x), where t is any term (subject to our conventions that the `bound variables' x; y, etc. do not occur (`free') in it). Note well that because of our convention that `bound variables' do not occur free, the usual proviso that x does not occur free in A in (26) and (27) is automatically satis ed. (27) is the obvious `in nite' analogy of Distribution, and as such it causes as many technical problems for RQ as does Distribution for R (cf. Section 4.8). Finally, as an additional rule corresponding to adjunction, we need: A(a=x) Generalisation: (28) 8xA There are various more or less standard ways of varying this formulation. Thus, e.g. (cf. Meyer, Dunn and Leblanc [1974]) one can take all universal generalisations of axioms, thus avoiding the need for the rule of Generalisation. Also (26) can be `split' into two parts:

8x(A ! B ) ! (8xA ! 8xB ) A ! 8xA

Vacuous Quanti cation

(26a) (26b)

(again note that if we allowed x to occur free we would have to require that x not be free in A). The most economical formulation is due to Meyer [1970]. It uses only the axiom scheme of 8-elimination and the rule. A ! B _ C (a=x) (a cannot occur in A or B ) (29) A ! B _ 8xC which combines (26){(28).

1.4 Deduction Theorems in Relevance Logic Let X be a formal system, with certain formulas of X picked out as axioms and certain ( nitary) relations among the formulas of X picked out as rules.

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(For the sake of concreteness, X can be thought of as any of the Hilbertstyle systems of the previous section.) Where is a list of formulas of X (thought of as hypotheses ) it is customary to de ne a deduction from to be a sequence B1 ; : : : ; Bn , where for each Bi (1  i  n), either (1) Bi is in , or (2) B is an axiom of X, or (3) Bi `follows from' earlier members of the sequence, i.e. R(Bj1 ; : : : ; Bjk ; Bi ) holds for some (k + 1)|any rule R of X and Bj1 ; : : : ; Bjk all precede Bi in the sequence B1 ; : : : ; Bn . A formula A is then said to be deducible from just in case there is some deduction from terminating in A. We symbolise this as `X A (often suppressing the subscript). A proof is of course a deduction from the empty set, and a theorem is just the last item in a proof. There is the well-known (Herbrand). If A1 ; : : : ; An ; A have also A1 ; : : : ; An `H! A ! B . Deduction Theorem

`H! B ,

then we

This theorem is proven in standard textbooks for classical logic, but the standard inductive proof shows that in fact the Deduction Theorem holds for any formal system X having modus ponens as its sole rule and H!  X (i.e. each instance of an axiom scheme of H! is a theorem of X). Indeed H! can be motivated as the minimal pure implicational calculus having modus ponens as its sole rule and satisfying the Deduction Theorem. This is because the axioms of H! can all be derived as theorems in any formal system X using merely modus ponens and the supposition that X satis es the Deduction Theorem. Thus consider as an example: (1) A; B ` A De nition of ` (2) A ` B ! A (1), Deduction Theorem (3) ` A ! (B ! A) (2), Deduction Theorem: Thus the most problematic axiom of H! has a simple `a priori deduction', indeed one using only the Deduction Theorem, not even modus ponens (which is though needed for more sane axioms like Self-Distribution). It might be thought that the above considerations provide a very powerful argument for motivating intuitionistic logic (or at least some logic having he same implicational fragment) as The One True Logic. For what else should an implication do but satisfy modus ponens and the Deduction Theorem? But it turns out that there is another sensible notion of deduction. This is what is sometimes called a relevant deduction.(Anderson and Belnap [1975, Section 22.2.1] claim that this is the only sensible notion of deduction, but we need not follow them in that). If there is anything that sticks out in the a priori deduction of Positive Paradox above it is that in (1), B was not used in the deduction of A. A number of researchers have been independently bothered by this point and have been motivated to study a relevant implication that goes hand in

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hand with a notion of relevant deduction. This, in this manner Moh [1950] and Church [1951b] came up with what is in eect R! . And Anderson and Belnap [1975, p. 261] say \In fact, the search for a suitable deduction theorem for Ackermann's systems : : : provided the impetus leading us to the research reported in this book." This research program begun in the late 1950s took its starting point in the system(s) of Ackermann [1956], and the bold stroke separating the Anderson{Belnap system E from Ackermann's system 0 was basically the dropping of Ackermann's rule so as to have an appropriate deduction theorem (cf. Section 2.1). Let us accordingly de ne a deduction of B from A1 ; : : : ; An to be relevant with respect to a given hypothesis Ai just in case Ai is actually used in the given deduction of B in the sense (paraphrasing [Church, 1951b]) that there is a chain of inferences connecting Ai with the nal formula B . This last can be made formally precise in any number of ways, but perhaps the most convenient is to ag Ai with say a ] and to pass the ag along in the deduction each time modus ponens is applied to two items at least one of which is agged. It is then simply required that the last step of the deduction (B ) be agged. Such devices are familiar from various textbook presentations of classical predicate calculus when one wants to keep track whether some hypothesis Ai (x) was used in the deduction of some formula B (x) to which one wants to apply Universal Generalisation. We shall de ne a deduction of B from A1 ; : : : ; An to be relevant simpliciter just in case it is relevant with respect to each hypothesis Ai . A practical way to test for this is to ag each Ai with a dierent ag (say the subscript i) and then demand that all of the ags show up on the last step B. We can now state a version of the Relevant Deduction Theorem (Moh, Church). If there is a deduction in R! of B from A1 ; : : : ; An ; A that is relevant with respect to A, then there is a deduction in R! of A ! B from A1 ; : : : ; An . Furthermore the new deduction will be `as relevant' as the old one, i.e. any Ai that was used in the given deduction will be used in the new deduction.

Proof. Let the given deduction be B1 ; : : : ; Bk , and let it be given with a particular analysis as to how each step is justi ed. By induction we show for each Bi that if A was used in obtaining Bi (Bi is agged), then there is a deduction of A ! Bi from A1 ; : : : ; An , and otherwise there is a deduction of Bi from those same hypotheses. The tedious business of checking that the new deduction is as relevant as the old one is left to the reader. We divide up cases depending on how the step Bi is justi ed. Case 1. Bi was justi ed as a hypothesis. Then neither Bi is A or it is some Aj . But A ! A is an axiom of R! (and hence deducible from A1 ; : : : ; An ), which takes care of the rst alternative. And clearly on the second alternative Bi is deducible from A1 ; : : : ; An (being one of them).

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Case 2. Bi was justi ed as an axiom. Then A was not used in obtaining Bi , and of course Bi is deducible (being an axiom). Case 3. Bi was justi ed as coming from preceding steps Bj ! Bi and Bj by modus ponens. There are four subcases depending on whether A was used in obtaining the premises. Subcase 3.1. A was used in obtaining both Bj ! Bi and Bj . Then by inductive hypothesis A1 ; : : : ; An `R! A ! (Bj ! Bi ) and A1 ; : : : ; An `R! A ! Bj . So A ! B may be obtained using the axiom of Self-Distribution. Subcase 3.2. A was used in obtaining Bj ! Bi but not Bj . Use the axiom of Permutation to obtain A ! Bi from A ! (Bj ! Bi ) and Bj . Subcase 3.3. A was not used in obtaining Bj ! Bi but was used for Bj . Use the axiom of Pre xing to obtain A ! Bi from Bj ! Bi and A ! Bj . Subcase 3.4. A was not used in obtaining either Bj ! Bi nor Bj . Then Bi follows form these using just modus ponens. Incidentally, R! can easily be veri ed to be the minimal pure implicational calculus having modus ponens as sole rule and satisfying the Relevant Deduction Theorem, since each of the axioms invoked in the proof of this theorem can be easily seen to be theorems in any such system (cf. the next section for an illustration of sorts). There thus seem to be at least two natural competing pure implicational logics R! and H! , diering only in whether one wants one's deductions to be relevant or not.5 

Where does the Anderson{Belnap's [1975] preferred system E! t into all of this? The key is that the implication of E! is both a strict and a relevant implication (cf. Section 1.3 for some subtleties related to this claim). As such, and since Anderson and Belnap have seen t to give it the modal structure of the Lewis system S4, it is appropriate to recall the appropriate deduction theorem for S4. [Barcan Marcus, 1946] If A1 ! B1 ; : : : ; An ! Bn ; A `S4 B (! here denotes strict implication), then A1 ! B1 ; : : : ; An ! Bn `S4 A ! B . Modal Deduction Theorem

The idea here is that in general in order to derive the strict (necessary ) implication A ! B one must not only be able to deduce B from A and some other hypotheses but furthermore those other hypotheses must be supposed to be necessary. And in S4 since Ai ! Bj is equivalent to (Ai ! Bj ), requiring those additional hypotheses to be strict implications at least suÆces for this. Thus we could only hope that E! would satisfy the 5 This seems to dier from the good-humoured polemical stand of Anderson and Belnap [1975, Section 22.2.1], which says that the rst kind of `deduction', which they call (pejoratively) `OÆcial deduction', is no kind of deduction at all.

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[Anderson and Belnap, 1975] If there is a deduction in E! of B from A1 ! B1 ; : : : ; An ! Bn ; A that is relevant with respect to A, then there is a deduction in E! of A ! B from A1 ! B1 ; : : : ; An ! Bn that is as relevant as the original. Modal Relevant Deduction Theorem

The proof of this theorem is somewhat more complicated than its unmodalised counterpart which we just proved (cf. [Anderson and Belnap, 1975, Section 4.21] for a proof). We now examine a subtle distinction (stressed by Meyer|see, for example, [Anderson and Belnap, 1975, pp. 394{395]), postponed until now for pedagogical reasons. We must ask, how many hypotheses can dance on the head of a formula? The question is: given the list of hypotheses A, A, do we have one hypothesis or two? When the notion of a deduction was rst introduced in this section and a `list' of hypotheses was mentioned, the reader would naturally think that this was just informal language for a set. And of course the set fA; Ag is identical to the set fAg. Clearly A is relevantly deducible from A. The question is whether it is so deducible from A; A. We have then two dierent criteria of use, depending on whether we interpret hypotheses as grouped together into lists that distinguish multiplicity of occurrences (sequences)6 or sets. This issue has been taken up elsewhere of late, with other accounts of deduction appealing to `resource consciousness' [Girard, 1987; Troelstra, 1992; Schroeder-Heister and Dosen, 1993] as motivating some non-classical logics. Substructural logics in general appeal to the notion that the number of times a premise is used, or even more radically, the order in which premises are used, matter. At issue in R and its neighbours is whether A ! (A ! A) is a correct relevant implication (coming by two applications of `The Deduction Theorem' from A; A ` A). This is in fact not a theorem of R, but it is the characteristic axiom of RM (cf. Section 1.3). So it is important that in the Relevant Deduction Theorem proved for R! that the hypotheses A1 ; : : : ; An be understood as a sequence in which the same formula may occur more than once. One can prove a version of the Relevant Deduction Theorem with hypotheses understood as collected into a set for the system RMO! , obtained by adding A ! (A ! A) to R! (but the reader should be told that Meyer has shown that RMO!, is not the implicational fragment of RM, cf. [Anderson and Belnap, 1975, Section 8.15]).7 6 Sequences are not quite the best mathematical structures to represent this grouping since it is clear that the order of hypotheses makes no dierence (at least in the case of R). Meyer and McRobbie [1979] have investigated ` resets' ( nitely repeatable sets) as the most appropriate abstraction. 7 Arnon Avron has defended this system, RMO! , as a natural way to characterise relevant implication [Avron, 1986; Avron, 1990a; Avron, 1990b; Avron, 1990c; Avron, 1992]. In Avron's system, conjunction and disjunction are intensional connectives, de ned in terms of the implication and negation of RMO! . As a result, they do

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Another consideration pointing to the naturalness of R! is its connection to the I -calculus. A formula is a theorem of R! if and only if it is the type of a closed term of the I -calculus as de ned by Church. A I term is a  term in which every lambda abstraction
binds at least one free variable. So, x:y:xy has type A ! (A ! B ) ! B , and so, is a theorem of R! , while x:y:x, has type A ! (B ! A), which is an intuitionistic theorem, but not an R! theorem. This is re ected in the  term, in which the y does not bind a free variable. We now brie y discuss what happens to deduction theorems when the pure implication systems R! and E! are extended to include other connectives, especially ^. R will be the paradigm, its situation extending straight-forwardly to E. The problem is that the full system R seems not to be formulable with modus ponens as the sole rule; there is also need for adjunction (A; B ` A ^ B ) (cf. Section 1.3). Thus when we think of proving a version of the Relevant Deduction Theorem for the full system R, it would seem that we are forced to think through once more the issue of when a hypothesis is used, this time with relation to adjunction. It might be thought that the thing to do would be to pass the

ag ] along over an application of adjunction so that A ^ B ends up agged if either of the premises A or B was agged, in obvious analogy with the decision concerning modus ponens. Unfortunately, that decision leads to disaster. For then the deduction A; B ` A ^ B would be a relevant one (both A and B would be `used'), and two applications of `The Deduction Theorem' would lead to the thesis A ! (B ! A ^ B ), the undesirability of which has already been remarked. A more appropriate decision is to count hypotheses as used in obtaining A^B just when they were used to obtain both premises. This corresponds to the axiom of Conjunction Introduction (C ! A) ^ (C ! B ) ! (C ! A ^ B ), which thus handles the case in the inductive proof of the deduction theorem when the adjunction is applied. This decision may seem ad hoc (perhaps `use' simpliciter is not quite the right concept), but it is the only decision to be made unless one wants to say that the hypothesis A can (in the presence of the hypothesis B ) be `used' to obtain A ^ B and hence B (passing on the

ag from A this way is something like laundering dirty money). This is the decision that was made by Anderson and Belnap in the context of natural deduction systems (see next section), and it was applied by Kron [1973; 1976] in proving appropriate deduction theorems for R, E (and T). It should be said that the appropriate Deduction Theorem requires simultaneous agging of the hypothesis (distinct ags being applied to each formula occurrence, say using subscripts in the manner of the `practical suggestion' after our de nition of relevant deduction for R! ), with the requirement that all of the subscripts are passed on to the conclusion. So the not have all of the distributive lattice properties of traditional relevance logics.

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J. MICHAEL DUNN AND GREG RESTALL

Deduction Theorem applies only to fully relevant deductions, where every premise is used (note that no such restriction was placed on the Relevant Deduction Theorem for R! ). An alternative stated in Meyer and McRobbie [1979] would be to adjust the de nition of deduction, modifying clause (2) so as to allow as a step in a deduction any theorem (not just axiom) of R, and to restrict clause (3) so that the only rule allowed in moving to later steps is modus ponens.8 This is in eect to restrict adjunction to theorems, and reminds one of similar restrictions in the context of deduction theorems of similarly restricting the rules of necessitation and universal generalisation. It has the virtue that the Relevant Deduction Theorem and its proof are the same as for R! . (Incidentally, Meyer's and Kron's sense of deduction coincide when all of A1 ; : : : ; An are used in deducing B ; this is obvious in one direction, and less than obvious in the other.) There are yet two other versions of the deduction theorem that merit discussion in the context of relevance logic (relevance logic, as Meyer often points out, allows for many distinctions). First in Belnap [1960b] and Anderson and Belnap [1975], there is a theorem (stated for E, but we will state it for our paradigm R) called The Entailment Theorem, which says that A1 ; : : : ; An `entails' B i `R (A1 ^ : : : ^ An ) ! B . A formula B is de ned in eect to be entailed by hypothesis A1 ; : : : ; An just in case there is a deduction of B using their conjunction A1 ^ : : : ^ An . Adjunction is allowed, but subject to the restriction that the conjunctive hypothesis was used in obtaining both premises. The Entailment Theorem is clearly implied by Kron's version of the Deduction Theorem. The last deduction theorem for R we wish to discuss is the Enthymematic Deduction Theorem (Meyer, Dunn and Leblanc [1974]). If A1 ; : : : ; An ; A `R B , then A1 ; : : : ; An `R A ^ t ! B . Here ordinary deducibility is all that is at issue (no insistence on the hypotheses being used). It can either be proved by induction, or cranked out of one of the more relevant versions of the deduction theorem. Thus it falls out of the Entailment Theorem that

`R X ^ A ^ T ! B; where X is the conjunction of A1 ; : : : ; An , and T is the conjunction of all the axioms of R used in the deduction of B . But since `R t ! T , we have `R X ^ A ^ t ! B . 8 Of course this requires we give an independent characterisation of proof (and theorem), since we can no longer de ne a proof as a deduction from zero premisses. We thus de ne a proof as a sequence of formulas, each of which is either an axiom or follows from preceding items by either modus ponens or adjunction (!).

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However, the following R theorem holds:

`R (X ^ A ^ t ! B ) ! (X ^ t ! (A ^ t ! B )): So `R X ^ t ! (A ^ t ! B ), which leads (using `R t) to X `R A ^ t ! B , which dissolving the conjunction gives the desired

A1 ; : : : ; An `R A ^ t ! B: In view of the importance of the notion, let us symbolise A ^ t ! B as A !t B . This functions as a kind of `enthymematic implication' (A and some truth really implies B ) and there will be more about Anderson, Belnap and Meyer's investigations of this concept in Section 1.7. Let us simply note now that in the context of deduction theorems, it functions like intuitionistic implication, and allows us in R! to have two dierent kinds of implication, each well motivated in its relation to the two dierent kinds of deducibility (ordinary and relevant).9 For a more extensive discussion of deduction theorems in relevance logics and related systems, more recent papers by Avron [1991] and Brady [1994] should be consulted.

1.5 Natural Deduction Formulations We shall be very brief about these since natural deduction methods are amply discussed by Anderson and Belnap [1975], where such methods in fact are used s a major motivation for relevance logic. Here we shall concentrate on a natural deduction system NR for R. The main idea of natural deduction (cf. Chapters [[were I.1 and I.2]] of the Handbook ) of course is to allow the making of temporary hypotheses, with some device usually being provided to facilitate the book-keeping concerning the use of hypotheses (and when their use is `discharged'). Several textbooks (for example, [Suppes, 1957] and [Lemmon, 1965])10 have used the device of in eect subscripting each hypothesis made with a distinct numeral, and then passing this numeral along with each application of a rule, thus keeping track of which hypothesis are used. When a hypothesis is discharged, the subscript is dropped. A line obtained with no subscripts is a `theorem' since it depends on no hypotheses. Let us then let ; , etc. range over classes of numerals. The rules for ! are then naturally:

A ! B A B[

[!E ]

Afkg .. . B A ! B fkg

[!I ] (provided k 2 )

9 In E enthymematic implication is like S4 strict implication. See [Meyer, 1970a]. 10 The idea actually originates with [Feys and Ladriere, 1955].

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J. MICHAEL DUNN AND GREG RESTALL

Two fussy, really incidental remarks must be made. First, in the rule !E it is to be understood that the premises need not occur in the order listed, nor need they be adjacent to each other or to the conclusion. Otherwise we would need a rule of `Repetition', which allows the repeating of a formula with its subscripts as a later line. (Repetition is trivially derivable given our `non-adjacent' understanding of !E |in order to repeat A , just prove A ! A and apply !E .) Second, it is understood that we have what one might call a rule of `Hypothesis Introduction': anytime one likes one can write a formula as a line with a new subscript (perhaps most conveniently, the line number). Now a non-fussy remark must be made, which is really the heart of the whole matter. In the rule for !I , a proviso has been attached which has the eect of requiring that the hypothesis A was actually used in obtaining B . This is precisely what makes the implication relevant (one gets the intuitionistic implication system H! if one drops this requirement). The reader should nd it instructive to attempt a proof of Positive Paradox (A ! (B ! A)) and see how it breaks down for NR! (but succeeds in NH! . The reader should also construct proofs in NR! of all the axioms in one of the Hilbert-style formulations of R! from Section 1.3. Then the equivalence of R! in its Hilbert-style and natural deduction formulations is more or less self-evident given the Relevant Deduction Theorem (which shows that the rule ! I can be `simulated' in the Hilbert-style system, the only point at issue). Indeed it is interesting to note that Lemmon [1965], who seems to have the same proviso on !I that we have for NR! (his actual language is a bit informal), does not prove Positive Paradox until his second chapter adding conjunction (and disjunction) to the implication-negation system he developed in his rst chapter. His proof of Positive Paradox depends nally upon an `irrelevant' ^I rule. The following is perhaps the most straightforward proof in his system (diering from the proof he actually gives): (1) (2) (3) (4) (5) (6)

A1 B2 A ^ B1;2 A1;2 B ! A1 A ! (B ! A)

Hyp Hyp 1; 2; ^I ? 3; ^E 2; 4; ! I 1; 5; ! I .

We think that the manoeuvre used in getting B 's 2 to show up attached to A in line (4) should be compared to laundering dirty money by running it through an apparently legitimate business. The correct `relevant' versions

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of the conjunction rules are instead

A B A ^ B

[^I ]

A ^ B A

A ^ B B

[^E ]

What about disjunction? In R (also E, etc.) one has de Morgan's Laws and Double Negation, so one can simply de ne A _ B = :(:A ^ :B ). One might think that settling down in separate int-elim rules for _ would then only be a matter of convenience. Indeed, one can nd in [Anderson and Belnap, 1975] in eect the following rules:

A A _ B

B A _ B

[_I ]

A _ B .. . Ak .. . C [fkg Bh .. . C [fhg C[

[_E ]

But (as Anderson and Belnap point out) these rules are insuÆcient. >From them one cannot derive the following

A ^ (B _ C ) Distribution: (A ^ B ) _ C And so it must be taken as an additional rule (even if disjunction is de ned from conjunction and negation). This is clearly an unsatisfying, if not unsatisfactory, state of aairs. The customary motivation behind int-elim rules is that they show how a connective may be introduced into and eliminated from argumentative discourse (in which it has no essential occurrence), and thereby give the connective's role or meaning. In this context the Distribution rule looks very much to be regretted. One remedy is to modify the natural deduction system by allowing hypotheses to be introduced in two dierent ways, `relevantly' and `irrelevantly'. The rst way is already familiar to us and is what requires a subscript to keep track of the relevance of the hypothesis. It requires that the hypotheses introduced this way will all be used to get the conclusion. The second way involves only the weaker promise that at least some of the hypotheses so introduced will be used. This suggestion can be formalised by

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allowing several hypotheses to be listed on a line, but with a single relevance numeral attached to them as a bunch. Thus, schematically, an argument of the form (1) A; B1 (2) C; D2 .. . (k) E1;2 should be interpreted as establishing

A ^ B ! (C ^ D ! E ): Now the natural deduction rules must be stated in a more general form allowing for the fact that more than one formula can occur on a line. Key among these would be the new rule:

; A _ B .. . ; Ak .. . [_E 0 ]
[fkg ; Bl .. .
[flg
[ It is fairly obvious that this rule has Distribution built into it. Of course, other rules must be suitably modi ed. It is easiest to interpret the formulas on a line as grouped into a set so as not to have to worry about `structural rules' corresponding to the commutation and idempotence of conjunction. The rules !I; !E; _I; _E; ^I , and ^E can all be left as they were (or except for !I and !E , trivially generalised so as to allow for the fact that the premises might be occurring on a line with several other `irrelevant' premises), but we do need one new structural rule: 


 ;


[Comma I ]

Once we have this it is natural to take the conjunction rules in `Ketonen form': ; A; B [^I 0 ] ; A ^ B

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; A ^ B [^E 0 ] ; A; B with the rule

;
 

[Comma E ]

It is merely a tedious exercise for the reader to show that this new system N 0 R is equivalent to NR. Incidentally, N 0 R was suggested by re ection upon the Gentzen System LR+ of Section 4.9. Before leaving the question of natural deduction for R, we would like to mention one or two technical aspects. First, the system of Prawitz [1965] diers from R in that it lacks the rule of Distribution. This is perhaps compensated for by the fact that Prawitz can prove a normal form theorem for proofs in his system. A dierent system yet is that of [Pottinger, 1979], based on the idea that the correct ^I rule is A B A ^ B[ He too gets a normal form theorem. We conjecture that some appropriate normal form theorem is provable for the system N 0 R+ on the well-known analogy between cut-elimination and normalisation and the fact that cutelimination has been proven for LR+ (cf. Section 4.9). Negation though would seem to bring extra problems, as it does when one is trying to add it to LR+ . One last set of remarks, and we close the discussion of natural deduction. The system NR above diers from the natural deduction system for R of Anderson and Belnap [1975]. Their system is a so-called `Fitch-style' formalism, and so named F R. The reader is presumed to know that in this formalism when a hypothesis is introduced it is thought of as starting a subproof, and a line is drawn along the left of the subproof (or a box is drawn around the subproof, or some such thing) to demarcate the scope of the hypothesis. If one is doing a natural deduction system for classical or intuitionistic logic, subproofs or dependency numerals can either one be used to do essentially the same job of keeping track the use of hypotheses (though dependency numerals keep more careful track, and that is why they are so useful for relevant implication). Mathematically, a Fitch-style proof is a nested structure, representing the fact that subproofs can contain further subproofs, etc. But once one has dependency numerals, this extra structure, at least for R, seems otiose, and so we have dispensed with it. The story for E is more complex, since

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on the Anderson and Belnap approach E diers from R only in what is allowed to be `reiterable' into subproof. Since implication in E is necessary as well as relevant, the story is that in deducing B from A in order to show A ! B , one should only be allowed to use items that have been assumed to be necessarily true, and that these can be taken to be formulas of the form C ! D. So only formulas of this form can be reiterated for use in the subproof from A to B . Working out how best to articulate this idea using only dependency numerals (no lines, boxes, etc.) is a little messy. This concern to keep track of how premises are used in a proof by way of labels has been taken up in a general way by recent work on Labelled Deductive Systems [D'Agostino and Gabbay, 1994; Gabbay, 1997]. We would be remiss not to mention other formulations of natural deduction systems for relevance logics and their cousins. A dierent generalisation of Hunter's natural deduction systems (which follows more closely the Gentzen systems for positive logics | see Section 4.9) is in [Read, 1988; Slaney, 1990].11

1.6 Basic Formal Properties of Relevance Logic This section contains a few relatively simple properties of relevance logics, proofs for which can be found in [Anderson and Belnap, 1975]. With one exception (the `Ackermann Properties'|see below), these properties all hold for both the system R and E, and indeed for most of the relevance logics de ned in Section 1.3. For simplicity, we shall state these properties for sentential logics, but appropriate versions hold as well for their rst-order counterparts. First we examine the Replacement Theorem For both R and E,

` (A $ B ) ^ t ! ((A) $ (B )): Here (A) is any formula with perhaps some occurrences of A and (B ) is the result of perhaps replacing one or more of those occurrences by B . The proof is by a straightforward induction on the complexity of (A), and one clear role of the conjoined t is to imply  !  when (= (A)) contains no occurrences of A, or does but none of them is replaced by B . It might be thought that if these degenerate cases are ruled out by requiring that some actual occurrence of A be replaced by B , then the need for t would vanish. This is indeed true for the implication-negation (and of course the pure implication) fragments of R and E, but not for the whole systems in virtue of the non-theoremhood of what V. Routley has dubbed `Factor': 11 The reader should be informed that still other natural deduction formalisms for R of various virtues can be found in [Meyer, 1979b] and [Meyer and McRobbie, 1979].

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1. (A ! B ) ! (A ^  ! B ^ ). Here the closest one can come is to 2. (A ! B ) ^ t ! (A ^  ! B ^ ),

the conjoined g giving the force of having  !  in the antecedent, and the theorem (A ! B ) ^ ( ! ) ! (A ^  ! B ^ ) getting us home. (2) of course is just a special case of the Replacement Theorem. Of more `relevant' interest is the Variable Sharing Property. If A ! B is a theorem of R (or E), then there exists some sentential variable p that occurs in both A and B . This is understood by Anderson and Belnap as requiring some commonality of meaning between antecedent and consequent of logically true relevant implications. The proof uses an ingenious logical matrix, having eight values, for which see [Anderson and Belnap, 1975, Section 22.1.3]. There are discussed both the original proof of Belnap and an independent proof of Doncenko, and strengthening by Maksimova. Of modal interest is the Ackermann Property. No formula of the form A ! (B ! C ) (A containing no !) is a theorem of E. The proof again uses an ingenious matrix (due to Ackermann) and has been strengthened by Maksimova (see [Anderson and Belnap, 1975, Section 22.1.1 and Section 22.1.2]) (contributed by J. A. Coa) on `fallacies of modality'.

1.7 First-degree Entailments

A zero degree formula contains only the connectives ^; _, and :, and can be regarded as either a formula of relevance logic or of classical logic, as one pleases. A rst degree implication is a formula of the form A ! B , where both A and B are zero-degree formulas: Thus rst degree implications can be regarded as either a restricted fragment of some relevance logic (say R or E) or else as expressing some metalinguistic logical relation between two classical formulas A and B . This last is worth mention, since then even a classical logician of Quinean tendencies (who remains unconverted by the considerations of Section 1.2 in favour of nested implications) can still take rst degree logical relevant implications to be legitimate. A natural question is what is the relationship between the provable rstdegree implications of R and those of E. It is well-known that the corresponding relationship between classical logic and some normal modal logic, say S4 (with the ! being the material conditional and strict implication, respectively), is that they are identical in their rst degree fragments. The same holds of R and E (cf. [Anderson and Belnap, 1975, Section 2.42]). This fragment, which we shall call Rfde (Anderson and Belnap [1975] call it Efde ) is stable (cf. [Anderson and Belnap, 1975, Section 7.1]) in the sense

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that it can be described from a variety of perspectives. For some semantical perspectives see Sections 3.3 and 3.4. We now consider some syntactical perspectives of more than mere `orthographic' signi cance. The perhaps least interesting of these perspectives is a `Hilbert-style' presentation of Rfde (cf. [Anderson and Belnap, 1975, Section 15.2]). It has the following axioms: 3. A ^ B ! A; A ^ B ! B 4. A ! A _ B; B ! A _ B

Conjunction Elimination Disjunction Introduction

5. A ^ (B _ C ) ! (A ^ B ) _ C 6. A ! ::A; ::A ! A

Distribution Double Negation

It also has gobs of rules: 7. A ! B; B ! C ` A ! C

8. A ! B; A ! C ` A ! B ^ C 9. A ! C; B ! C ` A _ B ! C

Transitivity Conjunction Introduction Disjunction Introduction

10. A ! B ` :B ! :A

Contraposition. More interesting is the characterisation of Anderson and Belnap [1962b; 1975] of Rfde as `tautological entailments'. The root idea is to consider rst the `primitive entailments'. 11. A1 ^ : : : ^ Am ! B1 _ : : : _ Bn , where each Ai and Bj is either a sentential variable or its negate (an `atom') and make it a necessary and suÆcient criterion for such a primitive entailment to hold that same Ai actually be identically the same formula as some Bj (that the entailment be `tautological' in the sense that Ai is repeated). This rules out both 12. p ^ :p ! q, 13. p ! q _ :q,

where there is no variable sharing, but also such things as 14. p ^ :p ^ q ! :q, where there is (of course all of (12){(14) are valid classically, where a primitive entailment may hold because of atom sharing or because either the antecedent is contradictory or else the consequent is a logical truth). Now the question remains as to which non-primitive entailments to count as valid. Both relevance logic and classical logic agree on the standard

RELEVANCE LOGIC

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count as valid. Both relevance logic and classical logic agree on the standard `normal form equivalences': commutation, association, idempotence, distribution, double negation, and de Morgan's laws. So the idea is, given a candidate entailment A ! B , by way of these equivalences, A can be put into disjunctive normal form and B may be put into conjunctive normal form, reducing the problem to the question of whether the following is a valid entailment: 15. A1 _


_ Ak ! B1 ^


^ Bh . But simple considerations (on which both classical and relevance logic agree) having to do with conjunction and disjunction introduction and elimination show that (15) holds if for each disjunct Ai and conjunct Bj , the primitive entailment Ai ! Bj is valid. For relevance logic this means that there must be atom sharing between the conjunction Ai and the disjunction Bj . This criterion obviously counts the Disjunctive Syllogism 16. :p ^ (p _ q) ! q, as an invalid entailment, for using distribution to put its antecedent into disjunctive normal form, (16) is reduced to 160 (:p ^ p) _ (:p ^ q) ! q. But by the criterion of tautological entailments, 17. :p ^ p ! q,

which is required for the validity of (160 ), is rejected. Another pleasant characterisation of Rfde is contained in [Dunn, 1976a] using a simpli cation of Jerey's `coupled trees' method for testing classically valid entailments. The idea is that to test A ! B one works out a truth-tree for A and a truth tree for B . One then requires that every branch in the tree for A `covers' some branch in the tree for B in the sense that every atom in the covered branch occurs in the covering branch. This has the intuitive sense that every way in which A might be true is also a way in which B would be true, whether these ways are logically possible or not, since `closed' branches (those containing contradictions) are not exempt as they are in Jerey's method for classical logic. This coupled-trees approach is ultimately related to the Anderson{Belnap tautological entailment method, as is also the method of [Dunn, 1980b] which explicates an earlier attempt of Levy to characterise entailment (cf. also [Clark, 1980]).

1.8 Relations to Familiar Logics There is a sense in which relevance logic contains classical logic.

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J. MICHAEL DUNN AND GREG RESTALL

(Anderson and Belnap [1959a]). The zero-degree formulas (those containing only the connectives ^; _; :) provable in R (or E) are precisely the theorems of classical logic. ZDF Theorem

The proof went by considering a `cut-free' formulation of classical logic whose axioms are essentially just excluded middles (which are theorems of R / E) and whose rules are all provable rst-degree relevant entailments (cf. Section 2.7). This result extends to a rst-order version [Anderson and Belnap Jr., 1959b]. (The admissibility of (cf. Section 2) provides another route to the proof to the ZDF Theorem.) There is however another sense in which relevance logic does not contain classical logic: (Anderson and Belnap [1975, Section 25.1]). R (and E) lack as a derivable rule Disjunctive Syllogism: Fact

:A; A _ B ` B: This is to say there is no deduction (in the standard sense of Section 1.4) of B from :A and A _ B as premises. This is of course the most notorious feature of relevance logic, and the whole of Section 2 is devoted to its discussion. Looking now in another direction, Anderson and Belnap [1961] began the investigation of how to translate intuitionistic and strict implication into R and E, respectively, as `enthymematic' implication. Anderson and Belnap's work presupposed the addition of propositional quanti es to, let us say R, with the subsequent de nition of `A intuitionistically implies B ' (in symbols A  B ) as 9p(p ^ (A ^ p ! B )). This has the sense that A together with some truth relevantly implies B , and does seem to be at least in the neighbourhood of capturing Heyting's idea that A  B should hold if there exists some `construction' (the p) which adjoined to A `yields' (relevant implication) B . Meyer in a series of papers [1970a; 1973] has extended and simpli ed these ideas, using the propositional constant t in place of propositional quanti cation, de ning A  B as A ^ t ! B . If a propositional constant F for the intuitionistic absurdity is introduced, then intuitionistic negation can be de ned in the style of Johansson as :A =df A  F . As Meyer has discovered one must be careful what axiom one chooses to govern F . F ! A or even F  A is too strong. In intuitionistic logic, the absurd proposition intuitionistically implies only the intuitionistic formulas, so the correct axiom is F  A , where A is a translation into R of an intuitionistic formula. Similar translations carry S4 into E and classical logic into R.

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2 THE ADMISSIBILITY OF

2.1 Ackermann's Rule The rst mentioned problem for relevance logics in Anderson's [1963] seminal `open problems' paper is the question of `the admissibility of '. To demystify things a bit it should be said that is simply modus ponens for the material conditions (:A _ B ): 1.

A

:A _ B : B

It was the third listed rule of Ackermann's [1956] system of strenge Implikation (; ; ; 1st, 2nd, 3rd). This was the system Anderson and Belnap `tinkered with' to produce E (Ackermann also had a rule Æ which they replaced with an axiom). The major part of Anderson and Belnap's `tinkering' was the extremely bold step of simply deleting as a primitive rule, on the well- motivated ground that the corresponding object language formula 2. A ^ (:A _ B ) ! B is not a theorem of E. It is easy to see that (2) could not be a theorem of either E or R, since it is easy to prove in those systems 3. A ^ :A ! A ^ (:A _ B ) (largely because :A ! :A _ B is an instance of an axiom), and of course (3) and (2) yield by transitivity the `irrelevancy' 4. A ^ :A ! B . The inference (1) is obviously related to the Stoic principle of the disjunctive syllogism :

:A 5. A _ B : B Indeed, given the law of double negation (and replacement) they are equivalent, and double negation is never at issue in the orthodox logics. Thus E and R reject 6. :A ^ (A _ B ) ! B

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J. MICHAEL DUNN AND GREG RESTALL

as well as (2). This rejection is typically the hardest thing to swallow concerning relevance logics. One starts o with some pleasant motivations about relevant implication and using subscripts to keep track of whether a hypothesis has actually been used (as in Section 1.5), and then one comes to the point where one says `and of course we have to give up the disjunctive syllogism' and one loses one's audience. Please do not stop reading! We shall try to make this rejection of disjunctive syllogism as palatable as we can. (See [Belnap and Dunn, 1981; Restall, 1999] for related discussions, and also discussion of [Anderson and Belnap, 1975, Section 16.1]); see Burgess [1981] for an opposing point of view.

2.2 The Lewis `Proof' One reason that disjunctive syllogism has gured so prominently int he controversy surrounding relevance logic is because of the use it was put to by C. I. Lewis [Lewis and Langford, 1932] in his so-called `independent proof': that a contradiction entails any sentence whatsoever (taken by Anderson and Belnap as a clear breakdown of relevance). Lewis's proof (with our notations of justi cation) goes as follows: (1) (2) (3) (4) (5)

p ^ :p p :p p_q q

2, ^-Elimination 1, ^-Elimination 2, _-Introduction 3, 4 disjunctive syllogism

Indeed one can usefully classify alternative approaches to relevant implication according to how they reject the Lewis proof. Thus, e.g. Nelson rejects ^-Elimination and _-Introduction, as does McCall's connexive logic. Parry, on the other hand, rejects only _-Introduction. Geach, and more recently, Tennant [1994],accept each step, but says that `entailment' (relevant implication) is not transitive. It is the genius of the Anderson{Belnap approach to see disjunctive syllogism as the culprit and the sole culprit.12 Lewis concludes his proof by saying, \If by (3), p is false; and, by (4), at least one of the two, p and q is true, then q must be true". As is told in [Dunn, 1976a], Dunn was saying such a thing to an elementary logic class one time (with no propaganda about relevance logic) when a student yelled out, \But p was the true one|look again at your assumption". 12 Although this point is complicated, especially in some of their earlier writings (see, e.g. [Anderson and Belnap Jr., 1962a]) by the claim that there is a kind of fallacy of ambiguity in the Lewis proof. the idea is that if _ is read in the `intensional' way (as :A ! B), then the move from (3) and (4) to (5) is ok (it's just modus ponens for the relevant conditional), but the move from (2) to (4) is not (now being a paradox of implication rather than ordinary disjunction introduction).

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That student had a point. Disjunctive syllogism is not obviously appropriate to a situation of inconsistent information|where p is assumed (given, believed, etc.) to be both true and false. This point has been argued strenuously in, e.g. [Routley and Routley, 1972; Dunn, 1976a] and Belnap [1977b; 1977a]. The rst two of these develop a semantical analysis that lets both p and :p receive the value `true' (as is appropriate to model the situation where p ^ :p has been assumed true), and there will be more about these ideas in Section 3.4. The last is particularly interesting since it extends the ideas of Dunn [1976a] so as to provide a model of how a computer might be programmed as to make inferences from its (possibly inconsistent) database. One would not want trivially inconsistent information about the colour of your car that somehow got fed into the fbi's computer (perhaps by pooled databases) to lead to the conclusion that you are Public Enemy Number One. We would like to add yet one more criticism of disjunctive syllogism, which is sympathetic to many of the earlier criticisms. We need as background to this criticism the natural deduction framework of [Gentzen, 1934] as interpreted by [Prawitz, 1965] and others. the idea (as in Section 1.5) is that each connective should come with rules that introduce it into discourse(as principal connective of a conclusion) and rules that eliminate it from discourse (as principal connective of a premise). further the `normalisation ideas of Prawitz, though of great technical interest and complication, boil down philosophically to the observation that an elimination rule should not be able to get out of a connective more than an introduction rule can put into the connective. This is just the old conservation Principle, `You can't get something for nothing', applied to logic. The paradigm here is the introduction and elimination rules for conjunction. The introduction rule, from A; B to infer A ^ B packs into A ^ B precisely what the elimination rule, from A ^ B to infer either A or B (separately), then unpacks. Now the standard introduction rule for disjunction is this: from either A or B separately, infer A _ B . We have no quarrel with an introduction rule. an introduction rule gives meaning to a connective and the only thing to watch out for is that the elimination rule does not take more meaning from a connective than the introduction rule gives to it (of course, one can also worry about the usefulness and/or naturalness of the introduction rules for a given connective, but that (pace [Parry, 1933]) seems not an issue in the case of disjunction. In the Lewis `proof' above, it is then clear that the disjunctive syllogism is the only conceivably problematic rule of inference. Some logicians (as indicated above) have queried the inferences from (1) to (2) and (4), and from (2) to (3), but from the point of view that we are now urging, this is simply wrongheaded. Like Humpty Dumpty, we use words to mean what we say. So there is nothing wrong with introducing connectives ^ and _

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via the standard introduction rules. Other people may want connectives for which they provide dierent introduction (and matching elimination) rules, but that is their business. We want the standard (`extensional') senses of ^ and _. Now the d.s. is a very odd rule when viewed as an elimination rule for _ parasitical upon the standard introduction rules (whereas the constructive dilemma, the usual _-Elimination rule is not at all odd). Remember that the introduction rules provide the actual inferences that are to be stored in the connective's battery as potential inferences, perhaps later to be released again as actual inferences by elimination rules. The problem with the disjunctive syllogism is that it can release inferences from _ that it just does not contain. (In another context, [Belnap, 1962] observed that Gentzenstyle rules for a given connective should be `conservative', i.e. they should not create new inferences not involving the given connective.) Thus the problem with the disjunctive syllogism is just that p _ q might have been introduced into discourse (as it is in the Lewis `proof') by _Introduction from p. So then to go on to infer q from p _ q and :p by the disjunctive syllogism would be legitimate only if the inference from p; :p to q were legitimate. But this is precisely the point at issue. At the very least the Lewis argument is circular (and not independent).13

2.3 The Admissibility of Certain rules of inference are sometimes `admissible' in formal logics in the sense that whenever the premises are theorems, so is the conclusion a theorem, although these rules are nonetheless invalid in the sense that the premises may be true while the conclusion is not. Familiar examples are the rule of substitution in propositional logic, generalisation in predicate logic, and necessitation in modal logic. Using this last as paradigm, although the inference from A to A (necessarily A) is clearly invalid and would indeed vitiate the entire point of modal logic, still for the (`normal') modal logics, whenever A is a theorem so is A (and indeed their motivation would be somehow askew if this did not hold). Anderson [1963] speculated that something similar was afoot with respect to the rule and relevance logic. Anderson hoped for a `sort of lucky accident', but the admissibility of seems more crucial to the motivation of E and R than that. Kripke [1965] gives a list of four conditions that a propositional calculus must meet in order to have a normal characteristic matrix, one of which is the admissibility of .14 `Normal' is meant in the 13 This is a new argument on the side of Anderson and Belnap [1962b, pp. 19, 21]. 14 The other conditions are that it be consistent, that it contain all classical tautologies, and that it be `complete in the sense of Hallden'. R and E can be rather easily seen to have the rst two properties (see Section 1.8 for the bit about classical tautologies), but the last is rather more diÆcult (see Section 3.11).

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sense of Church, and boils down to being able to divide up its elements into the `true' and the `false' with the operations of conjunction, disjunction, and negation treating truth and falsity in the style of the truth tables (a conjunction is true if both components are true, etc.). If one thinks of E (as Anderson surely did) as the logic of propositions with the logical operations, and surely this should divide itself up into the true and the false propositions.15

2.4 Proof(s) of the Admissibility of There are by now at least four variant proofs of the admissibility of for E and R. The rst three proofs (in chronological order: [Meyer and Dunn, 1969], [Routley and Meyer, 1973] and [Meyer, 1976a]) are all basically due to Meyer (with some help from Dunn on the rst, and some help from Routley on the second), and all depend on the same rst lemma. The last proof was obtained by Kripke in 1978 and is unpublished (see [Dunn and Meyer, 1989]). All of the Meyer proofs are what Smullyan [1968] would call `synthetic' in style, and are inspired by Henkin-style methods. The Kripke proof is `analytic' in style, and is inspired by Kanger{Beth{Hintikka tableau-style methods. In actual detail, Kripke's argument is modelled on completeness proofs for tableau systems, wherein a partial valuation for some open branch is extended to a total valuation. As Kripke has stressed, this avoids the apparatus of inconsistent theories that has hitherto been distinctive of the various proofs of 's admissibility. We shall sketch the third of Meyer's proofs, leaving a brief description of the rst and second for Section 3.11. Since they depend on semantical notions introduced there. The strategy of all the Meyer proofs can be divided into two segments: The Way Up and The Way Down. Of course we start with the hypotheses that ` A and ` :A _ B , yet assume not ` B for the sake of reduction. We shall be more precise in a moment, but The Way Up involves constructing in a Henkin-like manner a maximal theory T (containing all the logical theorems) with B 62 T . The problem though is that T may be inconsistent in the sense of having both C; :C 2 T for some formula C . (Of course this could not happen in classical logic, for by virtue of the paradox of implication C ^ :C ! B; B would be a member of T contrary to construction.) The Way Down xes this by nding in eect some subtheory T 0  T that is both complete and consistent, and indeed is a `truth set' in the sense of [Smullyan, 1968] (Meyer has labelled it the Converse Lindenbaum Lemma). Thus for all formulas X and Y , :X 2 T 0 i X 62 T 0 , and X _ Y 2 T 0 i at 15 This would be less obvious to Routley and Meyer [1976], and Priest [1987; 1995] who raise the `consistency of the world' as a real problem.

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J. MICHAEL DUNN AND GREG RESTALL

least one of X and Y is in T 0 . So since :A _ B 2 T 0, at least one of :A and B is in T 0. But since A 2 T 0, then :A is not in T 0. So B must be in T 0.16 But T 0 is a subset of T , which was constructed to keep B out. So B cannot be in T 0, and so by reductio we obtain B as desired. Enough of strategy! We now collect together a few notions needed for a more precise statement of The Way Up Lemma. Incidentally, we shall from this point on in our discussion of consider only the case of R. Results for E (and a variety of neighbours) hold analogously. By an `R-theory' we mean a set of formulas T of R closed under adjunction And logical relevant implication, i.e. such that 1. if A; B 2 T , then A ^ B 2 T ; 2. if `R A ! B and A 2 T , then B 2 T . Note that an arbitrary R-theory may lack some or all of the theorems of R (in classical logic and most familiar logics this would be impossible because of the paradox of strict implication which says that a logical theorem is implied by everything). We thus need a special name for those R{theories that contain all of the R-theorems|those are called regular.17 In this section, since we have no use of irregular theories and shall be talking only of R, by a theory we shall always mean a regular R theory (irregular Rtheories however play a great role in the completeness theorems of Section 3 below and there we shall have to be more careful about our distinctions). A theory T is called prime if whenever A _ B 2 T , then A 2 T or B 2 T . The converse of this holds for any theory T in virtue of the Raxioms A ! A _ B and B ! A _ B and property (2). A theory T is called complete if for every formula A; A 2 T or :A 2 T , and called consistent if for no formula A do we have both A; :A 2 T . In virtue of the R-theorem A _ :A, we have that all prime theories are complete. A consistent prime theory is called normal, and it should by now be apparent that a normal theory is a truth set in the sense of Smullyan given above. Where is a set of formulas, we write `R A to mean that A is deducible from in the `oÆcial sense' of there being a nite sequence B1 ; : : : ; Bn , 16 The proof as given here would appear to use disjunctive syllogism in the metalanguage at just this point, but it can be restructured (indeed we so restructured the original proofs [Meyer and Dunn, 1969]) so as to avoid at least such an explicit use of disjunctive syllogism. The idea is to obtain by distribution (A 2 T 0 and A 62 T 0 ) or (B 2 T 0 and B 62 T 0 ) from the hypothesis B 62 T 0 . The whole question of a `relevant' version of the admissibility of is a complicated one, and admits of various interpretations. See [Belnap and Dunn, 1981; Meyer, 1978]. 17 It is interesting to note for regular theories, condition (2) may be replaced with the condition (20 ) if A 2 T and (A ! B ) 2 T , then B 2 T , in virtue of the R-theorem A ^ (A ! B ) ! B.

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with Bn = A and each Bi being either a member of , or an axiom of R, or a consequence of earlier terms by modus ponens or adjunction (in context we shall often omit the subscript R). We write `
A to mean that [
`R A, and quite standardly we write things like ; A `R B in place of the more formal [ fAg `R B . Note that for any theory T , writing `T A in place of  `T A boils down to saying that A is a theorem of T (A 2 T ). Where
is a set of formulas not necessarily a theory, `
A can be thought of as saying that A is deducible from the `axioms'
. The set fA : `
Ag is pretty intuitively the smallest theory containing the axioms
, and we shall label it as Th(
). We can now state and sketch a proof of the Way Up Lemma. Suppose not `R A. Then there exists a prime theory T such that not `T A.

Proof. Enumerate the formulas of R : X1 ; X2 ; : : : . De ne a sequence of sets of formulas by induction as follows. T0 = set of theorems of R. Ti+1 = Th(Ti [ fXi+1 g) if it is not the case that Ti ; Xi+1 Ti , otherwise.

` A;

Let T be the union of all these Tn 's. It is easy to see as is standard that T is a theory not containing A. Also we can show that T is prime. Thus suppose `T X _ Y and yet X; Y 62 T . Then it is easy to se that since neither X nor Y could be added to the construction when their turn came up without yielding A, we have both 1. X `T A, 2. Y `T A. But by reasonably standard moves (R has distribution), we get 3. X _ Y `T A, and so `T A contrary to the construction.



. Let T 0 be a prime theory. Then there exists a 0 normal theory T  T . The concept we need is that of a `metavaluation' (more precisely as we use it here a `quasi -metavaluation', but we shall not bother the reader with such detail). The concept and its use re may be found in [Meyer, 1976a]. (See also Meyer [1971; 1976b] for other applications.) For simplicity we assume for a while that the only primitive connectives are :; _ and ! (^ can be de ned via de Morgan). A metavaluation v is a function from the set of formulas into the truth values f0; 1g, such that The Way Down Lemma

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J. MICHAEL DUNN AND GREG RESTALL

1. for a propositional variable p; v(p) = 1 i p 2 T ; 2. v(:A) = 1 i both (a) v(A) = 0 and (b) :A 2 T ; 3. v(A _ B ) = 1 i either v(A) = 1 or v(B ) = 1. 4. v(A ! B ) = 1 i both (a) v(A) = 0 or v(B ) = 1, and (b) A ! B 2 T . One surprising aspect of these conditions is the double condition in (2) that must be met for :A to be assigned the value 1. Not only must (a) A be assigned 0 (the usual `extensional condition'), but also (b) :A must be a theorem of T (the `intensional condition'). and of course there are similar remarks about (4). The condition in (1) also relies upon G (actually to a lesser extent than it might seem|when both p; :p 2 T , it would not hurt to let v(p) = 0). We now set T 0 = fA : v(A) = 1g. The following lemma is useful, and has an easy proof by induction on complexity of formulas (the case when A is a negation evaluated as 0 uses the completeness of T ). Completeness Lemma. If v (A) = 1, then A 2 T . If v (A) = 0, then :A 2 T . It is reasonably easy to see that T 0 is in fact a truth set. That it behaves ok with respect to disjunction can be read right o of clause (3) in the de nition of v, so we need only look at negation where the issue is whether T 0 is both consistent and complete. It is clear from clause (2) that T 0 is consistent, but T 0 is also complete. Thus, suppose A 62 T 0 , then by the Completeness Lemma :A 2 T . This is the intensional condition for v(:A) = 1, but our supposition that A 62 T 0 is just the extensional condition that v(A) = 0. Hence v(:A) = 1, i.e. :A 2 T 0 as desired. It is also reasonably easy to check that T 0 is an R-theory. It is left to the reader to do the easy calculation that T 0 is closed under adjunction and R-implication, i.e. that these preserve assignments by v of the value 1. Here we will illustrate the more interesting veri cation that the R-axioms all get assigned the value 1. We shall not actually check all of them, but rather consider several typical ones. First we check suÆxing: (A ! B ) ! [(B ! C ) ! (A ! C )]. Suppose v assigns it 0. Since it is a theorem of R and a fortiori of T , then it satis es the intensional condition and so must fail to satisfy the extensional condition. So v(A ! B ) = 1 and v((B ! C ) ! (A ! C )) = 0. By the Completeness Lemma, then (A ! B ) 2 T , and so by modus ponens from the very axiom in question (SuÆxing) we have that (B ! C ) ! (A ! C ) 2 T . So v((B ! C ) ! (A ! C )) satis es the intensional condition, and so must fail to satisfy the extensional condition since it is 0. So v(B ! C ) = 1 and v(A ! C ) = 0. By reasoning analogous to that above (one more modus ponens ) we derive that v(A ! C ) must nally fail to satisfy the

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extensional condition, i.e. v(A) = 1 and v(C ) = 0. But clearly since all of v(A ! B ) = 1, v(B ! C ) = 1, v(A) = 1, then by the extensional condition, v(C ) = 1, and we have a contradiction. The reader might nd it instructive in seeing how negation is handled to verify rst the intuitionistically acceptable form of the Reductio axioms (A ! :A) ! :A, and then to verify its classical variant (used in some axiomatisations of R), (:A ! A) ! A. The rst is easier. Also Classical Double Negation, ::A ! A is fun. This completes the sketch of Meyer's latest proof of the admissibility of

for R.

2.5 for First-order Relevance Logics The rst proof of the admissibility of for rst-order R, E, etc. (which we shall denote as RQ, etc.) was in Meyer, Dunn and Leblanc [1974], and uses algebraic methods analogous to those used for the propositional relevance logic in [Meyer and Dunn, 1969]. The proof we shall describe here though will again be Meyer's metavaluation-style proof. The basic trick needed to handle rst-order quanti ers is to produce this time a rst-order truth set. Assuming that only the universal quanti er 8 is primitive (the existential can be de ned: 9x =df :8x:), this means we need (8)

8xA 2 T i A(a=x) 2 T

for all parameters (free variables) a:

This is easily accommodated by adding a clause to the de nition of the metavaluation v so that 5. v(8xA) = 1 i v(A(a=x)) = 1 for all parameters a.

This does not entirely x things, for in proving the Completeness Lemma we have now in the induction to consider the case when A is of the form 8xB . If v(8xB ) = 1, then (by (5)), va(B (a=x)) = 1 for all parameters a. By inductive hypothesis, for all a; B (a=x) 2 T . But, and here's the rub, this does not guarantee that 8xB (a=x) 2 T . We need to have constructed on The Way Up a theory T that is `!-complete' in just the sense that this guarantee is provided. ([Meyer et al., 1974] call such a theory `rich'.) Of course it is understood by `theory' we now mean a `regular RQ-theory', i.e. one containing all of the axioms of RQ and closed under its rules (see Section 1.3). Actually things can be arranged as in [Meyer et al., 1974] so that generalisation is in eect built into the axioms so that the only rules can continue to be adjunction and modus ponens. Thus we need the following Way Up Lemma for RQ. Suppose A is not a theorem of rst-order RQ. Then there exists a prime rich theory T so that A 62 T .

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J. MICHAEL DUNN AND GREG RESTALL

This lemma is Theorem 3 of [Meyer et al., 1974], and its proof is of basically a Henkin style with one novelty. In usual Henkin proofs one can assure !-completeness by building into the construction of T that whenever :8xB is put in, then so is :B (a=x) for some new parameter a. This guarantees !-completeness since if B (a) 2 T for all a, but 8xB 62 T , then by completeness :8xB 2 T and so by the usual construction :B (a) 2 T for some a, and so by consistency (??) B (a) 62 T for some a, contradicting the hypothesis for !-completeness. But we of course have for relevance logics no guarantee that T is consistent, as has been remarked above. The novelty then was to modify the construction so as to keep things out as well as put things in, though this last still was emphasised. Full symmetry with respect to `good guys' and `bad guys' was nally obtained by Belnap, 18 in what is called the Belnap Extension Lemma, which shall be stated after a bit of necessary terminology. We shall call an ordered pair (
; ) of sets of formulas of RQ and `RQpair'. We shall say that one RQ pair (
1 ; 1 ) extends another (
0 ; 0) if
0 
1 and 0  1 . An RQ pair is de ned to be exclusive if for no A1 ; : : : ; Am 2
; B1 ; : : : ; Bn 2  do we have ` A1 ^


^Am ! B1 _


_Bn . It is called exhaustive if for every formula A, either A 2
or A 2 .19 It is now easiest to assume that ^ and 9 are back as primitive. We call a set of formulas _-prime (^-prime ) if whenever A _ B 2 (A ^ B 2 ), at least one of A or B 2 (clearly _-primeness is the same as primeness). Analogously, we call 9-prime (8- prime ) if whenever 9xA 2 (8xA 2 ), then A(a=x) 2 for some a. Given an RQ pair (
; ) we shall call
() completely prime if
is both _- and 9-prime ( is both ^- and 8-prime). the pair (
; ) is called completely prime if both
and  are completely prime. We can now state the . Let (
; ) be an exclusive RQ pair. Then (
; ) can be extended to an exclusive, exhaustive, completely prime RQ pair (T; F ) in a language just like the language of RQ except for having denumerably many new parameters. Belnap Extension Lemma

We shall not prove this lemma here, but simply remark that it is a surprisingly straightforward application of Henkin methods to construct a maximal RQ-pair and show it has the desired properties (indeed it simply symmetrises the usual Henkin construction of rst-order classical logic).

18 Belnap's result is unpublished, although he communicated it to Dunn in 1973. Dunn circulated a write-up of it about 1975. It is cited in some detail in [Dunn, 1976d]. Gabbay [1976] contains an independent but precise analogue for the rst-order intuitionistic logic with constant domain. 19 We choose our terminology carefully, not calling (
; ) a `theory', not using `consistency' for exclusiveness, and not using `completeness' for exhaustiveness. We do this so as to avoid con ict with our earlier (and more customary) usage of these terms and in this we dier on at least one term from usages on other occasions by Gabbay, Belnap, or Dunn.

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In order to derive the RQ Way Up Lemma we simply set
= RQ and  = fAg and extend it to the pair (T; F ) using the Belnap Extension Lemma. It is easy to see that T is a (regular ) RQ-theory, and clearly G is prime. but also T is !-complete. Thus suppose B (a=x) 2 T for all a, but 8xB 62 T . Then by exhaustiveness 8xB 2 F R. Then by 8primeness, B (a=x) 2 F R for some a. But since `RQ B (a=x) ! B (a=x), this contradicts the exclusiveness of the pair (T; F ).

2.6 for Higher-order Relevance Logics and Relevant Arithmetic The whole point about being merely an admissible rule is that it might not hold for various extensions of F (cf. [Dunn, 1970] for actual counter examples). Thus, as we just saw, it was an achievement to show that continues to e admissible in R when it is extended to include rst-order quanti cation. The question of the admissibility of naturally has great interest when R is further extended to include theories in the foundations of mathematics such as type theory (set theory) and arithmetic. Meyer [1976a] contains investigations of the admissibility of for relevant type theory (R! ). We shall report nothing in the way of detail here except to observe that Meyer's result is invariant among various restrictions of the formulas A in the Comprehension Axiom scheme: 9X x+18yn(X n+1 (yn ) $ A): As for relevantly formulated arithmetic, most work has gone on in studying Meyer's systems R] , R]] and their relatives, based on Peano arithmetic, though Dunn has also considered a relevantly formulated version of Robinson Arithmetic [Anderson et al., 1992]. Here we will recount the results for R] and R]] for they are rather surprising. In a nutshell, is admissible in relevant arithmetics with the in nitary !-rule (from A(0), A(1), A(2), : : : to infer 8xA(x)), but not without it [Friedman and Meyer, 1992; Meyer, 1998]. The system R] is given by rewriting the traditional axioms of Peano arithmetic with relevant implication instead of material implication in the natural places. You get the following list of axioms Identity y = z ! (x = y ! x = z ) Successor x0 = y0 ! x = y x = y ! x0 = y0 0 6= x0 Addition x + 0 = x x + y 0 = (x + y ) 0 Multiplication x0 = 0 xy0 = xy + x
Induction A(0) ^ 8x A(x) ! A(x0 ) ! 8xA(x)

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which you add to those of RQ in order to obtain an arithmetic theory. The question about the admissibility of was open for many years, until Friedman teamed up with Meyer to show that it is not [Friedman and Meyer, 1992]. The proof does not provide a direct counterexample to . Instead, it takes a more circuitous route. First, we need Meyer's classical containment result for R] . When we map formulae in the extensional vocabulary of arithmetic to the language of R] by setting  (x = y) to (x = y) _ (0 6= 0) and leaving the rest of the map to respect truth functions (so  (A ^ B ) =  (A) ^  (B ),  (:A) = : (A) and  (8xA) = 8x (A)) then we have the following theorem:

 (A) is a theorem of R] i A is a theorem of classical Peano arithmetic. This is a subtle result. The proof goes through by showing, by induction, that  (A) is equivalent either to (A ^ (0 = 0)) _ (0 6= 0) or to (A _ (0 6= 0)) ^ (0 = 0), and then that and the classical form of induction (with material implication in place of relevant implication) is valid for formulae of this form in R] . Then, if we had the admissibility of for R] , we could infer A from  (A). (If  (A) is equivalent to (A ^ (0 = 0)) _ (0 6= 0), then we can use 0 = 0 and to derive A ^ (0 = 0), and hence A. Similarly for the other case). The next signi cant result is that not all theorems of classical Peano arithmetic are theorems of R] . Friedman provided a counterexample, which is simple enough to explain here. First, we need some simple preparatory results.

 R]

is a conservative extension of the theory R]+ axiomatised by the negation free axioms of R] [Meyer and Urbas, 1986].



If classical Peano theorem is to be provable in R] and if it contains no negations, then it must be provable in R]+ .



Any theorem provable in R]+ must be provable in the classical positive system PA+ which is based on classical logic, instead of R.

The proofs of these results are relatively straightforward. The next result is due to Friedman, and it is much more surprising.



The ring of complex numbers is a model of PA+ .

The only diÆcult thing to show is that it satis es the induction axiom. For any formula A(x) in the vocabulary of arithmetic, the set of complex numbers  such that A() is true is either nite or co nite. If A(x) is atomic, then it is equivalent to a polynomial of the form f (x) = 0, and f must either have nitely many roots or be 0 everywhere. But the set of either nite or co nite sets is closed under boolean operations, so no A(x)

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we can construct will have an extension which is neither nite or co nite.) As a result, the induction axiom must be satis ed. For if A(0) holds and if A(x)  A(x0 ) holds then there are in nitely many complex numbers  such that A(). So the extension of A is at least co nite. But if there is a point  such that A() fails, then so would A( 1), A( 2) and so on by the induction step A(x)  A(x0 ), and this contradicts the con nitude of the extension of A. As a result, A() holds for every . We can then use this surprising model of positive Peano arithmetic to construct a Peano theorem which is not a theorem of R] . It is known that for any odd prime p, there is a positive integer y which is not a quadratic residue mod p. That is, 9y8z :(y  z 2 mod p) is provable in Peano arithmetic. This formula can be rewritten in the language or arithmetic with a little work. However, the corresponding formula is false in the complex numbers, so it is not a theorem of PA+ . Therefore it isn't a theorem of R] + , and by the conservative extension result, it is not a theorem of R] . As a consequence, R] is not closed under . Where is the counterexample to ? Meyer's containment result provides a proof of  (B ), where B is the quadratic residue formula. The rule would allow us to derive B from  (B ), and it is here that must fail. If we replace the induction axiom by the in nitary rule !, we can prove the admissibility of using a modi cation of the Belnap Extension Lemma for the Way Up and using the standard metavaluation technique for the Way Down. The modi cation of the Belnap Extension Lemma is due to Meyer [1998]. Belnap Extension Lemma, with Witness Protection:

Let (
; ) be an exclusive R]] pair in the language of arithmetic (that is, with 0 as the only constant). Then (
; ) can be extended to an exclusive, exhaustive, completely prime R]] pair (T; F ) in the same language. This lemma requires the !-rule for its proof. Consider the induction stage in which you wish to place 8xA(x) in i . The witness condition dictates that there be some term t such that A(t) also appear in i . The !-rule ensures that we can do this without the need for a new term, for if no term 000


0 could be consistently added to i , then each A(000


0 ) is a consequence of
i , and by the !-rule, so is 8xA(x), contradicting the fact that we can add 8xA(x) to i . So, we know that some 000


0 will do, and as a result, we need add no new constants to form the complete theory T . The rest of the way up lemma and the whole of the way down lemma can then be proved with little modi cation. (for details, see [Meyer, 1998]). Consequently, is admissible in R]] . These have been surprising results, and important ones, for relevant arithmetic is an important `test case' for accounts of relevance. It is a theory in which we can have some fairly clear idea of what it is for one formulae to

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J. MICHAEL DUNN AND GREG RESTALL

properly follow from another. In R] and R]] , we have 0 = 2 ! 0 = 4 because there is an `arithmetically appropriate' way to derive 0 = 4 from 0 = 2 | by multiplying both sides by 2. However, we cannot derive 0 = 2 ! 0 = 3, and, correspondingly, there is no way to derive 0 = 3 from 0 = 2 using the resources of arithmetic. The only way to do it within the vocabulary is to appeal to the falsity of 0 = 3, and this is not a relevantly acceptable move. 0 6= 3 ! (0 = 3 ! 0 = 2) does not have much to recommend as pattern of reasoning which respects the canons of relevance. We are left with important questions. Are there axiomatisable extensions of R] which are closed under ? Can theories like R] and R]] be extended to deal with more interesting mathematical structures, while keeping account of some useful notion of relevance? Early work on this area, from a slightly dierent motivation (paraconsistency, not relevance) indicates that there are some interesting results at hand, but the area is not without its diÆculties [Mortensen, 1995]. The admissibility of would also seem to be of interest for relevant type theory (even relevant second-order logic) with an axiom of in nity (see [Dunn, 1979b]). One of the chief points of philosophical interest in showing the admissibility of for some relevantly formulated version of a classical theory relates to the question of the consistency of the classical theory (this was rst pointed out in Meyer, Dunn and Leblanc [1974]). As we know from Godel's work, interesting classical theories cannot be relied upon to prove their own consistency. To exaggerate perhaps only a little, the consistency of systems like Peano (even Robinson) arithmetic must be taken in faith. But using relevance logic in place of classical logic in formulating such theories gives us a new strategy of faith. It is conceivable that since relevance logic is weaker than classical logic, the consistency of the resultant theory might be easier to demonstrate. This has proved true at least in the sense of absolute consistency (some sentence is unprovable) as shown by [Meyer, 1976c] for Peano arithmetic using elementary methods. Classically of course there is no dierence between absolute consistency and ordinary (negation) consistency (for no sentence are both A and :A provable), and if is admissible for the theory, then this holds for relevance logic, too. The interesting thing then would be to produce a proof of the admissibility of

, which we know from Godel would itself have to be non-elementary. One could then imagine arguing with a classical mathematician in the following Pascal's Wager sort of way [Dunn, 1980a]. Look. You have equally good reason to believe in the negation consistency of the classical system and the (relative) completeness of the relevant system. In both cases you have a nonelementary proof which secures your belief, but which might be mistaken. Consider the consequences in each case if it is mis-

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taken. If you are using the classical system, disaster! Since even one contradiction classically implies everything, for each theorem you have proven, you might just as well have proven its negation. But if you are using the relevant system, things are not so bad. For at least large classes of sentences, it can be shown by elementary methods (Meyer's work) that not both the sentences and their negations are theorems.

2.7 Ackermann's and Gentzen's Cut: Gentzen Systems as Relevance Logic In [Meyer et al., 1974] an analogy was noted between the role that the admissibility of plays in relevance logic and the role that cut elimination plays in Gentzen calculi (even those for classical systems). For the reader unfamiliar with Gentzen calculi, this subsection will make more sense after she has read Sections 4.6 and 4.7. The Gentzen system for the classical propositional calculus LK with the material conditional and negation as primitive (as is well-known, all of the other truth-functional connectives can be de ned from these) may be obtained by adding to the rules of LR:! of Section 4.7. the rule of Thinning (see Section 4.6) on both the left and right. Gentzen also had as a primitive rule:

 ` A; B ; A ` Æ ; ; ` B; Æ

(Cut)

which has as a special case

` A`B (1) : `B Since A ` B is derivable just when ` A ! B is derivable, and since in classical logic A ! B is equivalent to :A _ B , (1) above is in eect ` A ` :A _ B (10 ) ; `B which is just . All of the Gentzen rules except Cut have the Subformula Property: Every formula that occurs in the premises also occurs in the conclusion, though perhaps there as a subformula. Gentzen showed via his Hauptsatz that Cut was redundant|it could be eliminated without loss (hence this is often called the Elimination Theorem). Later writers have tended to think of Gentzen systems as lacking the Cut Rule, and so the Elimination Theorem is stated as showing that Cut is admissible in the sense that whenever the premises are derivable so is the conclusion. There is thus even a parallel

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historical development with Ackermann's rule in relevance logic, since writers on relevance logic have tended to follow Anderson and Belnap's decision to drop as a primitive rule. Note that the Subformula Property can be thought of as a kind of relation of relevance between premises and conclusion. Thus Cut as primitive destroys a certain kind of relevance property of Gentzen systems, just as

as primitive destroys the relevance of premises to conclusion in relevance logic. The analogies become even clearer if we reformulate Gentzen's system according to the following ideas of [Schutte, 1956]. The basic objects of Gentzen's calculus LK were the sequents A1 ; : : : ; Am ` B1 ; : : : ; Bn , where the Ai 's and Bj 's are formulas (any or all of which might be missing). Such a sequent may be interpreted as a statement to the effect that either one of the Ai 's is false or one of the Bj 's is true. To every such sequent there corresponds what we might as well call its `right-handed counterpart':

` :A1 ; : : : ; :Am ; B1 ; : : : ; Bn It is possible to develop a calculus parallel to Gentzen's using only `righthanded' sequents, i.e. those with empty left side. This is in eect what Schutte did, but with one further trick. Instead of working with a righthanded sequent ` A1 ; : : : ; Am , which can be thought of as a sequence of formulas, he in eect replaced it with the single formula A1 _


_ Am .20 With these explanations in mind, the reader should have no trouble in perceiving Schutte's calculus K1 as `merely' a notational variant of Gentzen's original calculus LK (albeit, a highly ingenious one). Also Schutte's system had the existential quanti er which we have omitted here purely for simplicity. Dunn and Meyer [1989] treats it as well. The axioms of K1 are all formulas of the form A _ :A. The inference rules divide themselves into two types: Structural rules: M_A_B_N N _A_A [Interchange] [Contraction] M_B_A_N N _A Operational rules:

N _ :A N _ :B N N _A [de Morgan] [Thinning] [Double Negation] N _ :(A _ B ) N _B N _ ::A It is understood in every case but that of Thinning that either both of M and N may be missing. Also there is an understanding in multiple disjunctions that parentheses are to be associated to the right.

20 It ought be noted that similar \single sided" Gentzen systems nd extensive use in the proof theory for Linear Logic [Girard, 1987; Troelstra, 1992].

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In [Meyer et al., 1974] it was said that the rule Cut is just `in peculiar notation'. In the context of Schutte's formalism the notation is not even so dierent. Thus: M _ A :A _ N A :A _ B [Cut] [ ]: M_N B Since either M or N may be missing, obviously is just a special case of Cut. It is pretty easy to check that each of the rules above corresponds to a provable rst-degree relevant implication. Indeed [Anderson and Belnap Jr., 1959a] with their `Simple Treatment' formulation of classical logic (extended to quanti ers in [Anderson and Belnap Jr., 1959b]) independently arrived at a Cut-free system for classical logic much like Schutte's (but with some improvements, i.e. they have more general axioms and avoid the need for structural rules). They used this to show that E contains all the classical tautologies as theorems, the point being that the Simple Treatment rules are all provable entailments in E (unlike the usual rule for axiomatic formulations of classical logic, modus ponens for the material conditional, i.e. ). Thus the later proven admissibility of was not needed for this purpose, although it surely can be so used. Schutte's system can also clearly be adapted to the purpose of showing that classical logic is contained in relevance logic, and indeed [Belnap, 1960a] used K1 (with its quanti cational rules) to show that EQ contains all the theorems of classical rst-order logic. It turns out that one can give a proof of the admissibility of Cut for a classical Gentzen-style system, say Schutte's K1 , along the lines of a Meyer-style proof of the admissibility of (see [Dunn and Meyer, 1989], rst reported in 1974).21 We will not give many details here, but the key idea is to treat the rules of K1 as rules of deducibility and not merely as theorem generating devices. Thus we de ne a deduction of A from a set of formulas as a nite tree of formulas with A as its origin, members of or axioms of K1 at its tips, and such that each point that is not a tip follows from the points just above it by one of the rules of K1 (this de nition has to be slightly more complicated if quanti ers are present due to usual problems caused by generalisation). We can then inductively build a prime complete 21 We hasten to acknowledge the nonconstructive character of this prof. In this our proof compares with that of Schutte [1956] (also proofs for related formalisms due to Anderson and Belnap, Beth, Hintikka, Kanger) in its uses of semantical (model-theoretic) notions, and diers from Gentzen's. Like the proofs of Schutte et al. this proof really provides a completeness theorem. We may brie y label the dierence between this proof and those of Schutte and the others by using (loosely) the jargon of Smullyan [1968]. Calling both Hilbert-style formalisms and their typical Henkin-style completeness proofs `synthetic', and calling both Gentzen-style formalisms and their typical Schutte-style completeness proof `analytic', it looks as if we can be said to have given an synthetic completeness proof for an analytic formalism.

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theory (closed under deducibility) on The Way up, which will clearly be inconsistent since because of the `Subformula Property' clearly, e.g. q is not deducible from p; :p. but this can be xed on The Way Down by using metavaluation techniques so as to nd a complete consistent subtheory. In 1976 E. P. Martin, Meyer and Dunn extended and analogised the result of Meyer concerning the admissibility of for relevant type theory described in the last subsection, in much the same way as the argument for the rst-order logic has been analogised here, so as to obtain a new proof of Takeuti's Theorem (Cut-elimination for simple type theory). This unpublished proof dualises the proof of Takahashi and Prawitz (cf. [Prawitz, 1965]) in the same way that the proof here dualises the usual semantical proofs of Cut-elimination for classical rst-order logic. This dualisation is vividly described by saying that in place of `Schutte's Lemma' that every semi- (partial-) valuation may be extended to a (total) valuation, there is instead the `Converse Schutte Lemma' that every `ambi-valuation' (sometimes assigns a sentence both the values 0, 1) may be restricted to a (consistent) valuation. 3 SEMANTICS

3.1 Introduction In Anderson's [1963] `open problems' paper, the last major question listed, almost as if an afterthought, was the question of the semantics of E and EQ. Despite this appearance Anderson said (p. 16) `the writer does not regard this question as \minor"; it is rather the principle large question remaining open'. Anderson cited approvingly some earlier work of Belnap's (and his) on providing an algebraic semantics for rst-degree entailments, and said (p. 16), `But the general problem of nding a semantics for the whole of E, with an appropriate completeness theorem, remains unsolved'. It is interesting to note that Anderson's paper appeared in the same Acta Filosphica Fennica volume as the now classic paper of Kripke [1963] which provided what is now simply called `Kripke-style' semantics for a variety of modal logics (Kripke [1959a] of course provided a semantics for S5, but it lacked the accessibility relation R which is so versatile in providing variations). When Anderson was writing his `open problems' paper, the paradigm of a semantical analysis of a non-classical logic was probably still something like the work of McKinsey and Tarski [1948], which provided interpretations for modal logic and intuitionistic logic by way of certain algebraic structures analogous to the Boolean algebras that are the appropriate structures for classical logic. But since then the Kripke-style semantics (sometimes referred to as `possible-worlds semantics', or `set-theoretical semantics') seems

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to have become the paradigm. Fortunately, E and R now have both an algebraic semantics and a Kripke-style semantics. We shall rst distinguish in a kind of general way the dierences between these two main approaches to semantics, before going on to explain the particular details of the semantics for relevant logics (again R will be our paradigm).

3.2 Algebraic vs. Set-theoretical Semantics It is convenient to think of a logical system as having two distinct aspects syntax (well-formed strings of symbols, e.g. sentences) and semantics (what, e.g. these sentences mean, i.e. propositions). These double aspects compete with one another as can be seen in the competing usages `sentential calculus' and `propositional calculus', but we should keep rmly in mind both aspects. Since sentences can be combined by way of connectives, say he conjunction sign ^, to form further sentences, typically there is for each logical system at least one natural algebra arising at the level of syntax, the algebra of sentences (if one has a natural logical equivalence relation there is yet another that one obtains by identifying logically equivalent sentences together into equivalence classes|the so-called `Lindenbaum algebra'). And since propositions can be combined by the corresponding logical operations, say conjunction, to form propositions, here is an analogous algebra of propositions. Now undoubtedly some readers, who were taught to `Quine' propositions from an early age, will have troubles with the above story. The same reader would most likely not nd compelling any particular metaphysical account we might give of numbers. We ask that reader then to at least suspend dis belief in propositions so that we can get on with the mathematics. There is an alternative approach to semantics which can be described by saying that rather than taking propositions as primitive, it `constructs' them out of certain other semantical primitives. Thus there is as a paradigm of this approach the so-called `UCLA proposition' as a set of `possible worlds'.22 We here want to stress the general structural idea, not placing much emphasis upon the particular choice of `possible worlds ' as the semantical primitive. Various authors have chosen `reference points', `cases', `situations', `set-ups', etc.|as the name for the semantical primitive varying for sundry subtle reasons from author to author. We have both in relevance logic contexts have preferred `situations', but in a show of solidarity we shall here join forces with the Routley's [1972] in their use of `set-ups'. Such `set-theoretical' semantical accounts do not always explicitly verify such a construction of propositions. Indeed perhaps the more common approach is to provide an interpretation that says whether a formula A is 22 Actually the germ of this idea was already in Boole (cf. [Dipert, 1978]), although apparently he thought of it as an analogy rather than as a reduction.

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true and false at a given set-up S writing (a; S ) = T or S  A or some such thing. Think of Kripke's [1963] presentation of his semantics for modal logic. But (unless one has severe ontological scruples about sets) one might just as well interpret A by assigning it a class of set-ups, writing (A) or jAj or some such thing. One can go from one framework to the other by way of equivalence

S 2 jAj i S  A:

3.3 Algebra of First-degree Relevant Implications Given two propositions a and b, it is natural to consider the implication relation among them, which we write as a  b (`a implies b'). It might be thought to be natural to write this the other way around as a  b on some intuition that a is the stronger or `bigger' one if it implies b. Also it suggests a  b (`b is contained in a'), which is a natural enough way to think of implication. There are good reasons though behind our by now almost universal choice (of course at one level it is just notation, and it doesn't matter what your convention is). Following the idea that a proposition might be identi ed with the set of cases in which it is true, a implies b corresponds to a  b, which has the same direction as a  b. Then conjunction ^ corresponds to intersection \, and they have roughly the same symbol (and similarly for _ and [). It is also natural to assume, as the notation suggests, that implication is a partial order, i.e. (p.o.1) a  a (Re exivity), (p.o.2) a  b and b  a ) a = b (Antisymmetry), (p.o.3) a  b and b  c ) a  c (Transitivity).

It is natural also to assume that there are operations of conjunction ^ and disjunction _ that satisfy (^lb) (^glb) (_ub) (_lub)

a ^ b  a, a ^ b  b, x  a and x  b ) x  a ^ b, a  a _ b, b  a _ b, a  x and b  x ) a _ b  x.

Note that (^lb) says that a ^ b is a lower bound both of a and of b, and (^glb) says it is the greatest such lower bound. Similarly a _ b is the least upper bound of a and b. A structure (L; ; ^; _) satisfying all the properties above is a well-known structure called a lattice. Almost any logic would be compatible with the assumption that propositions form a lattice (but there are exceptions, witness Parry's [1933] Analytic Implication which would reject (_ub)).

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Lattices can be de ned entirely operationally as structures (L; ^; _) with the relation a  b de ned as a ^ b = a. Postulates characterising the operations are: Idempotence: a ^ a = a, a _ a = a Commutativity: a ^ b = b ^ a, a _ b = b _ a Associativity: a ^ (b ^ c) = (a ^ b) ^ c, a _ (b _ c) = (a _ b) _ c Absorption: a ^ (a _ b) = a, a _ (a ^ b) = a. An (upper) semi-lattice is a structure (S; _), with _ satisfying Idempotence, Commutativity, and Associativity. Given two lattices (L; ^; _) and (L0 ; ^0 ; _0 ), a function h from L into 0 is called a (lattice ) homomorphism if both h(a ^ b) = h(a) ^0 h(b) and h(a _ b) = h(a) _0 h(b). If h is one{one, h is called an isomorphism. Many logics (certainly orthodox relevance logic) would insist as well that propositions form a distributive lattice, i.e. that

a ^ (b _ c)  (a ^ b) _ c:

This implies the usual distributive laws a ^ (b _ c) = (a ^ b) ^ (a ^ c) and a _ (b ^ c) = (a _ b) ^ (a _ c). (Again there are exceptions, important ones being quantum logic with its weaker orthomodular law, and linear logic with its rejection of even the orthomodular law.) The paradigm example of a distributive lattice is a collection of sets closed under intersection and union (a so-called `ring' of sets). Stone [1936] indeed showed that abstractly all distributive lattices can be represented in this way. Although we will not argue this here, it is natural to think that if propositions correspond to classes of cases, then conjunction should carry over to intersection and disjunction to union, and so productions should form a distributive lattice. Certain subsets of lattices are especially important. A lter is a nonempty subset F such that 1. a; b 2 F ) a ^ b 2 F , 2. a 2 F and a  b ) b 2 F . Filters are like theories. Note by easy moves that a lter satis es 10. a; b 2 F , a ^ b 2 F , 20. a 2 F or b 2 F ) a _ b 2 F . When a lter also satis es the converse of (20 ) it is called prime, and is like a prime theory. A lter that is not the whole lattice is called proper. Stone [1936] showed (using an equivalent of the Axiom of Choice) the Prime Filter Separation Property. In a distributive lattice, if a 6 b, then there exists a prime lter P with a 2 P and b 62 P .

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This is related to the Belnap Extension Lemma of Section 2.5. So far we have omitted discussion of negation. This is because there is less agreement among logics as to what properties it should have.23 There is, however, widespread agreement that it should at least have these: 1. (Contraposition) a  b ) :b  :a, 2. (Weak Double Negation) a  ::a.

These can both be neatly packaged in one law: 3. (Intuitionistic Contraposition) a  :b ) b  :a.

We shall call any unary function : satisfying (3) (or equivalently (1) and (2)) a minimal complement. The intuitionists of course do not accept 4. (Classical Contraposition) :a  :b ) b  a, or 5. (Classical Double Negation) ::a  a.

If one adds either of (4) or (5) to the requirements for a minimal complement one gets what is called a de Morgan complement (or quasi-complement), because, as can be easily veri ed, it satis es all of de Morgan's laws (deM1) :(a ^ b) = :a _ :b, (deM2) :(a _ b) = :a ^ :b. Speaking in an algebraic tone of voice, de Morgan complement is just a (one{one) order-inverting mapping (a dual automorphism ) of period two. De Morgan complement captures many of the features of classical negation, but it misses (Irrelevance 1) a ^ :a  b, (Irrelevance 2) a  b _ :b. If (either of) these are added to a de Morgan complement it becomes a Boolean complement. If Irrelevance 1 is added to a minimal complement, it becomes a Heyting complement (or pseudo-complement ). A structure (L; ^; _; :), where (L; ^; _) is a distributive lattice and : is a de Morgan (Boolean) complement is called a de Morgan (Boolean ) lattice. Note that we did not try to extend this terminological framework to `Heyting lattices', because in the literature a Heyting lattice requires an operation called `relative pseudo-complementation' in addition to Heyting complementation (plain pseudo-complementation). As an example of de Morgan lattices consider the following (here we use ordinary Hasse diagrams to display the order; a  b is displayed by putting a in a connected path below b): 23 Cf. Dunn [1994; 1996] wherein the various properties below are related to various ways of treating incompatibility between states of information.

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1

1

@@

s

s

3: p = :p

2:

53

s

s

4: p

s s

0

@@

@ :p s

s

0 1

@@ @

@ s

0

s

4: p = :p

@ s

@@

s

q = :q

s

0

The backwards numeral labelling the third lattice over is not a misprint. It signi es that not only has the de Morgan complement been obtained by inverting he order of the diagram, as in the order three (of course :I =  and vice versa ), but also by rotating it from right to left at the same time. 2 and 4 are Boolean lattices. A homomorphism (isomorphism ) h between de Morgan Lattice with de Morgan complements : and :0 respectively is a lattice homomorphism (isomorphism) such that h(:a) = :0 h(a). A valuation in a lattice out tted with one or the other of these `complementations' is a map v from the zero-degree formulas into its elements satisfying

v(:A) = :v(A); v(A ^ B ) = v(A) ^ v(B ); v(A _ B ) = v(A) _ v(B ): Note that the occurrence of `:' on the left-hand side of the equation denotes the negation connective, whereas the occurrence on the right-hand side denotes the complementation operation in the lattice (similarly for ^ and _). Such ambiguities resolve themselves contextually. A valuation v can be regarded as in interpretation of the formulas as propositions. De Morgan lattices have become central to the study of relevance of logic, but they were antecedently studied, especially in the late 1950s by Moisil and Monteiro, by Bialynicki-Birula and Rasiowa (as `quasi-Boolean algebras'), and by Kalman (as `distributive i-lattices') (see Anderson and Belnap [1975] or Rasiowa [1974] for references and information).

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Belnap seems to have rst recognised their signi cance for relevance logic, though his research favoured a special variety which he called an intensionally complemented distributive lattice with truth lter (`icdlw/tf'), shortened in Section 18 of [Anderson and Belnap, 1975] to just intensional lattice. An intensional lattice is a structure (L; ^; _; :; T ), where (L; ^; _; :) is a de Morgan lattice and T is a truth- lter, i.e. T is a lter which is complete in the sense T contains at least one of a and :a for each a 2 L, and consistent in the sense that T contains no more than one of a and :a. Belnap and Spencer [1966] showed that a necessary and suÆcient condition for a de Morgan lattice to have a truth lter is that negation have no xed point, i.e. for no element a; a = :a (such a lattice was called an icdl ). For Boolean algebras this is a non-degeneracy condition, assuring that the algebra has more than one element, the one element Boolean algebra being best ignored for many purposes. But the experience in relevance logic has been that de Morgan lattices where some elements are xed points are extremely important (not all elements can be xed points or else we do have the one element lattice). The viewpoint of [Dunn, 1966] was to take general de Morgan lattices as basic to the study of relevance logics (though still results were analogised wherever possible to the more special icdl's). Dunn [1966] showed that upon de ning a rst-degree implication A ! B to be (de Morgan ) valid i for every valuation v in a de Morgan lattice, v(A)  v(B ); A ! B is valid i A ! B is a theorem of Rfde (or Efde ). The analogous result for icdl's (in eect due to Belnap) holds as well. Soundness (`Rfde A ! B ) A ! B is valid) is a more or less trivial induction on the length of proofs in Rfde fragment|cf. [Anderson and Belnap, 1975, Section 18]. Completeness (A ! B valid ) `Rfde A ! B ) is established by proving the contrapositive. We suppose not `Rfde A ! B . We then form the `Lindenbaum algebra', which has as an element for each zero degree formula (zdf) X; [X ] =df fY : Y is a zdf and `Rfde X $ Y g. Operations are de ned so that :[X ] = [:X ], [X ] ^ [Y ] = [X ^ Y ], and [X ] _ [Y ] = [X _ Y ], and we set [X ]  [Y ] whenever `Rfde X ! Y . It is more or less transparent, given Rfde formulated as it is, that the result is a de Morgan lattice. It is then easy to see that A ! B is invalidated by the canonical valuation vc (X ) = [X ], since clearly [A] 6 [B ]. The above kind of soundness and completeness result is really quite trivial (though not unimportant), once at least the logic has had its axioms chopped so that they look like the algebraic postulates merely written in a dierent notation. The next result is not so trivial. Characterisation Theorem of Rfde with Respect to 4. `Rfde A ! B i A ! B is valid in 4, i.e. for every valuation v in 4, v(A)  v(B ).

Proof. Soundness follows from the trivial fact recorded above that Rfde is

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sound with respect to de Morgan lattices in general. For completeness we need the following: 4-Valued Homomorphism Separation Property. Let D be a de Morgan lattice with a 6 b. Then there exists a homomorphism h of D into 4 so that h(a) 6 h(b). Completeness will follow almost immediately from this result, for upon supposing that not `Rfde A ! B , we have v(A) = h[A] 6 h[B ] = v(B ) (the composition of a homomorphism with a valuation is transparently a valuation). So we go on to establish the Homomorphism Separation Property. Assume that a 6 b. By the Prime Filter Separation Property, we know there is a prime lter P with a 2 P and b 62 P . for a given element x, we de ne h(x) according to the following four possible `complementation patterns' with respect to P . 1. x 2 P; :x 62 P : set h(x) = 1; 2. :x 2 P; x 62 P : set h(x) = 0;

3. x 2 P; :x 2 P : set h(x) = p; 4. x 62 P; :x 62 P : set h(x) = q.

It is worth remarking that if D is a Boolean lattice, (3) (inconsistency) and (4) (incompleteness) can never arise, which explains the well-known signi cance of 2 for Boolean homomorphism theory. Clearly these speci cations assure that h(a) 2 fp; I g and h(b) 2 fq; 0g, and so by inspection h(a) 6 h(b). It is left to the reader to verify that h in fact is a homomorphism. (Hint to avoid more calculation: set [p) = fp; I g and [q) = fq; I g (the principal lters determined by p and q). Observe that the de nition of h above is equivalent to requiring of h that h(x) 2 [p) i x 2 P , and h(x) 2 [q) i :a 62 P . Observe that if whenever i = p; q; y 2 [i) i z 2 [i), then y = z . Show for i = p; q; h(a ^ b) 2 [i) i h(a) ^ H (b) 2 [i), h(a _ b) 2 [i) i h(a) _ h(b) 2 [i), and h(:a) 2 [i) i :h(a) 2 [i). 

3.4 Set-theoretical Semantics for First-degree Relevant Implication Dunn [1966] (cf. also [Dunn, 1967]) considered a variety of (eectively equivalent) representations of de Morgan lattices as structures of sets. We shall here discuss the two of these that have been the most in uential in the development of set-theoretical semantics for relevance logic. The earliest one of these is due to Bialynicki-Birula and Rasiowa [1957] and goes as follows. Let U be a non-empty set and let g : U ! U be such that it is of period two, i.e. 1. g(g(x)) = x, for all x 2 U .

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(We shall call the pair (U; g) and involuted set |g is the involution, and is clearly 1{1). Let Q(U ) be a `ring' of subsets of U (closed under \ and [) closed as well under the operation of `quasi-complement' 2. :X = U

g[X ](X  U ).

(Q(U ); [; \; :) is called a quasi- led of sets and is a de Morgan lattice. Quasi-fields of Sets Theorem [Bialynicki-Birula and Rasiowa, 1957]. Every de Morgan lattice D is isomorphic to a quasi- eld of sets.

Proof. Let U be the set of all prime lters of D, and let P range over U . Let :P ! f:a : a 2 P g, and de ne g(P ) = D :P . We leave to the reader to verify that U is closed under g. For each element a 2 D, set f (a) = fP : a 2 P g: Clearly f is one{one because of the Prime Filter Separation Property, so we need only check that f preserves the operations. ad ^: P 2 f (a ^ b) , a ^ b 2 P , ((10 ) of Section 3.3) a 2 P and b 2 P , P 2 f (a) and P 2 f (b) , P 2 f (a) \ f (b). So f (a ^ b) = f (a) \ f (b) as desired. ad _: The argument that f (a _ b) = f (a) [ f (b) is exactly parallel using (20 ) (or alternately this can be skipped using the fact that a _ b = :(:a ^ :b). ad :: P 2 f (:a) , :a 2 P , a 2 :P g[f (a)] , P 2 U g[f (a)].

, a 62 g(P ) , g(P ) 62 f (a) , P 62

We shall now discuss a second representation. Let U be a non-empty set and let R be a ring of subsets of U (closed under intersection and union, but not necessarily under complement, quasi-complement, etc.). by a polarity in R we mean an ordered pair X = (X1 ; X2 ) such that X1 ; X2 2 R. We de ne a relation and operations as follows, given polarities X = (X1 ; X2 ) and Y = (Y1 ; Y2 ):

X  Y , X1  Y1 and Y2  X2 X ^ Y = (X1 \ Y1 ; X2 [ Y2 ) X _ Y = (X1 [ Y1 ; X2 \ Y2 ) :X = (X2 ; X1 ): By a eld of polarities we mean a structure (P (R); ; ^; _; :) where P (R) is the set of all polarities in some ring of sets R, and the other components are de ned as above. We leave to the reader the easy veri cation that every eld of polarities is a de Morgan lattice. 

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We shall prove the following Polarities Theorem [Dunn, 1966]. Every de Morgan lattice is isomorphic to a eld of polarities.

Proof. Given he previous representation, it clearly suÆces to show that every quasi- eld of sets is isomorphic to a eld of polarities. The idea is to set f (X ) = (X; U g[X ]). Clearly f is one{one. We check that it preserves operations. ad ^: f (X \ Y ) = (X \ Y; U g[X \ Y ]) = (X \ Y; (U g[X ]) \ (U g[Y ])) = (X; U g[X ]) ^ (Y; U g[Y ]) = f (X ) ^ f (Y ). ad _: Similar.

ad :: f (:X ) = (:X; U g(:X )) = (U g[X ]; X ) = :f (X ).

g[X ]; U

g(U

g[X ])) = (U

 We now discuss informal interpretations of the representation theorems that relate to semantical treatments of relevant rst-degree implications familiar in the literature. Routley and Routley [1972] presented a semantics for Rfde, the main ingredients of which were a set K of `atomic set-ups' (to be explained) on which was de ned an involution . An `atomic set-up' is just a set of propositional variables, and it is used to determine inductively when complex formulas are also `in' a given set-up. A set-up is explained informally as being like a possible world except that it is not required to be either consistent or complete. The Routley's [1972] paper seems to conceive of set-ups very syntactically as literally being sets of formulas, but the Routley and Meyer [1973] paper conceives of them more abstractly. We shall think of them this latter way here so as to simplify exposition. The Routleys' models can then be considered a structure (K; ; ), where K is a non-empty set,  is an involution on K , and  is a relation from K to zero-degree formulas. We read `a  A' as the formula A holds at the set-up a: 1. (^ ) a  A ^ B , a  A and a  B 2. (_ ) a  A _ B , a  A or a  B 3. (: ) a  :A , not a  A.

