Logica Paraconsistente

ACKNOWLEDGEMENTS v

Acknowledgements

This book owes a great deal to a great many people. So many indeedthat we have probably forgotten some of them and to those we apologizein advance.First, since the work reported in this volume sprang from some initialwork of Ray Jennings and Peter Schotch, the Social Science and Hu-manities Research Council of Canada deserves a particularly hearty voteof thanks. The Council made that collaboration possible under a numberof operating grants over those first crucial years. Jennings and Schotchhad a chance not only to meet face to face, but also to make their workknown to a larger audience by giving talks and attending conferencesover much of the learned world. Without that support, it is hard to seeany of this having happened.Second, and nearly tied for first, we must thank our students over theyears. They have been many but we owe them a great debt not only fortheir enthusiasm and hard work, but also because many of them wereresponsible for proving things that needed proving. At that head of thatlist is Bryson Brown, a one-time research assistant who rose to becomea famous philosopher. Along the way, he with Peter Apostoli provedthe first representation theorem and many graph theoretic results whichdo not appear in this volume.1

Apart from Bryson the names Blaine D’Entremont, [fill in] were amongthe early adopters of our ideas and each has made a signal contribution.The latest generation includes Dorian Nicholson, Darko Sarenac andGillman Payette. Two of those have contributed to this volume and allhave contributed to the project in ways too valuable to describe in aparagraph.It also goes without saying that we owe an enormous debt of gratitudeto our wives and families who have had to put up with us while wesearched for a proof or even just a way of explaining something thatseemed clear to us alone.

1But which will appear in a projected companion volume focused on applications of thepreservationist approach.

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . v

1 Introduction to the EssaysPETER SCHOTCH 31.1 The Origins of Preservationism . . . . . . . . . . . . . 31.2 Paraconsistency and Modal Semantics . . . . . . . . . 71.3 The Concept of Level . . . . . . . . . . . . . . . . . . 101.4 Preservation . . . . . . . . . . . . . . . . . . . . . . . 111.5 The Essays . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Paraconsistency: Who Needs It?RAY JENNINGS AND PETER SCHOTCH 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 152.2 The Strange Case of C. I. Lewis . . . . . . . . . . . . . 182.3 The Inconsistency of Belief . . . . . . . . . . . . . . . 212.4 Inconsistency and Ethics . . . . . . . . . . . . . . . . . 262.5 Summing Up . . . . . . . . . . . . . . . . . . . . . . . 28

3 Weakly Additive Algebras and a Completeness ProblemALASDAIR URQUHART 313.1 Introduction and Brief History . . . . . . . . . . . . . . 313.2 Weakly additive operators . . . . . . . . . . . . . . . . 353.3 A Further Generalization . . . . . . . . . . . . . . . . 383.4 Duality theory for weakly additive operators . . . . . . 39

vi

vii

4 A Dualization of Neighborhood StructuresDORIAN NICHOLSON 474.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 474.2 Hyperframes . . . . . . . . . . . . . . . . . . . . . . . 484.3 Normal Hyperframes . . . . . . . . . . . . . . . . . . . 524.4 Weakly Aggregative Modal Logic . . . . . . . . . . . . 55

5 Polyadic Modal Logics and Their Monadic FragmentsKAM SING LEUNG AND R.E. JENNINGS 595.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 595.2 Polyadic modal languages and multi-ary relational frames 625.3 Normal polyadic modal logics . . . . . . . . . . . . . . 655.4 Frame definability . . . . . . . . . . . . . . . . . . . . 715.5 Soundness and completeness . . . . . . . . . . . . . . 735.6 Diagonal fragments of normal polyadic modal logics . . 78

6 Preserving What?GILLMAN PAYETTE AND PETER SCHOTCH 816.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 816.2 Making A Few Things Precise . . . . . . . . . . . . . . 846.3 What’s Wrong With This Picture? . . . . . . . . . . . . 886.4 Speak of the Level . . . . . . . . . . . . . . . . . . . . 896.5 Level Preservation . . . . . . . . . . . . . . . . . . . . 926.6 Yes, But Is It Inference? . . . . . . . . . . . . . . . . . 976.7 Other Level-preserving Relations . . . . . . . . . . . . 99

7 Preserving Logical StructureGILLMAN PAYETTE 1017.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1017.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . 1047.3 Substitutions . . . . . . . . . . . . . . . . . . . . . . . 1137.4 Algebraic Concerns . . . . . . . . . . . . . . . . . . . 1177.5 Denials . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.6 Forcing and Preservationism . . . . . . . . . . . . . . . 1237.7 Structural Rules and Compactness . . . . . . . . . . . 1247.8 First-Order Logic and Forcing . . . . . . . . . . . . . . 128

viii CONTENTS

7.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 137

8 Representation of ForcingDORIAN NICHOLSON AND BRYSON BROWN 1398.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1398.2 Background . . . . . . . . . . . . . . . . . . . . . . . 1408.3 Representation . . . . . . . . . . . . . . . . . . . . . . 1428.4 More Definitions and motivation . . . . . . . . . . . . 1468.5 �

�`X � . . . . . . . . . . . . . . . . . . . . . . . . . 1518.6 �

� X � . . . . . . . . . . . . . . . . . . . . . . . . . 152

8.7 The Representation of Forcing . . . . . . . . . . . . . . 152

9 Forcing and Practical InferencePETER SCHOTCH 1559.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1559.2 Dirty Hands . . . . . . . . . . . . . . . . . . . . . . . 1579.3 †-Forcing . . . . . . . . . . . . . . . . . . . . . . . . 1589.4 A-Forcing . . . . . . . . . . . . . . . . . . . . . . . . 1649.5 Wrap-up . . . . . . . . . . . . . . . . . . . . . . . . . 166

10 Ambiguity Games and Preserving Ambiguity MeasuresBRYSON BROWN 16710.1 Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . 16810.2 The Logic of Paradox . . . . . . . . . . . . . . . . . . 16910.3 Restoring Symmetry . . . . . . . . . . . . . . . . . . . 17210.4 Another Approach to FDE . . . . . . . . . . . . . . . . 17410.5 Ambiguity and quantification . . . . . . . . . . . . . . 17910.6 Echoes of supervaluation . . . . . . . . . . . . . . . . 17910.7 Final Remarks on Preservation . . . . . . . . . . . . . 180

Nomenclature 183

References 189

Index 193Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . 197

ON PRESERVING

One

Introduction to the Essays

PETER SCHOTCH

The essays in this volume are intended to serve as an introduction to aresearch project that started a long time ago, back in the 20th Century.But it didn’t start as a project about paraconsistent logic or about whatwe have come to call preservationism.

1.1 The Origins of Preservationism

Instead it started in the summer of 1975, at Dalhousie University, whenRay Jennings1 and Peter Schotch decided to write a ‘primer’ of modallogic—a project that never quite made it into the dark of print. So it wasmodal and decidedly not paraconsistent logic which was uppermost inthe minds of Jennings and Schotch. In this connection one should keepin mind that neither had so much as heard the word ‘paraconsistent’ atthe time we are considering.2

1Jennings was then (and now) a member of the philosophy department at Simon Fraser Uni-versity, but he had accepted a summer-school job at Dalhousie in order to work with Schotch.

2In fact the first time they heard that word was at the 1978 meeting of the Society for ExactPhilosophy, held in Pittsburgh. And in all truth, they paid little enough attention to the word thefirst time they did hear it.

3

4 CHAPTER 1. INTRODUCTION TO THE ESSAYS

What got the ball rolling, was the discovery by Jennings, which hecommunicated to Schotch, that so-called normal modal logic was notsuitable for certain philosophical applications of a deontic cast. In par-ticular, there seemed to be distinctions which one might draw intuitivelythat evaporate in normal modal logic.3 Now the only restriction on thekind of modal logic involved was that it be normal, so enabling distinc-tions which cannot be made in that logic was not going to be a simplematter of dropping some modal axioms or, on the semantic side, relax-ing a frame condition.

If we are talking about normal modal logic in general, then thereare no frame conditions to relax. The fundamental normal modal logic,often called K, is determined by the class of all frames. Period.