The connection of the Routleys' semantics with quasi- elds of sets will become clear if we let (K; ) induce a quasi- eld of sets Q with quasi- complement :, and let j j interpret sentences in Q subject to the following conditions:

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10 . 20 .

j ^ j jA ^ B j = jAj \ jB j j _ j jA _ B j = jAj [ jB j 30 . j:j j:Aj = :jAj. Clause (^ ) results from clause j ^ j by translating a 2 jX j as a  X (cf. Section 3.2). Thus clause j ^ j says a 2 jA ^ B j , a 2 jAj and a 2 jB j; i.e. it translates as clause (^ ). The case of disjunction is obviously the same. The case of negation is clearly of special interest, so we write it out. Thus clause j:j says

a 2 j:Aj , a 2 :jAj; , a 2 K jAj ; , a 62 jAj ; , a 62 jAj: But the translation of this last is just clause (: ). Of course the translation works both ways, so that the Routleys' semantics is just an interpretation in the quasi- elds of sets of Bialynicki-Birula and Rasiowa written in dierent notation. Incidentally soundness and completeness of Rfde relative to the Routleys' semantics follows immediately via the translation above from the corresponding theorem of the previous section vis a vis de Morgan lattices together with their representation as quasi- elds of sets. Of course the Routleys' conceived their results and derived them independently from the representation of Bialynicki-Birula and Rasiowa. We will not say very much here about what intuitive sense (if any) can be attached to the Routleys' use of the -operator in their valuational clause for negation. Indeed this question has had little extended discussion in the literature (though see [Meyer, 1979a; Copeland, 1979]). The Routleys' [1972] paper more or less just springs it on the reader, which led Dunn in [Dunn, 1976a] to describe the switching of a with a as `a feat of prestidigitation'. Routley and Meyer [1973] contains a memorable story about how a `weakly asserts', i.e. fails to deny, precisely what a asserts, but one somehow feels that this makes the negation clause vaguely circular. Still, semantics often gives one this feeling and maybe it is just a question of degree. One way of thinking of a and a is to regard them as `mirror images' of one another reversing `in' and `out'. Where one is inconsistent (containing both A and :A), the other is incomplete (lacking both A and :A), and vice versa (when a = a , a is both consistent and complete and we have a situation appropriate to classical logic). Viewed this way the Routleys' negation clause

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makes sense, but it does require some anterior intuitions about inconsistent and incomplete set-ups. More about the interpretation of this clause will be discussed in Section 5.1. Let us now discuss the philosophical interpretation(s) to be placed on the representation of de Morgan lattices as elds of polarities. In Dunn [1966; 1971] the favoured interpretation of a polarity (X1 ; X2 ) was as a `proposition surrogate', X1 consisting of the `topics' the proposition gives de nite positive information about and, X2 of the topics the proposition gives de nite negative information about. A valuation of a zero degree formula in a de Morgan lattice can be viewed after a representation of the elements of the lattice as polarities as an assignment of positive and negative content to the formula. The `mistake' in the `classical' Carnap/Bar-Hillel approach to content is to take the content of :A to be the set-theoretical complement of the content of A (relative to a given universe of discourse). In general there is no easy relation between the content of A and that of :A. They may overlap, they may not be exhaustive. Hence the need for the double-entry bookkeeping done by proposition surrogates (polarities). If A is interpreted as (X1 ; X2 ), :A gets interpreted as the interchanged (X2 ; X1 ). Another semantical interpretation of the same mathematics is to be found in Dunn [1969; 1976a]. There given a polarity X = (X1 ; X2 ); X1 is thought of as the set of situations in which X is true and X2 as the set of situations in which X is false. These situations are conceived of as maybe inconsistent and/or incomplete, and so again X1 and X2 need not be set-theoretic complements. This leads in the case when the set of situations being assessed is a singleton fag to a rather simple idea. The eld of polarities looks like this (fag; ;) (fag; fag)

B (= fT; F g)

Ts (= fT g)

@@ @

@@ @ s

(;; fag)

s

(;; ;)

N (= ;)

s

F (= fF g)

We have taken the liberty of labelling the points so as to make clear the informal meaning. (Thus the top is a polarity that is simply true in a and the bottom is one that is simply false, but the left-hand one is both true and false, and the right-hand one is neither.) Note that the de Morgan complement takes xed points on both B and N. This is of course our old friend 4, which we know to be characteristic for Rfde . This leads to the idea of an `ambi-valuation' as an assignment to sentences of one of the four values T, F, B, N, conceived either as primitive or realised

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as sets of the usual two truth values as suggested by the labelling. On this latter plan we have the valuation clauses (with double entry bookkeeping): (^) T 2 v(A ^ B ) , T 2 v(A) and T 2 v(B ); F 2 v(A ^ B ) , F 2 v(A) or F 2 v(B ); (_) T 2 v(A _ B ) , T 2 v(A) or T 2 v(B ); F 2 v(A _ B ) , F 2 v(A) and F 2 v(B ); (:) T 2 v(:A) , F 2 v(A); F 2 v(:A) , T 2 v(A): We stress here (as in [Dunn, 1976a]) that all this talk of something's being both true and false or neither is to be understood epistemically and not ontologically. One can have inconsistent and or incomplete assumptions, information, beliefs, etc. and this is what we are trying to model to see what follows from them in an interesting (relevant!) way. Belnap [1977b; 1977a] calls the elements of the lattice `told values' to make just this point, and goes on to develop (making connections with Scott's continuous lattices) a theory of `a useful four-valued logic' for `how a computer should think' without letting minor inconsistencies in its data lead to terrible consequences. Before we leave the semantics of rst-degree relevant implications, we should mention the interesting semantics of van Fraassen [1969] (see also Anderson and Belnap [1975, Section 20.3.1] and van Fraassen [1973]), which also has a double-entry bookkeeping device. We will not mention details here, but we do think it is an interesting problem to try to give a representation of de Morgan lattices using van Fraassen's facts so as to try to bring it under the same umbrella as the other semantics we have discussed here.

3.5 The Algebra of R This section is going to be brief. Dunn has already exposited on this topic in Section 28.2 of [Anderson and Belnap, 1975] and the interested reader should consult that and then Meyer and Routley [1972] for information about how to algebraise related weaker systems and how to give set-theoretical representations. De Morgan monoids are a class of algebras that are appropriate to R in the sense that (i) the Lindenbaum algebra of R is one of them and (ii) all R theorems are valid in them ((ii) gives soundness, and of course (i) delivers completeness by way of the canonical valuation). In thinking about de Morgan monoids it is essential that R be equipped with the sentential constant t. Also it is nice to think of fusion (Æ) as a primitive connective, with even perhaps ! de ned (A ! B =df :(A Æ :B )) but this is not essential since in R (but not the weaker relevance logics) fusion can be de ned as A Æ B =df :(A ! :B ). A de Morgan monoid is a structure D = (D; ^; _; :; Æ; e) where

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(I) (D; ^; _; :) is a de Morgan lattice,

(II) (D; Æ; e) is an Abelian monoid, i.e. Æ is a commutative, associative binary operation on D with e its identity, i.e. e 2 D and e Æ a = a for all a 2 D,

(III) the monoid is ordered by the lattice, i.e. a Æ (b _ c) = (a Æ b) _ (a Æ c), (IV) (V)

Æ is upper semi-idempotent (`square increasing'), i.e. a  a Æ a, a Æ b  c i a Æ :c  :b (Antilogism).

De Morgan monoids were rst studied in [Dunn, 1966] (although [Meyer, 1966] already had isolated some of the key structural features of fusion that they abstract). They also were described in [Meyer et al., 1974] and used in showing admissible. Similar structures were investigated quite independently by Maksimova [1967; 1971]. The key trick in relating de Morgan monoids to R is that they are residuated, i.e. there is a `residual' operation ! so that (VI) a Æ b  c i a  b ! c.

Indeed this operation turns out to be :(b Æ :c) (with the weaker systems or with positive R it is important to postulate this law of the residual). Thus (1) a Æ b  c , b Æ a  c Commutativity (2) a Æ b  c , b Æ :a 1, (V) (3) a Æ b  c , a  :(b Æ :c) 2, de Morgan lattice. As an illustration of the power of (VI) we show how the algebraic analogue of the Pre xing axiom follows from Associativity. First note that one can get from (III) the law of (Monotony)

a  b ) c Æ a  c Æ b.

Now getting down to Pre xing: 1. a ! b  a ! b

2. (a ! b) Æ a  b 1, (VI)

3. (c ! a) Æ c  a 2, Substitution

4. (a ! b) Æ ((c ! a) Æ c)  b 2,3, Monotony

5. ((a ! b) Æ (c ! a)) Æ c  b 4, Associatively 6. (a ! b) Æ (c ! a)  c ! b 5, (VI)

7. a ! b  ((c ! a) ! (c ! b)) 6, (VI).

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Incidentally, something better be said at this point about how validity in de Morgan monoids is de ned. Unlike the case with Rfde, there are theorems which are of the form A ! B , e.g. A _ :A. We need some way of de ning validity which is broader than insisting that always v(A)  v(B ). The identity e interprets the sentential constant t. By virtue of the R axiom A $ (t ! A) characterising t, it makes sense to count all de Morgan monoid elements a such that e  a as `designated', and to de ne A as valid i v(A)  e for all valuations in all de Morgan monoids. We have the following law

a  b , e  a ! b; which follows immediately from (VI) and the fact that e is the identity element. This means that (7) just above can be transformed into

e  (a ! b) ! ((c ! a) ! (c ! b)) validating pre xing as promised. Other axioms of R can be validated by similar moves. Commutativity validates Assertion, that e is the identity validates self-implication, squareincreasingness validates Contraction, antilogism validates Contraposition, and the other axioms fall out of de Morgan lattice properties with lattice ordering and the residual law pitching in. We shall not here investigate the `converse' questions about how the fusion connective in R is associative, etc. (that the Lindenbaum algebra of R is indeed a de Morgan monoid (cf. Dunn's Section 28.2.2 of [Anderson and Belnap, 1975])), but the proof is by ` ddling' with contraposition being the key move. Not as much is known about the algebraic properties of de Morgan monoids as one would like. Getting technical for a moment and using unexplained but standard terminology from universal algebra, it is known that de Morgan monoids are equationally de nable (replace (V) with a Æ:(a Æ:b)  b, which can be replaced by the equation (a Æ :(a Æ :b)) _ b = b). So by a theorem of Birkho the class of de Morgan monoids is closed under sub-algebras, homomorphic images, and subdirect products. Further, given a de Morgan monoid D with a prime lter P with e 2 P , the relation a  b , (a ! b) ^ (b ! a) 2 P is a congruence, and the quotient algebra D= is subdirectly irreducible, and every de Morgan monoid is a subdirect product of such. It would be nice to have some independent interesting characterisation of the subdirectly irreducibles. One signi cant recent result about the algebra of R has been provided by John Slaney. He has shown that there are exactly 3088 elements in the free De Morgan monoid generated by the identity e. Or equivalently, in the language of R including the constant t, there are exactly 3088 nonequivalent formulae free of propositional variables. The proof technique is

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quite subtle, as generating a large algebra of 3088 elements is not feasible, even with computer assistance. Instead, Slaney attacked the problem using a \divide and conquer" technique [Slaney, 1985]. Since R contains all formulae of the form A _ :A, for any A, whenever L is a logic extending R, L = (L + A) \ (L + :A), where L + A is the result of adding A as an axiom to L and closing under modus ponens and adjunction. Given this simple result, we can proceed as follows. R is (R + f ! t) \ (R + :(f ! t)). Now it is not diÆcult to show that the algebra of R + (f ! t) generated by t is the two element boolean algebra. Then you can restrict your attention to the algebra generated by t in the logic R + :(f ! t). If this has some characteristic algebra, then you can be sure that the elements freely generated by t in R are bounded above by the number of elements in the direct product of the two algebras. To get the characteristic algebra of R + :(f ! t), Slaney goes on to divide and conquer again. He ends up considering six matrices, characterising six dierent extensions of R. This would give him an upper bound on the number of constants (the matrices were size 2, 4, 6, 10, 10 and 14, so the bound was their product, 67200, well above 3088). Then you have to consider how many of these elements are generated by the identity in the direct product algebra. A reasonably direct argument shows that there are exactly 3088 elements generated in this way, so the result is proved.

3.6 The Operational Semantics (Urquhart) This set-theoretical semantics is based upon an idea that occurred independently to Urquhart and Routley in the very late 1960s and early 70s. We shall discuss Routley's contribution (as perfected by Meyer) in the next section and also just mention some related independent work of [Fine, 1974]. Here we concentrate upon the version of Urquhart [1972c] (cf. also Urquhart [1972b; 1972a; 1972d]). Common to all the versions is the idea that one has some set K whose elements are `pieces of information', and that there is a binary operation Æ on K that combines pieces of information. Also there is an `empty piece of information' 0 2 K . We shall write x  A to mean intuitively `A holds according to the piece of information x'. The whole point of the semantics is disclosed in the valuational clause (!) x  A ! B i 8y 2 K (if y  A, then x Æ y  B ):

The idea of the clause from left-to-right is immediately clear: if A ! B is given by the information x, then if A is given by y, then the combined piece of information x Æ y ought to give B (by modus ponens ). The idea of the clause from right-to-left is to say that if this happens for all pieces of information y, this can only be because x gives us the information that A ! B.

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Perhaps saying the whole point of the semantics is given in the clause (!) along is an exaggeration. There are at least two quick surprises. The rst is that we do not require (or want) a certain condition analogised from a condition required by Kripke's (relational) semantics for intuitionistic logic: (The Hereditary Condition) If x  A, then x Æ y  A: This would yield that if x  A, then x  B ! A, i.e. if y  B , ten x Æ y  A. This would quickly involve us in irrelevance. The other surprise is related to the failure of the Hereditary Condition: Validity cannot be de ned as a formula's holding at all pieces of information in all models, since even A ! A would not then turn out to be valid. Thus x  A ! A requires that if y  A then x Æ y  A. But this last is just a commuted form of the rejected Hereditary Condition, and there is no more reason to think it holds. We shall see in a moment that the appropriate de nition of validity is to require that 0  A for the empty piece of information in all models. Enough talk of what properties Æ does not have! What property does it have? We have just been irting with one of them. Clearly 0  A ! A requires that if x  A then 0 Æ x  A, and how more naturally would that be obtained than requiring that 0 be a (left) identity? 0 Æ x = x:

(Identity)

This then seems the minimal algebraic condition on a model. Urquhart in fact requires others, all naturally motivated by the idea that Æ is the `union' of pieces of information.

xÆy =yÆx x Æ (y Æ z ) = (x Æ y) Æ z x Æ x = x:

(Commutativity) (Associativity) (Idempotence)

These conditions combined may be expressed by saying that (K; Æ; 0) is a `(join) semi-lattice with least element 0', and accordingly Urquhart's semantics is often referred to as the `semi-lattice semantics'. It is well-known that every semi-lattice is isomorphic to a collection of sets with union as Æ and the empty set as 0 (map x to fy : x Æ y = yg so that henceforward Æ will be denoted by [. Each of the conditions above of course corresponds to an axiom of R! when it is nicely axiomatised. Thus commutativity plays a natural role in verifying the validity of assertion. The following use of natural deduction

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in the metalanguage makes this point nicely (we write `A; x' rather than x  A for a notational analogy): 1. A; x Hypothesis 2. A ! B; z Hypothesis 3. B; x [ z 1, 2, (!) 4. B; z [ x 3, Comm. 5. (A ! B ) ! B; x 2, 4(!) 6. (A ! B ) ! B; 0 [ x 5, Identity 7. A ! ((A ! B ) ! B ); 0. The reader may nd it amusing to write out an analogous pair of proofs for Pre xing, seeing how Associativity of [ enters in, and for Contraction watching the Idempotence.24 The game has now been given away. There is some ddling to be sure in proving a completeness theorem for R! re the semi-lattice semantics, but basically the idea is that the semi-lattice semantics is just the system F R! `written in the metalanguage'. There is not a problem in extending the semi-lattice semantics so as to accommodate conjunction. The clause

x  A ^ B i x  A and x  B

(^)

does nicely. Somewhat strangely, the `dual' clause

x  A _ B i x  A of x  B

(_)

causes trouble. It is analogous to having the rule of _-Elimination NR read:

A _ B; x A; x Hyp. .. . c; x [ y B; x Hyp. .. . C; x [ y C; x [ y:

With this rule we can prove (]) (A ! B _ C ) ^ (B ! C ) ! (A ! C );

24 Though unfortunately veri cation of this last does not depend purely on Idempotence, but rather on (xy)y = xy, which of course is equivalent to Idempotence given Associativity and Identity. The veri cation of the formula A ^ (A ! B ) ! B `exactly' uses Idempotence, but of course this is hardly a formula of the implicational fragment.

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which is not a theorem of R (see [Urquhart, 1972c]|the observation is Meyer's and Dunn's.)25 And of course one can analogously verify that it is valid in the semi-lattice semantics. Note that the condition (_) is not nearly as intuitive as the condition (^). The condition (^) is plausible for any piece of information x, at least if the relation x  c does not require that C be explicitly contained in x. On the other hand the condition (_) is much less than natural. Does not it happen all the time that a piece of information x determines A _ B to hold, without saying which? Is not this one of the whole points of disjunctions? Pieces of information x that satisfy (_) might be called `prime' (in analogy with this epithet applied to theories of Section 2.4), and they have a kind of completeness or eeminateness that is rare in ordinary pieces of information. This by itself counts as no criticism of the semantics, since it is quite usual in semantical treatments to work with such idealised notions. The condition (_) is not really as `dual' to the condition (^) as one might think. Thus the formula (]d) (B ^ C ! A) ^ (C ! B ) ! (C ! A); which is the dual of (]) is easily seen not to be valid in the semantics. This seems to be connected with another feature (problem?) of the semantics, to wit, no one has ever gured out how to add a natural semantical treatment of classical negation to the semantics (although it is straightforward to add a species of constructive negation|see [Urquhart, 1972c]).26 The point of the connection is that (]d) would follow from (]) given classical contraposition principles, and yet the rst is valid and the second one invalid in the positive semantics. So something about the positive semantics would have to be changed as well to accommodate negation. The semi-lattice semantics has been extensively investigated in Charlewood [1978; 1981]. He ts it out with (two) natural deduction systems one with subscripts and one without. This last is in fact the (positive) system of Prawitz [1965], which Prawitz wrongly conjectured to be the same as Anderson and Belnap's. Charlewood proves normalisation theorems (something that was anticipated by Prawitz for his system|incidentally the problem of normalisation for the Anderson{Belnap R seems still open). Incidentally, one advantage of these natural deduction systems is that, unlike the Anderson{Belnap one for their system R (cf. Section 1.5), they allow for a proof of distribution. Charlewood also carries out in detail the engineering needed to implement K. Fine's axiomatisation of the semi-lattice semantics. What is needed is 25 It would be with C ! C as an additional conjunct in the antecedent. 26 Charlewood and Daniels have investigated a combination of the semi-lattice semantics

for the positive connectives and a four-valued treatment of negation in the style of [Dunn, 1976a]. they avoid the problem just described by in eect building into their de nition of a model that it must satisfy classical contraposition. This does not seem to be natural.

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to add to the Anderson{Belnap's R+ the following rule: R1: From B0 ^ ((A1 ^ q1 ; : : : ; qn ^ An ) ! X ) ! ((B1 ^ q1 ; : : : ; Bn ^ qn ) ! E ) for X = B; C , and n  0 infer the same thing with B _ C put in place of the displayed X , provided that he qi are distinct and occur only where shown. We forbear taking cheap shots at such an ungainly rule, the true elegance of which is hidden in the details of the completeness proof that we shall not be looking into. Obviously Anderson and Belnap's R is to be preferred when the issue is simplicity of Hilbert-style axiomatisations.27

3.7 The Relational Semantics (Routley and Meyer) As was indicated in the last section, Routley too had the basic idea of the operational semantics at about the same time as Urquhart. Priority would be very hard to assess. At any rate Dunn rst got details concerning both their work in early 1971, although J. Garson told him of Urquhart's work in December of 1970 and he has seen references made to a typescript of Routley's with a 1970 date on it (in [Charlewood, 1978]). Meyer and Dunn were colleagues at the time, and Routley sent Meyer a somewhat incomplete draft of his ideas in early 1971. This was a courageous and open communication in response to our keen interest in the topic (instead he might have sat on it until it was perfected). The draft favoured the operational semantics, indeed the semi-lattice semantics, and was not clear that this was not the way to go to get Anderson and Belnap's R. But the draft started with a more general point of view suggesting the use of a 3-placed accessibility relation Rxyz (of course a 2-placed operation like [ is a 3-placed relation, but not always conversely), with the following valuation clause for !: (!) x  A ! B i 8y; z 2 K (if Rxyz and y  A, then z  B ): Forgetting negation for the moment, the clauses for ^ and _ are `truth functional', just as for the operational semantics. Meyer, having observed with Dunn the lack of t between the semilattice semantics and R, was all primed to make important contributions to Routley's suggestion. In particular he saw that the more general 3-placed relation approach could be made to work for all of R. In interpreting Rxyz perhaps the best reading is to say that the combination of the pieces of information x and y (not necessarily the union) is a piece of information in z (in bastard symbols, x Æ y  z ). Routley himself called the x; y, etc. 27 However, the semi-lattice semantics has been taken up and generalised in the eld of substructural logics in the work of [Dosen, 1988; Dosen, 1989] and [Wansing, 1993].

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`set-ups', and conceived of them as being something like possible worlds except that they were allowed to be inconsistent and incomplete (but always prime). On this reading Rxyz can be regarded as saying that x and y are compatible according to z , or some such thing. Before going on we want to advertise some work that we are not going to discuss in any detail at all because of space limitations. The work of Fine [1974] independently covers some of the same ground as the Routley-Meyer papers, with great virtuosity making clear how to vary the central ideas for various purposes. The book of Gabbay [1976, see chapter 15] is also deserving of mention. We now set out in more formal detail a version of the Routley{Meyer semantics for R+ (negation will be reserved for the next section). The techniques are novel and the completeness proof quite complicated, so we shall be reasonably explicit about details. The presentation here is very much indebted to work (some unpublished) of Routley, Meyer and Belnap. By an (R+ ) frame (or model structure) is meant a structure (K; R; 0), where K is a non-empty set (the elements of which are called set-ups ), R is a 3-placed relation on K; 0 2 K , all subject to some conditions we shall state after a few de nitions. We de ne for a; b 2 K; a  b (Routley and Meyer used >) i R0ab, and R2 abcd i 9x (Rabx and Rxcd). We also write this last as R2 (ab)cd and distinguish it from R2a(bc)d =df 9x(Raxd ^ Rbcx). The variables a; b, etc. will be understood as ranging over the elements of some K xed by the content of discussion. Transcribing the conditions on the semi-lattice semantics as closely as we can into this framework we get the requirements 1. (Identity) R0aa, 2. (Commutativity) Rabc ) Rbac, 3. (Associativity) R2 (ab)cd ) R2 a(bc)d,28 4. (Idempotence) Raaa. It should be remarked that these conditions fail to pick up the whole strength of the corresponding semi-lattice conditions. Thus, e.g. Identity here only picks up 0
a  a and not conversely, and similarly for Idempotence (also of course Commutativity and Associativity do not require any identity, but this is a slightly dierent point). We need for technical reasons one more condition: 5. (Monotony) Rabc and a0  a ) Ra0 bc.

28 In the original equivalent conditions of Routley and Meyer [1973] this was instead `Pasch's Law': R2 abcd ) R2 acbd. Also Monotony (condition (5) below) was misprinted there.

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By a model we mean a structure M = (K; R; 0; ), where (K; R; 0) is a frame and  is a relation from K to sentences of R+ satisfying the following conditions: (1) (Atomic Hereditary Condition). For a propositional variable p, if a  p and a  b, then b  p. (2) (Valuational Clauses). For formulas A; B (!) a  A ! B i 8b; c 2 K (if Rabc and b  A, then c  B ); (^) a  A ^ B i a  A and a  B ; (_) a  A _ B i a  or a  B . We shall say that A is veri ed on M if 0  A, and that A entails B on M if 8a 2 K (if a  A, then a  B ). We say that A is valid if A is veri ed on all models. It is easy to prove by an induction on A, the following (note how Monotony enters in): Hereditary Condition. For an arbitrary formula A, if a  A and a  b, then b  A. Verification Lemma. If in a given model (K; R; 0; )A entails B in the sense that for every a 2 K; a  A only if a  B , then A ! B is veri ed in the model, i.e. 0  A ! B .

Proof. suppose that R0ab and a  A. By the hypothesis of the Lemma, a  B , and by the Hereditary Condition, b  B , as is required for 0  A ! B.  We are now in a position to prove the Soundness Theorem. If `R A, then A is valid.

Proof. Most of this will be left to the reader. We rst show that the axioms of R+ are valid. Since they are all of the form A ! B we can simplify matters a little by using the Veri cation Lemma. As an illustration we verify Assertion (the reader may wish to compare this to the corresponding veri cation vis a vis the semi-lattice semantics of the last section). To show A ! [(A ! B ) ! B ] is valid, it suÆces by the Veri cation Lemma to assume a  A and show a  A ! B ! B . For this last we assume Rabc and b  A ! B , and show c  B . By Commutativity, Rabc. By (!) since we have b  A ! B and a  A, we get c  B as desired. The veri cation of the implicational axioms of Self-Implication and Pre xing are equally routine, falling right out of the Veri cation Lemma and

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Associativity for the relation R. Unfortunately the veri cation of Contraction is a bit contrived (cf. note 24 above), so we give it here. To verify Contraction, we assume that (1) a  A ! :A ! B and show a  A ! B . To show this last we assume that (2) Rabc and (3) b  A, and show c  B . From (2) we get, by Commutativity, Rbac. But Rbbb holds by Idempotence. so we have R2 (bb)ac. By Associativity we get R2 b(ba)c, i.e. for some x, both (4) Rbxc and (5) Rbax. by Commutativity, from (5) we get Rabx. Using (!), we obtain from this, (1), and(3) hat (6) x  A ! B . by Commutativity from (4) we get Rxbc, and from this, (6), and (3) we at last get the desired c  B . Veri cation of the conjunction and disjunction axioms is routine and is safely left to the reader. It only remains to be shown then that the rules modus ponens and adjunction preserve validity. Actually something stronger holds. It is easy to se that for any a 2 K (not just 0), if a  A ! B and a  A, then a  B (by virtue of Raaa), and of course it follows immediately from (^) that if a  A and a  B , then a  A ^ B .  We next go about the business of establishing the Completeness Theorem. If A is valid, then R+ A. The main idea of the proof is similar to that of the by now well-known Henkin-style completeness proofs for modal logic. We suppose that no R+ A and construct a so-called `canonical model', the set-us of which are certain prime theories (playing the role of the maximal theories of modal logic). The base set-up 0 is constructed as a regular theory (for the terminology `regular', `prime', etc. consult Section 2.4; of course everything is relativised to R+ ). From this point on for simplicity we shall assume that we are dealing with R+ out tted with the optional extra fusion connective Æ and the propositional constant t (recall these can be conservatively added | cf. Section 1.3). We then de ne Rabc to hold precisely when for all formulas A and B , whenever A 2 a and B 2 b, then A Æ B 2 c.29 Let us look now at the details. Pick 0 as some prime regular theory T with A 62 T . We can derive that at least one such exists using the Belnap Extension Lemma (it was stated in Section 2.5 for RQ, but it clearly holds for R+ as well). thus set
= R+ and  = fAg. De ne K = set of prime theories,30 and de ne the accessibility relation R canonically as above. 29 The use of Æ and t is a luxury to make things prettier at least at the level of description. Thus, e.g. as we shall see, the associativity of R follows from the associativity of Æ, and other mnemonically pleasant things happen. We could avoid its use by de ning Rabc to hold whenever if A 2 a and A ! B 2 b, then B 2 c. Incidentally, the valuational clause for fusion is : x  A Æ B i for some a; b such that Rabx; a  A and b  B . The valuational clause for t is x  t i 0  x. 30 One actually has a choice here. We have required of theories that they be closed under implications provable in 0, i.e. require of T that whenever A 2 T and A ! B 2 0,

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THEOREM 4. The canonically de ned structure (K; 0; R) is an R+ frame. LEMMA 5. The relation R de ned canonically above satis es Identity, Commutativity, Idempotence, and Associativity. Proof.

ad Identity. We need to show that R0aa, i.e. if X 2 0 and A 2 a, then X Æ A 2 a. By virtue of the R -theorem A ! t Æ A, we have t Æ A 2 a. But using the R-theorem X ! :t ! x, we have t ! X 2 0. By Monotony we have X Æ A 2 a as desired. ad Commutativity. Suppose Rabc. We need show Rbac, i.e. if B 2 b and A 2 a, then B Æ A 2 c. From Rabc, it follows that A Æ B 2 c. But by virtue of the R-theorem A Æ B ! B Æ A (commutativity of Æ) we have B Æ A 2 C , as desired. ad Idempotence. We need show Raaa, i.e. if A 2 a and B 2 a, then A Æ B 2 a. This follows from the R-theorem A ^ B ! A Æ B , which follows ultimately from the square increasingness of Æ; (X ! X Æ X ), as the proof sketch below makes clear.

1. A ^ B ! A Axiom 2. A ^ B ! B Axiom 3. (A ^ B ) Æ (A ^ B ) ! A Æ B 1, 2, Monotony 4. A ^ B ! A Æ B 3, square increasingness ad Associativity. This is by far the least trivial property. Let us then assume that R2 (ab)cd, i.e. 9x(Rabx and Rxcd). We need then show that there is a prime theory y such that Rayd and Rbcy, i.e. R2 a(bc)d. Set y0 = fY : 9B 2 b; C 2 c :`R B Æ C ! Y g. (This is sometimes referred to as b Æ c). Clearly the de nition of y0 assures that Rbcy0. Observe that y0 is a theory.31 Thus it is clear that y0 is closed under provable R-implication, since this is just transitivity. We show it is also closed under adjunction. Thus suppose for some B; B 0 2 b; C; C 0 2 c; R B Æ C ! Y and `R B 0 Æ C 0 ! Y 0 . Then `R (B Æ C ) ^ (B 0 Æ C 0 ) ! Y ^ Y 0 using easy properties of conjunction. But we have the R-theorem (B ^ B 0 ) Æ (C ^ C 0 ) ! (B Æ C ) ^ (B 0 Æ C 0 ) (which follows basically from the one-way distribution of Æ over ^; X Æ (Y ^ Z ) ! (X Æ Y ) ^ (X Æ Z ), which then B 2 T . The latter is a stronger requirement and leads to the `smaller' reduced models of [Routley et al., 1982], which are useful for various purposes. 31 The presentation of Routley{Meyer [1973] is more elegant than ours, developing as they do properties of what they call the calculus of `intensional R-theories', showing that it is a partially ordered (under inclusion) commutative monoid (Æ as de ned above) with identity 0. Further Æ is monotonous with respect to , i.e. if a  b then c Æ a  c Æ b, and Æ is square increasing, i.e. a  a Æ a. Then de ning Rabc to mean a Æ b  c, the requisite properties of R fall right out.

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follows basically from Monotony, Y1 ! Y2 ! X Æ Y1 ! X Æ Y2 , which is easy). So by transitivity we get `R (B ^ B 0 ) Æ (C ^ C 0 ) ! Y ^ Y 0 , from which it follows that Y ^ Y 0 2 y0 as promised (B ^ B 0 2 b; C ^ C 0 2 c of course, since b; c are closed under adjunction). We next verify that Ray0d. Suppose that A 2 a and Y 2 y0 . Then for some B 2 b; C 2 c; `R B Æ C ! Y . Since Rabx; A Æ B 2 x. And since Rxcd(A Æ B ) Æ C 2 d. By the associativity of Æ (since d is a theory), then A Æ (B Æ C ) 2 d. but by Monotony, since `R B Æ C ! Y , we have `R A Æ (B Æ C ) ! A Æ Y . Hence A Æ Y 2 d, as needed. The reader is excused if he has lost the thread a bit and thinks that we are now nished verifying the associativity of R. We wanted some prime theory y which lls in the blanks 1. Ra d and 2. Rbc , and we have just nished verifying that y0 is a theory that does ll in the blanks. The kicker is that y0 need not be prime. So we work next at pumping up y0 to make it prime while continuing to ll in the blanks. It clearly suÆces to prove . Let a0 and y0 be theories that need not be prime, and let d be a prime theory. If Ra0 y0 d, then there exists a prime theory y such that (i) y0  y and (ii) Ra0 yd. The Squeeze Lemma

This can be accomplished by a Lindenbaum-style construction like that of Section 2.3 (or alternatively Zorn's Lemma may be used as in Routley and Meyer [1973]). The idea is to de ne y as the union of a sequence of sets of formulas yn , where (relative to some xed enumeration of the formulas) yn+1 is de ned inductively as yn [ fAn+1 g if Ra(yn [ fAn+1 g)d, and otherwise yn+1 is just yn . But it is instructive to crank the existence of the given y out of the Belnap Extension Lemma for R. Thus set
= y0 and  = fA : 9B (A ! B ) 2 a and B 62 dg. We need check that (
; ) is exclusive. We observe rst that  is closed under disjunction. Thus suppose A1 ; A2 2 . Then for some B1 ; B )2; A1 ! B1 ; A2 ! B2 2 a, and yet B1 ; B2 62 d. Then (since d is prime) B1 _ B2 62 d. but since a is a theory, then A1 _ A2 ! B1 _ B2 by an appropriate theorem of R in the proximity of the disjunction axioms. So A1 _ A2 2  as desired. Since
is closed dually under adjunction (that was the point of observing above that y0 is a theory), this means that if the pair (
; ) fails to be exclusive, then for some X 2
; A 2 ; `R X ! A. So for some B; A ! B 2 a and B 62 d. But since a is a theory, by transitivity we derive that X ! B 2 a. But since Raxd

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and X 2 x, we get (X ! B ) Æ X 2 d. But since `R X Æ (X ! B ) ! B , we have B 2 d, contrary to the choice of B . Now that we know (
; ) is an exclusive pair we apply the Belnap Extension Lemma to get a pair (y; y0 ) with y0 =
 y and y a prime theory, completing the proof of the Squeeze Lemma, which actually does complete the proof that the relation R is Associativity. ad Monotony. (Yes, we still have something left to do.) Let us suppose that R0a0a and Rabc, and show Ra0 bc. Note that it follows from R0a0a that a0  a,32 from which it follows at once from Rabc and Ra0 bc. Thus if X 2 a0 then since X ! X 2 0, then (X ! X ) Æ X 2 a. But since `R+ (X ! X ) Æ X ! X , then X 2 a. Having now nally veri ed that the canonical (K; 0; R) has all the properties of an R+ -frame, we need now to de ne an appropriate relation  on it. The natural de nition is a  A i A 2 a, but we need now to verify that this has the properties (1) and (2) required of  above. Theorem 2. The canonically de ned (K; 0; R; ) is indeed an R-model.

Proof. ad (1) (the Hereditary Condition). Suppose a  b, i.e. R0ab. We show that a  b, from which the Hereditary Condition immediately follows. Suppose then that A 2 a. Since t 2 0; t Æ A 2 b. But via the R-theorem t Æ A ! A, we have A 2 b as desired. ad (2) (the valuation of clauses). The clauses (^) and (_) are more or less immediate (primeness is of course needed for half of (_)). The clause of interest is (!). Applying the canonical de nition of , this amounts to (!c ) A ! B 2 a i 8b; c(if Rabc and A 2 b; then B 2 c): Left-to-right is argued as follows. Suppose A ! B 2 a; Rabc; A 2 b, and show B 2 c. Rabc of course means canonically that whenever X 2 a and Y 2 b, then X Æ Y 2 c. Setting X = A ! B and Y = A, we get A Æ (A ! B ) ! C . Then using the R+ - theorem

A Æ (A ! B ) ! B; we obtain B 2 c: Right-to-left is harder, and in fact involves the third (and last) application of the Belnap Extension Lemma in the proof of Completeness. Thus suppose contrapositively that A ! B 62 a. We need to construct prime theories b and c, with A 2 b and B 62 c. We let
b = Th(fAg) and set
c = a Æ
b , i.e. fZ : 9X 2 a; 9Y 2
b `R+ X Æ Y ! Z g. This is the same as fZ : 9X 2 a `R+ X Æ A ! Z g. We set c = fB g. Clearly (
c ; c ) is an exclusive pair, for otherwise `R+ X Æ A ! B , i.e. `R+ X ! (A ! B ) for some X 2 a, and so A ! B 2 a contrary to our supposition. We apply Belnap's Extension Lemma to get an exclusive pair (c; c0 ) with
c  c and c prime theory. Note 32 In the `reduced models (cf. note 46) one can show that R0a0 a i a0  a.

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that by de nition of
b and
c ; Ra
b
c , and so Ra
b c. We are now in a position to apply the Squeeze Lemma getting a prime theory b 
b such that Rabc. Clearly A 2 b, but also B 62 c since B 2 c  c0 (c and c0 are exclusive). This at last completes the proof of the Completeness Theorem for R+ .



. It is fashionable these days to always prove strong completeness. This could have been done. Thus de ne A to be a logical consequence of a set of formulas i for every R+ -model M , if 0  B for every B 2 , then 0  A. This is a kind of classical notion and should not be confused with some kind of relevant consequence. Thus, e.g. where B is a theorem of R+ , since always 0  B , B will be a logical consequence of any set . De ne B to be deducible from (again in a neo-classical sense) to mean B 2 Th( [ R+ ). Appropriate modi cations of the work above will show that logical consequence is equivalent to deducibility. Remark

3.8 Adding Negation to R+ We now discuss the Routley{Meyer semantics for the whole system R. The idea is simply to add the Routley's treatment of negation using the -operator (discussed in Section 3.4). (This is not diÆcult and there is very little reason to segregate it o into this separate section, except that we thought that the treatment of R+ was complicated enough.) Thus an R-frame is a structure, (K; R; 0; ) where (K; R; 0) is an R+ frame and K is closed under the unary operation  satisfying: (Period two) A = a, (Inversion) Rabc ) Rac b For an R-model the valuational clauses for the positive connectives are as for an R-model, and we of course add (:) a  :A i a 6 A: The soundness and completeness results are relatively easy modi cations of those for R+ . That  is of period two naturally is used n the veri cation of Double Negation and Inversion is central to the veri cation of Contraposition. For completeness, a is de ned canonically as fA : :A 62 ag (cf. the de nition of the analogue g[P ] in the proof of Bialynicki{Birula and Rasiowa's representation of de Morgan lattices in Section 3.4), and one of course has to show that a is a prime theory when a is. One also has to show that canonical  is of period two and satis es (Inversion), and that canonical  satis es (:) above, i.e. A 2 a , :A 62 a , i.e. :A 62 fB : :B 62 ag, i.e. ::A 2 a, which of course just uses Double Negation.

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It is worth remarking that since the canonical 0 is a prime regular theory, then since `R A _ :A, then 0 is complete (but not necessarily consistent| this is relevant to the development in Section 3.9). For your garden variety Routley{Meyer model (not necessarily canonical) notice also that 0  A or 0  :A. This follows ultimately from 0  0, i.e. R000, proven below. 1. R00 0 2. R000

1, (Inversion), (Period two)

3. R000

2, (Commutation).

Now 0  0 means by the Hereditary Condition that if 0  A then 0  A, i.e. 0  :A as desired. It should be said that although either the four-valued treatment or the -operator treatment of negation work equally well for rst-degree relevant implications (at least from a technical point of view), the -operator treatment seems to win hands down in the context of all of R. Meyer [1979a] has succeeded in giving a four-valued treatment of all of R, but at the price of great technical complexity (e.g. the accessibility relation has to be made four-valued as well, and that is just for starters). Further, as Meyer points out, one's models still have to be closed under , so it still can be said to sneak in the back door.

3.9 Routley-Meyer Semantics for E and other Neighbours of R Once one sets down a set of conditions on an accessibility relation, they can be played with n various ways so as to produce semantics for a wide variety of systems as the experience with modal logic has taught us. Also other features of the frames can be generalised. We can here only give the avour of a whole range of possible and actual results. In all the results below  will satisfy the same conditions as for R+ (or R ) models (as appropriate). To begin with we follow Routley and Meyer [1973] with the description of a series of conditions on positive frames and corresponding axioms for propositional logic. They begin by requiring of a B+ -frame (K; R; 0) B1. a  a B2. a  b and b  x ) a  c B3. a0  a and Rabc ) Ra0 bc. B1{B2 of course say that that it is monotone.

 is a quasi-order, and B3 says something like

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`B' appears to be for `Basic', for they regard the above postulates as a natural minimal set on their approach.33 Gabbay [1976] investigates even weaker logics where no conditions at all are placed on the frame, but these have no theorems and are characterised only by rules of deducibility (unless Boolean negation and/or the Boolean material conditional is present, options which he does explore). The sense in which the above postulates are minimal goes something like this. B3 is needed in proving the Hereditary Condition for implications, and the Hereditary Condition is needed in turn for verifying 0  A ! A (indeed anything) so we have at least some minimal theorems. The Hereditary Condition is used in showing the equivalence of the veri cation of an implication in a model and entailment in that mode, i.e. 0  A ! B i 8x 2 K (x  A ) x  B ) (cf. Section 3.7 to see how these conditions were used to establish these facts about R+ -models). What about B2? We think it is just a `freebie'. It seems to play no role in verifying axioms or rules, but the completeness proof can be made to yield canonical (`reduced') models (cf. note 3.7) that satisfy it, so why not have it? This seems to be what Routley et. al. [1982] say. It appears that B1 is even more a freebie. It may be shown that A is a theorem of the system B+ (formulated in Section 1.3) i A is valid in all B+ models. Routley and Meyer establish the following correspondence between conditions on the accessibility relation R and axioms: (1) Raaa A ^ (A ! B ) ! B (2) Rabc ) R2 a(ab)c (A ! B ) ^ (B ! C ) ! (A ! C ) (3) R2 abcd ) R2 a(bc)d A ! B ! ([B ! C ] ! [A ! C ]) (4) R2 abcd ) R2 b(ac)d A ! B ! ([C ! A] ! [C ! B ]) (5) Rabc ) R2 abbc (A ! [A ! B ]) ! (A ! B ) (6) Ra0a ([A ! A] ! B ) ! B (7) Rabc ) Rbac A ! ([A ! B ] ! B ) (8) 0a A ! (B ! B ) (9) Rabc ) b  c A ! (B ! A). Routley and Meyer connect these conditions on accessibility relations to axioms extending the basic logic B. The correspondence is more perspicuous when you consider the structural rules corresponding to each axiom or condition. We can express these as conditions on fusion: 33 However, some notational confusion is possible, with Fine's use of `B' as another basic relevance logic diering slightly from Routley and Meyer's usage [Fine, 1974]. For Fine, B includes the law of the excluded middle, and for Routley and Meyer, it does not.

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(1) Raaa A`AÆA (2) Rabc ) R2 a(ab)c A Æ B ` A Æ (A Æ B ) (3) R2 abcd ) R2 a(bc)d (A Æ B ) Æ C ` A Æ (B Æ C ) (4) R2 abcd ) R2 b(ac)d (A Æ B ) Æ C ` B Æ (A Æ C ) (5) Rabc ) R2 abbc A Æ B ` (A Æ B ) Æ B (6) Ra0a AÆt`A (7) Rabc ) Rbac AÆB `BÆA (8) 0a B Æ A ` B (or A ` t) (9) Rabc ) b  c A Æ B ` B. General recipes for translating between structural rules and conditions on accessibility relations are to be found in Restall [1998; 2000]. If one wants to add to B+ any of the axioms on the right to get a sentential logic X, one merely adds the corresponding conditions to those for a B+ model to get the appropriate notion of an X-model, with a resultant sound and complete semantics. Some logics of particular interest arising in this way are (nomenclature as in [Anderson and Belnap, 1975]) (note well that T has nothing to do with Feys' t of modal logic fame):

TW+ : T+ : E+ : R+ : H+ : S4+ :

B+ + (3; 4) TW+ + (5) T+ + (6) E+ + (7) R+ + (8) E+ + (8):

These are far from the most elegant formulations from a postulational point of view, being highly redundant (in particular the Pre xing and Suf xing rules of B+ are supplanted already in TW+ by the corresponding axioms. further the rule of Necessitation (A ` (A ! A) ! A) is also redundant already in TW+ (this is not so obvious|proof is by browsing through [Anderson and Belnap, 1975]). What minimal conditions should be imposed on the -operator when it is added to a B+ -frame so as to give a B-frame? Routley et. al. [1982] choose B4. a = a, and B5. a  b ) b  a . The minimality of B5 can be defended in terms of its being needed for showing that negations satisfy the Hereditary Condition. B4 would seem to have little place in a minimal system except for the fact that the dominant trend in relevance logic has been to keep classical double negation.34

34 In fact, B5 is too strong for a purely minimal logic of negation. See Section 5.1 for more discussion on this.

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One can get semantics for the full systems TW, T, etc. simply by adding the appropriate postulates to the conditions on a B-model. We could go on, but will instead refer the reader to Routley et al. [1982], Fine [1974] and Gabbay [1976] for a variety of variations producing systems in the neighbourhood of R. Some nd the conditions on the \base point" 0 on frames rather puzzling or unintuitive. Why should the basic conditions on frames include conditions such as the fact that a  b de ned as R0ab generate a partial order? Some recent work by Priest and Sylvan and extended by Restall has shown that these conditions can be done away with and the frames given an interpretation rather reminiscent of that of non-normal modal logics [Priest and Sylvan, 1992; Restall, 1993]. The idea is as follows. We have two sorts of set-ups in a frame | normal ones and non-normal ones. Then we split the treatment of implication along this division. Normal points are given an S5-like interpretation.

 x  A ! B i for every y if y  A then y  B and non-normal points are given the condition which appeals to the ternary relation R

 x  A ! B i for every y and z where Rxyz if y  A then z  B The other connectives are treated in just the same way as in the original relational semantics. To prove soundness and completeness for this semantics, it is simplest to go through the original semantics | for it is not too diÆcult to show that this account is merely a notational variant, where we have set Rxyz i y = z when x is a normal set-up. This satis es all of the conditions in the original semantics, for we have set a  b to be simply a = b. We turn now to one such system RM deserving of special treatment.

3.10 Algebraic and Set-theoretical Semantics for RM RM has been described by Meyer as `the laboratory of relevance logic'. It plays a role somewhat like S5 among modal logics, being a place where conjectures can be tested relatively easily (e.g. the admissibility of was rst shown for RM). This then could be a very long section because RM is by far the best understood of the Anderson{Belnap style systems. We shall try to keep it short by being dogmatic. The interested reader can verify the results claimed by consulting Meyer's Section 29.3 and Section 29.4 of [Anderson and Belnap, 1975] (see also [Dunn, 1970; Tokarz, 1980]). In the rst place the appropriate algebras for RM are the idempotent de Morgan monoids (strengthening a  a Æ a to a = a Æ a). The subdirectly irreducible ones are all chains with de Morgan complement where aÆb = a^b

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if a  :b, and a Æ b = a _ b otherwise. The designated elements are all elements a such that :a  a, and of course these must have a greatest lower bound to serve as the identity e. (This is just another description with Æ as primitive instead of ! of the `Sugihara matrices' described in the publications cited above.) Meyer showed that if `RM A, then A is valid in all the nite Sugihara matrices, establishing the nite model property for RM. Dunn showed that every extension of RM closed under substitution and the rules of R has some nite Sugihara matrix as a characteristic matrix (RM is `pretabular'). A similar result was shown by Scroggs to hold for the modal logic S5, and researchers (particularly Maksimova) have obtained results characterising all such pretabular extensions of S4 and of the intuitionistic logic. Curiously enough there are only nitely many, and it is an interesting open problem to nd some similar results for R. RM corresponds to the super-system of the intuitionistic propositional calculus LC (indeed LC can be translated into RM; see [Dunn and Meyer, 1971]). Much study has been done of the `superintuitionistic' calculi (with an emphasis on the decision problem), and it would be good to see some of the ideas of this carried over to the `super-relevant' calculi. A small start was begun in [Dunn, 1979a]. Routley and Meyer [1973] add the postulate 0  a or 0  a to the requirement on an R-frame to get an RM-frame. Dunn [1979a] instead adds the requirement

Rac ) a  c or b  c; which neatly generalised to give a family of postulates yielding set-theoretical semantics for a denumerable family of weakenings of RM which are algebraised by adding various weakenings of idempotence (an+1 = an ). It is an open problem whether R itself is the intersection of this family and whether they all have the nite model property (if so, R is decidable). Since R is undecidable, one of these must be false. However, it is unknown at the time of writing which one fails. Proof Sketch

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1. :X ! (:X ! :X ) 2. X ! (:X ! X )

Mingle Axiom, Subst. 1, Permutation and Contraposition 2, Subst.

3. (A _ :A) ^ (B _ :B ) ! :((A _ :A) ^ (B _ :B )) ! ((A _ :A) ^ (B _ :B )) 4. :(A _ :A) _ :(B _ :B ) ! 3, MP, de M (A _ :A) ^ (B _ :B ) 5. A ^ :A ! B _ :B 4, _I; ^E , de M. Kalman [1958] especially investigated de Morgan lattices with the property a ^ :a  b _ :b. We will call these Kalman lattices. he showed that every Kalman lattice is isomorphic to a subdirect product of the de Morgan lattice 3. This implies a three- valued Homomorphism Separation Property for Kalman lattices (which also can be proven by modifying the proof of its four-valued analogue, noting that each `side' of 4 is just a copy of 3). The representation in terms of polarities uses polarities X = (X1 ; X2 ) where X1 [ X2 = U , i.e. X1 and X2 are exhaustive. This means informally that X always receives at least one of the values true and false. This leads to a semantics using ambivaluations into the left-hand side of 4: s

T = ftg

s

B = ft; f g

s

F = ff g.

This idea leads to a simpler Kripke-style semantics for RM using an ordinary binary accessibility relation instead of the Routley{Meyer ternary one (actually this semantics antedates the Routley{Meyer one, the results having been presented in [Dunn, 1969]|cf. [Dunn, 1976b] for a full presentation. No details will be supplied here. This semantics has been generalised to rst-order RM with a constant domain semantics [Dunn, 1976c]). The analogous question with Routley{Meyer semantics is has now been closed in the negative in the work of [Fine, 1989], which we consider in Section 3.12. Meyer [1980] has used this `binary semantics' to give a proof of an appropriate Interpolation Lemma for RM. (Unfortunately, interpolation fails for E and R [Urquhart, 1993].)

3.11 Spin Os from the Routley{Meyer Semantics The Routley{Meyer semantical techniques can be used to prove a variety of results concerning the system R and related logics which were either more

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complicated using other methods (usually algebraic or Gentzen methods), or even impossible. Thus (cf. [Routley and Meyer, 1973]), it is possible to give a variety of conservative extension results (being careful in constructing the canonical model to use only connectives and sentential constant available in the fragment being extended). Also it is possible to give a proof of the admissibility of (see [Routley and Meyer, 1973]) that is easier than the original algebraic proof (though not as easy as Meyer's latest proof using metavaluations|cf. Section 2.4). Admissibility of amounts to showing that if A is refutable in a given R-model (K; R; 0; ) then A is refutable in a normal R-model (K 0 ; R0 ; 00; 0 ) (one where 00 = 00) gotten by adding a new `zero' and rede ning R0 and 0 in a certain way from R and . Perhaps the most interesting new property to emerge this way is `Hallden completeness', i.e. if `R A _ B and A and B share no propositional variables in common, then `R A or `R B ([Routley and Meyer, 1973, Section 2.3]). Another direction that the Routley{Meyer semantics has taken quickly ends up in heresy: classical (Boolean) negation  can be added to R with horrible theorems resulting like A^  A ! B , and yet R does not collapse to classical logic. Indeed no new theorems emerge in the original vocabulary of R. The idea is to take a normal R-model (K; R; 0; ; ) and turn it in for a new R-model (K 0 ; R0 ; 00; 0 ; 0 ) , whose 00 is a new element K 0 = K [f00 g, 0 is like  but with 000 = 00 , and R0 is like R with the additional features: 1. R0 00 ab i R; a00 b i a = b, 2. R0 ab00 i a = b . Also 0 is just like  but with 00  A if 0  A. The whole point of this exercise is to provide refuting R-models for all non-R-theorems that have the property a  b (i.e. R00ab) ) a = b: These are called `classical R-models' ( rst studied in Meyer and Routley [1973a; 1973b]) and upon them one can de ne

a  A , not a  A: One could not do this on ordinary R-models without things coming apart at the seams, because in order to have the theorem p ! p valid, one would need the Hereditary Condition to hold for  p, i.e. if a  b, then if a  p then b  p, i.e. if a  p then b  p. But one has no reason to think that this is the case, since all one has is the converse coming from the fact that the Hereditary condition holds for p. The inductive proof the Hereditary condition breaks down in the presence of Boolean negation, but of course with classical R-models the Hereditary Condition becomes vacuous and there is no need for a proof.

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This leads to certain technical simplicities, e.g. it is possible to give Godel{Lemmon style axiomatisations of relevance logics like the familiar ones for modal logics, where one takes among one's axioms all classical tautologies (using )|cf. [Meyer, 1974]. But it also leads to certain philosophical perplexities. For example, what was all the fuss Anderson and Belnap made against contradictions implying everything and disjunctive syllogism? Boolean negation trivially satis es them, so what is the interest in de Morgan negation failing to satisfy them. Will the real negation please stand up? A certain schism developed in relevance logic over just how Boolean negation should be regarded. See [Belnap and Dunn, 1981; Restall, 1999] for the `con' side and [Meyer, 1978] for the `pro' side. Belnap and Dunn [1981] point out that although Meyer's axiomatisations of R with Boolean negation do not lead to any new theorems in the standard vocabulary of R, they do lead to new derivable rules, e.g. A ^ :A ` B and :A ^ (A _ B ) ` B (note well that the negation here is de Morgan negation). This can be seen quite readily if one recognises that the semantic correlate of X ` Y is that 0  X ) 0  Y in all classical R-models, and that since all such are normal, : behaves at 0 in these just like classical negation. We both think this point counts against enriching R with Boolean negation, but Meyer [1978, note 21] thinks otherwise.

3.12 Semantics for RQ The question of how to extend these techniques to handle quanti ed relevance logics was open for a long time. The rst signi cant results were by Routley, who showed that the obvious constant domain semantics were suÆcient to capture BQ, the natural rst-order extension of B [Routley, 1980b]. However, extending the result to deal with systems involving transitivity postulates in the semantics (such as Rabc ^ Rcde ) R2 abde) proved diÆcult. To verify that the frame of prime theories on some constant domain actually satis es this condition (given that the logic satis es a corresponding condition, here the pre xing axiom) requires constructing a new prime theory x such that Rabx and Rxde. And there seems to be no general way to show that such a theory can be constructed using the domain shared by the other theories. This is not a problem for logics like BQ, in which the frame conditions do not have conditions which, to be veri ed in the completeness proof, require the construction of new theories. Fine showed that this is not merely a problem with our proof techniques. Logics like RQ, EQ, TQ and even TWQ are incomplete with respect to the constant domain semantics on the frames for the propositional logics [Fine, 1989]. He has given a technical argument for this, constructing a formula in the language of RQ which is true in all constant domain models, but which is not provable. The argument is too detailed to give here. It consists of a

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simple part, which shows that the formula

83




(p ! 9xEx) ^ 8x((p ! F x) _ (Gx ! Hx))
! 8x(Ex ^ F x ! q) ^ 8x((Ex ! q) _ Gx) ! 9xHx _ (p ! q) is valid in the constant domain semantics. This is merely a tedious veri cation that there is no counterexample. The subtle part of his argument is the construction of a countermodel. Clearly the countermodel cannot be a constant domain frame. Instead, he constructs a frame with variable domains, in which each of the axioms of RQ is valid (and in which the rules preserve validity) but the oending formula fails. This is quite a tricky argument, for variable domain semantics tend not to verify RQ's analogue to the Barcan formula

8x(p ! F x) ! (p ! 8xF x) But Fine constructs his example in such a way that this formula is valid, despite the variable domains. Despite this problem, Fine has found a semantics with respect to which the logic RQ is sound and complete. This semantics rests on a dierent view of the quanti ers. For Fine's account, a statement of the form 8xA(x) is true at a set-up not only when A(c) is true for each individual c in the domain of the set-up, but instead, when A(c) is true for an arbitrary individual c. In symbols,

a  8xA(x) i (9a")(9c 2 Da" Da )(a"  A(c)). That is, for every set-up a there are expansions of the form a" where we add new elements to the domain, but these are totally arbitrary. The frames Fine de nes are rather complex, needing not only the " operator but also a corresponding # operator which cuts down the domain of a set-up, and an across operator which identi es points in setups (! (a; fc; dg) is the minimal extension of the set-up a in which the individuals c and d are identi ed. Instead of discussing the details of Fine's semantics, we refer the reader to his paper which introduced them [Fine, 1988]. Fine's work has received some attention, from Mares, who considers options for the semantics of identity [Mares, 1992]. However, it must be said that while the semantic structure pins down the behaviour of RQ and related systems exactly, it is not altogether clear whether the rich and complex structure of Fine's semantics is necessary to give a semantics for quanti ed relevance logics. Whatever one's thoughts about the theoretical adequacy of Fine's semantics, they do raise some important issues for anyone who would give a semantic structure for quanti ed relevance logics. There are a number of issues to be faced and a number of options to be weighed up. One option is to give complete primacy to the frames for the propositional logics, and

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to use the constant domain semantics on these frames. The task then is to axiomatise this extension. The task is also to give some interpretation of what the points in these semantic structures might be. For if they are theories (or prime theories) then the evaluation clauses for the quanti ers do not make a great deal of sense without further explanation. No-one thinks that a claim of the form 9xA(x) can be a member of a theory only if there is an object in the language of the theory which satis es A according to that theory. Nor are we so readily inclined to think that all theories need share the same domain of quanti cation. If, on the other hand, we take the set-ups in frames to be quite like (some class of) theories, then we must face the issue of the relationships between these theories. No doubt, if 8xA(x) is in some theory, then A(c) will be in that theory for any constant c in the language of the theory. However, the converse need not be the case. Anyway, it is clear that there is a lot of work to be done in the semantics of relevance logics with quanti ers. One area which hasn't been explored at any depth, but which looks like it could bring some light is the semantics of positive quanti ed relevance logics. Without the distribution of the universal quanti er over disjunction, these systems are subsystems of intuitionistic logic. 4 THE DECISION PROBLEM

4.1 Background When the original of this Handbook article was published back in 1985, without a doubt the outstanding open problem in relevance logics was the question as to whether there exists a decision procedure for determining whether formulas are theorems of the system E or R. Anderson [1963] listed it second among his now historic open problems (the rst was the admissibility of Ackermann's rule discussed in Section 2). Through the work of Urquhart [1984], we now know that there is no such decision procedure. Harrop [1965] lends interest to the decision problem with his remark that `all \philosophically interesting" propositional calculi for which the decision problem has been solved have been found to be decidable : : : '.35 We now have a very good counterexample to Harrop's claim. In this section we shall examine Urquhart's proof, but before we get there we shall also consider various fragments and subsystems of R for which there are decision procedures. R will be our paradigm throughout this discussion, though we will make clear how things apply to related systems. 35 He continues somewhat more technically ` : : : and none is known for which it has been proved that it does not possess the nite model property with recursive bound.'

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4.2 Zero-degree Formulas

These are formulas containing only ^; _, and :. As was explained in Section 1.7, the zero-degree theorems of R (or E) are precisely the same as those of the classical propositional calculus, so of course the usual two valued truth tables yield a decision procedure.

4.3 First-degree Entailments Two dierent (though related) `syntactical' decision procedures were described for these in Section 1.7 (the method of `tautological entailments' and the method of `coupled trees'). A `semantical' decision procedure using a certain four element matrix 4 is described in Section 3.3. The story thus told leaves out the historically (and otherwise) very important role of a certain eight element matrix M0 (cf. [Anderson and Belnap, 1975, Section 22.1.3]). This matrix is essential for the study of rst-degree formulas and higher (see Section 4.4 below), in so much as it is impossible to de ne an implication operation on 4 and pick out a proper subset of designated elements so as to satisfy the axioms of E (a fortiori R). Indeed M0 was used in [Anderson and Belnap Jr., 1962b] and [Belnap, 1960b] to isolate the rstdegree entailments of R, and the formulation of Section 1.7 presupposes this use.

4.4 First-degree Formulas These are `truth functions' of rst-degree entailments and/or formulas containing no ! at all (the `zero-degree formulas'). Belnap [1967a] gave a decision procedure using certain nite `products' of M0 . No one such product is characteristic for Dfdf, but every non-theorem of Efdf is refutable in some such products M0n (where n may in fact be computed as the largest number of rst-degree entailments occurring in a disjunction once the candidate theorem has been put in conjunctive normal form). Hence Efdf has the nite model property which suÆces of course for decidability (cf. [Harrop, 1965]). This is frankly one of the most diÆcult proofs to follow in the whole literature of relevance logics. A sketch may be found in [Anderson and Belnap, 1975, Section 19].

4.5 `Career Induction' This is what Belnap has labelled the approach, exempli ed in Sections 4.1{ 4.3 above of extending the positive solution to the decision problem `a degree at a time'. The last published word on the Belnap approach is to be found in his [1967b] where he examines entailments between conjunctions of rstdegree entailments and rst degree entailments.

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Meyer [1979c], by an amazingly general and simple proof, shows that a positive answer to the decision problem for `second-degree formulas' (no ! within the scope of an arrow within the scope of an !) is equivalent to nding a decision procedure for all of R.

4.6 Implication Fragment We now start another tack. Rather than looking at fragments of the whole system R delimited by complexity of formulas, we instead consider fragments delimited by the connectives which they contain. The earliest result of this kind is due to [Kripke, 1959b], who gave a Gentzen system for the implicational fragments of E and R, and showed them decidable. We shall here examine the implicational fragment of R (R! ) in some detail as a kind of paradigm for this style of argument.36 The appropriate Gentzen calculus37 LR! is the same as that given by Gentzen [1934] except for two trivial dierences and one profound dierence. The rst trivial dierence is the obvious one that we take only the operational rules for implication, and the second trivial dierence consequent on this (with negation it would have to be otherwise) is that we can restrict our sequents to those with a single formula in the consequent. The profound dierence is that we drop the structural rule variously called `thinning' or `weakening'. This leaves: Axioms.

A ` A: Structural Rules. Permutation

; A; B; ; ` C ; B; A; ; ` C

Contraction

; a; A ` B ; A ` B

Operational Rules.

(`!)

; A ` B `A!B

(!`)

 ` A ; B ` C : ; ; A ! B ` C

It is easy to see why thinning would be a disaster for relevant implication. 36 Actually this and various other results discussed below using Gentzen calculi presupposes `separation theorems' due to Meyer, showing, e.g. as is relevant to this case, that all of the theorems containing only ! are provable from the axioms containing only !. 37 We do not follow Anderson and Belnap [1975] in calling Gentzen systems `consecution calculi', much as their usage has to recommend it.

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Thus:

87

A`A Thinning A; B ` A (`!) A`B!A (`!) ` A ! (B ! A)

It is desirable to prove `The Elimination Theorem', which says that the following rule would be redundant (could be eliminated). (Cut)

 ` A ; A ` B : ; ` B

This is needed to show the equivalence of LR! to its usual Hilbert-style (axiomatic system `HR! ' R! one of the formulations of Section 1.3). We will not pause on details here, but the principal question regarding the equivalence is whether modus ponens (The sole rule for HR! ) is admissible in the sense that whenever ` A and ` A ! B are both derivable in LR! , so is ` B (let  and be empty). The strategy of the proof of the Elimination Theorem can essentially be that of Gentzen with one important but essentially minor modi cation. Thus, Gentzen actually proved something stronger than Cut elimination, namely, (Mix)

`A `B ; ; [ A] ` B

where [ A] is the result of deleting all occurrences of A from . This is useful in the induction, but sometimes it takes out too many occurrences of A. In Gentzen's framework these could always be thinned back in, but of course this is not available with LR! . We thus instead generalise Cut to the rule `A `B ; (Fusion) ; ( A) ` B where contains some occurrences of A and ( A) is the result of deleting as many of those occurrences as one wishes (but at least one). The main strategy of the decision procedure for LR! is to limit applications of the contraction rule so as to prevent a proof search from running on forever in the following manner: `Is p ` q derivable? Well it is if p; p ` q is derivable. Is p; p ` q derivable? Well it is if p; p; p ` q is, etc.'. We need one simple notion before strategy can be achieved. We shall say that the sequent of 0 ` A is a contraction of sequent  ` A just in

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case 0 ` A can be derived from  ` A by (repeated) applications of the rules Contraction and Permutation (with respect to this last it is helpful not even to distinguish two sequents that are mere permutations of one another). The idea that we now want to put in eect is to drop the rule Contraction, replacing it by building into the operational rules a limited amount of contraction (in the generalised sense just explained). More precisely, the idea is to allow a contraction of the conclusion of an operational rule only in so far as the same result could not be obtained by rst contracting the premises. A little thought shows that this means no change for the rule (`!), and that the following will suÆce for

 ` A ; B ` C (!`0 ) [; ; A ! B ] ` C

where [; ; A ! B ] is any contraction of ; ; A ! B such that :

1. A ! B occurs only 0, 1, or 2 times fewer than in ; ; A ! B ; 2. Any formula other than A ! B occurs only 0 or 1 time fewer.

It is clear that after modifying LR! by building some limited contraction into (!`) in the manner just discussed, the following is provable by an induction on length of derivations: 38 If a sequent 0 is a contraction of a sequent and Curry's Lemma. has a derivation of length n, then 0 has a derivation of length  n. Clearly this lemma shows that the modi cation of LR! leaves the same sequents derivable (since the lemma says the eect of contraction is retained). So henceforth we shall by LR! always mean the modi ed version. Besides the use just adverted to, Curry's Lemma clearly shows that every derivable sequent has an irredundant derivation in the following sense: one containing no branch with a sequent 0 below a sequent of which it is a contraction. We are nally ready to begin explicit talk about the decision procedure. Given a sequent , one begins the test for derivability as follows (building

38 This is named (following [Anderson and Belnap, 1975]) after an analogous lemma in [Curry, 1950] in relation to classical (and intuitionistic) Gentzen systems. There, with free thinning available, Curry proves his lemma with (!`) (in its singular version) stated as: ; A ! B ` A ; A ! B; B ` C : ;A ! B ` C This in eect requires the maximum contraction permitted in our statement of (!`) above, but this is ok since items contracted `too much' can always be thinned back in. Incidentally, our statement of (!`) also diers somewhat from the statement of Anderson and Belnap [1975] or Belnap and Wallace [1961], in that we build in just the minimal amount of contraction needed to do the job.

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a `complete proof search tree'): one places above all possible premises or pairs of premises from which follows by one of the rules. Note well that even with the little bit of contraction built into (!`) this will still be only a nite number of sequents. Incidentally, one draws lines from those premises to . One continues in this way getting a tree. It is reasonably clear that if a derivation exists at all, then it will be formed as a subtree of this `complete proof search there', by the paragraph just above, the is complete proof search tree can be constructed to be irredundant. But the problem is that the complete proof search tree may be in nite, which would tend to louse up the decision procedure. There is a well-known lemma which begins to come to the rescue:  nig's Lemma. A tree is nite i both (1) there are only nitely many Ko points connected directly by lines to a given point (` nite fork property') and (2) each branch is nite (` nite branch property'). By the `note well' in the paragraph above, we have (1). The question remaining then is (2), and this is where an extremely ingenious lemma of Kripke's plays a role. To state it we rst need a notion from Kleene. Two sequents  ` A and 0 ` A are cognate just when exactly the same formulas (not counting multiplicity) occur in  as in 0 . Thus, e.g. all of the following are cognate to each other:

`A X; X; Y ` A X; Y; Y ` A X; X; Y; Y ` A X; X; X; Y; Y ` A.

(1) X; Y (2) (3) (4) (5)

We call the class of all sequents cognate to a given sequent a cognation class. Kripke's Lemma. Suppose a sequence of cognate sequents 0 ; 1 ; : : : ; is irredundant in the sense that for no i ; j with i < j , is i a contraction of j . Then the sequence is nite. We postpone elaboration of Kripke's Lemma until we see what use it is to the decision procedure. First we remark an obvious property of LR! that is typical of Gentzen systems (that lack Cut as a primitive rule): Subformula Property. If is a derivable sequent of LR! , then any formula occurring in any sequent in the derivation is a subformula of some formula occurring in . This means that the number of cognation classes occurring in any derivation (and hence in each branch) is nite. But Kripke's Lemma further shows

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that only a nite number of members of each cognation class occur in a branch (this is because we have constructed the complete proof search tree to be irredundant). So every branch is nite, and so both conditions of Konig's lemma hold. Hence the complete proof search tree is nite and so there is a decision procedure.

7 6 5 4 3 2 0

1

2

3

4

5

6

7

Figure 1. Sequents in the Plane Returning now to Kripke's Lemma, we shall not present a proof (for which see [Belnap Jr. and Wallace, 1961] or [Anderson and Belnap, 1975]). Instead we describe how it can be geometrically visualised. For simplicity we consider sequents cognate to X; Y ` A ((1), (2), (3), etc. above). Each such sequent can be represented as a point in the upper right-hand quadrant of the co-ordinate plane (where origin is labelled with 1 rather than 0 since (1) is the minimal sequent in the cognation class). See Figure 1. Thus, e.g. (5) gets represented as `3 X units' and `2 Y units'. Now given any sequent, say ( 0 ) X; X; X; Y; Y

`A

as a starting point one might try to build an irredundant sequence by rst building up the number of Y 's tremendously (for purposes of keeping on the page we let this be to six rather than say a million). But in so doing one

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has to reduce the number of X 's (say, to be strategic, by one). The graph now looks like 2 for the rst two members of the sequence 0 ; 1 .

7 6 5 1

4 3 2 0

1

2

3

4

5

6

7

Figure 2. Descending Regions The purpose of the intersecting lines at each point is to mark o areas (shaded in the diagram) into which no further points of the sequence may be placed. Thus if 2 were placed as indicated at the point (6, 5), it would reduce to 0 . What this means is that each new point must march either one unit closer to the X axis or one unit closer to the Y axis. Clearly after a nite number of points one or the other of the two axes must be `bumped', and then after a short while the other must be bumped as well. When this happens there is no space left to play without the sequence becoming redundant. The generalisation to the case of n formulas in the antecedent to Euclidean n-space is clear (this is with n nite|with n in nite no axis need ever be bumped). Incidentally, Kripke's Lemma (as Meyer discovered) is equivalent to a theorem of Dickson about prime numbers: Let M be a set of natural numbers all of which are composed out of the rst m primes. Then every n 2 M is of the form P1n1
P2n2
: : : Pknk , and hence (by unique decomposition) can be regarded as a sequence of the Pi 's in which each Pi is repeated ni times. Divisibility corresponds then to contraction (at least neglecting the case

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ni = 0). Dickson's theorem says that if no member of M has a proper divisor in M , then M is nite. Before going on to consider how the addition of connectives changes the complexity, let us call the reader's attention to a major open problem: It is still unknown whether the implication fragment of T is decidable.

4.7 Implication{Negation Fragment

The idea of LR:! is to accommodate the classical negation principles presenting R in the same way that Gentzen [1934] accommodated them for classical logic: provide multiple right-hand sides for the sequents. this means that a sequent is of the form  ` , where  and are (possible empty) nite sequences of formulas. One adds structural rules for Permutation and Contraction on the right-hand side, reformulates (`!) and (!`) as follows ; A ` B;  ` A; ; B ` Æ (`!) (!`) ;  ` A ! B; ; ; A ! B ` ; Æ and adds ` ip and op' rules for negation: ; A `  ` A; (` :) : (: `) ; :A `  ` :A; LE:! is the same except that in the rule (`!) must be empty and  must consist only of formulas whose main connective is !. The decision procedure for LE:! was worked out by Belnap and Wallace [1961] along basically the lines of the argument of Kripke just reported in the last section, and is clearly reported in [Anderson and Belnap, 1975, Section 13]. the modi cation to LR:! is straightforward (indeed LR:! is easier because one need not prove the theorem of p. 128 of [Anderson and Belnap, 1975], and so one can avoid all the apparatus there of `squeezes'). McRobbie and Belnap [1979] have provided a nice reformulation of LR:! in an analytic tableau style, and Meyer has extended this to give analytic tableau for linear logic and other systems in the vicinity of R [Meyer et al., 1995].

4.8 Implication{Conjunction Fragment, and R Without Distribution This work is to be found in [Meyer, 1966]. The idea is to add to LR! the Gentzen rules: ; A ` C ; ` C `A `B (` ^) (^ `) : ; A ^ B ` C ; A ^ B ` C `A^B Again the argument for decidability is a simple modi cation of Kripke's.

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Note that it is important that the rule (^ `) is stated in two parts, and not as one `Ketonen form' rule: ; A; B ` C (K ^ `) : ; A ^ B ` C The reason is that without thinning it is impossible to derive the rule(s) (^ `) from (K ^ `). Early on it was recognised that the distribution axiom

A ^ (B _ C ) ! (A _ B ) _ C was diÆcult to derive from Gentzen-style rules for E and R. Thus Anderson [1963] saw this as the sticking point for developing Gentzen formulations, and Belnap [1960b, page 72]) says with respect to LE! that `the standard rules for conjunction and disjunction could be added : : : the Elimination Theorem (suitably modi ed) remaining provable. However, [since distribution would not be derivable], the game does not seem worth the candle'. Meyer [1966] carried out such an addition to LR:! , getting a system he called LR , whose Hilbert-style version is precisely R without the distribution axiom. He showed using a Kripke-style argument that this system is decidable. This system is now called LR, for \lattice R". Meyer [1966] also showed how LR can be translated into R!;^ rather simply. Given a formula A in the language of LR+ , let V be the set of variables in A, and let two atomic propositions pt and pf not in V . Set :A for the moment to be A ! pf , to de ne a translation A0 of A as follows. p0 = p t0 = pt (A ! B )0 = A0 ! B 0 (A ^ B )0 = A0 ^ B 0 (A _ B )0 = :(:A0 ^ :B 0 ) (A Æ B )0 = :(A0 ! :B 0 ) V

then setting t(A) = fpt ! (p ! p) : p 2 V [ fpt; pf gg and f (A) = V f::p ! p : p 2 V [ fpt; pf gg, we get the following theorem: + then A is Translation Theorem (Meyer). If A is a formula in LR + 0 provable in LR if and only if (t(A) ^ f (A) ^ pt ) ! A is provable in R!;^ . The proof is given in detail in [Urquhart, 1997], and we will not present it here. Some recent work of Alasdair Urquhart has shown that although R!;^ is decidable, it is only just decidable [Urquhart, 1990; Urquhart, 1997].

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More formally, Urquhart has shown that given any particular formula in the language of R!;^ , there is no primitive recursive bound on either the time or the space taken by a computation of whether or not that formula is a theorem. Presenting the proof here would take us too far away from the logic to be worthwhile, however we can give the reader the kernel of the idea behind Urquhart's result. Urquhart follows work of [Lincoln et al., 1992] by using a propositional logic to encode the behaviour of a branching counter machines. A counter machine has a nite number of registers (say, ri for suitable i) which each hold one non-negative integer, and some nite set of possible states (say, qj for suitable j ). Machines are coded with a list of instructions, which enable you to increment or decrement registers, and test for registers' being zero. A branching counter machine dispenses with the test instructions and allows instead for machines to take multiple execution paths, by way of forking instructions. The instruction qi + rj qk means \when in qi , add 1 to register rj and enter stage qk ," and qi rj qk means \when in qi , subtract 1 to register rj (if it is non-empty) and enter stage qk ," and qi fqj qk is \when in qi , fork into two paths, one taking state qj and the other taking qk ." A machine con guration is a state, together with the values of each register. Urquhart uses the logic LR to simulate the behaviour of a machine. For each register ri , choose a distinct variable Ri , for each state qj choose a distinct variable Qj . The con guration hqi ; n1 ; : : : ; nl i, where ni is the value of ri is the formula

Qi Æ R1n1 Æ


Æ Rlnl and the instructions are modelled by sequents in the Gentzen system, as follows: Instruction Sequent qi + rj qk Qi ` Qk Æ Rj qi rj qk Qi ; Rj ` Qk qi fqj qk Qi ` Qj _ Qk Given a machine program (a set of instructions) we can consider what is provable from the sequents which code up those instructions. This set of sequents we can call the theory of the machine. Qi Æ R1n1 Æ


Æ Rlnl ` Qj Æ R1m1 Æ


Æ Rlml is intended to mean that from state con guration hqi ; n1 ; : : : ; nl i all paths will go through con guration hqj ; m1 ; : : : ; ml i after some number of steps. A branching counter machine accepts an initial con guration if when run on that con guration, all branches terminate at the nal state qf , with all registers taking the value zero. The corresponding condition in LR will be the provability of

Qi Æ R1n1 Æ


Æ Rlnl ` Qm

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This will nearly do to simulate branching counter machines, except for the fact that in LR we have A ` A Æ A. This means that each of our registers can be incremented as much as you like, provided that they are non-zero to start with. This means that each of our machines need to be equipped with every instruction of the form qi >0 + rj qi , meaning \if in state qi , add 1 to rj , provided that it is already nonzero, and remain in state qi ." Given these de nitions, Urquhart is able to prove that a con guration is accepted in branching counter machine, if and only if the corresponding sequent is provable from the theory of that machine. But this is equivalent to a formula ^

Theory(M ) ^ t ! (Q1 ! Qm)

in the language of LR. It is then a short step to our complexity result, given the fact that there is no primitive recursive bound on determining acceptability for these machines. Once this is done, the translation of LR into R!^ gives us our complexity result. It is still unknown if R! has similar complexity or whether it is a more tractable system. Despite this complexity result, Kripke's algorithm can be implemented with quite some success. The theorem prover Kripke, written by McRobbie, Thistlewaite and Meyer, implements Kripke's decision procedure, together with some quite intelligent proof-search pruning, by means of nite models. If a branch is satis able in RM3, for example, there is no need to extend it to give a contradiction. This implementation works in many cases [Thistlewaite et al., 1988]. Clearly, work must be done to see whether the horri c complexity of this problem in general can be transferred to results about average case complexity. Finally, before moving to add distribution, we should mention that Linear Logic (see Section 5.5) also lacks distribution, and the techniques used in the theorem prover Kripke have application in that eld also.

4.9 Positive R In this section we will examine extensions of the Gentzen technique to cover all of positive relevance logic. We know (see Section 4.12) that this will not provide decidability. However, they provide another angle on R and cousins. Dunn and Minc independently developed a Gentzen-style calculus (with some novel features) for R without negation (LR+ ).39 Belnap 39 Dunn's result was presented by title at a meeting of the Association for Symbolic Logic, December, 1969 (see [Dunn, 1974]), and the full account is to be found in [Anderson and Belnap, 1975, Section 28.5]). Minc [1972, earliest presentation said to there to be February 24] obtained essentially the same results (but for the system with a necessity operator). See also [Belnap Jr. et al., 1980].

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[1960b] had already suggested the idea of a Gentzen system in which antecedents were sequences of sequences of formulas, rather than just the usual sequences of formulas (in this section `sequence' always means nite sequence). The problem was that the Elimination Theorem was not provable. LR+ goes a step `or two' further, allowing an antecedent of a sequent instead to be a sequence of sequence of : : : sequences of formulas. More formally, we somehow distinguish two kinds of sequences, `intensional sequences' and `extensional sequences' (say pre x them with an `I ' or an `E '). an antecedent can then be an intensional sequence of formulas, an extensional sequence of the last mentioned, etc. or the same thing but with `intensional' and `extensional' interchanged. (We do not allow things to `pile up', with, e.g. intensional sequences of intensional sequences|there must be alternation).40 Extensional sequences are to be interpreted using ordinary `extensional' conjunction ^, whereas intensional sequences are to be interpreted using `intensional conjunction' Æ, which may be de ned in the full system R as A Æ B = :(A ! :B ), but here it is taken as primitive|see below). We state the rules, using commas for extensional sequences, semicolons for intentional sequences, and asterisks ambiguously for either; we also employ an obvious substitution notation.41 Permutation

Thinning

[  ] ` A [  ] ` A [ ] ` A ; [ ; ] ` A

; A ` B (`!) `A!B

Contraction

provided



[  ] ` A [ ] ` A :

is non-empty

 ` A [B ] ` C (!`) [; A ! B ] ` C

[A; B ] ` C `A `B (^ `) (` ^) [A ^ B ] ` C  `A^B

40 This diers from the presentation of [Anderson and Belnap, 1975] which allows such `pile ups', and then adds additional structural rules to eliminate them. Belnap felt this was a clearer, more explicit way of handling things and he is undoubtedly right, but Dunn has not been able to read his own section since he rewrote it, and so return to the simpler, more sloppy form here. 41 With the understanding that substitutions do not produce `pile ups'. Thus, e.g. a `substitution' of an intensional sequence for an item in an intensional sequence does not produce an intensional sequence with an element that is an intensional sequence formed by juxtaposition. Again this diers from the presentation of [Anderson and Belnap, 1975, cf. note 28].

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`A `B (` _) (` _) `A_B  `A_B `A `B (` Æ) ; ` A Æ B

97

[a] ` C [B ] ` C (_ `) [A _ B ] ` C [a; B ] ` C (Æ `)  Æ B] ` C

For technical reasons (see below) we add the sentential constant t with the axiom ` t and the rule: [B ] ` A (t `) [ ; t] ` A The point of the two kind of sequences can now be made clear. Let us examine the classically (and intuitionistically) valid derivation: (1) A ` A Axiom (2) A; B ` A Thinning (3) A ` B ! A (`!): It is indierent whether (2) is interpreted as (2^) (A ^ B ) ! A; or (2!) A ! (B ! A); because of the principles of exportation and importation. In LR+ however we may regard (2) as ambiguous between (2; ) A; B ` A (extensional), and (2; ) A; B ` A (intensional). (2,) continues to be interpreted as (2^), but (2;) is interpreted as (2Æ) (A Æ B ) ! A:

Now in R, exportation holds for Æ but not for ^ (importation holds for both). Thus the move from (2;) to (3) is valid, but not from (2,) to (3). On the other hand, in R, the inference from A ! C to (A ^ B ) ! C is valid, whereas the inference to (A Æ B ) ! C is not. Thus the move from (1) to (2,) is valid, but not the move from (1) to (2;). the whole point of LR+ is to allow some thinning, but only in extensional sequences. This allows the usual classical derivation of the distribution axiom to go through, since clearly

A ^ (B _ C ) ` (A ^ B ) _ C can be derived with no need of any but the usual extensional sequence. The following sketch of a derivation of distribution in the consequent is even

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more illustrative of the powers of LR+ (permutations are left implicit; also the top half is left to the reader);

A; B ` (A ^ B ) _ C A; C ` (A ^ B ) _ C (_ `) X`X A; B _ C ` (A ^ B ) _ C (!`) X ` X A; (X ; X ! B _ C ) ` (A ^ B ) _ C (!`) (X ; X ! A); (X ; X ! A; X ! B _ C ) ` (A ^ B ) _ C X ! A; X ! B _ C ; X ` (A ^ B ) _ C (X ! A) ^ (X ! B _ C ); X ` (A ^ B ) _ C ` (X ! A) ^ (X ! B _ C ) ! [X ! (A ^ B ) _ C ] + LR is equivalent to R+ in the sense that for any negation-free sentence A of R; ` A is derivable in LR+ i A is a theorem of R. The proofs of both halves of the equivalence are complicated by technical details. Right-to-left (the interpretation theorem) requires the addition of intensional conjunction as primitive, and then a lemma, due to R. K. Meyer, to the eect that this is harmless (a conservative extension). Left-to-right (the Elimination Theorem) is what requires the addition of the constant true sentence t. This is because the `Cut' rule is stated as:  ` A (A) ` B ; () ` B where () is the result of replacing arbitrarily many occurrences of A in (A) by  if  is non-empty, and otherwise by t.42 Without this emendation of the Cut rule one could derive B ` A whenever ` A is derivable (for arbitrary B , relevant or not) as follows A`A Thinning ` A A; B ` A Cut B`A Discussing decidability a bit, one problem seems to be that Kripke's Lemma (appropriately modi ed) is just plain false. The following is a sequence of cognate sequents in just the two propositional variables X and Y which is irredundant in the sense that structural rules will not get you from a later member to an earlier member: X ; Y ` X (X ; Y ); X ` X ((X ; Y ); X ); Y ` X : : : 43 42 Considerations about the eliminability of occurrences of t are then needed to show the admissibility of modus ponens. This was at least the plan of [Dunn, 1974]. A dierent plan is to be found in [Anderson and Belnap, 1975, Section 28.5], where things are arranged so that sequents are never allowed to have empty left-hand sides (they have t there instead).

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4.10 Systems Without Contraction Gentzen systems without the contraction rule tend to be more amenable to decision procedures than those with it. Clearly, all of the work in Kripke's Lemma is in keeping contraction under control. So it comes as no surprise that if we consider systems without contraction for intensional structure, decision procedures are forthcoming. If we remove the contraction rule from LR we get the system which has been known as LRW (R without W without distribution), and which is equivalent to the additive and multiplicative fragment of Girard's linear logic [Girard, 1987]. It is well known that this system is decidable. In the Gentzen system, de ne the complexity of a sequent to be the number of connectives and commas which appear in it. It is trivial to show that complexity never increases in a proof and that as a result, from any given sequent there are only a nite number of sequents which could appear in a proof of the original sequent (if there is one). This gives rise to a simple decision procedure for the logic. (Once the work has already been done in showing that Cut is eliminable.) If we add the extensional structure which appears in the proof theories of traditional relevance logics then the situation becomes more diÆcult. However, work by Giambrone has shown that the Gentzen systems for positive relevance logics without contraction do in fact yield decision procedures [Giambrone, 1985]. In these systems we do have extensional contraction, so such a simple minded measure of complexity as we had before will not yield a result. In the rest of this section we will sketch Giambrone's ideas, and consider some more recent extensions of them to include negation. For details, the reader should consult his paper. The results are also in the second volume of Entailment [Anderson et al., 1992]. Two sequents are equivalent just when you can get from one to the other by means of the invertible structural rules (intensional commutativity, extensional commutativity, and so on). A sequent is super-reduced if no equivalent sequent can be the premise of a rule of extensional contraction. A sequent is reduced if for any equivalent sequents which are the premise of a rule of extensional contraction, the conclusion of that rule is super-reduced. So, intuitively, a super-reduced sequent has no duplication in it, and a reduced sequent can have one part of it `duplicated', but no more. Clearly any sequent is equivalent to a super-reduced sequent. The crucial lemma is that any super-reduced sequent has a proof in which every sequent appearing is 43 Further, this is not just caused by a paucity of structural rules. Interpreting the sequents of formulas of R+ (^ for comma, Æ for semicolon, ! for `) no later formula provably implies an earlier formula. Incidentally, one does need at least two variables (cf. R. K. Meyer [1970b]).

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reduced. This is clear, for given any proof you can transform it into one in which every sequent is reduced without too much fuss. As a result, we have gained as much control over extensional contraction as we need. Giambrone is able to show that only nitely many reduced sequent can appear in the proof of a given sequent, and as a result, the size of the proof-search tree is bounded, and we have decidability. This technique does not work for intensional contraction, as we do not have the result that every sequent is equivalent to an intensionally super-reduced sequent, in the absence of the mingle rule. While A ^ A ` B is equivalent to A ` B , we do not have the equivalence of A Æ A ` B and A ` B , without mingle. These methods can be extended to deal with negation. Brady [1991] constructs out of signed formulae T A and F A instead of formulae alone, and this is enough to include negation without spoiling the decidability property. Restall [1998] uses the techniques of Belnap's Display Logic (see Section 5.2) to provide an alternate way of modelling negation in sequent systems. These techniques show that the decidability of systems without intensional contraction are decidable, to a large extent independently of the other properties of the intensional structure.

4.11 Various Methods Used to Attack the Decision Problem Decision procedures can basically be subdivided into two types: syntactic (proof-theoretic) and semantic (model-theoretical). A paradigm of the rst type would be the use of Gentzen systems, and a paradigm of the second would be the development of the nite model property. It seems fair to say, looking over the previous sections, that syntactic methods have dominated the scene when nested implications have been afoot, and that semantical methods have dominated when the issue has been rst-degree implications and rst-degree formulas.44 There are two well-known model-theoretic decision procedures used for such non-classical logics as the intuitionistic and modal logics. One is due to McKinsey and Tarski and is appropriate to algebraic models (matrices) (cf. [Lemmon, 1966, p. 56 .]), and the other (often called ` ltration') is due to Lemmon and Scott and is appropriate to Kripke-style models (cf. [Lemmon, 1966, p. 208 ]). Actually these two methods are closely connected (equivalent?) in the familiar situation where algebraic model sand Kripke models are duals. The problem is that neither seems to work with E and R. The diÆculty is most clearly stated with R as paradigm. For the algebraic models the problem is given a de Morgan monoid (M; ^; _; ; Æ) 44 As something like `the exception that proves the rule' it should be noted that Belnap's [1967a] work on rst-degree formulas and slightly more complex formulas has actually been a subtle blend of model-theoretic (algebraic) and proof-theoretic methods.

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0 and a nite de Morgan sublattice (D; ^0 ; _; ), how to de ne a new multiplicative operation Æ0 on D so as to make it a de Morgan monoid and so for x; y 2 D, if x Æ y 2 D then x Æ y = x Æ0 y. the chief diÆcult is in satisfying the associative law. For the Kripke-style models (say the Routley{Meyer variety) the problem is more diÆcult to state (especially if the reader has skipped Section 3.7) but the basic diÆculty is in the satisfying of certain requirements on the three-placed accessibility relation once set-ups have been identi ed into a nite number of equivalence classes by ` ltration'. Thus, e.g. the requirement corresponding to the algebraic requirement of associativity is Rayx & Rbcy ) 9y(Raby & Rycx)45 the problem in a nutshell is that after ltration one does not know that there exists the appropriate equivalence class y needed to feed such an existentially hungry postulate. The McKinsey-Tarski method has been used successfully by Maksimova [1967] with respect to a subsystem of R, which diers essentially only in that it replaces the nested form of the transitivity axiom

(A ! B ) ! [(B ! C ) ! (A ! C )] by the `conjoined' form (A ! B ) ^ (B ! C ) ! (A ! C ):46 Perhaps the most striking positive solution to the decision problem for a relevance logic is that provided for RM by Meyer (see [Anderson and Belnap, 1975, Section 29.3], although the result was rst obtained by Meyer [1968].47 Meyer showed that a formula containing n propositional variables is a theorem of RM i it is valid in the `Sugihara matrix' de ned on the nonzero integers from n to +n. this result was extended by [Dunn, 1970] to show that every `normal' extension of RM has some nite Sugihara matrix (with possibly 0 as an element) as a characteristic matrix. So clearly RM and its extensions have at least the nite model property. Cf. Section 3.10 for further information about RM. Meyer [private communication] has thought that the fact that the decidability of R is equivalent to the solvability of the word problem for de Morgan monoids suggests that R might be shown to be undecidable by some suitable modi cation of the proof that the word problem for monoids is unsolvable. It turns out that this is technique is the one which pays o | although the proof is very complex. The complexity arises because there is an important disanalogy between monoids and de Morgan monoids in 45 This is suggestively written (following Meyer) as Ra(bc)x ) R(ab)cx. 47 In fact neither McKinsey-Tarski methods nor ltration was used in this proof. We are

no clearer now that they could not be used, and we think the place to start would be to try to apply ltration to the Kripke-style semantics for RM of [Dunn, 1976b], which uses a binary accessibility relation and seems to avoid the problems caused by `existentially hungry axioms' for the ternary accessibility relation.

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that in the latter the multiplicative operation is necessarily commutative (and the word problem for commutative monoids is solvable).48 Still it has occurred to both Meyer and Dunn that it might be possible to de ne a new multiplication operation  for both Æ and ^ in such a way as to embed the free monoid into the free de Morgan monoid. This suspicion has turned out to be right, as we shall see in the next section.

4.12

,

R E

and Related Systems

As is quite well known by now, the principal systems of relevance logic, R, E and others, are undecidable. Alasdair Urquhart proved this in his ground breaking papers [Urquhart, 1983; Urquhart, 1984]. We have recounted earlier attempts to come to a conclusion on the decidability question. The insights that helped decide the issue came from an unexpected quarter | projective geometry. To see why projective geometry gave the necessary insights, we will rst consider a simple case, the undecidability of the system KR. KR is given by adding A ^ :A ! B to R. A KR frame is one satisfying the following conditions (given by adding the clause that a = a to the conditions for an R frame).

R0ab i a = b Rabc i Rbac i Racb (total permutation) Raaa for each a R2 abcd only if R2 acbd The clauses for the connectives are standard, with the proviso that a  :A i a 6 A, since a = a . Urquhart's rst important insight was that KR frames are quite like projective spaces. A projective space P is a set P of points, and a collection L of subsets of P called lines, such that any two distinct points are on exactly one line, and any two distinct lines intersect in exactly one point. But we can de ne projective spaces instead through the ternary relation of collinearity. Given a projective space P , its collinearity relation C is a ternary relation satisfying the condition:

Cabc i a = b = c, or a, b and c are distinct and they lie on a common line. If P is a projective space, then its collinearity relation C satis es the following conditions,

Caaa for each a. Cabc i Cbac i Cacb. C 2 abcd only if C 2 acbd. 48 In this connection two things should be mentioned. First, Meyer [unpublished typescript, 1973] has shown that not all nitely generated de Morgan monoids are nitely presentable. Second, Meyer and Routley [1973c] have constructed a positive relevance logic Q+ (the algebraic semantics for which dispenses with commutativity) and shown it undecidable.

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provided that every line has at least four points (this last requirement is necessary to verify the last condition). Conversely, if we have a set with a ternary relation C satisfying these conditions, then the space de ned with the original set as points and the sets lab = fc : Cabcg [ fa; bg where a 6= b as lines is a projective space. Now the similarity with KR frames becomes obvious. If P is a projective space, the frame F (P ) generated by P is given by adjoining a new point 0, adding the conditions C 0aa, Ca0a, and Caa0, and by taking the extended relation C to be the accessibility relation of the frame. Projective spaces have a naturally associated undecidable problem. The problem arises when considering the linear subspaces of projective spaces. A subspace of a projective space is a subset which is also a projective space under its inherited collinearity relation. Given any two linear subspaces X and Y , the subspace X + Y is the set of all points on lines through points in X and points in Y . In KR frames there are propositions which play the role of linear subspaces in projective spaces. We need a convention to deal with the extra point 0, and we simply decree that 0 should be in every \subspace." Then linear subspaces are equivalent to the positive idempotents in a frame. That is, they are the propositions X which are positive (so 0 2 X ) and idempotent (so X = X Æ X ). Clearly for any sentence A and any KR model M, the extension of A, jjAjj in M is a positive idempotent i 0  A ^ (A $ A Æ A). It is then not too diÆcult to show that if A and B are positive idempotents, so are A Æ B and A ^ B , and that t and > are positive idempotents. Given a projective space P , the lattice algebra hL; \; +i of all linear subspaces of the projective space, under intersection and + is a modular geometric lattice. That is, it is a complete lattice, satisfying these conditions:


Modularity a  c ) (8b) a \ (b + c)  (a \ b) + c

Geometricity Every lattice element is a join of atoms, and if a is an atom and X is a set where a  X then there's some nite Y  X , where a  Y . The lattice of linear subspaces of a projective space satis es these conditions, and that in fact, any modular geometric lattice is isomorphic to the lattice of linear subspaces of some projective space. Furthermore the lattice of positive idempotents of any KR frame is also a modular geometric lattice. The undecidable problem which Urquhart uses to prove the undecidability of KR is now simple to state. Hutchinson [1973] and Lipshitz [1974] proved that The word problem for any class of modular lattices which includes the subspace lattice of an in nite dimensional projective space is undecidable.