But later that afternoon, it started to seem that perhaps the semanticsof normal modal logic did impose some restrictions after all. To beginwith, there was the restriction that the universe of the frame, its set of‘possible worlds,’ must be non-empty. That didn’t seem a very promis-ing condition to relax, though there certainly have been those who wereprepared to give it up, if only to see what happens to modal logic.4 No,it would have to be a condition on the frame relation which did the trick.Which condition was it? There can be only one, and that is the conditionthat the frame relation be binary. And there it was.

Jennings and Schotch then began trying to work out what the truth-condition for necessity would have to be when the frame relation wasnot binary but say, ternary. Later that evening they had narrowed it downto only one possibility:

�˛ is true at a point x in a model M if and only if for everyordered pair hy; zi such that Rxyz, either ˛ is true at y or ˛is true at z.

The generalization to n-ary frame relations is trivial.3The particular example mooted by Jennings had to do with two modal formulas, one of which

had a necessity operator outside of a conditional and the other having the necessity distributed overthe conditional. Perhaps the easiest version of this kind of distinction, and the one which becameabsolutely central to the program, is the one between�˛ � :�:˛ and:�?. The latter claimsthat no contradiction can be necessary while the former asserts that if any formula is necessary, itsnegation is not.

4Charles Morgan has constructed this kind of modal logic.

1.1. THE ORIGINS OF PRESERVATIONISM 5

In the hot flush of discovery, Jennings and Schotch imagined thatthey were treading on terra incognita, but as Goethe reminds us,

All the great thoughts have already been thought. Our job ismerely to think them again.

n-ary frames had already been introduced by Bjarni Jonsson and Al-fred Tarski (Tarski and Jonsson, 1951). In that work n-ary frames area way of representing Boolean Algebras with n-1-ary operators. Modallogic following Kripke had rediscovered (a decade after the original dis-covery) the special case of this idea, for binary frames.

The problem for Jennings and Schotch was how to axiomatize a frag-ment of the Tarski and Jonsson n-ary frames, the diagonal fragment. Inother words, in the original work the extra operator is (n-1)-ary, let’scall it �. So for every n-1-tuple of formulas ˛1; : : : ; ˛n�1 the ‘modal’formula is �.˛1; : : : ; ˛n�1/. This notion, as interesting and useful asit no doubt is, isn’t the modal notion selected by Jennings and Schotch.Theirs, the unary �˛, is represented in Tarski and Jonsson’s terms as�.˛; : : : ; ˛/ where the . . . represents n-3 iterations of the formula ˛.

This was not a problem which had ever been considered before soJennings and Schotch were happily refuting Goethe, or at least attempt-ing to do that. The thing is: Using the resources of normal modal logic,the problem is extremely difficult.

If we stick to the ternary case for ease of exposition, one quicklyfinds that the principle of complete modal aggregation

[K] ` �˛ ^�ˇ � �.˛ ^ ˇ/

doesn’t hold, although the rule of monotonicity (sometimes calledregularity) and the rule of (unrestricted) neccessitation (also called therule of normality):

[RM] ` ˛ � ˇ H) ` �˛ � �ˇ[RN] ` ˛ H) ` �˛

both hold in ternary frames (and indeed in n-ary frames).


So now comes the problem: How do you axiomatize this semantics?It turns out to be quite a trick without [K]. In fact, it turns out to beimpossible without some kind of aggregation principle, as Jennings andSchotch realized early on in the project. The correct version of aggre-gation seemed to be the principle:

[K3] ` .�˛1 ^�˛2 ^�˛3/ � �..˛1 ^ ˛2/_ .˛2 ^ ˛3/_ .˛1 ^ ˛3//

But how to prove this conjecture? The situation is entirely differentin K, where all the instances of the principle of complete modal aggre-gation that you might need follow transparently from [K]. In the generalcase, one needs to show that every aggregation principle which holds invirtue of ‘pigeonhole reasoning’5, (the way [K3] does) follows from[K3]. That is a non-trivial piece of combinatorial work.

While pondering this question and writing a few papers on modalaggregation (none of which solved the problem), Jennings and Schotchchanced to attend the meeting of the Society for Exact Philosophy inPittsburgh which was held in June of 1978. It was there that they firstheard of paraconsistent logic via a paper on the subject presented byRobert Wolf. It was also there that they first saw normal modal logicaxiomatized by the rule:

[N] � ` ˛ H) �Œ�� ` �˛

(where�Œ�� stands for f� j 2 �g) when it was used in a paperpresented by Brian Chellas.6

In the question period following the presentation by Jennings andSchotch, Barbara Partee happened to remark that it all seemed to her likean attempt to represent a kind of non-trivial reasoning from inconsistentdata, since, without complete modal aggregation one cannot construct�? from �˛ and �:˛. So in particular, if one thought of the � assome sort of belief operator then . . . . Jennings and Schotch took note of

5By this is meant merely the principle that if there are k objects to distribute over k � 1containers, at least one container must contain at least two objects

6Chellas later attributed the rule to Dana Scott.

1.2. PARACONSISTENCY AND MODAL SEMANTICS 7

how the audience perked up at this, and kindly thanked Professor Parteefor the observation.

It was on the flight back to Halifax7 that Jennings urged Schotchto consider the Chellas axiomatization in connection with their n-arymodal logic. What Jennings had seen is that the rule [N] characterizesnecessity in terms of some notion of inference.8 For the necessity ofnormal modal logic in the usual sense, the ‘corresponding’9 inferencerelation is simply the classical one. But Jennings presumed, since the n-ary semantics lacked complete modal aggregation, the inference relationthat characterized the new notion of necessity cannot be classical.

This turned out to be an absolutely crucial insight, and one which ledultimately to the program now called preservationism. Schotch begantrying to discover an inference relation which bears the same relation ton-ary modal logic as the classical relation bears to normal modal logic.There will of course be infinitely many such relations corresponding tothe different orders of the frame relations. The project didn’t take longsince it was possible to, in effect, read off the relation from the modalsemantics.

1.2 How Parconsistent Inference Fell Out of ModalSemantics

The trick is to begin as though you already have the relation in questionand then start the ‘hard’ direction of the fundamental theorem for all,let’s say, ternary modal logics. These will be all the extensions of thebase logic K3 which logic is axiomatized by the single rule (on top ofclassical sentence logic):

[N3] � B3 ˛ H) �Œ�� ` �˛In order to discover just what propertiesB3 has to have, we see what

will be needed to prove:

7Where Jennings was teaching summer school again.8An algebraist would say that the two concepts inference and necessity, are Galois connected.9Depending upon how precise one wishes to make the notion of correspondence used here,

one might be restricted to the necessity operator of the logic K, rather than the necessity operatorof any normal modal logic.


�˛ 62 † H) hMK3; †i ² �˛Where MK3 is the K3 canonical model and † is an element of theK3-canonical domain, which is to say a maximal K3-consistent set offormulas. The converse of this is trivial so long as we define theK3-canonical relation, RK3 over the elements of K3-canonicaldomain, in the obvious way as:RK3��1�2 ” .8˛/.�˛ 2 � H) ˛ 2 �1 or ˛ 2 �2/

So assume �˛ 62 †. We need to find a pair of maximally K3-consistent sets to which † is related, neither of which contains ˛. Theexistence of such a pair amounts to the failure of�˛ at the element† ofthe K3-canonical model, which is what we are trying to show. To showthat a pair of maximally K3-consistent sets exists to which † stands inthe canonical relation, it suffices to show that there is some partition ofthe set �.†/ D fˇj�ˇ 2 †g both cells of which are K3-consistent. Insuch a case each cell can be extended to maximal K3-consistency in theusual way. We also require that ˛ not belong to either of the maximallyK3-consistent sets thus constructed. The only way make sure of this isto add :˛ to each cell of the partition. From this point we argue byreductio:

Suppose that no such partition can be found, which is to say thatfor every partition of �.†/ into two K3-consistent cells, at least oneof the cells is not K3-consistent with :˛. By classical reasoning, thisamounts to saying that at least one cell of every partition of �.†/ intotwo K3-consistent sets (classically) proves ˛. If our axiomatization ofK3 is going to work, what we just wrote must amount to a statementthat �.†/ B3 ˛. This is because if and only if the inference relation isdefined in that particular way then, by appeal to the rule [N3], we musthave:

�Œ�.†/� ` �˛.

But the set on the left of ` is a subset of † since it collects thoseformulas in † which are ‘guarded’ by �. More precisely, the first (in-ner) operation forms the set of those formulas in† which begin with�and strips off the leading � while the second (outer) operation prefixes

1.2. PARACONSISTENCY AND MODAL SEMANTICS 9

each of those formulas with �. This operation leaves us with exactlythose formulas in † which have an initial �. This subset will usuallybe proper, since an arbitrary set like† will normally contain lots of for-mulas ˛ which do not begin with � such that �˛ does not belong to†.

Now if a subset of † proves �˛ then † itself must likewise provethat formula by monotonicity of inference.10 But, by classical reason-ing, every maximally K3-consistent set is a theory, which is to say thatit contains every formula that it proves. So �˛ 2 †, which contradictsthe hypothesis. Thus K3 is determined by the class of all ternary frames.

In order for B3 to be a paraconsistent inference relation, it mustbe an inference relation. There is cheerful news on that matter: Therelation satisfies all the classical structural rules. So yes, it is inferenceof some kind, but is it really paraconsistent?

Well, yes and no. It is at least a partially paraconsistent relation,since, if the premises set contains only two contradictory premises with-out either being self-contradictory, then the closure of the set under therelationB3 will not contain every formula.

On the other hand the premise set might contain a subset like:

f˛1;:˛1 ^ ˛2;:˛1 ^ :˛2g

for which any individual member is consistent by itself but each is in-consistent with the other two. Any set containing a subset like this can-not be partitioned into only two consistent cells. So, contrary to thespirit of paraconsistency perhaps, all such sets prove (in the sense ofB3 every formula.

SoB3 allows sets to explode inferentially even though they may berepresentable as the union of sets all of which are consistent. Elsewherethis condition is vilified by branding it as a kind of inconsistency.11

Now, of course we can move to a relation defined in terms of parti-tions containing three cells. In that situation there will be an inferencerelation, call it B4 which does what we want but it will, in turn, fall

10This is, in effect, an appeal to the classical structural rule of dilution.11In (Schotch and Jennings, 1989) inference relations which allow sets to explode without their

containing an absurd formulas are called inconsistent*.


prey to sets of 4 formulas each consistent in isolation but explosive incombination with any of the others.12

Are we to be driven to infinity here? No, but before we see whynot, let us come clean. There are premise sets such that no partition intoconsistent cells (even allowing infinite partitions) is possible. These arethe sets that contain formulas which are, by themselves, inconsistent.Such formulas are often called absurd, with ˛^:˛ serving as the mostpopular classical example.

Those inconsistent sets, the ones containing absurd formulas, can-not be fixed. No matter which version of our new relation we take,the consequences of such a a set under that relation will be the set ofall formulas. If the goal of paraconsistent logic is prevent even absurdformulas from exploding, then our relations are, none of them, paracon-sistent. Fortunately, neither Jennings nor Schotch (nor the angels, weare tempted to say) believe that paraconsistent logic has that goal.

Thus was a great gulf fixed between the preservationist approach,and the ‘dialethic’ approach championed by several people in the South-ern Hemisphere.13 It is characteristic of the latter view that both a sen-tence and its negation might be true, and that there could be a way ofmaking ˛^:˛ true while at the same time making ˇ false. Hence, evenabsurd formulas might be redeemed.

It is tempting to refer to this program as ‘non-negative’ on the groundthat whatever else may be going on, if you succeed in making both ˛ and:˛ true, then you must be using the symbol : to represent somethingother than negation. This temptation is reinforced by the fact that thedialethists insist on referring to the preservationist program as ‘non-adjunctive.’

1.3 The Concept of Level

In the previous section it seemed as though there was no limit upon howlarge we might have to make a partition in order to ensure that everypremise set (not containing an absurdity) could be partitioned into that

12B4 and the earlierB3 are examples of what we call fixed-level forcing relations.13The name Graham Priest springs to mind.

1.4. PRESERVATION 11

many consistent cells. That is no doubt true but not entirely to the point.What we clearly need to do is to treat every premise set on its ownmerits, rather than assuming that there is some fixed number n such thatthe premises can be partitioned into no more than n consistent cells.

Rather, let us say that the level of a set, say � of formulas, modulosome technicalities which will be explained in detail below, is the leastnumber of cells (if there is such a thing) such that � can be partitionedinto that many consistent cells. If there is no such number, which is tosay if � contains an absurd formula, then assign the symbolic value1to be the level.

And now we have what we might regard as a whole-hearted paracon-sistent inference relation. We refer to this relation as the forcing relationindicated by

� and it is defined not in reference to some fixed number

of cells, but to partitions into the number of consistent cells equal to thelevel of the premise set. If, on every such partition, at least one of thecells proves ˛, then �

� ˛.

1.4 Preservation

At this point the modal logical motivation has fallen away. In the firstplace, the version of the generalized form of the rule [N], which hasforcing on the left of H) , doesn’t correspond to any particular n-arymodal semantics, but rather to a semantics in which the frame relationis allowed to vary. This is a bit of an odd duck, modally speaking. Inthe second place, the problem which started the whole thing off, that ofshowing that the n-ary semantics can be axiomatized using [RN], [RM],and the appropriate n-ary form of the aggregation principle [KN] is stillunsolved!14

So where does preserving come in? During several presentationsthat Jennings and Schotch made in the early 1980’s more than once thecomplaint was heard that the notion of inference based on forcing andits near relations seemed at odds with the ‘truth-preservation’ account.15

14As a matter of historical fact, several years had to pass before that problem was finally solved.See chapter 4 in this Volume.

15This paradigm is considered at much greater length in the essay ‘Preserving What? (chapter6) in this volume.


When this complaint was considered outside of heated debate, it seemedto come apart in one’s hands.

To say that forcing (or any of its fixed-level modal cousins) does notaccord with truth-preservation is to say that some inferences licensedby the latter are not licensed by the former. This is without doubt true,but those inferences are precisely the ones that forcing is designed toblock. In other words the complaint seems to come down to the factthat forcing doesn’t behave as stupidly as classical inference in the faceof (many) inconsistent premise sets. To which one might well reply‘And . . . ?’

But since merely preserving truth opens the door to a lot of infer-ences we’d rather not draw, perhaps forcing & Co. can still be thoughtof as preserving something worthwhile. And it is relatively easy to seethat forcing preserves level, which includes preserving truth when thepremises taken all together are capable of being true.16 So level preser-vation can be thought of as a way of fixing truth-preservation at theprecise place where it is broken—when there is no truth to preserve. Inthat case, forcing finds something else, or rather something in addition,that can still be preserved even while truth is on vacation.

Now the dialethists can and do argue that they are the ones who re-ally fix truth-preservation, since their notion of paraconsistent reasoningpreserves truth and not something other or over and above, like level.Unfortunately this claim stands or falls with the claim that both a sen-tence and its negation can be true on the same occasion of evaluation.This is a major sticking point, even a deal breaker, for many of thoseoutside the dialethic school. Not to put too fine a point on the matter,an uncharitable person might remark that dialethism, rather than fixingtruth-preservation, breaks it beyond any hope of repair.

1.5 The Essays

In the essay ‘Paraconsistency: Who Needs It?’ Ray Jennings and Pe-ter Schotch present an informal account of the core motivating exam-

16Which is of course to say that the set of premises is consistent. This is taken up in muchmore detail in ’On Preserving’ chapter 6in this Volume.

1.5. THE ESSAYS 13

ples that lead to a preservationist approach to paraconsistency. Theytake up the usual examples of belief and obligation and also a not sousual one in recasting the motivation of C.I. Lewis for his notion ofstrict-implication. In effect, they suggest that Lewis was a pioneer ofparaconsistency.

In the essay ‘Weakly Additive Algebras and a Completeness Prob-lem’ Alasdair Urquhart presents a solution to the problem of showingthat theKn logics are complete with respect to their intended semantics.This is the problem mentioned above that had to wait several years forits solution. At the same time, Urquhart presents his result as an ex-tension of the original algebraic approach of Tarski and Jonsson, whichapproach in effect ‘started the whole thing off.’