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Given an in nite dimensional projective space in which every line includes at least four points P , the logic of the frame (P ) is said to be a strong logic. Our undecidability theorem then goes like this: Any logic between KR and a strong logic is undecidable. The proof is not too diÆcult. Consider a modular lattice problem If v1 = w1 : : : vn = wn then v = w stated in a language with variables xi (i = 1; 2; : : : ) constants 1 and 0, and the lattice connectives \ and +. Fix a map into the language of KR by setting xti = pi for variables, 0t = t, 1t = >, (v \ w)t = vt ^ wt and (v + w)t = vt Æ wt . The translation of our modular lattice problem is then the KR sentence


B ^ (v1t $ w1t ) ^


^ (vnt $ wnt ) ^ t

! (vt $ wt ) where the sentence B is the conjunction of all sentences pi ^ (pi $ pi Æ pi )

for each pi appearing in the formulae vjt or wjt . We will show that given a particular in nite dimensional projective space (with every line containing at least four points) P , then the word problem is valid in the lattice of linear subspaces of P if and only if its translation is provable in L, for any logic L intermediate between KR and the logic of the frame F (P ). If the translation of the word problem is valid in L, then it holds in the frame F (P ). Consider the word problem. If it were invalid, then there would be linear subspaces x1 ; x2 ; : : : in the space P such that each vi = wi would be true while v 6= w. Construct a model on the frame F (P ) as follows. Let the extension of pi be the space xi together with the point 0. It is then simple to show that 0  B , as each pi is a positive idempotent. In addition, 0  t, and 0  vit $ wit , for the extension of each vit and wit will be the spaces picked out by vi and wi (both with the obligatory 0 added). However, we would have 0 6 vt $ wt , since the extensions of vt and wt were picked out to dier. This would amount to a counterexample to the translation of the word problem, which we said was valid. As a result, the word problem is valid in the space P . The converse reasoning is similar. Unfortunately, these techniques do not work for systems weaker than KR. The proof that positive idempotents are modular uses essentially the special properties of KR. Not every positive idempotent in R need be modular. But nonetheless, the techniques of the proof can be extended to apply to a much wider range of systems. Urquhart examined the structure of the of the modular lattice undecidability result, and he showed that you could make do with much less. You do not need to restrict your attention to modular lattices to construct an undecidable word problem. But to do that, you need to examine Lipshitz and Hutchinson's proof more carefully. In the rest of

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this section, we will sketch the structure of Urquhart's undecidability proof. The techniques are quite involved, so we do not have the space to go into detail. For that, the reader is referred to Urquhart [1984]. Lipshitz and Hutchinson proved that the word problem for modular lattices was undecidable by embedding into that problem the already known undecidable word problem for semigroups. It is enough to show that a structure can de ne a \free associative binary operation", for then you will have the tools for representing arbitrary semigroup problems. (A semigroup is a set with an associative binary operation. An operation is a \free associative" operation if it satis es those conditions satis ed by any associative operation but no more.) We will sketch how this can be done without resorting to a modular lattice. The required structure is what is called a 0-structure, and a modular 4-frame de ned within a 0-structure. An 0-structure is a set equipped with the following structure  A semilattice with respect to u.

 With a binary operator + which is associative and commutative.  And x  y ) x + z  y + z .  0 + x = x.  y  0 ) x u (x + y) = x. A 4-frame in a 0-structure is a set fa1 ; a2 ; a3 ; a4 g [ fcij : i = 6 j; i; j = 1; : : : ; 4g such that  The ai s are independent. If G; H  fa1 ; : : : ; a4 g then (G) u (H ) = (G \ H ) (where ; = 0)  If G  fa1 ; : : : ; a4 g then G is modular  ai + ai = ai  cij = cji  ai + aj = ai + cik ; cij u aj = 0, if i = 6 j  (ai + ak ) u (cij + cjk ) = cik for distinct i; j; k Now, we are nearly at the point where we can de ne a semigroup structure. First, for each distinct i; j , we de ne the set Lij to be fx : x + aj = ai + aj and x u aj = 0g. Then if b 2 Lij and d 2 Ljk where i; j; k are distinct, then we set b d = (b + d) u (ai + ak ), and it is not diÆcult to show that b d 2 Lik . Then, we can de ne a semigroup operation `:' on L12 by setting

x:y = (x c23 ) (c31 y)

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Now it is quite an involved operation to show that this is in fact an associative operation, but it can be done. And in fact, in certain circumstances, the operation is a free associative operation. Given a countably in nitedimensional vector space V , its lattice of subspaces is a 0-structure, and it is possible to de ne a modular 4-frame in this lattice of subspaces, such that any countable semigroup is isomorphic to a subsemigroup of L12 under the de ned associative operation. (Urquhart gives the complete proof of this result [Urquhart, 1984].) The rest of the work of the undecidability proof involves showing that this construction can be modelled in a logic. Perhaps surprisingly, it can all be done in a weak logic like TW[^; _; !; >; ?]. We can do without negation by picking out a distinguished propositional atom f , and by de ning A to be A ! f , t to be f , and A : B to be (A ! B ). A is a regular proposition i A $ A is provable. The regular propositions form an 0-structure, under the assumption of the formula  = fR(t; f; >; ?); N (t; f; >; ?); > $ ?g. where R(A) is A $ A, N (A) is (t ! A) ! A, and R(A; B; : : : ) is R(A) ^ R(B ) ^


and similarly for N . In other words, we can show that the conditions for an 0-structure hold in the regular propositions, assuming  as an extra premise. To interpret the 0-structure conditions we interpret u by ^, + by : and 0 by t. Now we need to model a 4-frame in the 0-structure. This can be done as we get just the modularity we need from another condition which is simple to state. De ne K (A) to be R(A) ^ (A ^ A $ ?) ^ (A _ A $ >) ^ (A : A $ A) ^ (A $ A : A). Then we can show the following

K (A); R(B; C ); C ! A

` A ^ (B : C ) $ (A ^ B ) : C

In other words, if K (A), then A is modular in the class of regular propositions. Then the conditions for a 4-frame are simple to state. We pick out our atomic propositions A1 ; : : : ; A4 and C12 ; : : : ; C34 which will do duty for a1 ; : : : ; a4 and c12 ; : : : ; c34 . Then, for example, one independence axiom is (A1 : A2 : A3 ) ^ (A2 : A3 : A4 ) $ (A2 : A3 ) and one modularity condition is

K (A1 : A3 : A4 ) We will let  be the conjunction of the statements that express that the propositions Ai and Cij form a 4-frame in the 0-structure of regular propositions. So,  [  is a nite (but complex) set of propositions. In any algebra in which  [  is true, the lattice of regular propositions is a 0-structure, and the denotations of the propositions Ai and Cij form a 4-frame. Finally, when coding up a semigroup problem with variables x1 ; x2 ; : : : ; xm , we will need formulae in the language which do duty for these variables. Thus we

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need a condition which picks out the fact that pi (standing for xi ) is in L12 . We de ne L(p) to be (p : A2 $ A1 : A2 ) ^ (p ^ A2 $ t). Then the semigroup operation on elements of L12 can be de ned in terms of ^ and : and the formulae Ai and Cij . We assume that done, and we will simply take it that there is an operation
on formulae which picks out the algebraic operation on L12 . This is enough for us to sketch the undecidability argument. The deducibility problem for any logic between TW[^; _; ! ; >; ?] and KR is undecidable.

Take a semigroup problem which is known to be undecidable. It may be presented in the following way If v1 = w1 : : : vn = wn then v = w where each term vi ; wi is a term in the language of semigroups, constructed out of the variables x1 ; x2 ; : : : ; xm for some m. The translation of that problem into the language of TW[^; _; !; >; ?] is the deducibility problem ; ; L(p1 ; : : : ; pm ); v1t $ w1t ; : : : ; vnt $ wnt ` vt $ wt where each the translation ut of each term u is de ned recursively by setting xti to be pi , and (u1 :u2 )t to be ut1
ut2 . Now the undecidability result will be immediate once we show that for any logic between TW and KR the word problem in semigroups is valid if and only if its translation is valid in that logic. For left to right, if the word problem is valid in the theory of semigroups, its translation must be valid, for given the truth of  and  and L(p1; : : : ; pm ), the operator
is provably a semigroup operation on the propositions in L12 in the algebra of the logic, and the terms vi and wi satisfy the semigroup conditions. As a result, we must have vt and wt picking out the same propositions, hence we have a proof of vt $ wt . Conversely, if the word problem is invalid, then it has an interpretation in the semigroup S de ned on L12 in the lattice of subspaces of an in nite dimensional vector space. The lattice of subspaces of this vector space is the 0-structure in our countermodel. However, we need a countermodel for our | the 0-structure is not a model of the whole of the logic, since it just models the regular propositions. How can we construct this? Consider the argument for KR. There, the subspaces were the positive idempotents in the frame. The other propositions in the frame were arbitrary subsets of points. Something similar can work here. On the vector space, consider the subsets of points which are closed under multiplication (that is, if x 2 , so is kx, where k is taken from the eld of the vector space). This is a De Morgan algebra, de ning conjunction and disjunction by means of intersection and union as is usual. Negation is modelled by set dierence. The fusion  Æ of two sets of points is the set fx + y : x 2  and y 2 g. It is not too diÆcult

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to show that this is commutative and associative, and square increasing, when the vector space is in a eld of characteristic other than 2, since if x 2  then x = 12 x + 21 x 2  Æ . Then  ! is simply ( Æ ). It is not too diÆcult to show that this is an algebraic model for KR, and that the regular propositions in this model are exactly the subspaces of the vector space. It follows that our counterexample in the 0-structure is a counterexample in a model of KR to the translation of the word problem. As a result, the translation is not provable in KR or in any weaker logic. This result applies to systems between TW and KR, and it shows that the deducibility problem is undecidable for any of these systems. In the presence of the modus ponens axiom A ^ (A ! B ) ^ t ! B , this immediately yields the undecidability of the theoremhood problem, as the deducibility problem can be rewritten as a single formula.


 ^  ^ L(p1 ; : : : ; pm) ^ (v1t $ w1t ) ^


^ (vnt $ wnt ) ^ t

! (vt $ wt )

As a result, the theoremhood problem for logics between T and KR is undecidable. In particular, R, E and T are all undecidable. The restriction to TW is necessary in the theorem. Without the pre xing and suÆxing axioms, you cannot show that the lattice of regular propositions is closed under the `fusion-like' connective ` : '. Before moving on to our next section, let us mention that these geometric methods have been useful not only in proving the undecidability of logics, but also in showing that interpolation fails in R and related logics [Urquhart, 1993]. 5 LOOKING ABOUT A lot of the work in relevance logics taking place in the late 1980's and in the 1990's has not focussed on Anderson's core problems. Now that these have been more or less resolved, work has proceeded apace in other directions. In this section we will give an undeniably indiosyncratic and personal overview of what we think are some of the strategic directions of this recent research. The rst two sections in this part deal with generalisations | rst of semantics, and second of proof theory | which situate relevance logic into a wider setting. The next sections deal with neighbouring formal theories, and we end with one philosophical application of the machinery of relevance logics.

5.1 Gaggle Theory

The fusion connective Æ has played an important part in the study of relevance logics. This is because fusion and implication are tied together by

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the residuation condition

a  b ! c i a Æ b  c In addition, in the frame semantics, fusion and implication are tied to the same ternary relation R, implication with the universal condition and fusion with the existential condition. This is an instance of a generalised Galois connection. Galois studied connections between functions on partially ordered sets. A Galois connection between two partial orders  on A and 0 on B is a pair of functions f : A ! B and g : B ! A such that b 0 f (a) i a  g(b) The condition tying together fusion and implication is akin to that tying together f and g for Galois. So, gaggle theory (for `ggl': generalised Galois logic ) studies these connections in their generality, and it turns out that relevance logics like R, E and T are a part of a general structure which not only includes other relevance logics, but also traditional modal logics, Jonsson and Tarski's Boolean algebras with operators [Jonsson and Tarski, 1951] and many other formal systems. Dunn has shown that if a logic has a family of nary connectives which are tied together with a generalised galois connection, then the logic has a frame semantics in which those connectives are modelled using the one n +1-ary relation, in the way that fusion and implication are modelled by the same ternary relation in relevance logics [Dunn, 1991; Dunn, 1993a; Dunn, 1994]. In general, an n-ary connective f has a trace (1 ; : : : ; n ) 7! + if

   

f (c1 ; : : : ; 1; : : : ; cn ) = 1, if i = + (where the 1 is in position i). f (c1 ; : : : ; 0; : : : ; cn ) = 1, if i =

(where the 0 is in position i).

If a  b, and if i = + then f (c1 ; : : : ; a; : : : ; cn )  f (c1 ; : : : ; b; : : : ; cn ). If a  b, and if i =

then f (c1 ; : : : ; b; : : : ; cn )  f (c1 ; : : : ; a; : : : ; cn ).

We write this as T (f ) = (1 ; : : : ; n ) 7! +. On the other hand, the connective f has trace (1 ; : : : ; n ) 7! if

   

f (c1 ; : : : ; 1; : : : ; cn ) = 0, if i = + (where the 0 is in position i). f (c1 ; : : : ; 0; : : : ; cn ) = 0, if i =

(where the 1 is in position i).

If a  b, and if i = + then f (c1 ; : : : ; b; : : : ; cn )  f (c1 ; : : : ; a; : : : ; cn ). If a  b, and if i =

then f (c1 ; : : : ; a; : : : ; cn )  f (c1 ; : : : ; b; : : : ; cn ).

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We write this as T (c) = (1 ; : : : ; n ) 7! . Here are a few examples of traces of connectives. Conjunction-like connectives tend to be ( ; ) 7! , disjunction-like connectives tend to be (+; +) 7! +, necessity-like connectives tend to be + 7! +, possibility-like connectives tend to be 7! , and negations can be either + 7! or 7! + (and in many cases they are both). Now we are nearly able to state the abstract law of residuation. First, we de ne S (f; a1 ; : : : ; an ; b) as follows. If T (f ) = (


) 7! +, then S (f; a1 ; : : : ; an ; b) is the condition f (a1 ; : : : ; an )  b. If, on the other hand, T (f ) = (


) 7! , then S (f; a1 ; : : : ; an ; b) is b  f (a1 ; : : : ; an ). Then, two connectives f and g are contrapositives in place j i, if T (f ) = (1 ; : : : ; j ; : : : ; n ) 7!  , then T (g) = (1 ; : : : ; ; : : : ; n ) 7! j . (Where we de ne + as and as +.) Two operators f and g satisfy the abstract law of residuation i f and g are contrapositives in place j , and S (f; a1 ; : : : ; aj ; : : : ; an ; b) i S (g; a1; : : : ; b; : : : ; an ; aj ). A collection of connectives in which there is some connective f such that every element of the collection satis es the abstract law of residuation with f , is called a founded family of connectives. Dunn's major result is that if you have an algebra in which every connective is in a founded family, then the algebra is isomorphic to a subalgebra of the collection of propositions in a model in which each founded family of n-ary connectives shares an n + 1ary relation. The soundness and completeness of the Routley{Meyer ternary relational semantics is for the implication-fusion fragment of relevance logics is an instance of this more general result. The gaggle theoretic account of negation in relevance logics is interesting. We do not automatically get negation modelled by the Routley star | instead, being a unary connective, negation is modelled with a binary relation. One way negation can be modelled along gaggle theoretic lines is as follows. The De Morgan negation connective has trace 7! +, so the gaggle theoretic result is that there is a binary relation C between set-ups such that

 x  :A i for each y where xCy, y 6 A This is the general semantic structure which models negation connectives with trace 7! +. Given a relation C , which we may read as `compatibility', we can de ne another negation connective , using C 's converse:

 x  A i for each y where yCx, y 6 A Then it follows that A ` B i B ` :A. For the De Morgan negation of relevance logics,  and : are the same, for the compatibility relation C is

symmetric. But in more general settings, this need not hold. The general perspective of gaggle theory not only opens up new formal systems to study | it also helps with interpreting the semantics. The

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condition for : above can be read as follows: :A is true at x i for each y compatible with x, A is not true at y. This certainly sounds like a more palatable condition for negation than that using Routley star. We have an understanding of what it is for two set-ups (theories, worlds or situations) to be compatible, and the notion of compatibility is tied naturally to that of negation. Furthermore, the Routley star condition is an instance of this more general `compatibility' condition. For any set-up a, a can be seen as the set-up which `wraps up' all set-ups compatible with a. We can argue whether there is such an all-encompassing set-up, but if there is, then the semantics for negation in terms of the compatibility relation is equivalent to that of the Routley star. And in addition, we have another means of explaining it. Furthermore, once we have this generalised position from which to view negation, we can tinker with the binary accessibility relation in just the same way that modal logics are studied. Clearly if Boolean negation (written ` ') is present, then :A is simply  A for the positive modal operator  which uses C as its accessibility relation; and the study of these negation is dealt with using the techniques of modal logic. However, in relevance logics and other related systems, boolean negation is not present. And in this case the theory of negations arising from compatibility clauses like the one we have seen is a young and interesting subject in its own right. This perspective is pursued in Dunn [1994], and Restall [1999] develops a philosophical interpretation of the semantics.

5.2 Display Logic Nuel Belnap has developed proof theoretical techniques which are quite similar to those from gaggle theory. Consider the general problem of providing a sequent calculus for logics like R and others. We have the choice of how to formulae sequents. If they are of the form X ` A, where X is a structured collection of formulae, and A is a formula, then we have the problem of how to state the introduction and elimination of negation rules in such a way as to make ::A equivalent to A. It is unclear how to do this while maintaining that the succedent of every sequent is a single formula. On the other hand, if we allow that sequents are of the form X ` Y , where now both X and Y are structured complexes of formulae, it is unclear how to state a cut rule which is both valid and admits of a cut-elimination proof in the style of Gentzen. If we are restricted to single formulae in the succedent position the rule is easy to state: X ` A Y (A) ` B Y (X ) ` B but in the presence of multiple succedents it is unclear how to state the rule generally enough to be eliminable yet strictly enough to be valid under

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interpretation. If there is only one sort of structuring in the consequent this might be possible, in the way used in the proof theories of classical or linear logic, for example: X ` A; Y X 0 ; A ` Y 0 X; X 0 ` Y; Y 0

But if we have X ` Y (A) and X 0 (A) ` Y 0 where the indicated instances of A are buried under multiple sorts of structure, then what is the appropriate conclusion of a cut rule? X 0 (X ) ` Y (Y 0 ) will not do in general, for it is invalid in many instances. For example, in R if Y (A) is A ^ B and X 0 (A) is A Æ D, then we have A ^ B ` A ^ B and A Æ D ` A Æ D, but we don't have (A ^ B ) Æ D ` (A Æ D) ^ B in general. (Consider the case where B = A. A Æ D needn't imply A.) The alternative examined by Belnap is to make do with Cut where the cut formula is \displayed" in both premises of the rule.

X`A A`Y X`Y In order to get away with this, a system needs to be such that whenever you need to use a cut you can. The way Belnap does this is by requiring what he calls the \display condition". The display condition is satis ed i for every formula, every sequent including that formula is equivalent (using invertible rules) to one in which that formula is either the entire antecedent or the entire succedent of the sequent. For Belnap's original formulation, this is achieved by having a binary structuring connective Æ (not to be confused with the sentential connective Æ) and a unary connective . The display rules were as follows: X Æ Y ` Z () X ` Y Æ Z X ` Y Æ Z () X Æ Y ` Z () X ` Z Æ Y X ` Y () Y ` X ()   X ` Y A structure is in antecedent position if it is in the left under an even number of stars, or in the right under an odd number of stars. If it is not in antecedent position, it is in succedent position. The star is read as negation, and the circle is read as conjunction in antecedent position, and disjunction in succedent position. The display postulates are a reworking of conditions like the residuation condition for fusion and implication. Here we have the conditions that a Æ b  c i a  b + c (where x + y is the ssion of x and y). Belnap's system allows that dierent families of structural connectives can be used for dierent families of connectives in the language. For example, when Æ and  are read intensionally, we can have the following rules for

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implication:

X ÆA`B X `A B`Y X`A!B A ! B ` X Æ Y If the properties of Æ vary, so do the properties of the connective !. We can give Æ properties of extensional conjunction in order to get a material conditional. Or conditions can be tightened, to give ! modal properties. It is clear that the family of structural connectives (here Æ and ) act in analogously to accessibility relations on frames. However, the connections with gaggle theory run deeper, however. It can be shown a connective introduced in with rules without side conditions, and in a way which `mimics' structural connectives (just as here A ! B mimics X Æ Y in consequent position) must have a de nable trace. Any implication satisfying those rules will have trace ( ; +) 7! +, for example. For more details of this connection and a general argument, see Restall's paper [Restall, 1995a]. Display logic gives these systems a natural cut-free proof theory, for Belnap has shown that under a broad set of conditions, any proof theory with this structure will satisfy cut-elimination. So again, just as with gaggle theory, we have an example of the way that the study of relevance logics like R and E have opened up into a more general theory of logics with similar structures.

5.3 Paraconsistency

Relevance logics are paraconsistent, in that argument forms such as A^:A ` B are taken to be invalid. As a result, relevance logics have been seen to be important for the study of paraconsistent theories. [[See Priest's article in this volume]]. Relevance logics are suited to applications for which a paraconsistent notion of consequence is needed however, not all logics are equal in this regard. For example, paraconsistentists have often considered the topic of nave theories of sets and of truth (any predicate yields the set of things satisfying that predicate, the proposition p is true if and only if p). With a relevance logic at hand, you can avoid the inference to triviality from contradictions such as that arising from the liar This proposition is not true. (from which you can deduce that it is true, and hence that it isn't) and Russell's paradox (fx : x 62 xg both is and is not a member of itself). However, the Curried forms of these paradoxes If this proposition is true then there is a Santa Claus. and fx : (x 2 x) ! P g are more diÆcult to deal with. These yield arguments for the existence of Santa Claus and the truth of P (which was

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arbitrary) in logics like R, or any others with theorems related to the rule of
contraction. The theoremhood of propositions such as A ! (A ! B ) ! (A ! B ) and A ^ (A ! B ) ! B rule out a logic for service in the cause of paraconsistent theories like these [Meyer et al., 1979]. However, this has not deterred some hardier souls in considering weaker relevance logics which do not allow one to deduce triviality in these theories. Some work has been done to show that in some logics these theories are consistent, and in others, though inconsistent, not everything is a theorem [Brady, 1989]. Another direction of paraconsistency in which techniques of relevance logics have borne fruit is in the more computational area of reasoning with inconsistent information. The techniques of rst degree entailment have found a home in the study of \bilattices" by Melvin Fitting and others, who seen in them a suitable framework for reasoning under the possibility of inconsistent information [Fitting, 1989].

5.4 Semantic Neighbours Another area in which research has grown in the recent years has been toward connections with other elds. It has turned out that seemingly completely unrelated elds have studied structures remarkably like those studied in relevance logics. These neighbours are helpful, not only for giving independent evidence for the fact that relevance logicians have been studying something worthwhile, but also because of the dierent insights they can bring to bear on theorising. In this section we will see just three of the neighbours which can shed light on work in relevance logics. The rst connection comes with Barwise and Perry's situation semantics [1983]. For Barwise and Perry, utterances classify situations (parts of the world) which may be incomplete with regard to their semantic `content'. Consider the claim that Max saw Queensland win the SheÆeld Shield". How is this to be understood? For the Barwise and Perryof Situations and Attitudes [Barwise and Perry, 1983], this was to be parsed as expressing a relationship between Max and a situation, where a situation is simply a restricted part of the world. Situations are parts of the world and they support information. Max saw a situation and in this situation, Queensland won the SheÆeld Shield. If, in this very situation, Queensland beat South Australia, then Max saw Queensland beat South Australia. This shows why for this account situations have to be (in general) restricted bits of the world. The situation Max saw had better not be one in which Paul Keating lost the 1996 Federal Election, lest it follow from the fact that Max witnessed Queensland's victory that he also witnessed Keating's defeat, and surely that would be an untoward conclusion. Let's denote this relationship between situations an the information they support as follows. We'll abbreviate the claim that the situation s supports the

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information that A by writing `s  A', and we'll write its negation, that s doesn't support the information that A by writing `s 6 A'. This is standard in the situation theoretic literature. The information carried by these situations has, according to Barwise and Perry, a kind of logical coherence. For them, infons are closed under conjunction and disjunction, and s  A ^ B if and only if s  A and s  B , and s  A _ B if and only if s  A or s  B . However, negation is a dierent story | clearly situations don't support the traditional equivalence between s  :A and s 6 A (where :A is the negation of A), for our situation witnessed by Max supports neither the infon \Keating won the 1996 election" nor its negation. What to do? Well, Barwise and Perry suggest that negation interacts with conjunction and disjunction in the familiar ways | :(A _ B ) is (equivalent to) :A ^ :B , and :(A ^ B ) is (equivalent to) :A _ :B . And similarly, ::A is (equivalent to) A. This gives us a logic of sorts of negation | it is rst degree entailment. Now for Barwise and Perry, there are no actual situations in which s  A ^ :A (the world is not self-contradictory). However, they agree that it is helpful to consider abstract situations which allow this sort of inconsistency. So, Barwise and Perry have an independent motivation for a semantic account of rst-degree entailment. (More work has gone on to consider other connections between situation theory and relevance logics [Mares, 1997; Restall, 1994; Restall, 1995b].) Another connection with a parallel eld has come from completely dierent areas of research. The semantic structures of relevance logics have close cousins in the models for the Lambek Calculus and in Relation algebras. Let's consider relation algebras rst. A relation algebra is a Boolean algebra with some extra operations, a binary operation which denotes composition of relation, a unary operation ^ ,for the converse of a relation, and a constant 1 for the identity relation. There is a widely accepted axiomatisation of the variety RA of relation algebras. A relation algebra is set R with operations ^; _; ; 1; Æ; ^ such that

 hR; ^; _; i is a boolean algebra.  ^ is an automorphism on the algebra, satisfying a^^ = a, (a ^ b)^ = a^ ^ b^ , (a^ ) = ( a)^ .  Æ is associative, with a left and right identity 1, satisfying (a _ b) Æ c = (a Æ c) _ (b Æ c), a Æ (b _ c) = (a Æ b) _ (a Æ c).  ^ and Æ are connected by setting (a Æ b)^ = b^ Æ a^ .

These conditions are satis ed by the class of relations on any base set (that is, by any concrete relation algebra). However, not every algebra satisfying these equations is isomorphic to a subalgebra of a concrete relation algebra.

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These algebras are quite similar to de Morgan monoids. If we de ne :A to be (a)^ or ( a)^ then the conjunction, disjunction, :, 1 fragment is that of rst degree entailment. We do not have a  b _ :b, and nor do we have a ^ :a  b. Consider the relation a:

a x y x 1 1 y 0 1 Then :a is the following relation

a x y x 0 1 y 0 0 So we don't have b  a _ :a for every b, and nor do we have a ^ :a _ b. (However, we do have 1  a _ :a.) The class of relation algebras have a natural form of implication to go along with the fusionlike connective Æ. If we de ne a ! b to be :(:b Æ a), then we have the residuation condition a Æ b  c i a  b ! c. However, that is not the only implication-like connective we may de ne. If we set b a to be :(a Æ:b), then a Æ b  c i b  c a. Since Æ is not, in general, commutative, we have two residuals. In logics like R this is not possible, for the left and the right residuals of fusion are the same connective. However, in systems in the vicinity of E, these implication operations come apart. This is mirrored by the behaviour on frames, since we can de ne B A by setting x  B A i for each y; z where Ryxz if y  A then z  B . This will be another residual for fusion, and it will not agree with ! in the absence of commutativity of R (if Rxyz then Ryxz ).49 It was hoped for some time that relation algebras would give an interesting model for logics like R. However, there does not seem to be a natural class of relations for which composition is commutative and square increasing. (The class of symmetric relations will not do. Even if a = a^ and b = b^ , it does not follow that a Æ b = b Æ a. You merely get that a Æ b = a^ Æ b^ = (b Æ a)^ .) Considered as a logic, RA is a sublogic of R (ignoring boolean negation for the moment). It is not a sublogic of E, since in RA, a = 1 ! a. Another dierence between RA and typical relevance logics is the behaviour of contraposition. We do not have a ! b = :b ! :a. Instead, a ! b = :a :b. A nal connection between RA and relevance logics is in the issue of semantics. As we stated earlier, not all relation algebras are representable as subalgebras of concrete relation algebras. However, Dunn has shown 49 We should ag here that in the relevance logic literature, [Meyer and Routley, 1972] seems to have been the rst to consider both left- and right-residuals for fusion.

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that all relation algebras are representable by algebras of propositions on a particular class of Routley{Meyer frames [Dunn, 1993b]. This is the rst representation theorem for RA, and it shows that the semantical techniques of relevance logics have a wider scope than applications to R, E and their immediate neighbours. In a similar vein, Dunn and Meyer [1997] have provided a Routley{Meyer style frame semantics for combinatory logic. The key idea here is that the ternary relation R satis es no special conditions, but these properties are encoded by combinators, which are modelled by special propositions on frames. Lambek's categorial grammar is also similar to relevance logics, though this time it is introduced with frames, not algebras [Lambek, 1958; Lambek, 1961]. Here, the points in frames are pieces of syntax, and the `propositions' are syntactic classi cations of various kinds. For example, the classi cations into noun phrases, verbs, and sentences. The interest comes with the way in which these classi cations can be combined. For example A Æ B can be de ned, where we say x  A Æ B i x is a concatenation of two strings y and z , where y  A and z  B . We can also de ne `slicing' operations, setting x  AnB i for each y where y  A, yx  B ; and x  B=A i for each y where y  A, xy  B . These are obviously analogues for Æ and ! in relevance logics, and again, we have a `left' and `right' residuals for fusion. In these frames Rxyz i xy = z . So the Lambek calculus gives us an independently motivated interpretation of a class of Routley{Meyer frames. This connection has been explored by Kurtonina [1995], which is a helpful sourcebook of some recent work on ternary frames in connection with the Lambek calculus and related logics. If you like, you can enrich the logic of strings with conjunction and disjunction, and if you do it in the obvious way (using the same clauses as in relevance logics) you get a formal logic quite like RA [Restall, 1994]. But more importantly, the conditions for conjunction and disjunction may be independently motivated. A string is of type A _ B just when it is of type A or of type B . A string is of type A ^ B just when it is of type A and of type B . The resulting logic is clearly interpretable, but it was a number of years before a proof theory was found for it. Here the techniques for the Gentzenisation for positive relevance logics are appropriate, and the proof theory can be found by utilising the proof theory for R+, and removing the commutativity and contraction of the intensional bunching operation. The resulting proof theory captures exactly the Lambek calculus enriched with conjunction and disjunction. In addition, the techniques of Giambrone show that the resulting logic is decidable [Restall, 1994].

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5.5 Linear Logic The burgeoning phenomenon of linear logic is one which has a number of formal similarities to relevance logics [Girard, 1987; Troelstra, 1992]. Linear logic is the study of systems in the vicinity of LRW (R without contraction, without distribution). This is proof-theoretically a very stable system. It is simple to show that it is decidable. Girard's innovation, however, is to extend the proof theory with a modal operator ! which allows intuitionistic logic to be modelled inside linear logic. This operation in given as follows, in single-succedent Gentzen systems.

X ` B X ; A ` B !X ` B X; !A; !A ` B X; !A ` B X ; !A ` B !X ` !B X; !A ` B Given this proof theory it is possible to show that A ) B de ned as !A ! B is an intuitionistic implication. This is similar to Meyer's result that A ^ t ! B is an intuitionistic implication in R (indeed, !A de ned as A ^ t satis es each of the conditions for ! above in R, but not in systems without contraction). However, nothing like it holds in relevance logics without contraction. Linear logic also brings with it many new algebraic structures and models in category theory. None of these models have been mined to see if they can bring any `relevant' insight. However, some transfer has gone on in the other direction | Allwein and Dunn [1993] have shown that the multiplicative and additive fragment of linear logic can be given a Routley{Meyer style semantics. This is not a simple job, as the absence of the distribution of (additive) conjunction over disjunction means that at least one of these connectives (in this case, disjunction) must take a non-standard interpretation.

5.6 Relevant Predication There has been one major way in which relevance logics have been used in application to philosophical issues, and this application makes a good topic to end this article. The topic is Dunn's work on relevant predication [Dunn, 1987].50 The guiding idea is that a theory of relevant implication will give you some way of marking out the distinction between the way that Socrates' wisdom is a property of Socrates, in the way that Socrates' wisdom is not a property of Bill Clinton. Classical rst order logic is not good at marking out such a distinction, for if W x stands for `x is wise', and s stands for Socrates, and c stands for 50 The reference [Dunn, 1987] is of course \Relevant Predication": Of course all work has precursors, in this instance (largely unpublished) thoughts in the 1970's by N. Belnap, J. Freeman, and most importantly R. K. Meyer and A. Urquhart (and Dunn). Cf. Sec. 9 of [Dunn, 1987] for some history.

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Bill Clinton, then W x is true of x i it is wise, and (W s ^ x = x) _ W s is true of something i Socrates is wise. Why is one a `real' property and the other not? The guiding idea for relevant predication is the following distinction. It is true that if x is Socrates then x is wise. However, it is not true that if x is Bill Clinton then Socrates is wise. At least, it is plausible that this conditional fail, when read `relevantly'. This can be cashed out formally as follows. F is a relevant property of a (written (xF x)a) if and only if (8x)(x = a ! F x). Given this de nition, if F is a relevant property of a then F a holds (quite clearly) and if F and G are relevant properties of a then so is their conjunction, and the disjunction of any relevant property with anything at all is still a relevant property. Furthermore, one can de ne what it is for a relation to truly be a relation between objects. If Hx is `x's height is over 1 meter', and Ly is `y is a logician' then, it is true that Greg's height is over 1 meter and Mike is a logician. However, it would be bizarre to hold that in this there is a real relation that holds between Greg and Mike because of this fact. We would have the following

8x8y(x = g ^ y = m ! Hx ^ Ly) (assuming that (xHx)g and (yLy)m) but it need not follow that

8x8y(x = g ! (y = m ! Hx ^ Ly)) for there is no reason that Hx should follow from y = m, even given that x = g holds. There is no connection between `y's being m' and Hg. This latter proposition is a good candidate for expressing that there is a real relationship holding between g and m. In other words, we can de ne (xyLxy)ab to be


8x8y x = a ! (y = b ! Lxy)

to express the holding of a relevant relation. For more on relevant predication, consult Dunn's series of papers [Dunn, 1987; Dunn, 1990a; Dunn, 1990b] Relevance logics are very good at telling you what follows from what as a matter of logic | and in this case, the logical structure of relevant predication and relations. However, more work needs to be done to see in what it consists to say that a relevant implication is true. For that, we need a better grip on how to understand the models of relevance logics. It is our hope that this chapter will help people in this aim, ant to bring the technique of relevance logics to a still wider audience.

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ACKNOWLEDGEMENTS Dunn's acknowledgements from the rst edition : I wish to express my thanks and deep indebtedness to a number of fellow toilers in the relevant vineyards, for information and discussion over the years. These include Richard Routley and Alasdair Urquhart and especially Nuel D. Belnap, Jr. and Robert K. Meyer, and of course Alan Ross Anderson, to whose memory I dedicate this essay. I also wish to thank Yong Auh for his patient and skilful help in preparing this manuscript, and to thank Nuel Belnap, Lloyd Humberstone, and Allen Hazen for corrections, although all errors and infelicities are to be charged to me. Our acknowledgements from the second edition : Thanks to Bob Meyer, John Slaney, Graham Priest, Nuel Belnap, Richard Sylvan, Ed Mares, Rajeev Gore, Errol Martin, Chris Mortensen, Uwe Petersen and Pragati Jain for helpful conversations and correspondence on matters relevant to what is discussed here. Thanks too to Jane Spurr, who valiantly typed in the rst edition of the article, to enable us to more easily create this version. This edition is dedicated to Richard Sylvan, who died while this essay was being written. Relevance (and relevant ) logic has lost one of its most original and productive proponents.

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[Restall, 1995a] Greg Restall. Display logic and gaggle theory. Reports in Mathematical Logic, 29:133{146, 1995. [Restall, 1995b] Greg Restall. Information ow and relevant logics. In Jerry Seligman and Dag Westerstahl, editors, Logic, Language and Computation: The 1994 Moraga Proceedings, pages 463{477. csli Publications, 1995. [Restall, 1998] Greg Restall. Displaying and deciding substructural logics 1: Logics with contraposition. Journal of Philosophical Logic, 27:179{216, 1998. [Restall, 1999] Greg Restall. Negation in Relevant Logics: How I Stopped Worrying and Learned to Love the Routley Star. In Dov Gabbay and Heinrich Wansing, editors, What is Negation?, volume 13 of Applied Logic Series, pages 53{76. Kluwer Academic Publishers, 1999. [Restall, 2000] Greg Restall. An Introduction to Substructural Logics. Routledge, 2000. [Routley and Meyer, 1973] Richard Routley and Robert K. Meyer. Semantics of entailment. In Hugues Leblanc, editor, Truth Syntax and Modality, pages 194{243. North Holland, 1973. Proceedings of the Temple University Conference on Alternative Semantics. [Routley and Meyer, 1976] Richard Routley and Robert K. Meyer. Dialectal logic, classical logic and the consistency of the world. Studies in Soviet Thought, 16:1{25, 1976. [Routley and Routley, 1972] Richard Routley and Valerie Routley. Semantics of rstdegree entailment. No^us, 3:335{359, 1972. [Routley et al., 1982] Richard Routley, Val Plumwood, Robert K. Meyer, and Ross T. Brady. Relevant Logics and their Rivals. Ridgeview, 1982. [Routley, 1977] Richard Routley. Ultralogic as universal. Relevance Logic Newsletter, 2:51{89, 1977. Reprinted in Exploring Meinong's Jungle [Routley, 1980a]. [Routley, 1980a] Richard Routley. Exploring Meinong's Jungle and Beyond. Philosophy Department, RSSS, Australian National University, 1980. Interim Edition, Departmental Monograph number 3. [Routley, 1980b] Richard Routley. Problems and solutions in the semantics of quanti ed relevant logics | I. In A. I. Arrduda, R. Chuaqui, and N. C. A. da Costa, editors, Proceedings of the Fourth Latin American Symposium on Mathematical Logic, pages 305{340. North Holland, 1980. [Schroeder-Heister and Dosen, 1993] Peter Schroeder-Heister and Kosta Dosen, editors. Substructural Logics. Oxford University Press, 1993. [Schutte, 1956] K. Schutte. Ein system des verknupfenden schliessens. Archiv fur Mathematische Logik, 2:55{67, 1956. [Scott, 1971] Dana Scott. On engendering an illusion of understanding. Journal of Philosophy, 68:787{807, 1971. [Shaw-Kwei, 1950] Moh Shaw-Kwei. The deduction theorems and two new logical systems. Methodos, 2:56{75, 1950. [Slaney, 1985] John K. Slaney. 3088 varieties: A solution to the Ackermann constant problem. Journal of Symbolic Logic, 50:487{501, 1985. [Slaney, 1990] John K. Slaney. A general logic. Australasian Journal of Philosophy, 68:74{88, 1990. [Smullyan, 1968] R. M. Smullyan. First-Order Logic. Springer-Verlag, Berlin, 1968. Reprinted by Dover Press, 1995. [Stone, 1936] M. H. Stone. The theory of representation for Boolean algebras. Transactions of the American Mathematical Society, 40:37{111, 1936. [Suppes, 1957] P. Suppes. Introduction to Logic. D. Van Nostrand Co, Princeton, 1957. [Tennant, 1994] Neil Tennant. The transmission of truth and the transitivity of deduction. In Dov Gabbay, editor, What is a Logical System?, volume 4 of Studies in Logic and Computation, pages 161{177. Oxford University Press, Oxford, 1994. [Thistlewaite et al., 1988] Paul Thistlewaite, Michael McRobbie, and Robert K. Meyer. Automated Theorem Proving in Non-Classical Logics. Wiley, New York, 1988. [Tokarz, 1980] M. Tokarz. Essays in Matrix Semantics of Relevant Logics. The Polish Academy of Science, Warsaw, 1980. [Troelstra, 1992] A. S. Troelstra. Lectures on Linear Logic. csli Publications, 1992. [Urquhart, 1972a] Alasdair Urquhart. The completeness of weak implication. Theoria, 37:274{282, 1972.

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[Urquhart, 1972b] Alasdair Urquhart. A general theory of implication. Journal of Symbolic Logic, 37:443, 1972. [Urquhart, 1972c] Alasdair Urquhart. Semantics for relevant logics. Journal of Symbolic Logic, 37:159{169, 1972. [Urquhart, 1972d] Alasdair Urquhart. The Semantics of Entailment. PhD thesis, University of Pittsburgh, 1972. [Urquhart, 1983] Alasdair Urquhart. Relevant implication and projective geometry. Logique et Analyse, 26:345{357, 1983. [Urquhart, 1984] Alasdair Urquhart. The undecidability of entailment and relevant implication. Journal of Symbolic Logic, 49:1059{1073, 1984. [Urquhart, 1990] Alasdair Urquhart. The complexity of decision procedures in relevance logic. In J. Michael Dunn and Anil Gupta, editors, Truth or Consequences, pages 77{ 95. Kluwer, 1990. [Urquhart, 1993] Alasdair Urquhart. Failure of interpolation in relevant logics. Journal of Philosophical Logic, 22:449{479, 1993. [Urquhart, 1997] Alasdair Urquhart. The complexity of decision procedures in relevance logic ii. Available from the author, University of Toronto, 1997. [van Fraassen, 1969] Bas van Fraassen. Facts and tautological entailments. Journal of Philosophy, 66:477{487, 1969. Reprinted in Entailment Volume 1 [Anderson and Belnap, 1975]. [van Fraassen, 1973] Bas van Fraassen. Extension, intension and comprehension. In M. Munitz, editor, Logic and Ontology, pages 101{103. New York University Press, New York, 1973. [Wansing, 1993] Heinrich Wansing. The Logic of Information Structures. Number 681 in Lecture Notes in Arti cial Intelligence. Springer-Verlag, 1993. [Wolf, 1978] R. G. Wolf. Are relevant logics deviant. Philosophia, 7:327{340, 1978.

MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

QUANTUM LOGICS

1 INTRODUCTION The oÆcial birth of quantum logic is represented by a famous article of Birkho and von Neumann \The logic of quantum mechanics" [Birkho and von Neumann, 1936]. At the very beginning of their paper, Birkho and von Neumann observe: One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes .... The object of the present paper is to discover what logical structures one may hope to nd in physical theories which, like quantum mechanics, do not conform to classical logic. In order to understand the basic reason why a non classical logic arises from the mathematical formalism of quantum theory (QT), a comparison with classical physics will be useful. There is one concept which quantum theory shares alike with classical mechanics and classical electrodynamics. This is the concept of a mathematical \phase-space". According to this concept, any physical system S is at each instant hypothetically associated with a \point" in a xed phase-space ; this point is supposed to represent mathematically, the \state" of S , and the \state" of S is supposed to be ascertainable by \maximal" observations. Maximal pieces of information about physical systems are called also pure states . For instance, in classical particle mechanics, a pure state of a single particle can be represented by a sequence of six real numbers hr1 ; : : : ; r6 i where the rst three numbers correspond to the position -coordinates, whereas the last ones are the momentum -components. As a consequence, the phase-space of a single particle system can be identi ed with the set IR6 , consisting of all sextuples of real numbers. Similarly for the case of compound systems, consisting of a nite number n of particles. Let us now consider an experimental proposition P about our system, asserting that a given physical quantity has a certain value (for instance: \the value of position in the x-direction lies in a certain interval"). Such D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 6, 129{228.

c 2002, Kluwer Academic Publishers. Printed in the Netherlands.

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a proposition P will be naturally associated with a subset X of our phasespace, consisting of all the pure states for which P holds. In other words, the subsets of  seem to represent good mathematical representatives of experimental propositions. These subsets are called by Birkho and von Neumann physical qualities (we will say simply events ). Needless to say, the correspondence between the set of all experimental propositions and the set of all events will be many-to-one. When a pure state p belongs to an event X , we will say that our system in state p veri es both X and the corresponding experimental proposition. What about the structure of all events? As is well known, the power-set of any set is a Boolean algebra . And also the set F () of all measurable subsets of  (which is more tractable than the full power-set of ) turns out to have a Boolean structure. Hence, we may refer to the following Boolean algebra:

B = hF (); ; \; [; ; 1; 0i; where: 1)

;\;[;

are, respectively, the set-theoretic inclusion relation and the operations intersection, union, relative complement;

2) 1 is the total space , while 0 is the empty set. According to a standard interpretation, \ ; [ ; can be naturally regarded as a set-theoretic realization of the classical logical connectives and , or , not . As a consequence, we will obtain a classical semantic behaviour:



a state p veri es a conjunction X \ Y i p 2 X \ Y i p veri es both members;

 p veri es a disjunction X [ Y member;

i p 2 X [ Y i p veri es at least one

 p veri es a negation X i p 2= X i p does not verify X . To what extent can such a picture be adequately extended to QT? Birkho and von Neumann observe: In quantum theory the points of  correspond to the so called \wave-functions" and hence  is : : : a function-space, usually assumed to be Hilbert space. As a consequence, we immediately obtain a basic dierence between the quantum and the classical case. The excluded middle principle holds in

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classical mechanics. In other words, pure states semantically decide any event: for any p and X ,

p 2 X or p 2 X: QT is, instead, essentially probabilistic. Generally, pure states assign only probability-values to quantum events. Let represent a pure state (a wave function) of a quantum system and let P be an experimental proposition (for instance \the spin value in the x-direction is up"). The following cases are possible: (i)

assigns to P probability-value 1 ( (P) = 1);

(ii)

assigns to P probability-value 0 ( (P) = 0);

(iii)

assigns to P a probability-value dierent from 1 and from 0 ( (P) 6= 0; 1).

In the rst two cases, we will say that P is true (false ) for our system in state . In the third case, P will be semantically indeterminate. Now the question arises: what will be an adequate mathematical representative for the notion of quantum experimental proposition? The most important novelty of Birkho and von Neumann's proposal is based on the following answer: \The mathematical representative of any experimental proposition is a closed linear subspace of Hilbert space" (we will say simply a closed subspace ).1 Let H be a (separable) Hilbert space, whose unitary vectors correspond to possible wave functions of a quantum system. The closed subspaces of H are particular instances of subsets of H that are closed under linear combinations and Cauchy sequences. Why are mere subsets of the phase-space not interesting in QT? The reason depends on the superposition principle , which represents one of the basic dividing line between the quantum and the classical case. Dierently from classical mechanics, in quantum mechanics, nite and even in nite linear combinations of pure states give rise to new pure states (provided only some formal conditions are satis ed). Suppose three pure states ; 1 ; 2 and let be a linear combination of 1 ; 2 : = c1

1 + c2 2 :

1 A Hilbert space is a vector space over a division ring whose elements are the real or the complex or the quaternionic numbers such that (i) An inner product ( : ; :) that transforms any pair of vectors into an element of the division ring is de ned; (ii) the space is metrically complete with respect to the metrics induced by the inner product ( : ; :). A Hilbert space H is called separable i H admits a countable basis.

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According to the standard interpretation of the formalism, roughly this means that a quantum system in state might verify with probability jc1 j2 those propositions that are certain for state 1 (and are not certain for ) and might verify with probability jc2 j2 those propositions that are certain for state 2 (and are not certain for ). Suppose now some pure states 1 ; 2 ; : : : each assigning probability 1 to a given experimental proposition P, and suppose that the linear combination =

X

i

ci i (ci 6= 0)

is a pure state. Then also will assign probability 1 to our proposition P. As a consequence, the mathematical representatives of experimental propositions should be closed under nite and in nite linear combinations. The closed subspaces of H are just the mathematical objects that can realize such a role. What about the algebraic structure that can be de ned on the set C (H) of all mathematical representatives of experimental propositions (let us call them quantum events )? For instance, what does it mean negation , conjunction and disjunction in the realm of quantum events? As to negation, Birkho and von Neumann's answer is the following: The mathematical representative of the negative of any experimental proposition is the orthogonal complement of the mathematical representative of the proposition itself. The orthogonal complement X 0 of a subspace X is de ned as the set of all vectors that are orthogonal to all elements of X . In other words, 2 X 0 i ? X i for any  2 X : ( ; ) = 0 (where ( ; ) is the inner product of and ). From the point of view of the physical interpretation, the orthogonal complement (called also orthocomplement ) is particularly interesting, since it satis es the following property: for any event X and any pure state , (X ) = 1 i (X 0 ) = 0; (X ) = 0 i (X 0 ) = 1; In other words, assigns to an event X probability 1 (0, respectively) i assigns to the orthocomplement of X probability 0 (1, respectively). As a consequence, one is dealing with an operation that inverts the two extreme probability-values, which naturally correspond to the truth-values truth and falsity (similarly to the classical truth-table of negation). As to conjunction, Birkho and von Neumann notice that this can be still represented by the set-theoretic intersection (like in the classical case). For,

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the intersection X \ Y of two closed subspaces is again a closed subspace. Hence, we will obtain the usual truth-table for the connective and : veri es X \ Y i

veri es both members.

Disjunction, however, cannot be represented here as a set-theoretic union. For, generally, the union X [ Y of two closed subspaces is not a closed subspace. In spite of this, we have at our disposal another good representative for the connective or : the supremum X t Y of two closed subspaces, that is the smallest closed subspace including both X and Y . Of course, X t Y will include X [ Y . As a consequence, we obtain the following structure C (H) = hC (H) ; v ; u ; t ; 0 ; 1 ; 0i ; where v ; u are the set-theoretic inclusion and intersection; t ; 0 are de ned as above; while 1 and 0 represent, respectively, the total space H and the null subspace (the singleton of the null vector, representing the smallest possible subspace). An isomorphic structure can be obtained by using as a support, instead of C (H), the set P (H) of all projections P of H. As is well known projections (i.e. idempotent and self-adjoint linear operators ) and closed subspaces are in one-to-one correspondence, by the projection theorem. Our structure C (H) turns out to simulate a \quasi-Boolean behaviour"; however, it is not a Boolean algebra. Something very essential is missing. For instance, conjunction and disjunction are no more distributive. Generally,

X u (Y

t Z ) 6= (X u Y ) t (X u Z ):

It turns out that C (H) belongs to the variety of all orthocomplemented orthomodular lattices , that are not necessarily distributive. The failure of distributivity is connected with a characteristic property of disjunction in QT. Dierently from classical (bivalent) semantics, a quantum disjunction X t Y may be true even if neither member is true. In fact, it may happen that a pure state belongs to a subspace X t Y , even if belongs neither to X nor to Y (see Figure 1). Such a semantic behaviour, which may appear prima facie somewhat strange, seems to re ect pretty well a number of concrete quantum situations. In QT one is often dealing with alternatives that are semantically determined and true, while both members are, in principle, strongly indeterminate. For instance, suppose we are referring to some one-half spin particle (say an electron) whose spin may assume only two possible values: either up or down . Now, according to one of the uncertainty principles , the spin in the x direction (spinx) and the spin in the y direction (spiny ) represent two strongly incompatible quantities that cannot be simultaneously

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI O

Y

o

    o7 jj X   ooojjjjjjj o  o ooojjjj ojojojojjj  jjojoj jjjj  j j j  jjj  jjjj j j  j j jjjj       /



Figure 1. Failure of bivalence in QT measured. Suppose an electron in state veri es the proposition \spinx is up". As a consequence of the uncertainty principle both propositions \spiny is up" and \spiny is down" shall be strongly indeterminate. However the disjunction \either spiny is up or spiny is down" must be true. Birkho and von Neumann's proposal did not arouse any immediate interest, either in the logical or in the physical community. Probably, the quantum logical approach appeared too abstract for the foundational debate about QT, which in the Thirties was generally formulated in a more traditional philosophical language. As an example, let us only think of the famous discussion between Einstein and Bohr. At the same time, the work of logicians was still mainly devoted to classical logic. Only twenty years later, after the appearance of George Mackey's book Mathematical Foundations of Quantum Theory [Mackey, 1957], one has witnessed a \renaissance period\ for the logico-algebraic approach to QT. This has been mainly stimulated by the contributions of Jauch, Piron, Varadarajan, Suppes, Finkelstein, Foulis, Randall, Greechie, Gudder, Beltrametti, Cassinelli, Mittelstaedt and many others. The new proposals are characterized by a more general approach, based on a kind of abstraction from the Hilbert space structures. The starting point of the new trends can be summarized as follows. Generally, any physical theory T determines a class

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of event-state systems hE ; S i, where E contains the events that may occur to our system, while S contains the states that a physical system described by the theory may assume. The question arises: what are the abstract conditions that one should postulate for any pair hE ; S i? In the case of QT, having in mind the Hilbert space model, one is naturally led to the following requirement:



the set E of events should be a good abstraction from the structure of all closed subspaces in a Hilbert space. As a consequence E should be at least a -complete orthomodular lattice (generally non distributive).



The set S of states should be a good abstraction from the statistical operators in a Hilbert space, that represent possible states of physical systems. As a consequence, any state shall behave as a probability measure , that assigns to any event in E a value in the interval [0; 1]. Both in the concrete and in the abstract case, states may be either pure (maximal pieces of information that cannot be consistently extended to a richer knowledge) or mixtures (non maximal pieces of information).

In such a framework two basic problems arise: I) Is it possible to capture, by means of some abstract conditions that are required for any event-state pair hE ; S i, the behaviour of the concrete Hilbert space pairs? II) To what extent should the Hilbert space model be absolutely binding? The rst problem gave rise to a number of attempts to prove a kind of representation theorem . More precisely, the main question was: what are the necessary and suÆcient conditions for a generic event-state pair hE ; S i that make E isomorphic to the lattice of all closed subspaces in a Hilbert space? Our second problem stimulated the investigation about more and more general quantum structures. Of course, looking for more general structures seems to imply a kind of discontent towards the standard quantum logical approach, based on Hilbert space lattices. The fundamental criticisms that have been moved concern the following items: 1) The standard structures seem to determine a kind of extensional collapse. In fact, the closed subspaces of a Hilbert space represent at the same time physical properties in an intensional sense and the extensions thereof (sets of states that certainly verify the properties in question). As happens in classical set theoretical semantics, there is no mathematical representative for physical properties in an intensional

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sense. Foulis and Randall have called such an extensional collapse \the metaphysical disaster" of the standard quantum logical approach. 2) The lattice structure of the closed subspaces automatically renders the quantum proposition system closed under logical conjunction. This seems to imply some counterintuitive consequences from the physical point of view. Suppose two experimental propositions that concern two strongly incompatible quantities, like \the spin in the x direction is up", \the spin in the y direction is down". In such a situation, the intuition of the quantum physicist seems to suggest the following semantic requirement: the conjunction of our propositions has no definite meaning; for, they cannot be experimentally tested at the same time. As a consequence, the lattice proposition structure seems to be too strong. An interesting weakening can be obtained by giving up the lattice condition: generally the in mum and the supremum are assumed to exist only for countable sets of propositions that are pairwise orthogonal. In the recent quantum logical literature an orthomodular partially ordered set that satis es the above condition is simply called a quantum logic . At the same time, by standard quantum logic one usually means the complete orthomodular lattice based on the closed subspaces in a Hilbert space. Needless to observe, such a terminology that identi es a logic with a particular example of an algebraic structure turns out to be somewhat misleading from the strict logical point of view. As we will see in the next sections, dierent forms of quantum logic, which represent \genuine logics" according to the standard way of thinking of the logical tradition, can be characterized by convenient abstraction from the physical models. 2 ORTHOMODULAR QUANTUM LOGIC AND ORTHOLOGIC We will rst study two interesting examples of logic that represent a natural logical abstraction from the class of all Hilbert space lattices.These are represented respectively by orthomodular quantum logic (OQL) and by the weaker orthologic (OL), which for a long time has been also termed minimal quantum logic . In fact, the name \minimal quantum logic" appears today quite inappropriate, since a number of weaker forms of quantum logic have been recently investigated. In the following we will use QL as an abbreviation for both OL and OQL. The language of QL consists of a denumerable set of sentential literals and of two primitive connectives: : (not ), ^ (and ). The notion of formula of the language is de ned in the expected way. We will use the following metavariables: p; q; r; : : : for sentential literals and , , ; : : : for formulas.

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The connective disjunction (_ ) is supposed de ned via de Morgan's law:

 _ := : (: ^ : ) :

The problem concerning the possibility of a well behaved conditional connective will be discussed in the next Section. We will indicate the basic metalogical constants as follows: not, and, or, y (if...then), i (if and only if), 8 (for all ), 9 (for at least one). Because of its historical origin, the most natural characterization of QL can be carried out in the framework of an algebraic semantics. It will be expedient to recall rst the de nition of ortholattice : DEFINITION 1 (Ortholattice). An ortholattice is a structure B = hB ; v ;0 ; 1 ; 0i, where (1.1) hB ; v ; 1 ; 0i is a bounded lattice, where 1 is the maximum and 0 is the minimum . In other words: (i) v is a partial order relation on B (re exive, antisymmetric and transitive); (ii) any pair of elements a; b has an in mum aub and a supremum a t b such that: a u b v a; b and 8c: c v a; b y c v a u b; a; b v a t b and 8c: a; b v c y a t b v c; (iii) 8a: 0 v a; a v 1. (1.2) the 1-ary operation 0 (called orthocomplement ) satis es the following conditions: (i) a00 = a (double negation); (ii) a v b y b0 v a0 (contraposition); (iii) a u a0 = 0 (non contradiction). Dierently from Boolean algebras, ortholattices do not generally satisfy the distributive laws of u and t. There holds only (a u b) t (a u c) v a u (b t c)

and the dual form

a t (b u c) v(a t b) u (a t c): The lattice hC (H) ; v ; 0 ; 1 ; 0i of all closed subspaces in a Hilbert space H is a characteristic example of a non distributive ortholattice. DEFINITION 2 (Algebraic realization for OL). An algebraic realization for OL is a pair A = hB ; vi, consisting of an ortholattice B = hB ; v ; 0 ; 1 ; 0i and a valuation -function v that associates to any formula  of the language an element (truth-value ) in B , satisfying the following conditions:

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

(i) v(: ) = v( )0 ;

(ii) v( ^ ) = v( ) u v( ). DEFINITION 3 (Truth and logical truth). A formula  is true in a realization A = hB ; vi (abbreviated as j=A ) i v() = 1;  is a logical truth of OL (j=OL) i for any algebraic realization A = hB ; vi, j=A . When j=A , we will also say that A is a model of ; A will be called a model of a set of formulas T (j=A T ) i A is a model of any 2 T . DEFINITION 4 (Consequence in a realization and logical consequence). Let T be a set of formulas and let A = hB ; vi be a realization. A formula  is a consequence in A of T (T j=A ) i for any element a of B : if for any 2 T , a v v( ) then a v v(). A formula  is a logical consequence of T (T j=OL ) i for any algebraic realization A: T j=A . Instead of fg j=OL we will write  j=OL . If T is nite and equal to f1 ; : : : ; n g, we will obviously have: T j=OL i v(1 ) u


u v(n ) v v(). One can easily check that j=OL i for any T , T j=OL. OL can be equivalently characterized also by means of a Kripke-style semantics, which has been rst proposed by [Dishkant, 1972]. As is well known, the algebraic semantic approach can be described as founded on the following intuitive idea: interpreting a language essentially means associating to any sentence  an abstract truth-value or, more generally, an abstract meaning (an element of an algebraic structure). In the Kripkean semantics, instead, one assumes that interpreting a language essentially means associating to any sentence  the set of the possible worlds or situations where  holds. This set, which represents the extensional meaning of , is called the proposition associated to  (or simply the proposition of ). Hence, generally, a Kripkean realization for a logic L will have the form: D E K = I ; R!i ; ! oj ;  ;  ;

where (i) I is a non-empty set of possible worlds possibly correlated by relations !i and operations in the sequence o!j . In most cases, in the sequence R we have only one binary relation R, called accessibility relation. (ii)  is a set of sets of possible worlds, representing possible propositions of sentences. Any proposition and the total set of propositions  must satisfy convenient closure conditions that depend on the particular logic. (iii)  transforms sentences into propositions preserving the logical form.

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The Kripkean realizations that turn out to be adequate for OL have only one accessibility relation, which is re exive and symmetric. As is well known, many logics, that are stronger than positive logic , are instead characterized by Kripkean realizations where the accessibility relation is at least re exive and transitive. As an example, let us think of intuitionistic logic . From an intuitive point of view, one can easily understand the reason why semantic models with a re exive and symmetric accessibility relation may be physically signi cant. In fact, physical theories are not generally concerned with possible evolutions of states of knowledge with respect to a constant world, but rather with sets of physical situations that may be similar , where states of knowledge must single out some invariants . And similarity relations are re exive and symmetric, but generally not transitive. Let us now introduce the basic concepts of a Kripkean semantics for OL. DEFINITION 5 (Orthoframe). An orthoframe is a relational structure F = hI; R i, where I is a non-empty set (called the set of worlds ) and R (the accessibility relation ) is a binary re exive and symmetric relation on I . Given an orthoframe, we will use i; j; k; : : : as variables ranging over the set of worlds. Instead of Rij (not Rij ) we will also write i ? = j (i ? j ). DEFINITION 6 (Orthocomplement in an orthoframe). Let F = hI; R i be an orthoframe. For any set of worlds X  I , the orthocomplement X 0 of X is de ned as follows: X 0 = fi j 8j (j 2 X y j ? i)g : In other words, X is the set of all worlds that are unaccessible to all elements of X . Instead of i 2 X 0 , we will also write i ? X (and we will read it as \i is orthogonal to the set X "). Instead of i 2= X 0 , we will also write i? = X. DEFINITION 7 (Proposition). Let F = hI; R i be an orthoframe. A set of worlds X is called a proposition of F i it satis es the following condition:

8i [i 2 X i 8j (i ? =jyj? = X )] : In other words, a proposition is a set of worlds X that contains all and only the worlds whose accessible worlds are not unaccessible to X . Notice that the conditional i 2 X y 8j (i ? =jyj? = X ) trivially holds for any set of worlds X . Our de nition of proposition represents a quite general notion of \possible meaning of a formula", that can be signi cantly extended also to other logics. Suppose for instance, a Kripkean frame F = hI; R i, where the accessibility relation is at least re exive and transitive (as happens in the Kripkean semantics for intuitionistic logic). Then a set of worlds X turns out to be a proposition (in the sense of De nition 7) i it is R-closed (i.e.,

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8ij (i 2 X and Rij y j 2 X )). And R-closed sets of worlds represent precisely the possible meanings of formulas in the Kripkean characterization of intuitionistic logic. LEMMA 8. Let F be an orthoframe and X a set of worlds of F . (8.1) (8.2)

X is a proposition of F i 8i [i 2= X y 9j (i ? = j and j ? X )] 00 X is a proposition of F i X = X .

LEMMA 9. Let F = hI; R i be an orthoframe. (9.1) (9.2) (9.3)

I and ; are propositions. If X is any set of worlds, then X 0 is a proposition. T If C is a family of propositions, then C is a proposition.

DEFINITION 10 (Kripkean realization for OL). A Kripkean realization for OL is a system K = hI; R ;  ; i, where: (i) hI; Ri is an orthoframe and  is a set of propositions of the frame that contains ;; I and is closed under the orthocomplement 0 and the set-theoretic intersection \; (ii)  is a function that associates to any formula  a proposition in , satisfying the following conditions: (: ) = ( )0 ; ( ^ ) = ( ) \ ( ).

Instead of i 2 (), we will also write i j=  (or, i j=K , in case of possible confusions) and we will read: \ is true in the world i". If T is a set of formulas, i j= T will mean i j= for any 2 T . THEOREM 11. For any Kripkean realization K and any formula :

i j=  i 8j ? = i 9k ? = j (k j= ): Proof. Since the accessibility relation is symmetric, the left to right implication is trivial. Let us prove i j==  y not8j ?= i 9k ?= j (k j= ), which is equivalent to i 2= () y 9j ?= i 8k ?= j (k 2= ()). Suppose i 2= (). Since () is a proposition, by Lemma 8.1 there holds for a certain j : j ?= i and j ? (). Let k ?= j , and suppose, by contradiction, k 2 (). Since j ? (), there follows j ? k, against k ? = j . Consequently, 9j ? = i 8k ? = j (k 2= ()).  LEMMA 12. In any Kripkean realization K: (12.1)

i j= : i 8j ? = i (j j= = );

QUANTUM LOGICS

(12.2)

141

i j= ^ i i j= and i j= .

DEFINITION 13 (Truth and logical truth). A formula  is true in a realization K = hI; R ;  ; i (abbreviated j=K ) i () = I ;  is a logical truth of OL (j=OL) i for any realization K, j=K . When j=K , we will also say that K is a model of . Similarly in the case of a set of formulas T . DEFINITION 14 (Consequence in a realization and logical consequence). Let T be a set of formulas and let K be a realization. A formula  is a consequence in K of T (T j=K ) i for any world i of K, i j= T y i j= . A formula  is a logical consequence of T (T j=OL) i for any realization K: T j=K . When no confusion is possible we will simply write T j= . Now we will prove that the algebraic and the Kripkean semantics for OL characterize the same logic. Let us abbreviate the metalogical expressions \ is a logical truth of OL according to the algebraic semantics", \ is a logical consequence in OL of T according to the algebraic semantics", \ is a logical truth of OL according to the Kripkean semantics", \ is a logical consequence in OL of T according to the Kripkean semantics", by j=A  ; T OL j=A  , OL j=K  ; T OL j=K , respectively. OL A

K

THEOREM 15. OL j=  i OL j= ; for any . The Theorem is an immediate corollary of the following Lemma: LEMMA 16.

For any algebraic realization A there exists a Kripkean realization KA such that for any , j=A  i j=KA . (16.2) For any Kripkean realization K there exists an algebraic realization AK such that for any , j=K  i j=AK .

(16.1)

Sketch of the proof (16.1) The basic intuitive idea of the proof is the following: any algebraic realization can be canonically transformed into a Kripkean realization by identifying the set of worlds with the set of all non-null elements of the algebra, the accessibility-relation with the non-orthogonality relation in the algebra, and nally the set of propositions with the set of all principal quasi-ideals (i.e., the principal ideals, devoided of the zero-element). More precisely, given A = hB ; vi, the Kripkean realization KA = hI; R ;  ; i is de ned as follows: I = fb 2 B j b 6= 0g; Rij i i 6v j 0 ;  = ffb 2 B j b 6= 0 and b v ag j a 2 B g;

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(p) = fb 2 I j b v v(p)g. One can easily check that KA is a \good" Kripkean realization; further, there holds, for any  : () = fb 2 B j b 6= 0 and b v v()g. Consequently, j=A  i j=KA . (16.2) Any Kripkean realization K = hI; R ;  ; i can be canonically transformed into an algebraic realization AK = hB ; vi by putting: B = ; for any a; b 2 B : a v b i a  b; a0 = fi 2 I j i ? ag; 1 = I ; 0 = ;; v(p) = (p). It turns out that B is an ortholattice. Further, for any , v() = (). Consequently: j=K  i j=AK .  THEOREM 17.

A

K

T OL j=  i T OL j= .

Proof. In order to prove the left to right implication, suppose by conA tradiction: T jOL =  and T OL j==K. Hence there exists a Kripkean realization K = hI; R ;  ; i and a world i of K such that i j= T and i j== . One can easily see that K can be transformed into KÆ = hI; R ; Æ ; i where Æ is the smallest subset of the power-set of I , that includes  and is closed under in nitary intersection. Owing to Lemma 9.3, KÆ is a \good" Kripkean realization for OL and for any , ( ) turns out to be the same proposition in K and in KÆ . Consequently, also in KÆ , there holds: i j= T and i j= = . Æ K Let us now consider A . The algebra B of T AKÆ is complete, because Æ is closed under in nitary intersection. Hence, f( )Tj 2 T g is an element of B . Since i j= for any 2 T , we will have i 2 f( ) j 2 T g. Thus there is an element of B , which is less or equal than v( )(= ( )) for any 2 T , but is not less or equal than v()(= ()), because i 2= (). This contradicts the hypothesis T OL j=A . The right to left implication is trivial.  Let us now turn to a semantic characterization of OQL. We will rst recall the de nition of orthomodular lattice. DEFINITION 18 (Orthomodular lattice). An orthomodular lattice is an ortholattice B = hB ; v ;0 ; 1 ; 0i such that for any a; b 2 B : a u (a0 t (a u b)) v b: Orthomodularity clearly represents a weak form of distributivity. LEMMA 19. Let B be an ortholattice. The following conditions are equivalent:

QUANTUM LOGICS

(i) (ii) (iii) (iv)

B is orthomodular. For any a; b 2 B : a v b For any a; b 2 B : a v b For any a; b 2 B : a v b

143

b = a t (a0 u b). i a u (a u b)0 = 0. and a0 u b = 0 y a = b. y

The property considered in (19(iii)) represents a signi cant weakening of the Boolean condition: a v b i a u b0 = 0: DEFINITION 20 (Algebraic realization for OQL). An algebraic realization for OQL is an algebraic realization A = hB; vi for OL, where B is an orthomodular lattice. The de nitions of truth, logical truth and logical consequence in OQL are analogous to the corresponding de nitions of OL. Like OL, also OQL can be characterized by means of a Kripkean semantics. DEFINITION 21 (Kripkean realization for OQL). A Kripkean realization for OQL is a Kripkean realization K = hI; R ;  ; i for OL, where the set of propositions  satis es the orthomodular property : X 6 Y y X \ (X \ Y )0 6= ;: The de nitions of truth, logical truth and logical consequence in OQL are analogous to the corresponding de nitions of OL. Also in the case of OQL one can show: THEOREM 22. OQL j=A  i OQL j=K . The Theorem is an immediate corollary of Lemma 16 and of the following lemma: LEMMA 23. (23.1) If A is orthomodular then KA is orthomodular; (23.2) If K is orthomodular then AK is orthomodular.

Proof. (23.1) We have to prove X 6 Y y X \ (X \ Y )0 6= ; for any propositions X; Y of KA . Suppose X 6 Y . By de nition of proposition in KA : X = fb j b 6= 0 and b v xg for a given x; Y = fb j b 6= 0 and b v yg for a given y; Consequently, x 6v y, and by Lemma 19: x u (x u y)0 6= 0 , because A is orthomodular. Hence, x u (x u y)0 is a world in KA . In order to prove

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X \ (X \ Y )0 6= ;, it is suÆcient to prove x u (x u y)0 2 X \ (X \ Y )0 . There holds trivially x u (x u y)0 2 X . Further, x u (x u y)0 2 (X \ Y )0 , because (x u y)0 is the generator of the quasi-ideal (X \ Y )0 . Consequently, x u (x u y)0 2 X \ (X \ Y )0 . (23.2) Let K be orthomodular. Then for any X; Y 2 : X 6 Y y X \ (X \ Y )0 6= ;: One can trivially prove:

X \ (X \ Y )0 6= ; y X 6 Y: Hence, by Lemma 19, the algebra B of AK is orthomodular.



As to the concept of logical consequence, the proof we have given for OL (Theorem 17) cannot be automatically extended to the case of OQL. The critical point is represented by the transformation of K into KÆ whose set of propositions is closed under in nitary intersection: KÆ is trivially a \good" OL-realization; at the same time, it is not granted that KÆ preserves the orthomodular property. One can easily prove: THEOREM 24. T OQL j=K  y T OQL j=A : The inverse relation has been proved by [Minari, 1987]: THEOREM 25. T OQL j=A  y T OQL j=K :

AK and Are there any signi cant structural relations between A and K between K and AKA ? The question admits a very strong answer in the case K of A and KA . A THEOREM 26. A = hB; vi and AK = hB; v i are isomorphic realizations. Sketch of the proof Let us de ne the function : B ! B  in the following way: (a) = fb j b 6= 0 and b v ag for any a 2 B . One can easily check that: (1) is an isomorphism (from B onto B ); (2) v (p) = (v(p)) for any atomic formula p.  K At the same time, in the case of K and KA , there is no natural correspondence between I and . As a consequence, one can prove only the weaker relation: K THEOREM 27. Given K = hI ; R ;  ; i and KA = hI  ; R ;  ;  i, there holds:  () = fX 2  j X  ()g ; for any :

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145

In the class of all Kripkean realizations for QL, the realizations KA (which have been obtained by canonical transformation of an algebraic realization A) present some interesting properties, which are summarized by the following theorem. THEOREM 28. In any KA = hI ; R ;  ; i there is a one-to-one correspondence  between the set of worlds I and the set of propositions  f;g such that: (28.1) i 2 (i); (28.2) i ? = j i (i) 6 (j )0 ; (28.3) 8X 2 : i 2 X i 8k 2 (i)(k 2 X ).

Sketch of the proof Let us take as (i) the quasi-ideal generated by i.



Theorem 28 suggests to isolate, in the class of all K, an interesting subclass of Kripkean realizations, that we will call algebraically adequate . DEFINITION 29. A Kripkean realization K is algebraically adequate i it satis es the conditions of Theorem 28. When restricting to the class of all algebraically adequate Kripkean realizations one can prove: K THEOREM 30. K = hI ; R ;  ; i and KA = hI  ; R ;  ;  i are isomorphic realizations; i.e., there exists a bijective function from I onto I  such that: (30.1) Rij i R (i) (j ), for any i; j 2 I ; (30.2)  = f (X ) j X 2 g, where (X ) := f (i) j i 2 X g; (30.3)  (p) = ((p)), for any atomic formula p.

One can easily show that the class of all algebraically adequate Kripkean realizations determines the same concept of logical consequence that is determined by the larger class of all possible realizations. The Kripkean characterization of QL turns out to have a quite natural physical interpretation. As we have seen in the Introduction, the mathematical formalism of quantum theory (QT) associates to any physical system S a Hilbert space H, while the pure states of S are mathematically represented by unitary vectors of H. Let us now consider an elementary sublanguage LQ of QT, whose atomic formulas represent possible measurement reports (i.e., statements of the form \the value for the observable Q lies in the Borel set
") and suppose LQ closed under the quantum logical connectives. Given a physical system S (whose associated Hilbert space is H), one can de ne a natural Kripkean realization for the language LQ as follows: KS = hI ; R ;  ; i ;

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where:

 I is the set of all pure states of S .  R is the non-orthogonality relation between vectors (in other words, two pure states are accessible i their inner product is dierent from zero).



 is the set of all propositions that is univocally determined by the set of all closed subspaces of H (one can easily check that the set of all unitary vectors of any subspace is a proposition).



For any atomic formula p, (p) is the proposition containing all the pure states that assign to p probability-value 1.

Interestingly enough, the accessibility relation turns out to have the following physical meaning: Rij i j is a pure state into which i can be transformed after the performance of a physical measurement that concern an observable of the system. 3 THE IMPLICATION PROBLEM Dierently from most weak logics, QL gives rise to a critical \implicationproblem". All conditional connectives one can reasonably introduce in QL are, to a certain extent, anomalous; for, they do not share most of the characteristic properties that are satis ed by the positive conditionals (which are governed by a logic that is at least as strong as positive logic ). Just the failure of a well-behaved conditional led some authors to the conclusion that QL cannot be a \real" logic. In spite of these diÆculties, these days one cannot help recognizing that QL admits a set of dierent implicational connectives, even if none of them has a positive behaviour. Let us rst propose a general semantic condition for a logical connective to be classi ed as an implication-connective.  is called an DEFINITION 31. In any semantics, a binary connective ! implication-connective i it satis es at least the two following conditions:   is always true (identity ); (31.1)  !  is true then is true (modus ponens ). (31.2) if  is true and  ! In the particular case of QL, one can easily obtain:  to be an implicationLEMMA 32. A suÆcient condition for a connective ! connective is:  (i) in the algebraic semantics: for any realization A = hA; vi, j=A  ! i v() v v( );

QUANTUM LOGICS

(ii) in the Kripkean semantics: for any realization j=K  ! i ()  ( ).

147

K = hI; R; ; i,

In QL it seems reasonable to assume the suÆcient condition of Lemma 32 as a minimal condition for a connective to be an implication-connective. Suppose we have independently de ned two dierent implication-connectives in the algebraic and in the Kripkean semantics. When shall we admit that they represent the \same logical connective"? A reasonable answer to this question is represented by the following convention: A

DEFINITION 33. Let  be a binary connective de ned in the algebraic K A semantics and  a binary connective de ned in the Kripkean semantics:  K and  represent the same logical connective i the following conditions are satis ed: (33.1) given any A = hB; vi and given the corresponding KA = (33.2)

hI; R; ; i, ( K ) is the quasi-ideal generated by v( A ); given any K = hI; R; ; i and given the corresponding AK = hB ; vi, there holds: v( A ) = ( K ).

We will now consider dierent possible semantic characterizations of an implication-connective in QL. Dierently from classical logic, in QL a material conditional de ned by Philo-law ( ! := : _ ), does not give rise to an implication-connective. For, there are algebraic realizations A = hB; vi such that v(: _ ) = 1, while v() 6v v( ). Further, ortholattices and orthomodular lattices are not, generally, pseudocomplemented lattices: in other words, given a; b 2 B , the maximum c such that a u c v b does not necessarily exist in B . In fact, one can prove [Birkho, 1995] that any pseudocomplemented lattice is distributive. We will rst consider the case of polynomial conditionals , that can be de ned in terms of the connectives ^ ; _ ; :. In the algebraic semantics, the minimal requirement of Lemma 32 restricts the choice only to ve possible candidates [Kalmbach, 1983]. This result follows from the fact that in the orthomodular lattice freely generated by two elements there are only ve polynomial binary operations Æ satisfying the condition a v b i a Æ b = 1. These are our ve candidates: (i) v( !1 ) = v()0 t (v() u v( )). (ii) (iii) (iv)

v( !2 ) = v( ) t (v()0 u v( )0 ). v( !3 ) = (v()0 u v( )) t (v() u v( )) t (v()0 u v( )0 ).

v( !4 ) = (v()0 u v( )) t (v() u v( )) t ((v ()0 t v( )) u v( )0 ).

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v( !5 ) = (v()0 u v( )) t (v()0 u v( )0 ) t (v() u (v()0 t v( ))). The corresponding ve implication-connectives in the Kripkean semantics can be easily obtained. It is not hard to see that for any i (1  i  5), !i represents the same logical connective in both semantics (in the sense of De nition 33). THEOREM 34. The polynomial conditionals !i (1  i  5) are implicationconnectives in OQL; at the same time they are not implication-connectives in OL. (v)

Proof. Since !i represent the same connective in both semantics, it will be suÆcient to refer to the algebraic semantics. As an example, let us prove the theorem for i = 1 (the other cases are similar). First we have to prove v() v v( ) i 1 = v( !1 ) = v()0 t (v() u v( )), which is equivalent to v() v v( ) i v() u (v() u v( ))0 = 0. From Lemma 19, we know that the latter condition holds for any pair of elements of B i B is orthomodular. This proves at the same time that !1 is an implication-connective in OQL, but cannot be an implication-connective in OL.  Interestingly enough, each polynomial conditional !i represents a good weakening of the classical material conditional. In order to show this result, let us rst introduce an important relation that describes a \Boolean mutual behaviour" between elements of an orthomodular lattice. DEFINITION 35 (Compatibility). Two elements a; b of an orthomodular lattice B are compatible i a = (a u b0 ) t (a u b): One can prove that a; b are compatible i the subalgebra of B generated by fa; bg is Boolean. THEOREM 36. For any algebraic realization A = hB; vi and for any ; : v( !i ) = v()0 t v( ) i v() and v( ) are compatible. As previously mentioned, Boolean algebras are pseudocomplemented lattices. Therefore they satisfy the following condition for any a; b; c:

c u a v b i c v a

b;

where: a b := a0 t b. An orthomodular lattice B turns out to be a Boolean algebra i for any algebraic realization A = hB; vi, any i (1  i  5) and any ; the following import-export condition is satis ed:

v( ) u v() v v( ) i v( ) v v( !i ):

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149

In order to single out a unique polynomial conditional, various weakenings of the import-export condition have been proposed. For instance the following condition (which we will call weak import-export ):

v( ) u v() v v( ) i v( ) v v() !i v( ); whenever v() and v( ) are compatible. One can prove [Hardegree, 1975; Mittelstaedt, 1972] that a polynomial conditional !i satis es the weak import-export condition i i = 1. As a consequence, we can conclude that !1 represents, in a sense, the best possible approximation for a material conditional in quantum logic. This connective (often called Sasaki-hook ) was originally proposed by [Mittelstaedt, 1972] and [Finch, 1970], and was further investigated by [Hardegree, 1976] and other authors. In the following, we will usually write ! instead of !1 and we will neglect the other four polynomial conditionals. Some important positive laws that are violated by our quantum logical conditional are the following:

 ! ( ! ); ( ! ( ! )) ! (( ! ) ! ( ! )); ( ! ) ! (( ! ) ! ( ! )); ( ^ ! ) ! ( ! ( ! )); ( ! ( ! )) ! ( ! ( ! )): This somewhat \anomalous" behaviour has suggested that one is dealing with a kind of counterfactual conditional . Such a conjecture seems to be con rmed by some important physical examples. Let us consider again the class of the Kripkean realizations of the sublanguage LQ of QT (whose atomic sentences express measurement reports). And let K S = hI; R; ; i represent a Kripkean realization of our language, which is associated to a physical system S . As [Hardegree, 1975] has shown, in such a case the conditional ! turns out to receive a quite natural counterfactual interpretation (in the sense of Stalnaker). More precisely, one can de ne, for any formula , a partial Stalnaker-function f in the following way:

f : Dom(f ) ! I; where:

Dom(f ) = fi 2 I j i ? = ()g : In other words, f is de ned for all and only the states that are not orthogonal to the proposition of .

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If i 2 Dom(f ), then:

f (i) = j P() i ; where P() is the projection that is uniquely associated with the closed subspace determined by (), and j P() i is the normalized vector determined by P() i. There holds:

i j=  ! i either 8j ? = i(j j= = ) or f (i) j= : From an intuitive point of view, one can say that f (i) represents the \pure state nearest" to i, that veri es , where \nearest" is here de ned in terms of the metrics of the Hilbert space H. By de nition and in virtue of one of the basic postulates of QT (von Neumann's collapse of the wave function ), f (i) turns out to have the following physical meaning: it represents the transformation of state i after the performance of a measurement concerning the physical property expressed by , provided the result was positive. As a consequence, one obtains:  ! is true in a state i i either  is impossible for i or the state into which i has been transformed after a positive -test, veri es . Another interesting characteristic of our connective !, is a weak non monotonic behaviour. In fact, in the algebraic semantics the inequality

v( ! ) v v( ^ ! ) can be violated (a counterexample can be easily obtained in the orthomodular lattice based on IR3 ). As a consequence:

 ! j= =  ^ ! : Polynomial conditionals are not the only signi cant examples of implicationconnectives in QL. In the framework of a Kripkean semantic approach, it seems quite natural to introduce a conditional connective (, that represents a kind of strict implication . Given a Kripkean realization K = hI; R; ; i one would like to require:

i j=  ( i 8j ? = i (j j=  y j j= ): However such a condition does not automatically represent a correct semantic de nition, because it is not granted that ( ( ) is an element of . In order to overcome this diÆculty, let us rst de ne a new operation in the power-set of an orthoframe hI; Ri. DEFINITION 37 (Strict-implication operation ( ( ). Given an orthoframe hI; Ri and X; Y  I :

X

(Y

:= fi j 8j (i ? = j and j 2 X y j 2 Y )g :

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151

If X and Y are sets of worlds in the orthoframe, then X ( Y turns out to be a proposition of the frame. When the set  of K is closed under ( , we will say that K is a realization for a strict-implication language . DEFINITION 38 (Strict implication ((). If K = hI; R; ; i is a realization for a strict-implication language, then

( ( ) := () ( ( ): One can easily check that immediately:

(

is a \good" conditional. There follows

i j=  ( i 8j ? = i (j j=  y j j= ): Another interesting implication that can be de ned in QL is represented by an entailment-connective. DEFINITION 39 (Entailment (). Given K = hI; R; ; i,

(  ) :=

(

I; if ()  ( ); ;; otherwise:

Since I; ; 2 , the de nition is correct. One can trivially check that  is a \good" conditional. Interestingly enough, our strict implication and our entailment represent \good" implications also for OL. The general relations between !; ( and  are described by the following theorem: THEOREM 40. For any realization K for a strict-implication language of OL: j=K (  )  ( ( ):

K for a strict-implication language of OQL: j=K (  )  ( ! ); j=K ( ( )  ( ! ):

For any realization

But the inverse relations do not generally hold! Are the connectives( and  de nable also in the algebraic semantics? The possibility of de ning  is straightforward. DEFINITION 41 (Entailment in the algebraic semantics). Given A = hB; vi,

v(  ) :=

(

1; if v() v v( ); 0; otherwise:

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One can easily check that  represents the same connective in the two semantics. As to (, given A = hB; vi, one would like to require: F v( ( ) = fb 2 B j b 6= 0 and 8c(c 6= 0 and b 6v c0 and c v v() y c v v( ))g: However such a de nition supposes the algebraic completeness of B. Further we can prove that ( represents the same connective in the two semantics only if we restrict our consideration to the class of all algebraically adequate Kripkean realizations. 4 METALOGICAL PROPERTIES AND ANOMALIES Some metalogical distinctions that are not interesting in the case of a number of familiar logics weaker than classical logic turn out to be signi cant for QL (and for non distributive logics in general). We have already de ned (both in the algebraic and in the Kripkean semantics) the concepts of model and of logical consequence . Now we will introduce, in both semantics, the notions of quasi-model , weak consequence and quasi-consequence . Let T be any set of formulas. DEFINITION 42 (Quasi-model). Algebraic semantics A realization A = hB; vi is a quasi-model of T i 9a[a 2 B and a 6= 0 and 8 2 T (a v v( ))].

Kripkean semantics A realization K = hI; R; ; i is a quasi-model of T i 9i(i 2 I and i j= T ).

The following de nitions can be expressed in both semantics. DEFINITION 43 (Realizability and veri ability). T is realizable (Real T ) i it has a quasi-model; T is veri able (Verif T ) i it has a model. DEFINITION 44 (Weak consequence). A formula  is a weak consequence of T (T j ) i any model of T is a model of . DEFINITION 45 (Quasi-consequence). A formula  is a quasi-consequence of T (T j ) i any quasi-model of T is a quasi-model of . One can easily check that the algebraic notions of veri ability, realizability, weak consequence and quasi-consequence turn out to coincide with the corresponding Kripkean notions. In other words, T is Kripke-realizable i T is algebraically realizable. Similarly for the other concepts. In both semantics one can trivially prove the following lemmas. LEMMA 46. Verif T y Real T. LEMMA 47. Real T i for any contradiction ^ : , T j= = ^ : .

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153

LEMMA 48. T j=  y T j ; T j=  y T j . LEMMA 49.  j i : j :. Most familiar logics, that are stronger than positive logic, turn out to satisfy the following metalogical properties, which we will call Herbrand{ Tarski , veri ability and Lindenbaum , respectively.



Herbrand{Tarski T j=  i T



Veri ability Ver T i Real T



Lindenbaum Real T y 9T  [T  T  and Compl T ], where Compl T i 8 [ 2 T or : 2 T ].

j 

i T

j 

The Herbrand{Tarski property represents a semantic version of the deduction theorem. The Lindenbaum property asserts that any semantically non-contradictory set of formulas admits a semantically non-contradictory complete extension. In the algebraic semantics, canonical proofs of these properties essentially use some versions of Stone-theorem, according to which any proper lter F in an algebra B can be extended to a proper complete lter F  (such that 8a(a 2 F  or a0 2 F  )). However, Stonetheorem does not generally hold for non distributive orthomodular lattices! In the case of ortholattices, one can still prove that every proper lter can be extended to an ultra lter (i.e., a maximal lter that does not admit any extension that is a proper lter). However, dierently from Boolean algebras, ultra lters need not be complete. A counterexample to the Herbrand{Tarski property in OL can be obtained using the \non-valid" part of the distributive law. We know that (owing to the failure of distributivity in ortholattices):

 ^ ( _ ) j= = ( ^ ) _ ( ^ ): At the same time

 ^ ( _ ) j ( ^ ) _ ( ^ ); since one can easily calculate that for any realization A = hB; vi the hypothesis v( ^ ( _ )) = 1, v(( ^ ) _ ( ^ )) 6= 1 leads to a contradiction 2 .

2 In OQL a counterexample in two variables can be obtained by using the failure of the contraposition law for !. One has:  ! j= = : ! :. At the same time  ! j : ! :; since for any realization A = hB; vi the hypothesis v( ! ) = 1, implies v() v v( ) and therefore v(: ! :) = v( ) t (v()0 u v( )0 ) = v( ) t v( )0 = 1.

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A counterexample to the veri ability-property is represented by the negation of the a fortiori principle for the quantum logical conditional !:

:= :( ! ( ! )) = :(: _ ( ^ (: _ ( ^ )))): This has an algebraic quasi-model. For instance the realization A = hB; vi, where B is the orthomodular lattice determined by all subspaces of the plane (as shown in Figure 2). There holds: v( ) = v() 6= 0. But one can easily check that cannot have any model, since the hypothesis that v( ) = 1 leads to a contradiction in any algebraic realization of QL. O

v()

o

    j v ( )   jjjj j j  j jjjj jjjjjjj  jj j jjjj  j j j  jjj  jjjj j j  j j  jjjj       /



Figure 2. Quasi-model for The same also represents a counterexample to the Lindenbaum-property. Let us rst prove the following lemma. LEMMA 50. If T is realizable and T  T , where T  is realizable and complete, then T is veri able. Sketch of the proof Let us de ne a realization A = hB; vi such that (i) B = f1; 0g; (ii) ( 1; if T  j= ; v() = 0; otherwise:

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Since T  is realizable and complete, A is a good realization and is trivially a model of T .  Now, one can easily show that violates Lindenbaum. Suppose, by contradiction, that has a realizable and complete extension. Then, by Lemma 50, must have a model, and we already know that this is impossible. The failure of the metalogical properties we have considered represents, in a sense, a relevant \anomaly" of quantum logics. Just these anomalies suggest the following conjecture: the distinction between epistemic logics (characterized by Kripkean models where the accessibility relation is at least re exive and transitive) and similarity logics (characterized by Kripkean models where the accessibility relation is at least re exive and symmetric) seems to represent a highly signi cant dividing line in the class of all logics that are weaker than classical logic. 5 A MODAL INTERPRETATION OF OL AND OQL

QL admits a modal interpretation [Goldblatt, 1974; Dalla Chiara, 1981] which is formally very similar to the modal interpretation of intuitionistic logic. Any modal interpretation of a given non-classical logic turns out to be quite interesting from the intuitive point of view, since it permits us to associate a classical meaning to a given system of non-classical logical constants. As is well known, intuitionistic logic can be translated into the modal system S4. The modal basis that turns out to be adequate for OL is instead the logic B. Such a result is of course not surprising, since both the B-realizations and the OL-realizations are characterized by frames where the accessibility relation is re exive and symmetric. Suppose a modal language LM whose alphabet contains the same sentential literals as QL and the following primitive logical constants: the classical connectives  (not ), f (and ) and the modal operator  (necessarily ). At the same time, the connectives g (or ),  (if ... then ),  (if and only if ), and the modal operator  (possibly ) are supposed de ned in the standard way. The modal logic B is semantically characterized by a class of Kripkean realizations that we will call B-realizations. DEFINITION 51. A B-realization is a system M = hI; R; ; i where: (i) hI; Ri is an orthoframe; (ii)  is a subset of the power-set of I satisfying the following conditions: I; ; 2 ;  is closed under the set-theoretic relative complement , the set-theoretic intersection \ and the modal operation , which is de ned as follows:

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

for any X  I; X := fi j 8j (Rij y j 2 X )g; (iii)  associates to any formula  of LM a proposition in  satisfying the conditions: ( ) = ( ); ( f ) = ( )\( ); ( ) = ( ). Instead of i 2 (), we will write i j= . The de nitions of truth, logical truth and logical consequence for B are analogous to the corresponding de nitions in the Kripkean semantics for QL. Let us now de ne a translation  of the language of QL into the language LB . DEFINITION 52 (Modal translation of OL).

  (p) = p;   (: ) =    ( );   ( ^ ) =  ( ) f  ( ). In other words,  translates any atomic formula as the necessity of the possibility of the same formula; further, the quantum logical negation is interpreted as the necessity of the classical negation, while the quantum logical conjunction is interpreted as the classical conjunction. We will indicate the set f ( ) j 2 T g by  (T ). THEOREM 53. For any  and T of OL: T j=OL i  (T ) j=B  () Theorem 53 is an immediate corollary of the following Lemmas 54 and 55. LEMMA 54. Any OL-realization K = hI; R; ; i can be transformed into a B-realization MK = hI  ; R ;  ;  i such that: I  = I ; R = R; 8i (i j=K  i i j=MK  ()). Sketch of the proof Take  as the smallest subset of the power-set of I that contains (p) for any atomic formula p and that is closed under I; ;; ; \; . Further, take  (p) equal to (p).  LEMMA 55. Any B-realization M = hI; R; ; i can be transformed into a OL-realization KM = hI  ; R;  ;  i such that: I  = I ; R = R; 8i (i j=KM  i i j=M  ()).

Sketch of the proof Take  as the smallest subset of the power-set of I that contains (p) for any atomic formula p and that is closed under I; ;;0 ; \ (where for any set X of worlds, X 0 := fj j not Rij g). Further take  (p) equal to (p). The set  (p) turns out to be a proposition in the orthoframe hI  ; R i, owing to the B-logical truth:   . 

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The translation of OL into B is technically very useful, since it permits us to transfer to OL some nice metalogical properties such as decidability and the nite-model property . Does also OQL admit a modal interpretation? The question has a somewhat trivial answer. It is suÆcient to apply the technique used for OL by referring to a convenient modal system Bo (stronger than B) which is founded on a modal version of the orthomodular principle. Semantically Bo can be characterized by a particular class of realizations. In order to determine this class, let us rst de ne the concept of quantum proposition in a B-realization. DEFINITION 56. Given a B-realization M = hI; R; ; i the set Q of all quantum propositions of M is the smallest subset of the power-set of I which contains (p) for any atomic p and is closed under 0 and \. LEMMA 57. In any B-realization M = hI; R; ; i, there holds Q  . Sketch of the proof The only non-trivial point of the proof is represented by the closure of  under 0 . This holds since one can prove: 8X 2  (X 0 = X ).  LEMMA 58. Given M = hI; R; ; i and KM = hI; R;  ;  i, there holds Q =  . LEMMA 59. Given K = hI; R; ; i and MK = hI; R;  ;  i, there holds   Q . DEFINITION 60. A Bo -realization is a B-realization hI; R; ; i that satis es the orthomodular property: 8X; Y 2 Q : X 6 Y y X \ (X \ Y )0 6= ;: We will also call the Bo-realizations orthomodular realizations . THEOREM 61. For any T and  of OQL: T j=OL i  (T ) j=Bo (). The Theorem is an immediate corollary of Lemmas 54, 55 and of the following Lemma: LEMMA 62. (62.1) If K is orthomodular then MK is orthomodular. (62.2) If M is orthomodular then KM is orthomodular. Unfortunately, our modal interpretation of OQL is not particularly interesting from a logical point of view. Dierently from the OL-case, Bo does not correspond to a familiar modal system with well-behaved metalogical properties. A characteristic logical truth of this logic will be a modal version of orthomodularity: f    [ f   ( f )] ;

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where ; are modal translations of formulas of OQL into the language LM. 6 AN AXIOMATIZATION OF OL AND OQL

QL is an axiomatizable logic. Many axiomatizations are known: both in the Hilbert-Bernays style and in the Gentzen-style (natural deduction and sequent-calculi ).3 We will present here a QL-calculus (in the natural deduction style) which is a slight modi cation of a calculus proposed by [Goldblatt, 1974]. The advantage of this axiomatization is represented by the fact that it is formally very close to the algebraic de nition of ortholattice; further it is independent of any idea of quantum logical implication. Our calculus (which has no axioms) is determined as a set of rules . Let T1 ; : : : ; Tn be nite or in nite (possibly empty) sets of formulas. Any rule has the form T1 j 1 ; : : : ; Tn j n Tj  ( if 1 has been inferred from T1 ; : : : ; n has been inferred from Tn, then  can be inferred from T ). We will call any T j  a con guration . The con gurations T1 j 1 ; : : : ; Tn j n represent the premisses of the rule, while T j  is the conclusion . As a limit case, we may have a rule, where the set of premisses is empty; in such a case we will speak of an improper rule . Instead of T j;  we will write T j ; instead of ; j , we will write j . Rules of OL

(OL1)

T [ fg j 

(OL2)

T j ; T  [ fg j T [ T j

(OL3)

T [ f ^ g j 

(^-elimination)

(OL4)

T [ f ^ g j

(^-elimination)

(OL5) (OL6)

T j ; T j T j ^ T [ f; g j T [ f ^ g j

(identity) (transitivity)

(^-introduction) (^-introduction)

3 Sequent calculi for dierent forms of quantum logic will be described in Section 17.

QUANTUM LOGICS

(OL8)

fg j ; fg j : : T [ fg j ::

(OL9)

T [ f::g j 

(OL7)

(OL10)

T [ f ^ :g j

(OL11)

fg j f: g j :

159

(absurdity) (weak double negation) (strong double negation) (Duns Scotus) (contraposition)

DEFINITION 63 (Derivation). A derivation of OL is a nite sequence of con gurations T j , where any element of the sequence is either the conclusion of an improper rule or the conclusion of a proper rule whose premisses are previous elements of the sequence. DEFINITION 64 (Derivability). A formula  is derivable from T (T j OL ) i there is a derivation such that the con guration T j  is the last element of the derivation. Instead of fg jOL we will write  j OL . When no confusion is possible, we will write T j  instead of T jOL . DEFINITION 65 (Logical theorem). A formula  is a logical theorem of OL ( j OL ) i ; jOL . One can easily prove the following syntactical lemmas. LEMMA 66. 1 ; : : : ; n j  i 1 ^


^ n j . LEMMA 67. Syntactical compactness. T j  i 9T   T (T  is nite and T  j ). LEMMA 68. T j  i 91 ; : : : ; n (1 2 T and : : : and n 2 T and 1 ^


^ n j ). DEFINITION 69 (Consistency). T is an inconsistent set of formulas if 9 (T j  ^ :); T is consistent , otherwise. DEFINITION 70 (Deductive closure). The deductive closure T of a set of formulas T is the smallest set which includes the set f j T j g. T is called deductively closed i T = T . DEFINITION 71 (Syntactical compatibility). Two sets of formulas T1 and T2 are called syntactically compatible i

8 (T1 j  y T2 j= :): The following theorem represents a kind of \weak Lindenbaum theorem".