In the essay ‘Polyadic Modal Logics and Their Monadic Fragments’,Kam Sing Leung and Ray Jennings present the long awaited treatmentof n-ary modal logics, and in a particularly elegant way.

In ‘Preserving What?’ Gillman Payette and Peter Schotch exam-ine the somewhat shaky status of the truth-preservation paradigm forright reason and suggest alternatives. They treat the concepts of leveland forcing with much greater precision and generality than we manageabove.

In ‘Preserving Logical Structure’ Gillman Payette explores proper-ties of the structure of consequence relations and languages that arepreserved in the ‘move up’ to the forcing relation. A number of cru-cial things about compactness in the context of level are proved, andthese results are used to derive a very important theorem concerning theplace of forcing among all other similar inference relations. The workon compactness is based on joint work with Blaine d’Entremont.

In ‘Representation of Forcing’ Dorian Nicholson and Bryson Brownshow how to represent the forcing relation in terms of a more syntacticalpresentation of the logic. In a traditional context, what we here call rep-resentation would be axiomatization. They then extend these results tothe more general sequent logic-like case, in which the relation is recon-ceived as a relation between sets of formulas (premise sets) and sets offormulas (conclusion sets). This important essay corrects a number ofmistakes in the earlier ‘On Detonating’ (Schotch and Jennings, 1989).


‘Forcing and Practical Inference’ is the essay in which Peter Schotchtakes up the issue of modifying the definition of forcing in two differ-ent ways. In the first modification, the notion of level upon which theforcing relation is based, is required to respect certain natural ‘clumps’of premises in the sense that no logical cover of a set of premises canbreak up any of these clumps. The second variation recognizes that log-ical consistency is a very weak idea and that in ordinary and technicalreasoning both, we recognize that the collection of ‘bad guys’ extendsfar beyond the logically absurd. This recognition requires an obviouschange in the definition of logical cover.

In ‘Preserving Ambiguity Measures’ Bryson Brown steers preserva-tionism into previously uncharted waters. By defining ambiguity mea-sures as an alternative to partitions for getting something consistent outof an inconsistent set of sentences. Brown not only produces a propertyworthy of preservation but also a way of connecting preservationismto what was once thought of as an incompatible way of dealing withinconsistency—dialethism. This is rather startling to say the least.

The essays are, generally speaking,17 capable of standing alone thoughthere is a considerable overlap. All of them give short weight to someor other important feature (or features) of the preservationist programthough none performs such disservice to all the important features. Infine, the essays act together to fix each others’ expository flaws. Whilethe flaws themselves make the essays more readable than completenesswould allow.

17J. S. Minas used always to add the gloss: ‘which is to say, in no particular case.’

Two

Paraconsistency: Who Needs It?

RAY JENNINGS AND PETER SCHOTCH

Abstract

In this essay Ray Jennings and Peter Schotch present some of the centralmotivation for what has come to be called the preservationist approach toparaconsistent logic.

2.1 Introduction

The classical account of consistency is not adequate in many situationsin which we certainly require a notion like it. This is tantamount tosaying that there are situations in which classical logic is not adequate.While such an assertion might have enjoyed a certain amount of shock-value in the early 20th Century, things have changed since those headydays. Now most people agree that some changes are often necessary inthe classical account of inference, though there is nothing approachingwide-spread agreement on the precise nature of those changes.

One can distinguish two broad strategies for fixing a misbehavinglogic. What we count as misbehavior here is the licensing of inferenceswhich we imagine ought not to be licensed, or the refusal to recognize

15

16 CHAPTER 2. PARACONSISTENCY: WHO NEEDS IT?

some inferences as valid which we ordinary reasoners find entirely con-genial and unproblematic. The strategies referred to just now we mayterm replacement and revision.

When a logic misbehaves in the second way, we most frequentlyrevise it. We don’t need to look far for an example: The logic of ‘un-analysed sentences’ fails to recognize the correctness of the move from‘All women are mortal’ and ‘Xanthippe is a woman’ to ‘Xanthippe ismortal.’ Annoyed by this denial of such an obvious example of correctreasoning, we call for the logic of sentences to be extended to includeterms and individual quantifiers. We are inclined to take as characteris-tic of the revision strategy that the revised logic contain the one whichhad revision visited upon it.

When a logic misbehaves in the first way, things aren’t quite so clear.At first blush it seems that all we can do is sue for replacement of theoffending logic. This is certainly what C.I. Lewis seemed to be doingin the early 20th Century, when he proposed that so-called material im-plication be replaced with what he called strict implication. Nor was healone in pointing to perceived flaws in the classical account of reason-ing and suggesting that it be given up at least partially, if not root andbranch. The purveyors of many-valued logics and the intuitionists toname just two, were also early adopters of the replacement strategy.

Now ‘replace’ is a strong word, and we must temper our understand-ing of it in this context with the realization that the classical account ofinference is as inclusive as it could be, barring triviality. If we wereto take any inference from propositional logic which is not classicallyvalid, and add it to classical logic as an extra principle, the logic soconstituted would allow any formula to be derived from any premiseset—which is to say it would be trivial. The upshot of this fact, firstproved by Post is that any non-trivial logic over the classical language,is almost sure to be a sublogic of classical logic. So a logic offered forreplacement is not normally going to depart from the classical canon inthe sense of being disjoint from it. This is certainly the case with in-tuitionistic logic and many-valued logics, which are easily shown to beincluded in classical logic.

This leaves all those logics which, like the Lewis ones, wish to re-

2.1. INTRODUCTION 17

place the classical � with a connective more like what we really meanwhen we use words like ‘implication.’ In many, but not all of thesecases replacement is the order of the day. But even in these cases, it hassometimes turned out that the proposed replacement is actually a revi-sion, with the new notion of implication representable as some speciesof modalized classical conditional. This befell the C.I. Lewis program,much to his chagrin one suspects, but we shall revisit his motivation abit later in this essay to see if any lessons remain to be learned from it.

There is an important socio-historical point to be made here. Clas-sical logic is not so-called because of its antiquity;1 it is in fact an in-vention of the late 19th and early 20th Century. When the first callsfor replacement went forth, philosophers and mathematicians were notbeing asked to jettison some cherished theory handed down by their re-spected forefathers. Instead, they were being asked to reconsider theirallegiance to a relatively new and untried theory. That situation haschanged. By the turn of the 21st Century classical logic had become en-trenched. By this is meant that the scientific community now has a verylarge investment in the theory. In practical terms it means that replace-ment theorists are going to have uphill sledding, and even revisionistswill have to endure a certain lack of goodwill if not downright surliness.

Of all the replacement theories, one must give pride of place to in-tuitionistic logic. There is no single reason for this notable success,but one may discern at least two strands to the story. The first is thatintuitionism began early enough not to have to fight an entrenched op-ponent. Trench warfare being what it is, this is a weighty consideration.The second is that what we might call the structures which match theformalized account of the intuitionist calculus2, seem to arise in manydifferent parts of formal science.3 This is often taken to be an outwardand visible sign of the worthiness of the logic displaying it.

1As one wag has put it, classical logic is so-called because of its name.2But what irony that the formalized part of the intuitionistic program carry so much of the

weight of promoting the popularity of that program. Intuitionism arose historically as a protestagainst formalism.

3Topology, category theory, and quantum physics are three examples.


2.2 The Strange Case of C. I. Lewis

Of all the replacement theories which have been co-opted by classicallogic into revision theories, the best known is modal logic. C.I. Lewislived long enough to see his recommendations concerning strict impli-cation dissolve, by turns, into a proposal to treat the modals as newconnectives to be added to the classical store. Of course the proposaleven in this new revisionist garb was far from non-controversial. Quineand his students were the most prominent adversaries of this idea andthey (and others) were able to cast a pall over modal logic as an areaof research. This pall had dispersed by the late 1960’s however and thearea has been as respectable as any other since then.

To put the early controversy in a nutshell, Lewis thought that theclassical account of implication was wrong. By this he meant that someof the classical theorems were wrong when we read the symbol � as‘implies.’ In defence of this view he would produce what had come tobe called ‘paradoxes of material implication.’ Most prominent amongthese were:

˛ � .ˇ � ˛/

:˛ � .˛ � ˇ/

where these were required to bear the respective interpretations ‘A trueproposition is implied by any proposition’ and ’A false propositionimplies any proposition.’