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THEOREM 72. Weak Lindenbaum theorem. If T j= :, then there exists a set of formulas T  such that T  is compatible with T and T  j . Proof. Suppose T j= :. Take T  = fg. There holds trivially: T  j . Let us prove the compatibility between T and T  . Suppose, by contradiction, T and T  incompatible. Then, for a certain , T  j and T j : . Hence (by de nition of T ),  j and by contraposition, : j :. Consequently, because T j : , one obtains by transitivity: T j :, against our hypothesis.  We will now prove a soundness and a completeness theorem with respect to the Kripkean semantics. THEOREM 73. Soundness theorem.

T j  y T j= :



Proof. Straightforward. THEOREM 74. Completeness theorem.

T j=  y T j : Proof. It is suÆcient to construct a canonical model that: T j  i T j=K : As a consequence we will immediately obtain:

K = hI; R; ; i such

T j=  y T j= = K  y T j= = : De nition of the canonical model

(i) I is the set of all consistent and deductively closed sets of formulas; (ii) R is the compatibility relation between sets of formulas; (iii)  is the set of all propositions in the frame hI; Ri; (iv) (p) = fi 2 I

j p 2 ig.

In order to recognize that K is a \good" OL-realization, it is suÆcient to prove that: (a) R is re exive and symmetric; (b) (p) is a proposition in the frame hI; Ri. The proof of (a) is immediate (re exivity depends on the consistency of any i, and symmetry can be shown using the weak double negation rule).

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In order to prove (b), it is suÆcient to show (by Lemma 8.1): i 2= (p) y 9j ?= i (j ? (p)). Let i 2= (p). Then (by de nition of (p)): p 2= i; and, since i is deductively closed, i j= p. Consequently, by the weak Lindenbaum theorem (and by the strong double negation rule), for a certain j : j ? = i and :p 2 j . Hence, j ? (p). LEMMA 75. Lemma of the canonical model. For any  and any i 2 I , i j=  i  2 i.

Sketch of the proof By induction on the length of . The case  = p holds by de nition of (p). The case  = : can be proved by using Lemma 12.1 and the weak Lindenbaum theorem. The case  = ^ can be proved using the ^-introduction and the ^-elimination rules.  Finally we can show that T j  i T j=K . Since the left to right implication is a consequence of the soundness-theorem, it is suÆcient to prove: T j=  y T j= = K . Let T j= ; then, by Duns Scotus, T is consistent. Take i := T . There holds: i 2 I and T  i. As a consequence, by the Lemma of the canonical model, i j= T . At the same time i j= = . For, should i j=  be the case, we would obtain  2 i and by de nition of i, T j , against our hypothesis.  An axiomatization of OQL can be obtained by adding to the OL-calculus the following rule: (OQL)

 ^ :( ^ :( ^ )) j .

(orthomodularity)

All the syntactical de nitions we have considered for OL can be extended to OQL. Also Lemmas 66, 67, 68 and the weak Lindenbaum theorem can be proved exactly in the same way. Since OQL admits a material conditional, we will be able to prove here a deduction theorem : THEOREM 76.  j OQL i j OQL  ! . This version of the deduction-theorem is obviously not in contrast with the failure in QL of the semantical property we have called Herbrand{ Tarski. For, dierently from other logics, here the syntactical relation j does not correspond to the weak consequence relation! The soundness theorem can be easily proved, since in any orthomodular realization K there holds:

 ^ :( ^ :( ^ )) j=K : As to the completeness theorem, we need a slight modi cation of the proof we have given for ` OL. In fact, should we try and construct the canonical model K, by taking  as the set of all possible propositions of the

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frame, we would not be able to prove the orthomodularity of K. In order to obtain an orthomodular canonical model K = fI; R; ; g, it is suÆcient to de ne  as the set of all propositions X of K such that X = () for a certain . One immediately recognizes that (p) 2  and that  is closed under 0 and \. Hence K is a \good" OL-realization. Also for this K one can easily show that i j=  i  2 i. In order to prove the orthomodularity of K, one has to prove for any propositions X; Y 2 , X 6 Y y X \ (X \ Y )0 6= ;; which is equivalent (by Lemma 19) to X \ (X \ (X \ Y )0 )0  Y . By construction of , X = () and Y = ( ) for certain ; . By the orthomodular rule there holds  ^ :( ^ :( ^ )) j . Consequently, for any i 2 I; i j=  ^ :( ^ :( ^ )) y i j= . Hence, () \ (() \ (() \ ( ))0 )0  ( ). Of course, also the canonical model of OL could be constructed by taking  as the set of all propositions that are \meanings" of formulas. Nevertheless, in this case, we would lose the following important information: the canonical model of OL gives rise to an algebraically complete realization (closed under in nitary intersection). 7 THE INTRACTABILITY OF ORTHOMODULARITY As we have seen, the proposition-ortholattice in a Kripkean realization K = hI; R;  ; i does not generally coincide with the (algebraically) complete ortholattice of all propositions of the orthoframe hI; Ri.4 When  is the set of all propositions, K will be called standard . Thus, a standard orthomodular Kripkean realization is a standard realization, where  is orthomodular. In the case of OL, every non standard Kripkean realization can be naturally extended to a standard one (see the proof of Theorem 17). In particular,  can be always embedded into the complete ortholattice of all propositions of the orthoframe at issue. Moreover, as we have learnt from the completeness proof, the canonical model of OL is standard. In the case of OQL, instead, there are variuos reasons that make signi cant the distinction between standard and non standard realizations: (i) Orthomodularity is not elementary [Goldblatt, 1984]. In other words, there is no way to express the orthomodular property of the ortholattice  in an orthoframe hI; Ri as an elementary ( rst-order) property. (ii) It is not known whether every orthomodular lattice is embeddable into a complete orthomodular lattice. (iii) It is an open question whether OQL is characterized by the class of all standard orthomodular Kripkean realizations.

4 For the sake of simplicity, we indicate brie y by  the ortholattice h ; v ; 0 ; 1 ; 0i. Similarly, in the case of other structures dealt with in this section.

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(iv) It is not known whether the canonical model of OQL is standard. Try and construct a canonical realization for OQL by taking  as the set of all possible propositions (similarly to the OL-case). Let us call such a realization a pseudo canonical realization . Do we obtain in this way an OQL-realization, satisfying the orthomodular property? In other words, is the pseudo canonical realization a model of OQL? In order to prove that OQL is characterized by the class of all standard Kripkean realizations it would be suÆcient to show that the canonical model belongs to such a class. Should orthomodularity be elementary, then, by a general result proved by Fine, this problem would amount to showing the following statement: there is an elementary condition (or a set thereof) implying the orthomodularity of the standard pseudo canonical realization. Result (i), however, makes this way de nitively unpracticable. Notice that a positive solution to problem (iv) would automatically provide a proof of the full equivalence between the algebraic and the Kripkean consequence relation (T OQL j=A  i T OQL j=K ). If OQL is characterized by a standard canonical model, then we can apply the same argument used in the case of OL, the ortholattice  of the canonical model being orthomodular. By similar reasons, also a positive solution to problem (ii) would provide a direct proof of the same result. For, the orthomodular lattice  of the (not necessarily standard) canonical model of OQL would be embeddable into a complete orthomodular lattice. We will now present Goldblatt's result proving that orthomodularity is not elementarity. Further, we will show how orthomodularity leaves defeated one of the most powerful embedding technique: the MacNeille completion method.

Orthomodularity is not elementary Let us consider a rst-order language L2 with a single predicate denoting a binary relation R. Any frame hI; Ri (where I is a non-empty set and R any binary relation) will represent a classical realization of L2 . DEFINITION 77 (Elementary class). (i) Let be a class of frames. A possible property P of the elements of is called rst-order (or elementary ) i there exists a sentence  of L2 such that for any hI; Ri 2 :

hI; Ri j=  (ii)

i hI; Ri has the property P :

is said to be an elementary class i the property of being in is an elementary property of .

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

is an elementary class i there is a sentence  of L2 such that

Thus,

= fhI; Ri

j hI; Ri j= g : DEFINITION 78 (Elementary substructure). Let hI1 ; R1 i ; hI2 ; R2 i be two frames.

(a) hI1 ; R1 i is a substructure of hI2 ; R2 i i the following conditions are satis ed: (i) I1  I2 ; (ii) R1 = R2 \ (I1  I1 );

(b) hI1 ; R1 i is an elementary substructure of hI2 ; R2 i i the following conditions hold: (i) hI1 ; R1 i is a substructure of hI2 ; R2 i; (ii) For any formula (x1 ; : : : ; xn ) of L2 and any i1 ; : : : ; in of I1 :

hI1 ; R1 i j= [i1 ; : : : in]

i hI2 ; R2 i j= [i1 ; : : : in ]:

In other words, the elements of the \smaller" structure satisfy exactly the same L2 -formulas in both structures. The following Theorem [Bell and Slomson, 1969] provides an useful criterion to check whether a substructure is an elementary substructure. THEOREM 79. Let hI1 ; R1 i be a substructure of hI2 ; R2 i. Then, hI1 ; R1 i is an elementary substructure of hI2 ; R2 i i whenever (x1 ;


; xn ; y) is a formula of L2 (in the free variables x1 ;


; xn ; y) and i1 ;


; in are elements of I1 such that for some j 2 I2 , hI2 ; R2 i j= [i1 ;


; in; j ], then there is some i 2 I1 such that hI2 ; R2 i j= [i1 ;


; in ; i]: Let us now consider a pre-Hilbert space 5 H and let H+ := f 2 H j 6= 0g, where 0 is the null vector. The pair

+  ;=

H ? where 8 ;  2 H+ : ?=  i

is an orthoframe, the inner product of and  is dierent from the null vector 0 (i.e., ( ; ) 6= 0). Let (H) be the ortholattice of all propositions of hH+ ; ? = i, which turns out to be isomorphic to the ortholattice C (H) of all (not necessarily closed) subspaces of H (a proposition is simply a subspace devoided of the null vector). The following deep Theorem, due to Amemiya and Halperin [Varadarajan, 1985] permits

5 A pre-Hilbert space is a vector space over a division ring whose elements are the real or the complex or the quaternionic numbers such that an inner product (which transforms any pair of vectors into an element of the ring) is de ned. Dierently from Hilbert spaces, pre-Hilbert spaces need not be metrically complete.

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us to characterize the class of all Hilbert spaces in the larger class of all pre-Hilbert spaces, by means of the orthomodular property. THEOREM 80 (Amemiya{Halperin Theorem). C (H) is orthomodular i H is a Hilbert space. In other words, C (H) is orthomodular i H is metrically complete. As is well known [Bell and Slomson, 1969], the property of \being metrically complete" is not elementary. On this basis, it will be highly expected that also the orthomodular property is not elementary. The key-lemma in Goldblatt's proof is the following: LEMMA 81. Let Y be an in nite-dimensional (not necessarily closed) subspace of a separable Hilbert space H. If  is any formula of L2 and 1 ;


; n are vectors of Y such that for some  2 H, hH+ ; ? = i j= [ 1 ;


; = i j= [ 1 ;


; n ; ]. n ; ], then there exists a vector 2 Y such that hH+ ; ? As a consequence one obtains: THEOREM 82. The orthomodular property is not elementary.

Proof. Let H be any metrically incomplete pre-Hilbert space. Let H be its metric completion. Thus H is an in nite-dimensional subspace of the Hilbert space H. By DLemma E81 and by Theorem 79, hH+ ; ? = i is an elemen+ tary substructure of H ; ? = . At the same time, by Amemiya{Halperin's Theorem, C (H) cannot be orthomodular, because H is metrically incomplete. However, C (H) is orthomodular. As a consequence, orthomodularity cannot be expressed as an elementary property.  The embeddability problem As we have seen in Section 2, the class of all propositions of an orthoframe is a complete ortholattice. Conversely, the representation theorem for ortholattices states that every ortholattice B = hB; v ; 0 ; 1 ; 0i is embeddable into the complete ortholattice of all propositions of the orthoframe hB + ; ? = i, where: B + := B f0g and 8a; b 2 B : a ? = b i a 6v b0 . The embedding is given by the map h : a 7! ha ]; where ha ] is the quasi-ideal generated by a. In other words: ha ] = fb 6= 0 j b v ag. One can prove the following Theorem: THEOREM 83. Let B = hB; v ; 0 ; 1 ; 0i be an ortholattice. 8X  B , X is a proposition of hB + ; ? = i i X = l(u(X )), where: 







u(Y ) := b 2 B + j 8a 2 Y : a v b and l(Y ) := b 2 B + j 8a 2 Y : b v a :

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

Accordingly, the complete ortholattice of all propositions of the orthoframe

hB + ; ?= i is isomorphic to the MacNeille completion (or completion by cuts ) of B [Kalmbach, 1983].6 At the same time, orthomodularity (similarly to

distributivity and modularity) is not preserved by the MacNeille completion, as the following example shows [Kalmbach, 1983]. 0 (IR) be the class of all continuous complex-valued functions f on Let C(2) IR such that Z +1 j f (x) j2 dx < 1 1 0 (IR)  C 0 (IR) ! Cj Let us de ne the following bilinear form (: ; :) : C(2) (2) (representing an inner product): Z +1 (f; g) = f  (x)g(x)dx; 1  0 (IR), where f (x) is the complex conjugate of f (x). It turns out that C(2) equipped with the inner product (: ; :), gives rise to a metrically incomplete in nite-dimensional pre-Hilbert space. Thus, by Amemiya{Halperin's 0 (IR)) Theorem (Theorem 80), the algebraically complete ortholattice C (C(2) 0 (IR) cannot be orthomodular. Now consider the of all subspaces of C(2) 0 sublattice FI of C (C(2) (IR)), consisting of all nite or co nite dimensional subspaces. It is not hard to see that FI is orthomodular. One can prove 0 (IR)) is sup-dense in FI ; in other words, any X 2 C (C 0 (IR)) is that C (C(2) (2) the sup of a set of elements of FI . Thus, by a theorem proved by McLaren 0 (IR)) is isomorphic to [Kalmbach, 1983], the MacNeille completion of C (C(2) 0 (IR)) is algebraically complete, the MacNeille completion of FI . Since C (C(2) 0 0 (IR)) itself. the MacNeille completion of C (C(2)(IR)) is isomorphic to C (C(2) As a consequence, FI is orthomodular, while its MacNeille completion is not. 8 HILBERT QUANTUM LOGIC AND THE ORTHOMODULAR LAW As we have seen, the prototypical models of OQL that are interesting from the physical point of view are based on the class H of all Hilbert lattices, whose support is the set C (H) of all closed subspaces of a Hilbert space H. Let us call Hilbert quantum logic (HQL) the logic that is semantically characterized by H . A question naturally arises: do OQL and HQL represent one and the same logic? As proved by [Greechie, 1981],7 this question 6 The MacNeille completion of an ortholattice B = hB; v ; 0 ; 1 ; 0i is the lattice whose support consists of all X  B such that X = l(u(X )), where: u(Y ) := fb 2 B j 8a 2 Y : a v bg and l(Y ) := fb 2 B j 8 a 2+Y :b v ag. Clearly the only dierence between the proposition-lattice of the frame B ; =? and the Mac Neille completion of B is due to the fact that propositions do not contain 0. 7 See also [Kalmbach, 1983].

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has a negative answer: there is a lattice-theoretical equation (the so-called orthoarguesian law ) that holds in H , but fails in a particular orthomodular lattice. As a consequence, OQL does not represent a faithful logical abstraction from its quantum theoretical origin. DEFINITION 84. Let be a class of orthomodular lattices. We say that OQL is characterized by i for any T and any  the following condition is satis ed:

T j=OQL i for any B 2 and any A = hB; vi : T j=A : In order to formulate the orthoarguesian law in an equational way, let us rst introduce the notion of Sasaki projection . DEFINITION 85 (The Sasaki projection). Let B be an orthomodular lattice and let a; b be any two elements of B . The Sasaki projection of a onto b, denoted by a e b, is de ned as follows: a e b := (a t b0 ) u b: It is easy to see that two elements a; b of an orthomodular lattice are compatible (a = (a u b0 ) t (a u b)) i a e b = a u b. Consequently, in any Boolean lattice, e coincides with u. DEFINITION 86 (The orthoarguesian law).

a v b t f(a e b0 ) u [(a e c0 ) t ((b t c) u ((a e b0) t (a e c0 )))]g

(OAL)

Greechie has proved that (OAL) holds in H but fails in a particular nite orthomodular lattice. In order to understand Greechie's counterexample, it will be expedient to illustrate the notion of Greechie diagram . Let us rst recall the de nition of atom . DEFINITION 87 (Atom). Let B = hB; v; 1; 0i any bounded lattice. An atom is an element a 2 B f0g such that:

8b 2 B : 0 v b v a

y

b = 0 or a = b:

Greechie diagrams are hypergraphs that permits us to represent particular orthomodular lattices. The representation is essentially based on the fact that a nite Boolean algebra is completely determined by its atoms. A Greechie diagram of an orthomodular lattice B consists of points and lines. Points are in one-to-one correspondence with the atoms of B; lines are in oneto-one correspondence with the maximal Boolean subalgebras8 of B. Two lines are crossing in a common atom. For example, the Greechie diagram pictured in Figure 3. represents the orthomodular lattice G12 (Figure 4).

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

a

Æ ??

?? ?? ??

b

e

Æ ??

?? ?? ??

c

d

Æ

    

Æ

    

Æ

Figure 3. The Greechie diagram of G12

1

a0 a

o  ?OO ooo ??O?OOOO o o ?? OOO o  ?? OOO ooo   ooo o 0 0 0  ?O?OOOb   ?? ocoo  ?O?OOOd   ?? oeo0oO  ?? OOO oo?o? ?? OOO oo?o? ?? ooOoOO?? Oo ?  ??o o ? ?oo OO ? O  ooo ?  OO? ? oooo ? OOO?O?  O o   oOOOOb  ?? c  o d oeoo o OOO ??  ooooo OOO ? o OOO??   ooo OO?   oo

0

Figure 4. The orthomodular lattice G12

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a

n c i

Æ??  ???   ??  ?  o Æ Æ Æ ??r ? ??   l ???   ?? ??     ?? ??     Æ Æ Æ Æ Æ ? ?  g ??? m ???  k   ?? ??   ?? ??     Æ  Æ Æ ?? Æ  f ????  h  ??   Æ

s b

Æe

Figure 5. The Greechie diagram of B30 Let us now consider a particular nite orthomodular lattice, called B30 , whose Greechie diagram is pictured in Figure 3. THEOREM 88. (OAL) fails in B30 . Proof. There holds: a e b0 = (a t b) u b0 = s0 u b0 = e; a e c0 = (a t c) u c0 = n0 u c0 = i and b t c = l0 . Thus, b t f(a e b0 ) u [(a e c0 ) t ((b t c) u ((a e b0 ) t (a e c0 )))]g = b t fe u [i t (l0 u (e t i))]g = b t fe u [i t (l0 u g0)]g = b t (e u (i t 0)) = b t (e u i) =b 6w a:

 Hence, there are two formulas  and (whose valuations in a convenient realization represent the left- and right- hand side of (OAL), respectively) such that  j==OQL . At the same time, for any C (H) 2 H and for any realization A = hC (H); vi, there holds:  j=A . As a consequence, OQL is not characterized by H . Accordingly, HQL is de nitely stronger than OQL. We are faced with the problem of nding

8 A maximal Boolean subalgebra of an ortholattice B is a Boolean subalgebra of B, that is not a proper subalgebra of any Boolean subalgebra of B.

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out a calculus, if any, that turns out to be sound and complete with respect to H . The main question is whether the class of all formulas valid in H is recursively enumerable. In order to solve this problem, it would be suÆcient (but not necessary) to show that the canonical model of HQL is isomorphic to the subdirect product of a class of Hilbert lattices. So far, very little is known about this question.

Lattice characterization of Hilbert lattices As mentioned in the Introduction, the algebraic structure of the set E of all events in an event-state system hE ; S i is usually assumed to be a complete orthomodular lattice. Hilbert lattices, however, satisfy further important structural properties. It will be expedient to recall rst some standard lattice theoretical de nitions. Let B = hB; v; 1; 0i be any bounded lattice. DEFINITION 89 (Atomicity). A bounded lattice B is atomic i 8a 2 B f0g there exists an atom b such that b v a. DEFINITION 90 (Covering property). Let a; b be two elements of a lattice B. We say that b covers a i a v b ; a 6= b, and 8c 2 B : a v c v b y a = c or b = c. A lattice B satis es the covering property i 8a; b 2 B : a covers a u b y a t b covers b. DEFINITION 91 (Irreducibility). Let B be an orthomodular lattice. B is said to be irreducible i fa 2 B j 8b 2 B : a is compatible with bg = f0; 1g. One can prove the following theorem: THEOREM 92. Any Hilbert lattice is a complete, irreducible, atomic orthomodular lattice, which satis es the covering property. Are these conditions suÆcient for a lattice B to be isomorphic to (or embeddable into) a Hilbert lattice? In other words, is it possible to capture lattice-theoretically the structure of Hilbert lattices? An important result along these lines is represented by the so-called Piron{McLaren's coordinatization theorem [Varadarajan, 1985]. THEOREM 93 (Piron{McLaren coordinatization theorem). Any orthomodular lattice B (of length9 at least 4) that is complete, irreducible, atomic with the covering property, is isomorphic to the orthomodular lattice of all (: ; :)closed subspaces of a Hilbertian space hV ; ; (: ; :); Di.10 9 The length of a lattice B is the supremum over the numbers of elements of all the chains of B, minus 1. 10 A Hilbertian space is a 4-tuple hV ; ; (: ; :); Di, where V is a vector space over a division ring D,  is an involutive antiautomorphism on D, and (: ; :) (to be interpreted

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Do the properties of the coordinatized lattice B restrict the choice to one j and therefore of the real, the complex or the quaternionic numbers ( Q) to a classical Hilbert space? Quite unexpectedly, [Keller, 1980] proved a negative result: there are lattices that satisfy all the conditions of PironMcLaren's Theorem; at the same time, they are coordinatized by Hilbertian spaces over non-archimedean division rings. Keller's counterexamples have been interpreted by some authors as showing the de nitive impossibility for the quantum logical approach to capture the Hilbert space mathematics. This impossibility was supposed to demonstrate the failure of the quantum logic approach in reaching its main goal: the \bottom-top" reconstruction of Hilbert lattices. Interestingly enough, such a negative conclusion has been recently contradicted by an important result proved by Soler [1995]: Hilbert lattices can be characterized in a lattice-theoretical way. Soler's result is essentially based on the following Theorem: THEOREM 94. Let hV ; ; (: ; :); Di be an in nite-dimensional Hilbertian space over a division ring D. Suppose our space includes a k-orthogonal set f i gi2IN , i.e., a family of vectors of V such that 8i : ( i ; i ) = k and 8i; j (i 6= j ) : ( i ; j ) = 0. Then hV ; ; (: ; :); Di is a classical Hilbert space. As a consequence, the existence of k-orthogonal sets characterizes Hilbert spaces in the class of all Hilbertian spaces. The point is that the existence of such sets admits of a purely lattice-theoretic characterization, by means of the so-called angle bisecting condition [Morash, 1973]. Accordingly, every lattice which satis es the angle bisecting condition (in addition to the usual conditions of Piron{McLaren's Theorem) is isomorphic to a classical Hilbert lattice. 9 FIRST-ORDER QUANTUM LOGIC The most signi cant logical and metalogical peculiarities of QL arise at the sentential level. At the same time the extension of sentential QL to a rstorder logic seems to be quite natural. Similarly to the case of sentential QL, we will characterize rst-order QL both by means of an algebraic and a Kripkean semantics. Suppose a standard rst-order language with predicates Pmn and individual constants am .11 The primitive logical constants are the connectives :; ^ and the universal quanti er 8. The concepts of term , formula and sentence as an inner product) is a de nite symmetric -bilinear form on V . Let X be any subset of V and let X 0 := f 2 V j 8 2 X; ( ; ) = 0g; X is called (: ; :)-closed i X = X 00 . The following condition is required to hold: for any (: ; :)-closed set X of V ; V = X + X 0 := f +  : 2 X;  2 X 0 g. If D is either IR or Cj or Qj and the antiautomorphism  is continuous, then hV ; ; (: ; :); Di turns out to be a classical Hilbert space. 11 For the sake of simplicity, we do not assume functional symbols.

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are de ned in the usual way. We will use x; y; z; x1 ;


; xn ;


as metavariables ranging over the individual variables, and t; t1 ; t2 ;


as metavariables ranging over terms. The existential quanti er 9 is supposed de ned by a generalized de Morgan law:

9x := :8x:: DEFINITION 95 (Algebraic realization for rst-order OL). An algebraic

realization for ( rst-order) OL is a system A = BC ; D ; v where:

 (i) BC = B C ; v ; 0 ; 1 ; 0 is anF ortholattice closed under in nitary in mum ( ) and supremum ( ) for any F  B C such that F 2 C (C being a particular family of subsets of B C ). F

(ii) D is a non-empty set (disjoint from B ) called the domain of A. (iii) v is the valuation -function satisfying the following conditions:

 

for any constant am : v(am ) 2 D; for any predicate Pmn , v(Pmn ) is an n-ary attribute in A, i.e., a function that associates to any n-tuple hd1 ;


; dn i of elements of D an element (truth-value ) of B c ; for any interpretation  of the variables in the domain D (i.e., for any function from the set of all variables into D) the pair hv; i (abbreviated by v and called generalized valuation ) associates to any term an element in D and to any formula a truth-value in B c , according to the conditions: v (am ) = v(am ) v (x) = (x) v (Pmn t1 ;


; tn ) = v(Pmn )(v (t1 );


; v (tn )) v (: ) = v ( )0 v ( ^ ) = v ( ) u v ( ) v (8x ) = v=d ] ( ) j d 2 D ; where  =d] v ( ) j d 2 D 2 C [ x= d ] ( is the interpretation that associates to x the individual d and diers from  at most in the value attributed to x). F

DEFINITION 96 (Truth and logical truth). A formula  is true in A =

C  B ; D; v (abbreviated as j=A ) i for any interpretation of the variables , v () = 1;  is a logical truth of OL (j=OL) i for any A, j=A . DEFINITION 97 (Consequence in a realization and logical consequence).

 Let A = BC ; D; v be a realization. A formula  is a consequence of T in A (abbreviated T j=A  ) i for any element a of B c and any interpretation

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: if for any 2 T , a v v ( ), then a v v ();  is a logical consequence of T (T j=OL ) i for any realization A: T j=A . DEFINITION 98 (Kripkean realization for ( rst-order) OL). A Kripkean

realization for ( rst-order) OL is a system K = I ; R ; C ; U ;  where:



(i) I ; R ; C satis es the same conditions as in the sentential case; further C is closed under in nitary intersection for any F  c such that F 2 C (where C is a particular family of subsets of C ); (ii) U , called the domain of K, is a non-empty set, disjoint from the set of worlds I . The elements of U are individual concepts u such that for any world i: u(i) is an individual (called the reference of u in the world i). An individual concept u is called rigid i for any pairs of worlds i, j : u(i) = u(j ). The set Ui = fu(i) j u 2 U g represents the domain of individuals in the world i . Whenever Ui = Uj for all i,j we will say that the realization K has a constant domain . (iii)  associates a meaning to any individual constant am and to any predicate Pmn according to the following conditions:

(am ) is an individual concept in U . (Pmn ) is a predicate-concept , i.e. a function that associates to any n-tuple of individual concepts hu1 ;


; un i a proposition in C ; (iv) for any interpretation of the variables  in the domain U , the pair h ; i (abbreviated as  and called valuation ) associates to any term t an individual concept in U and to any formula a proposition in C according to the conditions:

 (x) = (x)  (am ) = (am )  (Pmn t1 ;


; tn ) = (Pmn )( (t1 );


;  (tn ))  (: ) =  ( )0  ( ^ ) =  ( ) \  ( ) T  (8x ) = =u ] ( ) j u 2 U ; where =u] ( ) j u 2 U 2 C . For any world i and any interpretation  of the variables, the triplet h ; i ; i (abbreviated as i ) will be called a world-valuation . DEFINITION 99 (Satisfaction). i j=  (i satis es ) i i 2  (). DEFINITION 100 (Veri cation). i j=  (i veri es ) i for any : i j= .

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DEFINITION 101 (Truth and logical truth). j=K  ( is true in K) i for any i: i j= ; j=OL ( is a logical truth of OL) i for any K: j=K . DEFINITION 102 (Consequence in a realization and logical consequence). T j=K  i for any i of K and any : i j= T y i j= ; T j=OL i for any realization K: T j=K . The algebraic and the Kripkean characterization for rst-order OQL can be obtained, in the obvious way, by requiring that any realization be orthomodular. In both semantics for rst-order QL one can prove a coincidence lemma:



 LEMMA 103. Given A = BC ; D ; v and K = I ; R ; C ; U;  : (103.1) If  and  coincide in the values attributed to the variables oc  curring in a term t, then v (t) = v (t);  (t) =  (t). (103.2) If  and  coincide in the values attributed  to the free variables  occurring in a formula , then v () = v ();  () =  (). One can easily prove, like in the sentential case, the following lemma: LEMMA 104. (104.1) For any algebraic realization A there exists a Kripkean realization KA such that for any : j=A  i j=KA . Further, if A is orthomodular then KA is orthomodular. (104.2) For any Kripkean realization K, there exists an algebraic realization AK such that for any for any : j=K  i j=AK . Further, if K is orthomodular then AK is orthomodular. An axiomatization of rst-order OL (OQL) can be obtained by adding to the rules of our OL (OQL)-sentential calculus the following new rules: (PR1) (PR2)

T [ f8xg j (x=t), where (x=t) indicates a legitimate substitution). Tj  (provided x is not free in T ). T j 8x

All the basic syntactical notions are de ned in the expected way. One can prove that any consistent set of sentences T admits of a consistent inductive extension T , such that T  j 8x(t) whenever for any closed term t, T  j (t). The \weak Lindenbaum theorem" can be strengthened as follows: for any sentence , if T j= : then there exists a consistent and

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inductive T  such that:

T is syntactically compatible with T  and T  j :12 One can prove a soundness and a completeness theorem of our calculus with respect to the Kripkean semantics. THEOREM 105. Soundness. T j  y T j= :



Proof. Straightforward. THEOREM 106. Completeness. T j=  y T j :

Sketch of the proof Like case, it is suÆcient to construct

in the sentential  a canonical model K = I ; R ; C ; U ;  such that T j  i T j=K . De nition of the canonical model

(i) I is the set of all consistent, deductively closed and inductive sets of sentences expressed in a common language LK , which is an extension of the original language; (ii) R is determined like in the sentential case; (iii) U is a set of rigid individual concepts that is naturally determined by the set of all individual constants of the extended language LK . For any constant c of LK , let uc be the corresponding individual concept in U . We require: for any world i, uc (i) = c. In other words, the reference of the individual concept uc is in any world the constant c. We will indicate by cu the constant corresponding to u. (iv)

(am ) = uam ; (Pmn )(uc11 ; : : : ; ucnn ) = fi j Pmn c1 ; : : : ; cn 2 ig : Our  is well de ned since one can prove for any sentence  of LK : i j=  y 9j ? =i :j

j ::

As a consequence,  (Pmn t1 ; : : : ; tn ) is a possible proposition.

(v) C is the set of all \meanings" offormulas (i.e., X 2 C i 9 9(X =  ()); C is the set of all sets =u]( ) j u 2 U for any formula .

12 By De nition 71, T is syntactically compatible with T  i there is no formula  such that T j  and T  j :.

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One can easily check that domain.

K

is a \good" realization with a constant

LEMMA 107. Lemma of the canonical model. For any , any i 2 I and any :

i j=  i  2 i; where  is the sentence obtained by substituting in  any free variable x with the constant c(x) corresponding to the individual concept (x). Sketch of the proof. By induction on the length of . The cases  = Pmn t1 ;


; tn ,  = : ,  = ^ are proved by an obvious transformation of the sentential argument. Let us consider the case  = 8x and suppose x occurring in (otherwise the proof is trivial). In order to prove the left to right implication, suppose i j= 8x . Then, for any u in U , =u] j= (x). Hence, by inductive hypothesis, 8u 2 U , [ (x)]=u] 2 i. In other words, for any constant cu of i: [ (x)] (x=cu ) 2 i. And, since i is inductive and deductively closed: 8x (x) 2 i. In order to prove the right to left implication, suppose [8x (x)] 2 i. Then, [by (PR1)], for any constantcc of i: [ (x=c)] 2 i. Hence by inductive hypothesis: for any uc 2 U , i [x=u ] j= (x), i.e., i j= 8x (x). On this ground, similarly to the sentential case, one can prove T j  i T j=K . 

First-order QL can be easily extended (in a standard way) to a rst-order logic with identity. However, a critical problem is represented by the possibility of developing, within this logic, a satisfactory theory of descriptions . The main diÆculty can be sketched as follows. A natural condition to be required in any characterization of a -operator is obviously the following:

9x f (x) ^ 8y [( (y) ^ x = y) _ (: (y) ^ :x = y)] ^ (x)g is true y (x (x)) is true:

However, in QL, the truth of the antecedent of our implication does not generally guarantee the existence of a particular individual such that x can be regarded as a name for such an individual. As a counterexample, let us consider the following case (in the algebraic semantics): let A be hB ; D ; vi where B is the complete orthomodular lattice based on the set of all closed subspaces of the plane IR2 , and D contains exactly two individuals d1 ; d2 . Let P be a monadic predicate and X; Y two orthogonal unidimensional subspaces of B such that v(P )(d1 ) = X , v(P )(d2 ) = Y . If the equality predicate = is interpreted as the standard identity relation (i.e., v (t1 = t2 ) = 1, if v (t1 ) = v (t2 ); 0, otherwise), one can easily calculate:

v (9x [P x ^ 8y((P y ^ x = y) _ (:P y ^ :x = y))]) = 1:

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However, for both individuals d1 ; d2 of the domain, we have:

v=d1 ] (P x) 6= 1; v=d2 ] (P x) 6= 1: In other words, there is no precise individual in the domain that satis es the property expressed by our predicate P ! 10 QUANTUM SET THEORIES AND THEORIES OF QUASISETS An important application of QL to set theory has been developed by [Takeuti, 1981]. We will sketch here only the fundamental idea of this application. Let L be a standard set-theoretical language. One can construct ortho-valued models for L, which are formally very similar to the usual Boolean-valued models for standard set-theory, with the following dierence: the set of truth-values is supposed to have the algebraic structure of a complete orthomodular lattice, instead of a complete Boolean algebra. Let B be a complete orthomodular lattice, and let  , ,... represent ordinal numbers. An ortho-valued (set-theoretical ) universe V is constructed as follows: S V B =  2On V ( ) , where: V (0) = ;.

V ( +1) = fg j g is a function and Dom(g)  V ( ) and Rang(g)  B g. S

V () = < V ( ), for any limit-ordinal . ( Dom(g) and Rang(g) are the domain and the range of function g, respectively). Given an orthovalued universe V B one can de ne for any formula of L the truth-value [ ]  in B induced by any interpretation  of the variables in the universe V B . F





[ x 2 y]  = g2Dom((y)) (y)(g) u [ x = z ] =g]  [ x = y]  = g2Dom((x)) (x)(g) [ z 2 y] =g] u  [ z 2 x] =g] . g2Dom((y))  (y )(g ) F

F

where is the quantum logical conditional operation (a b := a0 t (a u b), for any a; b 2 B ). A formula  is called true in the universe V B (j=V B ) i [ ]  = 1, for any . Interestingly enough, the segment V (!) of V B turns out to contain some important mathematical objects, that we can call quantum-logical natural numbers .

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The standard axioms of set-theory hold in B only in a restricted form. An extremely interesting property of V B is connected with the notion of identity. Dierently from the case of Boolean-valued models, the identity relation in V B turns out to be non-Leibnizian. For, one can choose an orthomodular lattice B such that:

6j=V B x = y ! 8z (x 2 z $ y 2 z ): According to our semantic de nitions, the relation = represents a kind of \extensional equality". As a consequence, one may conclude that two quantum-sets that are extensionally equal do not necessarily share all the same properties. Such a failure of the Leibniz-substitutivity principle in quantum set theory might perhaps nd interesting applications in the eld of intensional logics. A completely dierent approach is followed in the framework of the theories of quasisets (or quasets ). The basic aim of these theories is to provide a mathematical description for collections of microobjects, which seem to violate some characteristic properties of the classical identity relation. In some of his general writings, Schrodinger discussed the inconsistency between the classical concept of physical object (conceived as an individual entity) and the behaviour of particles in quantum mechanics. Quantum particles { he noticed { lack individuality and the concept of identity cannot be applied to them, similarly to the case of classical objects. One of the aims of the theories of quasisets (proposed by [da Costa et al., 1992]) is to describe formally the following idea defended by Schrodinger: identity is generally not de ned for microobjects. As a consequence, one cannot even assert that an \electron is identical with itself". In the realm of microobjects only an indistinguishability relation (an equivalence relation that may violate the substitutivity principle) makes sense. On this basis, dierent formal systems have been proposed. Generally, these systems represent convenient generalizations of a Zermelo{Fraenkel like set theory with urelements . Dierently from the classical case, an urelement may be either a macro or a microobject . Collections are represented by quasisets and classical sets turn out to be limit cases of quasisets. A somewhat dierent approach has been followed in the theory of quasets (proposed in [Dalla Chiara and Toraldo di Francia, 1993]). The starting point is based on the following observation: physical kinds and compound systems in QM seem to share some features that are characteristic of intensional entities. Further, the relation between intensions and extensions turns out to behave quite dierently from the classical semantic situations. Generally, one cannot say that a quantum intensional notion uniquely determines a corresponding extension. For instance, take the notion of electron , whose intension is well de ned by the following physical property: mass = 9:1  10 28 g, electron charge = 4:8  10 10e.s.u., spin

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= 1=2. Does this property determine a corresponding set , whose elements should be all and only the physical objects that satisfy our property at a certain time interval? The answer is negative. In fact, physicists have the possibility of recognizing, by theoretical or experimental means, whether a given physical system is an electron system or not. If yes, they can also enumerate all the quantum states available within it. But they can do so in a number of dierent ways. For example, take the spin. One can choose the x-axis and state how many electrons have spin up and how many have spin down. However, we could instead refer to the z -axis or any other direction, obtaining dierent collections of quantum states, all having the same cardinality. This seems to suggest that microobject systems present an irreducibly intensional behaviour: generally they do not determine precise extensions and are not determined thereby. Accordingly, a basic feature of the theory is a strong violation of the extensionality principle. Quasets are convenient generalizations of classical sets, where both the extensionality axiom and Leibniz' principle of indiscernibles are violated. Generally a quaset has only a cardinal but not an ordinal number, since it cannot be well ordered. 11 THE UNSHARP APPROACHES The unsharp approaches to QT ( rst proposed by [Ludwig, 1983] and further developed by Kraus, Davies, Mittelstaedt, Busch, Lahti, Bugajski, Beltrametti, Cattaneo and many others) have been suggested by some deep criticism of the standard logico-algebraic approach. Orthodox quantum logic (based on Birkho and von Neumann's proposal) turns out to be at the same time a total and a sharp logic. It is total because the meaningful propositions are represented as closed under the basic logical operations: the conjunction (disjunction) of two meaningful propositions is a meaningful proposition. Further, it is also sharp, because propositions, in the standard interpretation, correspond to exact possible properties of the physical system under investigation. These properties express the fact that \the value of a given observable lies in a certain exact Borel set". As we have seen, the set of the physical properties, that may hold for a quantum system, is mathematically represented by the set of all closed subspaces of the Hilbert space associated to our system. Instead of closed subspaces, one can equivalently refer to the set of all projections , that is in one-to-one correspondence with the set of all closed subspaces. Such a correspondence leads to a collapse of dierent semantic notions, which Foulis and Randall described as the \metaphysical disaster" of orthodox QT. The collapse involves the notions of \experimental proposition", \physical property", \physical event" (which represent empirical and intensional concepts), and the notion of proposition as a set of states (which corresponds

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to a typical extensional notion according to the tradition of standard semantics). Both the total and the sharp character of QL have been put in question in dierent contexts. One of the basic ideas of the unsharp approaches is a \liberalization" of the mathematical counterpart for the intuitive notion of \experimental proposition". Let P be a projection operator in the Hilbert space H, associated to the physical system under investigation. Suppose P describes an experimental proposition and let W be a statistical operator representing a possible state of our system. Then, according to one of the axioms of the theory (the Born rule ), the number Tr(W P ) (the trace of the operator W P ) will represent the probability-value that our system in state W veri es P . This value is also called Born probability . However, projections are not the only operators for which a Born probability can be de ned. Let us consider the class E (H) of all linear bounded operators E such that for any statistical operator W , Tr(W E ) 2 [0; 1]: It turns out that E (H) properly includes the set P (H) of all projections on H. The elements of E (H) represent, in a sense, a \maximal" mathematical representative for the notion of experimental proposition, in agreement with the probabilistic rules of quantum theory. In the framework of the unsharp approach, E (H) has been called the set of all eects .13 An important dierence between projections and proper eects is the following: projections can be associated to sharp propositions having the form \the value for the observable A lies in the exact Borel set
", while eects may represent also fuzzy propositions like \the value of the observable A lies in the fuzzy Borel set ". As a consequence, there are eects E , dierent from j such that no state W can verify E with probability the null projection O, 1. A limit case is represented by the semitransparent eect 21 1I (where 1I is the identity operator), to which any state W assigns probability-value 12 . From the intuitive point of view, one could say that moving to an unsharp approach represents an important step towards a kind of \second degree of fuzziness". In the framework of the sharp approach, any physical event E can be regarded as a kind of \clear" property. Whenever a state W assigns to E a probability value dierent from 1 and 0, one can think that the semantic uncertainty involved in such a situation totally depends on the ambiguity of the state ( rst degree of fuzziness). In other words, even a pure state in QT does not represent a logically complete information , that is able to decide any possible physical event. In the unsharp approaches, instead, one take into account also \genuine ambiguous properties". This second degree of fuzziness may be regarded as depending on the accuracy 13 It is easy to see that an eect E is a projection i E 2 := EE = E . In other words, projections are idempotent eects.

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of the measurement (which tests the property), and also on the accuracy involved in the operational de nition for the physical quantities which our property refers to. 12 EFFECT STRUCTURES Dierent algebraic structures can be induced on the class E (H) of all eects. Let us rst recall some de nitions. DEFINITION 108 (Involutive bounded poset (lattice)). An involutive bounded poset (lattice) is a structure B = hB ; v ; 0 ; 1 ; 0i, where hB ; v ; 1 ; 0i is a partially ordered set (lattice) with maximum (1) and minimum (0); 0 is a 1-ary operation on B such that the following conditions are satis ed: (i) a00 = a; (ii) a v b y b0 v a0 . DEFINITION 109 (Orthoposet). An orthoposet is an involutive bounded poset that satis es the non contradiction principle: a u a0 = 0: DEFINITION 110 (Orthomodular poset). An orthomodular poset is an orthoposet that is closed under the orthogonal sup (a v b0 y a t b exists) and satis es the orthomodular property: a v b y 9c such that a v c0 and b = a t c. DEFINITION 111 (Regularity). An involutive bounded poset (lattice) B is regular i a v a0 and b v b0 y a v b0 . Whenever an involutive bounded poset B is a lattice, then B is regular i it satis es the Kleene condition : a u a0 v b t b0 : The set E (H) of all eects can be naturally structured as an involutive bounded poset: E (H) = hE (H) ; v ; 0 ; 1 ; 0i ; where (i) E v F i for any state (statistical operator) W , Tr(W E )  Tr(W E ) (in other words, any state assigns to E a probability-value that is less or equal than the probability-value assigned to F ); j projection, respectively; (ii) 1, 0 are the identity (1I) and the null (O) (iii) E 0 = 1 E .

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One can easily check that v is a partial order, 0 is an order-reversing involution, while 1 and 0 are respectively the maximum and the minimum with respect to v. At the same time this poset fails to be a lattice. Dierently from projections, some pairs of eects have no in mum and no supremum as the following example shows [Greechie and Gudder, n.d.]: EXAMPLE 112. Let us consider the following eects (in the matrix-representation) on the Hilbert space IR2 : 

1 E = 02 01 2





3 F = 04 01 4





1 G = 02 01 4



It is not hard to see that G v E; F . Suppose, by contradiction, that L = E u F exists in E (IR2 ). An easy computation shows that L must be equal to G. Let 7 1 16 M = 1 38 8

16

Then M is an eect such that M v E; F ; however, M 6v L, which is a contradiction. In order to obtain a lattice structure, one has to embed E (H) into its MacNeille completion E (H).

The MacNeille completion of an involutive bounded poset Let hB ; v ; 1 ; 0i be an involutive bounded poset. For any non-empty subset X of B , let l(X ) and u(X ) represent respectively the set of all lower bounds and the set of all upper bounds of X . Let MC (B ) := fX  B j X = u(l(X ))g. It turns out that X 2 MC (B ) i X = X 00 , where X 0 := fa 2 B j 8b 2 X : a v b0 g. Moreover, the structure B = hMC (B ) ;  ; 0 ; f0g ; B i is a complete involutive bounded lattice (which is regular if B is regular), where X u Y = X \ Y and X t Y = (X [ Y )00 . It turns out that B is embeddable into B, via the map h : a ! ha], where ha] is the principal ideal generated by a. Such an embedding preserves the in mum and the supremum , when existing in B. The Mac Neille completion of an involutive bounded poset does not generally satis es the non contradiction principle (a u a0 = 0 ) and the excluded middle principle (a t a0 = 1 ). As a consequence, dierently from the projection case, the Mac Neille completion of E (H) is not an ortholattice. Apparently, our operation 0 turns out to behave as a fuzzy negation , both in the case of E (H) and of its Mac Neille completion. This is one of the reasons why proper eects (that are not projections) may be regarded as

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183

representing unsharp physical properties , possibly violating the non contradiction principle. The eect poset E (H) can be naturally extended to a richer structure, equipped with a new complement  , that has an intuitionistic-like behaviour: E  is the projection operator PKer(E) whose range is the kernel Ker(E ) of E , consisting of all vectors that are transformed by the operator E into the null vector. By de nition, the intuitionistic complement of an eect is always a projection. In the particular case, where E is a projection, it turns out that: E 0 = E  . In other words, the fuzzy and the intuitionistic complement collapse into one and the same operation. The structure hE (H) ; v ; 0 ;  ; 1 ; 0i turns out to be a particular example of a Brouwer Zadeh poset [Cattaneo and Nistico, 1986]. DEFINITION 113. A Brouwer{Zadeh poset (simply a BZ-poset ) is a structure hB ; v ; 0 ;  ; 1 ; 0i, where (113.1) hB ; v ; 0 ; 1 ; 0i is a regular involutive bounded poset; (113.2)  is a 1-ary operation on B , which behaves like an intuitionistic complement: (i) a u a = 0. (ii) a v a. (iii) a v b y b v a . (113.3) The following relation connects the fuzzy and the intuitionistic complement: a0 = a . DEFINITION 114. A Brouwer Zadeh lattice is a BZ-poset that is also a lattice.

The Mac Neille completion of a BZ-poset Let B = hB ; v ; 0 ;  ; 1 ; 0i be a BZ-poset and let B the Mac Neille completion of the regular involutive bounded poset hB ; v ; 0 ; 1 ; 0i. For any non-empty subset X of B , let X  := fa 2 B j 8b 2 X : a v b g : It turns out that B = hMC (B );  ; 0 ;  ; f0g ; B i is a complete BZlattice [Giuntini, 1991], which B can be embedded into, via the map h de ned above.

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

Another interesting way of structuring the set of all eects can be obtained by using a particular kind of partial structure, that has been called effect algebra [Foulis and Bennett, 1994] or unsharp orthoalgebra [Dalla Chiara and Giuntini, 1994]. Abstract eect algebras are de ned as follows: DEFINITION 115. An eect algebra is a partial structure A = hA ;
; 1 ; 0i where
is a partial binary operation on A. When
is de ned for a pair a ; b 2 A, we will write 9 (a
b). The following conditions hold: (i) Weak commutativity 9(a
b) y 9(b
a) and a
b = b
a. (ii) Weak associativity [9(b
c) and 9(a
(b
c))] y [9(a
b) and 9((a
b)
c) and a
(b
c) = (a
b)
c]. (iii) Strong excluded middle For any a, there exists a unique x such that a
x = 1. (iv) Weak consistency 9(a
1) y a = 0. From an intuitive point of view, our operation
can be regarded as an exclusive disjunction (aut ), which is de ned only for pairs of logically incompatible events. An orthogonality relation ?, a partial order relation v and a generalized complement 0 can be de ned in any eect algebra. DEFINITION 116. Let A = hA ;
; 1 ; 0i be an eect algebra and let a; b 2 A. (i) a ? b i a
b is de ned in A. (ii) a v b i 9c 2 A such that a ? c and b = a
c. (iii) The generalized complement of a is the unique element a0 such that a
a0 = 1 (the de nition is justi ed by the strong excluded middle condition). The category of all eect algebras turns out to be (categorically) equivalent to the category of all dierence posets , which have been rst studied in [K^opka and Chovanec, 1994] and further investigated in [Dvurecenskij and Pulmannova, 1994]. Eect algebras that satisfy the non contradiction principle are called orthoalgebras : DEFINITION 117. An orthoalgebra is an eect algebra B = hB ;
; 1 ; 0i such that the following condition is satis ed:

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185

Strong consistency 9 (a
a) y a = 0. In other words: 0 is the only element that is orthogonal to itself.

In order to induce the structure of an eect algebra on E (H), it is suÆcient to de ne a partial sum
as follows:

9 (E
F ) i E + F

2 E (H);

where + is the usual sum-operator. Further:

9 (E
F ) y E
F = E + F: It turns out that the structure hE (H) ;
; 1I ; Oj i is an eect algebra, where the generalized complement of any eect E is just 1I E . At the same time, this structure fails to be an orthoalgebra. Any abstract eect algebra

A = hA ;
; 1 ; 0i can be naturally extended to a kind of total structure, that has been termed quantum MV-algebra (abbreviated as QMV-algebra) [Giuntini, 1996]. Before introducing QMV-algebras, it will be expedient to recall the de nition of MV-algebra. As is well known, MV-algebras (multi-valued algebras ) have been introduced by Chang [1957] in order to provide an algebraic proof of the completeness theorem for Lukasiewicz' in nite-many-valued logic L@ . A \privileged" model of this logic is based on the real interval [0; 1], which gives rise to a particular example of a totally ordered (or linear) MV-algebra. Both MV-algebras and QMV-algebras are total structures having the following form: M = (M ;  ;  ; 1; 0) where: (i) 1 ; 0 represent the certain and the impossible propositions (or alternatively the two extreme truth values); (ii)  is the negation-operation; (iii)



represents a disjunction (or ) which is generally non idempotent (a  a 6= a).

A (generally non idempotent) conjunction (and ) is then de ned via de Morgan law: a b := (a  b) :

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

On this basis, a pair consisting of an idempotent conjunction et (e) and of an idempotent disjunction vel (d) is then de ned: a e b := (a  b ) b

a d b := (a b)  b: In the concrete MV-algebra based on [0; 1], the operations are de ned as follows: (i) 1 = 1; 0 = 0; (ii) a = 1 a; (iii)

 is the truncated sum : ab=

(

a + b; if a + b  1; 1; otherwise:

In this particular case, it turns out that:

a e b = Minfa; bg (a et b is the minimum between a and b): a d b = Maxfa; bg (a vel b is the maximum between a and b): A standard abstract de nition of MV-algebras is the following [Mangani, 1973]: DEFINITION 118. An MV-algebra is a structure M = (M ;  ;  ; 1; 0), where  is a binary operation,  is a unary operation and 0 and 1 are special elements of M , satisfying the following axioms: (MV1) (MV2) (MV3) (MV4) (MV5) (MV6) (MV7) (MV8)

(a  b)  c = a  (b  c) a0=a ab=ba a1=1 (a ) = a 0 = 1 a  a = 1 (a  b)  b = (a  b )  a

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187

In other words, an MV-algebra represents a particular weakening of a Boolean algebra, where  and are generally non idempotent. A partial order relation can be de ned in any MV-algebra in the following way: a  b i a e b = a: Some important properties of MV-algebras are the following: (i) the structure hM ;  ; ; 1 ; 0i is a bounded involutive distributive lattice, where a e b (a d b) is the inf (sup) of a; b; (ii) the non-contradiction principle and the excluded middle principles for  ; e; d are generally violated: a d a 6= 1 and a e a 6= 0 are possible. As a consequence, MV algebras permit us to describe fuzzy and paraconsistent situations; (iii) a  b = 1 i a  b. In other words: similarly to the Boolean case, \not-a or b" represents a good material implication; (iv) every MV-algebra is a subdirect product of totally ordered MV-algebras [Chang, 1958]; (v) an equation holds in the class of all MV-algebras i it holds in the concrete MV-algebra based on [0; 1] [Chang, 1958]. Let us now go back to our eect-structure hE (H) ;
; 1 ; 0i. The partial operation
can be extended to a total operation  that behaves like a truncated sum. For any E; F 2 E (H):

EF = Further, let us put:

(

E + F; if 9(E
F ); 1; otherwise:

E  = 1I E: The structure E (H) = hE (H) ;  ; ; 1 ; 0i turns out to be \very close" to an MV-algebra. However, something is missing: E (H) satis es the rst seven axioms of our de nition (MV1-MV7); at the same time one can easily check that the axiom (MV8) (usually called \Lukasiewicz axiom") is violated. For instance, let us consider two non trivial projections P; Q such that P is not orthogonal to Q and Q is not orthogonal to P  . Then, by de nition of , we have that P  Q = 1I and Q  P  = 1I. Hence: (P   Q)  Q = Q 6= P = (P  Q)  P . As a consequence, Lukasiewicz axiom must be conveniently weakened to obtain a representation for our concrete eect structure. This can be done by means of the notion of QMV-algebra.

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

DEFINITION 119. A quantum MV-algebra (QMV-algebra ) is a structure M = (M ;  ;  ; 1; 0) where  is a binary operation,  is a 1-ary operation, and 0; 1 are special elements of M . For any a; b 2 M : a b := (a  b ) ; a e b := (a  b ) a ; a d b := (a b )  b. The following axioms are required: (QMV1) a  (b  c) = (b  a)  c, (QMV2) a  a = 1, (QMV3) a  0 = a, (QMV4) a  1 = 1, (QMV5) a = a, (QMV6) 0 = 1, (QMV7) a  [(a e b) e (c e a )] = (a  b) e (a  c). The operations e and d of a QMV-algebra M are generally non commutative. As a consequence, they do not represent lattice-operations. It is not diÆcult to prove that a QMV-algebra M is an MV-algebra i for all a; b 2 M : a e b = b e a. At the same time, any QMV-algebra M = (M ;  ;  ; 1; 0) gives rise to an involutive bounded poset hM ;  ;  ; 1 ; 0i, where the partial order relation is de ned like in the MV case. One can easily show that QMV-algebras represent a \good abstraction" from the eect-structures: THEOREM 120. The structure E (H) = hE (H) ;  ; ; 1 ; 0i (where  ; ; 1 ; 0 are the operations and the special elements previously de ned) is a QMValgebra. The QMV-algebra E (H) cannot be linear. For, one can easily check that any linear QMV-algebra collapses into an MV-algebra. In spite of this, our algebra of eects turns out to satisfy some weak forms of linearity. DEFINITION 121. A QMV-algebra M is called weakly linear i 8a; b 2 M : a e b = b or b e a = a. DEFINITION 122. A QMV-algebra M is called quasi-linear i 8a; b 2 M : a e b = a or a e b = b. It is easy to see that every quasi-linear QMV-algebra is weakly linear, but not the other way around (because e is not commutative). A very strong relation connects the class of all eect algebras with the class of all quasi-linear QMV-algebras: every eect algebra can be uniquely transformed into a quasi-linear QMV-algebra and viceversa. Let B = hB ;
; 1 ; 0i be an eect algebra. The operation
can be extended to a total operation


:BB !B

QUANTUM LOGICS

in the following way:

a
b :=

(



189

a
b; if 9(a
b); 1; otherwise: 

The resulting structure B ;
; 0 ; 1 ; 0 will be denoted by Bqmv . Viceversa, let M = (M ;  ;  ; 1; 0) be a QMV-algebra. Then, one can de ne a partial operation  on M such that Dom() := fha; bi 2 M  M j a  b g :

9(a  b) y a  b = a  b: The resulting structure hM ;  ; 1 ; 0i will be denoted by Mea . THEOREM 123. [Gudder, 1995; Giuntini, 1995] Let B = hB ;
; 1 ; 0i be an eect algebra and let M = (M ;  ;  ; 1; 0) be a QMV-algebra.

(i) Bqmv is a quasi-linear QMV-algebra; (ii) Mea is an eect algebra; (iii) (Bqmv )ea = B; (iv) M is quasi-linear i (Mea )qmv = M; (v) Bqmv is the unique quasi-linear QMV-algebra such that
extends
and a  b in Bqmv implies a v b in B.

As a consequence, the eect algebra E (H) of all eects on a Hilbert space H determines a quasi-linear QMV-algebra E (H)qmv = hE (H) ;  ; ; 1 ; 0i, where

EF =

(

E + F; if 9(E
F ); 1; otherwise;

and

E = 1 E = E0: These dierent ways of inducing a structure on the set of all unsharp physical properties have suggested dierent logical abstractions. In the following sections, we will investigate some interesting examples of unsharp quantum logics. 13 PARACONSISTENT QUANTUM LOGIC Paraconsistent quantum logic (PQL) represents the most obvious unsharp weakening of orthologic. In the algebraic semantics, this logic is characterized by the class of all realizations based on an involutive bounded lattice, where the non contradiction principle (a u a0 = 0) is possibly violated.

190

MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

In the Kripkean semantics, instead, PQL is characterized by the class of all realizations hI ; R ;  ; i, where the accessibility relation R is symmetric (but not necessarily re exive), while  behaves like in the OL - case. Any pair hI; Ri, where R is a symmetric relation on I , will be called symmetric frame . Dierently from OL and OQL, a world i of a PQL realization may verify a contradiction. Since R is generally not re exive, it may happen that i 2  () and i ?  (). Hence: i j=  ^ :. All the other semantic de nitions are given like in the case of OL, mutatis mutandis . On this basis, one can show that our algebraic and Kripkean semantics characterize the same logic. An axiomatization of PQL can be obtained by dropping the absurdity rule and the Duns Scotus rule in the OL calculus. Similarly to OL, our logic PQL satis es the nite model property and is consequently decidable. Hilbert-space realizations for PQL can be constructed, in a natural way, both in the algebraic and in the Kripkean semantics. In the algebraic semantics, take the realizations based on the Mac Neille completion of an involutive bounded poset having the form

hE (H) ; v ; 0 ; 1 ; 0i ; where H is any Hilbert space. In the Kripkean semantics, consider the realizations based on the following frames

hE (H) f0g ; 6?i ; where ? = represents the non orthogonality relation between eects (E 6? F i E 6v F 0 ). Dierently from the projection case, here the accessibility relation is symmetric but generally non-re exive. For instance, the semitransparent eect 21 1I (representing the prototypical ambiguous property) is a xed point of the generalized complement 0 ; hence 12 1I ? 21 1I and ( 12 1I)0 ? ( 12 1I)0 . From the physical point of view, possible worlds are here identi ed with possible pieces of information about the physical system under investigation. Any information may be either maximal (a pure state) or non maximal (a mixed state); either sharp (a projection) or unsharp (a proper eect). Violations of the non contradiction principle are determined by unsharp (ambiguous) pieces of knowledge. Interestingly enough, proper mixed states (which cannot be represented as projections) turn out to coincide with particular eects. In other words, within the unsharp approach, it is possible to represent both states and events by a unique kind of mathematical object, an eect. PQL represents a somewhat rough logical abstraction from the class of all eect-realizations. An important condition that holds in all eect realizations is represented by the regularity property (which may fail in a generic PQL-realization).

QUANTUM LOGICS

191

DEFINITION 124. An algebraic PQL realization hB ; v i is called regular i the involutive bounded lattice B is regular (a u a0 v b t b0 ). The regularity property can be naturally formulated also in the framework of the Kripkean semantics: DEFINITION 125. A PQL Kripkean realization hI; R ;  ; i is regular i its frame hI ; R i is regular . In other words, 8i; j 2 I : i ? i and j ? j y i ? j. One can prove that a symmetric frame hI; Ri is regular i the involutive bounded lattice of all propositions of hI; Ri is regular. As a consequence, an algebraic realization is regular i its Kripkean transformation is regular and viceversa (where the Kripkean [algebraic] transformation of an algebraic [Kripkean] realization is de ned like in OL). On this basis one can introduce a proper extension of PQL: regular paraconsistent quantum logic (RPQL). Semantically RPQL is characterized by the class of all regular realizations (both in the algebraic and in the Kripkean semantics). The calculus for RPQL is obtained by adding to the PQL-calculus the following rule:

 ^ : j _ :

(Kleene rule )

A completeness theorem for both PQL and RPQL can be proved, similarly to the case of OL. Both logics PQL and RPQL admit a natural modal translation (similarly to OL). The suitable modal system which PQL can be transformed into is the system KB, semantically characterized by the class of all symmetric frames. A convenient strengthening of KB gives rise to a regular modal system, that is suitable for RPQL. An interesting question concerns the relation between PQL and the orthomodular property. Let B = hA; v ; 0 ; 1 ; 0i be an ortholattice. By Lemma 19 the following three conditions (expressing possible de nitions of the orthomodular property) turn out to be equivalent: (i) 8a; b 2 B : a v b y b = a t (a0 u b); (ii) 8a; b 2 B : a v b and a0 u b = 0 y a = b; (iii) 8a; b 2 B : a u (a0 t (a u b)) v b. However, this equivalence breaks down in the case of involutive bounded lattices. One can prove only: LEMMA 126. Let B be an involutive bounded lattice. If B satis es condition (i), then B satis es conditions (ii) and (iii).

Proof. (i) implies (ii): trivial. Suppose (i); we want to show that (iii) holds. Now, a0 v a0 t b0 = (a u b)0 . Therefore, by (i), (a u b)0 = a0 t (a u (a u b)0 ). By de Morgan law: a u b = (a u (a0 t (a u b)) v b. 

192

MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

1



0

a0 a

o  ?OfO ooo ??O?OOOO o o ?? OOO o  ?? OOO ooo   ooo o 0 0  ?O?OOO   ?b? ooo  ?O?cOOO   ?d?0 oooO  ?? OOO oo?o? ?? OOO oo?o? ?? oooOOO??  o??o?o?ooOOOOO??? ? O  o ?   O o ?  O ?    O? o o ooOoOOO ??b O ?ooco  d oooO ? OOO ??  oo OOO ??  oo OOO?? ooooo OO ooo

e0 e

f 

0 Figure 6.

G14

LEMMA 127. Any involutive bounded lattice B that satis es condition (iii) is an ortholattice.

Proof. Suppose (iii). It is suÆcient to prove that 8a; b 2 B : a u a0 v b. Now, a u a0 v a; a0 . Moreover, a0 v a0 t (a u b). Therefore, by (iii), a u a0 v a u (a0 t (a u b)) v b. Thus, 8a 2 B : a u a0 = 0.  As a consequence, we can conclude that there exists no proper orthomodular paraconsistent quantum logic when orthomodularity is understood in the sense (i) or (iii). A residual possibility for a proper paraconsistent quantum logic to be orthomodular is orthomodularity in the sense (ii). In fact, the lattice G14 (see Figure 6) is an involutive bounded lattice which turns out be orthomodular (ii) but not orthomodular (i). Since f u f 0 = f 6= 0, G14 cannot be an ortholattice. Hence, G14 is neither orthomodular (i) nor orthomodular (iii). However, G14 is trivially orthomodular (ii) since the premiss of condition (ii) is satis ed only in the trivial case where both a; b are either 0 or 1. Hilbert space realizations for orthomodular paraconsistent quantum logic can be constructed in the algebraic semantics by taking as support the following proper subset of the set of all eects:

I (H) := fa1I j a 2 [0; 1]g [ P (H):

QUANTUM LOGICS

193

In other words, a possible meaning of the formula is either a sharp property (projection) or an unsharp property that can be represented as a multiple of the universal property (1I). The set I (H) determines an orthomodular involutive regular bounded lattice, where the partial order is the partial order of E (H) restricted to I (H), while the fuzzy complement is de ned like in the class of all eects (E 0 := 1I E ). An interesting feature of PQL is represented by the fact that this logic turns out to be a common sublogic in a wide class of important logics. In particular, PQL is a sublogic of Girard's linear logic [Girard, 1987], of Lukasiewicz' in nite many-valued logic and of some relevant logics. As we will see in Section 17, PQL represents the most natural quantum logical extension of a quite weak and general logic, that has been called basic logic . 14 THE BROUWER{ZADEH LOGICS The Brouwer{Zadeh logics (called also fuzzy intuitionistic logics ) represent natural abstractions from the class of all BZ-lattices (de ned in Section 12). As a consequence, a characteristic property of these logics is a splitting of the connective \not" into two forms of negation: a fuzzy-like negation, that gives rise to a paraconsistent behaviour and an intuitionistic-like negation. The fuzzy \not" represents a weak negation, that inverts the two extreme truth-values (truth and falsity), satis es the double negation principle but generally violates the non-contradiction principle. The second \not" is a stronger negation, a kind of necessitation of the fuzzy \not". We will consider two forms of Brouwer{Zadeh logic: BZL (weak Brouwer{ Zadeh logic ) and BZL3 (strong Brouwer{Zadeh logic ). The language of both BZL and BZL3 is an extension of the language of QL. The primitive connectives are: the conjunction (^), the fuzzy negation (:), the intuitionistic negation (). Disjunction is metatheoretically de ned in terms of conjunction and of the fuzzy negation:  _ := :(: ^ : ) : A necessity operator is de ned in terms of the intuitionistic and of the fuzzy negation: L := : : A possibility operator is de ned in terms of the necessity operator and of the fuzzy negation: M := :L: : Let us rst consider our weaker logic BZL. Similarly to OL and PQL, also BZL can be characterized by an algebraic and a Kripkean semantics.

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

DEFINITION 128 (Algebraic realization for BZL). An algebraic realization of BZL is a pair hB ; vi, consisting of a BZ-lattice hB ; v ; 0 ; ; 1 ; 0i and a valuation-function v that associates to any formula  an element in B , satisfying the following conditions: (i) v(: ) = v( )0 (ii) v( ) = v( ) (iii) v( ^ ) = v( ) u v( ).

The de nitions of truth, consequence in an algebraic realization for BZL, logical truth and logical consequence are given similarly to the case of OL. A Kripkean semantics for BZL has been rst proposed in [Giuntini, 1991]. A characteristic of this semantics is the use of frames with two accessibility relations. DEFINITION 129. A Kripkean realization for BZL is a system K = hI ; 6? ; 6? ;  ; i where: (i) hI ; 6? ;  6?i is a frame with a non empty set I of possible worlds and two accessibility relations: 6? (the fuzzy accessibility relation) and  6? (the intuitionistic accessibility relation). Two worlds i ; j are called fuzzy-accessible i i 6? j . They are called intuitionistically-accessible i i  6? j . Instead of not(i 6? j ) and not(i  6? j ), we will write i ? j and i ? j , respectively. The following conditions are required for the two accessibility relations: (ia) hI; 6?i is a regular symmetric frame; (ib) any world is fuzzy-accessible to at least one world:

8i 9j : i 6? j : (ic) hI;  6?i is an orthoframe; (id) Fuzzy accessibility implies intuitionistic accessibility:

i 6? j y i  6? j: (ie) Any world i has a kind of \twin-world" j such that for any world k: (a) i  6? k i j 6? k (b) i  6? k y j 6? k. For any set X of worlds, the fuzzy-orthogonal set X 0 is de ned like in OL: X 0 = fi 2 I j 8j 2 X : i ? j g :

QUANTUM LOGICS

195

Similarly, the intuitionistic orthogonal set X  is de ned as follows: X  = fi 2 I j 8j 2 X : i  ? jg : The notion of proposition is de ned like in OL. It turns out that a set of worlds X is a proposition i X = X 00 . One can prove that for any set of worlds X , both X 0 and X  are propositions. Further, like in OL, X u Y (the greatest proposition included in the propositions X and Y ) is X \ Y , while X t Y (the smallest proposition including X and Y ) is (X [ Y )00 . (ii)  is a set of propositions that contains I , and is closed under 0 ; ; u. (iii)  associates to any formula a proposition in  according to the following conditions: (: ) = ( )0 ; ( ) = ( ) ; ( ^ ) = ( ) u ( ). THEOREM 130. Let hI ; 6? ;  6? i be a BZ-frame (i.e. a frame satisfying the conditions of De nition 129(i)) and let 0 be theset of all propositions of the frame. Then, the structure 0 ;  ; 0 ;  ; ; ; I is a complete BZ-lattice such that for any set  0 : [ 00 \ G inf ( ) := = and sup ( ) := = : F

As a consequence, the proposition-structure h ;  ; 0 ;  ; ; ; I i of a BZL realization, turns out to be a BZ-lattice. The de nitions of truth, consequence in a Kripkean realization, logical truth and logical consequence, are given similarly to the case of OL. One can prove, with standard techniques, that the algebraic and the Kripkean semantics for BZL characterize the same logic. We will now introduce a calculus that represents an adequate axiomatization for the logic BZL. The most intuitive way to formulate our calculus is to present it as a modal extension of the axiomatic version of regular paraconsistent quantum logic RPQL. (Recall that the modal operators of BZL are de ned as follows: L := :; M := :L:). Rules of BZL.

The BZL-calculus includes, besides the rules of RPQL the following modal rules:

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

(BZ1)

L j 

(BZ2)

L j LL

(BZ3)

ML j L

(BZ4)

j L j L

(BZ5)

L ^ L j L( ^ )

(BZ6)

; j :(L ^ :L)

The rules (BZ1)-(BZ5) give rise to a S5 {like modal behaviour. The rule (BZ6) (the non-contradiction principle for necessitated formulas) is, of course, trivial in any classical modal system. One can prove a soundness and completeness Theorem with respect to the Kripkean semantics (by an appropriate modi cation of the corresponding proofs for QL). Characteristic logical properties of BZL are the following: (a) like in PQL, the distributive principles, Duns Scotus, the noncontradiction and the excluded middle principles (for the fuzzy negation) break down; (b) like in intuitionistic logic, we have:

j=BZL (^  ); =j=BZL_   ;  j=BZL  ;   =j=BZL ;   j=BZL  ;  j=BZL y  j=BZL  ; (c) moreover, we have:

  j=BZL: ; : =j=BZL  ; :   j=BZL  ; One can prove that BZL has the nite model property; as a consequence it is decidable [Giuntini, 1992].

The ortho-pair semantics Our stronger logic BZL3 has been suggested by a form of fuzzy-intuitionistic semantics, that has been rst studied in [Cattaneo and Nistico, 1986]. The intuitive idea, underlying this semantics (which has some features in common with Klaua's partielle Mengen and with Dunn's polarities ) can be sketched as follows: one supposes that interpreting a language means associating to any sentence two domains of certainty : the domain of the situations

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where our sentence certainly holds, and the domain of the situations where our sentence certainly does not hold. Similarly to Kripkean semantics, the situations we are referring to can be thought of as a kind of possible worlds. However, dierently from the standard Kripkean behaviour, the positive domain of a given sentence does not generally determine the negative domain of the same sentence. As a consequence, propositions are here identi ed with particular pairs of sets of worlds, rather than with particular sets of worlds. Let us again assume the BZL language. We will de ne the notion of realization with positive and negative certainty domains (shortly ortho-pair realization ) for a BZL language. DEFINITION 131. An ortho-pair realization is a system O = hI ; R ; ; vi ; where: (i) hI ; R i is an orthoframe. (ii) Let 0 be the set of all propositions of the orthoframe hI ; Ri. As we already know, this set gives rise to an ortholattice with respect to the operations u; t and 0 (where u is the set-theoretic intersection). An orthopairproposition of hI ; R i is any pair hA1 ; A0 i, where A1 ; A0 are propositions in 0 such that A1  A00 . An orthopairproposition hA1 ; A0 i is called exact i A0 = A01 (in other words, A0 is maximal). The following operations and relations can be de ned on the set of all orthopairpropositions: (iia) The fuzzy complement: hA ; A i 0 := hA ; A i : 1

0

0

1

(iib) The intuitionistic complement: hA1 ; A0 i  := hA0 ; A00 i : (iic) The orthopairpropositional conjunction:

hA1 ; A0 i u hB1 ; B0 i := hA1 u B1 ; A0 t B0 i : (iid) The orthopairpropositional disjunction:

hA1 ; A0 i t hB1 ; B0 i := hA1 t B1 ; A0 u B0 i : (iie) The in nitary conjunction: n n n fhA1 ; A0 ig :=

* \

n

fAn1 g ;

G

n

+

fAn0 g

:

F

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

(iif) The in nitary disjunction: G

n

fhAn1 ; An0 ig :=

*

G

\

n

n

+

fAn1 g ; fAn0 g :

(iig) The necessity operator:

(hA1 ; A0 i) := hA1 ; A01 i : (iih) The possibility operator:

(hA1 ; A0 i) := ((hA1 ; A0 i 0)) 0: (iik) The order-relation:

hA1 ; A0 i v hB1 ; B0 i

i A1  B1 and B0  A0 :  ;u;t (iii) is a set of orthopairpropositions, that is closed under 0; and 0 := h; ; I i : (iv) v is a valuation-function that maps formulas into orthopairpropositions according to the following conditions: v(: ) = v( ) 0; ; v( ) = v( ) v( ^ ) = v( )uv( ). The other basic semantic de nitions are given like in the algebraic semantics. One can prove the following Theorem: THEOREM 132. Let hI ; R i be an orthoframe and let 0 be the set of all  ; h;; I i; orthopairpropositions of hI ; R i. Then, the structure h 0 ;  ; 0 ; hI; ;ii is a complete BZ-lattice with respect to the in nitary conjunction and disjunction de ned above. Further, the following conditions are satis ed: for any hA1 ; A0 i ; hB1 ; B0 i 2 0 : . (i)  hA1 ; A0 i = hA1 ; A0 i 0  = (hA ; A i 0 ). (ii) hA1 ; A0 i 1 0  0. (iii)  hA1 ; A0 i = hA1 ; A0 i (iv) (hA ; A i u hB ; B i)  = hA ; A i  t hB ; B i  : 1

0

1

0

1

0

1

0

(Strong de Morgan law) (v) (hA1 ; A0 i u hB1 ; B0 i  )  (hA1 ; A0 i 0  t hB1 ; B0 i):

Accordingly, in any ortho-pair realization the set of all orthopairpropositions 0 gives rise to a BZ-lattice. As a consequence, one can immediately

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prove a soundness theorem with respect to the ortho-pair semantics. Does perhaps the ortho-pair semantics characterizes the logic BZL? The answer to this question is negative. As a counterexample, let us consider an instance of the fuzzy excluded middle and an instance of the intuitionistic excluded middle applied to the same formula :

 _ : and

 _  :

One can easily check that they are logically equivalent in the ortho-pair semantics. For, given any ortho-pair realization O, there holds::

 _ : j=O _   and  _   j=O  _ : : However, generally

 _ : =j=BZL _   : For instance, let us consider the following algebraic BZL{realization A = hB ; vi, where the support B of is the real interval [0 ; 1] and the algebraic structure on B is de ned as follows: a v b i a  b; a0 = 1 a; ( 1 ; if a = 0; a = 0 ; otherwise: 1 = 1; 0 = 0. Suppose for a given sentential literal p: 0 < v(p) < 1=2. We will have v(p _  p) = Max(v(p) ; 0) = v(p) < 1=2. But v(p _ :p) = Max(v(p) ; 1 v(p)) = 1 v(p)  1=2. Hence: v(p _  p) < v(p _ :p). As a consequence, the orthopair-semantics characterizes a logic stronger than BZL. We will call this logic BZL3 . The name is due to the characteristic three-valued features of the ortho-pair semantics. Our logic BZL3 is axiomatizable. A suitable calculus can be obtained by adding to the BZL-calculus the following rules. Rules of BZL3 .

(BZ3 1)

L( _ ) j L _ L

(BZ3 2)

L j ;  j M j

The following rules turn out to be derivable: L j ; M j M (DR1) j

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

(DR2)

M ^ M j M ( ^ )

(DR3)

 ( ^ ) j  _ 

The validity of a strong de Morgan's principle for the connective  (DR3) shows that this connective represents, in this logic, a kind of strong \superintuitionistic" negation (dierently from BZL, where the strong de Morgan law fails, like in intuitionistic logic). One can prove a soundness and a completeness theorem of our calculus with respect to the ortho-pair semantics. THEOREM 133 (Soundness theorem).

T

j BZL  3

y

T j=BZL3:



Proof. By routine techniques. THEOREM 134. Completeness theorem.

T j=BZL3 y T

j BZL : 3

Sketch of the proof Instead of T j=BZL3 and T j BZL3 , we will shortly write T j=  and T j . It is suÆcient to construct a canonical model O = hI ; R ; ; vi such that: T j=O  y T j  : (The other way around follows from the soundness theorem). De nition of the canonical model

(i) I is the set of all possible sets i of formulas satisfying the following conditions: (ia) i is non contradictory with respect to the fuzzy negation :: for any , if  2 i, then : 62 i; (ib) i is L-closed : for any , if  2 i, then L 2 i; (ic) i is deductively closed : for any , if i j , then  2 i. (ii) The accessibility relation R is de ned as follows: Rij i for any formula :  2 i y : 62 j . (In other words, i and j are not contradictory with respect to the fuzzy negation). Instead of not Rij , we will write i ? j . (iii) is the set of all orthopairpropositions of hI; Ri.

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(iv) For any atomic formula p:

v(p) = hv1 (p) ; v0 (p)i ; where:

v1 (p) = fi j i j pg and v0 (p) = fi j i j

:pg :

O is well de ned since one can prove the following Lemmas:

LEMMA 135. R is re exive and symmetric. LEMMA 136. For any  ; fi j i j g is a proposition of the orthoframe hI ; Ri. LEMMA 137. For any , fi j i j g  fi j i j :g0 . Further, one can prove LEMMA 138. For any , v() = hv1 () ; v0 ()i, where:

v1 () = fi j i j g v0 () = fi j i j :g LEMMA 139. For any formula : 0 := h;; I i = hfi j i j L ^ :Lg ; fi j i j :(L ^ :L)gi. LEMMA 140. Let T = f1 ; : : : ; n ; : : : g be a set of formulas and let  be any formula. T fv1 (n ) j n 2 T g  v1 () y L1 ; : : : ; Ln ; : : : j . As a consequence, one can prove: LEMMA 141. Lemma of the canonical model

T j=O  y T j : Suppose T j=O . Hence (by de nition of consequence in a given realization): for any orthopairproposition hA1 ; A0 i 2 , if for all n 2 T , hA1 ; A0 i v v(n ), then hA1 ; A0 i v v(). The propositional lattice, consisting of all orthopairpropositions of O is complete (see Theorem 132). Hence: n fv(n ) j n 2 T g v v(). In other words, by de nition of v: F

T

(i) fv1 (n ) j n 2 T g  v1 (); F (ii) v0 ()  fv0 (n ) j n 2 T g. Thus, by (i) and by Lemma 140: L1 ; : : : ; Ln ; : : : j . Consequently, there exists a nite subset fn1 ; : : : ; nk g of T such that Ln1 ^ : : : ^ Lnk j . Hence, by the rules for ^ and L: L(n1 ^ : : : ^ nk ) j .

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

At F the same time, we obtain from (ii) and by Lemma 138: v1 (:) v fv1 (:n ) j n 2 T g. Whence, by de Morgan, h\ i0 v1 (:)  f(v1 (:n ))0 j n 2 T g : Now, one can easily check Tthat in any realization: v1 (:)0 = v1 (M). As a consequence: v1 (:)  [ f(v1 (Mn ) j n 2 T g]0 : Hence, by contraposition: \ fv1 (Mn ) j n 2 T g  (v1 (:))0 and \ fv1 (Mn ) j n 2 T g  v1 (M): Consequently, by Lemma 140 and by the S5 -rules:

LM1 ; : : : ; LMn ; : : : j M ;

M1 ; : : : ; Mn ; : : : j M :

By syntactical compactness, there exists a nite subset fm1 ; : : : ; mh g of T such that Mm1 ; : : : ; Mmh j M. Whence, by the rules for ^ and M : M (m1 ^ : : : ^ mh ) j M. Let us put 1 = n1 ^ : : : ^ nk and

2 = m1 ^ : : : ^ mh . We have obtained: L 1 j  and M 2 j M. Whence, L 1 ^ L 2 j , L( 1 ^ 2 ) j , M 1 ^ M 2 j M, M ( 1 ^ 2 ) j M. From L( 1 ^ 2 ) j , and M ( 1 ^ 2 ) j M we obtain, by the derivable rule (DR1):

1 ^ 2 j . Consequently: T j .  Similarly to other forms of quantum logic, also BZL3 admits an algebraic semantic characterization [Giuntini, 1993] based on the notion of BZ3 -lattice. DEFINITION 142. A BZ3 -lattice is a BZ-lattice B = hB ; v ; 0 ; ; 1 ; 0i, which satis es the following conditions: (i) (a u b) = a t b ; 0 (ii) a u b v a  t b. By Theorem 132, the set of all orthopairpropositions of an orthoframe determines a complete BZ3-lattice. One can prove the following representation theorem: THEOREM 143. Every BZ3 -lattice is embeddable into the (complete) BZ3 lattice of all orthopairpropositions of an orthoframe. A slight modi cation of the proof of Theorem 17 permits us to show that ortho-pair semantics and the algebraic semantics strongly characterize the same logic. One can prove that BZL3 can be also characterized by means of a non standard version of Kripkean semantics [Giuntini, 1993]. Some problems concerning the Brouwer-Zadeh logics remain still open:

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1) Is there any Kripkean characterization of the logic that is algebraically characterized by the class of all de Morgan BZ-lattices (i.e. BZ-lattices satisfying condition (i) of De nition 142)? In this framework, the problem can be reformulated in this way: is the (strong) de Morgan law elementary? 2) Is it possible to axiomatize a logic based on an in nite manyvalued generalization of the ortho-pair semantics? 3) Find possible conditional connectives in BZL3 . 4) Find an appropriate orthomodular extension of BZL3 .

Unsharp quantum models for BZL3 The ortho-pair semantics has been suggested by the eect-structures in Hilbert-space QT. In this framework, natural quantum ortho-pair realizations for BZL3 can be constructed. Let us refer again to the language LQ (whose atoms express possible measurement reports) and let S be a quantum system whose associated Hilbert space is H. As usual, E (H) will represent the set of all eects of H. Now, an ortho-pair realization MS = hI ; R ; ; vi (for the system S ) can be de ned as follows: (i) I is the set of all pure states of S in H. (ii) Rij i for any eect E 2 E (H) the following condition holds: whenever i assigns to E probability 1, then j assigns to E a probability dierent from 0. In other words, i and j are accessible i they cannot be strongly distinguished by any physical property represented by an eect. (iii) The propositions of the orthoframe hI; R i are determined by the set of all closed subspaces of H (sharp properties), like in OQL. (iv) is the set of all orthopairpropositions of hI; R i. Any eect E can be transformed into an orthopairproposition f (E ) := hX1E ; X0E i of , where: X1E := fi

j i assigns to E probability 1g ; X0E := fi j i assigns to E probability 0g :

In other words, X1E ; X0E represent the positive and the negative domain of E , respectively. The map f turns out to preserve the order relation and the two complements:

E v F i f (E ) v f (F ):

  0

f (E 0 ) = f (E ) 0 = X1E ; X0E = X0E ; X1E :  E E 0  = X E ; X E  f (E  ) = f (E ) = X ; (X ) : 1

0

0

0

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

(v) The valuation-function v follows the intuitive physical meaning of the atomic sentences. Let p express the assertion \the value for the observable A lies in the sharp (or fuzzy) Borel set
" and let E p be the eect that is associated to p in H. We de ne v as follows: D p pE v(p) = f (E p ) = X1E ; X0E : It is worthwhile to notice that our map f is not injective: dierent eects will be transformed into one and the same orthopairproposition. As a consequence, moving from eects to orthopairpropositions clearly determines a loss of information. In fact, orthopairpropositions are only concerned with the two extreme probability value (0,1), a situation that corresponds to a three-valued semantics. 15 PARTIAL QUANTUM LOGICS In Section 12, we have considered examples of partial algebraic structures, where the basic operations are not always de ned. How to give a semantic characterization for dierent forms of quantum logic, corresponding respectively to the class of all eect algebras, of all orthoalgebras and of all orthomodular posets? We will call these logics: unsharp partial quantum logic (UPaQL), weak partial quantum logic (WPaQL) and strong partial quantum logic (SPaQL). Let us rst consider the case of UPaQL, that represents the \logic of eect algebras" [Dalla Chiara and Giuntini, 1995]. The language of UPaQL consists of a denumerably in nite list of atomic sentences and of two primitive connectives: the negation : and the exclusive disjunction _+ (aut). The set of sentences is de ned in the usual way. A conjunction is metalinguistically de ned, via de Morgan law:

 ^: := :(: _+ : ): The intuitive idea underlying our semantics for UPaQL is the following: disjunctions and conjunctions are always considered \legitimate" from a mere linguistic point of view. However, semantically, a disjunction  _+ will have the intended meaning only in the \well behaved cases" (where the values of  and are orthogonal in the corresponding eect orthoalgebra). Otherwise,  _+ will have any meaning whatsoever (generally not connected with the meanings of  and ). As is well known, a similar semantic \trick" is used in some classical treatments of the description operator  (\the unique individual satisfying a given property"; for instance, \the present king of Italy").

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DEFINITION 144. A realization for UPaQL is a pair A = hB ; vi, where B = hB ;
; 1 ; 0i is an eect algebra (see De nition 115); v (the valuationfunction) associates to any formula  an element of B , satisfying the following conditions: (i) v(: ) = v( )0 , where 0 is the generalized complement (de ned in B). (ii)

v( _+ ) =

(

v( )
v( ); if v( )
v( ) is de ned in B; any element; otherwise.

The other semantic de nitions (truth, consequence in a given realization, logical truth, logical consequence) are given like in the QL-case. Weak partial quantum logic (WPaQL) and strong partial quantum logic (SPaQL) (formalized in the same language as UPaQL) will be naturally characterized mutatis mutandis . It will be suÆcient to replace, in the de nition of realization, the notion of eect algebra with the notion of orthoalgebra and of orthomodular poset (see De nition 117 and De nition 110). Of course, UPaQL is weaker than WPaQL, which is, in turn, weaker than SPaQL. Partial quantum logics are axiomatizable. We will rst present a calculus for UPaQL, which is obtained as a natural transformation of the calculus for orthologic. Dierently from QL, the rules of our calculus will always have the form:

1 j 1 ; : : : ; n j n j In other words, we will consider only inferences from single formulas. Rules of UPaQL

(UPa1)

j 

(UPa2)

j j j

(UPa3)

j

::

(identity)

(transitivity)

(weak double negation)

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

(UPa4)

:: j 

(strong double negation)

(UPa5)

j : j :

(contraposition)

(UPa6)

j  _+ :

(UPa7)

j

(UPa8)

j ity)

(UPa9)

(excluded middle)

:  _ : j  _ : j +

+

(unicity of negation)

:  j 1 1 j  j 1 1 j  _ j 1 _ 1 +

+

 j :  _+ j _+ 

(weak substitutiv-

(weak commutativity)

(UPa10)

j

:  j :( _ )  j :

(weak associativity)

(UPa11)

j

:  j :( _ )  _ j :

(weak associativity)

+

+

+

(UPa12)

j

:  j :( _ )  _ ( _ ) j ( _ ) _

(weak associativity)

(UPa13)

j :  j :( _+ ) ( _+ ) _+ j  _+ ( _+ )

(weak associativity)

+

+

+

+

+

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The concepts of derivation and of derivability are de ned in the expected way. In order to axiomatize weak partial quantum logic (WPaQL) it is suÆcient to add a rule, which corresponds to a Duns Scotus-principle :

 j : j

(WPaQL)

(Duns Scotus)

Clearly, the Duns Scotus-rule corresponds to the strong consistency condition in our de nition of orthoalgebra (see De nition 117). In other words, dierently from UPaQL, the logic WPaQL forbids paraconsistent situations. Finally, an axiomatization of strong partial quantum logic (SPaQL) can be obtained, by adding the following rule to (UPa1)-(UPa13), (WPa):

j

(SPaQL)

:  j j _ j +

In other words, (SPaQL) requires that the disjunction _+ behaves like a supremum, whenever it has the \right meaning". Let PaQL represent any instance of our three calculi. We will use the following abbreviations. Instead of  j PaQL we will write  j . When  and are logically equivalent ( j and j ) we will write   . Let p represent a particular sentential literal of the language: T will be an abbreviation for p _+ :p; while F will be an abbreviation for : (p _+ :p). Some important derivable rules of all calculi are the following: (D1)

Fj ; j T

(D2)

 j :  j  _+

(Weak Duns Scotus) (weak sup rule)

j (orthomodular-like rule)   _+ : ( _+ : )

(D3)

j

(D4)

: j :  _  _  +

+

(cancellation)

As a consequence, the following syntactical lemma holds: LEMMA 145. For any  ; :  j i there exists a formula such that (i)  j : ; (ii)   _+ .

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

In other words, the logical implication behaves similarly to the partial order relation in the eect algebras. The following derivable rule holds for WPaQL and for SPaQL:  j :  j j j  _+ (D5)  _+ j Our calculi turn out to be adequate with respect to the corresponding semantic characterizations. Soundness proofs are straightforward. Let us sketch the proof of the completeness theorem for our weakest calculus (UPaQL). THEOREM 146. Completeness.

 j= y  j : Proof. Following the usual procedure, it is suÆcient to construct a canonical model A = hB ; vi such that for any formulas ; :  j y  j=A :

De nition of the canonical model. (i) The algebra B = hB ;
; 1 ; 0i is determined as follows: (ia) B is the class of all equivalence classes of logically equivalent formulas: B := f[] j  is a formulag. (In the following, we will write [] instead of [] ). (ib) []
[ ] is de ned i  j : . If de ned, []
[ ] := [ _+ ]. (ic) 1 := [T]; 0 := [F]. (ii) The valuation function v is de ned as follows: v() = []. One can easily check that A is a \good" model for our logic. The operation
is well de ned (by the transitivity, contraposition and weak substitutivity rules). Further, B is an eect algebra:
is weakly commutative and weakly associative, because of the corresponding rules of our calculus. The strong excluded middle axiom holds by de nition of
and in virtue of the following rules: excluded middle, unicity of negation, double negation. Finally, the weak consistency axiom holds by weak Duns Scotus (D1) and by de nition of
. LEMMA 147. Lemma of the canonical model

[] v [ ] i  j :

Sketch of the proof. By de nition of has to prove:

v (in any eect algebra) one

 j i for a given such that [] ? [ ] : []
[ ] = [ ]:

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209

This equivalence holds by Lemma 145 and by de nition of
. Finally, let us check that v is a \good" valuation function. In other words: (i) v(: ) = v( )0 (ii) v( _+ ) = v( )
v( ), if v( )
v( ) is de ned.

(i) By de nition of v, we have to show that [: ] is the unique [ ] such that [ ]
[ ] = 1 := [T]. In other words, (ia) [T] v [ ]
[: ]. (ib) [T] v [ ]
[ ] y

:  .

This holds by de nition of the canonical model, by de nition of
and by the following rules: double negation, excluded middle, unicity of negation. (ii) Suppose v( )
v( ) is de ned. Then j : . Hence, by de nition of
and of v: v( )
v( ) = [ ]
[ ] = [ _+ ] = v( _+ ). As a consequence, we obtain:

 j i [] v [ ] i v() v v( ) i  j=A

 The completeness argument can be easily transformed, mutatis mutandis for the case of weak and strong partial quantum logic. 16 LUKASIEWICZ QUANTUM LOGIC As we have seen in Section 12, the class E (H) of all eects on a Hilbert space H determines a quasi-linear QMV-algebra. The theory of QMValgebras suggests, in a natural way, the semantic characterization of a new form of quantum logic (called Lukasiewicz quantum logic (LQL)), which generalizes both OQL and L@ . The language of LQL contains the same primitive connectives as WPaQL ( _+ ; :). The conjunction (^: ) is de ned via de Morgan law (like in WPaQL). Further, a new pair of conjunction ( ^^ ) and disjunction ( __ ) connectives are de ned as follows:

 ^^ := ( _+ : ) ^:  __ := :(: ^^ : ) DEFINITION 148. A realization for LQL is a pair A = hM ; vi, where (i) M = hM ;  ; ; 1 ; 0i is a QMV-algebra.

(ii) v (the valuation-function) associates to any formula  an element of M , satisfying the following conditions:

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

v(: ) = v( ) . v( _+ ) = v( )  v( ). The other semantic de nitions (truth, consequence in a given realization, logical truth, logical consequence) are given like in the QL-case. LQL can be easily axiomatized by means of a calculus that simply mimics the axioms of QMV-algebras. The quasi-linearity property, which is satis ed by the QMV-algebras of eects, is highly non equational. Thus, the following question naturally arises: is LQL characterized by the class of all quasi-linear QMV-algebras (QLQMV)? In the case of L@ , Chang has proved that L@ is characterized by the MV-algebra determined by the real interval [0; 1]. This MV-algebra is clearly quasi-linear, being totally ordered. The relation between LQL and QMV algebras turns out to be much more complicated. In fact one can show that LQL cannot be characterized even by the class of all weakly linear QMV-algebras (WLQMV). Since WLQMV is strictly contained in QLQMV, there follows that LQL is not characterized by QLQMV. To obtain these results, something stronger is proved. In particular, we can show that:



the variety of all QMV-algebras (QMV ) strictly includes the variety generated by the class of all weakly linear QMV-algebras (H SP(WLQMV)).



H SP(WLQMV) strictly includes the variety generated by the class of all quasi-linear QMV-algebras (H SP(QLQMV)).

So far, little is known about the axiomatizability of the logic based on

H SP(QLQMV). In the case of H SP(WLQMV), instead, one can prove that

this variety is generated by the QMV-axioms together with the following axiom: a = (a  c b ) e (a  c b): The problem of the axiomatizability of the logic based on H SP(QLQMV) is complicated by the fact that not every (quasi-linear) QMV-algebra M = hM ;  ; ; 1 ; 0i admits of a \good polynomial conditional", i.e., a polynomial binary operation Æ such that

a Æ b = 1 i a  b: Thus, it might happen that the notion of logical truth of the logic based on H SP(QLQMV) is ( nitely) axiomatizable, while the notion of \logical entailment" ( j= ) is not. We will now show that the QMV-algebra M4 (see Figure 7 below) does not admit any good polynomial conditional. The operations of M4 are

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211

de ned as follows:

0 0 0 0 a a a a b b b b 1 1 1 ` 1

0 a b 1 0 a b 1 0 b a 1 0 a b 1

 0 a b 1 a 1 1 1 b 1 1 1 1 1 1 1



0 a b 1

1 a b 0

1

a

?  ???  ??  ??    ??  ??   ??  ??  

b



0 Figure 7.

M4

Let us consider the three-valued MV-algebra M3 , whose operations are

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

1

 

1 2

 0 Figure 8.

M3

de ned as follows: 0 0 0 1 2 1 2 1 2

1 1 1

0 1 2

1 0 1 2

1 0 1 2

1

 0 1 2

1 1 2

1 1 1 1 1

It is easy to see that the map h :

h(x) :=

 0

1

1

0

1 2

1 2

M4 ! M3 such that 8x 2 M4

8 > <0;

if x = 0; if x = a or x = b; : 1; otherwise 1; >2

is a homomorphism of M4 into M3. Suppose, by contradiction, that M4 admits of a good polynomial conditional !M4 . Since a 6 b, we have h(a !M4 b) 6= 1. Thus, 1 6= h(a !M4 b) = h(a) !M3 h(b) = contradiction.

1 2

!M

3

1 = 1; 2

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17 QUANTUM LOGIC AND THE CUBE OF LOGICS (BY GIULIA BATTILOTTI AND CLAUDIA FAGGIAN)

Dierent forms of quantum logic can be axiomatized as sequent calculi [Dummett, 1976; Nishimura, 1980; Cutland and Gibbins, 1982; Tamura, 1988; Nishimura, 1994]. This permits us to investigate such logics more and more deeply from the proof-theoretical point of view. A sequent calculus for orthologic can be obtained from a calculus for classical logic, by requiring a special restriction on contexts in the rules that would permit to derive the distributive laws. The critical rules are the following: the introduction of disjunction on the left side, the introduction of conjunction on the right side, the rules concerning implication and negation. Such a restriction, however, determines some serious proof-theoretical diÆculties, because quantum logic has a suÆciently strong negation that satis es de Morgan's laws. The shortcoming becomes apparent when we try to prove the cornerstone result, represented by a cut-elimination theorem (which, essentially depends on the formulation of the rules that appear in our proofs). A simple and compact sequent calculus for orthologic [Faggian and Sambin, 1997; Battilotti and Sambin, 1999], which admits cut-elimination by means of a neat procedure, can be obtained by a convenient strengthening of basic logic . This is a new logic that has been introduced in order to investigate a general structure for the space of logics [Sambin et al., 1998]. In the framework of basic logic, constraints on contexts are not considered a limitation; on the contrary, they are regarded as a positive feature, which is called visibility . At the same time, negation is treated by exploiting the symmetry of the calculus: the main idea is to use Girard's linear negation, which can be interpreted as an orthocomplement in a quite natural way. This approach shows that orthologic (and non-distributive logics, in general) admits a proof-theory, which turns out to be simpler than the proof-theory for classical logic. Describing quantum logic in the framework of a uniform and general setting gives many advantages, since it permits us to study various logics and their mutual relations. In particular, we obtain a whole system of quantum logics (including linear orthologic ); and for each of these logics we have a proof of the cut-elimination theorem. All this gives rise to a new formulation of classical logic [Faggian, 1997], with respect to which orthologic and the other quantum-like logics (created by this method) turn out to be characterizable as substructural logics . On this basis it is easy to compare dierent logics, and to prove embedding results [Battilotti, 1998]. Basic logic and the cube of logics. As we already know, quantum logic represents a weakening of classical logic, obtained by dropping the distributive laws. There are at least two other important logics that are weaker than classical logic: intuitionistic logic and linear logic [Girard, 1987]. The situation can be sketched as pictured in Figure 9.

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I

          

C ??

?? ?? ?? ?? ?? ?? ?? ?

QL L

Figure 9. The most important weakenings of classical logic It is natural to ask whether there exists a logic that represents a kind of common denominator for Q, I and L, in the same way as classical logic theoretically includes all the other logics. A solution to this problem has been found in terms of a suitable sequent calculus B, that represents a basic logic . Dierently from the calculi we have considered in the previous sections, a sequent calculus for a given logic L is based on axioms and rules that govern the behaviour of sequents . Any sequent has the form

M

`N

where M; N are (possibly empty) nite multisets of formulas.14 Axioms are particular sequents. Any rule has the form

M1 ` N1 : : : Mn ` Nn M `N where M1 ` N1 ; : : : ; Mn ` Nn are the premisses , while M ` N is the conclusion of the rule. Rules can be structural or operational . Operational rules introduce a new connective, while structural rules deal only with the structure of the sequents (orders, repetitions, etc.). A derivation is a sequence of sequents where any element is either an axiom or the conclusion of a rule whose premisses are previous elements of the sequence. Basic logic has been introduced in [Battilotti and Sambin, 1999], and substantially reformulated in [Sambin et al., 1998]. According to the sec14 A multiset is a set of pairs such that the rst element of every pair denotes any object, while the second element denotes the multiplicity of the occurrences of our object. Two multisets are equal if and only if all their pairs are equal, that is all their objects together with their multiplicities are equal.

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ond formulation (we will follow here),15 our logic is characterized by three strictly linked principles: re ection , symmetry , visibility . The re ection principle states the fact that logical constants are the result of importing into a formal system metalinguistic links between assertions, considered preexisting. There is a method that leads to the rules of the calculus, starting from metalinguistic links between assertions. Such a method analyses the following equivalences, which assert a correspondence between language and metalanguage:

M

`


if and only if M

`  ÆR


` N if and only if  ÆL ` N Here the generic sign \
", corresponding to a metalinguistic link between assertions, is translated respectively into the connective ÆR , when it appears on the right of the sign `, and into the connective ÆL, when it appears on the left. In B, the operational rules are completely determined by such equivalences. As a consequence, the meaning of a connective turns out to be semantically determined by the correspondence with a metalinguistic link, quite independently of any link with a context. Since every metalinguistic link is translated into a connective according to two specular ways, the system of rules, obtained by this method, turns out to be strongly symmetric. In fact B contains, for every axiom and for every (unary or binary) rule R Mi ` Ni M `N R its symmetric rule Rs , given by

Nis ` Mis s Ns ` Ms R where the map ( )s is de ned by induction (on the length of formulas), by putting ÆsR  ÆL and ÆsL  ÆR , given a suitable correspondence between propositional variables. The third principle that B satis es is the visibility property. A rule for a given connective is called visible when the principal formula and the corresponding secondary formulas appear in the rule without any context.16

15 The formulation of the rules of B presented in [Sambin et al., 1998] is based on nite lists rather than nite multisets of formulas; hence it contains in addition the structural rule of exchange. Here we prefer to use multisets, in order to obtain an easy comparison with sequent calculi for quantum logics. 16 In any operational rule, the formula in the conclusion that contains the connective introduced by the rule itself is called the principal formula; the formulas in the premisses that are the components of the formula introduced by the rule are called the secondary formulas.

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We shall see below the syntactical consequences of visibility; we stress here that semantically it corresponds to the fact that basic connectives have a primitive meaning, in accordance with the re ection principle. As an example let us refer to a rule that plays an important role in the case of quantum logic. As is well known, in classical logic, disjunction is introduced on the left according to the following rule:

M;  ` N M; ` N M;  _ ` N In the case of B, instead, disjunction is introduced according to the following visible form:

`N `N _ `N where the context M has disappeared. From the intuitive point of view, one can read the dierence between the two cases as follows: the rule typical of classical logic attaches a meaning to the connective _ in presence of the link \;" with M (such a link is to be interpreted as a conjunction), whereas the visible rule is intended to explain the meaning of the connective _ by referring only to the connective itself. In particular, the visible rule does not permit us to prove the equation that links conjunction and disjunction ( the distributive law of ^ with respect to _). As a consequence, any sequent calculus for a quantum logic shall adopt the visible form for the rule that concerns the introduction of disjunction on the left. As to the other rules, visibility is not strictly necessary in order to obtain an adequate sequent calculus for quantum logic. However, a more convenient strategy permits us to axiomatize quantum logic, by adding only structural rules to basic logic, without any change in the rules for the connectives. In this way, we can preserve the characteristic properties of symmetry and visibility of B, that turn out to be highly convenient from the proof-theoretical point of view (as we will see below). Basic logic B has no structural rules. As a consequence, B can be regarded as \the logic of connectives" from which various stronger logics can be obtained by adding suitable structural rules. Let us now present the sequent calculus for B. Similarly to linear logic, the language of B contains two pairs of conjunctions and disjunctions: the additive conjunction ^ and the multiplicative conjunction ; the additive disjunction _ and the multiplicative disjunction . Further there are two conditionals (!, ), and two pairs of propositional literals 1 and >, 0 and ?. &

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The basic sequent calculus B Axioms

`

Operational rules

&

(Re ection)

&

(Formation)

?` ? L

`N `N _`N `N `N (Re ection) ^ `N ^ `N (Formation)

(Formation) (Formation) (Re ection) (Order)

_L ^L

0 ` N 0L

` L ` ` `N !L ! `N ` `Æ !U ! `!Æ

M ` ; R M ` M2 ` M1 `  M2 ; M1 `  M` ?R M `? &

(Re ection)

;  ` N

L  `N  ` N1 ` N2 L  ` N1 ; N2 ` N 1L 1`N

&

(Formation)

R

` 1 1R

M ` M ` ^R M `^ M ` M ` M ` _ M ` _ M

_R

` > >R

` `! !R M ` ` R M ` 

`Æ ` U

`Æ 

We will distinguish three main kinds of structural rules, labelled by the letters L, R and S. The extensions of B obtained by the addition of any combination of such rules can be organized in a cube, which is conceived as an architecture whose basis is B (see Figure 9).

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

BRS

BLRS

          

BLS

          

BS

BLR

          

BL

BR

B

          

Figure 10. The cube of logics In the cube, every logic with S satis es the structural rules of weakening and contraction:17

M ` N weakening M; O; O ` N; N; P contraction M; O ` P; N M; O ` N; P Every logic with \L" allows left contexts in any inference rule; every logic with R allows right contexts in any inference rule. In particular, the cube solves our initial problem, sketched in Figure 11. In fact, vertex BLRS, opposed to B represents classical logic, vertex BLR and vertex BLS represent respectively Girard's linear logic and intuitionistic logic; nally, vertex BS corresponds to paraconsistent quantum logic (see below). Moreover, since logics with R are simply the symmetric copy of logics with L, logics containing both L and R (BLRS, BLR) or logics containing neither L nor 17 As we have seen, in B (as well as in linear logic) the connectives conjunction and disjunction are splitted into a multiplicative and an additive connective. Such a distinction depends on the fact that there are two ways of formulating contexts in any operational rule: this leads to a multiplicative and to an additive form for each rule. The multiplicative and additive formulation turn out to be equivalent, whenever the structural rules of weakening and contraction hold. Hence, the distinction plays an essential role in linear logic and a fortiori in basic logic (where weakening and contraction fail); at the same time, it vanishes in classical logic and in orthologic.

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R (BS, B), are symmetric. The study of quantum logics nds place in the diagonal of symmetric logics, where a ner distinction of structural rules can be obtained. Sequent calculus for Orthologic. The logic BS is non-distributive. Let us consider the fragment of BS restricted to the connectives ^ and _. If we want to obtain a quantum logic from it, what is still missing is an involutive negation, satisfying de Morgan. This aim can be reached by extending the language and by adopting Girard's negation. The key point is to assume as primitive symbols of the language both the propositional variables and their duals. In other words, the propositional literals are assumed to be given in pairs, consisting of a positive element (written p) and of a negative one (written p?). On this basis, the negation of a formula is de ned as follows: p?? := p ( ^ )? := ? _ ? ( _ )? := ? ^ ? By this choice, we obtain a calculus called basic orthologic and denoted by ? BS (where the symbol ? reminds us that our calculus is applied to a dual language). Basic orthologic turns out to be equivalent to paraconsistent quantum logic (PQL). As we already know, PQL represents a weakening of orthologic, that is obtained by dropping the non contradiction and the excluded middle principles. Hence, in order to have a calculus for orthologic, it will be suÆcient to add such principles to our ? BS. This can be done by means of two new structural rules called transfer . The result is a calculus for orthologic, which will be denoted by ? O. The rules of ? O are the following (where (i) -(v) are the rules of ? BS 18 while (vi) express the transfer rules).

`

(i) (ii)

`N `N  _ ` N _L

(iii) `N ^ `N (iv)

`N  ^ ` N ^L

M ` M ` M `  ^ ^R M ` M `_

M `N M; O ` P; N weakening

M ` M `  _ _R

18 Note that, in ? BS, weakening and contraction are redundant. In fact, one can show that such a calculus admits elimination of contraction. At the same time, weakening on the right and on the left can be simulated by ^L and _R, respectively. On this basis, PQL turns out to admit a very simple formulation, given by (i), (ii), (iii).

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(v) (vi)

MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

M; O; O ` N; N; P contraction M; O ` N; P M ` N tr1 M; N ? `

M `N ` M ? ; N tr2

It is not hard to prove: THEOREM 149. ? BS is a calculus for paraconsistent quantum logic. THEOREM 150. ? O is a calculus for orthologic. As we have seen, our calculus ? O contains both p; q; r::: and p?; q? ; r? ::: . Moreover, for any rule of the calculus, the calculus shall contain also the symmetric one. As a consequence, whenever the calculus produces a derivation , it will also produce the dual derivation ? , obtained substituting every axiom  `  with the axiom ? ` ? and every occurrence of a rule with an occurrence of its corresponding symmetric rule (e.g. ^R with _L). On this basis there holds: LEMMA 151. The following rule is derivable for ? O:

M `N N? ` M?

Sketch of the proof One can easily see that M ` N is derivable by a derivation  if and only if N ? ` M ? is derivable by the symmetric derivation ? .  The structure of the calculus ? O permits us to prove the following cutelimination result. THEOREM 152. ? O admits the elimination of the cuts. O `  M;  ` N cutL M; O ` N

O ` ; P  ` N cutR O ` N; P

Sketch of the proof Like in Gentzen, the cut-elimination procedure is obtained by induction on two parameters: the degree and the rank of the cut-formula19. 19 Suppose a derivation and a sequent where a formula  occurs. Let us consider the paths (i.e. the sequences of consecutive sequents) connecting that point with the point where the formula  has been introduced, (by an axiom, or by weakening, or by an operational rule whose principal formula was ). The rank of this particular occurrence of  is the maximum among the lengths of all these paths. In other words, the rank represents the `maximum length' between the formula- occurrence we are examining and the point where that occurrence has been introduced. The degree (or length) of a formula  is the number of the connectives occurring in .

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The calculus ? O permits us to overcome in a simple way two questions that usually make cut elimination for orthologic so complicated: (i) constraints on contexts and (ii) negation. We give a sketch of the proof, considering the two points. The rst problem is solved by visibility, while the second one is solved by symmetry. (i) As we have seen, in any calculus for quantum logic the rule that introduces _ on the left (here indicated with _L) must have an empty context on the left. Now consider, for a generic calculus, the derivation

 ` ^Æ ` ^Æ _L M;M; ^ `Æ `

_ ` ^Æ cutL M;  _ `
In this derivation, the cut-formula is principal on the right premiss; hence the right rank is 1. In such a situation, Gentzen's procedure to lower the rank must operate on the left; this would necessarily produce the two derivations

 ` ^ Æ M; ^ Æ `
` ^ Æ M; ^ Æ `
cutL cutL M;  `
M; `
Now, one would like to conclude by applying _L, in order to obtain M;  _ `
. However, this step is here not allowed, unless M is empty. Such a problem does not arise for the calculus ? O, because, by visibility, every principal formula has an empty context. (ii) In ? O the only rules about negation are the structural rules of transfer. Let us consider a derivation of the form: .. ..  M ` tr1 O ` ? M; ? ` cutL M; O ` We can reduce the rank in a quick way, by exploiting symmetry. In fact, Girard's negation has the nice property that every formula  and its dual ? have exactly the same degree. The same idea can be extended to derivations, and hence to the rank of a cut. As we have seen in Lemma 151, whenever we have a derivation  for the sequent M ` N , we also have the dual derivation ? , which derives N ? ` M ? . The two derivations  and ? have exactly the same (symmetrical) structure. Hence in particular, if  is principal, ? is principal. If  has rank r, then also ? will have the same rank r. In

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MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

C

?

tr

S

S

L

?O

?

? OL

BS S

tr

?B

Figure 11. The diagonal of the cube such a situation, in order to raise the cut rule, we can substitute ? by  ( ipping derivation). As a consequence, the initial derivation will be simply reduced to: .. ? ..  ? ? O `   ` M? cutL O ` M ? tr1 O; M `

 Quantum logics and classical logic We will now consider the symmetric diagonal of the cube in the diagram in Figure 11. In our diagram, the calculus ? O appears as an intermediate point between basic orthologic and classical logic. Similarly, we have another intermediate point between basic logic and linear logic: this is given by ? B + tr, which represents the common denominator for orthologic and linear logic (we will call it \ortholinear logic" ? OL). In the same way, ? B turns out to be the common denominator of basic orthologic and linear logic. On this basis, we obtain a whole system of quantum logics, which are all cutfree. The last of our logics, ? B + tr, seems to be a good candidate in order to represent a linear quantum logic in the sense of Pratt [1993]. So far we have only dealt with a fragment of basic logic, which has no implication connective. By means of this linguistic restriction, we have easily proved the equivalence between our calculi and the usual formulations of paraconsistent quantum logic and of orthologic. However, the same methods can be naturally applied to the complete versions of our calculi, preserving cut-elimination and ipping of derivations. In this way, we will have a primitive implication connective ! (together with its dual ) in all the logics we have considered. An interesting question to be investigated concerns the possibility of physical interpretations of such new connectives.

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In the diagram above, we have still a question mark concerning the path from orthologic to classical logic. Our question can be solved as follows: THEOREM 153. A calculus for classical logic is obtained from a calculus for orthologic by adding a pair of structural rules, named separation: (vii)

M; O ` sep1 M ` O?

` N; P sep2 N? ` P

It is easy to see that, in the framework of ? O, separation rules allow us to derive the following full cut

M1 ` ; N1 M2 ;  ` N2 cut M1 ; M2 ` N1 ; N2 The converse is also true: full cut allows us to derive separation (by cutting with ` A; A? ). In this sense, separation and cut rule are equivalent: adding either of them to orthologic gives rise to one and the same logic. Theorem 153 then expresses, with a more eective20 content, the well known fact that adding a full cut rule to orthologic yields classical logic (cf. [Dummett, 1976], [Cutland and Gibbins, 1982]). It is natural to ask what is the meaning of sep. In the same way as the tr rules are equivalent to tertium non datur and non contradiction, the sep rules turn out to be equivalent to reductio ad absurdum 21 M; ? ` M `  RAA Let us consider again our Figure 11, where the question marks have been substituted by sep. Given the logic B as a basic calculus, which contains the fundamental rules for the connectives, several structural rules can be added: each rule permits us to reach a \superior" logic. The strongest element is represented by classical logic, which can be characterized as ? B + S + tr + sep. With respect to our formulation of classical logic (denoted by ? C) all the other logics in the diagram can be described as `substructural logics: for, they can be obtained by dropping some structural rules. This situation holds in particular for quantum logics, which turn out to be simpler and more basic than classical logic, from the proof-theoretical point of view. As we have seen, the examples of quantum logic (we have considered so far) are, at the same time, substructural with respect to classical logic and substructural one with respect to the other. On this basis, on can prove

20 For, in a sequent calculus cut should represent a metarule, that is should be eliminable. 21 In [Gibbins, 1985], pag.361, Gibbins shows that dropping the rule RAA has a direct justi cation in terms of quantum mechanics, and this is the only case of direct justi cation, among all the rules which must be restricted in quantum logic.

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some embedding theorems, by convenient restriction of our structural rules to suitable kinds of formulas, by means of special modalities. In the case of linear logic, exponentials have been introduced in order to express weakening and contraction. In the case of quantum logics, instead, we should obtain rules of separation and of transfer in a suitable way. How to express the separation rules in orthologic, in order to obtain an embedding of classical logic into orthologic? Given ? O, let us rst assume in the language, besides the literals p and p?, two new kinds of literals, #p and #p? . This permits us to obtain a new kind of formulas, that will be named \separable formulas", formulated as follows: #(p) := #p #(p? ) := #p? #(#p) := #p #(#p? ) := #p? for literals;

#( Æ ) := # Æ # for every binary connective Æ.

Separable formulas are precisely those formulas that satisfy the separation rules, which are then de ned as follows: (vii0 )

M; # O ` M `# O?

# sep1

`# N; P # sep2 # N? ` P

where formulas in M , N are any kind of formulas, while formulas in #M , #N are separable formulas. We can now introduce the system #? O, which is de ned by the rules of ? O and by the rules #sep. In this system, the sign # plays the role of a modality (which behaves as an unary monotonic connective: if M ` N , is a derivable sequent in #? O, then also #M ` #N is a derivable sequent). Consider now the system ? C for classical logic, and let us describe # as a map from formulas of the language of ? C into formulas of the language of #? O. It is easy to show, by induction on the depth of the derivation, that: THEOREM 154. For every M , N , M ` N is derivable in ? C if and only if #M ` #N is derivable in #? O.

Sketch of the proof A proof can be obtained by a natural transformation of a similar proof, given in [Battilotti, 1998] for the case of classical logic and basic orthologic.  As a consequence we obtain an embedding of ? C in #? O. Formulas of the kind # can be interpreted as \the classical part of #? O". Similarly to ?, the sign # does not represent here a connective; therefore, there is no need of introduction rules. One can prove that sequents like # `  or like  ` # are not provable (dierently from the exponentials in linear logic).

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In this way, the system #? O is simply a way to represent the coexistence of classical and quantum logic: it does not assert that \classical" propositions are stronger or weaker than \quantum" propositions. 18 CONCLUSION Some general questions that have been often discussed in connection with (or against) quantum logic are the following: (a) Why quantum logics? (b) Are quantum logics helpful to solve the diÆculties of QT? (c) Are quantum logics \real logics"? And how is their use compatible with the mathematical formalism of QT, based on classical logic? (d) Does quantum logic con rm the thesis that \logic is empirical"? Our answers to these questions are, in a sense, trivial, and close to a position that Gibbins (1991) has de ned a \quietist view of quantum logic". It seems to us that quantum logics are not to be regarded as a kind of \clue", capable of solving the main physical and epistemological diÆculties of QT. This was perhaps an illusion of some pioneering workers in quantum logic. Let us think of the attempts to recover a realistic interpretation of QT based on the properties of the quantum logical connectives22. Why quantum logics? Simply because \quantum logics are there!" They seem to be deeply incorporated in the abstract structures generated by QT. Quantum logics are, without any doubt, logics . As we have seen, they satisfy all the canonical conditions that the present community of logicians require in order to call a given abstract object a logic . A question that has been often discussed concerns the compatibility between quantum logic and the mathematical formalism of quantum theory, based on classical logic. Is the quantum physicist bound to a kind of \logical schizophrenia"? At rst sight, the compresence of dierent logics in one and the same theory may give a sense of uneasiness. However, the splitting of the basic logical operations (negation, conjunction, disjunction,...) into dierent connectives with dierent meanings and uses is now a well accepted logical phenomenon, that admits consistent descriptions. Classical and quantum logic turn out to apply to dierent sublanguages of quantum theory, that must be sharply distinguished. Finally, does quantum logic con rm the thesis that \logic is empirical"? At the very beginning of the contemporary discussion about the nature of logic , the claim that the \right logic" to be used in a given theoretical situation may depend also on experimental data appeared to be a kind of 22 See for instance [Putnam, 1969]

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extremistic view, in contrast with a leading philosophical tradition according to which a characteristic feature of logic should be its absolute independence from any content. These days, an empirical position in logic is generally no longer regarded as a \daring heresy" . At the same time, as we have seen, we are facing not only a variety of logics, but even a variety of quantum logics . As a consequence, the original question seems to have turned to the new one : to what extent is it reasonable to look for the \right logic" of QT? ACKNOWLEDGEMENTS The authors are grateful to Giulia Battilotti and Claudia Faggian who wrote a section of this chapter. M. L. Dalla Chiara Universita Firenze, Italy. R. Giuntini Universita Cagliari, Italy. BIBLIOGRAPHY [Battilotti, 1998] G. Battilotti. Embedding classical logic into basic orthologic with a primitive modality, Logic Journal of the IGPL, 6, 383{402, 1998. [Battilotti and Sambin, 1999] G. Battilotti and G. Sambin. Basic logic and the cube of its extensions, in A. Cantini, E. Casari, and P. Minari (eds), Logic and Foundations of Mathematics, pp. 165{186. Kluwer, Dordrecht, 1999. [Bell and Slomson, 1969] J. L. Bell and A. B. Slomson. Models and Ultraproducts: An Introduction, North-Holland, Amsterdam, 1969. [Beltrametti and Cassinelli, 1981] E. Beltrametti and G. Cassinelli. The Logic of Quantum Mechanics, Vol. 15 of Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, 1981. [Birkho, 1995] G. Birkho. Lattice Theory, Vol. 25 of Colloquium Publications, 38 edn, American Mathematical Society, Providence, 1995. [Birkho and von Neumann, 1936] G. Birkho and J. von Neumann. The logic of quantum mechanics, Annals of Mathematics 37, 823{843, 1936. [Cattaneo and Laudisa, 1994] G. Cattaneo and F. Laudisa. Axiomatic unsharp quantum mechanics, Foundations of Physics 24, 631{684, 1984. [Cattaneo and Nistico, 1986] G. Cattaneo and G. Nistico. Brouwer{Zadeh posets and three-valued Lukasiewicz posets, Fuzzy Sets and Systems 33, 165{190, 1986. [Chang, 1957] C. C. Chang. Algebraic analysis of many valued logics, Transactions of the American Mathematical Society 88, 74{80. 1957. [Chang, 1958] C. C. Chang. A new proof of the completeness of Lukasiewicz axioms, Transactions of the American Mathematical Society 93, 467{490, 1958. [Dalla Chiara, 1981] M. L. Dalla Chiara. Some metalogical pathologies of quantum logic, in E. Beltrametti and B. V. Fraassen (eds), Current Issues in Quantum Logic, Vol. 8 of Ettore Majorana International Science Series, Plenum, New York, pp. 147{159, 1981. [Cutland and Gibbins, 1982] N. Cutland and P. Gibbins. A regular sequent calculus for quantum logic in which ^ and _ are dual, Logique et Analyse - Nouvelle Serie 25(45), 221{248, 1982.

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[da Costa et al., 1992] N. C. A. da Costa, S. French and D. Krause. The Schrodinger problem, in M. Bibtol and O. Darrigol (eds), Erwin Schrodinger: Philosophy and the Birth of Quantum Mechanics, Editions Frontieres, pp. 445{460, 1992. [Dalla Chiara and Giuntini, 1994] M. L. Dalla Chiara and R. Giuntini. Unsharp quantum logics, Foundations of Physics 24, 1161{1177, 1994. [Dalla Chiara and Giuntini, 1995] M. L. Dalla Chiara and R. Giuntini. The logics of orthoalgebras, Studia Logica 55, 3{22, 1995. [Dalla Chiara and Toraldo di Francia, 1993] M. L. Dalla Chiara and G. Toraldo di Francia. Individuals, kinds and names in physics, in G. Corsi, M. L. Dalla Chiara and G. Ghirardi (eds), Bridging the Gap: Philosophy, Mathematics, and Physics, Kluwer Academic Publisher, Dordrecht, pp. 261{283, 1993. [Davies, 1976] E. B. Davies. Quantum Theory of Open Systems, Academic, New York, 1976. [Dishkant, 1972] H. Dishkant. Semantics of the minimal logic of quantum mechanics, Studia Logica 30, 17{29, 1972. [Dummett, 1976] M. Dummett. Introduction to quantum logic, unpublished, 1976. [Dvurecenskij and Pulmannova, 1994] A. Dvurecenskij and S. Pulmannova. D-test spaces and dierence poset, Reports on Mathematical Physics 34, 151{170, 1994. [Faggian, 1997] C. Faggian. Classical proofs via basic logic, in Proceedings of CSL '97, pp. 203{219. Lectures Notes in Computer Science 1414, Springer, Berlin, 1997. [Faggian and Sambin, 1997] C Faggian and G. Sambin. From basic logic to quantum logics with cut elimination, International Journal of Theoretical Physics 12, 1997. [Finch, 1970] P. D. Finch. Quantum logic as an implication algebra, Bulletin of the Australian Mathematical Society 2, 101{106, 1970. [Foulis and Bennett, 1994] D. J. Foulis and M. K. Bennett. Eect algebras and unsharp quantum logics, Foundations of Physics 24, 1325{1346, 1994. [Gibbins, 1985] P. Gibbins. A user-friendly quantum logic, Logique-et-Analyse.Nouvelle-Serie 28, 353{362, 1985. [Gibbins, 1987] P. Gibbins. Particles and Paradoxes - The Limits of Quantum Logic, Cambridge University Press, Cambridge, 1987. [Girard, 1987] J. Y. Girard. Liner logic, Theoretical Computer Science 50, 1{102, 1987. [Giuntini, 1991] R. Giuntini. A semantical investigation on Brouwer-Zadeh logic, Journal of Philosophical Logic 20, 411{433, 1991. [Giuntini, 1992] R. Giuntini. Brouwer-Zadeh logic, decidability and bimodal systems, Studia Logica 51, 97{112, 1992. [Giuntini, 1993] R. Giuntini. Three-valued Brouwer-Zadeh logic, International Journal of Theoretical Physics 32, 1875{1887, 1993. [Giuntini, 1995] R. Giuntini. Quasilinear QMV algebras, International Journal of Theoretical Physics 34, 1397{1407, 1995. [Giuntini, 1996] R. Giuntini. Quantum MV algebras, Studia Logica 56, 393{417, 1996. [Goldblatt, 1984] R. H. Goldblatt. Orthomodularity is not elementary, Journal of Symbolic Logic 49, 401{404, 1984. [Goldblatt, 1974] R. Goldblatt. Semantics analysis of orthologic, Journal of Philosophical Logic 3, 19{35, 1974. [Greechie, 1981] R. J. Greechie. A non-standard quantum logic with a strong set of states, in E. G. Beltrametti and B. C. van Fraassen (eds), Current Issues in Quantum Logic, Vol. 8 of Ettore Majorana International Science Series, Plenum, New York, pp. 375{380, 1981. [Greechie and Gudder, n.d.] R. J. Greechie and S. P. Gudder. Eect algebra counterexamples, preprint, n. d. [Gudder, 1995] S. P. Gudder. Total extensions of eect algebras, Foundations of Physics Letters 8, 243{252, 1995. [Hardegree, 1975] G. H. Hardegree. Stalnaker conditionals and quantum logic, Journal of Philosophical Logic 4, 399{421, 1975. [Hardegree, 1976] G. M. Hardegree. The conditional in quantum logic, in P. Suppes (ed.), Logic and Probability in Quantum Mechanics, Reidel, Dordrecht, pp. 55{72, 1976.

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[Kalmbach, 1983] G. Kalmbach. Orthomodular Lattices, Academic Press, New York, 1983. [Keller, 1980] H. A. Keller. Ein nichtklassischer Hilbertscher Raum, Mathematische Zeitschrift 172, 41{49, 1980. [K^opka and Chovanec, 1994] F. K^opka and F. Chovanec. D-posets, Mathematica Slovaca 44, 21{34, 1994. [Kraus, 1983] K. Kraus. States, Eects and Operations, Vol. 190 of Lecture Notes in Physics, Springer, Berlin, 1983. [Ludwig, 1983] G. Ludwig. Foundations of Quantum Mechanics, Vol. 1, Springer, Berlin, 1983. [Mackey, 1957] G. Mackey. The Mathematical Foundations of Quantum Mechanics, Benjamin, New York, 1957. [Mangani, 1973] P. Mangani. Su certe algebre connesse con logiche a piu valori, Bollettino Unione Matematica Italiana 8, 68{78, 1973. [Minari, 1987] P. Minari. On the algebraic and Kripkean logical consequence relation for orthomodular quantum logic, Reports on Mathematical Logic 21, 47{54, 1987. [Mittelstaedt, 1972] P. Mittelstaedt. On the interpretation of the lattice of subspaces of Hilbert space as a propositional calculus, Zeitschrift fur Naturforschung 27a, 1358{ 1362, 1972. [Morash, 1973] R. P. Morash. Angle bisection and orthoautomorphisms in Hilbert lattices, Canadian Journal of Mathematics 25, 261{272, 1973. [Nishimura, 1980] H. Nishimura. Sequential method in quantum logic, Journal of Symbolic Logic 45, 339{352, 1980. [Nishimura, 1994] H. Nishimura. Proof theory for minimal quantum logic I and II, International Journal of Theoretical Physics 33, 102{113, 1427{1443, 1994. [Pratt, 1993] V. Pratt. Linear logic for generalized quantum mechanics. In Proceeings of the Worksho on Physics and Computation, pp. 166{180, IEEE, 1993. [Ptak and Pulmannova, 1991] P. Ptak and S. Pulmannova. Orthomodular Structures as Quantum Logics, number 44 in Fundamental Theories of Physics, Kluwer, Dordrecht, 1991. [Putnam, 1969] H. Putnam. Is logic empirical?, Vol. 5 of Boston Studies in the Philosophy of Science, Reidel, Dordrecht, pp. 216{241, 1969. [Sambin, 1996] G. Sambin. A new elementary method to represent every complete Boolean algebra, in A. Ursini, and P. Agliano (eds), Logic and Algebra, Marcel Dekker, New York, pp. 655{665, 1996. [Sambin et al., 1998] G. Sambin, G. Battilotti and C. Faggian. Basic logic: re ection, symmetry, visibility, The Journal of Symbolic Logic, to appear. [Soler, 1995] M. P. Soler. Characterization of Hilbert space by orthomodular spaces, Communications in Algebra, 23, 219{243, 1995. [Takeuti, 1981] G. Takeuti. Quantum set theory, in E. G. Beltrametti and B. C. van Fraassen (eds), Current Issues in Quantum Logic, Vol. 8 of Ettore Majorana International Science Series, Plenum, New York, pp. 303{322, 1981. [Tamura, 1988] S. Tamura. A Gentzen formulation without the cut rule for ortholattices, Kobe Journal of Mathematics 5, 133{150, 1988. [Varadarajan, 1985] V. S. Varadarajan. Geometry of Quantum Theory, 2 edn, Springer, Berlin, 1985.

MARTIN BUNDER

COMBINATORS, PROOFS AND IMPLICATIONAL LOGICS

1 INTRODUCTION In this chapter we rst look at operators called combinators. These are very simple but extremely powerful. They provide a means of doing logic and mathematics without using variables, are powerful enough to allow the de nition of all recursive functions and have more recently been used as a basis for certain \functional" computer languages. We are interested in another use here which involves the functional character or type possessed by many combinators. Each type can be interpreted as a theorem of the intuitionistic implicational logic H! and combinators possessing that type can be interpreted as Hilbert-style proofs of that theorem. Weaker sets of combinators can be used to represent proofs in sublogics of H! , these include the substructural logics, such as the relevance logics R! and T!. There is a further interpretation of combinators and types as programs and speci cations which we will not discuss here. Next we look at lambda calculus. This also allows the de nition of all recursive functions and has also been used in foundations of mathematics and computer language development. Many lambda terms also have types and these again are the theorems of H! . The lambda terms represent natural deduction style proofs of these theorems. In the third section of this chapter we look at translations from combinators to lambda terms and vice versa. For the combinators and lambda terms that represent proofs in H! these translations are well known, for those corresponding to proofs in weaker logics they are quite new. In a fourth section we develop a new algorithm which, given an implicational formula, allows us to nd lambda terms representing natural deduction style proofs of the formula or demonstrates that the formula has no proof. Most implicational substructural logics are speci ed by substructural rules or by axioms and not by rules in the natural deduction form. Our translation procedure, together with the algorithm, provides us with a simple constructive means of nding Hilbert-style proofs in many of these logics. As the translation procedure tells us which lambda terms are translatable into which sets of combinators, the algorithm can be directed to look only for the lambda terms of the appropriate kind. The algorithm is inherently nite; for any given formula, and for many logics, bounds for the proof searches can be written down. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 6, 229{286.

c 2002, Kluwer Academic Publishers. Printed in the Netherlands.

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The H! algorithm has been implemented by Anthony Dekker as Brouwer 7.9.0 (see [Dekker, 1996]) and, for any implicational formula, produces a -term proof (or even 50 alternative proofs) or a guarantee that there is no proof, virtually instantly. The implementation has more recently been extended by Martijn Oostdijk as LambdaCal2, (see [Oostdijk, 1996]) to cover the other implicational systems in this chapter as well as certain systems with other connectives. This implementation supplies combinator and lambda calculus proofs. 2 COMBINATORY LOGIC Combinators are operators which manipulate arbitrary expressions by cancellation, duplication, bracketing and permutation. Combinators were rst de ned by Schon nkel in his 1924 paper and rediscovered by Curry in [1930]. To illustrate their use we consider the following examples: let Axy (rather than A(x; y)) represent x + y. The commutative law for addition can then be written as

Axy = Ayx: Given a combinator C with the property:

Cxyz = xzy this becomes

Axy = CAxy which could be simply written, without variables, as A = CA: Given an identity combinator I, i.e. one such that

Ix = x we can write as or and so, without variables, as

0+x=x

A0x = x A0x = Ix A0 = I: x+0=x

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would be

CA0 = I: Schon nkel found that only two combinators, K and S, were enough to de ne all others. We will now introduce these, other combinators, and our method of writing functional expressions (such as Axy rather than A(x; y)) more formally.

2.1 Combinators and Application DEFINITION 1 (Combinator). 1. K and S are combinators. 2. If X and Y are combinators so is (XY ). (The operation in (2) is called application.) Though it is possible, it is often not convenient to work without variables; we therefore introduce terms which are made up of combinators and variables using application. Other constants could also be included in (1) below. DEFINITION 2 (Term). 1. K, S, x; y; z; : : : ; x1 ; x2 ; : : : are terms 2. If X and Y are terms so is (XY ).

Notation We use association to the left for terms. This means that our Axy is short for ((Ax)y). A binary function over the real numbers such as A is therefore interpreted as a unary function from real numbers into the set of unary functions from real numbers to real numbers. The process of going from CXY Z to XZY or from IX to X is called reduction. This is de ned as follows: DEFINITION 3 (Reduction). The relation X . Y (X reduces weakly to Y ) is de ned as follows: (K) KXY . X (S) SXY Z . XZ (Y Z ) () X . X

) UX . UY ( ) X . Y ) XU . Y U ( ) X . Y and Y . Z ) X . Z () X . Y

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(K) and (S) are called the reduction axioms for K and S. DEFINITION 4 (Weak equality). X = Y if this can be derived from the axioms and rules of De nition 3 with \=" instead of \ .", together with () X = Y

) Y

= X.

The formal system consisting of at least De nition 1 and the postulates in De nitions 3 we call combinatory logic. Other axioms and rules may be added. We now show how the combinators we met earlier, and others, can be de ned. We use \" for \equals by de nition". DEFINITION 5. I  SKK B  S(KS)K C  S(BBS)(KK) B0  CB W  SS(KI) S0  B(BW)(BBB0 ) Each of these de ned combinators has a characteristic reduction theorem: THEOREM 6. 1. IX . X 2. BXY Z . X (Y Z ) 3. CXY Z . XZY 4. B0 XY Z . Y (XZ ) 5. WXY . XY Y 6. S0 XY Z . Y Z (XZ )

Proof. 1. SKKX . KX (KX ) by (S) so SKKX . X by (K) and ( ) 2. S(KS)KXY Z . KSX (KX )Y Z by ( ) and ( ) . S(KX )Y Z by (K) and ( ) . KXZ (Y Z ) by (S) . X (Y Z ) by (K) and ( ). so S(KS)KXY Z . X (Y Z ) by ( )  If M (X1 ; : : : ; Xn ) is a term made up by application using zero or more occurrences of each of X1 ; : : : ; Xn ; we can nd a combinator Z such that ZX1 X2 : : : Xn . M (X1 ; : : : ; Xn ).

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This property is called the \combinatory completeness" of the combinatory logic based on K and S. The combinator Z above is represented by

Z

 [x1 ]([x2 ](: : : ([xn ]M (x1 ; : : : ; xn )) : : :))

where each [xi ](: : :) is called a bracket abstraction. Bracket abstractions can be de ned in various ways, a simple de nition, involving S and K, is as follows: DEFINITION 7 (Bracket abstraction [xi ]).

 I (k) [xi ] Y  KY if xi 62 Y () [xi ]Y xi  Y if xi 62 Y (s) [xi ]Y Z  S([xi ]Y )([xi ]Z ), where xi 62 Y stands for xi does not appear in Y . (i) [xi ] xi

The above clauses must be used in the order given i.e. (iks). In the order (iks), we would always obtain S([xi ]Y )I for [xi ]Y xi , if xi 62 Y , instead of the simpler Y . Repeated bracket abstraction as in [x1 ]([x2 ](: : : ([xn ]M ) : : :)) we will write as [x1 ; x2 ; : : : ; xn ]M . EXAMPLE 8. [x1 ; x2 ; x3 ] x3 (x1 x3 )

   

[x1 ; x2 ]S([x3 ]x3 )([x3 ](x1 x3 )) [x1 ; x2 ]SIx1 [x1 ]K(SIx1 ) S(KK)(SI)

S(KK)(SI)x1 x2 x3 . . . . .

by (s) by (i) and () by (k) by (k) and ():

KKx1 (SIx1 )x2 x3 K(SIx1 )x2 x3 SIx1 x3 Ix3 (x1 x3 ) x3 (x1 x3 ):

Bracket abstraction has the following property which we call ( ) for lambda abstraction in Section 3. THEOREM 9. ([x]X )Y . [Y=x]X where [Y=x]X is the result of substituting Y for all occurrences of x in X .

Proof. By a simple induction on the length of X .



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From this theorem we get: ([x]X ) x . X and if x1 ; : : : ; xn 62 X1 X2 : : : Xn , ([x1 ; : : : ; xn ]X ) X1 : : : Xn . [X1 =x1 ; : : : ; Xn=xn ]X ; where [X1 =x1 ; : : : ; Xn =xn ]X is the result of substituting simultaneously X1 for all occurrences of x1 in X; : : : ; Xn for all occurrences of xn in X .

2.2 Combinators, Types, Proofs and Theorems

If a term X is an element of a set  (which we write as X 2  or X :  below) we have as KXY = X; KXY 2 : If Y 2 we have that KX is a function from the set into the set , i.e. in the usual mathematical notation:

KX : !  ; where !  represents the set of all functions from into . From this it follows that K is a function from  into ! , so we can write: K :  ! ( ! ) : Sets such as the ; and  ! ( ! ) above will be denoted by expressions called types. Types are de ned as follows: DEFINITION 10 (Types). 1. a; b; c; : : : are (atomic) types. 2. If  and are types so is ( ! ). For types we use association to the right,  ! ( ! ) can therefore be written as  ! ! . The type variables or atomic types can be interpreted as arbitrary sets, the compound types then represent sets of functions. Above we arrived at K :  ! ! ; such a derivation we call a type assignment, we call  ! !  the type of K and K an inhabitant of  ! ! . Type assignments can be more formally derived from: DEFINITION 11 (Type Assignment). 1. Variables can be assigned arbitrary types

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2. If X :  ! and Y :  then (XY ) : . 3. If Xx : ; x 62 X and x : then X : ! . We will illustrate this by assigning a type to S. If we let x :  ! ! ; y :  ! and z :  we have by (2) xz : ! and yz : and so xz (yz ) : which is, by (S), Sxyz : . Now by (3) Sxy :  ! , Sx : ( ! ) !  ! and S : ( ! ! ) ! ( ! ) !  ! : In particular we have

S : (a ! b ! c)

! (a ! b) ! a ! c ;

and it can be seen, from the work above, that every type of S has to be a substitution instance of this. A type with this property we call the principal type scheme (PTS). The PTS of K is given by

K:a!b!a: Notice that the types of K and S are exactly the axioms of H! , intuitionistic implicational logic, when ! is read as implication and ; ; : : : are read as well formed formulas or propositions. De nition 11.2 and 3 can be rewritten as:

!e and

X:! Y : XY :

[x :. ] .. . Xx !i X :  :! (x 62 X )

We have, considering only the right hand sides of the :s, the rules of inference of a natural deduction formulation of H! . If we take the types of K and S as axiom schemes and use only the parts of !e to the right of the :s, we have a Hilbert-style formulation of H! . What appears on the left hand side of the nal step in such a proof gives us a unique representation of a proof of the theorem expressed on the right of the :.

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EXAMPLE 12. If we abbreviate ( ! ! Æ)  we have:

! ( ! ) ! ! Æ by

K :  ! ( ! Æ) !  S: KS : ( ! Æ) !  S : (( ! Æ) ! ) ! (( ! Æ) ! ! ! Æ) ! ( ! Æ) ! ( ! ) ! ! Æ S(KS) : (( ! Æ) ! ! ! Æ) ! ( ! Æ) ! ( ! ) ! ! Æ K : ( ! Æ) ! ! ! Æ S(KS)K : ( ! Æ) ! ( ! ) ! ! Æ We note that each S or K in S(KS)K represents the use of an axiom and each application a use of !e . We also note that the above represents a type for the combinator B of De nition 5. Some combinators do not have types, for example if we want to nd a type for SSS we would proceed as follows: let S : ( ! ! Æ) and S : ( !  ! )

!  !  = ;  !  = and  !  = Æ:

then we have putting 

SS : ((

! ( ! ) ! ! Æ ! ( ! ) !  ! 

!  ! ) !  ! ) ! ( !  ! ) !  !  :

Now with S : ( !  assign a type to SSS :

!  ) ! ( !  ) !  !  , we would need, to 

and

!!

= !! ;  = ! ;  = ! :

However this requires both  =  and  =  !  which is impossible. Also there are types that have no inhabitant, for example a ! b and ((a ! b) ! a) ! a. In fact: THEOREM 13. 1. If a type  has a combinator inhabitant,  is a theorem of the intuitionistic implicational logic H! . 2. If  is a theorem of H! , then it is the type of a combinator.

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Proof. 1. It can be proved by an easy induction on the length of X that

x1 : 1 ; : : : ; xn : n ` X :  implies

1 ; : : : ; n ` 

2. It can be proved by an induction on the length of the deduction leading to 1 ; : : : ; n `  that there are variables x11 ; : : : x1m1 ; : : : ; xnmn and a term X such that F V (X )  fx11 ; : : : ; xnmn g and

x11 : 1 ; : : : ; x1m1 : 1 ; x21 : 2 ; : : : ; xnmn : n ` X :  : For more details on this see Hindley [1997, Section 6B2 and Section 6B5].



The isomorphism between inhabitants and types and proofs and theorems of H! , which can be extended to t programs and speci cations, is called the Curry-Howard or Formulas-as types isomorphism. Curry was the rst to recognise the relation between types and theorems of H! (see [Curry and Feys, 1958]). The idea was taken up and extended to other connectives and quanti ers in Lauchli [1965; 1970], Howard [1980] (but written in 1969), de Bruin [1970; 1980] and Scott [1970]. Recently it was extended to include a large amount of mathematics in Crossley and Shepherdson [1993].

2.3 Types and Weaker Logics If we consider an arbitrary set of combinators Q, we can de ne the set of Q-combinators and Q-terms as follows: DEFINITION 14 (Q-terms). 1. Elements of Q and x; y; z; : : : ; x1 ; x2 ; : : : are Q-terms 2. If X and Y are Q-terms so is (XY ).

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DEFINITION 15 (Q-combinators). A Q-term containing no variables is a Q-combinator. The formal system consisting of at least De nition 15 and the postulates of De nition 3, with the reduction axioms (K) and (S) replaced by ones appropriate to Q, is called Q-combinatory logic. Q-logic is the implicational logic whose axioms are the types of the combinators in Q and whose rule is ! e. Thus combinatory logic, as de ned before, is fS; Kg-(or simply SK-) combinatory logic, terms are SK-terms and combinators are SK-combinators. DEFINITION 16 (Weaker sets of combinators). A set Q1 of combinators is said to be weaker than a set Q2 , if for every Q1 -combinator X there is a Q2 -combinator Y with the same reduction theorem. (In that case we say X is Q2 -de nable.) Also there must be a Q2 -combinator which is not Q1 -de nable. DEFINITION 17 (Weaker combinatory logics). Q1 -combinatory logic is weaker than Q2 -combinatory logic if Q1 is weaker than Q2 . BCKW-combinatory logic is just as strong as SK-combinatory logic as B,C,K and W are (by De nition 5) SK-de nable and S is also BCKWde nable (S  B(BW)(BC(BB))). BCW and BCIW-combinatory logics are both weaker than SK-combinatory logic as S is not de nable using B,C,I and W. It is clear that BCK- and BCIW-combinatory logics are not combinatorially complete. The set of types of the BCK-combinators can easily be seen to be the set of theorems generated by !e using the types of B,C and K. This we call BCK-(implicational) logic, which is a subsystem of H! . In general, if all the combinators of Q1 [ Q2 have types and Q1 is weaker than Q2 then Q1 -logic is weaker than Q2-logic. As before, given a Q-combinator, a Q-theorem and its proof can be read o. EXAMPLE 18. Determine the type of BC(BK) and the BCK-proof of this as a BCK-theorem. B : ( ! ) ! ( ! ) ! ! C : (Æ !  ! ) !  ! Æ !  BC : ( ! Æ !  ! ) ! !  ! Æ !  (Here  = Æ !  ! , =  ! Æ !  :) K: !  !  BK : ( ! ) !  !  !  (Here  = ; =  !  and =  .) BC(BK) : ( ! ) !  !  !  (Here =  ! , Æ = ;  =  and  = )

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2.4 Combinator Reduction and Proof Reduction We illustrate here what happens to (part of) a proof represented by a combinator KXY or SXY Z when this combinator is reduced by (K) or (S). The original proof involving KXY must look like:

D1 K :  ! !  X :  D2 KX : !  Y : KXY :  D3 with D1 ; D2 and D3 representing other proof steps. With the reduction of KXY to X the proof reduces (or normalises) to:

D1 X :  D3 If the proof involving SXY Z is:

D1 S : ( ! ! ) ! ( ! ) !  ! X :  ! ! D2 Y :! SX : ( ! ) !  ! D3 SXY :  ! Z: SXY Z : D4 With the reduction of SXY Z to XZ (Y Z ) the proof reduces (or normalises) to:

D1 D3 D2 D3 X:! ! Z: Y :! Z: XZ : ! YZ : XZ (Y Z ) : D4 It may be, by the way, if D3 is a particularly long part of the proof, that the \reduced proof" is actually longer than the original. In the same way, if Z is long, XZ (Y Z ) may be longer than SXY Z . DEFINITION 19. If a term has no subterms of the form KXY or SXY Z , the term is said to be in normal form. Not every combinator has a normal form, for example WI(WI) and WW(WW) do not, but every combinator that has a type also has a normal form.

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A combinator in normal form will represent a normalised proof, these however are not unique. For example: K:a!a!a and also KI : a ! a ! a: 3 LAMBDA CALCULUS Lambda calculus, like combinatory logic, provides a means of representing all recursive functions. It is, these days, much used as the basis for functional computer languages. Extensions of the typed lambda calculus we introduce in Section 3.2 below, also have applications in program veri cation. Lambda calculus was rst developed by Church in the early 30s (see [Church, 1932; Church, 1933]) as part of a foundation of logic and mathematics. This was also the aim of Curry's \illative combinatory logic", but both Church's and Curry's extended systems proved to be inconsistent. The use of the lambda calculus notation is best seen through an example such as the following: If the value of the sin function at x is sin x and the value of the log function at x is log x, what is the function whose value at x is x2 ? Usually this function is also called x2 . The lambda calculus allows us to eliminate this ambiguity by using x:x2 for the name of the function.

3.1 Lambda terms and lambda reductions We will now set up the system more formally: DEFINITION 20 (Lambda terms or -terms). 1. Variables are -terms. 2. If X is a -term and x a variable then, the abstraction of X with respect to x, (x:X ) is a -term. 3. If X and Y are -terms then (XY ), the application of X to Y , is a -term. (x:X ) is interpreted as the function whose value at x is X . Given this we would expect the following to hold: EXAMPLE 21. (x: sin x) = sin ((x: sin x)x) = sin x ((x: sin x)) = sin  ((x:x2 )2) = 4 ((x:2)x) = 2

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(x:2) is the constant function whose value is 2. For terms formed by application we use association to the left as for terms of combinatory logic. Repeated -abstraction as in (x1 :(x2 : : : (xn :X ) : : :)) we abbreviate to x1 :x2 : : : xn :X or to x1 x2 : : : xn :X . Note that while x1 x2 : : : xn :X represents a function of n variables, it is also a function of one variable whose value, x2 x3 : : : xn :X at x1 , is also a function (if n  2). The process of simplifying (x: sin x)x to sin x or (x:x2 )2 to 22 is called -reduction . To explain this we need to de ne free and bound variables and, using these, a substitution operator. As in combinatory logic we use  for equality by de nition or identity. DEFINITION 22 (Free and bound variables, closed terms). 1. x is a free variable in x. 2. If x is free in Y or Z then x is free in (Y Z ). 3. If x is free in Y and y 6 x, x is free in y:Y . 4. Every x that appears in x:Y is bound in x:Y . We write F V (X ) for the set of free variables of X . A closed term is one without free variables. EXAMPLE 23. 1. If X  xy:xyx; X is a closed term and x and y are bound in X . 2. If X  xy:xyzx(u:zx)w; x; y and u are bound in X and F V (X ) = fz; wg 3. If X  x(x:xy)x, the rst and last occurrences of x are free occurrences of x. Both the occurrences of x in x:xy are bound. DEFINITION 24. ([Y=x]X - the result of substituting Y for all free occurrences of x in X ) 1. [Y=x]x  Y 2. [Y=x]y  y if y is an atom, x 6 y 3. [Y=x](W Z )  ([Y=x]W )([Y=x]Z ) 4. [Y=x](x:Z )  x:Z 5. [Y=x](y:Z )  y:[Y=x]Z if y 6 x and, y 62 F V (Y ) or x 62 F V (Z ).

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6. [Y=x]y:Z  z:[Y=x][z=y]Z if y 6 x; y 2 F V (Y ); x 2 F V (Z ) and z 62 F V (Y Z ). As we saw in Example 21, a -term of the form (x:X )Y can be simpli ed or \reduced". This kind of reduction is speci ed by the following axiom: ( ) (x:X )Y . [Y=x]X : A -term of the form (x:X )Y is called a -redex. The following axiom allows a change of bound variables and is called an -reduction. () x:X . y:[y=x]X

if y 62 F V (X ):

Rules (), ( ) and ( ) of De nition 3 together with: ( ) X . Y

) x:X . x:Y ;

allow us to perform - and -reductions within a term. DEFINITION 25 ( - and -reduction). The reduction . speci ed by (), ( ), (), (), ( ), ( ) and ( ) is called -reduction. This becomes - (or just -) reduction if the following axiom is added: () x:Xx . X

if x 62 F V (X ):

We denote the two forms of reduction, when we wish to distinguish them, by . and .  . If in a -reduction ( ) is not used we sometimes write . instead of .  . We will call -, - and - reductions -reductions to distinguish them from combinator reductions. EXAMPLE 26. 1. x1 :x1 ((x2 :x3 x2 x2 )x1 ) . x1 :x1 (x3 x1 x1 ) 2. x3 :(x1 :x1 x2 (x2 :x3 x1 x2 ))(x1 :x1 ) . x3 :(x1 :x1 x2 (x3 x1 ))(x1 :x1 ) . x3 :(x1 :x1 )x2 (x3 (x1 :x1 )) . x3 :x2 (x3 (x1 :x1 )) so x3 :(x1 :x1 x2 (x2 :x3 x1 x2 ))(x1 :x1 ) .  x3 :x2 (x3 (x1 :x1 )) A term of the form x:Xx where x 62 F V (X ), is called an -redex. A term with no -redexes is said to be in -normal form. One without - and - redexes is in -normal form. One without -redexes is said to

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be in -normal form. -, - and -normal forms are unique (see [Hindley and Seldin, 1986]). DEFINITION 27 (-, - and -equality). The relation = is speci ed by (); (); (); ( ); ( ) and ( ), all with = for . and () X = Y

)Y

= X .

The relation = is speci ed by all the above postulates, as well as ( ), with = for . and = . The relation =  is speci ed by all the = postulates, as well as (), with =  replacing = and .. Note that the weak equality of combinatory logic obeys all the above postulates (with [..] for ..) except ( ) and (). It is possible to extend weak equality by means of some extra equations involving combinators to make ( ) and/or () admissible (see [Curry and Feys, 1958, Section 6C4]). We will call the corresponding equalities for combinatory logic = and =  respectively.

3.2 Lambda Terms, Types, Proofs and Theorems If a term X is an element of a set  and the variable y is in a set ; y:X will represent a function from into  i.e. y:X : ! . We will denote sets such as these by the types introduced in De nition 10. We assign types to -terms in a similar way to the assignment to combinators. DEFINITION 28 (Type Assignment). 1. Variables can be assigned arbitrary types. 2. If X :  ! and Y :  then (XY ) : 3. If X :  and y : then y:X : ! . EXAMPLE 29. 1. If x2 : a ! a ! b; x3 : a then x2 x3 x3 : b and x3 :x2 x3 x3 : a ! b. If x1 : (a ! b) ! c then x1 (x3 :x2 x3 x3 ) : c, so x1 x2 :x1 (x3 :x2 x3 x3 ) : ((a ! b) ! c) ! (a ! a ! b) ! c. 2. If x1 :  ! then for x1 x1 to have a type we must have  =  ! which is impossible. Hence x1 x1 and so x1 :x1 x1 have no types.

DEFINITION 30. If X is a closed lambda term and ` X :  is derived by the type assignment rules then X is said to be an inhabitant of  and  a type of X .

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Note that, as with combinatory logic, if we ignore the terms on the left of the :, we have in De nition 20.2 and 3 the elimination and introduction rules for ! of the natural deduction version of H! . For any Y : , the types of the variables free in Y constitute the uncancelled hypothesis which yield . When Y has no free variables,  is a theorem of H! and Y represents a natural deduction proof of . This can be best seen when the type assignment rules are written in tree form as in the example below. EXAMPLE 31. (6 1) (6 2) x : a 1 !e x!1 x2b :!b !c cx2 : a x3 : (b !(6c3)) ! a ! d (6 2) !e x3 (x1 x2 ) : a ! d !e x2 : a x3 (x1 x2 )x2 : d (6 4) !i x2 :x3 (x1 x2 )x2 : a ! d (2) x4 : (a ! d) ! e !e 2 :x3 (x1 x2 )x2 ) : e (3) !i x3 :x4 (x2 :xx3 (4x(x 1 x2 )x2 ) : ((b ! c) ! a ! d) ! e (1) !i x1 x3 :x4 (x2 :x3 (x1 x2 )x2 ) : (a ! b ! c) ! c) ! a ! d) ! e !i x4 x1 x3 :x4 (x2 :x3 (x1 x2 )x2(() b: ! (4) ((a ! d) ! e) ! (a ! b ! c) ! ((b ! c) ! a ! d) ! e

At a !i step the assumption cancelled is indicated by -(n). At this time the assumption is relabelled with (6 n). In the lambda term proof each application represents a use of !e and each -abstraction a use of !i . It follows that: THEOREM 32. 1. Every type of a -term is a theorem of H! . 2. Every natural deduction style proof of a theorem of H! can be represented by a closed -term with that theorem as its type.

Proof. 1. We will prove the following more general result by induction on X :  If F V (X )  x1 ; : : : ; xn and if then in H! .

x1 : 1 ; : : : ; xn : n ` X :

(a)

1 ; : : : ; n `

(b)

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If X is an atom, X = xi for some i; 1  i  n and = i , so (b) holds. If (a) is obtained from

x1 : 1 ; : : : ; xn : n ; xn+1 : n+1 ` Y :  ; where = n+1 hypothesis

!

and X = xn+1 :Y then by the induction

1 ; : : : ; n ; n+1 ` 

is valid in H! and so also (b).

X  UV where xi1 : i1 ; : : : ; xik : ik ` U :  ! ; xj1 : j1 ; : : : ; xim : jm ` V:  and xi1 : i1 ; : : : ; xin : ik [ xj1 : j1 ; : : : ; jm : jm = x1 : 1 ; : : : ; xn : n If

we have by the induction hypothesis in H! :

i1 ; : : : ; ik and

`!

j1 ; : : : ; jm

`

and so (b). 2. We show by induction on the length of a proof in H! of

1 ; : : : ; n ` 

(c)

that there is a term X with F V (X )  x1 ; : : : ; xn such that

x1 : 1 ; : : : ; xn : n ` X : If is one of the i s this is obvious with X = xi . If (c) is obtained by the !i rule from

1 ; : : : ; n ; ` Æ where = ! Æ, then by the induction hypothesis we have

x1 : 1 ; : : : ; xn : n ; xn+1 : ` Y : Æ

(d)

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where F V (Y )  x1 ; : : : ; xn ; xn+1 : Then (d) follows with X = xn+1 :Y . If (c) is obtained from

i1 : : : ; ik `  ! and   j1 : : : ; jm `    where i1 ; : : : ; ik [ j1 ; : : : ; jm = 1 ; : : : ; n We have, by the induction hypothesis:

xi1 : i1 ; : : : ; xik : ik ` Y :  ! xj1 : j1 ; : : : ; xjm : jm ` Z :    where F V (Y )  xik ; : : : ; xik and F V (Z )  xj1 ; : : : ; xjm . (d) then follows with X = Y Z .



3.3 Long Normal Forms In Section 5 we will develop an algorithm, which, when given a type, will produce an inhabitant of this type if it has one. The inhabitant that is produced is in long normal form. This is de ned below. DEFINITION 33 (Long normal form). A typed -term x1 : : : xn :xi X1 ::Xk (n  0; k  0) is said to be in long normal form (lnf) if X1 ; : : : ; Xk are in lnf and have types 1 ; : : : ; k and xi has type 1 ! : : : ! k ! a where a is an atom. THEOREM 34. If X is a -term such that ` X : , then there is a term Y in lnf such that ` Y :  and Y . X .

Proof. By induction on the number of parts xi X1 : : : Xk of X that are not in lnf and are not the initial part of a term xi X1 : : : Xm, with m > k. Consider the shortest of these. X1 ; : : : ; Xk must then be in lnf and if each Xj has type j , xi must have type 1 ! : : : ! m ! a where m > k. Let xp+1 ; xp+2 ; : : : ; xp+m k be variables not free in xi X1 : : : Xk with types k+1 ; k+2 ; : : : ; m respectively. Then xi X1 : : : Xk xp+1 : : : xp+m k has type a and xp+1 : : : xp+m k :xi X1 : : : Xk xp+1 : : : xp+m k has type k+1 : : : ! m ! a, the same as xi X1 : : : Xk . This new term with the same type as xi X1 ; : : : ; Xk is in lnf, so when it replaces xi X1 : : : Xk in X , there is one fewer part not in lnf. Hence X can be expanded to a term Y in lnf such that Y . X . The types of the parts of X being changed are not aected so the type of Y will be the same as the type of X . 

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3.4 Lambda Reductions and Proof Reductions We illustrate here what happens to (a part of) a proof represented by a -term (x:X )Y or y:Xy with y 62 F V (X ) when this is reduced by ( ) or (). (6 1) x: D1 X : D2 (1) x:X :  ! Y : (x:X )Y : D3 reduces to: D2 Y : [Y=x] D1 [Y=x]X : [[Y=x]X=(x:X )Y ] D3 and

D1 (6 1) X :! x: !e Xx : (x 62 F V (X )) !i (1) x:Xx :  ! D2

reduces to

D1 X:! [X=x:Xx] D2 An expansion of a typed -term to lnf reverses the second reduction. The eect of these reductions on a type assignment shown by the above is stated in the theorem below. THEOREM 35 (Subject Reduction Theorem). If x1 : 1 ; : : : ; xn : n ` X :  and then

X .  Y ; x1 : 1 ; : : : ; xn : n ` Y : 

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Proof. A full proof of this is lengthy. A good outline appears in [Hindley and Seldin, 1986, Chapter 15].  The converse of this is not true, for example (uv:v)(x:xx) has no type but it -reduces to v:v which has type a ! a. xyz:(u:y)(xz ) has type (c ! d) ! b ! c ! b, but not type a ! b ! c ! b; it reduces to xyz:y which has type a ! b ! c ! b. The -expansion used in forming a lnf in the proof of Theorem 34, illustrates the following limited form of the Subject Expansion Theorem. THEOREM 36. If

x1 : 1 ; : : : ; xn : n ` X :  ! then if x 62 F V (X ).

x1 : 1 ; : : : ; xn : n ` x:Xx :  !

Proof. See Hindley and Seldin [1986, Chapter 15].



References Much more detail on the work in this section can be found in [Hindley, 1997, Chapter 2]. The system T A discussed there is eectively the system we have introduced. See also [Barendregt, 1984, Appendix A]. 4 TRANSLATIONS As -terms and combinators describe the same set of (recursive) functions it is not surprising that for each -term there is a combinator representing the same function and, in a simple case, with the same reduction theorems (as in (K), (S) or Theorem 6) and the same types. For every combinator there is also a similar -term given by: DEFINITION 37 ((X )).

S K (XY )

 xyz:xz (yz )  xy:x  X Y :

Any -term X can be translated into a combinator by taking parts of X of the form xi : : : xj :Y where Y contains no xk s and changing these to [xi ; : : : ; xj ]Y as de ned in De nition 7, then further parts of the form xi : : : xj :Y can be changed in the same way until there are no xk s left. If we call this translation  we might hope to have

X  X

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and

Y  Y for any combinator X and -term Y . The former identity holds, but the latter one does not, in general. EXAMPLE 38. 1. (SKK)  (xyz:xz (yz )) (uv:u) (ts:t)  ([x; y; z ]:xz (yz ))([u:v]u)([t; s]t)  ([x; y]:Sxy)([u]:Ku)([t]:Kt)  SKK 2. (z (xyz:xyyz ))  z ([x; y; z ]:xyyz )  z ([x; y]:xyy)  z ([x]:SxI)  z (SS(KI))  z (xyz:xz (yz ))(uvw:uw(vw))((st:s)(r:r)) . z (yzw:zw(yzw))(tr:r) . z (zw:zw((tr:r)zw)) . z (zw:zww) In the second example above we have only Y =  Y : This turns out to be about as much as we can hope to have. Weaker sets of combinators can be translated into -calculus in the same way as above with the translations of S and K in De nition 37 replaced by appropriate translations such as:

B  xyz:x(yz ) ; for each element of the given basis set Q. The reverse process however is not so simple. It is not clear which terms can be translated, say, into BCIW combinators, nor how to perform this translation. We will resolve this problem later for several basis sets Q. The process is important as, as was mentioned in the introduction, it provides a decision procedure and a constructive proof nding algorithm for axiomatic logics.

4.1 Q-Translation Algorithms Trigg, Bunder and Hindley in [Trigg et al., 1994] have extended the notion of abstractibility from that of De nition 7 for SK-combinators to abstractibility for Q-combinators for various basis sets Q. These de nitions will form part of our translation procedures. First we will give the de nition, from [Bunder, 1996], of a translation from -terms into Q-combinators. DEFINITION 39. A mapping  from -terms to Q-terms is de ned by:

250

1. X  X

MARTIN BUNDER

(X an atom)

2. (XY )  X Y 3. (x:X )   x:X ;

 x varies with Q but is, in simple cases, like the bracket abstraction [x] of De nition 7, a sequence of some of the following clauses: (i)  x:x  I (k)  x:X  KX if x 62 F V (X ) ()  x:Xx (s)  x:XY

 X if x 62 F V (X )  S( x:X )( x:Y ) (b)  x:XY  BX ( x:Y ) if x 62 F V (X ) (c)  x:XY  C( x:X ) if x 62 F V (Y )

Note that, as before, the clauses in an algorithm  are used strictly in the order in which they appear. Note also that the mapping starts by assigning a  to each  in the term starting from the outermost ones and working inwards. Then, to the innermost terms  x:Z i.e. those for which Z contains no  s, we apply the appropriate abstraction clause. Later steps now evaluate terms  x:Z1 where Z1 is a Q-term. The Q-combinators in Z1 arise from the evaluation of previous  x:Z s. EXAMPLE 40. (iks) xy:xzy  (iks) x:xz  SI(Kz ) (isk) xy:xzy  (isk) x:S(S(Kx)(Kz ))I  S(S(KS)S(S(KS)(S(KK)I))(S(KK)(Kz ))))(KI) DEFINITION 41. A mapping  from -terms to Q-terms is said to be a Q-translation algorithm if: (A) For every Q-combinator X , X is de ned and

X  X :

(B) If for a -term Y; Y is de ned and is a Q-term, then there is a -term Y1 such that: Y .Q Y1 / Y where .Q means that only full or partial reductions involving the versions of Q-combinators are used and . involves only  and reductions.

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(again) In this example we had,

EXAMPLE 38.2

with

  (iks);  z (xyz:xyyz );  z (zw:zww);

Y and Y so (B) holds with Y1 = Y .

EXAMPLE 42. (BCC)  (xyz:x(yz )) (uvw:uwv) (rst:rts) If  is (ibc) we have (BCC)  BCC as required by (A). If however  is (ikc) (xyz:x(yz )) is not de nable so (ikc) is not a BCI-translation algorithm. If  is (ibc) we have

B

 (xyz:x(yz ))  C(BB(BBI))(C(BBI)I) 6 B;

so (ibc) is also not a BCI-translation algorithm. (note that we do have B =  B.) EXAMPLE 43. When  is (ibc) (yz:z (x:yx))  ( yz:z (x:yx))  ( yz:zy)  ( z:CIz )  (CI)  (uvw:uwv)(x:x) .BCI vw:(x:x)wv .BCI vw:wv: while also yz:z (x:yx) . vw:wv The following theorem lists some properties that are preserved under the operations  and . THEOREM 44. 1. If X and Y are Q-terms then

X .Q Y

) X .Q Y :

If  is a Q-translation algorithm then: 2. for every Q-term X , X is de ned and

X  X

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3. For every pair of -terms U and V for which U and V are de ned and are Q-terms, U =  V , U =  V 4. If Y is de ned and is a Q-term then

(x:Y ) x =  Y

Proof. 1. It suÆces to prove the result for X  P X1 : : : Xn where P is any Q-combinator and Y  f (X1 ; : : : ; Xn ) which results by a single P reduction step. Then

X

 P X1 : : : Xn  (x1 : : : xn :f (x1 ; : : : ; xn ))X1 : : : Xn .Q f (X1 ; : : : ; Xn )  Y :

2. By an easy induction using (A) and De nition 39.2. 3. By Curry and Feys [1958, Section 6C4, Theorem 1]:

U =  V () U =  V so the result follows by (B). 4. By (3) ((x:Y )x) =  Y so the result holds by De nition 39.2.



4.2 Q-de nability We now come to the important question as to which -terms can be translated into Q-combinators, for a given Q. We rst de ne Q-de nability. DEFINITION 45. A -term Y is Q-de nable if there is a Q-translation algorithm  for which Y is de ned and is a Q-term. The following theorem relates the ( ) operation to Q-de nability. THEOREM 46. 1. If Z is Q-de nable there is a Q-term X such that Z =  X . 2. If X is a Q-term and there is a Q-translation algorithm then X is Q-de nable.

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Proof. 1. If Z is Q-de nable there is a Q-translation algorithm  so that, by (B)

Z =  Z: thus Z is the required X . 2. If X is a Q-term and  is a Q-translation algorithm, by Theorem 44.2 X  X; so clearly X is Q-de nable.  The importance of Q-de nability, for implicational logics is that a typed -term Y is Q-de nable if and only if its type is a theorem of Q-logic. In all the cases we deal with below we nd that, for a given Q, we can nd a Q-translation algorithm which translates all Q-de nable -terms. We now give translations algorithms for a number of sets of combinators. First we need a lemma. LEMMA 47. If U is a -term for which U is de ned then 1. if  is the (iks) algorithm ( x:U )x .KS U : 2. if  is the (ikbc) algorithm 3. if  is the (ibc) algorithm 4. if  is the (ibcs) algorithm

( x:U )x .BCK U: ( x:U )x .BCI U :

( x:U )x .BCIW U :

Proof. 1. By induction on the length of U . If U  x ( x:U )x  Ix  SKKx . KS x  U : If U  U1 x where x 2=F V (U1 ); ( x:U )x  U1 x  U : If x 2=F V (U );

( x:U )x  KU x . KS U If U  U1 U2 , where x 2 F V (U1 U2 ) and either x 2 F V (U1 ) or U2 6 x, ( x:U )x = S( x:U1 )( x:U2 )x . KS ( x:U1 ) x(( x:U2 )x) . KS U1 U2  U by the induction hypothesis. Cases 2. to 4. are similar.



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THEOREM 48. 1. (iks) is an SK-translation algorithm. 2. (ikbc) is a BCK-translation algorithm. 3. (ibc) is a BCI-translation algorithm. 4. (ibcs) is a BCIW-translation algorithm.

Proof. 1. It is easy to check that (A) holds for the (iks) abstraction algorithm. We prove (B) by induction on the length of the -term Y . If Y is an atom Y  Y: If Y  UV; Y  U V and by the inductive hypothesis we have a U1 and V1 such that Y  U V .KS U1V1 / UV  Y: If Y  x:Xx, where x 62 F V (X ),

Y  (x:Xx)  X : By the induction hypothesis there is an X1 such that

Y  X .KS X1 / X / Y: If Y = x:UV where x 2 F V (U ) or x 6 V; but x 2 F V (UV ):

Y .SK x:((x:U ) x)((x:V ) x)

 S (x:U ) (x:V )  (uvx:ux(vx)) (x:U ) (x:V )

 x:(( x:U )x) (( x:V )x) .SK x:U V

by Lemma 47 and Theorem 44.1. Now by the induction hypothesis there exist U1 and V1 such that

x:U V .SK x:U1 V1 / x:UV

Y :

Note that if x had not been chosen as the third bound variable in S , an extra -reduction would have been required from x:U1 V1 to reach the term obtained by the SK-reduction. We will use similar simpli cations below. If Y  x:x; Y  I = Y :

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If Y

255

 x:X; where x 62 F V (X ), Y  K X  (yx:y) X .SK x:X :

By the induction hypothesis there exists an X1 such that

X .SK X1 / X so

Y .SK x:X1 / Y



2., 3. and 4. are similar. EXAMPLE 49.
1. ikbc x1 x2 :x2 ikbc x3 :x1 x4  ikbc x1 x2 :x2 (K(x1 x4 ))  ikbc x1 :CI(K(x1 x4 ))  B(CI)(BK(CIx4 ))
2. ibc x1 x2 :x2 ibc x3 :x3 x1  ibc x1 x2 :x2 (CIx1 )  ibc x1 :CI(CIx1 )  B(CI)(CI):




3. ikbcs x1 x2 :x2 ikbcs x3 :x2 (x1 x3 )  ikbcs x1 x2 :x2 (Bx2 x1 )  ikbcs x1 :SI(CBx1 )  B(SI)(CB):

4.3 SK, BCI, BCK and BKIW De nable Terms For many sets Q we can delineate the Q-de nable terms. First we need some notation. DEFINITION 50. 1.  is the set of all -terms. 2. A -term is in Once (i1 ; : : : ; in ) if each xi1 ; : : : ; xin appears free exactly once in the term and if in every subterm xj1 : : : xjk :Y of the term, Y is in Once (j1 ; : : : ; jk ). 3. A -term is in Once (i1 ; : : : ; in ) if each xi1 ; : : : ; xin appears free at most once in the term and if in every subterm xj1 : : : xjk :Y of the term, Y is in Once (j1 ; : : : ; jk ).

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4. A -term is in Once+(i1 ; : : : ; in ) if each of xi1 ; : : : ; xin appears at least once in the term and if in every subterm xj1 : : : xjk :Y of the term, Y is in Once+ (j1 ; : : : ; jk ). THEOREM 51. 1. The set of SK-de nable terms is . 2. The set of BCI-de nable terms is Once( ). 3. The set of BCK-de nable terms is Once ( ). 4. The set of BCIW-de nable terms is Once+ ( ).

Proof. 1. is trivial. 2. If Y 2 Once(i1 ; : : : ; in ) for some i1 ; : : : ; in we show by induction on Y that Y is BCI-de nable using the (ibc) algorithm. Case 1 Y is an atom Y(ibc)  Y . Case 2 Y  UV; U 2 Once(j1 ; : : : ; jr ) and V 2 Once(m1 ; : : : ; ms ), where (j1 ; : : : ; jr ) and (m1 ; : : : ; ms ) are disjoint subsequences of (i1 ; : : : ; in ) and r + s = n: Then by the induction hypothesis U and V are BCI-de nable using the (ibc) algorithm and

Y(ibc) = U(ibc) V(ibc) : Case 3 Y  xp :Zxp where xp 2=F V (Z ). Zxp 2 Once(i1 : : : ; in ; p) and by the induction hypothesis Zxp is BCI-de nable as Z(ibc) xp and Y(ibc) = Z(ibc) : Case 4 Y  xp :UV; UV 2 Once(i1 : : : ; in ; p) and so xp 2 F V (V ) F V (U ) or xp 2 F V (U ) F V (V ): Hence, for similar disjoint sequences to the above we have U 2 Once (j1 ; : : : ; jr ) and V 2 Once(m1 ; : : : ; ms ; p) or U 2 Once(j1 ; : : : ; jr ; p) and V 2 Once(m1 ; : : : ; ms ): In the former case U; V and xp :V are BCI de nable and Y(ibc)  BU(ibc) ((ibc) xp :V(ibc) ):

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In the latter case U; V and xp :U are BCI-de nable and

Y(ibc)  C((ibc) xp :U(ibc) )V(ibc) : 3. and 4. are similar, except that in 4. (m1 ; : : : ; ms ) and (j1 ; : : : ; jr ) need not be disjoint.  Note that in each of the four cases above the set of de nable -terms leads us to a natural deduction style system for the logic. In the case of BCIW-logic (the relevance logic R! ), the only -terms allowable are those in Once+ ( ). These can be generated by allowing x:X to be de ned only when x 2 F V (X ). This restriction when translated to typed terms becomes: (6 1) x: D1 X : (1) x:X :  ! only if x :  is used in the proof D1 . This therefore gives the appropriate restriction on an R! ! introduction rule.

4.4 Bases Without

C

We now look at some basis sets that do not include (as de ned or primitive) the combinator C. Its lack causes a problem. Previously the algorithms we used in evaluating  x2 :X did not aect the de nability of  x1 : x2 :X . When we are dealing with algorithms that do not include (c) (or (s)), this de nability may fail depending on a choice of . If, for example, we de ne  x3 :x4 x2 (x1 x3 ) to be B(x4 x2 )x1 , using clause (b), we cannot easily de ne  x2 :B(x4 x2 )x1 . If however we de ned  x3 :x4 x2 (x1 x3 ) as B0 x1 (x4 x2 ), using a clause (b0 ), we can de ne  x2 : x3 :x4 x2 (x1 x3 ) as B(BB0 )x1 x4 using (b). Clearly if (b) and (b0 ) (and so B and B0 ) are both available in our translation algorithm, the choice of which to use would depend on the variables to be abstracted later. If a subterm xin+1 :Y of a -term X is to be translated by  and is in the scope of (from left to right in X ): xi1 ; : : : ; xin , we will write this in translation as xxii1n;:::;x :Y , so that the variables with respect to which we +1 need to abstract later are agged as: xin to be done next, then xin 1 etc. These abstractions are of course tied to some set Q and to algorithm clauses which we still denote by . To ensure that the agged xij s are distinct we will assume that any term X being translated has, if necessary, rst been altered so that no xk appears more than once in X .

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MARTIN BUNDER

For Q-translation algorithms , where C is not in Q, such as BB0 I and BB0 IW, we replace De nition 39 by: DEFINITION 52. (xi1 ; : : : ; xin ; Y ) and in particular ( ; Y )  Y are given by: (xi1 ; : : : ; xin ; P ) (xi1 ; : : : ; xin ; P Q) (xi1 ; : : : ; xin ; xin+1 :R) ( ; xi1 :R)

   

P if P is an atom (xi1 ; : : : ; xin ; P ) (xi1 ; : : : ; xin ; Q) ;:::;xin xxii1n+1 :(xi1 ; : : : ; xin ; xin+1 ; R)   xi1 (xi1 ; R) :

Note that Theorem 44 still holds under this de nition. EXAMPLE 53. ( ; x4 x1 :x3 (x5 x6 :x1 (x7 :x1 x5 ))(x2 :x2 (x9 :x2 )))   x4  xx41 :x3 ( xx4 x5 1 :( x4 xx16 x5 :x1 ( x4 xx1 x7 5 x6 :x1 x5 ))( xx4 x2 1 :x2 ( x4 xx19 x2 :x2 )) Before we can write down the algorithm clauses used for logics without C, we need the notion of the index of a term with respect to a set of variables. DEFINITION 54. (idx(M;ni1 ; : : : ; in )) o idx(M; i1 ; : : : ; in ) = max p j 1  p  n ^ xip 2 F V (M ) DEFINITION 55 ((xxini1+1:::xin :P )). This is de ned using some or all of the following clauses, depending on . (i)

in xxii1n:::x :xin+1 +1

I

in () xxii1n:::x :Uxin+1 +1

 U if xin 62 F V (U ) +1

in in (b) xxii1n:::x :P Q  BP (xxii1n:::x :Q) +1 +1 if idx(P; i1 ; : : : ; in )  idx(Q; i1 ; : : : ; in ) or xi1 : : : xin is replaced by ; and xi n+1 2= F V (P ). in in (b0 ) xxii1n:::x :P Q  B0 (xxii1n:::x :Q)P +1 +1 if idx(P; i1 ; : : : ; in ) > idx(Q; i1 ; : : : ; in) and xin+1 2= F V (P ).

:::xin in in (s) xxii1n:::x :P Q  S(xxii1n+1 :P )(xxii1n:::x :Q) +1 +1 if idx(P; i1 ; : : : ; in )  idx(Q; i1 ; : : : ; in ) or if xi1 : : : xin is replaced by



in in in (s0 ) xxii1n:::x :P Q  S0 (xxii1n:::x :Q)(xxii1n:::x :P ) +1 +1 +1 if idx(Q; i1 ; : : : ; in ) < idx(P; i1 ; : : : ; in)

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EXAMPLE 56. 0

1. ( ; x2 x4 :x2 (x3 :(x5 x6 :x5 (x4 x6 ))x3 )(ibb )

  x2 xx :x2 (xx x :(xx x x xx x x x :x5 (x4 x6 ))x3 )   x2 xx :x2 (xx x :(xx x x :B0 x4 x5 )x3 )   x2 xx x2 (xx x :B0 x4 x3 )   x2 xx :x2 (B0 x4 )  B0 B0 where  is (ibb0 ) 2 4 2 4 2 4 2 4

2 4 3 2 4 3 2 4 3

2 4 3 5 2 4 3 5

2 4 3 5 6

0 0

2. (; x2 x4 :x2 (x3 :x2 x4 (x1 :x3 (x4 x1 ))))(ikbb ss )

  x2 :xx :x2 (xx x :x2 x4 (xx x x :x3 (x4 x1 )))   x2 :xx :x2 (xx x :x2 x4 (B0 x4 x3 ))   x2 :xx :x2 (B(x2 x4 )(B0 x4 ))   x2 :Bx2 (S0 B0 (BBx2 ))  SB(B(S0 B0 )(BB)) where  is (ikbb0 ss0 ) : 2 4 2 4 2 4

2 4 3 2 4 3

2 4 3 1

4.5 Translation Algorithms and De nable Terms for BB0 I and BB0 IW

Before we give the algorithms we need a lemma in LEMMA 57. If Q is (i) BB0 I or (ii) BB0 IW, X is a Q-term and xxii1n:::x :X +1 is de ned then   in xxii1n:::x :X xin+1 .Q X : +1

Proof. By induction on the length of X . Cases 1 to 4 apply where Q is BBI0 or BB0 IW, Cases 5 and 6 only for BB0 IW. Case 1 X  xin+1 . 



in xxii1n:::x :X xin+1 +1

 Ixin .Q xin  X +1

+1

Case 2 X  Uxin+1 where xin+1 62 F V (U ) 



in xxii1n:::x :X xin+1 +1

Case 3 X  UV where xin+1 idx(V; i1 ; : : : ; in ).

 Uxin  X : +1

62 F V (U ), xin 6 V , and idx(U; i1 ; : : : ; in )  +1

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MARTIN BUNDER 







in in :V x xxii1n:::x :X xin+1  BU xxiin;:::x in+1 +1   +1  xi1 :::xin .Q U xin+1 :V xin+1 .Q UV  X ; by the induction hypothesis. Case 4 X  UV where xin+1 62 F V (U ), xin+1 6 V and idx(V; i1 ; : : : ; in ) < idx(U; i1 ; : : : ; in ). Similar to Case 3. Case 5 X  UV where xin+1 2 F V (U ) \ F V (V ) and idx(U; i1; : : : ; in )  idx(V; i1 ; : : : ; in ). Similar to Case 3. Case 6 X  UV where xin+1 2 F V (U ) \ F V (V ) and idx(V; i1 ; : : : ; in ) < idx(U; i1 ; : : : ; in ). Similar to Case 3. 

THEOREM 58. The following are translation algorithms: 0 1. ( ; )(ibb ) , for BB0 I 0 0 2. ( ; )(ibb ss ) , for BB0 IW

Proof. In each case (A) is obvious. We will prove, for each algorithm  and each basis Q, if ((xi1 ; : : : ; xin ; Y ) is de ned, that there is a Y1 such that:    xi1 ; : : : ; xin ; Y .Q Y1 / Y : 

This we do by induction on the number k of clauses of  that are needed to evaluate (xi1 ; : : : ; xin ; Y ) . If k = 0 there are no s in Y and so ((xi ; : : : ; xi ; Y ) )  Y  Y1 n

1

If k > 0 it is suÆcient to consider a subterm (xi1 ; : : : ; xim ; xim+1 :Z ) :::xim ( xxim1+1 :Z ) of (xi1 ; : : : ; xin ; Y ) where Z contains no s and to show that there is a Z1 such that:


:::xim xxim1+1 :Z  .Q Z1 / xim+1 :Z We then have, by Theorem 44.1, ((xi ; : : : ; xi ; Y ) ) .Q ((xi ; : : : ; xi ; Y 0 ) ) / Y 0 / Y 1

n

1

n

where Y 0 is Y with Z1 for xim+1 :Z . Now Y 0 needs fewer than k clauses of  for its evaluation, so by the induction hypothesis we have a Y1 such that ((xi ; : : : ; xi ; Y 0 ) ) .Q Y1 / Y 0 ; 1

n

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which provides the result. To prove the above result for xin+1 :Z we consider 6 cases for BB0 IW. The rst 4 also apply to BB0 I. Note that in each case if U is Z or a subterm of Z , as this contains no s, we have U  U . (a)

If Z  xim+1 , then 

im xxii1m;:::;x :Z +1

(b)





 I  xim :Z +1

If Z  Uxi m+1 where xi m+1 2= F V (U ) then   im xxii1m;:::;x :Z  U : +1 

By the induction hypothesis there is a U1 , such that im (xxii1m:::x :Z ) .Q U .Q U1 / U / xim+1 :Z +1

If Z  UV , where xim+1 2= F V (U ); V im )  idx(V; i1 ; : : : ; im)

(c)







6 xim

+1

and idx(U; i1 ; : : : ;





im xxii1m;:::;x :Z  BU xximi1+1:::xim :V +1   xi ;:::;xim  B U :V )  (xi1m+1  im .Q xim+1 :U xxii1m;:::;x :V xim+1 +1 

so by Theorem 44.1 and Lemma 57,


1 ;:::;xim xxiim :Z  .Q xim+1 :U V  xin+1 :U V +1 Now by the induction hypothesis we have a U1 and V1 such that:




1 ;:::;xim xxiim :Z  .Q xim+1 :U1 V1 / xim+1 :UV +1

 xim :Z +1

(d), (e), (f) The cases Z  UV where xim+1 2= F V (U ) and idx(U; i1 ; : : : ; in ) > idx(V; i1 ; : : : ; im) or where xi m+1 2 F V (U ) \ F V (U ) are similar.  Our classi cation of the BB0 I and BB0 IW-de nable -terms involves a class HRM (i1; : : : ; in ) of heriditary right maximal terms with respect to xi1 : : : xin ; which we now de ne. DEFINITION 59 ((HRM (i1 ; : : : ; in ))). 1. Every variable and every basis combinator is in HRM (i1; : : : ; in ).

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MARTIN BUNDER

2. If M; N 2 HRM (i1; : : : ; in ) and idx(M; i1 ; : : : ; in )  idx(N; i1 ; : : : ; in) then MN 2 HRM (i1 ; : : : ; in ). 3. If M 2 HRM (i1; : : : ; in+1 ) then xin+1 :M 2 HRM (i1; : : : ; in ). Strictly we should write HRMQ(i1 ; : : : ; in ), but in each case below the basis Q will be clear from the context. HRMBB0 I (1; : : : ; n) is HRM (x1 ; : : : ; xn ) of Hirokawa 1996. Our HRM (1; : : : ; n) is also HRMn of Trigg et al. [1994] where the basis is also taken from the context. Before obtaining the classi cations we need a lemma. LEMMA 60. If X is a BB0 IW-term or X  Y where Y is a BB0 IW-term and X .BB0 IW Z or Z . X then, if X is in HRM (i1; : : : ; in ), so is Z . Proof. If X is a BB0 IW-term this is easy to prove by induction on the length of the BB0 IW-reduction or of the -expansion. If X  Y , where Y is a BB0 IW-term more single BB0 IW-reductions are possible. For example instead of B0 UV W .B0 V (UW ) we can have (uvw:v(uw)) U V W .B0 (vw:v(U w))V W

.B0 (w:V (U w))W .B0 V (U W ); however in each case the membership of HRM (i1; : : : ; in) is preserved. Similarly for -expansions.  THEOREM 61. 1. The set of BB0 I-de nable terms is HRM ( ) \ Once( ). 2. The set of BB0 IW-de nable terms is HRM ( ) \ Once+ ( ). Proof. Theorem 58 gives translation algorithms. 1. If Y 2 HRM (i1; : : : in ) \ Once(i1 ; : : : ; in), it is easy to show by induc-0 tion on the length of Y that Y is BB0 I de nable by (xi1 ; : : : ; xin ; Y )ibb . This holds in particular when n = 0. If Y is BB0 I-de nable then there is a BB0 I-translation algorithm  and a -term Y1 such that Y .BB0 I Y1 / Y Now Y is a BB0 I-term and is therefore in HRM ( ) \ Once( ). It follows by Lemma 60 that Y1 and Y are also in HRM ( ) \ Once( ).

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2. is similar.  The (ibb0)-algorithm for BB0 I was rst used (for abstraction only) in [Helman, 1977]. It was also used by Hirokawa in his proof of the ) half of (B) in [Hirokawa, 1996]. Hirokawa proved the result of Theorem 61.1 there. 5 THE BASES BB0 IK; BB0 ; BB0 W AND BB0 K The BB0 IK(i1 ; : : : ; in ) abstractable terms of Trigg et al 1994 were terms obtainable from terms of HRM (i1; : : : ; in ) \ Once(i1 ; : : : ; in ) by deleting certain variables. The BB0 IK(i1 ; : : : ; in )-translation algorithm that we develop here has as its rst stage, a \full ordering algorithm" which reverses the deletion process by building up elements of a subclass of Once (i1 ; : : : ; in ) to elements of HRM (i10; : : : ; in ) \ Once(i1 ; : : : ; in). Such elements can then be translated by ( ; )(bb i) . A partial ordering algorithm, which builds up to elements of HRM (i1; : : : ; in ) \ Once (i1 ; : : : ; in ) could also be used and requires only simple alterations to 1. and 2. below. THE FULL ORDERING ALGORITHM Aim To extend, if possible, a  BB0 IK-term Y 2 Once (i1 ; : : : ; in) to a -BB0 IK-term Y o 2 HRM (i1; : : : ; in) \ Once(i1 ; : : : ; in ) so that Y o .KI Y . 1. If Y  a, an atom not in fxi1 ; : : : ; xin g Y o  K a (xi1 xi2 : : : xin ) 2. If Y  xim , and 1  m < n then Y o  K xim (xi1 : : : xim 1 xim+1 : : : xin )

3. If Y  xin , Y o  K I (xi1 : : : xin 1 )xin 4. If Y  xin+1 :Z , nd, if possible, Z o such that

Z o 2 HRM (i1 ; : : : ; in+1 ) \ Once (i1 ; : : : ; in+1)

and Z o .KI Z then Y o  xin+1 :Z o . 5. If Y

 Zxin , nd, if possible, Z o such that Z o 2 HRM (i1; : : : ; in 1 ) \ Once (i1 ; : : : ; in

and Z o .KI Z then Y o  Z oxin .

1)

264

MARTIN BUNDER

6. If Y  UV , where V 6 xin nd, by going back to (1), a term U o and a subsequence (xj1 ; : : : ; xjr ) of (xi1 ; : : : ; xin ) such that:

(a) F V (U ) \ fxi1 ; : : : ; xin g  fxj1 ; : : : ; xjr g and F V (V ) \ fxj 1 ; : : : ; xj r g = ;. (b) U o 2 HRM (j1 ; : : : ; jr ) \ Once(j1 ; : : : ; jr ) (c) U o .KI U (d) jr 6= in (e) maxfpjxjp 2 fxj1 ; : : : ; xjr g F V (U )g is minimal. (f) Given (e), the number of variables in fxj1 ; : : : ; xjr g F V (U ) is minimal. Now let (xs1 ; : : : ; xst ) be the sequence obtained from (xi1 ; : : : ; xin ) by removing (xj1 ; : : : ; xjr ). Now if possible (i.e. by going back to (1)) nd V o such that (g) V o 2 HRM (s1 ; : : : ; st ) \ Once(s1 ; : : : ; st ) and (h) V o .KI V then Y o  U oV o .

Choosing the maximal p in xjp 2 F V (U o ) \ fxi1 ; : : : ; xin g to be minimal in (e) and then using as few as possible variables new to U in (f), gives us maximal exibility for expanding V to V o using the remaining, especially the higher subscripted, variables. These clauses also ensure that a unique Y o is produced by the algorithm. Other Y o s satisfying the above aim may exist as well. The algorithm is applied in two examples below. EXAMPLE 62. Y  x3 x2 x1 cannot be ordered relative to (1; 2; 3) or even (1; 2; 3; 4), but relative to (1; 2; 3; 4; 5)

Y o  x3 (K x2 x4 )(K x1 x5 ) EXAMPLE 63.

Y  x7 x0 (x9 :x5 (x4 (x3 x2 ))x1 x9 ) Relative to (0; 1; 2; : : : 7; 8; 10; 11) we have Yo

 x7 (K x0 x8 )(x9 :x5 (x4 (x3 (K x2 x6 ))(K x1 (x10 x11 ))x9 ) 2 HRM (0; 1; 2; : : : ; 8; 10; 11) \ Once(0; 1; 2; : : : ; 8; 10; 11)

The -terms that are BB0 IK-translatable will be represented in terms of a generalisation of the class HRM (i1; : : : ; in). If it is not possible to extend a -BB0 IK-term Y to a Y o 2 HRM (i1; : : : ; in ) \ Once(i1 ; : : : ; in ), it is always

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possible to choose variables xin+1 ; : : : ; xim and a Y o 2 HRM (i1; : : : ; im) \ Once(i1 ; : : : ; im) so that Y o .KI Y . If (xj1 ; : : : ; xjr ) is (xi1 ; : : : ; xin ) with the free variables of Y deleted and if we named the atom occurrences in Y from the leftmost to the rightmost a1 ; : : : ; ap then Y o could be de ned as: Y with a1 replaced by Ka1 (xj1 : : : xjr xin+1 ) and ai (1 < i  p) replaced by Kai xin+i . Repeatedly using the full reordering algorithm, with n increased by one each time, will produce a minimal set of extra variables that need to be added to form a Y o. DEFINITION 64 (Potentially Right Maximal (i1 ; : : : ; in )--terms). (P RM (i1; : : : ; in )--terms) 1. If X is an atom X 2 P RM ( ). 2. xe 2 P RM (e). 3. If X 2 P RM (i1 ; : : : ; in 1 ) and xj 2= F V (X ) then X 2 P RM (i1 ; : : : ; ik ; ij ; ik+1 ; : : : ; in ) for 0  k  n. 4. If X 2 P RM (i1 ; : : : ; in+1 ) then xi n+1 :X 2 P RM (i1; : : : ; in ). 5. If X 2 P RM (j1 ; : : : ; jp ) and Y 2 P RM (r1 ; : : : ; rq ) where p = q = n = 0 or rq = in ,

fj1 ; : : : ; jp g \ fr1 ; : : : ; rq g = ;; F V (X ) \ fxr ; : : : ; xr q g = ;; F V (Y ) \ fxj 1 ; : : : ; xj p g = ;; 1

and (i1 ; : : : ; in ) is a merge of (j1 ; : : : ; jp ) and (r1 ; : : : ; rq ) (i.e. n = p + q and (i1 ; : : : ; in ) has the elements of the two sequences with the orders preserved) then XY 2 P RM (i1 ; : : : ; in ). EXAMPLE 65. so Similarly and so

x1 2 P RM (1) x1 2 P RM (1; 5) x2 2 P RM (2; 4) x3 2 P RM (3) x3 x2 2 P RM (2; 3; 4)

266

MARTIN BUNDER

and Note:

x3 x2 x1 2 P RM (1; 2; 3; 4; 5) x3 x2 x1 2= P RM (1; 2; 3) [ P RM (1; 2; 3; 4)

Note that the variables in fxi 1 ; : : : ; xi n g F V (X ) are used in the ordering of a term Y relative to (i1 ; : : : ; in), in the same way these extra variable subscripts are needed to show X 2 P RM (i1; : : : ; in ). The connection is given by the following lemmas. LEMMA 66. If Y , ordered relative to (i1 ; : : : ; in ) by the full ordering algorithm, becomes Y o , then Y o .KI Y where each single K-reduction eliminates one or more of xi1 ; : : : ; xin .