According to Lewis, genuine or what he called strict implicationmust be interpreted in terms of deduction. In his informal semantics ofimplication then, we would only say that ˛ implies ˇ provided there issome way of deducing the latter from the former.

To briefly rehearse some of the more puissant of Quine’s criticismsof this position: Implication is a relation rather than a connective. Whenwe read � as ‘implies’, as Russell often seems to do, this is nothingmore substantial than a facon de parler. When we read, for instance˛ � ˇ, as ˛ implies ˇ, we intend nothing more mysterious than the

2.2. THE STRANGE CASE OF C. I. LEWIS 19

assertion that ˛ and ˇ (in that order) are connected by �. As for ’real’or ‘genuine’ implication who knows what that is?4

Actually, as devastating as the criticism might be, Quine could havedone rather better than this. Alas, he was one of those who had been sosteeped in the 19th Century approach to logic that he took the so-calledlogical truths to be what logic is about. Once we rid ourselves of thatunfortunate affliction, we may say to Lewis: ‘It’s a bit silly to insistthat we have a genuine implication (or strict implication if you must)between the propositions ˛ and ˇ only when one can deduce ˇ from˛. This is silly because we already have a representation for that in ourlogic, namely `. So your strict implication is simply another word forprovability—and why would we need to rename provability?’

Thus do we sound the death-knell for Lewis’ complaint about clas-sical logic. It is simply based on a mistake or, to be as charitable as pos-sible, a confusion between which matters belong to the object languageand which to the metalanguage.5 Once we see this, we can dismiss thecomplaint with its fear-mongering talk of paradox as airily as Quine andhis followers. Or can we?

As the old saying has it, there is no smoke without fire. Even if Lewiswas partially blinded by the smoke, there might yet be a hot coal or twowhich remains once the smoke has been contemptuously dispersed. Tosay this is to say that perhaps all this talk of paradox reverberates intothe metalanguage. We shall consider this radical idea more carefully.

Start with ‘a false proposition implies an arbitrary proposition’ whichis impressive enough to have the Latin name ex falso quodlibet. Thereis an exegesis of this which puts it squarely at the metalinguistic level:‘If all we know about some premise set is that it contains a falsehood,then that is sufficient to allow the inference of any formula at all as aconclusion.’ Symbolically:

f˛;:˛g ` ˇ

4This is the Quinean sense of not knowing whatX is, on which I don’t know whatX is, andif you think you do, you’re mistaken.

5This is understandable for Lewis, who never really accepted the concept of a metalanguage.At least there is no textual evidence that he did.


This is no longer about implication, it is now an observation and,dare we say it, a complaint, about the classical account of provability.When Lewis says, as he does of the ‘paradox,’ that there is no gen-uine sense of deduction in which this assertion makes sense, he echoesgenerations of beginning logic students who think that it is cheating touse reductio to derive some formula ˇ from a set of assumptions whichalready contains a contradiction before we assume (for reductio as weblithely say) :ˇ. There is no sense in which the assumption of the hy-pothesis leads to an absurdity. To assert the contrary is like saying of aman born with a single arm, that the first sight of his mother’s face ledto his losing an arm.

We must take care here, since Lewis would distinguish the case inwhich all we know is that a premise is false, from the case in which weknow that a premise is necessarily false. For Lewis, the paradigm ofa necessarily false premise is a self-contradictory or, as we often say,an absurd one. In this case, says Lewis, bolstered by a famous proof,we really and truly can deduce, using entirely non-controversial rules ofproof, anything at all.

The dual ‘paradox’ that if all we know is that a conclusion is true,then it may be derived from any premises at all, requires an approach toinference in which we allow sets of conclusions on the right hand sideof `. These right-handed sets are understood dually to the left-handedones. In other words while the comma in f˛1; : : : ; ˛ng is understood‘conjunctively’ if the set is left-handed, it is understood ‘disjunctively’6

if the set is right-handed. So to say that all we know is that a (right-handed) set contains a truth, is to say that the set in question containssomething like f˛;:˛g and in that case classical logic tells us that� ` f˛;:˛g for any set of premises � .

If we follow Lewis once more we must distinguish this case fromthe one in which the conclusion set contains a dual-self-contradiction,namely a tautology.

Now the fact is, as the complaints of our students indicate, classical

6The reason for placing these terms in quotes is that we cannot understand a set of formulasdisjunctively or conjunctively if the sets in question are infinite. Too, there may be a problem ifeither of disjunction or conjunction does not occur in the object language.

2.3. THE INCONSISTENCY OF BELIEF 21

logic does not permit us to make any distinction of this sort. And if wethink about it for a bit, we may be able to find other reasons to criticizeclassical logic on that ground.

2.3 The Inconsistency of Belief

We begin with an historical example—Hume’s Labyrinth:

I find myself in such a labyrinth, that, I must confess, Ineither know how to correct my former opinions, nor how torender them consistent.

It is no good telling Hume that if his inconsistent opinions were,all of them, true then every sentence would be true. Even if he couldaccept this startling claim, it would have brought him no comfort. Themost which might be wrung out of it is that not all of his opinions couldbe true. This, so far from being news to Hume, was precisely whatoccasioned much of the anguish which he evidently felt. What we needin such circumstances is a way to cope.

Nor is it merely in the realm of metaphysics that inconsistent opin-ions charm us. In the physical sciences as well we seem to have incon-sistent consequences thrust upon us by equally well confirmed hypothe-ses. And even if, as we reassure ourselves, the inconsistences are notineluctable, we must live with them whilst awaiting the crucial experi-ment or the new paradigm, or the new metaphor, or merely promotion.Eventualities may render the inconsistencies resolvable or illusory orunimportant. In the meantime however, we must draw inferences fromthem which do not depend upon their inconsistency, but which are in-formed by it.

A more humanistic logic is required. Such a logic would accordwith what we must frequently do, namely the best we can with datawhich, although inconsistent, are nevertheless the best data we are ableto command. We would like to be able to reflect but also judge thereasonings of ordinary doxastic agents. We wish to offer them standardsof correctness which survive conflicts of belief without triviality. Thishumane approach need not pander. It would accept classical consistency


as an ideal without the pretense that it is pervasive or even especiallycommon in actual belief sets.

We say nothing controversial when we claim that often, perhaps usu-ally, the set of sentences to which someone will actually assent would,if scrutinized, be discovered to be classically inconsistent. The questionis: What are we to make of this observation? There appear to be two dis-tinct reactions. The first is to say that some such apparently inconsistentbelief sets are real and constitute a genuine difficulty for the receivedtheory of inference. The second takes such inconsistencies to be mereappearance—a fanciful mask over the underlying consistent reality.

Our choice between these is important, for our ordinary understand-ing of rationality involves drawing inferences from our beliefs. We arenot positively required to draw these inferences but we are required toaccept the conclusion once the validity of such an inference has beenmade known to us. But how is non-trivial reasoning possible in the faceof inconsistency? We are like jurors made trusting by cruel penalties.To the extent that we accept the classical account of reasoning, to thatextent we resist the view that the set of a person’s beliefs can reallybe inconsistent in a full-blooded sense. We want to either to acquit orsomehow diminish responsibility.

We have recourse to such strategies as distinguishing between activebeliefs, which must indeed be consistent, and mothballed beliefs whichlie strewn about in their cells unregarded and out of mind. We willbe inclined to wink at inconsistencies among this disused bricabrac orbetween them and the brighter ornaments of present thought.

It is tempting to think of this distinction as being congruent with thedistinction sometimes made between explicit and implicit contradiction.We are happy to offer forgiveness, if not pardon, for the latter but visitupon the former harshness without mitigation. We must step carefullyhere for the implicit/explicit distinction can be variously applied to con-fusing effect. We may wish to say that there is an explicit contradictionin a person’s beliefs when she actively adheres to two or more mutuallycontradictory opinions. Implicit contradictions, on this score, would bethose between two inactive beliefs or between one which is active andanother which isn’t. This is not Hume’s predicament.


His problem lay in his active acceptance of contradictory opinions.On the latest account of the distinction, Hume’s contradictions wereexplicit. Yet he did not recognize whatever harsh penalties later logicwould have meted out to him. It did not trivialize his inquires. It sim-ply provided him with ‘a sufficient reason to entertain a diffidence andmodesty in all my decisions.’

There is an alternative account of the distinction according to whichHume’s contradictions would count as implicit rather than explicit. Onthis account, an explicit contradiction is a single sentence which is self-contradictory; while for one’s set of beliefs to contain an implicit con-tradiction one must maintain mutually inconsistent opinions, whetheractively or inactively. The two ways of drawing the distinction are notalways kept separate and the fear of explicit contradictions of the secondsort may make us wary of the first sort also.

The prohibition against explicit contradictions of the second sortmay be expressed as the requirement that every member of a set ofbeliefs must be logically capable of being true. This stricture seemsplausible enough but it does not preclude sets of beliefs which containimplicit contradictions of the second sort. Moreover, it seems likely thatimplicitly inconsistent belief sets of the second sort are quite common.7

On the other hand the notion of someone believing an explicit con-tradiction of the second kind seems devoid of content. One might arguethat it is part of the root meaning of ‘belief’ that the sentence express-ing it must be at least capable of being true. If the classical penaltiesof triviality are to be imposed anywhere, let them fall upon a transgres-sion which no one is capable of committing. We feel no qualms at theprospect of requiring anybody who believed an explicit contradiction tobelieve everything.

The stronger restriction upon belief sets corresponds to the require-ment that such sets be satisfiable, which is to say that all the membersbe capable of simultaneous truth. It is this condition that flies so directlyin the face of our ordinary experience of belief. We have now arrived ata crux.

7Some have argued that to be rational one must have such a belief set. See (Campbell, 1980).


If we let ordinary experience (not to say common sense) be our guideallowing that our belief sets to contain implicit contradiction (of the sec-ond kind) but not explicit contradictions, we must of necessity abandonclassical logic. For by means of the classical picture of inference wemay always derive an explicit contradiction from a set of beliefs whichis implicitly inconsistent. Thus classical modes of reasoning tramplewhat is manifestly a useful distinction—the one we earlier attributed toLewis.

How far ought this felt need for a non-classical logic to be indulged?Appeals to experience can easily lead us astray. After all, almost any-body who proposes some species of non-classical logic is tempted toargue that since the world is non-classical so should our logic be. Totravel too far along this road is to invite some form of the genetic fal-lacy.

If our arithmetic or our syllogisms are not in accord with the clas-sical canons, the fault might well lie in our inability to calculate or toconstruct Venn diagrams. The observation that the reasonings of actualpeople are sometimes non-classical, in order to bite, must be coupledwith evidence that such reasoning is correct, or at the very least notevidently mistaken. If invention is to be mothered let it be through ne-cessity and not through the failure to take suitable precautions.

Nevertheless some of the evidence is compelling. Even when ourstock of beliefs is inconsistent we routinely draw a distinction betweenwhat follows from it and what does not, and regard certain inferencesfrom our beliefs as improper, in spite of the circumstances. If our pro-cedures were genuinely classical we would not, indeed could not, makesuch a distinction. There is no classical issue to be taken with any infer-ence from an inconsistent set.

Failure to take this sort of evidence into account seems as misguidedas any of the worst excesses of those who would base logical theorydemocratically upon the actual inferential practice of the proletariat. Itis no doubt true that if the rule

If A then B; B; therefore A.

(sometimes known as Modus Morons) were from time to time invoked


in country districts or local government, we should nevertheless resistviewing it as a ratiocinative dialect, or alternative inferential lifestyle.But if humanity universally manages to distinguish, at least in somecases, valid arguments from inconsistent premises on the one handfrom invalid arguments from those same premises, this is another andweightier matter.

The view that everyone commits a logical error in making such adistinction is an adaptation for logic of the doctrine of original sin.

We have already considered one non-classical view of the matter,namely the view that when we argue from our beliefs we disregardsome, reasoning only from the active portion of our belief set. Thatthis is non-classical is evident from the fact that it is non-monotonic.Whatever sense of inferable we adopt, it is clear that we do not think ofour conclusions as inferable in that significant way from an inconsistentset. The reason for such a non-monotonic stance is precisely that thewhole of one’s belief might be seen to be inconsistent if examined, thatan inference drawn from the set taken as a whole might be trivial. Toescape this consequence we must take the further step of claiming thatthe only real beliefs are the active ones.

This protestant view, which is really an attempt to deny that beliefscan be inconsistent, suffers from two flaws. The first is that on thisdoctrine it is nigh on impossible to identify a belief set. For supposesomeone’s apparent belief set is inconsistent. Which subset of the set ofapparent beliefs constitutes the real belief set? (assuming that the realbeliefs are also apparent; otherwise the problem becomes even worse).Even if we insist that the real beliefs form a maximal consistent sub-set of the apparent beliefs we do not thereby guarantee uniqueness. Wemust elaborate more sophisticated side conditions to ensure this, per-haps invoking the probability calculus to assist. In this case we musthave handy some interpretation of probability other than one of its beliefinterpretations. The project seems a mare’s nest. But these difficultieswould not deter us from the task were it not for the second flaw.

The second flaw is that the basic premise is false. Try any thinkingbeyond the wallpaper or musak variety of daily life, and inconsistentpairs of beliefs do turn up, the one belief as active as the other.


2.4 Inconsistency and Ethics

It seems to be part of our ordinary moral experience that we can findourselves in a dilemma, which is to say a situation in which the demandsof morality cannot (all) be heeded. Let us suppose that such situationsmight extend beyond the world of everyday all the way into philosophy.8

Let us next ask the question of whether or not obligations9are ‘closedunder logical consequence.’ This is an example of a question which isboth of interest to ethical theory, and also to logic. Some philosophers,even some ethicists, might object that the question is one that only alogician could love, and that they, as ethicists, have not the slightestinterest in it. Such a position betrays a lack of thought.

The issue is really quite deep. If you deny that the logical conse-quences of obligations are also obligations, then you are saying thatmoral argument, at least moral argument of a certain standard kind, isimpossible. When you and I enter into a moral dispute, a dispute overwhether or not you ought to bring it about that P say, then what of-ten, perhaps usually, happens is that I try to demonstrate that P followsfrom something, some general moral principle, to which you subscribe.In stage one then, we obtain the subscription.

Don’t you think that we ought to help those who cannot helpthemselves?Not in every case, suppose somebody is both helpless andnot in need of help?Very well, we ought to help those in need who cannot helpthemselves?Wonderfully high minded, but impossible I’m afraid.Why impossible?We are overwhelmed by the numbers don’t you see?The numbers of what? Those in need?

8Such a supposition is by no means beyond the bounds of controversy. There are moralphilosophers who argue that such conflicts of obligation are merely apparent.

9We are sacrificing rigor in the cause of clarity in this example. Strictly speaking we are nottalking about ‘obligations’ at all, since those are typically actions. We should be talking ratherabout ‘sentences which ought to be true.’ This is a much more awkward form of words howeverand it would interfere with the flow without adding very much to what we are saying.

2.4. INCONSISTENCY AND ETHICS 27

Precisely! We would spend all our resources helping theneedy only to join their ranks before long.Ah, well how about this modification then: We ought to helpthose in need who cannot help themselves, but not to thepoint of harming ourselves or even seriously inconvenienc-ing ourselves?Yes, it doesn’t sound so high-minded, but now it’s somethingI can support.

Now we close in for the kill.

Would you say that Jones is in need of help?Yes, not much doubt about that, poor bloke he.We should pass the hat, don’t you think? Everybody couldput in what they can easily spare.I don’t see the point in that.Do you imagine then that Jones can pull himself up by thebootstraps, that all he needs is sufficient willpower?No, I wouldn’t say that, not in his present state at least.Then, we ought to help him, wouldn’t you say?Well somebody needs to help him if he’s going to be helped,but I don’t see how that got to be my problem.

What has been shown is that ‘We help Jones’ is a logical conse-quence of ‘We help those in need who cannot help themselves andwhose help is not a significant burden to us.’ This much is non-controversial.Having first agreed upon ‘We ought to help those in need ...’. But theremaining step, the step from the forgoing to ‘We ought to help Jones’will only follow if the logical consequences of obligations are them-selves obligations. It is just crazy to say that the issue is of no interest toethicists, unless the ethicists in question have no interest in argumentslike our sample.