Proof. Obvious from the algorithm.



LEMMA 67. Y 2 P RM (i1; : : : ; in ) \ Once (i1 ; : : : ; in ) () there is a Y o 2 HRM (i1; : : : ; in) \ Once(i1 ; : : : ; in ) de ned by the full ordering algorithm.

Proof. ) By induction on Y . The case where Y is an atom is obvious. If Y  xi n+1 :Z , then as each bounded variable of Y appears at most once in Y , we have Z 2 P RM (i1; : : : ; in+1 ) \ Once (i1 ; : : : ; in+1). By the induction hypothesis we have an appropriate Z o 2 HRM (i1; : : : ; in+1 ) \ Once(i1 ; : : : ; in+1 ) and so a Y o  xin+1 :Z o 2 HRM (i1; : : : ; in ) \ Once(i1 ; : : : ; in ). If Y  Zxi n , then as each bounded variable of Y appears at most once in Y , we have Z 2 P RM (i1; : : : ; in 1) \ Once (i1 ; : : : ; in 1 ). By the induction hypothesis we have an appropriate

Z o 2 HRM (i1; : : : ; in 1 ) \ Once(i1 ; : : : ; in 1 ) and so a Y o  Z oxi n 2 HRM (i1; : : : ; in ) \ Once(i1 ; : : : ; in). If Y  UV (V 6 xin ) then we have (j1 ; : : : ; jp ) and (r1 ; : : : ; rq ) such that

fj1 ; : : : ; jp g \ fr1 ; : : : ; rq g = ;; F V (U ) \ fxr1 ; : : : ; xrq g = ;; rq = in ; or n = p = q = 0; F V (V ) \ fxj 1 ; : : : ; xj p g = ; U 2 P RM (j1 ; : : : ; jp ) \ Once (j1 ; : : : ; jp ) and

V

2 P RM (r1 ; : : : ; rq ) \ Once (r1 ; : : : ; rq )

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Also the order of (j1 ; : : : ; jp ) and (r1 ; : : : ; rq ) is preserved in (i1 ; : : : ; in ), where fi1; : : : ; in g = fj1 ; : : : ; jp g [ fr1 ; : : : ; rq g. Of the sequences (j1 ; : : : ; jp ); (r1 ; : : : ; rq ) that satisfy these properties (and so (a), (b), (c), (d), (e) and (h) of (4) of the full ordering algorithm), choose those that also satisfy (e) and (f). Then, as by the induction hypothesis we have:

U o 2 HRM (j1 ; : : : ; jp ) \ Once(j1 ; : : : ; jp ) and

V;o 2 HRM (r1 ; : : : ; rq ) \ Once(r1 ; : : : ; rq );

we have

Y o  U oV o 2 HRM (i1; : : : ; in ) \ Once(i1 ; : : : ; in ):

(

To prove this we only need to show, by the previous lemma, that K or I -reductions in -terms eliminating some of xi 1 ; : : : ; xi n , preserve membership of P RM (i1; : : : ; in ) in a reduction T .BB0 IK R. We prove this by induction on T . If T is an atom or contains an I redex this is obvious. If T  K W xi s 2 P RM (i1 ; : : : ; in ) then K W 2 P RM (j1; : : : ; jp ) and xis 2 P RM (r1 ; : : : ; rq ) where in = rq and the other conditions apply. From there it follows that xr1 ; : : : ; xr p 2= F V (W ). Also W 2 P RM (j1 ; : : : ; jp ) and as (j1 ; : : : ; jp ) is a subsequence of (i1 ; : : : ; in ), by 3,

W

2 P RM (i1; : : : ; in ):

 xin :W; R  xin :S and W . S , W 2 P RM (i1; : : : ; in+1 ) and by the induction hypothesis S 2 P RM (i1; : : : ; in+1 ) and so R 2 P RM (i1; : : : ; in ). If T  UV , where R  W S; U .BB0 IK W and V .BB0 IK S , we have: for If T

+1

+1

some (j1 ; : : : ; jp ) and (r1 ; : : : ; rq )

U 2 P RM (j1 ; : : : ; jp ) V 2 P RM (r1 ; : : : ; rq )

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with the usual conditions. By the induction hypothesis

W and

2 P RM (j1 ; : : : ; jp )

S 2 P RM (r1 ; : : : ; rq )

and so

R  W S 2 P RM (i1; : : : ; in )



LEMMA 68. If A. B and A.KI C then there is a term D such that B.KI D and C . D.

Proof. By induction on the number of reduction steps in A . B and a secondary induction on the number of steps in A .KI C (i.e. a standard Church-Rosser theorem proof.)  0 THEOREM 69. ( ; Y o )(ibb ) is a BB0 IK-translation algorithm. Proof. (A)0 holds as before. By Theorem 58.1 there is a term Y1 such that (( ; Y o )(ibb ) ) .BB0 I Y1 / Y o. Now also by Lemma 66 Y o .KI Y; so by Lemma 68 there is a Y2 such that Y1 .KI Y2 / Y 0

So (( ; Y o )(ibb ) ) .BB0 IK Y2 / Y , i.e. (B) holds.



0 Thus ( ; Y o )(ibb ) is a BB0 IK-translation algorithm. THEOREM 70. The set of BB0 IK-translatable terms is P RM ( )\ Once ( ). Proof. We have a BB0 IK-translation algorithm by Theorem 69. If Y 2 P RM ( )0\ Once ( ), then by Lemma 67 Y o 2 HRM ( ) \ Once( ) and so ( ; Y o )(ibb ) is a BB0 IK-term, so Y is BB0 IK-de nable. If Y is BB0 IK-de nable, the proof of Y 2 P RM ( ) \ Once ( ) proceeds as for Theorem 61. 

We now consider some bases without I. We de ne  as the set of all -terms whose -normal forms do not have subterms of the form xi1 : : : xin :xij xij+1 : : : xin or xi1 : : : xin :xi1 xi1 xi2 xi3 : : : xin . THEOREM 71.

COMBINATORS, PROOFS AND IMPLICATIONAL LOGICS

1. HRM ( ) \ 

\ Once is the set of BB0 -de nable terms.

2. HRM ( ) \ 

\ Once+ is the set of BB0 W-de nable terms.

3. P RM ( ) \ 

\ Once

269

is the set of BB0 K-de nable terms.

Proof. 3. Any BB0 K-translation algorithm  must contain (), otherwise, for example. B would not be de nable. If a Q-translation algorithm  contains () it is easy to show, by induction on the length of a Q-term Y , with -normal form Z that Y  Z . The full ordering algorithm is such that the combinator I is used only when it is essential (i.e. just K, B0 and B won't do) and it is clear that the only terms in -normal form in P RM ( ) \  \ Once that can have an I in their translation are of the form:xi1 : : : xin :xij xij+1 : : : xin . Hence our result follows from Theorems 69 and 70. 1. and 2. are similar but simpler.  Note that in a BB0 IW abstraction of a term only terms of the form xi1 : : : xin :xi1 xi2 : : : xin and xi1 : : : xin :xi1 xi1 xi2 : : : xin can contain an I and in BB0 I abstraction only the former. 6 THE  PROOF FINDING ALGORITHMS We have shown that for each theorem of a wide range of implicational logics there is a -term of a particular kind, depending on the logic, that has that theorem as a type. We have also shown that we can expand any such -term into long normal form. If a theorem, or type,  takes the form 1 ! : : : ! n ! a, where a is an atom, we know that any lnf inhabitant must take the form x1 : : : xn :X , where X has type a and has F V (X )  fx1 ; : : : ; xn g. This is the basis for the Ben-Yelles algorithm (see [Ben-Yelles, 1979; Hindley, 1997; Bunder, 1995]), which constructs potential inhabitants of a given type from the outside in. As there may be several potential X s with type a the process can branch several times. There is, at least for SK and sets of combinators without S and W a simple bound to this inhabitant, or proof, nding procedure. The version of the algorithm given in [Bunder, 1995], which is very eÆcient, has been implemented in [Dekker, 1996]. In this section we will be looking at an alternative inhabitant building algorithm, generally even more eÆcient, which builds the inhabitants of a type from the inside outwards. The given type provides the building blocks we use for this. This algorithm has been implemented in [Oostdijk, 1996].

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6.1 The Variables and Subterms of an Inhabitant of a Type In order to describe the way in which a -term inhabitant of a type is built up we need some notation. DEFINITION 72. 1.  is a positive subtype of a type . 2. If is a positive subtype of  or a negative subtype of then is a negative subtype of  ! . 3. If is a negative subtype of  or a positive subtype of then is a positive subtype of  ! . DEFINITION 73. An occurrence of a positive (negative) subtype of a type  is said to be long if the occurrence of is not the right hand part of a positive (negative) subtype  ! of  . All types other than atomic types are said to be composite. DEFINITION 74. The rightmost atomic subtype of a type is known as its tail. EXAMPLE 75.  = (a ! b) ! (b ! c) ! ((a ! b) ! c) ! c has b ! c; (a ! b) ! c, the second occurrence of a and the rst occurrence of a ! b as long negative subtypes and  , the rst occurrence of a and the second occurrences of b and a ! b as long positive subtypes of  . DEFINITION 76.

dn( ) = the number of distinct long negative subtypes of : do( ) = the number of occurrences of long negative subtypes of : dcp( ) = the number of distinct long positive composite subtypes of : dapn( ) = the number of distinct atoms that are tails of both long positive and long negative subtypes of : F ( ) = 2dn( )(dapn( ) + dcp( )) + dn( ): G( ) = 2do( )(dapn( ) + dcp( )) + do( ): j j = the total number of subtypes of : DEFINITION 77. 1. X is of -depth 0 in X . 2. If an occurrence of a term Y is of -depth d in X then Y is of -depth d in UX and XU , provided these are in -normal form. 3. If an occurrence of a term Y is of -depth d in xi :U , it is of -depth d in xj xi :U .

COMBINATORS, PROOFS AND IMPLICATIONAL LOGICS

4. If an occurrence of a term Y is of -depth d in V or U , then Y is of -depth d + 1 in xj :V .

271

6 xi :U for any xi

The two lemmas that now follow will help to tell us with what a long normal form inhabitant of a given type  can be constructed and also how to restrict the search for components. LEMMA 78. If X is a normal form inhabitant of  , U is a subterm of X of type  and V is a term of type  with F V (V )  F V (U ), then the result of replacing U by V in X is another inhabitant of  .

Proof. By a simple induction on the length of X .



EXAMPLE 79.

x1 x2 :x2 (x3 :x2 (x4 :x1 x3 x4 )) : (a ! a ! b) ! ((a ! b) ! b) ! b

Here x1 : a ! a ! b, x2 : (a ! b) ! b, x3 : a and x4 : a, so

x3 :x2 (x4 :x1 x3 x4 ) : a ! b

and So we have by Lemma 77:

x3 :x1 x3 x3 : a ! b:

x1 x2 :x2 (x3 :x1 x3 x3 ) : (a ! a ! b) ! ((a ! b) ! b) ! b:

LEMMA 80. If ` Z :  , then there is a term X in long normal form, in which no two distinct variables have the same type, such that: 1. ` X :  2. For every long subterm Y of X with F V (Y ) = fxi1 ; : : : ; xik g we have:

xi1 : i1 ; : : : ; xik : ik

`Y :

where is either a long occurrence of a composite positive subtype of  or an atom which is the tail of both a long positive and a long negative subtype of  . i1 ; : : : ; ik are distinct long negative subtypes of  .

Proof. 1. By Theorem 34 there is an X 0 in lnf such that ` X 0 :  . If X 0 now contains two variables xk and xe with the same type we can change any part xe :B (xk ; xe ) of X 0 to xk :B (xk ; xk ) and any part xe :B (xe ) to xk :B (xk ). Let X be the term obtained when all possible changes of this kind have been made to X 0 . In X no two distinct variables will have the same type.

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2. We prove this by induction on the -depth d of Y in X . If X is formed by application, as it is in long normal form, it takes the form: X  xi X1 : : : Xm and the type of X must be dependent on that of xi , which does not agree with 1. Hence X is not formed by application and if  = 1 ! : : : ! n ! a takes the form X  x1 : : : xn :xi X1 : : : Xm : (Note: some of x1 ; : : : ; xn could be identical.) If d = 0 and Y is long, we must have Y  X and the result holds with k = 0 and =  . If d = 1 and X = x1 : : : xn :xi X1 : : : Xm ; Y will appear in xi X1 : : : Xm , not in the scope of any xj s. Thus it appears in a part xt Z1 : : : Zr Y Zr+2 : : : Zq of X for some t(1  t  n). If t = 1 ! : : : ! q ! b we have, x1 : 1 ; : : : xn : n ` Y : r+1. r+1 is a long negative subtype of t and so a long positive subtype of  . 1 ; : : : ; n are occurrences of long negative subtypes of  . If r+1 is an atom, then r+1 must also be the tail of a long negative subtype of  (namely one of 1 ; : : : ; n ). Leaving out the variables not free in Y and variables identical to others gives the result. If d > 1, X  x1 : : : xn :xi X1 : : : Xj 1 (xn+1 : : : xk :xs Z1 : : : Zp )Xj+1 : : : Xm, where Y is a long subterm of xs Z1 : : : Zp . As xs Z1 : : : Zp is long and only in the scope of x1 : : : xn : : : xk , we have, if i = 1 ! : : : ! m ! c and j = n+1 ! : : : ! k ! b:

x1 : 1 ; : : : ; xn : n ; : : : ; xk : k ` xs Z1 : : : Zp : b: Thus ` x1 : : : xk :xs Z1 : : : Zp : 1 ! : : : ! k ! b where Y is a long subterm of x1 : : : xk :xs Z1 : : : Zp of -depth d 1. Thus 2. holds by the induction hypothesis with a long positive subtype of 1 ! : : : ! k ! b and so of  or an atom which is the tail of a long positive as well as of a long negative subtype of 1 ! : : : ! k ! b and so of  .

 Note The rst inhabitant given in Example 79 is a counterexample, due to Ryo Kashima,to an earlier version of Lemma 80. This claimed property 2.

COMBINATORS, PROOFS AND IMPLICATIONAL LOGICS

273

for any inhabitant X of the given type  in long normal form, rather than just some X . It follows from Lemmas 78 and 80 that an X in long normal form such that

` X : ; can be built up from subterms of the form

xr : : : xs :xi X1 : : : Xn ; where xi : i ; i is a long negative subtype of  , and i has tail a, where a is an atom which is also the tail of a positive subtype of  . The compound types of these subterms in long normal form must be among the long positive subtypes of  . With this in mind we arrive at the following algorithm for nding inhabitants of types, i.e. proofs in intuitionistic implicational logic.

6.2

-logic (H !)

SK

The H ! Decision Procedure or the SK Long Inhabitant Search Algorithm

Aim Given a type  , to nd a closed X in long normal form (if any) such that

`X: Step 1 To each distinct long negative subtype i of  assign a variable xi giving a nite list: x1 : 1 ; : : : ; xm : m Step 2 For each atomic type b that is the tail of both a long negative and a long positive subtype of  , form by application, (if possible) a lnf inhabitant Y of b if we don't already have a Y 0 : b with F V (Y 0 )  F V (Y ) in Step 1 or this or earlier Steps 2.

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MARTIN BUNDER

Step 3 For each long positive composite subtype of  , form, by abstraction, with respect to some of x1 ; : : : ; xm , a term Y in lnf such that Y : if we don't already have a Y 0 : with F V (Y 0 )  F V (Y ) in Step 1 or earlier Steps 3. If one of these terms Y is closed and has type  we stop. If there is no such term we continue with Steps 2 and 3 until we obtain no more terms with a \new" set of free variables or a new (atomic tail or long positive) type. If there are no new terms there is no solution X . Note In the work below and in all examples we will select our variables in Step 1 in the following order: x1 is assigned to the leftmost shallowest long negative subtype of ; x2 to the next to leftmost shallowest long negative subtype etc. until the shallowest long negative subtypes are used up. The next variable xn+1 is assigned to the leftmost next shallowest long negative subtype and eventually xm to the rightmost deepest long negative subtype. In Example 81 below, x1 to x4 are assigned to the shallowest subtypes and x5 to the next shallowest (the deepest) subtype. Because of this ordering of the variables of X any subterm Y formed by the algorithm will be in the scope of x1 x2 : : : xn xp1 xp1 +1 : : : xp2 xp3 : : : xp4 : : : xpq where n < p1 < p2 < p3 : : : < pq and where each xp2i+1 : : : xp2i+2 represents a single set of abstractions. EXAMPLE 81.  = [((a ! b) ! d) ! d] ! [(a ! b) ! d ! e] ! [a ! a ! b] ! a ! b Step 1 x1 : ((a ! b) ! d) ! d; x2 : (a ! b) ! d ! e; x3 : a ! a ! b; x4 : a; x5 : a ! b, Step 2 x3 x4 x4 : b; x5 x4 : b;

Step 3 x4 :x3 x4 x4 : a ! b; x4 :x5 x4 : a ! b; x1 x2 x3 x4 :x3 x4 x4 : : EXAMPLE 82.

 = ((a ! b) ! a) ! a:

Step 1 x1 : (a ! b) ! a; x2 : a:

Step 2 No new terms can be formed by application. Step 3 x1 :x2 : : No new terms can be formed. x1 :x2 is not closed so  has no inhabitants and no proof in H !. EXAMPLE 83.

 = [((a ! a) ! a) ! a] ! [(a ! a) ! a ! a] ! [a ! a ! a] ! a ! a:

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275

Step 1 x1 : ((a ! a) ! a) ! a; x2 : (a ! a) ! a ! a, x3 : a ! a ! a; x4 : a; x5 : a ! a; Step 2 x3 x4 x4 : a; x5 x4 : a; Step 3 x4 :x4 : a ! a, x5 :x5 x4 : (a ! a) ! a, x1 x2 x3 x4 :x3 x4 x4 :  . THEOREM 84. Given a type  , the SK long inhabitant search algorithm will, in nite time, produce an inhabitant of  or will demonstrate that  has no inhabitants. The algorithm will produce at most F ( ) terms before terminating.

Proof. It follows from the Weak Normalisation Theorem (see [Turing, 1942; Hindley, 1997]) that if  has an inhabitant, this inhabitant has a normal form and this will also have type  . By Lemma 80,  will have an inhabitant X of the form prescribed there. We show that our SK-algorithm provides such an inhabitant X of  . Step 1 of the SK-algorithm provides us with the largest set of variables x1 ; : : : ; xm that, by Lemma 80, need appear in a solution for X . Step 2 of the algorithm considers terms U = xi X1 : : : Xn with an atomic type having a particular subset of x1 ; : : : ; xm as free variables. By Lemma 78 other terms xj Y1 : : : Yk with the same atomic type and a superset of these free variables can at most produce alternative inhabitants and so do not need to be considered. The total number of variables we can have is dn( ). These are the terms generated by Step 1. The number of subsets of these is at most 2dn( ), the number of atomic types we can have is at most dapn( ) so the number of terms generated by Steps 2 is at most 2dn( ):dapn( ). Step 3 forms terms in long normal form which have composite long positive subtypes of  as types. There are dcp( ) of these and we can form at most one of these terms for each set of variables. Hence the most terms we can form using Steps 3 is 2dn( ):dcp( ). The maximal number of terms formed using the algorithm is therefore dn( ) + 2dn( ):(dapn( ) + dcp( )) = F ( ).  Note As dapn( ) + dcp( )  dp( ) where dp( ) is the number of occurrences of distinct positive subtypes in  , F ( ) < dn( )+2dn( ) :dp( ) < 2dn( )+dp( )  2d( ) where d( ) is the number of long subtypes of  so F ( ) < 2j j. In Example 1, F ( ) = 197 while 2d( ) = 212 and 2j j = 225. The actual number of terms formed by the algorithm was 10.

276

6.3

MARTIN BUNDER BCK

Logic

We now adapt our long Inhabitant Search Algorithm to search for BCK-terms in long normal form. By Theorem 51(3), these need to be elements of Once ( ). The BCK Logic Decision Procedure or the BCK-Long Inhabitant Search Algorithm Aim Given a type  to nd a closed BCK-de nable -term in long normal form (if any) such that ` X : : Step 1 To each occurrence of a long negative subtype i of  assign a variable xi giving a nite list:

x1 : 1 ; : : : ; xm : m : Step 2 For each atomic type b that is the tail of both a long positive and a long negative subtype of  form (if possible), by application, from the terms we have so far, an inhabitant Y of b such that no free variables appear more than once in Y , if we don't already have a Y 0 : b with F V (Y 0 )  F V (Y ). Step 3 For each long positive subtype of  , form, by abstraction, with respect to some of x1 ; : : : ; xm , a term Y such that Y : , if we don't already have a Y 0 : where F V (Y 0 )  F V (Y ). EXAMPLE 85.  = [((a ! b) ! d) ! d] ! [(a ! b) ! d ! e] ! [a ! a ! b] ! a ! b Step 1 x1 : ((a ! b) ! d) ! d; x2 : (a ! b) ! d ! e; x3 : a ! a ! b; x4 : a; x5 : a ! b; x6 : a; Step 2 x5 x4 : b; x5 x6 : b; x3 x4 x6 : b;

Step 3 x4 :x5 x4 : a ! b; x4 :x3 x4 x6 : a ! b; x6 :x3 x4 x6 : a ! b; x1 x2 x3 x4 :x5 x4 : ; x1 x2 x3 x6 :x3 x4 x6 : ; x1 x2 x3 x4 :x3 x4 x6 : : No new terms are generated by further uses of steps 2 and 3 and as the terms with type  are not closed, there is no BCK inhabitant of  . Note that  does have an SK inhabitant x1 x2 x3 x4 :x3 x4 x4 , but this is not BCK because x4 is used twice. EXAMPLE 86.  = ((a ! a) ! a ! a ! b) ! a ! a ! b

COMBINATORS, PROOFS AND IMPLICATIONAL LOGICS

277

Step 1 x1 : (a ! a) ! a ! a ! b; x2 : a; x3 : a; x4 : a; Step 2 No new terms are formed. Step 3 x2 :x2 : a ! a; Step 2 x1 (x2 :x2 )x2 x3 : b; x1 (x2 :x2 )x2 x4 : b; x1 (x2 :x2 )x3 x4 : b; Step 3 x1 x2 x3 :x1 (x2 :x2 )x2 x3 : : Notes: 1. The SK algorithm would have produced only x1 x2 x3 :x1 (x2 :x2 ) x2 x2 :  which is not a BCK--term. 2. It was essential here to have a variable for each distinct long negative occurrence of a in  . LEMMA 87. If X is a BCK term which is a normal form inhabitant of ; U a subterm of X of type  and V a BCK term of type , in which no free variable appears more than once, with F V (V )  F V (U ), then the result of replacing U by V in X is another BCK inhabitant of  .



Proof. As for Lemma 78. LEMMA 88. If such that: 1.

`BCK Z :  , then there is a term X

is long normal form

`BCK X : 

2. For every long subterm Y of X with F V (Y ) = fxi1 ; : : : ; xin g we have

xi1 : i1 ; : : : ; xin : in `BCK Y : ; where is either a long occurrence of a composite positive subtype of  or an atom which is the tail of both a long positive and a long negative subtype of  . i1 ; : : : ; in are distinct occurrences of long negative subtypes of  .

Proof. 1. The formation of an X in long normal form is as in the proof of Lemma 80(1) except that the extra variables xm+1 ; : : : ; xn that are choosen must not be free in xi X1 ; : : : ; Xm, (otherwise xm+1 : : : xn :xi X1 ; : : : ; Xmxm+1 ; : : : ; xn would not be a BCK-term). Also for the same reason, we don't identify distinct variables with the same type. We let X be a term obtained by the expansion of Z to long normal form.

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2. As for Lemma 80(2) except that we have to show that we have at most one variable for each distinct occurrence of a long negative subtype of  . We extend the induction proof to include this. When d = 1, we clearly have one variable for every long negative subtype 1 : : : ; n of  = 1 ! : : : ! n ! a and no others. When d > 1 the subterm Y appears as Zj in a term

xt U1 : : : Us where Ue = xu : : : xv :xr Z1 : : : Zp xt : 1 ! : : : ! s ! c and

w = Æu ! : : : ! Æv ! d and 1  j  p; 1  e  s and 1  w  s: The extra typed variables added at this stage are xu : Æu ; : : : ; xv : Æv . By the inductive hypothesis we have that 1 ! : : : ! s ! c is an occurrence of a long negative subtype of  and it therefore follows that the same holds for Æu ; : : : ; Æv . Note that as in BCK (and BCI) logic there can be only one free occurrence of the variable xt in a term before we abstract with respect to xt , so there can be no other use of xt that might generate another set of variables with types Æu ; : : : ; Æv i.e. one occurrence of 1 ! : : : ! q ! c in  generates at most one occurrence of a variable for each occurrence of Æu ; : : : ; Æv which are long negative subtypes of depth one lower than 1 ! : : : ! q ! c in .  THEOREM 89. Given a type  , the BCK long inhabitant search algorithm will in nite time produce an inhabitant or will demonstrate that  has no BCK-inhabitants. The algorithm will produce at most G( ) terms before terminating.

Proof. As for Theorem 84, except that Lemmas 87 and 88 replace Lemmas 78 and 80.  It might be thought that the BCK algorithm might require fewer than the maximal F ( ) terms required for SK, in fact it requires more because there may be several variables with the same type. Even when this is not the case, both algorithms require at most one term for each given type and each set of variables. For SK some variables may appear several times, for BCK they may not. For BCI in addition all abstracted variables will have to appear in the term being abstracted. The bounds F ( ) and G( ) are not directly related to standard complexity measures. The algorithm will in fact generate some other terms but not record them because they have the same type and set of free variables to another already recorded.

COMBINATORS, PROOFS AND IMPLICATIONAL LOGICS

6.4

BCI

279

Logic

Again we can adapt the Long Inhabitant Search Algorithm. This time the inhabitants found must, by Theorem 51(2) be in Once( ). The BCI-Logic Decision Procedure or the BCI Long Inhabitant Search Algorithm

Aim Given a type  to nd a closed BCI--term X in long normal form (if any) such that: ` X : : Method As for the BCK-algorithm except that Step 2 and 3 end in \F V (Y 0 ) = F V (Y )" and in Step 3 we may only form xi : : : xj :Z if xi ; : : : ; xj occur free in Z exactly once each. EXAMPLE 90.  = ((a ! b) ! c) ! b ! d ! (c ! e) ! e Step 1 x1 : (a ! b) ! c; x2 : b; x3 : d; x4 : c ! e; x5 : a Step 2 No new terms are formed. Step 3 x5 :x2 is not a BCI--term, so no new terms are generated and so there is no BCI-proof of  . EXAMPLE 91. ((a ! b ! c) ! d) ! (b ! a ! c) ! d:

Step 1 x1 : (a ! b ! c) ! d; x2 : b ! a ! c; x3 : a; x4 : b; Step 2 x2 x4 x3 : c; Step 3 x3 x4 :x2 x4 x3 : a ! b ! c; Step 2 x1 (x3 x4 :x2 x4 x3 ) : d; Step 3 x1 x2 :x1 (x3 x4 :x2 x4 x3 ) : : Example 86 also produced a BCI-term. THEOREM 92. Given a type  , the BCI long inhabitant search algorithm will, in nite time, produce an inhabitant or will demonstrate that  has no inhabitants. The algorithm will produce at most G( ) terms before terminating.

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Proof. Lemma 87 holds provided that V is a BCI term such that F V (V ) = F V (U ) and each free variable of V appears exactly once in V . If Z in Lemma 87 is a BCI term so is X . The proof of the theorem now proceeds as for Theorem 89 except that in any substitution [V=U ]X the above restrictions apply. Hence F V (Y 0 ) = F V (Y ) was needed in Step 2 of the BCI-algorithm. In Steps 2 and 3 we must, for each subset of fx1 ; : : : ; xm g, consider a term with those free variables with a particular atomic negative or long positive subtype of T .



6.5

BCIW

Logic (R !)

The BCI-search algorithm can easily be extended to BCIW-logic. By Theorem 51(4), the -terms required are from Once+ (). The BCIW Long Inhabitant Search Algorithm

Aim Given a type  to nd a closed BCIW--term X in long normal form (if any) such that `X: Method As for the BCI algorithm except that in Steps 2 and 3 each variable must appear in Y and Z at least once. If the algorithm that we have to this stage fails, additional variables with the same types as the ones rst given in Step 1 are added and the previous algorithm is repeated. Note that, as it is not clear as to how many times new variables might need to be added, this method, while, as shown in Theorem 96, it leads to nding an inhabitant if there is one, does not constitute a decision procedure. The need for extra variables is illustrated in Example 95 below. This logic does have a decision procedure (see [Urquhart, 1990]), but its maximum complexity is related to Ackermann's function. EXAMPLE 93.  = [((a ! b) ! d) ! d] ! [(a ! b) ! d ! e] ! [a ! a ! b] ! a ! b Step 1 x1 : ((a ! b) ! d) ! d; x2 : (a ! b) ! d ! e; x3 : a ! a ! b; x4 : a; x5 : a ! b; x6 : a Step 2 x3 x4 x4 : b; x3 x4 x6 : b; x3 x6 x6 : b; x5 x4 : b; x5 x6 : b

Step 3 x4 :x3 x4 x4 : a ! b; x4 :x3 x4 x6 : a ! b; x6 :x3 x4 x6 : a ! b; x4 :x5 x4 : a ! b

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281

We cannot form x1 x2 x3 x4 :x3 x4 x4 as this is not a BCIW--term. We can form no more terms by Step 2 as we have no terms of type d or (a ! b) ! d and no new terms of type a. Hence  has no BCIW inhabitant. EXAMPLE 94.

 = (c ! a ! a ! a) ! c ! a ! a

Step 1 x1 : c ! a ! a ! a; x2 : c; x3 : a: Step 2 x1 x2 x3 x3 : a: Step 3 x1 x2 x3 :x1 x2 x3 x3 : : EXAMPLE 95.

 = c ! c ! (a ! a ! b) ! (c ! (a ! b) ! b) ! b

x1 : c; x2 : c; x3 : a ! a ! b; x4 : c ! (a ! b) ! b; x5 : a x3 x5 x5 : b x5 :x3 x5 x5 : a ! b x4 x1 (x5 :x3 x5 x5 ) : b; x4 x2 (x5 :x3 x5 x5 ) : b; Step 3 No new terms can be formed. (Add to) Step 1 x6 : a Step 2 x3 x5 x6 : b, x3 x6 x6 : b Step 3 x6 :x3 x5 x6 : a ! b; x5 :x3 x5 x6 : a ! b Step 2 x4 x1 (x6 :x3 x5 x6 ) : b; x4 x2 (x6 :x3 x5 x6 ) : b; x4 x1 (x5 :x3 x5 x6 ) : b; x4 x2 (x5 :x3 x5 x6 ) : b; Step 3 x5 :x4 x1 (x6 :x3 x5 x6 ) : a ! b; x5 :x4 x2 (x6 :x3 x5 x6 ) : a ! b; Step 2 x4 x1 (x5 :x4 x1 (x6 :x3 x5 x6 )) : b; x4 x2 (x5 :x4 x1 (x6 :x3 x5 x6 )) : b; x4 x2 (x5 :x4 x2 (x6 :x3 x5 x6 )) : b; Step 3 x1 x2 x3 x4 :x4 x2 (x5 :x4 x1 (x6 :x3 x5 x6 )) : : Note that in Example 95 x4 is used twice and so the one occurrence of a in c ! (a ! b) ! b, requires two variables of type a. If these were identi ed the resultant -term would no longer be BCIW-de nable. THEOREM 96. Given a type  , the BCIW long inhabitant search algorithm will, in nite time, produce an inhabitant. Step Step Step Step

1 2 3 2

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Proof. Lemma 87 holds for BCIW-terms provided that we have F V (U ) = F V (V ). If Z in Lemma 88 is a BCIW-term so is X . In the counterpart to Lemma 88 the word \distinct" must also be dropped for the reasons illustrated in Example 95 above. The proof of the theorem now proceeds as that of Theorem 91 except that multiple copies of variables may appear in substitutions and in terms formed by the algorithm. 

Logic (T !) 0 BB IW search algorithm nds -terms of a given type that are in HRM ( )\ Once+ ( ), as required by Theorem 61. 6.6

BB0 IW

The BB0 IW Long Inhabitant Search Algorithm Aim Given a type  to nd a closed BB0 IW--term X in long normal form (if any) such that `X: Step 1 As for BCIW.

Step 2 For each atom b and for each subsequence (j1 ; : : : ; jr ) of (1; : : : ; m) nd one BB0 IW--term Y  xj i X1 : : : Xk (k  0; 1  i  r), such that Y 2 HRM (j1 ; : : : ; jr ); Y : b and F V (Y ) = fxj1 ; : : : ; xjr g, if there is not already such a Y . Step 3 For each subsequence (j1 ; : : : ; jr ) of (1; : : : ; m) and for each long positive subtype of  , form a BB0 IW--term Y by abstraction so that Y : and F V (Y ) = fxj1 ; : : : ; xj r g, if we don't already have such a Y . Now repeat steps 2 and 3 and if needs be add extra variables as for BCIW. As with the BCIW-algorithm this does not, in general, provide a decision procedure. EXAMPLE 97.  = (a ! b ! c ! d) ! a ! c ! b ! d Step 1 x1 : a ! b ! c ! d; x2 : a; x3 : c; x4 : b Step 2 x1 x2 x4 x3 : d and x1 x2 x4 x3 2 HRM (1; 2; 4; 3) Step 3 The only term with a positive subtype of  that can be formed is x1 x2 x3 x4 :x1 x2 x4 x3 : ; but this is not a BB0 IW--term. Adding extra variables with the same types only allows us to generate this same (modulo- conversion) inhabitant of  .

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EXAMPLE 98.

 = (c ! a ! a ! a) ! c ! a ! a The only BCIW de nable--term inhabitant of  was x1 x2 x3 :x1 x2 x3 x3 ; this is also a BB0 IW de nable -term. THEOREM 99. Given a type  , the BB0 IW long inhabitant search algorithm will, in nite time, produce an inhabitant.

Proof. As for Theorem 96, except that we can replace subterms only by subterms belonging to the same class HRM (j1 ; : : : ; jr ) (1  j1 < : : : < jr  m). 

6.7

BB0 I

Logic (T-W, P-W)

The search algorithm for this logic nds elements of the appropriate type in HRM ( ) \ Once( ), as requried by Theorem 61(1). The BB0 I Logic or T-W(P-W) Decision Procedure or the BB0 I Long Inhabitant Search Algorithm Aim Given a type  to nd a closed BB0 I--term X in long normal form (if any) such that `X:

Method As for BB0 IW logic except that in the terms formed in Step 2 no free variable may appear twice and that no extra variables need be added. EXAMPLE 100.  = (c ! a ! a ! a) ! c ! a ! a The only BB0 IW inhabitant of  is x1 x2 x3 :x1 x2 x3 x3 and this is not a BB0 I-de nable -term. Thus  has no BB0 I inhabitants. EXAMPLE 101.

 = [(a ! a) ! a] ! (a ! a) ! a Step 1 x1 : (a ! a) ! a; x2 : a ! a; x3 : a Step 2 x2 x3 : a Step 3 x3 :x3 : a ! a

x3 :x2 x3 : a ! a

Step 2 x1 (x3 :x2 x3 ) : a; x1 (x3 :x3 ) : a

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Step 3 x1 x2 :x1 (x3 :x2 x3 ) :  THEOREM 102. Given a type  the BB0 I long inhabitant search algorithm will, in nite time, produce an inhabitant or will demonstrate that  has no inhabitants. The algorithm will produce at most G( ) terms before terminating.

Proof. As for Theorem 99, except that each variable xi must appear exactly once in Y in any xi :Y as with BCI logic. Also as in Theorem 89 and 92 the procedure can be bounded. Note that the number of subsequences of a sequence is the same as the number of subsets of the corresponding set. 

6.8

BB0 IK

Logic

The search algorithm for this logic nds -terms of the appropriate type in P RM ( ) \ Once ( ). The BB0 IK Logic Decision Procedure or the BB0 IK Long Inhabitant Search Algorithm

Aim Given a type  to nd a closed BB0 IK- term X in long normal form (if any) such that ` X : : Method As for BB0 I logic except that in Step 2 HRM is replaced by P RM . EXAMPLE 103.  = b ! (b ! c) ! a ! c Step 1 x1 : b; x2 : b ! c; x3 : a Step 2 x2 x1 : c (x2 x1 2 P RM (1; 2; 3)) Step 3 x1 x2 x3 :x2 x1 2 : THEOREM 104. Given a type  the BB0 IK long inhabitant search algorithm will, in nite time, produce an inhabitant or will demonstrate that  has no inhabitant. The algorithm will produce at most G( ) terms before terminating.

Proof. As for Theorem 99.



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6.9 Some Other Logics Bunder [1996] also gives the -terms de nable in terms of the combinators BB0 , BT, BB0 K, BIT, BITK, BITW, BTK, and BTW. Those for BCW and BB0 W can easily be found. Inhabitant nding algorithms for these logics can easily be obtained as above. Decision procedures can be obtained for those without W. University of Wollongong, Australia

BIBLIOGRAPHY [Barendregt, 1984] H. P. Barendregt. The Lambda Calculus, North Holland, Amsterdam, 1984. [Ben-Yelles, 1979] C.-B. Ben Yelles. Type Assignments in the Lambda Calculus, Ph.D. thesis, University College, Swansea, Wales, 1979. [Bunder, 1996] M. W. Bunder. Lambda terms de nable as combinators. Theoretical Computer Science, 169, 3{21, 1996. [Bunder, 1995] M. W. Bunder. Ben-Yelles type algorithms and the generation of proofs in implicational logics. University of Wollongong, Department of Mathematics, Preprint Series no 3/95, 1995. [Church, 1932] A. Church. A set of postulates for the foundation of logic. Annals of Mathematics, 33, 346{366, 1932. [Church, 1933] A. Church. A set of postulates for the foundation of logic. (second paper). Annals of Mathematics, 34, 839{864, 1933. [Crossley and Shepherdson, 1993] J. N. Crossley and J. C. Shepherdson. Extracting programs from proofs by an extension of the Curry-Howard process. In Logical Methods, J. N. Crossley, J. B. Remmel, R. A. Shore and M. E. Sweedler, eds. pp. 222{288. Birkhauser, Boston, 1993. [Curry, 1930] H. B. Curry. Grundlagen der Kombinatorischen Logik. American Journal of Mathematics, 52, 509{536, 789{834, 1930. [Curry and Feys, 1958] H. B. Curry and R. Feys. Combinatory Logic, Vol 1, North Holland, Amsterdam, 1958. [de Bruin, 1970] N. G. de Bruin. The Mathematical Language AUTOMATH, Its Usage, and Some of Its Extensions. Vol. 125 of Lecture Notes in Mathematics, Springer Verlag, Berlin, 1970. [de Bruin, 1980] N. G. de Bruin. A survey of the project AUTOMATH. In To H. B. Curry, Essays on Combinatory Logic, Lambda Calculus and Formalism, J. R. Hindley and J. P. Seldin eds. pp. 576{606. Academic Press, London, 1980. [Dekker, 1996] A. H. Dekker. Brouwer 0.7.9- a proof nding program for intuitionistic, BCI, BCK and classical logic, 1996. [Helman, 1977] G. H. Helman. Restricted lambda abstraction and the interpretation of some nonclassical logics, Ph.D. thesis, Philosophy Department, University of Pittsburgh, 1977 [Hindley, 1997] J. R. Hindley. Basic Simple Type Theory. Cambridge University Press, Cambridge, 1997. [Hindley and Seldin, 1986] J. R. Hindley and J. P. Seldin. Introduction to Combinators and -Calculus. Cambridge University Press, Cambridge, 1986. [Hirokawa, 1996] S. Hirokawa. The proofs of  !  in P W . Journal of Symbolic Logic, 61, 195{211, 1996. [Howard, 1980] W. A. Howard. The formulae-as-types notion of construction. In To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J. R. Hindley and J. P. Seldin eds. pp 479{490. Academics Press, London, 1980.

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[Lauchli, 1965] H. Lauchli. Intuitionistic propositional calculus and de nably non-empty terms (abstract). Journal of Symbolic Logic, 30, 263, 1965. [Lauchli, 1970] H. Lauchli. An abstract notion of realizability for which intuitionistic predicate calculus is complete. In Intuitionism and Proof Theory, A. Kino et al., eds. pp. 227{234. North Holland, Amsterdam, 1970. [Oostdijk, 1996] M. Oostdijk. LambdaCal. 2|proof nding algorithms for intuitionistic and various (sub)classical propositional logics. Available on: http://www.win.tue.nl/martijno/lambdacal/html, 1996.  die Bausteine der mathematische Logik. Math[Schon nkel, 1924] M. Schon nkel. Uber ematische Annale, 92, 305{316, 1924. English translation in From Frege to Godel, J. van Heijenoort, ed. pp. 355{366. Harvard University Press,1967. [Scott, 1970] D. S. Scott. Constructive validity, In Vol 125 of Lecture notes in Mathematics, pp. 237{275. Springer Verlag, Berlin, 1970. [Trigg et al., 1994] P. Trigg, J. R. Hindley and M. W. Bunder. Combinatory abstraction using B,B0 and friends. Theoretical Computer Science, 135, 405{422, 1994. [Turing, 1942] A. M. Turing. R. O. Gandy An early proof of normalization by A. M. Turing. In To H. B. Curry: Essays... , pp. pp 453-455, 1980. [Urquhart, 1990] A. Urquhart. The complexity of decision procedures in relevant logic. In Truth or Consequences: Essays in honour of Nuel Belnap. J. M. Dunn and A. Gupta, eds. pp. 61{75, Reidel, Dordrecht, 1990.

GRAHAM PRIEST

PARACONSISTENT LOGIC

Indeed, even at this stage, I predict a time when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from `consistency'. Ludwig Wittgenstein, 1930.1

1 INTRODUCTION Paraconsistent logics are those which permit inference from inconsistent information in a non-trivial fashion. Their articulation and investigation is a relatively recent phenomenon, even by the standards of modern logic. (For example, there was no article on them in the rst edition of the Handbook.) The area has grown so rapidly, though, that a comprehensive survey is already impossible. The aim of this article is to spell out the basic ideas and some applications. Paraconsist logic has interest for philosophers, mathematicians and computer scientists. As be ts the Handbook, I will concentrate on those aspects of the subject that are likely to be of more interest to philosopher-logicians. The subject also raises many important philosophical issues. However, here I shall tread over these very lightly|except in the last section, where I shall tread over them lightly. I will start in part 2 by explaining the nature of, and motivation for, the subject. Part 3 gives a brief history of it. The next three parts explain the standard systems of paraconsistent logic; part 4 explains the basic ideas, and how, in particular, negation is treated; parts 5 and 6 discuss how this basic apparatus is extended to handle conditionals and quanti ers, respectively. In part 7 we look at how a paraconsistent logic may handle various other sorts of machinery, including modal operators and probability. The next two parts discuss the applications of paraconsistent logic to some important theories; part 8 concerns set theory and semantics; part 9, arithmetic. The nal part of the essay, 10, provides a brief discussion of some central philosophical aspects of paraconsistency. In writing an essay of this nature, there is a decision to be made as to how much detail to include concerning proofs. It is certainly necessary to include many proofs, since an understanding of them is essential for anything other than a relatively modest grasp of the subject. On the other hand, to prove everything in full would not only make the essay extremely long, but distract from more important issues. I hope that I have struck a happy via media. 1 Wittgenstein [1975], p. 332.

D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 6, 287{393.

c 2002, Kluwer Academic Publishers. Printed in the Netherlands.

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Where proofs are given, the basic de nitions and constructions are spelled out, and the harder parts of the proof worked. Routine details are usually left to the reader to check, even where this leaves a considerable amount of work to be done. In many places, particularly where the material is a dead end for the purposes of this essay, and is easily available elsewhere, I have not given proofs at all, but simply references. Those for whom a modest grasp of the subject is suÆcient may, I think, skip all proofs entirely. Paraconsistent logic is strongly connected with many other branches of logic. I have tried, in this essay, not to duplicate material to be found in other chapters of this Handbook, and especially, the chapter on Relevant Logic. At several points I therefore defer to these. There is no section of this essay entitled `Further Reading'. I have preferred to indicate in the text where further reading appropriate to any particular topic may be found.2 2 DEFINITION AND MOTIVATION

2.1 De nition The major motivation behind paraconsistent logic has always been the thought that in certain circumstances we may be in a situation where our information or theory is inconsistent, and yet where we are required to draw inferences in a sensible fashion. Let ` be any relationship of logical consequence. Call it explosive if it satis es the condition that for all  and , f; :g ` , ex contradictione quodlibet (ECQ). (In future I will omit set braces in this context.) Both classical and intuitionist logics are explosive. Clearly, if ` is explosive it is not a sensible inference relation in an inconsistent context, for applying it gives rise to triviality: everything. Thus, a minimal condition for a suitable inference relation in this context is that it not be explosive. Such inference relationships (and the logics that have them) have come to be called paraconsistent.3 Paraconsistency, so de ned, is something of a minimal condition for a logic to be used as envisaged; and there are logics that are paraconsistent but not really appropriate for the use. For example, Johansson's minimal logic is paraconsistent, but satis es ; : ` : . One might therefore attempt a stronger constraint on the de nition of `paraconsistent', such as: for no syntactically de nable class of sentences (e.g., negated sentences), , do 2 The most useful general reference is Priest et al. [1989] (though this is already a little dated). That book also contains a bibliography of paraconsistency up to about the mid-1980s. 3 The word was coined by Miro Quesada at the Third Latin American Symposium on Mathematical Logic, in 1976. Note that a paraconsistent logic need not itself have an inconsistent set of logical truths: most do not. But there are some that do, e.g., any logic produced by adding the connexivist principle :( ! :) to a relevant logic at least as strong as B . See Mortensen [1984].

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we have ; : ` , for all  2 . This seems too strong, however. In many logics, ; : ` , for every logical truth, . If the logic is decidable, then there is a clear sense in which the set of logical truths is syntactically characterisable. Yet such logics would still be acceptable for many paraconsistent purposes. Hence, this de nition would seem to be too strong.4 In his [1974], da Costa suggests another couple of natural constraints on a paraconsistent logic, of a rather dierent nature. One is to the eect that the logic should not contain :( ^ :) as a logical truth. The rationale for this is not spelled out. However, I take it that the idea is that if one has information that contains  and : one does not want to have a logical truth that contradicts this. Why not though? Since one is not ruling out inconsistency a priori, there would seem to be nothing a priori against this (though maybe for particular applications one would not want the situation to arise). As a general condition, then, it seems too strong. And certainly a number of the logics that we will consider have :( ^:) as a logical truth. Another of the constraints that da Costa suggests is to the eect that the logic should contain as much of classical|or at least intuitionist|logic, as does not interfere with its paraconsistent nature. The condition is somewhat vague, though its intent is clear enough; and again, it is too strong. It assumes that a paraconsistent logician must have no objection to other aspects of classical or intuitionist logic, and this is clearly not true. For example, a relevant logician might well object to paradoxes of implication, such as  ! ( ! ).5 As an aside, let me clarify the relationship between relevant logics and paraconsistent logics. The motivating concern of relevant logic is somewhat dierent from that of paraconsistency, namely to avoid paradoxes of the conditional. Thus, one may take a relevant (propositional) logic to be one such that if  ! is a logical truth then  and share a propositional parameter. The interests of relevant and paraconsistent logics clearly converge at many points. Relevant logics and paraconsistent logics are not coextensive, however. There are many paraconsistent logics that are not relevant, as we shall see. The relationship the other way is more complex, since there are dierent ways of using a relevant logic to de ne a consequence relation. A natural way is to say that  ` i  ! is a logical truth. Such a consequence relation is clearly paraconsistent. Another is to de ne logical consequence as deducibility, de ned in the standard way, using some set of axioms and rules for the relevant logic. Such a consequence relation may, but need not, be relevant. For example, Ackermann's original formulation of E contained the rule : if `  and ` : _ then ` . This gives explo4 Further attempts to tighten up the de nition of paraconsistency along these lines can be found in Batens [1980] (in the de nition of `A-destructive', p. 201, clause (i) should read 6`L A), and Urbas [1990]. 5 Indeed, it is just this principle that ruins minimal logic for serious paraconsistent purposes. For  and  ! ? (i.e., :) give ?, and the principle then gives ! ?.

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sion by an argument often called the `Lewis Independent Argument', that we will meet in a moment. Anyway, and to return from the digression: the de nition of paraconsistency given here is weaker than suÆcient to guarantee sensible application in inconsistent contexts; but an elegant stronger de nition is not at hand, and since the one in question has become standard, I will use it to de ne the contents of this essay.

2.2 Inconsistency and Dialetheism Numerous examples of inconsistent information/theories from which one might want to draw inferences in a controlled way have been oered by paraconsistent logicians. For example: 1. information in a computer data base; 2. various scienti c theories; 3. constitutions and other legal documents; 4. descriptions of ctional (and other non-existent) objects; 5. descriptions of counterfactual situations. The rst of these is fairly obvious. As an example of the second, consider, e.g., Bohr's theory of the atom, which required bound electrons both to radiate energy (by Maxwell's equations) and not to (since they do not spiral inwards towards the nucleus). As an example of the third, just consider a constitution that gives persons of kind A the right to do something, x, and forbids persons of kind B from doing x. Suppose, then, that a person in both categories turns up. (We may assume that it had never occurred to the legislators that there might be such a person.) In the fourth case, the information (in, say, a novel or a myth) characterises an object, and turns out|deliberately or otherwise|to be inconsistent. To illustrate the fth, suppose, for example, that we need to compute the truth of the conditional: if you were to square the circle, I would give you all my money. Applying the Ramsey-test, we see what follows from the antecedent (which is logically impossible), together with appropriate background assumptions. (And I would not give you all my money!)6 There is no suggestion here that in every case one must remain content with the inconsistent information in question. One might well like to remove 6 Many of these examples are discussed further in Priest et al. [1989], ch. 18. The Bohr case is discussed in Brown [1993]. Another kind of example that is sometimes cited is the information provided by witnesses at a trial. I nd this less persuasive. It seems to me that the relevant information here is all of the form: witness x says so and so. (That a witness is lying, or making an honest mistake, is always a possibility to be taken into account.) And any collection of statements of this form is quite consistent.

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some of the inconsistent information in the data base; reject or revise the scienti c theory; change the law to eliminate the inconsistency. But this is not possible in all of the cases given, e.g., for counterfactual conditionals with impossible antecedents. And even where it is, this not only may take time; it is often not clear how to do so satisfactorily. (The matter is certainly not algorithmic.) While we gure out how to do it, we may still be in a situation where inference is necessary, perhaps for practical ends, e.g., so that we can act on the information in the data base; or manipulate some piece of scienti c technology; or make decisions of law (on other than an obviously inconsistent case). Moreover, since there is no decision procedure for consistency, there is no guarantee that any revision will achieve consistency. We cannot, therefore, be sure that we have succeeded. (This is particularly important in the case of the data base, where the deductions go on \behind our back", and the need to revise may never become apparent.) In cases of this kind, then, even though we may not, ideally, be satis ed with the inconsistent information, it may be desirable|indeed, practically necessary|to use a paraconsistent logic. Moreover, we know that many scienti c theories are false; they may still be important because they make correct predications in most, or even all, cases; they may be good approximations to the truth, and so on. These points remain in force, even if the theories in question contain contradictions, and so are (thought to be) false for logical reasons. Of course, this is not so if the theories are trivial; but that's the whole point of using a paraconsistent logic. One can thus subscribe to the use of paraconsistent logics for some purposes without believing that inconsistent information or theories may be true. The view that some are true has come to be called dialetheism, a dialetheia being a true contradiction.7 If the truth about some subject is dialetheic then, clearly, a paraconsistent logic needs to be employed in reasoning about that subject. (I take it to be uncontentious that the set of truths is not trivial. Why this is so, especially once one has accepted dialetheism is, however, a substantial question.) Examples of situations that may give rise to dialetheias, and that have been proposed, are of several kinds, including: 1. certain kinds of moral and legal dilemmas; 2. borderline cases of vague predicates; 3. states of change. Thus, one may suppose, in the legal example mentioned before, that a person who is A and B both has and has not the right to do x; or that in 7 The term was coined by Priest and Routley in 1981. See Priest et al. [1989], p. xx. Note that some writers prefer `dialethism'.

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a case of light drizzle it both is and is not raining; or that at the instant a moving object comes to rest, it both is and is not in motion.8 The most frequent and, arguably, most persuasive examples of dialetheias that have been given are the paradoxes of self-reference, such as the Liar Paradox and Russell's Paradox. What we have in such cases, are apparently sound arguments resulting in contradictions. There are many suggestions as to what is wrong with such arguments, but none of them is entirely happy. Indeed, in the case of the semantic paradoxes there is not (even after 2,000 years) any consensus concerning the most plausible way to go. This gives the thought that the arguments are, after all, sound, its appeal.9 Naturally, all the examples cited in this section are contestable. I will return to the issue of possible objections in the last part of this essay. 3 A BRIEF HISTORY OF PARACONSISTENT LOGIC

3.1 The Law of Non-contradiction and Paraconsistency During the history of Western Philosophy, there have been a number of gures who deliberately endorsed inconsistent views.10 The earliest were some Presocratics, including Heraklitus. In the middle ages, some Neo-Platonists, such as Nicholas of Cusa, endorsed contradictory views. In the modern period, the most notable advocate of inconsistent views was Hegel.11 These gures are relatively isolated, however. It is something of an understatement to say that the dominant orthodoxy in Western Philosophy has been strongly hostile to inconsistency.12 Consistency has been taken to be pivotal to a number of fundamental notions, such as truth and rational belief. This antipathy to contradiction is, historically, due in large part to Aristo8 Many of these examples are discussed further in Priest et al. [1989], ch. 18, 2.2. A discussion of transition states and legal dialetheias can be found in chs. 11 and 13 of Priest [1987]. Moral dilemmas are also discussed in Routley and Routley [1989]. The dialetheic nature of vagueness is advocated in Pe~na [1989]. It has also been suggested that some contradictions in the Hegel/Marx tradition are dialetheic. For a discussion of this, see Priest [1989a]. 9 For further discussion, see Routley [1979] and Priest [1987], chs. 1-3. 10 And nearly every great philosopher has unwittingly endorsed inconsistent views. 11 In each case, there is, of course, some|though, I would argue, misguided|possibility for exegetical attempts to render the views consistent. Other modern philosophers whose thought also appears to endorse inconsistency are Meinong and the later Wittgenstein. In their cases there is more scope for exegetical evasion. For further discussion on all these matters, see Priest et al [1989], chs. 1, 2. 12 Eastern philosophy has been notably less so|though there is, again, room for exegetical debate. The most natural interpretation of Jaina philosophy has them endorsing inconsistent positions. And major Buddhist logicians of the stature of Nargarjuna held that it was quite possible for statements to be both true and false. Signi cant elemements of inconsistency can also be found in Chinese philosophy. For further discussion of all this, see Priest et al [1989], ch. 1, sect. 2.

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tle's defense of the Law of Non-contradiction in the Metaphysics.13 Given this situation, it may therefore be surprising that the orthodoxy against paraconsistency is a relatively recent phenomenon.

3.2 Paraconsistency Before the Twentieth Century The major account of validity until this century was, of course, Aristotelian Syllogistic. Now, consider any sentences of the Syllogistic E and I forms; for example, `No women are white' and `Some women are white'. These are contradictories. But the inference from them to, e.g., `All cows are black', is not a valid syllogism. Syllogistic is not, therefore, explosive: it is paraconsistent. It might be suggested that it is more appropriate to look for explosion in accounts of propositional inference. Here the story is more complex, but the conclusion is similar. Aristotle had no elaborated account of propositional inference. However, there are comments that bear on the matter scattered through the Organon, and they have a distinctly paraconsistent avour. For example, in the Prior Analytics (57b3), Aristotle states that contradictories cannot both entail the same thing. It would seem to follow that Aristotle did not endorse at least one of (in modern notation) ^: `  and ^: ` :. For contraposing (a move that Aristotle endorses immediately before), we obtain  ` :( ^ :) and : ` :( ^ :). Hence, not everything can follow from a contradiction. In fact, there are reasons to suppose that Aristotle held a view of negation according to which the negation of any claim cancels that claim out. A contradiction has, therefore, no content, and entails nothing. This view of negation (which would now be called `connexivist') was endorsed by a number of subsequent logicians (notably Abelard) well into the late middle ages.14 A theory of propositional inference was worked out much more thoroughly by Stoic logicians, and the explosive nature of their theories is more plausible for the following reason. There is a famous argument for ECQ, often called the Lewis (independent) argument, after C. I. Lewis. This goes (in natural deduction form) as follows:





: : _

13 Book , ch 4. The historical success of this defence is, however, out of all proportion to its intellectual weight. See Priest [1998e]. 14 Much of this and the rest of the material in this subsection is documented in Sylvan [2000], ch. 4. The discussion there is carried out in terms of the conditional, though it is equally applicable to the consequence relation.

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The argument uses just two principles (three if you include the transitivity of deducibility): Addition ( `  _ ) and the Disjunctive Syllogism (; : _ ` ). As we shall see in due course, the Disjunctive Syllogism (DS) has, unsurprisingly, been rejected by most paraconsistent logicians. Now Stoic logicians endorsed just this principle. The explosive nature of their logic would therefore seem a good bet. Despite this, it probably was not: there is reason to suppose that their disjunction was an intensional one that required some kind of connection between  and for the truth of  _ . If this is the case, Addition fails in general, as does the Lewis argument. It is not known who discovered the Lewis argument. Martin [1985] conjectures that it was William of Soissons in the 12th Century. (It was certainly known to, and endorsed by, some later logicians, such as Scotus and Buridan.) At any rate, William was a member of a group of logicians called the Parvipontanians, who were known not only for living by a small bridge, but for defending ECQ. This group may therefore herald the arrival of explosion on the philosophical stage. Whether or not this is so, after this time, some logicians endorsed explosion, some rejected it, dierent orthodoxies ruling at dierent times and dierent places (though, possibly, the explosive view was more common). One group of logicians who rejected it is notable, since they very much pre gure modern paraconsistent logicians. This is the Cologne School of the late 15th Century, who argued against the DS on the ground that if you start by assuming that  and :, then you cannot appeal to  to rule out : as the DS manifestly does. Notoriously, logic made little progress between the end of the Middle Ages and the start of the third great period in logic, towards the end of the 19th Century. With the work of logicians such as Boole and Frege, we see the mathematical articulation of an explosive logical theory that has come to be know, entirely inappropriately, as `classical logic'. Though, in its early years, many objected to its explosive features, it has achieved a hegemony (though never a universality) in the logical community, in a (historically) very brief space of time. Whether this is because the truth was de nitively and transparently revealed, or because at the time it was the only game in town, history will tell.

3.3 The Twentieth Century A feature of paraconsistent logic this century is that the idea appears to have occurred independently to many dierent people, at dierent times and places, working in ignorance of each other, and often motivated by somewhat dierent considerations. Some, notably, for example, da Costa, have been motivated by the idea that inconsistent theories might be of intrinsic importance. Others, notably the early relevant logicians, were motivated simply by the idea that explosion, as a property of entailment, is

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just too counter-intuitive.15 The earliest paraconsistent logics (that I am aware of) were given by two Russians. The rst of these was Vasil'ev. Starting about 1910, Vasil'ev proposed a modi ed Aristotelian syllogistic, according to which there is a new form: S is both P and not P . How, exactly, this form was to be interpreted is contentious, though, a problem exacerbated by the fact that he was not in a position to employ the techniques of modern logic. This is not true of the second logician, Orlov, who, in 1929, gave the rst axiomatisation of the relevant logic R. Sadly, the work of neither Vasil'ev nor Orlov made any impact at the time.16 An important gure who did have a good deal of in uence was the Polish logician and philosopher Lukasiewicz. Partly in uenced by Meinong's account of impossible objects, Lukasiewicz clearly envisaged the construction of paraconsistent logics in his seminal 1910 critique of Aristotle on the Law of Non-contradiction.17 And it was his erstwhile student, Jaskowski, who, in 1948, produced the rst non-adjunctive paraconsistent logic.18 Paraconsistent logics were again, independently, proposed in South America in doctoral dissertations by Asenjo (1954, Argentina) and da Costa (1963, Brazil). Asenjo proposed the rst many-valued paraconsistent logic. Da Costa gave axiom systems for a certain family of paraconsistent logics (the C systems), and produced the rst quanti ed paraconsistent logic. Many co-workers, such as Arruda and Loparic, joined da Costa in the next 20 years, to produce an active school of paraconsistent logicians at Campinas (and later S~ao Paulo). They developed non-truth-functional semantics for the C systems, and articulated the subject in various other ways; this included \rediscovering" Vasil'ev, taking up the work of Jaskowski, and formulating various other paraconsistent systems.19 Guided by considerations of relevance, an entirely dierent approach to paraconsistency was proposed in England by Smiley in [1959], who articulated the rst lter logic. Starting at about the same time, and drawing on the earlier work of Ackermann and Church, Anderson and Belnap in the USA proposed a number of relevant paraconsistent logics of a dierent kind. A research school quickly grew up around them in Pittsburgh, which included co-workers such as Meyer and Dunn.20 The algebraic semantics 15 The later Wittgenstein was also sympathetic to paraconsistency for various reasons, though he never articulated a paraconsistent logic. See, e.g., Marconi [1984]. 16 On Vasil'ev see Priest et al. [1989], ch. 3, 2.2 and Arruda [1977]. On Orlov, see Anderson et al. [1992], p. xvii. 17 A synopsis of this is published in English in Lukasiewicz [1971]. 18 For a discussion of Lukasiewicz and Jaskowski, see Priest et al. [1989], ch. 3, 2.1, 2.3. Jaskowski's work is translated into English in his [1969]. 19 Discussion and bibliography can be found in Priest et al. [1989], 5.6. The most accessible introduction to Asenjo's work is his [1966], and to da Costa's is his [1974]. Da Costa and Marconi [1989] reports much of the work of da Costa and his co-workers. 20 The work of this school is recorded in Anderson and Belnap [1975], and Anderson et

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for relevant logics, in particular, was inaugurated largely by Dunn's 1966 doctoral thesis.21 Investigation of things paraconsistent in Australia took o in the early 1970s with the discovery of world (intensional) semantics for negation by R. Routley (now Sylvan) and V. Routley (now Plumwood). This was developed into an intensional semantics for the Anderson/Belnap logics|and many others|by Routley,22 Meyer (now in Australia), and a school that developed around them in Canberra, which included workers such as Brady and Mortensen. These semantics made the paraconsistent aspects of relevant logics plain.23 Later in the 1970s the cudgel for dialetheism was taken up by Priest (now Priest) and Routley.24 By the mid-1970s the paraconsistent movement was a fully international one, with workers in all countries cooperating (though not necessarily agreeing!), and with logicians working in numerous countries other than the ones already mentioned, including Belgium, Bulgaria, Canada and Italy. Some feel for the state of the subject at the end of the 70s can be obtained from Priest et al. [1989].25 The rest, as they say, is not history. 4 BASIC TECHNIQUES OF PARACONSISTENT LOGICS An understanding of most paraconsistent logics can be obtained by looking at the strategies employed in virtue of which ECQ fails. There are many techniques for achieving this end. In this part, I will describe the most fundamental. In the process, we will meet dozens of dierent systems of paraconsistent logic, often constructed along very dierent lines. It is therefore necessary to have some common medium for comparison. I have chosen to make this semantics, and will specify systems in terms of these. (I would warn straight away though, that many of the systems we will meet appeared rst in proof theoretic terms. Indeed, some of the authors of these systems|e.g. Tennant|would privilege proof theory over semantics.) When I give details of corresponding proof theories, I will use the sort of proof theory (natural deduction, sequent calculus, or axiomatic) that seems most natural for the logic. Because paraconsistency concerns only negation essentially, we can see the essentials of paraconsistent logics in languages with very little logical al. [1992]. 21 See Anderson and Belnap [1975], and also the article on Relevant Logic in this volume of the Handbook. 22 Whenever the name `Routley' is used without initial in this essay, it refers to Sylvan. 23 The work of this group is most accessible in Routley et al. [1982]. 24 See, e.g., Routley [1979]. Priest's early work on the area is most accessible in Priest [1987]. 25 Despite the date, all the work in the collection was nished by 1980. A number of papers produced at the same time, that were not included in this, were published in a special issue of Studia Logica on paraconsistent logics (43 (1984), nos. 1 & 2).

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apparatus. In this part, we will be concerned with a propositional language whose only connectives are negation, :, conjunction, ^, and disjunction, _. I will use lower case Roman letters, starting with p, for propositional parameters, lower case Greeks, starting with , for arbitrary formulas, and upper case Greeks for sets of formulas. I will use j=C , j=I , and j=S5 for the consequence relations of classical logic, intuitionist logic and S 5, respectively, and j= for the semantic consequence relation of whichever paraconsistent logic happens to be the topic of discussion. If a proof theory is involved, I will use ` for the corresponding notion of deducibility.

4.1 Filtration One of the simplest ways to prevent explosion is to lter it, and any other undesirables, out. Consider, for simplicity, the one-premise case. (Finite sets of premises can always be reduced to this by conjoining.) Let F (; ) be any relationship between formulas. De ne an inference from  to to be prevalid i  j=C and F (; ). The thought here is that for an inference to be correct, something more than classical truth-preservation is required, e.g., some connection between premise and conclusion. This is expressed by F . Usually, prevalidity is too weak as a notion of validity, since, in general, it is not closed under uniform substitution, and this is normally taken to be a desideratum for any notion of validity. However, closure can be ensured if we de ne an inference to be valid i it is a uniform substitution instance of a prevalid inference. What inferences are valid depends, of course, entirely on the lter, F . One that naturally and obviously gives rise to a paraconsistent logic is: F (; ) i  and share a propositional parameter. (This collapses the notions of validity and prevalidity, since if  and share a propositional parameter, so do uniform substitution instances thereof.) This logic is not a very interesting paraconsistent one, however, since, as is clear, p ^:p j=  where  is any formula containing the parameter p.26 A dierent lter, proposed by Smiley [1959] is: F (; ) i  is not a (classical) contradiction and is not a (classical) tautology.27 (Note that, according to this de nition,  ^ : =  is not prevalid, but it is valid, since it is an instance of p ^ q = p.) It is easy to see that on this account p ^ :p does not entail q. The major notable feature of lter logics is that, in general, transitivity of deducibility breaks down.28 For example, using 26 A stronger lter is one to the eect that all the variables of the premise occur in the conclusion. This gives rise to a logic in the family of analytic implications. On this family, see Anderson and Belnap [1975], sect. 29.6. 27 Filters of a related kind were also suggested by Geach and von Wright. See Anderson and Belnap [1975], sect. 20.1. 28 Though it need not. First Degree Entailment, where transitivity holds, can be seen as a lter logic. See Dunn [1980].

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Smiley's lter, it is easy to see that p ^ :p j= p ^ (:p _ q), p ^ (:p _ q) j= q, but p ^ :p 6j= q. One of the most interesting lter logics, given by Tennant [1984], is obtained by generalising Smiley's approach. Let  and  be sets of sentences, and let ` j=C ' be understood in the natural way (every classical evaluation that makes every member of  true makes some member of  true). De ne the inference from  to  to be prevalid i:  j=C  and for no proper subsets of  and , 0 and 0 , respectively, do we have 0 j=C 0 . Validity is then de ned by closing under substitution as before. In this account, a valid inference is one which is classically valid, and minimally so: there is no \noise" amongst premise and conclusion set.29 Tennant's j= is obviously non-monotonic (that is, adding extra premises may invalidate an inference). It also has the following property: if  j=C , then there are subsets of  and , 0 and 0 , respectively, such that 0 j= 0 . For if  j=C , we can simply throw out premises and/or conclusions until this is no longer true; the result is a prevalid, and so valid, inference. In particular, if  j=C  then for some 0  , 0 j=  or 0 j= . In the rst case,  follows validly from part of ; in the second, part of  can be shown to be inconsistent by valid reasoning. Filtration can also be applied proof theoretically: we start with classical proofs and throw out those that do not satisfy some speci c criteria. Tennant's logic can be characterised proof-theoretically in just this way. For nite premises and conclusions, the valid inferences are exactly those that are provable in the Gentzen sequent calculus for classical logic, but which do not use the structural rules of dilution (thinning) and cut. Speci cally, consider the sequent calculus whose basic sequents are of the form  : , and whose rules are as follows. (1 ; 2 means 1 [ 2 ; similarly, ;  means  [ fg, and if something of this form occurs as a premise of a rule, it is to be understood that  2= ). ;  :
 :
; :

 :
;  ; : :


;  :
;  ^ :


; :
;  ^ :


1 :
1 ;  2 :
2 ; 1 ; 2 :
1 ;
2 ;  ^

 :
;   :
;  _

 :
;  :
;  _

1 ;  :
1 2 ; :
2 1 ; 2 ;  _ :
1 ;
2

29 The restriction of Tennant's approach to the one-premise, one-conclusion, case obviously gives Smiley's account. Smiley himself, handles the multiple-premise case, simply by conjoining. As Tennant points out ([1984], p. 199), this generates a dierent account from his. It is not diÆcult to check that p _ q; :(p _ q) 6j= p ^ q for Smiley. (The conjoined antecedent is a contradiction; and any inference of which the conjoined form is a substitution instance is not classically valid.) But it is valid for Tennant, since it is a substitution instance of p _ q; r _ s; :(t _ q); :(r _ u) j= p ^ s.

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Then we have  j=  i the sequent  :  is provable. For the proof see Tennant [1984].30 Tennant's account of inference seems to capture very nicely what one might call the `essential core' of classical inference. As an inference engine to be applied to inconsistent information/theories, it could not be applied in the obvious way, however. This is because information is often heavily redundant. For example, for Tennant's j=, we do not have p; :p _ q; q j= q. Yet given the information in the premises, it would certainly seem that we are entitled to infer q. Presumably, then, we would take  to follow from  i for some 0  , 0 j= .31 If we do this then more than transitivity fails; so does Adjunction. For :p; p _ q j= q and :q; p _ q j= p, hence both p and q follow from f:p; p _ q; :qg = . But for no subset of , 0 , do we have 0 j= p ^ q. ( j= , and if 0 is a proper subset of , 0 6j=C p ^ q.) In this respect, Tennant's approach is similar to the next one that we will look at.

4.2 Non-adjunction All the other approaches that we will consider, except the last (algebraic logics) accept validity as de ned simply in terms of model-preservation. Thus, given some notion of interpretation, call it a model of a sentence if the sentence holds in the interpretation; an interpretation is a model of a set of sentences if it is a model of every member of the set; and an inference is valid i every model of the premises is a model of the conclusion. In particular, then, if explosion is to be avoided, it must be possible to have models for contradictions, which are not models of everything. Where the dierences in the following approaches lie is in what counts as an interpretation, and what counts as holding in it. For the next approach, an interpretation, I , is a Kripke interpretation of some modal logic, say S 5, employing the usual truth conditions. Each world in an interpretation may be thought of as the world according to some party in a debate or discussion. This gives the approach its common name, discussive (or discursive) logic. I is a (discursive) model of sentence  i  holds at some world in I , i.e., 3 holds in the model. Thus,  j=  i  holds, discursively, in every discursive model of , i.e., i 3 j=S5 3, where 3 is f3;  2 g. This approach is that of Jaskowski [1969].32 It is clear that discussive logic is paraconsistent, since we may have 3 and 3: 30 The proof theory can be given a ltered natural deduction form too. Essentially, classical deductions that have a certain \normal form" pass through the lter. See Tennant [1980]. 31 Though if we do this, symmetry suggests that we should take  to follow from  i for some 0   and 0  , 0 ` 0 ; in this case paraconsistency is lost since ; : ` . 32 Popper also seems to have had a similar idea in 1948. See his [1963], p. 321.

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in an S 5 interpretation, without having 3 . For similar reasons, Adjunction (; j=  ^ ) fails. It should be noted, however, that  ^ : j= , so the logic is not paraconsistent for conjoined contradictions. A closely related approach can be found in Rescher and Brandom [1979]. They de ne validity as truth preservation in all worlds, but they augment the worlds of standard modal logic by inconsistent and complete worlds, constructed using operators [_ and \_ . Speci cally, worlds are constructed recursively from standard worlds as follows. If W is a set of worlds, [_ W is a world such that  is true in [_ W i for some w in W ,  is true in w; and \_ W is a world such that  is true in \_ W i for all w in W ,  is true in w. As is intuitively clear, inconsistent worlds just provide another way of expressing what holds in a Jaskowski interpretation. Incomplete worlds appear more novel, but, in fact, add nothing. For if truth fails to be preserved in one of these, it fails to be preserved in one of the ordinary worlds which go into making it up. These ideas can be recast to show that the semantics of Rescher and Brandom, and of Jaskowski are inter-translatable, and deliver the same notion of validity.33 A notable feature of discussive logic is that  j=  i for some 2 , j=C . (The proof from right to left is obvious. From left to right, suppose that for every 2 , 6j=C . Let w be a classical world where holds but  does not. If we take the interpretation whose worlds are fw ; 2 g this is a counter-model for  j= .) Thus, single-premise discussive inference is classical, and there is no essentially multiple-premise inference. One way to avoid the second of these features is to add an appropriate conditional connective. We will look at this later. Another way is to allow a certain amount of conjoining of premises. The question is how to do this in a controlled way so that explosion does not arise. One suggestion, made by Rescher and Manor, is, in eect, to allow conjoining up to maximal consistency.34 Given a set of premises, , a maximally consistent subset (mcs) is any consistent subset, 0 , such that if 33 Proof: Suppose that, discursively,  6j= . Then there is an interpretation such that for each  2 , there is some world, w , such that  is true in w , but  is not true in _ w ;  2 g, then w is a Rescher/Brandom counter-model. Conversely, w . Let w = [f suppose that  6j=  for Rescher and Brandom. Then there is some world such that for every  2 ,  is true at w, but  is not. We show that there is a Jaskowski countermodel. The result is proved by recusion on the construction of Rescher/Brandom worlds. If w is a standard world, the result is clear. So suppose that w = \_ W , where the result holds for all members of W . By de nition, for every z 2 W , and every  2 ,  is true in z , but for some z ,  is not true in z . Consider that z . This is a Rescher/Brandom countermodel to the inference. Hence, the result holds by recursion hypothesis. Alternatively, suppose that w = [_ W , where the result holds for all members of W . By de nition, for every  2 , there is some w 2 W , such that  is true in w , but  is not. By recursion, there must be a Jaskowski countermodel for the inference =.  is true at some world in this, but  is not. If we form the collection of worlds for all such , this then gives us a Jaskowski counter-model to the original inference. 34 Rescher and Manor [1970-1]. This takes o from the earlier work of Rescher [1964].

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 2  0, 0 [ fg is inconsistent. We can now say that  follows from  i for some mcs 0 , 0 j=C . In possible world terms, we can rephrase this as follows. Let us say that an interpretation, I , respects  i for every mcs 0 , there is a world, w, in I such that 0 is true in w. Then it is not diÆcult to see that this policy is a variant of discussive logic:  j=  i  holds discussively in every interpretation that respects . (If  follows classically from some 0 , then it holds in every discussive interpretation that respects . Conversely, suppose that it follows from no 0 . Then for each 0 choose a world w0 where 0 is true, but  is not. The interpretation containing all such w0 is a countermodel.) This policy is certainly stronger than simple discussive consequence. For example, it gives: p; q j= p ^ q. In fact, if  is (classically) consistent then every classical consequence of  is a consequence. But it is still nonadjunctive: p; :p 6j= p ^ :p.35 A slightly dierent way of proceeding is provided by Schotch and Jennings.36 Given a nite set, , a partition is any family of disjoint sets, each of which is classically consistent, and whose union is . The level of , l(), is the least n such that  can be partitioned into n sets (or, conventionally, 1 if there is no such n).  j=  i l() = 1 or, l() = n and for any partition of  of size n, fi ; 1  i  ng, there is an i such that i j=C . As with the previous approach, this de nition can be converted into discussive terms, by taking our models to be those that respect the premise set. But this time, an interpretation respects  i for some partition of the level of , fi ; 1  i  ng, and every i, there is a world in the interpretation where i is true. Leaving aside the fact that Schotch and Jennings consider only nite premise sets, one dierence between their approach and the previous one concerns the consequences of sets, , with single inconsistent members. Such sets have no partitions, and so explode for Schotch and Jennings. They still have mcss though (e.g., ), and so do not explode for Rescher and Manor. If  has no single inconsistent member then Schotch and Jennings' consequence relation is included in that of Rescher and Manor. For if fi ; i 2 ng is a partition of the premises, , and for some i, i j=C , then i can be extended to an mcs of , and this classically entails . The converse is not true, however. Let  = fp; :p; q; rg. This has two mcss, fp; q; rg and f:p; q; rg. Hence, for Rescher and Manor, q ^ r follows. But  has level 2, and one partition is ffp; qg; f:p; rgg. Neither of these classically entails q ^ r, so this does not follow for Schotch and Jennings (which seems wrong, intuitively).37 35 Rescher and Manor also formulate a weaker policy of inference.  follows from  i for all mcs 0 , 0 j=C . This logic is clearly adjunctive. 36 See their [1980], where they also discuss appropriate proof theories and modal connections. 37 The same example shows that Schotch and Jennings' j=, unlike Rescher and Manor's,

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GRAHAM PRIEST

Despite the dierences, Schotch and Jennings' approach shares with that of Rescher and Manor the following features: for consistent sets, consequence coincides with classical consequence; Adjunction fails. For Schotch and Jennings, like Jaskowski,  ^ : explodes. For Rescher and Manor, it has no consequences (other than tautologies). The logics that we will look at in subsequent sections are more discriminating concerning conjoined contradictions.

4.3 Interlude: Henkin Constructions Before we move on to look at the other basic approaches to paraconsistent logic, I want to isolate a construction that we will have many occasions to use. In a standard Henkin proof for the completeness of an explosive logic, we construct a maximally consistent set of sentences, and use this to de ne an evaluation. In the construction of the set, we keep something out of it by putting its negation in. As might be expected, these techniques do not work in paraconsistent logic; but they can be generalised to do so. What plays the role of a maximally consistent set in a paraconsistent logic is a prime theory, where a set of sentences, , is a theory i it is closed under deducibility; and it is prime i  _ 2  ) ( 2  or 2 ). To keep something out in the construction of a prime theory, we have to exclude it explicitly. I now show how. Assume that the proof theory is to be given in natural deduction terms. For de niteness I adopt the notational conventions of Prawitz [1965].38 Consider the following rules for conjunction and disjunction:

_I

 ^

^E

^ 

^

_I

 _

_

is non-monotonic, since we have q; r j= q ^ r. 38 In particular, something of the form:

 .. . in a rule indicates a subproof with  as one assumption|though there may be others| and conclusion . If  is overlined, this means that the application of the rule discharges it.

PARACONSISTENT LOGIC

_E

_

 .. .

303

.. .

Let ` be any proof theory that includes these rules. Write  `  to mean that there are members of , 1 ; :::; n , such that  ` 1 _ ::: _ n . Then if  6` , there are sets
  and  , such that
6` , and
is a prime theory. To prove this, we enumerate the formulas of the language: 0 ; 1 ; 2 ; :::, and de ne a sequence of sets n , n (n 2 !) as follows. 0 = ; 0 = . If n [ f n g 6` n , then n+1 = n [ S f n g and n+1S= n . Otherwise n+1 = n and n+1 = n [ f ng.
= n2! n ; = n2! n . It is not diÆcult to check by induction that for all n, n 6` n . (Suppose this holds for n; if n [ f n g 6` n , the result for n + 1 is immediate. So suppose that n [ f n g ` n and n+1 ` n+1 . Then n ` f n g [ n . By a sequence of moves that amount to \cut", n ` n , contrary to induction hypothesis.) By compactness, it follows that
6` . It is also easy to check that
is a prime theory. Suppose that
` , but  62
. Then for some n, n [fg ` n . Hence,
` . Next, suppose that  _ 2
, but  62
and 62
. Then for some m and n, n [ fg ` n and m [ f g ` m . Hence
[ f _ g ` , and so
` .

4.4 Non-truth-functionality Let us now return to the other basic approaches to paraconsistent logics. On the rst of these, explosion is invalidated by employing a non-truthfunctional account of negation. Typically, this account of negation is imposed on top of an orthodox account of positive logic. Thus, let an interpretation be a map,  , from the set of formulas to f1; 0g, satisfying just the following conditions:

 ( ^ ) = 1 i  () = 1 and  ( ) = 1  ( _ ) = 1 i  () = 1 or  ( ) = 1 In particular, the truth value of : is independent of that of . Validity is de ned as truth preservation over all interpretations. It is obvious that explosion fails, since we may choose an evaluation that assigns both p and :p (and their conjunction) the value 1, whilst assigning q the value 0. These semantics can be characterised very simply in natural deduction terms by just the rules _I , _E , ^I and ^E . Soundness is easy to check. For completeness, suppose that  6` . Then put  = fg, and extend 

304

GRAHAM PRIEST

to a prime theory,
, with the same property, as in 4.3. De ne a map,  , as follows:

 () = 1 if  2
 () = 0 if  2=
It is easy to check that  is an interpretation. Hence, we have the result. This system contains no inferences that involve negation essentially. For this reason, : can hardly be thought of as a negation functor. Stronger paraconsistent systems, where this is more plausibly the case, can be obtained by adding conditions on the semantics. The following are some examples:39 (i) if  () = 0,  (:) = 1 (ii) if  (::) = 1,  () = 1 (iii) if  () = 1,  (::) = 1 (iv)  (:( ^ )) =  (: _ : ) (v)  (:( _ )) =  (: ^ : ) Sound and complete rule systems can be obtained by adding the corresponding rules, which are, respectively: (i)  _ :  (ii) ::   (iii) ::  : ( ^ ) (iv) : _ : :( _ ) (v) : ^ : (Double underlining indicates a two-way rule of inference, and a zero premise rule, as in (i), can be thought of as an assumption that discharges itself.) The corresponding soundness and completeness proofs are simple extensions of the basic arguments. These additions give the ^; _; :-fragments of various systems in the literature. (i) gives that of Batens' P I [1980]; (i) and (ii) that of da Costa's C! ;40 (i)-(v) that of Batens' P I S . In P I S every sentence is logically equivalent to one in Conjunctive Normal Form. This can be used to show that P I S 39 Some others can be found in Loparic and da Costa [1984], and Beziau [1990]. 40 Semantics of the present kind for the da Costa systems were rst proposed in da

Costa and Alves [1977].

PARACONSISTENT LOGIC

305

is a maximal paraconsistent logic, in the sense that any logic that extends it is not paraconsistent. (For details, see Batens [1980].) Observe, for future reference, that if we add to P I or an extension thereof the condition: if  () = 1,  (:) = 0 then all interpretations are classical, and so we have classical logic. As may easily be checked, adding this is sound and complete with respect to the rule of inference:

 ^ : Another major da Costa system, C1 , extends C! in accordance with the following idea. It should be possible to express in the language the idea that a sentence, , behaves consistently; and for consistent sentences classical logic should apply. Let us write :( ^:) as 0 . Then it is natural enough to suppose that 0 expresses the consistency of . It does not, in any of the above systems, since we may have  ^: ^ 0 true in an interpretation. This is exactly what is ruled out by the condition: (vi)  () =  (:) then  (0 ) = 041 ( () =  (:) i both are 1, by semantic condition (i). Note that (i) also guarantees the converse of (vi):  () 6=  (:) then  (0 ) = 1.) C1 is obtained by adding (vi) to C! , together with the following condition, which requires consistency to be preserved under syntactic constructions: (vii) if  (0 ) =  ( 0 ) = 1 then  ((:)0 ) =  (( ^ )0 ) =  (( _ )0 ) = 1 The deduction rules that correspond to (vi) and (vii) are, respectively: 0 (vi)  ^ :  ^ 

(vii)

0 (:)0

0 0 ( ^ )0

0 0 ( _ )0

Soundness and completeness of the extensions are easily checked. Now suppose that we have a piece of valid classical reasoning concerning formulas composed of parameters p1 ; :::; pn . If we assume p01 ; :::; p0n then for 41 Da Costa's actual condition is: if  (0 ) =  (  ( ^ :)) = 1 then  ( ) = 0. This is equivalent, given his account of the conditional.

306

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every such formula, , 0 follows by the appropriate applications of the rules of (vii). Hence, whenever we have established  ^ : we may apply rule (vi) to give . But the addition of this inference is suÆcient to give classical logic, as I have already observed. Hence any valid classical reasoning may be recaptured formally by adding the appropriate consistency assumptions.42 One nal comment on treating negation non-truth-functionally. It is a consequence of this that the substitutivity of provable equivalents breaks down in general. For example, even though  is logically equivalent to  ^  there is no guarantee that the negations of these formulas have the same truth value in an interpretation.43

4.5 Many Values The previous approach sticks with the traditional two truth values, and obtains a paraconsistent logic by making negation non-truth-functional. The next approach retains truth functionality, but drops the idea that there are exactly two truth values. That is, such logics are many-valued.44 A many-valued logic will be paraconsistent if it is possible for a formula and its negation both to take designated values (whilst not everything does). A natural way of obtaining this is to have a designated value that is a xed point for negation. The simplest such logic is a three valued one with values, t, b, and f , where t and b are designated, and the matrices are:

:

t f b b f t

^ t b f t b f

t b f

b b f

f f f

_ t b f t b f

t t t t b b t b f

It will be noted that these are just the matrices of Lukasiewicz and Kleene's 3-valued logics, where the middle value is normally thought of as undecidable, or neither true nor false, and so not designated. It was the thought that this value might be read as both true and false|a natural enough thought, given dialetheism|and so be designated, that marks the start of many-valued paraconsistent logic. This was the approach proposed by Asenjo (see his [1966]), and others, e.g. Priest [1979], where the logic is called LP , a nomenclature that I will stick with here.

42 It might be suggested that one ought not to take 0 as expressing consistency unless it, itself, behaves consistently. This thought motivates the weaker da Costa system C2 , which is the same as C1 , except that 0 is replaced everywhere by 0 ^ 00 . Of course, there is no reason to suppose that this expresses the consistency of  unless it, itself, behaves consistently. This thought motivates the da Costa system C3 where 0 is replaced everywhere by 0 ^ 00 ^ 000 . And so on for all the da Costa Systems Ci , for nite non-zero i. 43 For a discussion of this in the context of da Costa's logics, see Urbas [1989]. 44 For a general discussion of many-valued logics, see the articles on the topic in this Handbook. See also, Rescher [1969].

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LP may be generalised in various dierent ways. One is as follows. If we let t = +1, b = 0 and f = 1 then the truth conditions of LP are:  (:) =  ()  ( ^ ) = minf (a);  ( )g  ( _ ) = maxf (a);  ( )g The same conditions can be used for any set of integers, X , containing 0 and closed under . The designated values are the non-negative values. Let us call this a Sugihara generalisation, after the person who, in eect, rst proposed a matrix of this kind, where X was the set of all integers.45 Any Sugihara generalisation, though semantically dierent from LP , is essentially equivalent to it. Any LP countermodel is a Sugihara countermodel. But conversely, if we have a Sugihara countermodel, we can obtain an LP countermodel by mapping all positive values to +1, 0 to 0, and all negative values to 1. A little thought is suÆcient to establish that the mapping respects the matrices and preserves designated values, as required. A dierent way of generalising LP is as follows. If we let t = 1, b = 0:5 and f = 0 then the truth conditions of LP are:

 (:) = 1  ()  ( ^ ) = minf (a);  ( )g  ( _ ) = maxf (a);  ( )g The same conditions can be used for any set of reals f0; 0:5; 1g  X  [0; 1], which is closed under subtraction (of a greater by a lesser). For suitable choices of X , these are the matrices of the odd-numbered nite Lukasiewicz many-valued logics, and for X = [0; 1] they are the matrices of Lukasiewicz' continuum-valued logic. In Lukasiewicz' logics proper, the only designated value is 1, which does not give a paraconsistent logic. But if one takes the designated values to be fx; a < x  1g (or fx; a  x  1g) then the logic will be paraconsistent provided that 0 < a < 0:5 (or 0 < a  0:5). Let us call such logics Lukasiewicz generalisations. In a Lukasiewicz generalisation where the set of truth values is [0; 1], these may naturally be thought of as degrees of truth. Hence, such a logic is a natural candidate for a paraconsistent fuzzy logic (logic of vagueness).46 It is not diÆcult to see that any Lukasiewicz generalisation is, in fact, equivalent to LP . As with the Sugihara generalisations, any LP countermodel is a Lukasiewicz countermodel; and conversely, any Lukasiewicz 45 See Anderson and Belnap [1975], sect. 26.9. 46 A variation on this theme is given by Pe~na in a number of papers. (See, e.g., Pe~na

[1984].) Pe~na takes truth values to be an ordered set of more complex entities de ned in terms of the interval [0; 1].

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countermodel can be collapsed into an LP countermodel by the mapping, f , de ned thus:

f (x) = 1 = 0:5 =0

if 1 a  x  1 if a < x < 1 a if 0  x  a

or for the case where a is a designated value:

f (x) = 1 if 1 a < x  1 = 0:5 if a  x  1 a =0 if 0  x < a The generalisations of LP that we have considered in this section all, therefore, generate the same logic. What its proof theory is, we will see in the next.

4.6 Relational Valuations Standardly, semantic evaluations are thought of as functions from formulas to truth values, say, 0 and 1. Another way of invalidating explosion is to take them to be, not functions, but relations. A formula may then relate to both 0 and 1, another way of expressing the thought that a sentence is both true and false. Assuming that negation behaves as usual, this means that both p and :p may relate to 1, whilst an arbitrary formula may not. A natural way of spelling out this idea is as follows. If P is the set of propositional parameters, an evaluation, , is a subset of P  f0; 1g. The evaluation is extended to a relation for all formulas by the familiar looking recursive clauses:

:1 i 0 :0 i 1  ^ 1 i 1 and 1  ^ 0 i 0 or 0  _ 1 i 1 or 1  _ 0 i 0 and 0 Let us say that a formula, , is true in an interpretation, , i 1, and false i 0; then validity may be de ned as truth preservation in all interpretations. According to this account, classical logic is just the special case where multi-valued relations have been forgotten.

PARACONSISTENT LOGIC

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These semantics are the Dunn semantics for the logic of First Degree Entailment, F DE .47 In natural deduction terms, F DE can be characterised by the rules ^I , ^E , _I and _E , together with the rules:

:( ^ ) : _ :

: ^ : :( _ )



::

Soundness is easily checked. For completeness, suppose that  6` . Extend  to a prime theory,
, with the same property, as in 4.3. Now de ne an interpretation, , thus:

p1 i p 2
p0 i :p 2
A straightforward (joint) induction shows that this characterisation extends to all formulas. Completeness follows. There are two natural restrictions that one may place upon Dunn evaluations: #1 #2

for every p, there is at most one x such that px for every p there is at least one x such that px

Both conditions extend from propositional parameters to all formulas, by a simple induction. Thus, the rst condition ensures that the relation is functional; the second that it is total. A relation that satis es both conditions is just a classical evaluation. These extra conditions are sound and complete with respect to the extra rules:

 ^ :

 _ :

respectively, as simple extensions of the completeness proofs demonstrate. We can express the relational semantics in functional terms by taking an evaluation to be a function from formulas to subsets of f1; 0g, since there is an obvious isomorphism between relations, ; and functions,  , given by the condition:

x i x 2  () 47 Published in Dunn [1976], though he discovered them somewhat earlier than this. In the present context, it might be better to call the system `Zero Degree Entailment' since the language does not contain a conditional connective.

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In this way, F DE can be seen as a many- (in fact, four-) valued logic.48 Restriction #2, which ensures that no formula takes the value , gives a three-valued logic that is identical with LP . It is easy enough to check that the values f1g, f1; 0g, and f0g work the same way as t, b, and f , respectively. I will make this identi cation in the rest of this essay. Restriction #1, which ensures that no formula takes the value f1; 0g, obviously gives an explosive logic, which is, in fact, the strong Kleene three-valued logic. This is therefore a logic dual to LP .49 A feature of these semantics for LP and F DE is that they are monotonic in the following sense. Let 1 and 2 be functional evaluations. If for all propositional parameters, p, 1 (p)  2 (p) then for all , 1 ()  2 (). The proof of this is by a simple induction. One consequence of this for LP is worth remarking on. LP is clearly a sub-logic of classical logic, since it has the classical matrices as sub-matrices. The consequence relation of LP is weaker than that of classical logic, since it is paraconsistent. But the set of logical truths of LP is identical with that of classical logic. For suppose that  is not valid in LP . Let  be an evaluation such that 1 2=  (). Let  0 be the interpretation that is the same as  , except that for every parameter, p, if  (p) = f0; 1g,  0 (p) = f0g. This is a classical evaluation; and by monotonicity, 1 2=  0 (), as required. Another feature of these semantics is the evaluation that assigns every propositional parameter the value f1; 0g, vf1;0g ; and, in the four-valued case, the evaluation that gives every parameter the value , v . A simple induction shows that these properties extend to all formulas. Thus, vf1;0g makes all formulas true|and false|and v makes every formula neither. In particular, then, F DE has no logical truths.50

4.7 Possible Worlds Yet another, closely connected, way of invalidating explosion is to treat negation as an intensional operator. This way was proposed by the Routleys in [1972]. A Routley interpretation is a structure, hW; ;  i, where W is a set (of worlds),  is a map from W to W , and  maps sets of pairs comprising a world and propositional parameter to f1; 0g. (I will write  (w; ) as w ().) The truth conditions for conjunction and disjunction are the standard:

w ( ^ ) = 1 i w () = 1 and w ( ) = 1

48 In fact, the straight truth tables with values 1, 2, 3 and 4 were enunciated by Smiley.

See Anderson and Belnap [1975], p. 161. 49 I will usually use the functional semantic representation for F DE and LP in the rest of this essay. A word of warning, though: in the context of a dialetheic metatheory, the functional approach may have consequences that the relational approach, proper, does not have. See Priest and Smiley [1993], p. 49. 50 Further interesting properties of LP and FDE are established in Pynko [1995a] and [1995b].

PARACONSISTENT LOGIC

311

w ( _ ) = 1 i w () = 1 or w ( ) = 1 The truth conditions for negation are:

w (:) = 1 i w () = 0 Note that if w = w, these conditions just reduce to the classical ones. A natural understanding of the  operator is a moot point.51 I will return to the issue in a moment. Validity is de ned in terms of truth preservation at all worlds of all interpretations. In natural deduction terms, this system can be characterised by modifying that for F DE by dropping the rule for double negation, and replacing it with:

 .. .

:

:

where, in the subproof, there are no undischarged assumptions other than . Soundness is easily checked. For completeness, suppose that  6` . Extend  to a prime theory,
, with the same property, as in 4.3. Now de ne an interpretation, hW; ;  i, where W is the set of all prime theories,  is de ned by the condition:

 2
 i : 2=
and  is de ned by:


(p) = 1 i p 2


(#)

It is not diÆcult to check that if
is a prime theory, so is
 and hence that  is well de ned. First, suppose that  2=
 and 2=
 . Then : and : are in
. Since
is a theory, : ^ : 2
, and so :( _ ) 2
. Hence,  _ 2=
 . Next, suppose that
 ` , but  2=
 . Then for some 1 ; :::; n 2
 , 1 ^ ::: ^ n ` . Hence, by contraposition and De Morgan : ` : 1 _ ::: _ : n . But : 2
; hence : 1 _ ::: _ : n 2
. Since
is prime, for some 1  i  n, : i 2
, i.e., i 2=
 . Contradiction. An easy recursion shows that (#) extends to all formulas. The result follows. 51 For some discussion and references, see the article on Relevance Logic and Entailment in this Handbook.

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GRAHAM PRIEST

The logic can be made stronger without (necessarily) ruining its paraconsistency by adding further conditions on . The most notable is: w = w. This is sound and complete with respect to the additional rule:



:: as a simple extension of the completeness argument demonstrates. These semantics are, in fact, very closely related to the those for F DE of the previous section. Given an F DE interpretation,  , de ne a Routley evaluation on the worlds w and w , as follows:

w (p) = 1 i 1 2  (p) w (p) = 1 i 0 2=  (p) A simple induction shows that these conditions follow for all formulas. Conversely, we can turn the conditions into reverse. Given any Routley evaluation on a pair of worlds, w, w , de ne a Dunn evaluation by the conditions: 1 2  (p) i w (p) = 1 0 2  (p) i w (p) 6= 1 Essentially the same induction shows that these conditions hold for all formulas. Hence, the two semantics are inter-translatable, and validate the same proof theories.52 The translation also suggests a natural interpretation of the  operator. w is that world characterised by the set of unfalsehoods of w. (This is, of course, in general, distinct from the set of truths in a four-valued context.) Under the above translation, the condition: 1 2  (p) or 0 2  (p), which gives an LP interpretation, is equivalent to: w (p) = 1 or w (p) 6= 1; imposing which condition on an intensional interpretation therefore gives an intensional semantics for LP .

4.8 Algebraic Semantics Let us now turn to the nal approach to paraconsistent logics that we will consider, an algebraic one. In algebraic logic, an interpretation is a homomorphism,  , from sentences into some algebraic structure, A = hA; ^; _; :i; i.e.,  (:) = : (),  ( ^ ) =  () ^  ( ), etc. (I will use the same signs for the connectives and the algebraic operations. Context, and the style of variable, will serve to disambiguate.) If the algebra is a lattice|as it usually 52 Which shows that the contraposition rule is admissible in F DE , something that is not at all obvious.

PARACONSISTENT LOGIC

313

is, and will be in all the cases we consider|the consequence relation of the logic is represented by the lattice order relation, de ned in the usual way: a  b i a ^ b = a. Thus, a logic will be paraconsistent if it is possible in the algebra to have an a and b such that a ^ :a 6 b. Several of the logics that we have looked at can be algebraicised. Consider, for example, F DE . If we take the four-valued semantics for this, we can think of the values as a lattice whose Hasse diagram is as follows:

f1; 0g

%

f1g

-

%



f0g (^ is lattice-meet; _ is lattice join; :f1; 0g = f1; 0g, and : = .)

This generalises to a De Morgan algebra. A De Morgan algebra is a structure A = hA; ^; _i, where hA; ^; _; :i is a distributive lattice, and : is an involution of period 2, i.e.:

::a = a a  b ) :b  :a The structures take their name from the fact that in every such algebra :(a ^ b) = :a _ :b holds, as do the other De Morgan laws. De ne an inference 1 ; :::; n = to be algebraically valid i for every homomorphism,  , into a De Morgan algebra, A,  (1 ) ^ ::: ^  (n )   ( ). Then the algebraically valid inferences are exactly those of F DE . It is easy to check that the rule system for F DE is sound with respect to these semantics. Completeness follows from completeness in the four-valued case. Alternatively, we can give a direct argument as follows. Consider the relation   , de ned by:  ` and ` . One can check that this is an equivalence relation, and a congruence on the logical operators (i.e., if 1  1 and 2  2 then 1 ^ 2  1 ^ 2 , etc.).53 If F is the set of formulas, de ne the quotient algebra, A = hF= ; ^; _; :i, where, if [] is the equivalence class of , :[] = [:], [] ^ [ ] = [ ^ ], etc. One can check that A is a De Morgan lattice. Now, let  be the homomorphism that maps every  to []. If  (1 ) ^ ::: ^  (n )   ( ). Then [1 ^ ::: ^ n ]  [ ], i.e., [1 ^ ::: ^ n ^ ] = [1 ^ ::: ^ n ]. Hence, 1 ^ ::: ^ n ` 1 ^ ::: ^ n ^ and so 1 ^ ::: ^ n ` . Conversely, then, if 1 ^ ::: ^ n 6` then  (1 ) ^ ::: ^  (n ) 6  ( ), as required. 53 The only tricky point concerns negation. For this, we need to appeal to the fact, which we have already noted, that if  ` then : ` :. This can be established directly, by an induction on proofs.

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It should be noted that not all the logics we have considered in previous sections algebraicise. In particular, the non-truth-functional logics resist this treatment in general. This is for the same reason that the substitutivity of provable equivalents breaks down: the semantic value of : is entirely independent of that of . It cannot, therefore, correspond to any well-de ned algebraic operation. The point can be made more precise in many cases. Suppose that A is some algebraic structure for a logic, and consider any interpretation,  , with values in the algebra, such that for some p, q and r,  (p) =  (q) 6=  (r). Then the condition  () =  ( ) is a congruence relation on the set of formulas, and collapse by it gives a non-degenerate quotient algebra (i.e., an algebra that is neither a single-element algebra, nor the algebra of formulas). But many non-truth-functional logics can be shown to have no such thing. (See, e.g., Mortensen [1980].) One nal algebraic paraconsistent logic is worth noting. This is that of Goodman [1981]. A Heyting algebra can be thought of as a distributive lattice, with a bottom element, ?, and an operator, !, satisfying the condition:

a ^ b  c i a  b ! c (which makes ? ! ? the top element). We may de ne :a as a ! ?. Let T be a topological space. Then a standard example of a Heyting algebra is the topological Heyting algebra hX; ^; _; !; ?i, where X is the set of open sets in T , ^ and _ are intersection and union, respectively, ? is , and a ! b is (a _ b)o |overlining denotes complementation and o is the interior operator of the topology. :a is clearly ao . It is well known that for nite sets of premises, Intuitionistic logic is sound and complete with respect to the class of Heyting algebras, in fact, with respect to the topological Heyting algebras. That is, 1 ; :::n j=I i for every homomorphism,  , into such an algebra,  (1 ^ ::: ^ n )   ( ).54 The whole construction can be dualised in a natural way to give a paraconsistent logic. A dual Heyting algebra is a distributive lattice, with a top element, >, and an operator, , satisfying the condition:

a  b _ c i a

bc

(which makes > > the bottom element). We may de ne :a as > a. As may be checked, if T is a topological space, then the structure hX; ^; _; ; >i is a dual Heyting algebra, where X is the set of closed sets of T , ^ and _ are intersection and union, respectively, > is the whole space, 54 See, e.g., Dummett [1977], 5.3.

PARACONSISTENT LOGIC

315

and a cb is (a ^ b)c |where c is the closure operator of the topology. :b is clearly b . The logic generated by dual Heyting algebras is dual to Intuitionistic logic. In particular, in Intuitionistic logic we have  ^ : j= , but not  j= _ : ; and  j= ::, but not :: j= . Thus in dual Intuitionist logic, we have  j= _ : but not  ^ : j= ; and :: j=  but not  j= ::. For a topological counter-model to the rst, consider the real line with its usual topology, and an interpretation,  , that maps p to [ 1; +1], and q to . Then  (p ^ :p) = f 1; +1g 6  =  (q). (This illustrates how the points in the set represented by p ^ :p may be thought of as the points on the topological boundary between the set of points represented by p and the set of points represented by :p.) For a counter-model to the second, let  (p) = f0g. Then  (::p) =  6  (p). If j=  in dual Intuitionist logic, then j=C , since the two-element Boolean algebra is a dual Heyting algebra. Conversely, if  is any classical tautology, its dual, 0 , is a contradiction. Hence, j=C :0 . But then by a result of Glivenko, j=I :0 , and so 0 j=I . Thus by duality, in dual Intuitionist logic j= . The logical truths of dual Intuitionist logic are therefore the same as those of classical logic. It is worth noting that just as Intuitionist logic can be given an intensional semantics, namely Kripke semantics, so can dual Intuitionist logic; we simply dualise the Kripke construction. For further details of all the above, see Goodman [1981].55 5 CONDITIONAL CONNECTIVES We have now looked at most of the basic techniques of paraconsistent logic, applied to languages containing only negation, conjunction and disjunction.56 I will call this language the basic language. Next, we will look at some important extensions of these techniques (which do not ruin paraconsistency). In this part, we will start with the conditional, by which I mean some con55 It is well known that in a certain well de ned sense, Intuitionist logic can be seen as the \internal logic" of the category-theoretic structures called topoi. It is possible to dualise the construction involved there to show that dual Intuitionist logic has an equally good claim to that title. For details, see Mortensen [1995], who calls the ^, _, :-fragment of a dual Heyting algebra a `paraconsistent algebra'. 56 There are others, such as the use of the techniques of combinatorial logic, but I will not go into these here. For details, one can consult, e.g., Bunder [1984]. There ought to be yet more. The discussion of connexivism in 3.2 suggests that there ought to be a distinctive connexivist approach to paraconsistency. To date, this has not emerged. The most articulated modern connexivist logic is due to McCall (see sect. 29.8 of Anderson and Belnap [1975], which can also be consulted for references to other discussions). Although this provides a connexivist treatment of the connective !, the logic of the basic language is classical, and so explosive. Alternatively, one can formulate versions of relevant logic that contain connexivist principles. See Routley [1978] and Mortensen [1984].

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GRAHAM PRIEST

nective, ! (if necessary, added to the basic language), satisfying, at least, modus ponens: ;  ! j= . Although paraconsistency does not concern the conditional as such, many of the paraconsistent logics that we have looked at have distinctive approaches to the conditional. And this is no accident. If one identi es  ! with the material conditional,   , de ned in the usual way as : _ , then modus ponens reduces to the disjunctive syllogism. But in any logic where disjunction behaves normally and deducibility is transitive, the disjunctive syllogism must fail, or explosion would arise, due to the \Lewis independent argument". Speci cally, in all the logics we have looked at except lter logics and some of the non-adjunctive logics, the syllogism fails. In such logics, therefore, a distinct account of the conditional is required. For completeness' sake, we will start by considering the others.

5.1

! as 

In lter logics, we may simply identify ! with . Things then proceed as before. A one-premise inference in this language, = , is prevalid i it is classically valid and F (; ). It is valid i it is a substitution instance of a prevalid inference.57 In the natural extension of Tennant's semantic approach, an inference from  to  is prevalid i  j=C  and for no proper subsets of  and , 0 and 0 , respectively, 0 j=C 0 . The natural extension of the proof theory is to add the conditional rules: ;  : ;
 :  ! ;


1 ;  :
1 ; 2 :
2 1 ; 2 ;  ! :
1 ;
2

Unfortunately, the equivalence between these two approaches now fails. For, semantically, p j= :p ! q (though the system is still paraconsistent); but without dilution there is no proof of the sequent p : :p ! q. At this point, Tennant prefers to go with the proof theory rather than the semantics. He also prefers the intuitionist version, which allows at most one formula on the right-hand side of a sequent. For further details, including natural deduction versions of the proof theory, see Tennant [1987], ch. 23. In [1992] Tennant suggests modifying the rule for the introduction of ! on the right.58 The  in the premise sequent is made optional, and the following rule is added. 57 One can modify this approach, invoking the lter in the truth conditions of the conditional itself, to give logics of a more relevant variety. This is pursued in a number of the essays in Philosophical Studies 26 (1979), no. 2, a special issue on relatedness logics. 58 In fact, he gives the natural deduction rules. The sequent rules described are the obvious equivalents.

PARACONSISTENT LOGIC

317

;  :
 :  ! ;
The exact relationship between these rules and the above semantics is as yet unresolved. In the non-adjunctive logics of Rescher and Manor, and Schotch and Jennings: ! may again be identi ed with , producing no novelties. The machinery of maximally consistent subsets and partitions carries straight over.

5.2 Discursive Implication The situation is otherwise with discursive logic. Here a distinct approach is required, since, as we have already seen, the disjunctive syllogism fails discursively. Given an S 5 interpretation, Jaskowski adds a conditional, ! (often written as d , and called discursive implication), and de nes  ! as 3  .59 It is easy to check that in discursive logic ;  ! j= , since 3; 3(3  ) j=S5 3 (and so there are essentially multi-premise inferences). In fact, the logical truths of the pure ! fragment of discursive logic are the same as those of the pure  fragment of classical logic. For let  be any sentence containing only s, and let ! be the corresponding sentence containing only !s. In an S 5 interpretation with only one world,  and 3! are equivalent. So if  is not a classical logical truth, 3! is not a discursive one. Conversely, suppose that  is a classical logical truth. We need to show that 3! is valid in every S 5 model. As may easily be checked, in S 5, 3(3  ) is logically equivalent to 3  3 . Hence, given 3! , we may \drive the 3s inwards" to obtain a logically equivalent sentence where the modal operator applies only to propositional parameters. But this is a substitution instance of  , and hence valid in S 5. This result does not carry over to the full language. For example, 6j=  ! (: ! ), since, as may be checked, 6j=S5 3(3  (3:  )).60 Full discursive logic can naturally be generalised in two obvious ways. The rst is by using some modal logic other than S 5. The second is by changing the de nition of what it is for a sentence, , to hold discursively in an interpretation. We change this from 3 holding to M holding, where M is some other modality (i.e., string of 3s and 2s). For references and discussion, see Blaszczuk [1984] and Kotas and da Costa [1989]. 59 Given what amounts to Jaskowski's identi cation of truth with truth in some possible world, it might be more natural to de ne  ! as 3 ! 3 . This would have just the same consequences. 60 The natural de nition of the biconditional,  $ , is ( ! ) ^ ( ! ). For reasons not explained, Jaskowski de nes it as ( ! ) ^ ( ! 3). This asymmetric and counter-intuitive de nition would seem to have no signi cant advantages.

318

GRAHAM PRIEST

5.3 Da Costa's C -systems The natural way of extending the non-truth functional semantics of 4.4 to include a conditional connective, in keeping with the idea that such logics are just the addition of a non-truth-functional negation to a standard positive logic, is to give ! the classical truth conditions:

 ( ! ) = 1 i  () = 0 or  ( ) = 1 (Note that !, so de ned, is distinct from .) Adding this condition to the logics of 4.4 (except, C! , which we will come to in a moment) gives the full (propositional) versions of the logics mentioned there; in particular it gives the da Costa logic C1 (and the other Ci for nite non-zero i). In each case, a natural deduction system can be obtained by adding the rules:

!E



(a)

!



!

(b)  _ ( ! ) Soundness is proved as usual. The extension to the completeness proof amounts to checking that for a prime theory, ,  ! 2  i  2=  or 2 . From left to right, the result follows by (! E ). From right to left: if 2  then the result follows from (a); if  2=  then ( ! ) 2  by (b) and primeness. If instead of (a) and (b), we add to any of these systems|except the ones with a consistency operator; I will come to these in a second|the rule:

!I

 .. .

!

we obtain, not classical positive logic, but intuitionist positive logic. (These rules are well known to be complete with respect to this logic.) In particular, if we add ! I and ! E to the rule system for the basic language fragment of C! we obtain da Costa's C! . The intuitionist conditional is not, of course, truth functional, but a valuational semantics for C! can be obtained as follows. A semi-valuation is any function that satis es the conditions for conjunction, disjunction and negation, plus:

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319

if  ( ! ) = 1 then  () = 0 or  ( ) = 1 if  ( ! ) = 0 then  ( ) = 0 A valuation is any semi-valuation,  , satisfying the following condition. Let  be of the form 1 ! (2 ! (3 ::: ! n ):::), where n is not itself of the form ! . Then if  () = 0 there is a semi-valuation,  0 , such that for all 1  i < n,  0 (i ) = 1, and  0 (n ) = 0. C! is sound and complete with respect to this notion of valuation. For details, see Loparic [1986].61 Changing the deduction rules for ! to the intuitionist ones, makes no dierence for those logics that contain a consistency operator, and in particular, the da Costa logics Ci for nite i.62 The reason, in nuce, is that the consistency operator allows us to de ne a negation with the properties of classical negation. As is well known, the addition of such a negation to positive intuitionist logic is not conservative, but produces classical logic. In more detail, the argument for C1 is as follows.63 De ne :  as : ^ o . Then it is easy to check that:

 (: ) = 1 i  () = 0 In particular, then, :  satis es the rules for classical negation:

 _ : 

 ^ : 

Given these, it is easy to show that  ! a` :  _ . (Hint: from left to right, assume  _ :  and argue by cases. From right to left, assume  and :  _ , and argue to by cases.) Hence, ! has the classical truth conditions.

5.4 Many-valued Conditionals There are numerous ways to de ne a many-valued conditional operator. We will just look at two of the more systematic.64 Given a Sugihara generalisation of LP , one can de ne a conditional with the following truth conditions: 61 A Kripke-style semantics for C! can be found in Baaz [1986]. 62 This was rst observed, in eect, by da Costa and Guillaume [1965]. 63 The argument for the other Ci s is similar. 64 In the three-valued case, other de nitions give the system of Asenjo and Tamburino [1975], and the J systems of D'Ottaviano and da Costa [1970]. A natural many-valued conditional, given the four-valued semantics of F DE , produces the system BN 4 of Brady [1982].

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GRAHAM PRIEST

 ( ! ) =  (: _ ) =  (: ^ )

if  ()   ( ) if  () >  ( )

This de nition gives rise to \semi-relevant" logics, i.e., logics that avoid the standard paradoxes of relevance, but are still not relevant. In the case where the set of truth values is the set of all integers, this gives the Anderson/Belnap logic RM . Proof-theoretically, RM is obtained from the relevant logic R, which we will come to in the next section, by adding the \mingle" axiom:

`  ! ( ! ) For details of proofs, see Anderson and Belnap [1975], sect. 29.3. In the 3-valued case, where the set of truth values is f 1; 0; +1g, the conditions for ! give the matrix:

! +1 0 1 +1 +1 1 1 0 +1 0 1 1 +1 +1 +1 and the stronger logic called RM 3. This is sound and complete with respect to the axiomatic system obtained by augmenting the system R with the axioms:

` (: ^ ) ! ( ! ) `  _ ( ! ) For the proof, see Brady [1982]. Turning to the second systematic approach, consider any Lukasiewicz generalisation of LP . Lukasiewicz' truth conditions for his conditional, 7!, are as follows:

 ( 7! ) = 1 = 1 ( ()  ( ))

if  ()   ( ) if  () >  ( )

In the three-valued case, this gives the well known matrix:

7!

1 0.5 0 1 1 0.5 0 0.5 1 1 0.5 0 1 1 1

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321

Now the most notable feature of the Lukasiewicz de nition, given that 0:5 is designated, is that modus ponens fails. For example, consider a valuation,  , where  (p) = 0:5 and  (q) = 0. Then  (p 7! q) = 0:5. Hence p; p 7! q 6j= q. (Modus ponens is valid provided that the only designated value is 1, but then the logic is not paraconsistent.) Kotas and da Costa [1978] get around this problem by adding to the language a new operator,
, with the truth conditions:

 (
) = 1 =0

if  () is designated otherwise

and then de ne a conditional,  ! , as
 7! .65 They point out the similarity of this de nition to Jaskowski's de nition of discursive implication. (In fact, they use the symbol 3 instead of
because of this.)66 It is not diÆcult to check that modus ponens for ! holds. In fact, as Kotas and da Costa point out, the ^, _, !-fragment of the logic is exactly positive classical logic. The easiest way to see this is just to collapse the designated values to 1, and the others to 0, to obtain classical truth tables.

5.5 Relevant !s Given a Routley interpretation (say one for F DE , though the other cases will be similar), it is natural to treat ! intensionally. The simplest way is to give it the S 5 truth conditions:

w ( ! ) = 1 i for all w0 2 W (w0 () = 1 ) w0 ( ) = 1) Clearly, given an interpretation either  ! is true at all worlds, or at none. With the Routley  giving the semantics for negation, it follows that the same is true of negated conditionals. It also follows that w ( ! ) = 1 i w ( ! ) = 1 i w :( ! ) 6= 1. Thus, the semantics validate the rules: LEM!

( ! ) _ :( ! )

:( ! )

and so are unsuitable for serious paraconsistent purposes. Moreover, even though there may be worlds where  ^ : is true, or where  _ : is false, EFQ!

!

65 In fact, their treatment is more general, since they consider the case in which the extension of
may be other than the set of designated values. 66 Pe~na [1984] de nes an operator, F , on real numbers such that the value F  is 0 if that of  is greater than 0, and 1 otherwise; and then de nes a conditional operator, C , as F  _ . The result is similar.

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and so neither ( ^ :) ! nor  ! ( _ :) is valid, the system is not a relevant one since, e.g., j= p ! (q ! q). These facts may both be changed by modifying the semantics, by adding a class of non-normal worlds. Thus, an interpretation is a structure hW; N; ;  i. The worlds in N are called normal; the worlds in W N (NN ) are called non-normal. Truth conditions are the same as before, except that at nonnormal worlds, the truth value of a conditional is arbitrary. Technically,  assigns to every pair of world and propositional parameter a truth value, as before, but for every w 2 NN and every conditional  ! , it now also assigns  ! a value at w. This provides the value of  ! at nonnormal worlds (non-recursively). Validity is de ned as truth preservation at all normal worlds of all interpretations. If one thinks of the conditionals as entailments, then the non-normal worlds are those where the facts of logic may be dierent. Thus, one may think of non-normal worlds as logically impossible situations.67 The system described is called H in Routley and Loparic [1978].68 It is sound and weakly complete (i.e., theorem-complete) with respect to the following axiom system.

`! ` ( ^ ) !  ` ( ^ ) ! ` ! ( _ ) ` ! ( _ ) `  $ :: ` (: _ : ) $ :( ^ ) ` (: ^ : ) $ :( _ ) ` ( ^ ( _ )) ! (( ^ ) _ ( ^ )) If `  and `  ! then ` If `  and ` then `  ^ If `  ! and ` ! then `  ! If `  ! then ` : ! : If `  ! and `  ! then `  ! ( ^ ) If `  ! and ` ! then ` ( _ ) ! Strong (i.e., deducibility-) completeness requires also the rules in disjunctive form.69 The disjunctive form of the rst is: `  _ and ` ( ! ) _ then ` _ . The others are similar.70

67 For a further discussion of non-normality, see Priest [1992]. 68 There are several other systems in the vicinity here. Some are obtained by varying the conditions on . Others, sometimes called the Arruda - da Costa P systems, are

obtained by retaining the positive logic and adding a non-truth-functional negation. For details, see Routely and Loparic [1978]. 69 Which are known to be admissible anyway. 70 A sound and complete natural deduction system is an open question.

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Soundness is proved as usual. The (strong) completeness proof is as follows. We rst show by induction on proofs that if  ` then  _ ` _ . It quickly follows that if  ` and ` then  _ ` . Now suppose that 6` . Extend to a prime theory, , with the same property, as in 4.3. Call a set
a -theory if it is prime, closed under adjunction, and ! 2  ) ( 2
) 2
). Note that  is a -theory. De ne the interpretation hW; N; ;  i, where W is the set of -theories; N = fg, 2
 i : 62
(which is well-de ned). If
2 NN then 
( ! ) = 1 i ! 2
; and for all
:


(p) = 1 i p 2
Once it can be shown that this condition carries over to all formulas, the result follows as usual. This is proved by induction. The only diÆcult case concerns ! when
= . From right to left, the result follows from the de nition of W . From left to right, the result follows from the following lemma. If ! 62  then there is a -theory,
, such that 2
and

62
. To prove this, we proceed essentially as in 4.3, except that  `  is rede ned. Let ! be the set of conditionals in ; then  `  is now taken to mean that there are 1 ; :::; n 2  and 1 ; :::; m 2  such that ! ` (1 ^ ::: ^ n ) ! (1 _ ::: _ m ). Now set  = f g, and  = f g, and proceed as in 4.3. The rest of the details are left as a (lengthy) exercise.71 If we add the Law of Excluded Middle to the axiom system:

`  _ : we obtain a logic that we will call HX . In virtue of the discussion in 4.7, one might suppose that this would be sound and complete if we add the condition: for all w, and parameters, p, 1 = w (p) or 0 = w (p). This condition indeed makes  _ : true in all worlds; but for just that reason, it also veri es the irrelevant ! ( _ :). To obtain HX , we place this constraint on just normal worlds. The semantics are then just right, as may be checked. For further details, see Routley and Loparic [1978]. Since normal worlds are now, in eect, LP interpretations, HX veri es all the logical truths of LP and so of classical logic. A feature of this system is that substitutivity of equivalents breaks down. For example, as is easy to check, p $ q 6j= (r ! p) $ (r ! q). This can be changed by taking the valuation function to work on propositions (i.e., set of worlds), rather than formulas.72 The most signi cant feature of semantics of this kind is that there are no principles of inference that employ nested 71 Details can be found in Priest and Sylvan [1992]. 72 For details see Priest [1992].

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GRAHAM PRIEST

conditionals in an essential way. This is due entirely to the anarchic nature of non-normal worlds. In eect, any breakdown of logic is countenanced. One way of putting a little order into the anarchy without destroying relevance, proposed by Routley and Meyer,73 is by employing a ternary relation, R, to give the truth conditions of conditionals at non-normal worlds. An interpretation is now of the form hW; N; R; ;  i. All is as before, except that  no longer gives the truth values of conditionals at non-normal worlds. Rather, for any w 2 NN , the truth conditions are:

w ( ! ) = 1 i for all x; y 2 W; Rwxy ) (x () = 1 ) y ( ) = 1) Note that this is just the standard condition for strict implication, except that the worlds of the antecedent (x) and the consequent (y) have become distinguished. What, exactly, the ternary relation, R, means, is still a matter for philosophical deliberation. Validity is again de ned as truth preservation at all normal worlds. These semantics give the basic system of aÆxing relevant logic, B . An axiom system therefor can be obtained by replacing the last two rules for H by the corresponding axioms:

` (( ! ) ^ ( ! )) ! ( ! ( ^ )) ` (( ! ) ^ ( ! )) ! (( _ ) ! ) and adding a rule that ensures replacement of equivalents: If `  ! and ` ! Æ then ` ( ! ) ! ( ! Æ) The soundness and completeness proofs generalise those for H . Details can be found in Priest and Sylvan [1992]. We may form the system BX proof theoretically by adding the Law of Excluded Middle. Semantically, we proceed as with H , placing the appropriate condition on normal worlds. As with modal logics, stronger logics can be obtained by placing conditions on the accessibility relation, R. In this way, most of the logics in the Anderson/Belnap family can be generated. Details can be found in Restall [1993]. The strongest of these is the logic R, an axiom system for which is as follows:

`! ` ( ! ) ! (( ! ) ! ( ! )) `  ! (( ! ) ! )

73 Initially, this was in Routley and Meyer [1973]. For further discussion of all the following, see the article on Relevent Logic in this volume of the Handbook.

PARACONSISTENT LOGIC

325

` ( ! ( ! )) ! ( ! ) ` ( ^ ) ! ; ` ( ^ ) !  ` (( ! ) ^ ( ! )) ! ( ! ( ^ )) ` ! ( _ ); `  ! ( _ ) ` (( ! ) ^ ( ! )) ! (( _ ) ! ) ` ( ^ ( _ )) ! (( ^ ) _ ) ` ( ! : ) ! ( ! :) ` :: !  with the rules of adjunction and modus ponens. The equivalence between the Dunn 4-valued semantics and the Routley  operation that we noted in 4.7 suggests another way of obtaining an intensional conditional connective. In the simplest case, an interpretation is a structure hW; vi where W is a set of worlds and  is an evaluation of the parameters at worlds, but this time it is a Dunn 4-valued interpretation. The truth conditions for the basic language are as in 4.6, except that they are relativised to worlds. Thus, using the functional notation: 1 2 w (:) i 0 2 w () 0 2 w (:) i 1 2 w () 1 2 w ( ^ ) i 1 2 w () and 1 2 w ( ) 0 2 w ( ^ ) i 0 2 w () or 0 2 w ( ) 1 2 w ( _ ) i 1 2 w () or 1 2 w ( ) 0 2 w ( _ ) i 0 2 w () and 0 2 w ( ) The natural truth and falsity conditions for ! are: 1 2 w ( ! ) i for all w0 2 W; (1 2 w0 () ) 1 2 w0 ( )) 0 2 w ( ! ) i for some w0 2 W , 1 2 w0 () and 0 2 w0 ( ) These semantics do not validate the undesirable:

!

:( ! )

as their  counterparts do. But they are still not relevant. Relevant logics can be obtained by adding a class of non-normal worlds. The semantic values of conditionals at these may either be arbitrary, as with H , or, as with B , we may employ a ternary relation and give the conditions as follows:

326

GRAHAM PRIEST

1 2 w ( ! ) i for all x; y 2 W , Rwxy ) (1 2 x () )12 y ( )) 0 2 w ( ! ) i for some x; y 2 W; Rwxy; 1 2 x () and 0 2 y ( ) As usual, extra conditions may be imposed on R. This construction produces a family of relevant logics distinct from the usual ones, and one that has not been studied in great detail. One way in which it diers from the more usual ones is that contraposition of the conditional fails, though this can be recti ed by modifying the truth conditions for ! by adding the clause: `and 0 2 w0 ( ) ) 0 2 w0 ()' (or in the case of nonnormal worlds employing a ternary relation: `and 0 2 x( ) ) 0 2 y ()'). A more substantial dierence concerns negated conditionals. Because of the falsity conditions of the conditional, all logics of this family validate  ^ : j= :( ! ). This is a natural enough principle, but absent from many of the logics obtained using the Routley . The more usual relevant logics can be obtained with the 4-valued semantics, but only by using some ad hoc device or other, such as an extra accessibility relation, or allowing only certain classes of worlds. For details, see Routley [1984] and Restall [1995].

5.6

! as 

There is a very natural way of employing any algebra which has an ordering relation to give a semantics for conditionals. One may think of the members of the algebra as propositions, or as Fregean senses. The relation  on the algebra can be thought of as an entailment relation, and it is then natural to take  ! to hold in some interpretation,  , i  ()   ( ). The problem, then, is to express the thought that  ! holds in algebraic terms. We obviously need an algebraic operator, !, corresponding to the connective; but how is one to express the idea that a ! b holds when the algebra may have no maximal element? A way to solve this problem for De Morgan algebras is to employ a designated member of the lattice, e, and take the things that hold in the algebra to be those whose values are  e.74 While we are introducing new machinery, it is also useful algebraically to introduce another binary (groupoid) operator, Æ, often called `fusion', whose signi cance we will come back to in a moment. We may also enrich the basic language to one containing a constant, e, and an operator, Æ, expressing the new algebraic features. Thus, following Meyer and Routley [1972], let us call the structure A = hD; e; !; Æi a De Morgan groupoid i D is a De Morgan algebra, hA; ^; _; :i, and for any a; b; c 2 A: 74 A dierent way is to let T be a prime lter on the lattice, thought of as the set of all true propositions. We can then require that a ! b 2 T i a  b. For details, see Priest [1980].

PARACONSISTENT LOGIC

327

eÆa=a a Æ b  c i a  b ! c if a  b then a Æ c  b Æ c and c Æ a  c Æ b a Æ (b _ c) = (a Æ b) _ (a Æ c) and (b _ c) Æ a = (b Æ a) _ (c Æ a) The rst of these conditions ensures that e is a left identity on the groupoid. (Note that the groupoid may not be commutative.) And it, together with the second, ensure that a  b i e  a ! b. The third and fourth ensure that Æ respects the lattice operations in a certain sense. The sense is question in that of a sort of conjunction, and this makes it possible to think of fusion as a kind of intensional conjunction. An inference, 1 ; :::; n = ; is algebraically valid i for every homomorphism,  , into a De Morgan groupoid,  (1 ^ ::: ^ n )   ( ), i.e., e   ((1 ^ ::: ^ n ) ! )).75 These semantics are sound and complete with respect to the relevant logic B of 5.5. Soundness is shown in the usual way, and completeness can be proved, as in 4.8, by constructing the Lindenbaum algebra, and showing that it is a De Morgan groupoid. Stronger logics can be obtained, as usual, by adding further constraints. The condition: e  a _:a gives the law of excluded middle (and all classical tautologies). Additional constraints on Æ give the stronger logics in the usual relevant family, including R. Details of all the above can be found in Meyer and Routley [1972] (who also show how to translate between algebraic and world semantics).76 Before leaving the topic of conditionals in algebraic paraconsistent logics, a nal comment on dual intuitionist logic. Goodman [1981] proves that in this logic there is no conditional operator (i.e., operator satisfying modus ponens) that can be de ned in terms of _; ^ and :; and draws somewhat pessimistic conclusions from this concerning the usefulness of the logic. Such pessimism is not warranted, however. Exactly the same is true in relevant logic; this does not mean that a conditional operator cannot be added to the basic language. And as Mortensen notes,77 given any algebraic structure with top (>) and bottom (?) elements, the following conditions can always be used to de ne a conditional operator:

 ( ! ) = > =?

if  ()   ( ) otherwise

75 A dierent notion of validity can be formulated using fusion thus:  (1 Æ ::: Æ n )   ( ), i.e., e   ((1 ! (::: ! (n ! ):::). 76 See also Brink [1988]. A rather dierent algebraic approach which produces a relevant logic is given in Avro`n [1990]. This maintains an ordered structure, but dispenses with the lattice. The result is a logic closely related to the intensional fragment of RM . 77 Mortensen [1995], p. 95.

328

GRAHAM PRIEST

Though this particular conditional is not suitable for robust paraconsistent purposes since it satis es:  ! ; :( ! ) j= .

5.7 Decidability Before we leave the topic of propositional logics, let me review, brie y, the question of decidability for the logics that we have looked at. Unsurprisingly, most (though not all) are decidable, as the following decision procedures indicate. As will be clear, in many cases the procedures actually given could be greatly optimised. Any lter logic is decidable if the lter is. Given any inference, we can eectively nd the set of all inferences of which it is a uniform substitution instance. Provided that the lter is decidable, we can test each of these for prevalidity. If any of them is valid, the original inference is valid; otherwise not. Smiley's lter is clearly decidable. So is Tennant's semantic lter. Given an inference with nite sets of premises and conclusions,  and , respectively, we can test the inference for classical validity. We may then test the inferences for all subsets of  and . (There is only a nite number of these.) If the original inference is valid, but its subinferences are not, it passes the test; otherwise not. Tennant's proof theory of 5.1 is also decidable. Anything provable has a Cut-free proof (since Cut is not a rule of proof). Decidability then follows as it does in the case of classical logic. Turning to non-adjunctive logics: Jaskowski's discursive logic is decidable; we may simply translate an inference into the corresponding one concerning S 5, and use the S 5 decision procedure for this. The same obviously goes for any generalisation, provided only that the underlying modal logic is decidable. Rescher and Manor's logic is decidable in the obvious way. Given any nite set of premises, we can compute all its subsets, the classical consistency of each of these, and hence determine which of the sets are maximally consistent. Once we have these, we can determine if any of them classically entails the conclusion. Similar comments apply to Schotch and Jennings' logic. Given any premise set, we can compute all its partitions, and so determine its level. For every partition of that size, we can test to see if one of its members classically entails the conclusion. Non-truth-functional logics are also decidable by a simple procedure. Given an inference, we consider the set of all subformulas of the sentences involved (which is nite). We then consider all mappings from these to f0; 1g, the set of which is also nite. For each of these we go through and test whether it satis es the appropriate constraints in the obvious way. Throwing away all those that do not, we see whether the conclusion holds

PARACONSISTENT LOGIC

329

in all that remain.78 All nite many-valued logics are decidable by truth-tables. The in nite valued Lukasiewicz logics (and so their Kotas and da Costa augmentations) are not, in general, even axiomatisable, let alone decidable. (See Chang [1963].) This leaves RM . If there is a counter-model for an RM inference, there must be a number of maximum absolute value employed. Ignoring all the numbers in the model whose absolute size is greater than this gives a nite counter-model. Hence, RM has the nite model property. As is well known, any axiomatisable theory with this property is decidable. (Enumerate the theorems and the nite models simultaneously. Eventually we must nd either a proof of a countermodel.) Dual intuitionist logic is decidable since intuitionist logic is. We just compute the dual inference and test it with the intuitionist procedure. This just leaves the logics of the relevant family. As we saw, the semantics of these can take either a world form or an algebraic form. The question of decidability here is the hardest and most sensitive. The weaker logics in the family are decidable, and can be shown to be so by semantic methods (such as ltration arguments) and/or proof theoretic ones (such as Gentzenisation plus Cut elimination).79 The stronger ones, such as R, are not. Urquhart's [1984] proof of this fact contains one of the few applications of geometry to logic. A crucial principle in this context would seem to be contraction: ( ! ( ! )) ! ( ! ) (or various equivalent forms, such as ( ^ ( ! )) ! ). Speaking very generally, systems without this principle are decidable; systems with it are not. 6 QUANTIFIERS The novelty of paraconsistent logic lies, it is fair to say, almost entirely at the propositional level. However, if a logic is to be applied in any serious way, it must be quanti cational. Most of the paraconsistent logics that we have considered extend in straightforward ways to quanti ed logics. In this section I will indicate how. Let us suppose that the propositional language is now augmented to a language, L, with predicates, constants, variables and the quanti ers 8 and 9 in the usual way. I will let the adicity of a predicate be shown by the context. Propositional parameters can be identi ed with predicates of adicity 0. I will write (x=t) to mean the result of substituting the term t for all free occurrences of x, any bound variables in  having been relabelled, if necessary, to avoid clashes. I will reserve the word `sentence' for formulas without free variables. I will always de ne validity for inferences containing only sentences, though the accounts could always be extended to ones employing all formulas, in 78 For the method applied to the da Costa systems, see da Costa and Alves [1977]. 79 See, respectively, Routley et al. [1982], sect. 5.9, and Brady [1991].

330

GRAHAM PRIEST

standard ways. Where quanti ers have an objectual interpretation, and the set of objects is D, I will assume|for the rest of this essay|that the language has been augmented by a set of constants in such a way that each member of the domain has a name. In particular, I will always assume that the names are the members of D themselves, and that each object names itself. This assumption is never essential, but it simpli es the notation.

6.1 Filter and Non-adjunctive Logics In lter logics, we may simply take the lter to be a relation on the extended language. Smiley's lter works equally well, for example, when the notion of classical logical truth employed is that for rst order, not propositional, logic. Similarly for Tennant's. In his case (without the conditional operator), the semantics are sound and complete with respect to the sequent calculus of 4.1 for the basic language, together with the usual rules for the quanti ers: : (x=c);
: 8x;
: ;
: 9x;


; :
; 8x :
; (x=c) :
; 9x :


where in the rst and last of these, c does not occur in any formula in or
. For proofs, see Tennant [1984]. (With the conditional operator added, the situation is dierent, as we saw in 5.1.) Non-adjunctive logic accommodates quanti ers in an obvious way. Consider discursive logic. An inference in the quanti ed language is discursively valid i 3 j=CS5 3, where CS 5 is constant-domain quanti ed S 5. Clearly, any other quanti ed modal logic could be used to generalise this notion.80 Rescher and Manor's approach and Schotch and Jennings' also generalise in the obvious way, the classical notion of propositional consequence involved being replaced by the classical rst-order notion. In the quanti cational case, the usefulness of these logics is moot, since the computation of classically maximally consistent sets of premises, or partitions, is highly non-eective. In all these logics, except Smiley's, the set of logical truths (in the appropriate vocabulary) coincides with that of classical quanti er logic; hence these logics are undecidable.81 80 For details of quanti ed modal logic, see the article on that topic in this Handbook. 81 I do not know whether Smiley's logic is decidable, though I assume that it is not.

PARACONSISTENT LOGIC

331

6.2 Positive-plus Logics Let us turn now to the logics that augment classical or intuitionist positive logic with a non-truth-functional negation. Since the semantics of these are not truth functional, the most natural quanti er semantics are not objectual, but substitutional. Let me illustrate this with the simplest non-truth-functional logic, with a classical conditional operator, but no semantic constraints on negation. Extensions of this to other cases are left as an exercise. An interpretation is a pair hC;  i. C is a set of constants, and LC is the language L augmented by the constants C .  is a map from the sentences of LC to f1; 0g satisfying the same conditions as in the propositional case, together with:

 (8x) = 1 i for every constant of LC , c,  ((x=c)) = 1  (9x) = 1 i for some constant of LC , c,  ((x=c)) = 1 An inference is valid i it is truth-preserving in all interpretations. The semantics are sound and complete with respect to the quanti er rules:

8I

_ (x=c) _ 8x

provided that c does not occur in , or in any undischarged assumption on which the premise depends.

x 8E 8(x=c )

9I

9E

(x=c) 9x (x=c) .. . 9x

provided that c does not occur in or in any undischarged assumption in the subproof. Soundness is proved by a standard recursive argument. For completeness, call a theory, , saturated in a set of constants, C , i:

9x 2  i for some c 2 C , (x=c) 2 

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GRAHAM PRIEST

8x 2  i for every c 2 C , (x=c) 2  It is easy to check that if
is a prime theory, saturated in C , then hC;  i is an interpretation, where  is de ned by:  () = 1 i  2
. It remains to show that if  6`  then  can be extended to a prime theory,
, saturated in some set of constants, C , with the same property; and the result follows as in the propositional case, using
to de ne the interpretation. To show this, we augment the language with an in nite set of new constants, C , and then extend the proof of 4.3 as follows. Enumerate the formulas of LC : 0 , 1 ,... If 8x or 9x occurs in the enumeration, and the constant c does not occur in any preceding formula, we will call (x=c) a witness. Now, we run through the enumeration, as before, but this time, if we throw 9x into the  side, we also throw in a witness; and if we throw 8x into the  side, we also throw in a witness. In proving that n 6` n , the only novelty is when a witness is present; and these can be ignored, by 9E on the left, and 8I on the right. The rest of the proof is as in 4.3. The saturation of
in C follows from deductive closure and construction. I observe that all the logics in this family contain positive classical quanti er logic, and so are undecidable.

6.3 Many-valued Logics Most of the many-valued logics with numerical values that we considered in 4.5 and 5.4 had two particular properties. First, the truth value of a conjunction [disjunction] is the minimum [maximum] of the values of the conjuncts [disjuncts]. Second, the set of truth values is closed under greatest lower bounds (glbs) and least upper bounds (lubs), i.e., if Y  X then glb(Y ) 2 X and lub(Y ) 2 X . Any such logic can be extended to a quanti ed logic in a very natural way, merely by treating 8 and 9 as the \in nitary" generalisations of conjunction and disjunction, respectively. Speci cally, a quanti er interpretation adds to the propositional machinery, the pair hD; di where D is a non-empty domain of objects, d maps every constant into D, and if P is an n-place predicate, d maps P to a function from n-tuples of the domain into the set of truth-values. Every sentence, , can now be assigned a truth value,  (), in the natural way. For atomic sentences, P c1 :::cn :

 (P c1 :::cn ) = d(P ) hd(c1 ):::d(cn )i The truth conditions for propositional connectives are as in the propositional logic. The truth conditions for the quanti ers are:

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 (8x) = glbf(x=c); c 2 Dg  (9x) = lubf(x=c); c 2 Dg Validity is de ned in terms of preservation of designated values, as in the propositional case. I will make just a few comments about what happens when these de nitions are applied to the many-valued logics we have looked at. The quanti ed nite-valued logics of 4.5 all collapse into quanti ed LP (which we will come to in the next section), as extensions of the arguments given there, show. For a general theory of quanti ed nitely-many-valued logics, see Rosser and Turquette [1952]. Quanti ed RM we will come to in a later section. In nite-valued Lukasiewicz logics are proof-theoretically problematic. For a start, standard quanti er rules may break down. In particular, 8x may be undesignated, even though each substitution instance is designated. Thus, 8I may fail. (Similarly for existential quanti cation.) Worse, as for their propositional counterparts, such logics are not even axiomatisable in general.82

6.4 LP and F DE The technique of extending a many-valued logic to a quanti ed one can be put in a slightly dierent, and possibly more illuminating, way for the logics with relational semantics, LP and F DE . An interpretation, I , is a pair, hD; di, where D is the usual domain of quanti cation, d is a function that maps every constant into the domain, and every n-place predicate into a pair, hEP ; AP i, each member of which is a subset of the set of n-tuples of D; Dn . EP is the extension of P ; AP is the anti-extension. For LP interpretations, we require, in addition, that EP [ AP = Dn . Truth values are now assigned to sentences in accord with the following conditions. For atomic sentences: 1 2  (P c1 :::cn ) i hd(c1 ); :::; d(cn )i 2 EP 0 2  (P c1 :::cn ) i hd(c1 ); :::; d(cn )i 2 AP Truth/falsity conditions for connectives are as in the propositional case; and for the quanti ers: 1 2  (8x) i for every c 2 D; 1 2  ((x=c)) 0 2  (8x) i for some c 2 D, 0 2  ((x=c)) 1 2  (9x) i for some c 2 D; 1 2  ((x=c))

82 See Chang [1963] for details.

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0 2  (9x) i for every c 2 D; 0 2  ((x=c)) An inference is valid i it is truth-preserving in all interpretations. It should be noted that if for every predicate, P , EP and AP are exclusive and exhaustive, we have an interpretation of classical rst order logic. All classical interpretations are therefore F DE (and LP ) interpretations. These semantics are sound and complete if we add to the rules for LP or F DE , the rules 8I , 8E , 9I and 9E , plus:

8x: :9x

9x: :8x

Soundness is established by the usual argument. For completeness, suppose that  6` . Extend  to a set
, which is prime, deductively closed and saturated in a set of new constants, such that
6` , as in 6.2. Then de ne an interpretation hD; di where D is the set of constants of the extended language, d maps any constant to itself, and for any predicate, P , its extension and anti-extension are de ned as follows:

hc1 ; :::; cn i 2 EP hc1 ; :::; cn i 2 AP

i P c1 :::cn 2
i :P c1 :::cn 2


We now establish that for all formulas,  : 1 2  () i  2
0 2  () i : 2
The argument is a routine induction. Here are the cases for 8.

8x 2
, for all c; (x=c) 2
saturation , for all c; 1 2  ((x=c)) induction hypothesis , 1 2  (8x) truth conditions of 8 :8x 2
, 9x: 2
quanti er rules , for some c; :(x=c) 2
saturation , for some c; 0 2  ((x=c)) induction hypothesis , 0 2  (8x) truth conditions of 8 , 1 2  (:8x) truth conditions of : The monotonicity property of the propositional logics LP and F DE carries over to the quanti ed case. If I1 and I2 are any interpretations, with truth value assignments 1 and 2 , de ne I1  I2 to mean that I1 and I2 have the same domain, and for every predicate, P , the extension (antiextension) of P in I1 is a subset of the extension (anti-extension) of P in

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I2 .

A simple induction shows that if I1  I2 then for all formulas,  (in a language with a name for every member of the domain), 1 ()  2 (). As in 4.6, it follows that the set of logical truths of LP is exactly the same as that of classical rst order logic. And F DE has no logical truths (just consider an interpretation that makes the extension and anti-extension of every predicate empty). Since classical quanti er logic is not decidable, neither is quanti ed LP . If P is any n-place predicate, let PLEM be the sentence: 8x1:::8xn (P x1 :::xn _ :P x1 :::xn ). If PLEM is true in an interpretation, then the extension and anti-extension of P , exhaust the n-tuples of the domain. If  is any formula, let LEM be the conjunction of all formulas of the form PLEM , where P occurs in . It follows that LEM j=  in F DE i j=  in LP . Hence, quanti ed F DE is undecidable too.

6.5 Relevant Logics Turning to relevant logics, the issues are more complex. This is due to the fact that there are various approaches to these logics, the variety of the logics themselves, and the intrinsic complexities of the stronger logics. Let us start with the world semantics. As we saw in 5.5, a world semantics for a relevant logic with the Routely operator is a structure hW; N; ;  (; R)i, where W is a set of worlds, N is a subclass of normal worlds (the complement being NN ),  is the Routley operation (such that w = w ), and  assigns truth values to all propositional parameters at worlds. In the logic H , it also assigns values to conditionals at non-normal worlds. In stronger logics, the ternary relation R is present, and is used to specify the values of conditionals at non-normal worlds. When no constraints are placed on R, we have the logic B . The simplest way of extending such semantics to those of a quanti ed language is by removing  from the structure and adding a domain of quanti cation, D, and a denotation function d. d speci es a denotation for each constant (same at each world) and an extension for each n-place predicate at each world, dw (P )  Dn . Truth conditions are given in the standard way. In particular, for the quanti ers:

 (8x) = 1 i for every c 2 D;  ((x=c)) = 1  (9x) = 1 i for some c 2 D,  ((x=c)) = 1 An inference is valid i it is truth-preserving in all normal worlds of all interpretations.83

83 More complex semantics can be employed in the usual variety of ways employed in modal logic. (See the article on Quanti ed Modal Logic in this Handbook.) In particular, we might employ variable-domain semantics. This makes matters more complex.

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Consider the following quanti er axioms and rules (where ` is now taken to indicate universal closure):

` 8x ! (x=c) ` (x=c) ! 9x `  ^ 9x ! 9x( ^ ) x not free in  ` 8x( _ ) ! ( _ 8x ) x not free in If ` 8x( ! ) then ` 9x ! x not free in If ` 8x( ! ) then `  ! 8x x not free in  It is easy to check that these axioms/rules are valid/truth-preserving for H . If they are added to the propositional axioms/rules for H , they are also complete. For the proof, see Routley and Loparic [1980].84 If we strengthen the two rules to conditionals (so that the rst of these becomes ` 8x( ! ) ! (9x ! ), etc.) and add them to the rules for B , they are also sound and complete. The same is true for a number of the extensions of B , including BX . (For details, see Routley [1980a].) A notable exception to this fact is the system R. Though the system is sound, it is, perhaps surprisingly, not complete.85 In fact, a proof-theoretic characterisation of constant domain quanti ed R is still an open problem. The axioms and rules are complete for the stronger semi-relevant system RM of 5.4.86 Since every relevant logic in the above family contains F DE , and this is undecidable, it follows that all the logics in this family are also undecidable.

6.6 Algebraic Logics Given any algebraic logic, for which the appropriate algebraic structures are lattices, and in which conjunction and disjunction behave as lattice meet and join, there is, as with many-valued logics, a natural way to extend the machinery to quanti ers. An algebra is complete W V i it is closed under least upper bounds ( ) and greatestWlower boundsV( ), i.e., if the domain of the algebra is A and B  A then B 2 A and B 2 A. If A is any algebraic structure of the required kind, with domain A, then an interpretation is a triple hA; D; di, where D is the domain of quanti cation, d maps every constant into D and every n-place predicate into a function from Dn into (The philosophical gain, however, is dubious: world relativised quanti ers can always be de ned in constant-domain semantics, provided we have an Existence predicate.) 84 If one works with a free-variable notion of deducibility, as Routley and Loparic do, one also has to add the rule of universal generalisation: if `  then ` 8x. 85 As Fine showed. Fine also produced a rather dierent semantics with respect to which it is complete. See Anderson et al. [1992], sects. 52 and 53. 86 See Anderson et al. [1992], sect. 49.2.

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A. Algebraic values are then assigned to all formulas in the usual way. In particular, for quanti ed sentences the conditions are: V

 (8x) = Wf ((x=c)); c 2 Dg  (9x) = f ((x=c)); c 2 Dg I will comment on this construction for only two kinds of algebras. The rst is when A is a De Morgan groupoid, or strengthening thereof. In this case, the above semantics clearly give quanti ed relevant logics. Their relation to quanti ed relevant logics based on the intensional semantics has not, as far as I am aware, been investigated. The second is where A is a dual intuitionist algebra. In this case, the semantics give a quanti ed logic that is dual to quanti ed intuitionist logic. For details, see Goodman [1981].87

6.7 A Brief Look Back Now that we have surveyed a large number of paraconsistent logics up to a quanti ed level|some very brie y|it would seem appropriate to look back for a moment and put the systems into some sort of perspective. The logics we have looked at fall roughly and inexactly into four categories: non-transitive logics, non-adjunctive logics, non-truth-functional logics and relevant logics. (The most interesting many-valued systems are zero degree relevant logic, F DE , or closely related to it, like LP , and so may be classed in this family.) The non-transitive logics seem to be good for extracting the essential juice out of classical inferences, but do not really take inconsistent semantic structures seriously. Non-adjunctive logics may be just what one needs for certain applications (e.g., inferences in a data base, where one would not necessarily want to infer  ^: from  and :); they also take inconsistent structure seriously, though conjoined contradictions are handled indiscriminately, which makes them unsuitable for many applications. Non-truth-functional logics contain the whole of classical (or at least intuitionist) positive logic, and so are useful when strong canons of positive reasoning are required. However, this very strength is a weakness when it comes to some important applications, as we shall see in connection with set theory. Undoubtedly the simplest and most robust paraconsistent logic is the logic LP . When conditional operators are required, the relevant logic BX is a good all-purpose paraconsistent logic. Its conditional operator is satisfactory for many purposes, but may be considered relatively weak. It may be strengthened to give stronger relevant logics; but this, too, may cause a problem for some applications, as we shall see. 87 There is also a topos-theoretic account of quanti cation for dual intuitionistic logic. See Mortensen [1995], ch. 11.

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7 OTHER EXTENSIONS OF THE BASIC APPARATUS I now want to look at other extensions of the basic paraconsistent apparatus. One way or another, all the paraconsistent logics we have looked at can be extended appropriately. However, it is tedious to run through every case, especially when details are often obvious. Hence, I shall illustrate the extensions mainly with respect to just one logic. Since LP is simple and natural, it recommends itself for this purpose. I will comment on other logics occasionally, when there is a point to doing so.

7.1 Identity and Function Symbols LP |and all the other logics with objectual semantics that we have looked at|can be extended to include function symbols and identity in the usual way. The denotation function, d, maps each n-place function symbol, f , to an n-place function on the domain. A denotation for every (closed) term, t, is then obtained by the usual recursive condition: d(ft1 :::tn ) = d(f )(d(t1 ); :::; d(tn )) With functional terms present, the quanti er rules of proof are extended to arbitrary (closed) terms in the usual way. If we require the extension of the identity predicate to be fhx; xi ; x 2 Dg then this is suÆcient to validate the usual laws of identity:

t=t

t1 = t2 (x=t1 ) (x=t2 )

This does not require identity statements to be consistent. In LP the antiextension of identity is any set whose union with the extension exhausts D2 , and so a pair can be in both the extension and the anti-extension of the identity predicate. In other logics, negated identities can be taken care of by whatever mechanism is used for negation. The completeness proof for quanti ed LP can be extended to include function symbols and identity in the usual Henkin fashion. I note that description operators can be added in the obvious ways, with the same panoply of options as in the classical case.88

7.2 Second-order Logic Paraconsistent logics can also be extended to second order in the obvious ways. Consider LP . We add (monadic) second order variables, X , Y ,::: to 88 See the article of Free Logics in this Handbook.

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the rst-order language. Then, given an interpretation, hD; di, we extend the language to one, LD , such that every member of the domain has a name, and for every pair E , A such that E [ A = D there is a predicate, P , with E and A as extension and anti-extension, respectively.89 The truth/falsity conditions for the second order universal quanti er are then: 1 2  (8X) i for every P in LD , 1 2  ((X=P )) 0 2  (8X) i for some P in LD , 0 2  ((X=P )) The truth/falsity conditions for the existential quanti er are the dual ones. Appropriate monotonicity carries over to second order LP . Recall from 6.4 that if I1 and I2 are any interpretations, with truth value assignments 1 and 2 , I1  I2 means that I1 and I2 have the same domain, and for every predicate, P , the extension (anti-extension) of P in I1 is a subset of the extension (anti-extension) of P in I2 . The same sort of induction as in the rst-order case shows that if I1  I2 then for all formulas, , in LD , 1 ()  2 (). (The predicates added in forming LD have the same extension/anti-extension in both interpretations; and thus atomic sentences containing them satisfy the condition.) In the second order case, and unlike the rst order case, the logical truths of LP are distinct from their classical counterparts. For example, as is easy to check, in LP , j= 9X (Xa ^:Xa) (just consider the predicate which has D as both extension and anti-extension).90 In fact, the logical truths of second order LP are inconsistent, since it is also a logical truth that 8X (Xa_:Xa), which is equivalent by quanti er rules and De Morgan to :9X (Xa ^ :Xa).

7.3 Modal Operators All the logics may have modal operators added to them in one way or another. In the case of discursive logics, indeed, the semantics already provide for the possibility of alethic modal operators. Adding modal operators to intensional logics where negation is handled by the Routley  operator is very natural, but suers problems similar to those we witnessed at the start of 5.5 in connection with the conditional. Suppose we take an intensional interpretation and give the modal operators the natural S 5 conditions:

w (2) = 1 i for every w0 2 W; w0 () = 1

89 This is the natural policy, since properties are characterised semantically by an extension/anti-extension pair. As in the classical case, there are other policies, e.g., where only predicates corresponding to some restricted class of properties are added. 90 Second order F DE is constructed in the obvious way. The same sentence is a logical truth of this, showing that, unlike the rst order case, it has logical truths.

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w (3) = 1 i for some w0 2 W; w0 () = 1 (or even N instead of W ). Then the truth values of modalised statements are the same at all worlds. Hence, w (2) = 1 , w (2) = 1 , w (:2) = 0. Hence 2; :2 j= , and so the logic is not suitable for serious paraconsistent purposes. The problem does not arise if we attempt a modal logic weaker than S 5, for then the truth conditions of modal operators are given employing a binary accessibility relation in the usual way, and the truth values of modal statements will vary across worlds. But, at least for some purposes, an S 5 modality is desirable. These problems are avoided if we use the Dunn semantics for negation. The values of modalised formulas will still be the same at all worlds (in the S 5 case), but we may now have both 2 and :2 true at a world. I will illustrate, again, with respect to LP . Let us start with the case where the binary accessibility relation is arbitrary, the three-valued analogue of the modal system K . An interpretation is now a structure hW; R;  i, where W is a set of worlds; R is a binary relation on W ; and for each parameter, p, w (p) 2 ff1g; f1; 0g; f0gg. Truth/falsity conditions for the propositional connectives are as in 5.5. The conditions for 2 are: 1 2 w (2) i for every w0 such that wRw0 , 1 2 w0 () 0 2 w (2) i for some w0 such that wRw0 , 0 2 w0 () and dually for 3.91 It is easy to check that at every world of an interpretation 2: has the same truth value as :3, and dually. In fact, we can simply de ne 3 as :2:, and will do this in what follows. To obtain a proof-theoretic characterisation for the logic, we add to the rules for LP the following (chosen to make the completeness proof simple):

.. .

2

2

.. . Æ

3Æ

3

where there are no other undischarged assumptions in the sub-proofs.

2( _ Æ) 2 _ 3Æ

2 ^ 3 3( ^ )

91 If a conditional operator is required, we may add a class of non-normal worlds|and maybe a ternary accessibility relation|and proceed as in 5.5.

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2 1 ^ ::: ^ 2 n 2( 1 ^ ::: ^ n )

341

3(Æ1 _ ::: _ Æn ) 3Æ1 _ ::: _ 3Æn

Soundness is easily checked. For completeness, suppose that  6` . Extend  to a prime, deductively closed theory, , with the same property, as in 4.3. De ne an interpretation, hW; R;  i, where W is the set of prime deductively closed theories; R
i for all :

2 2 ) 2
2
) 3 2

and  is de ned by: 1 2  (p) i p 2 0 2  (p) i :p 2 All that remains is to show that these conditions extend to all formulas. Completeness then follows as usual. This is established by induction. The only diÆcult case is that for 2, which requires the following two-part lemma. If 2 62 then there is a
2 W such that R
and 62
. Proof: Let 2 = f ; 2 2 g and 3 = fÆ ; 3Æ 62 g. Then 2 6` ; 3 , by the rst and second pair of rules, and a bit of ddling with the third. Extend 2 to a prime, deductively closed set,
, with the same property, as in 4.3. The result follows. If 3 2 then there is a
2 W such that R
and 2
. Proof: Let 2 and 3 be as before. Then 2 ; 6` 3, by the rst and second pair of rules, and a bit of ddling with the third. Extend 2 ; to a prime, deductively closed set,
, with the same property, as in 4.3. The result follows. We can now prove the induction step for 2:

2 2

, 8
s.t. R
, 2


, 8
s.t. R
, 1 2 
( ) , 1 2  (2 ) :2 2 , 3: 2 , 9
( R
and : 2
) , 9
( R
and 0 2 
( )) , 0 2  (2 )

lemma in one direction de nition of R in the other induction hypothesis de nition of 3 lemma in one direction de nition of R in the other induction hypothesis

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Stronger modal logics can be obtained by placing conditions on R, and corresponding conditions on the proof theory. But even if we make R universal (so that for all x and y, xRy), and obtain the analogue of S 5, we still do not get 2; :2 j= . To see this, merely consider the interpretation with one world, w, which accesses itself; and where w (p) = f1; 0g and w (q) = f0g. It is easy to check that w (2p) = w (:2p) = f1; 0g. Hence, 2p; :2p 6j= q. The same treatment can be given to temporal operators. If we take these, as usual, to be F and G for the future, and P and H for the past, then (three-valued) tense logic gives F and G the same truth conditions as 3 and 2, respectively; and P and H are the same, except that R is replaced by its  i yRx). Appropriate soundness and completeness converse, R (where xRy proofs for the case where R is arbitrary are obtained by modifying the alethic modal argument,92 and stronger tense logics are obtained by adding conditions on R, in the usual way.93 Let me also mention conditional operators, >, of the Lewis/Stalnaker variety. These are modal (binary) operators, and can be given LP (or F DE ) semantics in the same way that they are given a more usual semantics. For example, for the Stalnaker version, one extends interpretations with a selection function, f (w; ), thought of as selecting the nearest world to w where  is true.  > is then true at w i is true at f (w; ). Details are left as a very non-trivial exercise.94

7.4 The Paraconsistent Importance of Modal Operators Let me digress from the technical details to say a little about why modal operators are important in the context of paraconsistency. The reason is simply that so many of the natural areas where one might want to apply a paraconsistent logic involve them. Take alethic modalities rst. Even though one might not think that there are any true contradictions, one might still take them to be possible, in the sense of holding in some situations, such as ctional or counterfactual ones. Thus, one might hold that for some p, 3(p ^ :p). This has a simple and obvious model in the above semantics. In this context, let me mention again the importance for counterfactual conditionals of worlds where the impossible holds; \impossible worlds" are just what one needs to evaluate such conditionals, according to the Lewis/Stalnaker semantics in whose direction I have just gestured. Some have been tempted not just by the view that some contradictions 92 See Priest [1982]. 93 See the article on Tense Logic in this Handbook. 94 For a paraconsistent theory of conditionals of this kind, and of many other modal

operators, that employs the Routley  to handle negation, see Routley [1989].

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are possible, but by the view that everything is possible.95 The valuation f1;0g assigns every formula the value f1; 0g. (See 4.6). Hence, any interpretation that contains f1;0g as one world will verify 3, for all , at any world that accesses it.96 If we interpret the modal operators 2 and 3 as the deontic operators O (it is obligatory that) and P (it is permissible that), respectively, then the thesis that everything is possible becomes the nihilistic thesis that everything is permissible|what, according to Dostoevski, would be the case if there is no God. Less exotically, standard deontic logic suers badly from explosion.97 Since in classical logic ; : j= it follows that O; O: j= O : if you have inconsistent obligations then you are obliged to do everything. This is surely absurd. People incur inconsistent obligations; this may give rise to legal or moral dilemmas, but hardly to legal or moral anarchy.98 And one does not have to believe in dialetheism to accept this. Unsurprisingly, deontic explosion fails, given the semantics of the previous section: just consider the interpretation where there is a single world, w; R is universal; w (p) = f1; 0g and w (q) = f0g. It is not diÆcult to check that w (Op) = w (O:p) = f1; 0g, whilst w (Oq) = f0g. What is often taken to be the basic possible-worlds deontic logic (called KD by Chellas [1980], p. 131) makes matters even worse, by requiring that in an interpretation the accessibility relation be serial: for all x, there is a y such that xRy. This validates the inference O=P . It also validates the inference O:=:O. Hence we have, classically, O; O: j= O ^ :O j= ; one who incurs inconsistent obligations renders the world trivial. Someone who believes that there are deontic dilemmas may just have to jettison the view that obligation entails permission, and so give up seriality. But on the above account one can retain seriality, and so both the above inferences; for O ^ :O 6j= , as the countermodel of the last paragraph shows.99 Another standard way of interpreting the modal operator 2 is as an epistemic operator, K (it is known that), or a doxastic operator B (it is believed that). In these cases, classically, one would almost certainly want to put extra constraints on the accessibility relation, though what these should be might be contentious: all can accept re exivity (xRx) for K (but not for B ) since this validates K j= . Whether one would want transitivity ((xRy&yRz ) ) xRz ) is much more dubious for B and K , since this gives the 95 E.g., Mortensen [1989]. 96 A similar, but slightly more complex, construction can be employed to the same eect if the logic has a conditional operator. 97 See the article on Deontic Logic in this Handbook for details of Deontic Logic, including the possible-worlds approach. 98 For further discussion, see Priest [1987], ch. 13. 99 We have just been dealing with some of the \paradoxes of deontic logic". There are many of these. Arguably, all of them|or at least all the serious ones|are avoided by using a paraconsistent logic with a relevant conditional. See Routley and Routley [1989].

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highly suspect K j= KK and B j= BB. All this applies equally to the semantics of the previous section. Moreover, the paraconsistent semantics solve problems for doxastic logic of the same kind as for deontic logic. It is clear that people sometimes have inconsistent beliefs (if not knowledge). Standard semantics give B; B : j= B . Yet patently someone may have inconsistent beliefs without believing everything.100 Observations such as this are particularly apt in the branch of AI known as knowledge representation, where it is common to use epistemic operators to model the information available to a computer. (See, e.g., a number of the essays in Halpern [1986].) Such information may well be inconsistent. Finally, to tense operators. Whilst one does not have to be a dialetheist to hold that inconsistencies may be believed, obligatory, or true in some counterfactual situation, one does have to be, to believe that they were or will be true. Such views have certainly been held, however. Following Zeno, the whole dialectical tradition holds that contradictions arise in a state of change. To see one of the more plausible examples of this, just consider a state described by p which changes instantaneously at time t0 to a state described by :p. What is the state of aairs at t0 ? One answer is that at t0 , p ^ :p is true. Indeed, the contradictory state is the state of change.101 This can be modeled by the paraconsistent interpretation hW; R;  i, where W is the set of real numbers (thought of as times); R is the standard ordering on the reals; and  is de ned by the condition:

t (p) = f1g = f1; 0g = f0g

if t < t0 if t = t0 if t > t0

It is easy to check that this interpretation veri es the inference: p ^ F :p=(p ^ :p) _ F (p ^ :p), which we might call `Zeno's Principle': change implies contradiction.

7.5 Probability Probability is not a modal notion. But it, too, has paraconsistent signi cance. One of the most natural ways of constructing a paraconsistent probability theory is to extract one from a class of paraconsistent interpretations, in the manner of Carnap.102

100 If you believe classical logic, then you might suppose that they are rationally committed to everything, but that is quite dierent. Even here, however, an explosive logic would seem to go astray. Dialetheism aside, situations such as the paradox of the preface, as well as more mundane things, would seem to show that one can be rationally committed to inconsistent propositions without being rationally committed to everything. See Priest [1987], sect. 7.4. 101 See Priest [1982] and Priest [1987], ch. 11. 102 See Carnap [1950].

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A probabilistic interpretation is a pair, hI; i, where I is a class of interpretations for LP 103 and  a nitely additive measure on I , that is, a function from subsets of I to non-negative real numbers such that:

() = 0 (X [ Y ) = (X ) + (Y ) if X \ Y =  If  is any sentence, let [] = f 2 I ; 1 2  ()g. For reasons that we will come to, we also require that for all , ([]) 6= 0. There certainly are such interpretations and measures. For example, let I be any nite class that contains the trivial interpretation, f1;0g , where all sentences are true, and let (X ) be the cardinality of X . Then this condition is satis ed. Given a probabilistic interpretation, we de ne a probability function, p, by:

p() = ([])=(I ) It is easy to see that p satis es all the standard conditions for a probability function, such as: 0  p()  1 if  j= then p()  p( ) if j=  then p() = 1 p( _ ) = p() + p( ) p( ^ ) except, of course: p(:) + p() = 1. Since we have p( ^ :) > 0, and p( _ :) = 1, it follows that p() + p(:) > 1. By the construction, we have, in fact, p() > 0 for all . It might be suggested that a person whose personal probability function gives nothing the value zero would have to be very stupid|or at least credulous. But since p() may be as small as one wishes, this hardly seems to follow. Moreover, giving nothing a zero probability signals an open-minded and undogmatic policy of belief. Arguably, this is the most rational policy. Given a probability function, conditional probability can be de ned in the usual way:

p(= ) = p( ^ )=p( ) A singular advantage of this paraconsistent probability theory over standard accounts is that conditional probability is always de ned, since the denominator is always non-zero. 103 Again, many other paraconsistent logics could be used instead.

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Perhaps the major application of probability theory is in framing an account of non-deductive inference. How, exactly, to do this is a moot question. But however one does it, a paraconsistent account of non-deductive inference can be framed in the same way, employing paraconsistent probability theory. For example, we may de ne the degree of (non-deductive) validity of the inference = to be p( =( ^ )), where  is our background evidence. As one would expect, deductively valid inferences come out as having maximal degree of inductive validity. To compute the degree of validity of an inference, so de ned, we would often need to employ Bayes' Theorem. Let us look at the paraconsistent two-hypothesis version of this. Suppose that we have two hypotheses, h1 and h2 , that are exclusive and exhaustive, in the sense that j= h1 _ h2 and j= :(h1 ^h2 ), and that we wish to compute the probability of h1 on evidence, e, given the inverse probabilities of these hypotheses on the evidence (all relative to some background evidence, , which we will ignore). Note rst that p(h1 =e) = p(h1 ^ e)=p(e) = p(e=h1):p(h1 )=p(e). It remains to compute p(e). Since h1 _ h2 entails e _ h1 _ h2 we have : 1 = p(e _ h1 _ h2 ) = p(e) + p(h1 _ h2 ) p(e ^ (h1 _ h2 )) = p(e) + 1 p(e ^ (h1 _ h2 )) Hence:

p(e) = = = =

p(e ^ (h1 _ h2 )) p((e ^ h1 ) _ (e ^ h2 )) p(e ^ h1 ) + p(e ^ h2 ) p(e ^ h1 ^ h2 ) p(h1 ):p(e=h1) + p(h2 ):p(e=h2 ) p(h1 ^ h2 ):p(e=(h1 ^ h2) )

Thus:

p(h1 =e) =

p(e=h1 ):p(h1 ) p(h1 ):p(e=h1 ) + p(h2 ):p(e=h2 ) p(h1 ^ h2 ):p(e=(h1 ^ h2 ))

This is the paraconsistent version of Bayes' Theorem. In the classical case, the last term of the denominator is zero, since j= :(h1 ^ h2 ); but this is not so in the paraconsistent case. The theorem illustrates a general fact about paraconsistent probability theory: everything works as normal, except that we have to carry round some extra terms concerning the probabilities of certain contradictions which may be neglected in the classical case. The extra complication may actually be a gain in some contexts. Let me mention one possible one; this concerns quantum mechanics. Quantum mechanics is known to suer from various phenomena often called `causal anomalies', a famous one of which is the two-slit experiment.104 In this, a 104 See, e.g., Haack [1974], ch. 8.

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light is shone onto a screen through a mask with two slits. The intensity of light on any point on the screen is proportional to the probability that a photon hits it, , given that it goes through one slit, , or goes through the other, . Let us write p( _ ) as q. Then:

p(=( _ )) = p( ^ ( _ ))=p( _ ) = p(( ^ ) _ ( ^ ))=q = p( ^ )=q + p( ^ )=q p( ^  ^ )=q Classically, we know that :( ^ ), and so the last term may be ignored. For similar reasons, q = p( _ ) = p() + p( ), and by symmetry we can arrange for p() and p( ) to be equal. Hence:

p(=( _ )) = p( ^ )=2p() + p( ^ )=2p( ) = 12 (p(=) + p(= )) Thus, the intensity of light on the screen should be the average of the intensities of light going each slit independently (which can be determined by closing o the other). Exactly this is what is not found. Standard quantum logic105 avoids the result by rejecting the inference of distribution (i.e., the equivalence between  ^ ( _ ) and ( ^ ) _ ( ^ ), and so faulting the second line of the above proof. A paraconsistent solution is just to note that we cannot ignore the third term in the computation of p(=( _ )), even though we know that :( ^ ). In qualitative terms, what this means is that the photon has a non-zero probability of doing the impossible, and going through both slits simultaneously! This application of paraconsistent probability theory to quantum mechanics is highly speculative. Whether it could be employed to resolve the other causal anomalies of quantum theory, let alone to predict the observations that are actually made, has not been investigated.106

7.6 The Classical Recapture Most paraconsistent logicians have supposed that reasoning in accordance with classical logic is sometimes legitimate. Most, for example, have taken it that classical logic is perfectly acceptable in consistent situations. They have therefore proposed ways in which classical logic can be \recaptured" from a paraconsistent perspective. The simplest such recapture occurs in non-adjunctive logics. As we noted in 4.2, single premise non-adjunctive reasoning is classical. Hence, classical 105 See the article on this in the Handbook. 106 For more on the above issues, including the eects of paraconsistent probability

theory on con rmation theory, see Priest [1987], sect. 7.6, and Priest et al. [1989], pp. 376-9, 385-8.

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reasoning can be regained simply by conjoining all premises. A dierent strategy is to employ a consistency operator, as is done in the da Costa logics Ci , for nite non-zero i. As we saw in 5.3, this can be employed to de ne a negation which behaves classically; hence classical reasoning can simply be interpreted within the system. This approach has problems for some applications, as we shall see when we come to look at set theory. Yet another way to recapture classical reasoning, provided that a conditional operator is available, is to employ an absurdity constant, ?, satisfying the condition j= ? ! , for all . Such a constant makes perfectly good sense paraconsistently. Algebraically, it corresponds to the minimal value of an algebra (which can usually be added if it is not present already). In truth-preservational terms, there are two ways of handling its semantics. One is to require that ? be untrue at every (world of every) evaluation. Its characteristic principle then holds vacuously. The other way (which may be preferable if one objects to vacuous reasoning) is simply to assign ? (at a world) the value of the (in nitary) conjunction of all other formulas (at that world). A bit of juggling then usually veri es the characteristic principle. (The de nition itself guarantees it only when  does not contain ?.) Now let C be the set of all formulas of the form ( ^ :) ! ?. Then an inference is classically valid i it is enthymematically valid with C as the set of suppressed premises, in most paraconsistent logics. For if every member of C holds at (a world of) an interpretation, then the (world of the) interpretation is a classical one|or at least the trivial one|and hence if the premises of a classically valid inference are true at it, so is the conclusion. Thus, we have an enthymematic recapture. Let us write  for  ! ?. In classical (and intuitionist) logic,  just is :. It might therefore be thought that provided a logic possesses ?, we could simply interpret classical logic in it by identifying : with . This thought would be incorrect, though. In many paraconsistent logics,  behaves quite dierently from classical (and intuitionist) negation. What properties it has depends, of course, on the properties of !. While it will always be the case that ;  j= , it will certainly not be true in general that j=  _ , that  j= , or even that  j= . As an example of the last, consider an intensional interpretation for the logic H . (See 5.5.) Suppose that p is true at some normal world, w, but that at some nonnormal world p ! ? is true (and ? is not). Then (p ! ?) ! ? fails at w.107 A nal, and much less brute-force, way of recapturing classical logic starts from the idea that consistency is the norm. It is implicit in the paraconsistent enterprise that inconsistency can be contained. Instead of spreading everywhere, inconsistencies can exist isolated, as do singularities in a eld 107 It might be thought that the existence of the explosive connective ` ' would cause problems for certain paraconsistent applications; notably, for example, for set theory. This is not the case, however, as we will see.

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(of the kind found in physics, not agriculture). This metaphor suggests that even if inconsistencies are present they will be relatively rare. If it is true inconsistencies we are talking about, these will be even rarer|something that the classical logician can readily agree with!108 This suggests that consistency should be a default assumption, in the sense of non-monotonic logic. Many non-monotonic logics can be formulated by de ning validity over some class of models, minimal with respect to violation of the default condition. In eect, we consider only those interpretations that are no more pro igate in the relevant way than the information necessitates. In the case where it is consistency that is the default condition, we may de ne validity over models that are minimally inconsistent in some sense. I will illustrate, as usual, with respect to LP .109 Let I = hD; di be an LP interpretation. Let  2 I ! i  is P d1 :::dn , where P is an n-place predicate and hd1 ; :::; dn i 2 EP \ AP in I . (Recall that I am using members of the domain as names for themselves.) I ! is a measure of the inconsistency of I . In particular, I is a classical interpretation i I ! = . If I1 and I2 are LP interpretations, I will write I1 < I2 , and say that I1 is more consistent than I2 , i I1 !  I2 !. (The containment here is proper.) I is a minimally inconsistent (mi) model of  i I is a model of  i I is a model of  and if J < I ; J is not a model of . Finally,  is an mi consequence of  ( j=m ) i every mi model of  is a model of . As is to be expected, j=m is non-monotonic. For if p and q are atomic sentences, it is easy to check that fp; :p _ qg j=m q, but f:p; p; :p _ qg 6j=m q. Moreover, since all classical models (if there are any) are mi models, and all mi models are models, it follows that  j=  )  j=m  )  j=C . The implications are, in general, not reversible. For the rst, note that fp; :p _ qg 6j= q; for the second, note that fp; :pg 6j=m q. But if  is classically consistent, its mi models are exactly its classical models, and hence we have  j=m  ,  j=C : classical recapture. j=m has various other interesting properties. For example, it can be shown that if the LP consequences of some set is non-trivial, so are its mi consequences Reassurance. For details, see Priest [1991a].110 108 Though this is not so obvious once one accepts dialetheism. For a defence of the view given dialetheism, see Priest [1987], sect. 8.4. 109 Though the rst paraconsistent logician to employ this strategy was Batens [1989], who employs a non-truth-functional logic. Batens also considers the dymanical aspects of such default reasoning. 110 In that paper, in the de nition of <, a clause stating that the domains of I1 and I2 are the same is added. With this clause, the result concerning classical recapture is false (and that paper is mistaken). For example, if  is 9xP x ^ 9x:P x, then hD; di is an mi model, wher D = fag; EP = AP = fag, though this is not a classical model. (This was rst noted by Diderik Batens, in correspondence.) As < is de ned here, f8x(P x ^ :P x)g m 9x8yx = y, which may be thought to be counter-intuitive. But if 8x(P x ^ :P x) is all the information we have, and inconsistencies are to be minimised, perhaps it is correct to infer that there is just one thing. Note that f8x(P x^:P x); 9xQx^9x:Qxg 6j=m 9x8yx = y. For hd; di is an mi model of the premises, where D = fa; bg; EP = AP =

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8 SEMANTICS AND SET THEORY The previous part gestured in the direction of various applications of paraconsistent logic. I want, in the next two parts, to look at some other applications in greater detail. These concern theories of particular mathematical signi cance. In this part I will deal with semantics and set theory. Semantic and set-theoretic notions appear to be governed by simple and apparently obvious principles. In semantics, these concern truth, T , satisfaction, S , and denotation, D, and are:

T -schema: T hi $  S -schema: Sx h i $ (y=x) D-schema: D hti x $ x = t where  is any sentence, is any formula with one free variable, y, and t is any closed term. Angle brackets indicate a name-forming device. In set theory the principle is the schema of set existence: Comprehension Schema: 9x8y(y 2 x $ ) where is any formula not containing x. What the connective $ is in the above schemas, we will have to come back to. Despite the fact that these schemas appear to be obvious, they all give rise to contradictions, as is well known: the paradoxes of self-reference, such as (respectively) the Liar Paradox, the Heterological Paradox, Berry's Paradox and Russell's Paradox. The usual approaches to set theory and semantics restrict the principles in some way. Such approaches are all unsatisfactory in one way or another, though I shall not discuss this here.111 A paraconsistent approach can simply leave the principles as they are, and allow the contradictions to arise. They need do no damage, because the logic is not explosive. Even so, not all paraconsistent logics are suitable as the underlying logics of these theories. For a start, if the above schemas are formulated with the material  they give rise to a conjoined contradiction, so using a non-adjunctive logic (except Rescher and Manor's) explodes the theory.112 And in the da Costa systems, Ci , for nite i, an operator behaving like classical negation, : can be de ned (see 5.3). The usual arguments establish contradictions of the form  ^ : , and so again

fa; bg; EQ = fag; AQ = fbg.

With the present de nition, the proof of Reassurance for the rst-order case, appropriately modi ed, still goes through. 111 See, for example, Priest [1987], chs. 1, 2. 112 Rescher and Brandom, [1980], p. 164, suggest splitting the biconditionals up into two non-conjoined conditionals.

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the theories explode. Fortunately, there are other paraconsistent logics that will do the job.113

8.1 Truth Theory in LP Let us start with the semantic case. I will deal with truth; similar remarks and constructions hold for the other semantic notions, but I will leave readers to ponder these for themselves. The rst question we need to address is what connective it is that occurs in the biconditional of the T -schema. The rst possibility is that it is a material biconditional, .114 Let us, then, suppose that we are dealing with the logic LP . We will need some machinery to handle self reference; a straightforward option is to let this be arithmetic. Hence, we suppose the language, L, to be that of rst order arithmetic augmented by a one place predicate, T . To make things easy, we will assume that L has a function symbol for each primitive recursive function (and only those function symbols). Let T0 be the LP theory in this language which comprises the truths of rst order arithmetic plus the T -schema. The assumption that T0 contains all of arithmetic is obviously a very strong one, and means that the theory is not axiomatic. We could, instead, consider an axiomatic theory with some suitable fragment of arithmetic, but since a major part of our concern will be with what cannot be proved, it is useful to have the arithmetic part as strong as possible. The rst thing to note is that T0 is inconsistent. Given the resources of arithmetic, for any formula, , of one free variable, x, one can nd, by the usual Godel construction, a xed-point formula, , of the form (x= h i).115 Now, let  be :T x and let be its xed point. Then the T -schema gives us: T h i  , i.e., T h i  :T h i. Unpacking the de nition of , in terms of ^, _, and : and ddling, gives exactly T h i ^ :T h i.116 Despite being inconsistent, T0 is non-trivial. An easy way to see this is to observe, rst of all, that if in any interpretation  () = f1; 0g then  (  ) = f1; 0g. Hence, an LP model for T0 can be obtained by letting the denotations of the arithmetic language be that of the standard interpretation of arithmetic|so that, in particular, the domain is N , the natural numbers; recall that classical interpretations are just special cases of LP 113 There are paraconsistent set theories based on da Costa's C systems. (See, e.g., Arruda [1980], da Costa [1986].) In these theories, the schemas have to be constrained, as they are classically. This takes away much of the appeal of a paraconsistent approach. 114 It is natural to suppose that it ought to be a detachable conditional. Goodship [1996] argues that it is only a material conditional. Whether or not this is the case, it is certainly interesting to explore the two possibilities. 115 See, e.g., Priest [1987], sect. 3.5. 116 It is worth noting that for the S -schema, the xed point machinery is unnecessary for the demonstration of inconsistency. For let  be :Sxx. Then an instance of the S -schema is: S hi hi  :S hi hi, and we can then proceed as before.

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interpretations|and setting ET and AT , the extension and anti-extension of T , both to N . Call this interpretation I0 . In I0 every sentence of the form T hi takes the value f1; 0g, and so by the observation concerning , I0 is a model for the T -schema, and so of all of T0 . The same interpretation shows that if  is any arithmetic formula false in the standard model, T0 6j= . T0 is a relatively weak theory. In particular, it does not legitimate the two way rule of inference:

 T hi

(just consider the south-north inference in I0 , where  is an arithmetic sentence false in the standard model).117 Let the theory obtained by replacing the material T -schema of T0 with this rule be called T1 . T1 is inconsistent. For choose an  of the form :T hi. The law of excluded middle gives T hi _ :T hi, i.e., T hi _ , which, applying the rule, gives T hi and , i.e., :T hi. We can construct a model for T1 as follows. If an interpretation assigns the standard denotations to all arithmetical language let us call it arithmetical. Any arithmetical interpretation is a model all of T1 except, perhaps, the T -schema. Let I1 and I2 be two arithmetical interpretations, with assignment functions 1 and 2 . De ne 1  2 to mean that for all atomic sentences in the language; :

1 () = t ) 2 () = t 1 () = f ) 2 () = f If 1  2 then this condition extends to all formulas of L. For suppose that 1  2 . If n is in the extension of T in I2 but not I1 ; then 2 (T n) = t or b, but 1 (T n) = f , violating the condition. Similarly for anti-extensions. Hence, I2  I1 . By monotonicity, for all , 2 ()  1 (). The conclusion follows. For suppose that 2 () 6= t. Then  is false (i.e., b or f ) in I2 ; hence  is false in I1 , i.e., 1 () 6= t. The argument for f is similar. This result is, in fact, just another version of monotonicity; I will call it the Monotonicity Lemma. Let I0 be any arithmetical interpretation, with evaluation function 0 . We now de ne a trans nite sequence of arithmetical interpretations,

117 Whether or not more follows with minimally inconsistent LP (see 7.6) is presently unknown. Another non-monotonic notion of inference also suggests itself here. According to this, the things that follow are the things that hold in all minimally inconsistent models where the arithmetic part is the standard model. Employing this would be appropriate if there were good reasons to believe that the only inconsistencies involve the truth predicate.

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hJi ; i 2 Oni (On

is the class of ordinals). I will make the construction slightly more complex than necessary, for the bene ts of the next section. It suÆces to de ne the evaluation function i of each interpretation. If i > 0 and n is not the code of a sentence, then i (T n) = 0 (T n). We therefore need to consider only atomic formulas of the form T hi. Let us say that  is eventually t by k i 9i > 08j (i  j < k, j () = t). Similarly for f . Then for k 6= 0:

k (T hi) = t =f =b

if  is eventually t by k if  is eventually f by k otherwise

We can now establish that if 0 < i  k then i  k . The proof is by trans nite induction. Suppose that the result holds for all j < k. We show it for k. Since the truth values of atomic formulas other than ones of the form T hi are constant, we need consider only these. So suppose that i (T hi) = t. Then  is eventually t by i. In particular, for some 0 < j < i, j () = t. By induction hypothesis, for all l such that j < l < k, j  l . Hence, by monotonicity l () = t. Hence,  is eventually t by k, i.e., k (T hi) = t. The case for f is similar. What this lemma shows is that once i > 0, and increases, sentences of the form T hi can change their truth value at most once. If they ever attain a classical value, they keep it. Since there is only a countable number of sentences of this form, there must be an ordinal, l, by which all the formulas that change value have done so. Hence l = l+1 . Call Jl , J ; and its corresponding evaluation function   . Then if   () = t,   (T hi) = t. Similarly for f and b. Hence   () =   (T hi), and so J is a model of T1. For the same reason, J also veri es the two-way rule:

: :T hi Yet the theory is not trivial: anything false in the standard model of arithmetic is untrue in J , and so T1 6j= . It is not diÆcult to see that the construction used to de ne J is, in fact, just a dualised form of Kripke's xed point construction for a logic with truth value gaps using the strong Kleene three-valued logic.118 (Provided we start with a suitable ground model, monotonicity is guaranteed from the beginning, and so we can just set k (T hi) to t (or f ) if  takes the value t (or f ) at some i < k.) Hence, if any sentence is grounded in Kripke's sense, it takes a classical value in J . In particular, if  is any false grounded sentence, T1 6j= . 118 See the article on Semantics and the Liar Paradox in this Handbook. One of the rst people to realise that the construction could be dualised for this end was Dowden [1984].

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8.2 Adding a Conditional Although T1 validates the two-way inferential T -schema, it does not validate the T -schema as formulated with a detachable conditional. This is for the simple reason that LP does not contain such a conditional. A natural thought is to augment the language with one to make this possible. Let the resulting language be L!. Not all conditionals are suitable here, however. This is due to Curry paradoxes. If the conditional satis es the inference of contraction:  ! ( ! ) j=  ! , then the theory collapses into triviality. For consider the xed-point formula, , of the form T h i ! ? (or if ? is not present, just an arbitrary ). The T -schema gives: T h i $ (T h i ! ?). Contraction gives us: T h i ! ? and then a couple of applications of modus ponens give ?.119 This fact rules out the use of all the non-transitive logics we looked at (since they validate  $ ( ! ) j= ), all the da Costa logics and discussive logic (using discussive implication for the T -schema), since these validate contraction, and those relevance logics that validate contraction, such as R.120 A relevant logic without contraction can be used for the purpose. Let T2 be as for T0 , except that the T -schema is formulated with !, and the underlying logic is BX (see 5.5, 6.5). T2 is inconsistent, since it is obviously stronger than T1 . But it can be shown to be non-trivial. If we try to generalise the proof for T1 in simple ways, attempts are stymied by the failure of anything like monotonicity once ! is involved. However, there is a way of building on the proof.121 This requires us to move from objectual semantics to simple evaluational semantics. For the purpose of this section (and this one only), an atomic formula will be any of the usual kind or any one of the form  ! . Clearly, any sentence of the language can be built up from atomic formulas using ^, _, :, 9 and 8. Call an evaluation of atomic formulas,  , arithmetical if it assigns to every identity its value in the standard model of arithmetic. Given an arithmetical evaluation, it is extended to an evaluation of all sentences by LP truth conditions, using substitutional quanti cation. A quick induction shows that any arithmetical evaluation assigns t to all the arithmetic truths of the standard model (which do not contain ! or T ), and f to all the falsehoods. Moreover, for this notion of valuation, we do have the Monotonicity Lemma. Finally, given any such evaluation, we 119 An argument of this kind rst appeared in Curry [1942]. Dierent versions that employ close relatives of contraction, such as ` ( ^ ( ! )) ! (but not  ^ ( ! ) ` ) can also be found in the literature. See, e.g., Meyer et al. [1979]. 120 For good measure, it also rules out using Rescher and Manor's non-adjunctive approach. Using this, every consistent sentence would follow, since if is consistent, so is  $ ( ! ). 121 The following is taken from Priest [1991b], which simply modi es Brady's proof for set theory in [1989].

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can construct a xed point,   , such that   () =   (T hi), as in 8.1. The construction is the same, except that in the de nition of k , we set T s to t if s is a (closed) term that evaluates to the code of , and  is eventually t by k. Similarly for f . (The values of atoms of the form  ! do not change in the process.) An induction shows that if    then for all i, i  i . Suppose the result for all i < k. We show it for k. We need consider only those atomic formulas of the form T s where s evaluates to the code of sentence . k (T s) = t i  is eventually t by k, for . By induction hypothesis, this implies that  is eventually t by k for  . Hence, k (T s) = t, as required. The case for f is similar. From this result it obviously follows that if    then     . Let ) be the conditional connective of RM 3 (see 5.4, identifying +1, 0, and 1 with t, b, and f , respectively). This also plays a role in the proof. Its relevant property is that if    then if  and are formulas of L! and ( ) ) = t,  ( ) ) = t. For if ( ) ) = t then () = f or ( ) = t. By monotonicity  () = f or  ( ) = t. Hence,  ( ) ) = t. Let 0 be the arithmetical interpretation that assigns every sentence of the form T s the value b. We now de ne a trans nite sequence of arithmetic valuations, hi ; i 2 Oni, as follows. (I write (j ) as j .) For k 6= 0:

k ( ! ) = t =f =b

if 8j < k, j ( ) ) = t if 9 j < k, j ( ) ) = f otherwise

And where  is of the form T s, where s is any closed term which evaluates to the code of a sentence:

k () = t =f =b

if 9i8j (i  j < k; j () = t) if 9i8j (i  j < k; j () = f ) otherwise

We can now establish that if i  k then i  k . The proof is by trans nite induction. Suppose that the result holds for all j < k. We need to consider cases where a formula is of the form  ! or T s, where s is a term that evaluates to the code of a sentence. Take them in that order. Suppose that i ( ! ) = t. Then 0 ( ) ) = t. By induction hypothesis, for 0 < j < k, 0  j . Thus, 0  j . Hence, j ( ) ) = t, by the observation concerning ). Thus, k ( ! ) = t, as required. The case for f is trivial. For the other case, suppose that i () = t. Then 9j < i, j () = t. By induction hypothesis, if j  l < k, j  l , and hence j  l . By monotonicity, l () = t. Thus, k () = t. The case for f is similar.

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What this lemma shows, as before, is that we must eventually reach an l such that l = l+1 . Let this evaluation be ~ . Then ~ is a model of all the extensional arithmetic apparatus. It also models the T -schema. For if i < l, i () = i (T hi), and so i ( , T hi) = t or b, and ~ ( $ T hi) = t or b. (For the same reason, ~ models the contraposed form: : $ :T hi. Since T h:i $ : is an instance of the T -schema, it also models T h:i $ :T hi.) It remains to check that ~ models the axioms and respects the rules of inference of BX . This requires no little checking. Most of it is routine. Here, for example, is one of the harder propositional axioms: (( ! ) ^ ( !

)) ! ( ! (( ^ )). Let the antecedent be ', and the consequent be . Then ~ (' ! ) = t or b i for no i < l, i (' ) ) = f . Now, suppose that i (' ) ) = f . Then one of:

i (') = t and (i ( ) = b or i ( ) = f ) i (') = b and i ( ) = f In the rst case, i ( ! ) = i ( ! ) = t. But then for all j < i, j ( ) ) = j ( ) ) = t, in which case j ( ) ( ^ )) = t, and so i ( ! ( ^ )) = t, which is impossible. In the second case, i ( ! ) = t or b, and i ( ! )) = t or b. But then for all j < i, j ( ) ) = t or b, and j ( ) ) = t or b, in which case j ( ) ( ^ )) = t or b, and so i ( ! (( ^ )) = t or b, which is also impossible. For further details, see Brady [1989].122 The construction shows that T1 is non-trivial, since if  is any arithmetic sentence false in the standard model ~ () = f . (Indeed, as with the previous construction, which is incorporated in this, if  is any false grounded sentence, the same is true.)

8.3 Advantages of a Paraconsistent Approach What we have seen is that it is possible to have a theory containing all the machinery of arithmetic, plus a truth predicate which satis es the T -schema for every sentence of the language|whether this is formulated as a material biconditional, a two-way rule of inference, or a detachable bi-conditional. It is inconsistent, but non-trivial; in fact, the inconsistencies do not spread 122 Brady shows that the construction veri es propositional logics that are a good deal stronger than BX . His treatment of identiy is dierent, though. To verify the substitutivity rule of 7.1, it suÆces to show that if t1 = t2 holds in an interpretation then (x=t1 ) and (x=t2 ) have the same truth value. A quick induction shows that if this is true for atomic  it is true for all . Hence, we need consider only these. Next, show by induction that if this holds for  it holds for all evaluations in the construction of   , and so of   itself. Finally, we show by induction that it holds for every i in the hierarchy, and hence for ~ .