Nor can the relevance of logic to ethics be exhausted by this oneissue. Suppose that in exasperation, our stubborn ethicist agrees that theissue is of interest after all. But the interest is over once we see that wemust answer ‘yes’ to the question of whether or not the consequences of


obligations are themselves obligations, isn’t it? Not at all. Recall howthis got started.

We were wondering about moral dilemma, about conflict of obliga-tions. In view of what we just said about obligations and logical con-sequences, we now have a problem. Suppose I am in a moral dilemma,that there are two sentences P and Q which I believe should both betrue (I believe that each, individually, ought to be true). But the twocannot be true together—hence the dilemma.

This is bad enough, but it gets worse. One of the consequencesof the set of sentences which we say ought to be the case, the set ofobligations, is the sentence P ^Q. But since the two sentences cannotboth be true at the same time, this consequence is equivalent to P ^:P .Since the consequences are also obligations, P ^ :P is an obligation.But everything follows from P ^:P so in the case of a moral dilemma,one is obliged to do everything. This seems a bit burdensome.

2.5 Summing Up

The unhappy fact of the matter is that sometimes, through no fault of ourown, we are simply stuck with bad data. This can happen in any of thevariety of circumstances in which we gather information from severalsources. Should the sources contradict each other, we may have a wayof measuring their relative reliability in some reasonable way. Equallyoften however either the means at our disposal will not settle the matterconclusively (as in the case of beliefs) or we simply have no means forthe adjudication of conflict between our sources.

The classical theory of inconsistency must retire in such a situation,since non-trivial classical reasoning is impossible in the face of incon-sistent premise sets. All that the classical logician can advise in thesecircumstances is to start over again with a consistent set of premises.Sometimes this is wise counsel but more often it is of a piece with whatthe doctor said when told by a patient: ‘It hurts when I do this.’

To get back to our earlier example, both belief and obligation seemto call for the Lewis distinction (at least the one that remains after we‘fix’ the original claim). We need to be able to distinguish premise

2.5. SUMMING UP 29

sets which contain full-blooded self-inconsistencies from those whichdo contain pairs of premises which are inconsistent although the indi-vidual premises are not.

This is the path which we shall follow—the one we shall eventuallycall preservationist. But we should point out that there is another pathwhich is quite distinct from ours but which claims for itself all the meritsof our approach and perhaps others besides.

This other account may be distinguished from ours by being a whole-hearted replacement theory, where ours is a revision theory. Accordingto the account dialethism, as its defenders have come to call it, therecan be premise sets in which both a sentence A and its negation not-Aappear without any ill effect. Without that is, the premises having as theset of consequences, the set of all sentences. What is more, such a setwill also contain the single conjunction ‘A and not-A’. Subscribers tothis account suppose that the true meaning of ‘not’ is such as to make‘A and not-A’ capable of being true. This move allows the satisfiabil-ity condition to reappear since many more sets (perhaps even every set)will be satisfiable.

As a replacement theory this has all the drawbacks of such when thetarget of replacement is classical logic. But quite apart from the resis-tance of vested interests, the dialethists must also endure the scorn ofeven the naive and uncommitted. The very idea of ‘true contradiction’seems to be about as counterintuitive as one could imagine, unless oneis addicted to continental philosophy of the more literary sort.

In mincing negation, this scheme avoids the classical consequencesof contradiction by changing the meaning of ‘contradiction.’ Let us sayat once that the dialethists have many virtues. The boldness of their ap-proach is admirable, and they have been far from lazy. Both the theoryunderlying the move and its inferential consequences have been ener-getically worked out and widely published. But where one ought to feelrelease and fill one’s lungs, one feels a kind of dull unease. For evenif classical negation is not primordial in human language, but merely acontrivance of latter days, it is nonetheless with us and even if not all,at least some of our contradictions seem to be on that scale.

Three

Weakly Additive Algebras and aCompleteness Problem

ALASDAIR URQUHART

Abstract

In this essay an extension of existing approaches to the algebra of modallogic is studied. Where the originating work by Jonnson and Tarskiconcerned the representation of Boolean algebras with additive opera-tors, the present effort considers the representation of bounded distribu-tive lattices with weakly additive operators. Where Jonsson and Tarskirely upon Stone’s representation theory for Boolean algebras, this reliesupon Priestley’s representation theory (usually called Priestley duality)for bounded distributive lattices.

3.1 Introduction and Brief History

In a series of papers,1 Ray Jennings and Peter Schotch have developeda generalized relational frame theory that goes beyond the standard ap-proach in modal logic Schotch and Jennings (1980a,d); Jennings and

1For further details concerning the origin of these papers see the the introductory chapter ofthis volume.

31

32 CHAPTER 3. WEAKLY ADDITIVE ALGEBRAS

Schotch (1981a, 1984). The key idea is to generalize the usual truthcondition for the necessity operator by using multi-place relations ratherthan the usual binary relation. Thus, if R is an nC 1-place relation de-fined on a setW , then Jennings and Schotch state the truth condition forthe generalized necessity operator as follows. If x 2 W , then

x ˆ �˛ ” 8y1; : : : ; ynŒRxy1; : : : ; yn) 9i.yi ˆ ˛/�:This definition validates the scheme Kn of n-ary aggregation:

�˛0 ^ � � � ^�˛n � �Œ_

0�i<j�n˛i ^ j �:

In Jennings and Schotch (1984), the authors claim without proof thatthe class of formulas valid in all frames based on an n C 1-place rela-tion is axiomatized by the scheme Kn, together with rules of necessi-tation and monotonicity. This claim is proved by Apostoli and Brown(1995a). An earlier completeness proof was given in Schotch and Jen-nings (1980a), which exploited that idea that necessity operators andinference relations are related in a certain way which is reminiscent ofa Galois connection.

As clever (and fecund) as this latter approach is, it is not in the main-stream of modal logic. It isn’t, in other words, pure since it brings inideas that are not part of what we might call classical modal logic. Thisis because it makes essential use of a non-classical inference relation(called n-forcing). The same holds for the proof by Apostoli and Brownsince it depends on an intermediate lemma stating chromatic compact-ness for hypergraphs. Though the Apostoli Brown proof was substan-tially simplified by Nicholson, Jennings and Sarenac (Nicholson et al.,2000b), the notion of chromatic compactness was still used. There isnow a proof due to Dorian Nicholson , detailed in chapter 4, which isprobably the best one can do in the way of a pure proof.

A proof of completeness for the logicKn is presented in �3.4 of thisessay, where it is derived as a corollary to the main representation theo-rem for weakly additive algebras. Is this also pure? It certainly brings insomething that seems external to classical modal logic, but how muchis in that seeming? One might well argue that the representation the-ory of certain sorts of algebras is the basis of classical modal semantics,

3.1. INTRODUCTION AND BRIEF HISTORY 33

whether or not modal logicians acknowledge the fact. Agree to that, andyou agree that the proof below is pure.

In the early 1950’s Bjarni Jonsson and his thesis advisor2 AlfredTarski began work on the representation of Boolean algebras with extraoperators. One is tempted to conjecture that Tarski was the one whosuggested this topic which was a kind of melding of his friend Mar-shall Stone’s ground-breaking work on the representation of Booleanalgebras3, and his own work with J.C.C. McKinsey on the algebra ofclosure operators in, for example, (McKinsey and Tarski, 1948). Theirjoint work was published in (Jonsson and Tarski, 1951, 1952).

The representations in question work like this: We can easily see thatevery finite Boolean algebra is ‘the same’ as an algebra of sets, namelythe set of all subsets of some finite set, called the underlying set, withthe Boolean operations of meet, join, complement, 1, and 0 representedby set intersection, set union, relative set complement, the underlyingset and the empty set. We can spell out the sameness in several different(and equivalent) ways. One of them is: There is a translation T whichmaps the atoms of the Boolean algebra (all finite Boolean algebras areatomic) to the unit subsets of the underlying set of the set algebra and theequation x ^ y D 1 holds in a given Boolean algebra, with underlyingset † if and only if T .x/ \ T .y/ D †.