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into the arithmetic machinery.123 Thus, it is possible to have a workable, if inconsistent, theory which respects the central intuition about truth. It is not my aim here to discuss the shortcomings of other standard approaches to the theory of truth,124 but none can match this. All restrict the T -schema in one way or another. The one that comes closest to having the full T -schema is Kripke's account of truth, which at least has it in the form of a two way rule of inference. However, this account has the singular misfortune of being self-referentially inconsistent. According to this account, if  is the Liar sentence it is neither true nor false, and so not true, but the theory pronounces :T hi itself neither true nor false. According to T2 ,  is both true and false (i.e., has a true negation), and this is exactly what it proves: T hi ^ :T hi entails T hi ^ T h:i. It might also show that  is not true (and so not both true and false). But paraconsistency shows you exactly how to live with this kind of contradiction. This is not unconnected with the matter of \strengthened" paradoxes. If someone holds the Liar sentence to be neither true nor false, one can invite them to consider the sentence, , `This sentence is not true' (as opposed to false). Whether is true, false or neither, a contradiction arises. It is sometimes suggested that a paraconsistent account of truth falls to the same problem, since can have no consistent truth-value on this account either. It should be clear that this argument is just an ignoratio. A paraconsistent account does not require it to have a consistent truth-value. In fact, according to T2, T h:i $ :T hi; if this is right, there is no distinction between the standard Liar and the \strengthened Liar" at all.125 Let me nish with a word of caution. We can construct non-trivial theories which incorporate the S -schema of satisfaction and the D-schema of denotation, in exactly the same way as we did the T -schema. If, however, we try to add descriptions to a theory with self-reference and the D-schema, trouble does arise. Suppose that we have a description operator, ", satisfying the Hilbertian principle: 9x ` (x="x). If t is any closed term, t = t, and so by the D-schema D hti t , and 9xD hti x. Thus, by the description principle, D hti "xD hti x, whence, by the D-schema again:

t = "xD hti x 123 Nor does the T -schema have to be taken as axiomatic. One can give truth conditions for atomic sentences and then prove the T -schema in the usual Tarskian fashion. See Priest [1987], ch. 7. 124 For this, see Priest [1987], ch. 2. 125 The advantages of a paraconsitent account of truth rub o onto any account of modal (deontic, doxastic, etc.) operators that treats them as predicates. For all such theories are just sub-theories of the theory of truth. See Priest [1991b]. We will have an application of this concerning provability in 9.6.

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Now in arithmetic, just as for any formula, , with one free variable, x, we can nd a sentence, , of the form (x= h i), so, for any term, t, with one free variable, x, we can nd a closed term; s, such that s is t(x= hsi). If f is any one place function symbol, apply this fact to the term f"yDxy, to obtain an s such that:

s = f"yD hsi y Since s = "yD hsi y, it follows that s = fs: any function has a xed point. This shows that the semantic machinery does have purely arithmetic consequences. In particular, for example, 9x x = x + 1. Arithmetic statements like this can be kept under control, as we will see later in the next part, but worse is to come. Let f be the parity function, i.e.:

fx = 0 =1

if x is odd if x is even

We have fs = 0 _ fs = 1. In the rst case s = fs = 0, which is even, and so fs = 1. Thus, 0 = 1. Similarly in the second case. This is unacceptable, even for someone who supposes that there are some inconsistent numbers. Where to point the nger of suspicion is obvious enough. As we saw, the D-schema entails 9xD hti x, for any closed term, t; and there is no reason why someone who subscribes to a paraconsistent account of semantic notions must believe that every term has a denotation: in particular, in the vernacular, `s' is `a number that is 1 if it is even and 0 if it is odd', which would certainly seem to have no denotation. This suggests that the D-schema should be subjected to the condition that 9xD hti x in some suitable way. The behaviour of resulting theories is a particularly interesting unsolved problem.126

8.4 Set Theory in LP Let us now turn to the second theory that we will look at, set theory. This is a theory of sets governed by the full Comprehension schema. This schema is structurally very similar to the T -schema, and many of the considerations of previous subsections carry over to set theory in a straightforward manner. The major element of novelty concerns the other axiom, the Extensionality axiom. Let us start with set theory in LP . The language here contains just the predicates = and 2, and the axioms are: 126 For a further discussion of all of these issues, see Priest [1997a].

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9x8y(y 2 x  ) 8x(x 2 y  x 2 z )  y = z where x does not occur free in . Call this theory S0 . S0 is inconsistent. For putting y 2= y for , and instantiating the quanti er we get: 8y(y 2 r  y 2= y), whence r 2 r  r 2= r. Cashing out  in terms of : and _ gives r 2 r ^ r 2= r. In constructing models of S0 , the following observation (due to Restall [1992]) is a useful one. First some de nitions. Given two vectors of LP values, (gm ; m 2 D), (hm ; m 2 D), the rst subsumes the second i for all m 2 D; gm  hm . Now consider a matrix of such values (em;n; m; n 2 D). This is said to cover the vector (gm ; m 2 D) i for some n 2 D, the vector (em;n ; m 2 D) subsumes it. A vector indexed by D is classical i all its members are t or f . (Recall that we are writing f1g, f1; 0g, f0g as t, b f , respectively.) Now the observation. Consider an LP interpretation, hD; di, and the matrix (em;n ; m; n 2 D), where em;n =  (m 2 n). If this covers every classical vector indexed by D it veri es the Comprehension principle. For let  be any formula not containing x, and consider the vector ( ((y=m)); m 2 D). This certainly subsumes some classical vector; choose one such, and let this be subsumed by (em;n ; m 2 D). Now consider any formula of the form m 2 n  (y=m). Where the two sides dier in value, one of them has the value b. Hence, the value of the biconditional is either t or b. Thus the same is true of 8y(y 2 n  ), and 9x8y(y 2 x  ). Using this fact, it is easy to construct models for S0 . Consider an LP interpretation, hD; di, where D = fm; ng, and em;n is given by the following matrix:

2 m n m b n b

t t

Each column is the membership vector of the appropriate member of D; and since that of m subsumes every classical vector indexed by D, this veri es the Comprehension axiom. In the Extensionality axiom, if y and z are the same, the axiom is obviously true. If they are distinct, one is n and the other is m, and for each x, the value of x 2 n  x 2 m is b. Hence, 8x(x 2 n  x 2 m) has the value b and Extensionality is veri ed. In this model, m 2= n and n 2= n have the value f , as, therefore do 9y y 2= n and 8x9y y 2= x. Hence, S0 is non-trivial. A characterisation of what can be proved in S0 (and of what its minimally inconsistent consequences are) is still an open question. There are, however, certainly theorems of Zermelo Fraenkel set theory, ZF , that are not provable

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in S0 . For example, in ZF there is provably no universal set: ZF ` 8x9y y 2= x. But this is not a consequence of S0 , as we have just seen.127 The simple model of S0 that we have just used to prove non-triviality is obviously pathological in some sense. An interesting question is what the \intended" interpretations of S0 are like. Whilst unable to give an answer to this, I note that for any classical model of ZF , M = hD; di, there is a model of S0 which has M as a substructure. Let a be some new object, let M+ = hD+ ; d+ i, where D+ = D [ fag, d+ is the same as d, except that for every c 2 D+ , the value of c 2 a is b; for every c 2 D, the value of a 2 c is f ; the value of a = a is t; and for every c 2 D, the value of a = c is f . M is clearly a substructure of M+ . The membership vector of a subsumes every classical vector, and hence M is a model of Comprehension. It remains to verify Extensionality: 8x(x 2 m  x 2 n)  m = n. If m and n are the same in M+ , then the consequent is true, as is the conditional. So suppose that they are distinct. If they are both in D, then, by extensionality in M, there is some c 2 D such that c 2 m is t and c 2 n is f , or vice versa. Whichever of these is the case, c 2 m  c 2 n is f , as is 8x(x 2 m  x 2 n). Hence the conditional is t. Finally, suppose that m 2 D and n is a (or vice versa, which is similar). Then if c 2 D+ , every sentence of the form c 2 n is b. Hence, every sentence of the form c 2 m  c 2 n is b, as therefore is 8x(x 2 n  x 2 m). Hence, the conditional is true.

8.5 Brady's Non-triviality Proof As a working set theory, S0 is rather weak. Since the Comprehension axiom is only a material one, we cannot infer that something is in a set from the fact that it satis es its de ning condition, and vice versa. This suggests strengthening the principle to a two-way rule of inference, as we did for truth theory. This, in turn, requires the addition of set abstracts to the language. So let us enrich the language with terms of the form fx; g for any variable, x, and formula, ; and trade in the Comprehension principle of S0 for the two-way rule:

x 2 fy; g (y=x) Call this theory S1 . S1 is inconsistent. For let r be fx; x 2= xg. Then:

r2r r 2= r The law of excluded middle then quickly gives us r 2 r ^ r 2= r.

127 For this, and some further observations in this direction, see Restall [1992].

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The non-triviality of S1 is presently an open question. But even though it is probably non-trivial, as a working set theory, it is still rather weak. This is because we have no useful way of establishing that two sets are identical. Even if we can show that 8x(  ), and so that 8x(x 2 fx; g  x 2 fx; g), we cannot infer that fx; g = fx; g since Extensionality does not support a detachable inference. We might hope to circumvent this problem by trading in the Extensionality principle for the corresponding rule:

8x(  ) fx; g = fx; g But if we do this, trouble arises.128 For let r be as before. Then since r 2 r must take the value b in any interpretation, we have, for any , 8x(  r 2 r), and so fx; g = fx; r 2 rg. Thus, for any  and , fx; g = fx; g; which is rather too much. The problem arises because the Extensionality rule of inference allows us to move from an equivalence that does not guarantee substitution (  ;  6j=LP   ) to one that does (identity). This suggests formulating Extensionality itself with a connective that legitimises substitution. So let us add a detachable connective to the language, !, and formulate Extensionality as:

8x( $ ) fx; g = fx; g The trouble then disappears. And now that we have a detachable conditional connective at our disposal, it is natural to formulate the Comprehension principle as a detachable biconditional, as follows:

8y(y 2 fx; g $ (x=y)) We have to be careful about the conditional connective here. As with truth, any conditional connective that satis es contraction would give rise to triviality. For let c be fx 2 x ! ?g. Then an instance of Comprehension is y 2 c $ (y 2 y ! ?). Instantiating with c, we get c 2 c $ (c 2 c ! ?), and we can then proceed, as with truth, to obtain ?. Even if the logic does not contain contraction, Curry-style paradoxes may still be forthcoming. For example, if we drop the contraction axiom from the relevant logic R 128 There are other cases where the full Comprehension principle by itself is alright, but throwing in extensionality causes problems; for example, set theory based on Lukasiewicz' continuum-valued logic. See White [1979].

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and add the law of excluded middle, the Comprehension principle still gives triviality.129 Again however, a relevant logic without contraction will do the job. Consider the set theory with Extensionality and Comprehension formulated as just described, and based on the underlying logic BX (with free variables, so that these may occur in the schematic letters of Extensionality and Comprehension). Call this S2 : The rst thing to note about S2 is that identity can be de ned in it, in Russellian fashion. Writing x = y for 8z (x 2 z $ y 2 z ), x = x follows. Substituting fw; g for z , and using the Comprehension principle gives (w=x) $ (w=y). Hence, we need no longer assume that = is part of the language. Since the Comprehension principle of S2 gives the two-way deduction version of S1 , S2 is inconsistent. It is also demonstrably non-trivial, as shown by Brady [1989].130 To prove this, we repeat the proof for T2 of 8.2 with three modi cations. The rst, a minor one, is that we add two propositional constants t and f to the language; their truth values are always what the letters suggest. (This is necessary to kick-start the generation of the xed point into motion. In the case of truth, this was done by the arithmetic sentences.) More substantially, in constructing   we replace the clause for T by:

k (s 2 fx; g) = t if (x=s) is eventually t by k =f if (x=s) is eventually f by k =b otherwise where s is any closed term, and  contains at most x free. The nal modi cation is that in extending evaluations to all formulas, we use substitutional quanti cation with respect to the closed set abstracts. Now; ~ veri es all the theorems of S2 , in the sense that if  is any closed substitution instance of a theorem, it receives the value t or b in ~ . This is shown by an induction on the length of proofs. That the logical axioms have this property, and the logical rules of inference preserve this property, is shown as in 8.2. This leaves the set theoretic ones. Given the construction of ~ , it is not diÆcult to see that it veri es the Comprehension principle. It is not at all obvious that Extensionality preserves veri cation. What needs to be shown is that if 8x( $ ) is veri ed, so is anything of the form a 2 c $ b 2 c, where a is fx; g and b is fx; g. Let c be fy; g. Then, given Comprehension, what needs to be shown is that

(y=a) $ (y=b) is veri ed. If this can be shown for atomic , the result will follow by induction. Given the premise of the inference and Comprehension, it is true if is of the form d 2 y. If it is of the form y 2 d, where 129 See Slaney [1989]. Other classical principles are also known to give rise to triviality in conjunction with the Comprehension schema. See Bunder [1986]. 130 A modi cation of the proof shows that the theory based on the logic B is, in fact, consistent. See Brady [1983].

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d is fz ; Æg, we need to show that Æ(z=a) $ Æ(z=b) is veri ed. We obviously have a regress. In fact, the regression grounds out in a suitable way in the construction of ~ . For details, see Brady [1989].131 The non-triviality of S2 is established since there are many sentences that are not veri ed by ~ . It is easy to check, for example, that any sentence of the form c 2 fx; f g takes the value f , as, therefore, does the formula 8x9y y 2 x. A notable feature of Brady's proof is the following. As formulated, the Comprehension principle entails: 9y8x(x 2 y $ ), where y does not occur in . (The y in question is fx; g and so cannot be a subformula of .) If we relax the restriction, we get an absolutely unrestricted version of the principle. Brady's proof can be extended to verify this version too, by adding a xed point operator to the language, and treating it suitably. Again, for details, see Brady [1989]. Finally, it is worth observing that the T -schema is interpretable in S2 . If  is any closed formula, let us write hi for fz ; g, where z is some xed variable. De ne T x to be a 2 x, where a is any xed term. Then T hi = a 2 fz ; g $ . Moreover, the absolutely unrestricted Comprehension principle gives us xed points of the kind required for self-reference. Let  be any formula of one free variable, x. By the principle, there is a set, s, such that 8x(x 2 s $ (x=fz ; a 2 sg)). It follows that a 2 s $ (x=fz ; a 2 sg). Thus, if is a 2 s, we have $ (x= h i). S2 (with the absolutely unrestricted Comprehension principle) therefore gives us a demonstrably non-trivial joint theory of truth, sethood and self-reference.

8.6 Paraconsistent Set Theory Despite the strong structural similarities between semantics and set theory, there is an important historical dierence. Set theory is a well developed mathematical theory in a way that semantics is not. In the case of set theory, it is therefore natural to ask how a paraconsistent theory such as S2 relates to this development. To answer this question (at least to the extent that the answer is known), it will be useful to divide set theory into three parts. The rst comprises that basic set-theory which all branches of mathematics use as a tool. The second is trans nite set theory, as it can be established in ZF . The third

131 Brady's treatment of identity is slightly dierent from the one given here. He de nes x = y as 8z (z 2 x $ z 2 y). Given Comprehension, this delivers the version of Extensionality used here straight away. What is lost is the substitution principle x = y; (w=x) ` (w=y). Given the Comprehension principle, this can be reduced to x = y; x 2 z ` y 2 z (which follows from our de nition of identity). Brady takes something stronger than this as his substitutivity axiom: ` (x = y ^ z = z ) ! (x 2 z $ y 2 z ). Hence, his construction certainly veri es the weaker principle. It is worth noting that the construction does not validate the simpler ` x = y ! (x 2 z $ y 2 z ), which, in any case, is known to be a Destroyer of Relevance. See Routley [1980b], sect. 7.

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concerns results about sets, like Russell's set and the universal set, that do not exist in ZF . Let us take these matters in turn. S2 is able to provide for virtually all of bread-and-butter set theory (Boolean operations on sets, power sets, products, functions, operations on functions, etc.), and so provide for the needs of working mathematics.132 For example, if we de ne the Boolean operators, x \ y, x [ y and x as fz ; z 2 x ^ z 2 yg, fz ; z 2 x _ z 2 yg and fz ; z 2= xg, respectively, and x  y as 8z (z 2 x ! z 2 y), then we can establish the usual facts concerning these notions. Some care needs to be taken over de ning a universal set, U , and empty set, , though. If we de ne , as fx; x 6= xg, we cannot show that for all y,   y, since the underlying logic is relevant and cannot prove x 6= x !  for arbitrary . (Dually for U .) If we de ne  as fx; 8z x 2 z g, this problem is solved, since 8z x 2 z ! x 2 y. (Dually for U .) The reason for the quali cation `virtually' in the rst sentence of the last paragraph, is as follows. The sets, as structured by union, intersection and complementation, are not a Boolean algebra, but a De Morgan algebra with maximum and minimum elements. Though we can show that 8y y 62 x \ x, we cannot show that x \ x  , since, relevantly, ( ^ :) ! fails. (Dually for U .) There are, in a sense, more than one universal and empty sets. Moreover, this is essential. If we had x \ x   then, taking fz ; g for x, we get ( ^ :) ! 8y z 2 y. Now take fz ; g for y, and we get ( ^ :) ! ; paraconsistency fails. In fact, Dunn [1988] shows that if the principles that there is a unique universal set, and a unique empty set, are added to any set theory such as S2 , full classical logic falls out. Turning to the second area, the question of how much of the usual trans nite set theory can be established in S2 is one to which the answer is currently unknown. What can be said is that the standard proofs of a number of results break down. This is particularly the case for results that are proved by reductio, such as Cantor's Theorem. Where  is an assumption made for the purpose of reductio, we may well be able to establish that ( ^ ) ! ( ^ : ), for some , where is the conjunction of other facts appealed to in deducing the contradiction (such as instances of the Comprehension principle). But contraposing and detaching will give us only : _ : , and we can get no further.133 Lastly, the third area: reasoning in S2 , one can prove various results about sets that are impossible inSZF . For example, as usual, let fxg be fy; y = xg, fx; yg be fxg [ fyg and x be fz ; 9y 2 x; z 2 yg. r = fx; x 2= xg, and we know that r 2 r and r 2= r. Then:134 (1) If x 2 r then fxg 2 r. For fxg 2 fxg or fxg 2= fxg. In the rst case, 132 Much of this is spelled out in Routley [1980b], sect. 8. 133 Interesting enough, however, it is possible to prove a version of the Axiom of Choice

using the completely unrestricted version of the Comprehension principle. See Routley [1980b], sect. 8. 134 The following is taken from Arruda and Batens [1982].

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fxg = x, and so fxg 2 r. In the second case, fxg 2 r by de nition. (2) If x; y 2 r then fx; yg 2 r. For fx; yg 2 fx; yg or fx; yg 2= fx; yg. In the rst case, fx; yg = x or fx; yg = y, and so fx; yg 2 r. In the second case, fx; yg 2 r by de nition. (3) ffx; rgg 2 r. For ffx; rgg 2 ffx; rgg or ffx; rgg 2= ffx; rgg. In the rst case, ffx; rgg = fx; rg, hence, x = fx; rg = r. But then x; r 2 r so fx; rg 2 r, by (2), and, ffx; rgg 2 r, by (1). In the second case, ffx; rgg 2 r

by de nition. S (4) 8x x 2 r. For suppose that fx; rg 2 fx; rg. Then fx; rg = x or fx; rg = r. In the rst case, fxg = Sffx; rgg, so fxg 2 r, by (3). In the second, fx; rg 2 r. In either case x 2 r. Suppose, on the S other hand, that fx; rg 2= fx; rg. Then fx; rg 2 r, by de nition, and so x 2 r. S That r is universal, is hardly a profound result. But it at least illustrates the fact that there are possibilities which transcend ZF . Let me end this section with a speculative comment on what all this shows. The discussion of this section, and especially the part concerning the non-Boolean properties of sets in S2 , shows that it is impossible to recapture standard set theory in its entirety in this theory. Sets are extensional entities par excellence; using an intensional connective in their identity conditions is bound to gum up the works. In fact, it seems to me that the most plausible way of viewing S2 is as a theory of properties, where intensional identity conditions are entirely appropriate. But what you call these entities does not really matter here. The important fact is that they are not the sets of standard modern mathematical practice. If we want a theory of such entities, the appropriate identity conditions must employ , and this means that we are back with the proof-theoretically weak S0 (or S1 ). Since this does not contain ZF , how should someone who subscribes to a paraconsistent theory of such sets view modern mathematical practice? One answer is as follows. The standard model of ZF is the cumulative hierarchy. As we saw in 8.2, there are models of S0 which contain this hierarchy. We may thus take it that the intended interpretation of S0 is a model of this kind (or if there are more than one, that they are all models of this kind). The cumulative hierarchy is therefore a (consistent) fragment of the set-theoretic universe, and modern set theory provides a description of it. There is, however, more to the universe than this fragment. A classical logician may well agree with that claim. For example, they may think that there are also non-well-founded sets. The paraconsistent logician agrees with this: after all, r is not well-founded; but they will think that sets outside the hierarchy may have even more remarkable properties: some of them are inconsistent.

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9 ARITHMETIC AND ITS METATHEORY In this part I want to look at the application of paraconsistent logic to another important mathematical theory: arithmetic. The situation concerning arithmetic is rather dierent from that concerning set theory and semantics. There are no apparently obvious and intrinsically arithmetical principles that give rise to contradiction, in the way that the Comprehension principle and the T -schema do|or if there are, this fact has not yet been discovered. In the rst instance, the paraconsistent interest in arithmetic arises because there is a class of inconsistent models of arithmetic. (It might be more accurate to say `models of inconsistent arithmetic'.) It may be supposed that these models are pathological in some sense.135 I will come back to this matter later. But even if it is so, the models nevertheless have an interesting and important mathematical structure, as do the classical non-standard models of arithmetic|which are, in fact, just a special case, as we will see. And just as one does not have to be an intuitionist to nd intuitionistic structures of intrinsic mathematical interest, so one does not have to be a dialetheist for the same to be true of inconsistent structures. One thing this part illustrates, therefore, is the existence of a new branch of mathematics which concerns the investigation of just such structures.136 The existence of inconsistent models of arithmetic bears, as might be expected, on the limitative theorems of Metamathematics. And whatever the status of the inconsistent models themselves, many have held that these theorems have important philosophical implications. This part will also look at the connection between the inconsistent models and the limitative theorems, and I will comment on the signi cance of this for the philosophical implications of Godel's incompleteness theorem.

9.1 The Collapsing Lemma Let us start with a theorem about LP on which much of the following depends: the Collapsing Lemma.137 Let I = hD; di be any interpretation for LP . Let  be any equivalence relation on D, that is also a congruence relation on the denotations of the function symbols in the language (i.e., if g is such a denotation, and di  ei for all 1  i  n, then g(d1 ; :::; dn )  g(e1 ; :::; en )). If d 2 D let [d] be the 135 Though this claim has certainly been queried. See Priest [1994]. 136 On this, see further, Mortensen [1995]. Perhaps surprisingly, the rst person to

investigate an inconsistent arithmetic was Nelson [1959], who gave a realisability-style semantics for the language of arithmetic, according to which the set of formulas realised was inconsistent (and closed under a logic somewhat weaker than intuitionist logic). 137 The theorem works equally well for F DE , but we will be concerned primarily with models of theories that contain the law of excluded middle, and so where there are no truth-value gaps.

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equivalence class of d under . De ne an interpretation, I = hD ; d i, to be called the collapsed interpretation, where D = f[d]; d 2 Dg; if c is a constant, d (c) = [d(c)]; if f is an n-place function symbol:

d (f )([d1 ]; :::; [dn ]) = [d(f )(d1 ; :::; dn )] (this is well de ned, since  is a congruence relation); and if P is an n-place predicate, its extension and anti-extension in I , EP and A P , are de ned by:

h[d1 ]; :::; [dn ]i 2 EP i for all 1  i  n, 9ei  di , he1 ; :::; en i 2 EP h[d1 ]; :::; [dn ]i 2 AP i for all 1  i  n, 9ei  di , he1 ; :::; en i 2 AP where EP and AP are the extension and anti-extension of P in I . It is easy to check that E= is fh[d]; [d]i ; d 2 Dg, as required for an LP interpretation. The collapsed interpretation, in eect, identi es all members of an equivalence class to produce a composite individual that has the properties of all of its members. It may, of course, be inconsistent, even if its members are not. A swift induction con rms that for any closed term, t, d (t) = [d(t)]. Hence: 1 2  (P t1 :::tn )

) hd(t1 ); :::; d(tn )i 2 EP ) h[d(t1 )]; :::; [d(tn )]i 2 EP ) hd (t1 ); :::; d (tn )i 2 EP ) 1 2  (P t1 :::tn )

Similarly for 0 and anti-extensions. Monotonicity then entails that for any formula, ,  ()   (). This is the Collapsing Lemma.138 The Collapsing Lemma assures us that if an interpretation is a model of some set of sentences, then any interpretation obtained by collapsing it will also be a model. This gives us an important way of constructing inconsistent models. In particular, if the language contains no function symbols, and I is a model of some set of sentences, then, by appropriate choice of equivalence relation, we can collapse it down to a model of any smaller size. Thus we have a very strong downward Lowenheim-Skolem Theorem: If a theory in a language without function symbols has a model, it has a model of all smaller cardinalities. I note that, since monotonicity holds for second order LP (section 7.2), the Collapsing Lemma extends to second order LP . Details are left as an exercise. 138 The result is proved in Priest [1991a]. A similar result was proved by Dunn [1979].

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9.2 Collapsed Models of Arithmetic From now on, let L be the standard language of rst-order arithmetic: one constant, 0, function symbols for successor, addition and multiplication, 0 , +, and , respectively, and one predicate symbol, =. If I is any interpretation, let T h(I ) (the theory of I ) be the set of all sentences true in I . Let N be the standard model of arithmetic, and A = T h(N ). Let M = hM; di be any classical model of A|which is just special cases of an LP model. (As is well known, there are many of these other than N .139 ) I will refer to the denotations of 0 , +, and  as the arithmetic operations of M, and since no confusion is likely, use the same signs for them.140 Let  be an equivalence relation on M , that is also a congruence relation with respect to the interpretations of the function symbols. Then we may construct the collapsed interpretation, M . By the Collapsing Lemma, M is a model of A. Provided that  is not the trivial equivalence relation, that relates each thing only to itself, then M will model inconsistencies. For suppose that , relates the distinct members of M , n and m, then in M, [n] = [m] and so h[n]; [m]i is in the extension of =. But since n 6= m in M; h[n]; [m]i is in the anti-extension too. Thus, 9x(x = x ^ x 6= x) holds in M . As an illustration of constructing an inconsistent model of A using the Collapsing Lemma, suppose that we partition M into n+1 successive blocks, C0 ; :::; Cn+1 , such that if x; z 2 Ci and x < y < z then y 2 Ci . And suppose that for 0 < i  n + 1, Ci is closed under the arithmetic operations of M. (The existence of such a partition follows from a standard result in the study of classical models of arithmetic. See Kaye [1991], sect. 6.1.) Let 1  k 2 C0 [ C1 and de ne x  y as: (x; y 2 C0 and x = y) or for some 0 < i  n + 1, x; y 2 Ci and x = y mod k where `x = y mod k' means that for some j 2 M , x + j  k = y, in M. It is not diÆcult to check that  is an equivalence relation on M , and, moreover, that it is a congruence relation on the arithmetic operations of M. Hence, we may use it to give a collapsed model. In this, C0 collapses into an initial tail of numbers, and each Ci (0 < i  n + 1) collapses into a block of period k. For example, if M is the standard model, n = 1 and C0 = , the collapsed model is a simple cycle of period k. The successor function in the model may be depicted as follows: 139 See, e.g., Kaye [1991]. 140 For a more detailed discussion of the material in this section, see Priest [1997a].

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0

"

!

1

k 1

! ::: ! :::

369

i

#

i+1

I will call such models cycle models. They were, in fact, the rst inconsistent models to be discovered.141 If M is any model, n = 1; and k = 1, we have a tail isomorphic to C0 , and then a degenerate single-point cycle. In particular, if M is a non-standard model and C0 comprises the standard numbers, we have the natural numbers with a \point at in nity", : 0

!

1

! :::



-

9.3 Inconsistent Models of Arithmetic Now that we have seen the existence of inconsistent models of arithmetic, let us look at their general structure. Take any LP model of arithmetic, M = hM; di. I will call the denotations of the numerals regular numbers. Let x  y be de ned in the usual way, as 9z x + z = y. It is easy to check that  is transitive. For if i  j  k then for some x, y, i + x = j and j + y = k. Hence (i + x) + y = k. But (i + x) + y = i + (x + y) (since it is a model of arithmetic). The result follows. If i 2 M , let N (i) (the nucleus of i) be fx 2 M ; i  x  ig. In a classical model, N (i) = fig, but this need not be the case in an inconsistent model. For example, in a cycle model the members of the cycle constitute a nucleus. If j 2 N (i) then N (i) = N (j ). For if x 2 N (j ) then i  j  x  j  i, so x 2 N (i), and similarly in the other direction. Thus, every member of a nucleus de nes the same nucleus. Now, if N1 and N2 are nuclei, de ne N1  N2 to mean that for some (or all, it makes no dierence) i 2 N1 and j 2 N2 , i  j . It is not diÆcult to check that  is a partial ordering. Moreover, since for any i and j , i  j or j  i, it is a linear ordering. The least member of the ordering is N (0). If N (1) is distinct from this, it is the next (since for any x, x  0 _ x  1), and so on for all regular numbers. Say that i 2 M has period p 2 M i i + p = i. In a classical model every number has period 0 and only 0. But again, this need not be the case in an inconsistent model, as the cycle models demonstrate. If i  j and i has period p so does j . For j = i + x; so p + j = p + i + x = i + x = j . In particular, if p is a period of some member of a nucleus, it is a period of 141 This was by Meyer [1978]. Things are spelled out in Meyer and Mortensen [1984]. The idea of collapsing non-standard classical models is to be found in Mortensen [1987]. Dierent structures can be collapsed to provide inconsistent models of other kinds of number, e.g., real numbers. See Mortensen [1995].

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every member. We may thus say that p is a period of the nucleus itself. It also follows that if N1  N2 and p is a period of N1 it is a period of N2 . If a nucleus has a regular non-zero period, m, then it must have a minimum (in the usual sense) non-zero period, since the sequence 0; 1; 2; :::; m is nite. If N1  N2 and N1 has minimum regular non-zero period, p, then p is a period of N2 . Moreover, the minimum non-zero period of N2 , q, must be a divisor (in the usual sense) of p. For suppose that q < p, and that q is not a divisor of p. For some 0 < k < q, p is some nite multiple of q plus k. So if x 2 N2 , x = x + q = x + p + ::: + p + k. Hence x = x + k, i.e., k is a period of N2 , which is impossible. If a nucleus has period p  1, I will call it proper. Every proper nucleus is closed under successors. For suppose that j 2 N with period p. Then j  j 0  j + p = j . Hence, j 0 2 N . In an inconsistent model, a number may have more than one predecessor, i.e., there may be more than one x such that x0 = j . (Although x0 = y0  x = y holds in the model, we cannot necessarily detach to obtain x = y.)142 But if j is in a proper nucleus, N , it has a unique predecessor in N . For let the period of N be q0 . Then (j + q)0 = j + q0 = j . Hence, j + q is a predecessor of j ; and j  j + q0 = j . Hence, j + q 2 N . Next, suppose that x and y are in the nucleus, and that x0 = y0 = j . We have that x  y _ y  x. Suppose, without loss of generality, the rst disjunct. Then for some z , x + z = y; so j + z = j , and z is a period of the nucleus. But then x = x + z = y. I will write the unique predecessor of j in the nucleus as 0 j . Now let N be any proper nucleus, and i 2 N . Consider the sequence :::;00 i;0 i; i; i0 ; i00 :::. Call this the chromosome of i. Note that if i, j 2 N , the chromosomes of i and j are identical or disjoint. For if they have a common member, z , then all the nite successors of z are identical, as are all its nite predecessors (in N ). Thus they are identical. Now consider the chromosome of i, and suppose that two members are identical. There must be members where the successor distance between them is a minimum. Let these be j and j 0:::0 where there are n primes. Then j = j + n, and n is a period of the nucleus|in fact, its minimum non-zero period|and the chromosome of every member of the nucleus is a successor cycle of period n. Hence, any proper nucleus is a collection of chromosomes, all of which are either successor cycles of the same nite period, or are sequences isomorphic to the integers (positive and negative). Both sorts are possible in an inconsistent model. Just consider the collapse of a non-standard model, of the kind given in the last section, by an equivalence relation which leaves all the standard numbers alone and identi es all the others modulo p. If p is standard, the non-standard numbers collapse into a successor cycle; if it 142 In fact, it is not diÆcult to show that there is at most one number with multiple predecessors; and this can have only two.

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is non-standard, the nucleus generated is of the other kind. To summarise so far, the general structure of a model is a liner sequence of nuclei. There are three segments (any of which may be empty). The rst contains only improper nuclei. The second contains proper nuclei with linear chromosomes. The nal segment contains proper nuclei with cyclical chromosomes of nite period. A period of any nucleus is a period of any subsequent nucleus; and in particular, if a nucleus in the third segment has minimum non-zero period, p, the minimum non-zero period of any subsequent nucleus is a divisor of p. Thus, we might depict the general structure of a model as follows (where m + 1 is a multiple of n + 1): 0; 1; :::

:::a ! a0 ::: :::b ! b0 ::: .. .

::: :::

d0 :::di

" #

e0 :::ei

" # :::

dm :::d0i em :::e0i f0 :::fi g0:::gi

" # " # ::: fn:::fi0 gn:::gi0

::: :::

Another obvious question is what possible orderings the proper nuclei can have. For a start, they can have the order-type of any ordinal. To prove this, one establishes by trans nite induction that for any ordinal, , there is a classical model of arithmetic in which the non-standard numbers can be partitioned into a collection of disjoint blocks with order-type , closed under arithmetic operations. One then collapses this interpretation in such a way that each block collapses into a nucleus. The proper nuclei need not be discretely ordered. They can also have the order-type of the rationals. To prove this, one considers a classical nonstandard model of arithmetic, where the order-type of the non-standard numbers is that of the rationals. It is possible to show that these can be partitioned into a collection of disjoint blocks, closed under arithmetic operations, which themselves have the order-type of the rationals. One can then collapse this model in such a way that each of the blocks collapses into a proper nucleus, giving the result. This proof can be extended to show that any order-type that can be embedded in the rationals in a certain way, can also be the order-type of the proper nuclei. This includes ! (the reverse of !) and ! + !, but not ! + ! . For details of all this, see Priest [1997b]. What other linear order-types proper nuclei may or may not have, is still an open question.

9.4 Finite Models of Arithmetic First-order arithmetic has many classical nonstandard models, but none of these is nite. One of the intriguing features of LP is that it permits

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nite models of arithmetic, e.g., the cycle models. For these, a complete characterisation is known. Placing the constraint of nitude on the results of the previous section, we can infer as follows. The sequence of improper nuclei is either empty or is composed of the singletons of 0; 1; :::; n, for some nite n. There must be a nite collection of proper nuclei, N1  :::  Nm ; each Ni must comprise a nite collection of successor cycles of some minimum non-zero nite period, pi . And if 1  i  j  m, pj must be a divisor of pi .143 Moreover, there are models of any structure of this form. To show this, we can generalise the construction of 9.2. Take any non-standard classical model of arithmetic. This can be partitioned into the nite collection of blocks: C0 ; C10 ; :::; C1k(1) ; :::Ci0 ; :::; Cik(i) ; :::; Cm0 ; :::; Cmk(m) where C0 is either empty or is of the form f0; :::; ng, each subsequent block is closed under arithmetic operations, and there are k(i) successor cycles in Ni . We now de ne a relation, x  y, as follows: (x; y 2 C0 and x = y) or for some 1  i  m: (for some 0 < j < k(i), x; y 2 Cij , and x = y mod pi ) or ( x; y 2 Ci0 [ Cik(i) and x = y mod pi ) One can check that  is an equivalence relation, and also that it is a congruence relation on the arithmetic operations. Hence we can construct the collapsed model.  leaves all members of C0 alone. For every i it collapses every Cij into a successor cycle of period pi , and it identi es the blocks Ci0 and Cik(i) . Thus, the sequence Ci0 ; :::Cik(i) collapses into a nucleus of size k(i). The collapsed model therefore has exactly the required structure.144 There are many interesting questions about inconsistent models, even the nite ones, whose answer is not known. For example: how many models of each structure are there? (The behaviour of the successor function in a model does not determine the behavior of addition and multiplication, except in the tail.) Perhaps the most important question is as follows. Not all inconsistent model of arithmetic are collapses of classical models. Let M be any model of arithmetic; if M0 is obtained from M by adding extra pairs to the anti-extension of =, call M0 an extension in M. If M0 is an extension of M, monotonicity ensures that it is a model of arithmetic. Now, consider the extension of the standard model obtained by adding h0; 0i to the anti-extension of =. This is not a collapsed model, since, if it were, 0 would have to have been identi ed with some x > 0. But then 1 would have 143 It is also possible to show that each nucleus is closed under addition and multiplication. 144 For further details, see Priest [1997a].

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been identi ed with x0 > 1. Hence, 00 6= 00 would also be true in the model, which it is not. Maybe, however, each inconsistent model is the extension of a collapsed classical model. If this conjecture is correct, collapsed models can be investigated via an analysis of the classical models of arithmetic and their congruence relations.

9.5 The Limitative Theorems of Metamathematics Let us now turn to the limitative theorems of Metamathematics in the context of LP . These are the theorems of Lowenheim-Skolem, Church, Tarski and Godel. I will take them in that order.145 In what follows, let P be the set of theorems of classical Peano Arithmetic, and let Q be any non-trivial theory that contains P . According to the classical Lowenheim-Skolem, Q has models of every in nite cardinality but has no nite models. Moving to LP changes the situation somewhat. Q still has a model of every in nite cardinality.146 But it has models of nite size too: any inconsistent model may be collapsed to a nite model merely by identifying all numbers greater than some cut-o.147 The situation with second order P is again dierent in LP . Classically, this is known to be categorical, having the standard model as its only interpretation. But as I noted in 9.1, the Collapsing Lemma holds for second order LP . Hence, second order P is not categorical in LP . For example, it has nite models. Turning to Church's theorem, this says, classically, that Q is undecidable. In LP , extensions of Q may be decidable. For example, let M be any nite model of A (= T h(N )), and let Q be T h(M). Then Q is a theory that contains P . Yet Q is decidable, as is the theory of any nite interpretation. In the language of M there is only a nite number of atomic sentences; their truth values can be listed. The truth values of truth functions of these can be computed according to (LP ) truth tables, and the truth values of quanti ed sentences can be computed, since 9x has the same truth value 145 For a statement of these in the classical context, see Boolos and Jerey [1974]. This section expands on the appendix of Priest [1994]. 146 The standard classical proof of this adds a new set of constants, fci , i 2 I g, to the language, and all sentences of the form ci 6= cj , i 6= j , to Q. It then uses the compactness theorem. Things are more complex in LP , since the fact that ci 6= cj holds in an interpretation does not mean that the denotations of these constants are distinct. After extending the language, we observe that ci = cj cannot be proved. We then construct a prime theory in the manner of 4.3, keeping things of this form out. This is then used to de ne an appropriate interpretation. 147 Let us say that M is an exact model of a theory i the truths of M are exactly the members of the theory. Classically, for complete theories, there is no dierence between modelling and exact modelling. The situation for LP is more complex. It can be shown that if Q has an in nite exact model it has exact models of every greater cardinality. On the other hand, if Q has a nite model, M, in which every number is denoted by a numeral, M can be shown to be the only exact model of Q (up to isomorphism).

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as the disjunction of all formulas of the form (x=a), where a is in the domain of M; dually for 8. Tarski's Theorem: this says that Q cannot contain its own truth predicate, in the sense that even if Q is a theory in an extended language, there is no formula, , of one free variable, x, such that (x= hi)   2 Q, for all closed formulas, , of the language. This, too, fails for LP . Let M be any (classical) model of P , let M0 be any nite collapse of M, and let Q be T h(M0). By the Collapsing Lemma, Q contains P . Since Q is decidable, it is representable in (classical) P by a formula, , of one free variable, x. That is, we have : If  2 Q then (x= hi) 2 P If  62 Q then : (x= hi) 2 P By the Collapsing Lemma, `P ' may be replaced by `Q'. If  2 Q, (x= hi) 2 Q, and so (x= hi)   2 Q (since ; Æ j=LP  Æ); and if  62 Q, : (x= hi) 2 Q, and so (x= hi)   2 Q (since : 2 Q and : ; :Æ j=LP

 Æ). There is no guarantee that  and (x= hi) have the same truth value in M0 . In particular, then, may not satisfy the T -schema in the form of a two-way rule of inference. So it might be said that is not really a truth predicate. Whether or not this is so, we have already seen that there are Qs where there is a predicate satisfying this condition (though this has to be added to the language of arithmetic): the theory T1 of section 8.1.148 Finally, let us turn to Godel's undecidability theorems. A statement of the rst of these is that if Q is axiomatisable then there are sentences true in the standard model that are not in Q. It is clear that this may fail in LP . Let M be any nite model of arithmetic. Then if Q is T h(M), Q contains all of the sentences true in the standard model of arithmetic, but is decidable, as we have noted, and hence axiomatisable (by Craig's Theorem). It is worth asking what happens to the \undecidable" Godel sentence in such a theory. Let be any formula that represents Q in Q. (There are such formulas, as we just saw.) Then a Godel sentence is one, , of the form : (x= hi). If  2 Q then : (x= hi) 2 Q, but (x= hi) 2 Q by representability. If  62 Q then : (x= hi) 2 Q by representability, i.e.,  2 Q, so (x= hi) 2 Q by representability. In either case, then, 148 The construction of 8.1 can be applied to any model of arithmetic|not just the standard model|as the ground model. However, if we apply it to a nite model care needs to be exercised. The construction will not work as given, since dierent formulas may be coded by the same number in the model, which renders the de nition of the sequence of interpretations illicit. We can switch to evaluational semantics, as in 8.2, though the construction then no longer validates the substitutivity of identicals. Alternatively, we can refrain from using numbers as names, but just augment the language with names for all sentences.

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 ^ : 2 Q. Godel's second undecidability theorem says that the statement that canonically asserts the consistency of Q is not in Q; this statement is usually taken to be : (x= h0 i), where 0 is 0 = 00 , and is the canonical proof predicate of Q. This also fails in LP .149 Let Q be as in the previous two paragraphs. Then Q is not consistent. However, it is still the case that 0 62 Q (provided that the collapse is not the trivial one). Consider the relationship: n is (the code of) a proof of formula (with code) m in Q. Since this is recursive, it is represented in A by a formula P rov(x; y). If  is provable in Q then for some n, P rov(n; hi) 2 A (where n is the numeral for n); thus, 9xP rov(x; hi) 2 A and so Q. If  is not provable in Q then for all n, :P rov(n; hi) 2 A; thus, 8x:P rov(x; hi) 2 A (since A is !-complete) and :9xP rov(x; hi) 2 A and so Q. Thus, 9xP rov(x; y) represents Q in Q. In particular, since 0 62 Q, :9xP rov(x; h0 i) 2 Q, as required.

9.6 The Philosophical Signi cance of Godel's Theorem People have tried to make all sorts of philosophical capital out of the negative results provided by the limitative theorems of classical Metamathematics. As we have seen, all of these, save the Lowenheim-Skolem Theorem, fail for arithmetic based on a paraconsistent logic. Setting this theorem aside, then, nothing can be inferred from these negative results unless one has reason to rule out paraconsistent theories. At the very least, this adds a whole new dimension to the debates in question. This is not the place to discuss all the philosophical issues that arise in this context, but let me say a little more about one of the theorems by way of illustration. Doubtless, the incompleteness result that has provoked most philosophical rumination is Godel's rst incompleteness theorem: usually in a form such as: for any axiomatic theory of arithmetic (with suÆcient strength, etc.), which we can recognise to be sound, there will be an arithmetic truth|viz., its Godel sentence|not provable in it, but which we can establish as true.150 This is just false, paraconsistently. If the theory is inconsistent, the Godel sentence may well be provable in the theory, as we have seen. An obvious thought at this point is that if we can recognise the theory to be sound then it can hardly be inconsistent. But unless one closes the question prematurely, by a refusal to consider the paraconsistent possibility, this is by no means obvious. What is obvious to anyone familiar with the subject, is that at the heart of Godel's theorem, is a paradox. The paradox concerns the sentence, , `This sentence is not provable', where `provable' 149 It is worth noting that there are consistent arithmetics based on some relevant logics, notably R, for which the statement of consistency is in the theory. See Meyer [1978]. 150 For example, the theorem is stated in this form in Dummett [1963]; it also drives Lucas' notorious [1961], though it is less clearly stated there.

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is not to be understood to mean being the theorem of some axiom system or other, but as meaning `demonstrated to be true'. If is provable, then it is true and so not provable. Thus we have proved . It is therefore true, and so unprovable. Contradiction. The argument can be formalised with one predicate, B , satisfying the conditions:

` B hi !  If `  then ` B hi for all closed |including sentences containing B . For if is of the form :B h i, then, by the rst, ` B h i ! :B h i, and so ` :B h i, i.e., ` . Hence, ` B h i, by the second. And we do recognise these principles to be sound. Whatever is provable is true, by de nition; and demonstrating  shows that  is provable, and so counts as a demonstration of this fact.151 B is a predicate of numbers, but we do not have to assume that B is de nable in terms of 0 , + and  using truth functions and quanti ers. The argument could be formalised in a language with B as primitive. As we saw in the previous part in connection with truth, it is quite possible to have an inconsistent theory with a predicate of this kind, where the sentences de nable in terms of 0 , + and  using truth functions and quanti ers behave quite consistently. Of course, if B is so de nable, which it will be if the set of things we can prove is axiomatic, then the set of things that hold in this language is inconsistent. And there are reasons for supposing that this is indeed the case.152 Even this does not necessarily mean that the familiar natural numbers behave strangely, however. As the model with the \point at in nity" of 9.2 showed, it is quite possible for inconsistent models to have the ordinary natural numbers as a substructure.153 There are just more possibilities in Heaven and Earth than are dreamt of in a consistent philosophy. 10 PHILOSOPHICAL REMARKS In previous parts I have touched occasionally on the philosophical aspects of paraconsistency. In this section I want to take up a few of the philosophical implications of paraconsistency at slightly greater length. Its major 151 The paradox is structurally the same as a paradox often called the `Knower paradox'. In this, B is interpreted as `It is known that'. For references and discussion of this paradox and others of its kind, see Priest [1991b]. 152 See Priest [1987], ch. 3. This chapter discusses the connection between Godel's theorem, the paradoxes of self-reference and dialetheism at greater length. 153 Though whether the theory of that particular model is axiomatisable is currently unknown.

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implication is very simple. As I noted in 3.1, the absolute unacceptability of inconsistency has been deeply entrenched in Western philosophy. It is an assumption that has hardly been questioned since Aristotle. Whilst the law of non-contradiction is a traditional statement of this fact, it is ECQ which expresses the real horror contradictionis: contradictions explode into triviality. Paraconsistency challenges exactly this, and so questions any philosophical claim based on this supposed unacceptability. This does not mean that consistency cannot play a regulative function: it may still be an expected norm, departure from which requires a justi cation; but it can no longer provide a constraint of absolute nature. Given the centrality of consistency to Western thought, the philosophical rami cations of paraconsistency are bound to be profound, and this is hardly the place to take them all|or even some|up at great length. What I will do here is consider various objections to employing a paraconsistent logic, and explore a little some of the philosophical issues that arise in this context. In the process we will need to consider not only the purposes of logic, but also the natures of negation, denial, rational belief and belief revision.154

10.1 Instrumentalism and Information Why, then, might one object to paraconsistent logic? Logic has many uses, and any objection to the use of a paraconsistent logic must depend on what it is supposedly being used for. One thing one may want a logic for is to draw out consequences of some information in a purely instrumental way. In such circumstances one may use any logic one likes provided that it gives appropriate results. And if the information is inconsistent, an explosive logic is hardly likely to do this. Referring back to the list of motivations for the use of a paraconsistent logic in 2.2, drawing inferences from a scienti c theory would fall into this category if one is a scienti c instrumentalist. Drawing inferences from the information in a computer data base could also fall into this category. If the logic gives the right results|or at least, does not give the wrong results|use it. The only objection that there is likely to be to the use of a paraconsistent logic in this context is that it is too weak to be of any serious use. One might note, for example, that most paraconsistent logics invalidate the disjunctive syllogism, a special case of resolution, on the basis of which many theoremprovers work.155 This objection carries little weight, however. Theorem154 Other philosophical aspects of paraconsistency are discussed in numerous places, e.g., da Costa [1982], Priest [1987], Priest et al. [1989], ch. 18. 155 It is worth noting, however, that some theorem-provers that use resolution are not complete with respect to classical semantics. For example, to determine whether  follows from the information in a data base, some theorem-provers employ a heuristic that requires them resolve : with something on the data base, and so on recursively. Em-

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provers can certainly be based on other mechanisms.156 Moreover, the inferential moves of the standard programming language PROLOG can all be interpreted validly in many paraconsistent logics (when `:-' is interpreted as `a'). One will often require a logic for something other than merely instrumental use. This does not mean that one is necessarily interested in truthpreservation, however. One might, for example, require a logic whose valid inferences preserve, not truth, but information. The computer case could also be an example of this. Other natural examples of this in the list of 2.2 are the ctional and counterfactual situations. By de nition, truth is not at issue here.157 Information-preservation implies truth preservation, presumably, but the converse is not at all obvious, and not even terribly plausible. The information that the next ight to Sydney leaves at 3.45 and does not leave at 3.45 would hardly seem to contain the information that there is life on Mars. A paraconsistent logic is therefore a plausible one in this context. What information, and so information-preservation, are, is an issue that is currently much discussed. One popular approach is based on the situation semantics of Barwise and Perry [1983].158 This takes a unit of information (an infon) to be something of the form hR; a1 ; :::; an ; si, where R is an nplace relation, the ai 's are objects, and s is a sign-bit (0 or 1). A situation is a set of infons. The situations in question do not have to be veridical in any sense. In particular, they may be both inconsistent and incomplete. In fact, it is easy to see that a situation, so characterised, is just a relational F DE evaluation. This approach to information therefore naturally incorporates a paraconsistent logic, which may be thought of as a logic of information preservation.159

10.2 Negation Another major use of logic (perhaps the one that many think of rst) is in contexts where we want inference to be truth-preserving; for example, ploying this procedure when the data base is fp; :pg and the query is q will result in a negative answer. Such inference engines are therefore paraconsistent, though they do not answer to any principled semantics that I am aware of. 156 For details of some automated paraconsistent logics, see, e.g., Blair and Subrahmanian [1988], Thistlewaite et al. [1988]. 157 One might also take the other example on that list, constitutions and other legal documents, to be an example of this. Such documents certainly contain information. And one might doubt that this information is the sort of thing that is true or false: it can, after all, be brought into eect by at|and may be inconsistent. However, if it is that sort of thing, legal reasoning concerning it would seem to require truth-preservation. 158 See, e.g., Devlin [1991]. 159 It is worth noting that North American relevant logicians have very often|if not usually|thought of the F DE valuations information-theoretically, as told true and told false. See, e.g., Anderson et al. [1992], sect. 81.

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where we are investigating the veridicality of some theory or other. And here, it is very natural to object to the use of a paraconsistent logic. Since truth is never inconsistent a paraconsistent logic is not appropriate. A paraconsistent logician who thinks that truth is consistent may agree with this, in a sense. We have already seen in 7.6 how a paraconsistent logic, applied to a consistent situation, may give classical reasoning. However, a dialetheist will object; not to the need for truth preservation, but to the claim that truth is consistent: some contradictions are true: dialetheias. This is likely to provoke the ercest objections. Let me start by dividing these into two kinds: local and global. Global objections attack the possibility of dialetheias on completely general grounds. Local objections, by contrast, attack the claim that some particular claims are dialetheic on grounds speci c to the situation concerned. Let us take the global objections rst. Why might one think that dialetheias can be ruled out quite generally, independently of the considerations of any particular case? A rst argument is to the eect that a contradiction cannot be true, since contradictions entail everything, and not everything is true. It is clear that in the context where the use of a paraconsistent logic is being defended, this simply begs the question. Of more substance is the following objection. The truth of contradictions is ruled out by the (classical) account of negation, which is manifestly correct. The amount of substance is only slightly greater here, though: the claim that the classical account of negation is manifestly correct is just plain false. An account of negation is a theory concerning the behaviour of something or other. It is sometimes suggested that it is an account of how the particle `not', and similar particles in other languages, behaves. This is somewhat naive. Inserting a `not' does not necessarily negate a sentence. (The negation of `All logicians do believe the classical account of negation' is not `All logicians do not believe the classical account of negation'.) And `not' may function in ways that have nothing to do with negation at all. Consider, e.g.: `I'm not a Pom; I'm English', where it is connotations of what is said that are being rejected, not the literal truth. It seems to me that the most satisfactory understanding of an account of negation is to regard it as a theory of the relationship between statements that are contradictories. Note that this by no means rules out a paraconsistent account of negation.160 Even supposing that we characterise contradictories as pairs of formulas such that at least one must be true and at most one can be true|with which an intuitionist would certainly disagree|it is quite possible to have both 2( _ :) and :3( ^ :) valid in a paraconsistent logic, as we saw in 7.3. Anyway, whatever we take a theory of negation to be a theory of, it is but 160 As Slater [1995] claims.

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a theory. And dierent theories are possible. As we have already observed, Aristotle gave an account of this relationship that was quite dierent from the classical account as it developed after Boole and Frege. And modern intuitionists, too, give a quite dierent account. Which account is correct is to be determined by the usual criteria for the rational acceptability of theories. (I will say a little more about this later.) The matter is not at all obvious. Quine is well known for his objection to non-classical logic in general, and paraconsistent logic in particular, on the ground that changing the logic (from the classical one) is `changing the subject', i.e., succeeds only in giving an account of something else ([1970], p. 81). This just confuses logic, qua theory, with logic, qua object of theory. Changing one's theory of logic no more changes what it is one is theorising about|in this case, relationships grounding valid reasoning|than changing one's theoretical geometry changes the geometry of the cosmos. Nor does it help to suppose that logic, unlike geometry, is analytic (i.e., true solely in virtue of meanings). Whether or not, e.g., `There will be a sea battle tomorrow or there will not' is analytic in this sense, is no more obvious than is the geometry of the cosmos. And changing from one theory, according to which it is analytic, to another, according to which it is not, does not change the facts of meaning.161 How plausible a paraconsistent account of negation is depends, of course, on which paraconsistent account of negation is given. As we saw in part 4, there are many. One of the simplest and most natural is provided by the relational semantics of 4.5. This is just the classical account, except that classical logic makes the extra assumption that all statements have exactly one truth value. And logicians as far back as Aristotle have questioned that assumption.162

10.3 Denial Another global objection to dialetheism goes by way of a supposed connection between negation and denial. It is important to be clear about the distinction between these two things for a start. Negation is a syntactic and/or semantic feature of language. Denial is a feature of language use: it is one particular kind of force that an utterance can have, one kind of illocutionary act, as Austin put it. Speci cally, it is to be contrasted with assertion.163 Typically, to assert something is to express one's belief in, or acceptance of, it (or some Gricean sophistication thereof). Typically, to deny something is to express ones rejection of it, that is, one's refusal to ac161 The analogy between logic and geomety is discussed further in Priest [1997a]. 162 The topics of this section and the next are discussed at greater length in Priest [1999]. 163 Traditional logic usually drew the distinction, not in terms of saying, but in terms of

judging. It can be found in these terms, for example, in the Port-Royal Logic of Arnauld and Nicole.

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cept it (or some Gricean sophistication thereof). Clearly, if one is uncertain about a claim, one may wish neither to assert nor to deny it. Although assertion and denial are clearly dierent kinds of speech act, Frege argued, and many now suppose, that denial may be reduced to the assertion of negation.164 If this is correct, then dialetheism faces an obvious problem. Even if some contradictions were true, no one could ever endorse a contradiction, since they could not express an acceptance of one of the contradictories without expressing a rejection of the other.165 Frege's reduction has no appeal if we take the negation of a statement simply to be its contradictory. In asserting `Some men are not mortal', I am not denying `All men are mortal'. I might not even realise that these are contradictories, and neither might anyone else. And if this does not seem plausible in this simple case, just make the example more complex, and recall that there is no decision procedure for contradictories. The reduction takes on more plausibility if we identify the negation of a sentence as that sentence pre xed by `It is not the case that'. But even in this case, the claim that to assert a negation is invariably to deny the sentence negated appears to be false. Dialetheists who asserts both, e.g., `The Liar sentence is true' and `It is not the case that the Liar sentence is true', are not expressing the rejection of the former with the latter: they are simply expressing their acceptance of a certain negated sentence.166 It may well be retorted that this reply just begs the question, since what is at issue is whether a dialetheist can do just this. This may be so; but it may now be fairly pointed out that the original objection just begs the question against the dialetheist too. In any case, there would appear to be plenty of other examples where to assert a negation is not to deny. For example, we may be brought to see that our views are inconsistent by being questioned in Socratic fashion and thus made to assert an explicit contradiction. When this happens, we are not expressing the rejection of any view. What the questioning exposes is exactly our acceptance of contradictory views. We may, in the light of the questioning, come to reject one of the contradictories, and so revise our views, but that is another matter.167 To assert a negated sentence is not, then, ipso facto to deny the sentence negated. Some, having taken this point to heart, object on the other side of the street: dialetheists have no way of expressing some of their views, specifically their rejection of certain claims: we need take nothing a dialetheist 164 See Frege [1919]. 165 For an objection along these lines, see Smiley in Priest and Smiley [1993]. 166 And even those who take negation to express denial must hold that there is more to

the meaning of negation than this. It cannot, for example, perform that function when it occurs attached to part of a sentence. 167 Some non-dialetheists have even argued that it may not even be rational to revise our views in some contexts. See, e.g., Prior on the paradox of the preface [1971], pp. 84f.

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says as a denial.168 This objection is equally awed. For a start, even if to assert a negated sentence is to deny it, it is certainly not the only way to deny it. One can do so by a certain shake of the head, or by the use of some other body language. A dialetheist may deny in this way. Moreover, just because the assertion of a negated sentence by a dialetheist (or even a non-dialetheist, as we have seen) may not be a denial, it does not follow that it is not. In denial, a person aims to communicate to a listener a certain mental state, that of rejection; and asserting a negated sentence with the right intonation, and in the right context, may well do exactly that|even if the person is a dialetheist.169

10.4 The Rational Acceptability of Contradictions This does not exhaust the possible global objections to dialetheism,170 but let us move on to the local ones. These do not object to the possibility of dialetheism in general, but to particular (supposed) cases of it. We noted in 2.2 that a number of these have been proposed, which include legal dialetheias, descriptions of states of change, borderline cases of vague predications and the paradoxes of self-reference. Though the detailed reasons for endorsing dialetheism in each case are dierent, their general form is the same: a dialetheic account of the phenomenon in question provides the most satisfactory way of handling the problems it poses. A local objection may therefore be provided by producing a consistent account of the phenomenon, and arguing this is rationally preferable. The precise issues involved here will, again, depend on the topic in question; but let us examine one issue in more detail. This will allow the illustration of a number of more general points. The case we will look at is that of the semantic paradoxes. The background to this needs no long explanation, since a logician or philosopher who does not know it may fairly be asked where they have been this century. Certain arguments such as the Liar paradox, and many others discovered in the middle ages and around the turn of this century, appear to be sound arguments to the eect that certain contradictions employing self-reference and semantic notions are true. A dialetheic account simply endorses the semantic principles in question, and thus the contradictions to which these give rise. A consistent account must nd some way of rejecting the reasoning, notably by giving a dierent account of how the semantic apparatus 168 Objections along these lines can be found in Batens [1990], and Parsons [1990]. A reply can be found in Priest [1995]. 169 That the same sentence may have dierent forces in dierent contexts is hardly a novel observation. For example, an utterance of `Is the door closed', can be a question, a request or a command, depending on context, intonation, power-relationships, etc. 170 Some others, together with appropriate discussion, can be found in Sainsbury [1995], ch. 6.

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functions. This account must both do justice to the data, and avoid the contradictions. Many such accounts have, of course, been oered. But they are all well known to suer from various serious problems. For example, they may provide no independent justi cation for the restrictions on the semantic principles involved, and so fail to explain why we should be so drawn to the general and contradiction-producing principles. They are often manifestly contrived and/or y in the face of other well established views. Perhaps most seriously, none of them seems to avoid the paradoxes: all seem to be subject to extended paradoxes of one variety or another.171 If the global objections to dialetheism have no force, then, the dialetheic position here seems manifestly superior.172 It might be said that the inconsistency of the theory is at least a prima facie black mark against it. This may indeed be so; but even if one of the consistent theories could nd plausible replies to it problems, as long as the theory is complex and ghting a rearguard action, the dialetheic account may still have a simplicity, boldness and mathematical elegance that makes it preferable. As orthodox philosophy of science realised a long time ago, there are many criteria which are good-making for theories: simplicity, adequacy to the data, preservation of established problem-solutions, etc.; and many which are bad-making: being contrived, handling the data in an ad hoc way, and, let us grant, being inconsistent, amongst others.173 These criteria are usually orthogonal, and may even pull in opposite directions. But when applied to rival theories, the combined eect may well be to render an inconsistent theory rationally preferable to its consistent rival. General conclusion: a theory in some area is to be rationally preferred to its rivals if it best satis es the standard criteria of theory choice, familiar from the philosophy of science. An inconsistent theory may be the only viable theory; and even if it is not, it may still, on the whole, be rationally preferable.174 171 All this is documented in Priest [1987], ch. 2. 172 One strategy that may be employed at this point is to argue that a dialetheic theory

is trivial, and hence that any other theory, even one with problems, is better. As we have seen, dialetheic truth-theory is non-trivial, but one might nonetheless hope to prove that it is trivial when conjoined with other unobjectionable apparatus. Such arguments have been put forward by Denyer [1989], Smiley, in Priest and Smiley [1993], and Everett [1995] and elsewhere. Replies can be found in, respectively, Priest [1989b], Priest and Smiley [1993], and Priest [1996]. Since my aim here is to illustrate general features of the situation, I will not discuss these arguments. 173 Though one might well challenge the last of these as a universal rule. There might be nothing wrong with some contradictions at all. See Priest [1987], sect. 13.6, and Sylvan [1992], sect. 2. 174 For a longer discussion of the relationship between paraconsistency and rationality, see Priest [1987], ch. 7.

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10.5 Boolean Negation Another sort of local objection to some dialetheic theories is based on the claim that, whatever one says about negation, there is certainly an operator that behaves in the way that Boolean negation does|call it what you like. Some paraconsistent logicians may even agree with this. (As we saw in 5.3, such an operator is de nable in some of the da Costa systems.) And if the point is correct, it suÆces to dispose of any dialetheic account of the semantic paradoxes which endorses the T -schema; similarly, any account of set theory that endorses the Comprehension schema. For as I observed in the introduction to part 8, these schemas will then generate Boolean contradictions, and so entail triviality. Someone who endorses such an account of semantics or set theory must therefore object to the claim that there is an operator that behaves as does Boolean negation. Why, after all, should we suppose this?175 It might be suggested that we can simply de ne an operator, , satisfying the proof theoretic principles of Boolean negation, and in particular: ;  ` . Such a suggestion would fail: the reason is simply that there is no guarantee that a connective, so characterised, has any determinate sense. The point was made by Prior [1960], who illustrated it with the operator \tonk", , supposedly characterised by the rules  `   ,   ` . Such an operator induces triviality and can make no sense. Similarly, a paraconsistent logician who endorses the T -schema may fairly point out that the supposition that there is an operator satisfying the proof-theoretic conditions of Boolean negation induces triviality, and so makes no sense.176 The claim is theory laden, in the sense that it presupposes that the T schema is correct. (The addition of such an operator need not produce triviality if only more limited machinery is present.) But any claim about what makes sense is bound to be theory-laden in a similar way. Prior's argument, for example, presupposes the transitivity of deducibility, which may be questioned, as we have seen. The thought that Boolean negation is meaningless may initially be somewhat shocking. But the point has been argued by intuitionist logicians for many years. And though the grounds are quite dierent,177 the paraconsistent logician sides with the intuitionist against the classical logician on this occasion. Can we not, though, characterise Boolean negation semantically, and so show that it is a meaningful connective? The answer is, again, no; not without begging the question. How one attempts to characterise Boolean negation semantically will depend, of course, on one's preferred sort of se175 The following material is covered in more detail in Priest [1990]. 176 There may, of course, be operators that behave like Boolean negation in a limited domain. That is another matter. 177 The intuitionist reason is that meaningful logical operators cannot generate statements with recognition-transcendent truth conditions, which Boolean negation does. See, e.g., Dummett [1975].

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mantics. Let me illustrate the matter with the Dunn semantics. Similar considerations apply to others. With these semantics, the natural attempt to characterise Boolean negation is:

1 i it is not the case that 1 0 i 1 And such a characterisation makes perfectly good semantic sense. However, it does not entail that satis es the Boolean proof-theoretic principles. Why should one suppose, crucially, that it validates ;  j= ? From the characterisation, it certainly follows that for all , it is not the case that 1 and 1; but to infer from this that for all , if 1 and 1 then 1 (which states that the inference is valid), just employs the principle of inference that a conditional is true if the negation of its antecedent is. And no sensible paraconsistent conditional validates this. In other words, to insist that , so characterised, is explosive, just begs the question against the paraconsistentist. And if it is claimed that the negation in the statement of the truth conditions is itself Boolean, and so the inference is acceptable, this again begs the question: whether there is a connective satisfying the Boolean proof-theoretic conditions is exactly what is at issue.178

10.6 Logic as an Organon of Criticism We have now noted three reasons why one might employ a logic: as a purely instrumental means of generating consequences, as an organon of information preservation, and as an organon of truth preservation. This does not exhaust the uses for which one might employ a logic. Another very traditional one is as an organon of criticism, to force others to revise their views. One may object to the use of a paraconsistent logic in this context as follows. If one subscribes to a paraconsistent logic, then there is nothing to stop a person from accepting any inconsistency to which their views lead. Hence, paraconsistency renders logic useless in this context.179 The move from the premise that contradictions do not entail everything to the claim that there is nothing to stop a person subscribing to a contradiction is a blatant non-sequitur. The threat of triviality may be a reason 178 I have sometimes heard it said that the logic of a metatheory must be classical. This is just false, as the existence of intuitionist metatheories serves to remind. For certain purposes a dialetheist may, in any case, use a classical metatheory. If, for example, we are trying to show a certain theory to be non-trivial, it suÆces to show all the theorems have some property which not all sentences have. This might well be shown using ZF . As we saw in 8.6, ZF makes perfectly good dialetheic sense. 179 An objection of this kind is to be found in Popper [1963], pp. 316-7. The following is discussed at greater length in Priest [1987], ch. 7.

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for revision; it is not the only reason. This is quite obvious in the case of non-dialetheic paraconsistency. If a contradiction is entailed by one's views, then even though they do not explode into triviality, they are still not true. One will still, therefore, wish to revise. One may not, as in the classical case, have to revise immediately. It may not be at all clear how to revise; and in the meantime, an inconsistent but non-trivial belief set is better than no belief set at all. But the pressure will still be there to revise in due course. The situation may be thought to change if one brings dialetheism into the picture. For the contradiction may then be true, and the pressure to revise is removed. Again, however, the conclusion is too swift. It is certainly true that showing that a person's views are inconsistent may not necessarily force a dialetheist to revise, but other things may well do so. For example, if a person is committed to something of the form  ! ?, and their views are shown to entail , there will be pressure to revise, for exactly the classical reason.180 Even if a dialetheist's views do not collapse into triviality, the inference to the claim that there is no pressure to revise is still too fast. The fact that there is no logical objection to holding on to a contradiction does not show there are no other kinds of objection. There is a lot more to rationality than consistency. Even those who hold consistency to be a constraint on rationality hold that there are many other such constraints. In fact, consistency is a rather weak constraint. That the earth is at, that Elvis is alive and living in Melbourne, or, indeed, that one is Kermit the Frog, are all views that can be held consistently if one is prepared to make the right kinds of move elsewhere; but these views are manifestly irrational. For a start, there is no evidence for them; moreover, to make things work elsewhere one has to make all kinds of ad hoc adjustments to other well-supported views. And whatever constraints there are on rational belief|other than consistency| these work just as much on a dialetheist, and may provide pressure to revise. Not, perhaps, pressure of the stand 'em up - knock 'em down kind. But such would appear to be illusory in any case. As the history of ideas has shown, rational debates may be a long and drawn out business. There is no magic strategy that will always win the debate|other than employing (or at least showing) the instruments of torture.181 180 Provided that one is not a person who believes that everything is true, then asserting  ! ? is a way of denying . A dialetheist might do this for a whole class of sentences, and so rule out contradictions occurring in certain areas, wholesale. 181 Avicenna, apparently, realised this. According to Scotus, he wrote that those who deny the law of non-contradiction `should be ogged or burned until they admit that it is not the same thing to be burned and not burned, or whipped and not whipped'. (The Oxford Commentary on the Four Books of the Sentences, Bk. I, Dist. 39. Thanks to Vann McGee for the reference.)

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11 CONCLUSION Let me conclude this essay by trying to put a little perspective into the development of paraconsistent logic. Paraconsistent and explosive accounts of validity are both to be found in the history of logic. The revolution in logic that started around the turn of the century, and which was constituted by the development and application of novel and powerful mathematical techniques, entrenched explosion on the philosophical scene. The application of the same techniques to give paraconsistent logics had to wait until after the second world war. The period from then until about the late 1970s saw the development of many paraconsistent logics, their proof theories and semantics, and an initial exploration of their possible applications. Though there are still many open problems in these areas, as I have indicated in this essay, the subject was well enough developed by that time to permit the beginning of a second phase: the investigation of inconsistent mathematical theories and structures in their own rights. Whereas the rst period was dominated by a negative metaphor of paraconsistency as damage control, the second has been dominated by a more positive attitude: let us investigate inconsistent mathematical structures, both for their intrinsic interest and to see what problems|philosophical, mathematical, or even empirical|they can be used to solve.182 Where this stage will lead is as yet anyone's guess. But let me speculate. Traditional wisdom has it that there have been three foundational crises in the history of mathematics. The rst arose around the p Fourth Century BC, with the discovery of irrational numbers, such as 2. It resulted in the overthrow of the Pythagorean doctrine that mathematical truths are exhausted by the domain of the whole numbers (and the rational numbers, which are reducible to these); and eventually, in the development of an appropriate mathematics. The second started in the Seventeenth Century with the discovery of the in nitesimal calculus. The appropriate mathematics came a little faster this time; and the result was the overthrow of the Aristotelian doctrine that truth is exhausted by the domain of the nite (or at least the potential in nite, which is a species of the nite). The third crisis started around the turn of this century, with the discovery of apparently inconsistent entities (such as the Russell set and the Liar sentence) in the foundations of logic and set theory|or at least, with the realisation that such entities could not be regarded as mere curiosities. This provided a major|perhaps the major|impetus for the development of paraconsistent logic and mathematics (as far as it has got). And the philosophical result may be the overthrow of another Aristotelian doctrine: that truth is 182 It must be said that both stages have been pursued in the face of an attitude sometimes bordering on hostility from certain sections of the establishment logicophilosophical, though things are slowly changing.

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exhausted by the domain of the consistent.183 University of Queensland, Australia.

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183 I am very grateful to those who read the rst draft of this essay, and gave me valuable comments and suggestions: Diderik Batens, Ross Brady, Otavio Bueno, Newton da Costa, Chris Mortensen, Nicholas Rescher, Richard Sylvan, Koji Tanaka, Neil Tennant and Matthew Wilson.

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393

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INDEX

F ( ), 270 G( ), 270 HRM (i1; : : : ; in ), 261 P RM (i1; : : : ; in )--terms, 265 R! , 229 T!, 229 [Y=x]X , 241  , 268 !, modality in linear logic, 118 -normal form, 242 -redex, 242 -reduction, 241, 242 -normal form, 242 -reduction, 242 Æ, combination of information, 63 Æ, fusion, 12 -normal form, 243 -redex, 242 -reduction, 242 8-elimination, 14 8-introduction, 14

, see disjunctive syllogism I -calculus, 19 -depth, 270 -terms, 240 ! as , 326 , necessity, 13 ^ , relational converse, 115 dapn( ), 270 dcp( ), 270 dn( ), 270 do( ), 270 Lukasiewicz generalisations, 307 Lukasiewicz, J., 295, 306 abstraction, 240 accessibility relation, 138{141, 146, 190, 194, 200

accessible, 139, 146 Ackermann, W., 16 adjunction, 10 algebraic, 147, 190 algebraic BZL-realization, 199 algebraic logics, 336 algebraic realization, 137, 141, 143, 145, 147, 148, 174, 191, 194 algebraic realization for rst-order OL, 172 algebraic semantics, 137, 147, 148, 152, 153, 176, 189, 195, 198, 312 algebraically adequate, 145 algebraically complete realization, 162 Amemiya, I., 164 Amemiya{Halperin Theorem, 165 Amemiya{Halperin's Theorem, 166 Anderson, A. R., 1{5, 7, 9, 11, 14{ 17, 86, 88 Anderson, C. A., 295 angle bisecting condition, 171 application, 231 arithmetic, 366 Peano, 42 relevant, 41{44 Arruda, A., 295 Asenjo, F. G., 295 assertion, 8 atom, 167 atomic, 170 atomic types, 234 atomicity, 170 attribute, 172 Austin, J. L., 4

396

Avron, A., 18 axiom, 86 axiomatizaable logic, 158

B, the logic, 76{78, 82 B, 155 B-realizaation, 155 B-realization, 156, 157 B+ , the logic, 12, 75, 77 Barwise, J., 114 basic logic, 193, 213, 214, 216, 222 basic orthologic, 219, 222 Batens, D., 289, 349 Battilotti, G., 213, 214, 224, 226 Bell, J. L., 164, 165 Belnap extension lemma, 40 Belnap, N. D., 1{5, 7, 9, 11, 14{ 17, 86, 88, 100 Beltrametti, E., 135, 179 Bennett, M, K., 184 Birkho, G., 129{132, 134 Birkho, S., 179 Bo -realization, 157 Bohr, N., 134 Boolean algebra, 130, 148, 153, 187 Boolean negation, 384 Boolean-valued models, 177, 178 Born probabilty, 180 Born rule, 180 bound variables, 241 BQ, the logic, 82 bracket abstraction, 233 Brady, R. T., 100 branching counter machine, 94 Brouwer{Zadeh lattice, 183 Brouwer{Zadeh poset, 183 Brouwer{Zaden logics, 193 Bugajski, S., 179 Buridan, J., 294 Busch, P., 179 BZ3 -lattice, 202 BZ-frame, 195 BZ-lattice, 193, 195

INDEX

BZ-poset, 183 calculus for orthologic, 220 canonical model, 160, 162, 163, 170, 200, 208 canonical realization, 163 Cassinelli, G., 135 Cattaneo, G., 179, 183, 196 Cauchy sequences, 131 Chang, C. C., 185, 210 Chovanec, F., 184 Church, A., 7, 8, 16, 19 classical relevance logics, 7 Clinton, Bill, 118 closed subspace, 131{133, 135{137, 146, 176, 179, 203 closed term, 241 coincidence lemma, 174 collinearity, 102 combinators, 229 combinatory completeness, 233 combinatory logic, 232 compatibility, 148, 160 compatible, 148, 167, 170 complete, 154, 170 complete Boolean algebra, 177 complete BZ-lattice, 198 complete extension, 153 complete involutive bounded lattice, 182 complete ortholattice, 165, 166 complete orthomodular lattice, 163, 177 completeness, 175 completeness theorem, 160, 161, 175, 191, 196, 200, 208 completion by cuts, 166 conditional, 2, 354 counterfactual, 6 conditional connectives, 315 con guration, 158 con nement, 14 confusion about `Æ', 112

INDEX

about the logic B, 76 between kinds of consequence, 74 use{mention, 3 connexive logic, 32 consecution calculus, 86 consequence, 141, 172, 194, 195, 210 consequence in a realization, 138, 141, 174 conservative extension, 34 consistency, 159 consquence in a given realization, 205 constant domain, 173 context, 213, 221 continuous lattice, 60 contraction, 86, 87, 100, 218, 219 contradiction, 7 converse of a relation, 115 Correspondence Thesis, 4 Costa, N. C. A. da, 289 counterfactual conditional, 149 covering property, 170 covers, 170 cube, 217 cube of logics, 213, 218 Curry's Lemma, 88 Curry{Howard isomorphism, 237 cut rule, 47, 87 cut-elimination, 87, 213, 221, 222 cut-elimination theorem, 213 cut-formula, 221 Cutland, N., 213, 223 Da Costa's C -system, 318 da Costa's C1 , 305 da Costa's C! , 304 da Costa, N. C. A., 178 Dalla Chaiara, M. L., 204 Dalla Chiara, M. L., 155, 178, 184 Davies, E. B., 179 de Morgan monoid, 60

397

word problem, 101 decidability, 100, 157 decidability (of paraconsistent logic), 328 decidable, 196 deduction `OÆcial', 17 relevant, 15 deduction theorem, 8, 14, 15, 153, 161 modal, 17 modal relevant, 18 relevant, 16 deductive closure, 159 de nition of the canonical model, 175 demodaliser, 9 denial, 380 derivability, 159, 207 derivable, 159 derivation, 159, 207 description operator, 204 dialetheism, 291 Dickson's theorem, 92 dierence poset, 184 discursive implication, 317 Dishkant, H., 138 disjunctive syllogism ( ) admissibility in R, 36 disjunctive syllogism ( ), 1, 7, 31, 33, 47 admissibility RQ, 39 admissibility in R! , 41 admissibility in R]] , 43 admissibility in R, 34, 35, 39, 45 and Boolean negation, 82 in the metalanguage, 36 inadmissibility in R] , 43 the Lewis `Proof', 32 Disjunctive syllogism (DS), 294 display logic, 100 distribution

398

of conjunction over disjunction, 10, 29, 37, 66, 93, 95, 97, 118 distributive laws, 137 distributivity, 133, 153 domain, 172, 173 domain of individuals, 173 domains of certainty, 196 double negation, 12 Dummett, M., 213, 223 Dunn, J. M., 46, 48, 196 Dvurecenskij, A., 184 Dwyer, R., 10

E, the logic, 1, 2, 5, 7, 9, 11{ 13, 16, 19{21, 23, 25{27, 30, 31, 34{36, 39, 47{49, 75, 77, 80, 84{86, 92, 93, 100, 102, 108, 109, 113, 116, 117 E, the logic undecidability, 102 E! , the logic, 9{11, 17{19 Efde , the logic, 27, 54 eect, 190, 203 eect algebra, 184, 189, 205 eect algebras, 204 eects, 180, 181, 192, 209 Einstein, A., 134 elementary ( rst-order) property, 162 elementary class, 163 elementary substructure, 164 elimination of the cuts, 220 elimination theorem, 87 entailment, 2, 3, 151 non-transitive, 32 entailment in the algebraic semantics, 151 entailment-connective, 151 epistemic logics, 155 EQ, the logic, 13, 47, 48, 82 event-state systems, 135 events, 130

INDEX

ex contradictione quodlibet (ECQ), 288 excluded middle principle, 130 existential quanti er, 14 expansion, 13 experiemental proposition, 130 experimental proposition, 129, 131, 132, 136, 179 exponentials, 224

f , the false constant, 12 Faggian, C., 213, 226 FDE, 333 ltration, 297 Finch,P. D., 149 Fine, K., 66, 163 nite model property, 157, 196 Finkelstein, A., 135 rst-degree entailment, 60 formula signed, 100 formulas-as-types isomorphism, 237 Foulis, D. J., 135, 136, 179, 184 free variable, 241 full cut, 223 functional character, 229 fusion, 12 fuzzy, 180 fuzzy accessibilty relation, 194 fuzzy complement, 183, 193, 197 fuzzy intuitionistic logics, 193 fuzzy negation, 182, 196, 200 fuzzy-accessible, 194 fuzzy-intuitionistic semantics, 196 fuzzy-like negation, 193 fuzzy-orthogonal set, 194 Godel's Theorem, 375 generalisation, 14 generalized complement, 205 Gentzen system, 86, 88, 89, 92 Gentzen, G., 158, 220 Gibbins, P., 213, 223 Girard's linear logic, 193

INDEX

Girard's linear negation, 213 Girard's negation, 221 Girard, J.-Y., 118, 213, 219 Giuntini, R., 183{185, 189, 194, 202, 204 Goldblatt, R. H., 155, 158, 162, 163, 165 grammar, 3 Greechie diagram, 167 Greechie, R. J., 135, 166, 167, 182 Grice, H. P., 6 Gudder, S. P., 135, 182, 189 guinea pigs, 5

H, the logic, 7, 12, 77 H+, the logic, 12 H! , the logic, 10, 15, 17, 22 Haack, S., 6 Halperin, I., 164 Hardegree, G. H., 149 Hegel, G., 292 Henkin constructions, 302 Heraklitus, 292 Herbrand{Tarski, 153, 161 heredity, 64 heriditary right maximal terms, 261 Heyting algebra, 314 Hilbert lattice, 171 Hilbert quantum logic, 166 Hilbert space, 131, 135{137, 145, 150, 165, 170, 180, 190, 203, 209 Hilbert space lattice, 135 Hilbert systems, 7 Hilbert{Bernays, 158 Hilbert-space realizations, 190 Hilbertian space, 171 identity and function symbols (paraconsistent logic), 338 identity axiom, 86 implication, 2 implication-connective, 146{148, 150 import-export condition, 148

399

incompatible quantities, 133, 136 indistinguishability relation, 178 individual, 173 individual concept, 173, 176 inductive extension, 174 inference ticket, 9 in mum, 136, 137, 172, 182 in nitary conjunction, 197 in nitary disjunction, 198 information, 63 inhabitant, 234, 243 inner product, 131, 132, 146, 164 interpretation, 172, 177 intuitionistic accessibility relation, 194 intuitionistic complement, 183, 197 intuitionistic logic, 12, 139, 155, 213 intuitionistic logic (H), 7 intuitionistic orthogonal set, 195 intuitionistically-accessible, 194 involutive bounded lattice, 189, 191 involutive bounded poset, 183, 188 involutive bounded poset (lattice), 181 irreducibility, 170 irreducible, 170 Jauch, J., 135 Jaskowski, S., 295

K, the logic, 45{47 K^opka, F., 184 Kalmbach, G., 147, 166 KB, 191 Keating, P., 115 Keller, H. A., 171 Klaua, D., 196 Kleene condition, 181 Kleene, S. C., 306 KR, the logic, 102{104, 107, 108 KR, the logic undecidability, 102 Kraus, S., 179

400

Kripke, S., 86 Kripke-style semantics, 138 Kripkean models, 155 Kripkean realization, 139{143, 145, 149, 174, 194 Kripkean realization for ( rst-order) OL, 173 Kripkean realizations, 155 Kripkean semantics, 138, 139, 143, 147, 148, 150, 152, 156, 160, 190, 191, 194{197 Lahti, P., 179 lambda calculus, 229 lambda reductions, 247 lambda terms, 240 lattice, 182 de Morgan, 59 LC, the logic, 79 Leibniz' principle of indiscernibles, 179 Leibniz-substitutivity principle, 178 lemma of the canonical model, 161, 176, 201 Lemmon, E. J., 22 length of a lattice, 170 Lewis, C. I., 293 Lewis, D. K., 6 Lindenbaum, 153 Lindenbaum property, 154 line, 102 linear combinations, 131 linear logic, 46, 118, 222, 224 linear orthologic, 213 lnf, see long normal form logic deviant, 6 the One True, 15 logic R, 324 logical consequence, 138, 141, 143{ 145, 152, 156, 172, 174, 194, 195, 205, 210 logical theorem, 159

INDEX

logical truth, 138, 141, 143, 156, 172, 174, 194, 195, 205, 210 long, 270 long normal form, 246 LP, 333 LR, the logic, 93{95, 99 LR+ , the logic, 93 LRW, the logic, 99, 118 Ludwig, G., 179 Lukasiewicz axiom, 187 Lukasiewicz quantum logic, 209 Lukasiewicz' in nite many-valued logic, 193 Lukasiewicz' in nite-many-valued logic, 185 Mackey, G., 134 MacNeille completion, 163, 166, 182, 183, 190 Mangani, P., 186 many-valued conditionals, 319 many-valued logics, 332 Martin, C., 294 Martin, Errol, 10, 48 maximal Boolean subalgebras, 167 maximal lter, 153 McLaren, 166 metavaluation, 37 metrically complete, 131 metrically complkete, 165 Meyer, R. K., 10, 46, 48, 86, 101 Minari, P., 144 mingle, 13 minimal quantum logic, 136 minimally inconsistent, 349 minimally inconsistnet, 349 Mittelstaedt, 179 Mittelstaedt, P., 135, 149 mix rule, 87 mixtures, 135 modal deduction theorem, 17 modal interpretation, 155, 157 modal operators, 339

INDEX

modal relevant deduction theorem, 18 modal translation, 156, 191 modality, 2 model, 138, 141, 152 modular lattice, 105 modus ponens, 20, 32 modus ponens, 15{17, 19, 31, 37{ 39 for the material conditional, 47 Moh Shaw-Kwei, 8, 16 monotonicity lemma, 352 Morash, R.P., 171 Mortensen, C., 288 multisets, 214, 215 MV-algebra, 185{188, 210, 211 natural deduction, 21, 158 necessity, 2, 11, 13 necessity operator, 198 negation, 7, 12, 378 Boolean, 12, 81, 82, 102 minimal, 12 negative domain, 197, 203 Nicholas of Cusa, 292 Nishimura, H., 213 Nistico, G., 183, 196 non-adjunction, 299 non-adjunctive logics, 330 normal form, 239 normal worlds, 322 NR, the natural deduction system, 21 null space, 133 observable, 146 `OÆcial deduction', 17 Once (j1 ; : : : ; jk ), 255 Once+(j1 ; : : : ; jk ), 256 Once (j1 ; : : : ; jk ), 255 One True Logic, 15 operational, 214 operational rule, 86

401

Organon, 293 Orlov, 295 Orlov, I., 7 orothvalued universe, 177 ortho-pair realization, 197, 203 ortho-pair semantics, 199 ortho-valued (set-theoretical)universe V , 177 ortho-valued models, 177 orthoalgebra, 184, 185, 204, 205 orthoarguesian law, 167 orthocomplement, 132, 137, 139, 213 orthocomplemented orthomodular lattice, 133 orthoframe, 139, 140, 150, 155, 156, 165, 166, 194, 197, 198, 201 orthogonal complement, 132 ortholattice, 137, 142, 147, 153, 162, 164, 166, 182, 191, 197 orthologic, 136, 189, 205, 213, 219, 221, 223, 224 orthomodular, 143, 144, 148, 165, 174 orthomodular canonical model, 162 orthomodular lattice, 142, 143, 147, 148, 153, 154, 162, 167, 176, 178 orthomodular poset, 181, 204, 205 orthomodular property, 191 orthomodular quantum logic, 136 orthomodular realization, 157, 161 orthomodularity, 162 orthopair semantics, 199 orthopairproposition, 197, 198, 200, 201, 203 orthopairpropositional conjunction, 197 orthopairpropositional disjunction, 197 othoposet, 181 Oxford English Dictionary, 5

402

P W, the logic, 10 PA, Peano arithmetic, 42 PA+ , positive Peano arithmetic, 42 paraconsistent modal operators, 342 paraconsistent quantum logic, 189, 219, 220 paraconsistent set theory, 363 permutation, 7, 86 total, 102 Perry, J., 114 phase-space, 129{131 Philo-law, 147 physical event, 179 physical property, 179 physical qualities, 130 Piron, C., 135 Piron{McLaren's coordinationization theorem, 170 Piron{McLaren's Theorem, 171 point, 102 polarities, 59 polynomial conditionals, 147, 150 positive conditionals, 146 positive domain, 197, 203 positive laws, 149 positive logic, 139, 146, 153 positive paradox, 7, 10, 22 positive-plus logics, 331 possibility operator, 198 possible worlds, 138 Powers, L., 10 Pratt, V., 222 Prawitz, D., 66 pre-Hilbert space, 164 predicate-concept, 173 pre xing, 7 premisses, 214 Priest, G., 349 principal formula, 215, 221 principal type scheme (PTS), 235 principle quasi-ideals, 141 prinicpal ideal, 182 Prior Analytics, 293

INDEX

probabiity measure, 135 probability, 344 projection, 133, 150, 179, 180, 193 projective space, 102 projetion, 183 proof, 15 proof reductions, 247 proper lter, 153 proposition, 138, 139, 141, 144, 146, 149, 151, 164, 165, 179, 195 propositional quanti cation, 11 pseudo canonical realization, 163 pseudocomplemented lattice, 147, 148 Pulmannova, S., 184 pure state, 130, 132, 133, 145, 146, 150, 190 pure states, 129, 131, 146, 203

Q-combinators, 237 Q-combinatory logic, 238 Q-de nable, 252 Q-logic, 238 Q-terms, 237 Q+ , the logic, 102 Q-translation algorithm, 249 QMV-algebra, 185, 187, 188, 209, 210 quanti cation propositional, 11 quanti er, 14 quanti ers (in paraconsistent logic), 329 quantum events, 132 quantum logical approach, 135, 171 quantum logical implication, 158 quantum MV-algebra, 185 quantum proposition, 157 quantum set theory, 178 quantum-logical natural numbers, 177 quantum-sets, 178 quasets, 178

INDEX

quasi-consequence, 152 quasi-ideal, 145, 147, 165 quasi-linear, 188 quasi-linear QMV-algebra, 189, 209 quasi-model, 152, 154 quasisets, 178 Quesada, M., 288

R], the arithmetic, 41{44 R]], the arithmetic, 41, 43, 44 R, the logic, 2, 5, 7, 9{14, 17{21, 23, 25{27, 30, 31, 34{39, 41, 42, 49, 54, 55, 57, 58, 60{63, 65{82, 84{89, 92, 93, 95{102, 104, 108, 109, 111{114, 116{118 R, the logic undecidability, 102 R+, the logic, 10{12 R, the logic, 13 R! , the logic decidability, 86 R! , the logic, 7{10, 16{22, 64, 65, 86, 87 R:! , the logic, 45 R!;^, the logic complexity, 93 Rfde, the logic, 27{29, 54, 59 R!;^;_, the logic, 10 R!;^, the logic, 93, 94 R-theory, 36 RA, the logic, 116, 117 Ramsey-test, 290 Randall, C., 135, 136, 179 rational acceptability of contradictions, 382 realizability, 152 realizable, 154 realization, 209 realization for, 205 realization for a strict-implication language, 151 reassurance, 349 reductio, 12

403

reduction, 231 reference, 173, 175 re ection, 215 regular, 182, 191 regular involutive bounded poset, 183 regular paraconsistent quantum logic, 191, 195 regular symmetric frame, 194 regularity, 181 regularity property, 190 relation binary, 80 ternary, 68, 78, 80, 102 relation algebra, 115 relational valuations, 308 relevance, 2, 6 relevant arithmetic, 41{44 relevant arrows, 321 relevant deduction, 15 relevant implication, 2 relevant logics, 193, 335 relevant predication, 118 representation theorem, 135, 165 residuation, 12 resource consciousness, 18 Restall, G., 100 restricted-assertion, 8, 9 rigid, 173 rigid individual concepts, 175 RM, the logic, 7, 13, 18, 78{80, 101 RM3, the logic, 95 RMO!, the logic, 18 Routley interpretation, 310 Routley, R., 63, 292 Routley, V., 292 RQ, the logic, 13, 14, 39{42, 70, 82, 83 rule, 158, 195, 199, 205, 214, 215

S5 , 196, 202 S4, 155 S! , the logic, 10

404

S4, the logic, 9, 13, 17, 21, 27, 30, 77, 79 S5, the logic, 48, 78, 79 Sambin, G., 213, 214 Sasaki projection, 167 Sasaki-hook, 149 satis cation, 173 Schutte, K., 46, 47 Schrodinger, E., 178 Scott, D., 60 Scotus, 294 second-order (paraconsistent) logic, 338 secondary formula, 215 self-distribution, 8 permuted, 9 self-implication, 7 semantics, 48 algebraic, 49 operational, 63 Routley{Meyer, 6, 57, 118 semi-lattice, 64, 66 set-theoretical, 49, 55, 63 semantics and set theory, 350 semi-transparent eect, 180, 190 separable, 131 separable Hilbert space, 165 separation theorem, 86 sequent, 46 contraction of, 87 right handed, 46 sequent calculus, 158, 213, 214 sequents, 214 set theory in LP , 358 set-theory, 177 set-up, 57, 68 sharp, 179, 180, 190 sharp property, 193, 203 SheÆeld Shield, The, 114 -complete orthomodular lattice, 135, 170 signed formula, 100 siilarity logics, 155 situation theorey, 114

INDEX

Slomson, A. B., 164, 165 Socrates, 118 Soler, M. P., 171 soundness, 175, 196, 208 soundness theorem, 160, 161, 199, 200 Stalnaker, , 149 Stalnaker, R., 6 Stalnaker-function, 149 standard orthomodular Kripkean realization, 162 standard quantum logic, 136 standard realization, 162 statistical operator, 180, 181 statistical operators, 135 Stone theorem, 153 strict implication, 150, 151 strict-implication operation, 150 strong Brouwer{Zadeh logic, 193 strong partial quantum logic, 204, 205 structural, 214 structural rule, 86, 216, 217, 223 subspaces, 154, 166 substructural logics, 1, 223, 229 substructure, 164 suÆxing, 8 Sugihara countermodel, 307 Sugihara generalisation, 307 Sugihara Matrix, 101 summetry, 215 superposition principle, 131 Suppes, P., 135 supremum, 133, 136, 137, 172, 182, 207 Sylvan, R., 293 symmetric frame, 190, 191 symmetry, 221 syntactical compactness, 159 syntactical compatibility, 159 syntactically compatible, 175

T , the true constant, 12

INDEX

T, the logic, 11, 19, 77, 78, 92, 108, 109 t, the true constant, 11 t, the logic, 77 T W, the logic, 10 T+ , the logic, 11 T! , the logic, 9{11 Takeuti, G., 177 Tamura, S., 213 Tennant, N., 32 ternary relation, see relation, ternary the classical recapture, 347 the limitative theorems of metamathematics, 373 theorems, 37 theorey of descriptions, 176 theories of quasisets, 178 theory, 35 thinning, 86 Thomason, R., 6 title, 1 told values, 60 Toraldo di Francia, G., 178 total space, 133 TQ, the logic, 82 transfer rule, 219 translations, 229 true, 177 truncated sum, 186 truth, 138, 141, 143, 156, 172, 174, 194, 195, 205 TW, the logic, 77, 78, 106{108 TW+ , the logic, 12, 77 TW! , the logic, 10 TWQ, the logic, 82 type, 229 type assignment, 234, 243 uncertainty principles, 133 undecidability, 102 unitary vector, 131, 145 universal quanti er, 14 unsharp, 190 unsharp approach, 180

405

unsharp orthoalgebra, 184 unsharp partial quantum logic, 204 unsharp physical properties, 183 unsharp property, 193 Urbas, I., 289 urelements, 178 Urquhart, A., 63, 93, 102 vacuous quanti cation, 14 validity, 64 valuation, 173 valuation-function, 137, 172 Varadarajan, V. S., 135, 164, 170 veri ability, 152, 153 veri able, 154 veri cation, 173 visibility, 213, 215, 221 von Neumann's collapse of the wave function, 150 von Neumann, J., 129{132, 134, 179

W, the logic, 10, 99 wave functions, 131 Way Down Lemma, 35 Way Up Lemma, 35 weak Brouwer{Zadeh logic, 193 weak consequence, 152 weak equality, 232 weak implication calculus, 7 weak import-export, 149 weak Lindenbaum theorem, 160, 161, 174 weak non monotonic behaviour, 150 weak partial quantum logic, 204, 205, 207 weakening, 86, 218, 219 weaker sets of combinators, 229 weakly linear, 188 Wittgenstein, L., 287, 292 Wolf, R. G., 1 world valuation, 173 X, the logic, 14, 15, 77

406

Zermelo{Fraenkel, 178 zero degree entailment, 309

INDEX