The algebraic point to make here, is that such a translation T can bereversed by the obvious inverse translation T �1 which takes unit sets toatoms, intersections to meets etc. Moreover, the existence of these twotranslations amounts to an isomorphism between the Boolean algebraand the set algebra.

This is pretty much old news by the 1930’s. But it raises a question:can this result extend to arbitrary Boolean algebras? In other wordsis it the case that every Boolean algebra whatever is isomorphic to analgebra of sets? There is a lot of content in the ‘whatever,’ since there areinfinite Boolean algebras and, also non-atomic ones. Stone proved thatthis question has a positive answer though the set algebra in question

2In fact, Jonsson was the very first graduate student to work with Tarski after the latter’s moveto the United States.

3Which work is reported in Stone (1936).


isn’t what the novice might expect. The sets, in general, are drawnfrom a topological space and have certain curious properties. They areclopen, which is to say both both open and closed.

Jonsson and Tarski consider the problem of what changes must occurto preserve this isomorphism when we introduce extra operations. Theyconsider the case of a single n-ary operation, which we shall call Þn(with the subscript suppressed when no confusion will result) whichis a generalization of the closure operation. What they show is thatsuch expanded Boolean operations can be represented by the original setalgebras provided that we introduce an n+1-ary relation to deal with theadded operator. The translation which makes everything work is that thegeneralized closure Þn.x1; : : : ; xn/ goes to the set of sets fb1; : : : ; bngsuch that a is an element of T .xi / and Rab1 : : : bn.

Example 1. The ordinary closure is a unary operation. In that case theclosureÞ.x/ is represented by the set of elements b such thata 2 T .x/&Rab. It is easy to see the connection between this conditionand the usual ‘Kripke’ truth condition for possibility.4

Of course not any old generalized closure operator will be repre-sentable in this way. In fact, Jonsson and Tarski restrict their operatorsto those they call additive. Their additivity condition was expressed bythe Boolean equality:

Þn.x1; : : : ; xi _ yi ; xiC1; : : : ; xn/ DÞn.x1; : : : ; xi ; xiC1; : : : ; xn/ _Þn.x1; : : : ; xi�1; yi ; xiC1; : : : ; xn/;

In addition to additivity, the normality condition which is to say thatÞ.x1; : : : ; xi ; 0; xiC1; : : : ; xn/ D 0, was also usually imposed on thegeneralized closure operators.

Sometimes conditions are expressed by means of inequalities in or-der to stay closer to logical formulas which are most often expressedby conditionals. If the above condition were so-expressed the condi-tion (upward) monotonicity, which is to say that Þn.x1; : : : ; xn/ �

4In his Kripke (1963) Kripke refers (in a footnote) to the work of Jonsson and Tarski morethan a decade before his publication as a surprising anticipation, though it isn’t clear to the casualreader who should be (most) surprised.

3.2. WEAKLY ADDITIVE OPERATORS 35

Þn.y1; : : : ; yn/ whenever xi � yi for 1 � i � n, would have to beadded in order to express additivity. This condition applies in particularto the algebras which correspond to the Jennings and SchotchKn logicsin which the modal formula is a conditional.

The present essay sets the modal-logical results in a broader frame-work, by presenting the theory of Jennings and Schotch as a general-ization of the work of Jonsson and Tarski. To be consistent with thisearlier work, the theory is couched in terms of the possibility operatorrather than the dual necessity operator used in the work of Jennings andSchotch (and in modal logic generally speaking). The emphasis on pos-sibility does not reflect some preference for this modality over necessity,but rather a desire to focus on the notion of closure which is more likelyto be familiar to mathematicians. The topological equivalent of a neces-sity operator is an interior operator which, while not unfamiliar, is notas popular—topologically speaking.

3.2 Weakly additive operators

Let L be a bounded distributive lattice. Such an object is a generaliza-tion of the (perhaps) more familiar Boolean algebra in the sense thatevery Boolean algebra is a bounded distributive lattice, but the converseis not true. To take a single (but prominent) example, a Heyting algebrais a bounded distributive lattice which is not a Boolean algebra.

We shall use vector notation Ea; Eb; : : : to refer to sequences of ele-ments from L, including the zero vectorƒ. Operations and relations onsuch vectors are to be understood in terms of the product lattices Lk . Inother words, a binary operation, let’s call it ^, on two vectors Ea and Ebis to be understood as resulting in a unique vector Ea ^ Eb in which theelements of this sequence are defined as the ^ of the corresponding ele-ments of Ea and Eb.5 Of course this assumes that the two vectors have thesame length.

We use jEaj for the length of the vector Ea; the vector of length k inwhich all of the entries are the element a is written ak (when k D 0,

5This is often described as defining the ^ of two vectors componentwise.


this is the empty vectorƒ). If Ek is a sequence of integers, then we writeP Ek for the sum of the elements in the sequence.In the succeeding section on duality theory, we need notations and

definitions for sequences of elements from an arbitrary set. We writex 2 Ey if x is a member of the sequence Ey. If Ex is a sequence of elementsfrom a set S and A � S , then we write Ex � A if every element in thesequence Ex is inA. If Ex is a sequence of subsets of a fixed set S , then weuse the notation

T Ex for the intersection of all the sets in the sequenceEx; in the case of the empty sequence ƒ of sets,

Tƒ D S .

Definition 1. If Ex and Ey are sequences of elements from a set Swith an ordering relation � defined on it, and jExj � j Eyj, then wewrite Ex � Ey if xi � yi for all i , where 1 � i � jExj:

LetÞ be an n-place operation defined on L.

Definition 2. For k > 0, we say thatÞ is k-additive in the i thplace if it is normal and satisfies the conditions

ÞŒ Ea;^

0�h<j�k.bh _ bj /; Ec � �

_

0�j�kÞŒ Ea; bj ; Ec �

for Ea 2 Li�1, Ec 2 Ln�i and b0; : : : ; bk 2 L.

Example 2. Suppose Þ is binary. Then Þ is 1-additive in the secondplace if and only if

Þ.x; .b0 _ b1// � Œ.Þ.x; b0/ _Þ.x; b1/�

WhileÞ is 2-additive in the first place if and only if

Þ..b0_b1/^ .b0_b2/^ .b1_b2/; y/ � ŒÞ.b0; y/_Þ.b1; y/_Þ.b2; y/�

3.2. WEAKLY ADDITIVE OPERATORS 37

In the following definition we define something a bit more generalthan our earlier notion of monotonicity, which applies only where therelation � assumes that the related vectors are of equal length.

Definition 3. Þ is said to be monotone if it satisfies the condition:ÞEa � ÞEb whenever Ea � Eb.

Definition 4. An operator on a bounded distributive lattice isdefined to be weakly additive if it is monotone and in all of itsargument places it is k-additive for some k > 0.

Definition 5. An additive operator in the sense of Jonsson andTarski (Jonsson and Tarski, 1951, 1952) is a monotone operatorthat is 1-additive in all of its argument places.

Definition 6. We say that an n-place weakly additive operatorÞis of type Ek if Ek is a vector of positive integers of length n so thatÞis ki -additive in its i th place.

For example:

Example 3. IfÞ.a; b; c/ is 1-additive in its first place, 3-additive in itssecond place and 2-additive in its third, then it is of type h1; 3; 2i.

Definition 7. We define a weakly additive algebra of type Ek to be abounded distributive lattice L together with a weakly additiveoperator of type Ek defined on L.

If we translate the Kn modal logics considered by Jennings andSchotch into algebraic form, the result is a special case of the general


framework outlined above. Consider a propositional logic satisfyingthe scheme Kn of n-ary aggregation, as well as the rules of necessita-tion and monotonicity. We can define a possibility operator in the usualway as the dual of the necessity operator; that is to say,Þ˛ $ :�:˛.Then the scheme Kn is provably equivalent to the dual scheme:

ÞŒ^

0�i<j�n˛i _ j � � Þ˛0 _ : : : _Þ˛n:

Hence, the algebraic version of the Jennings/Schotch system with theaxiom scheme Kn is the theory of Boolean algebras with a one placemonotone normal operatorÞ satisfying

ÞŒ^

0�i<j�n.bi _ bj /� �

_

0�i�nÞbi

that is to say, Boolean algebras with a one place weakly additive oper-ator of type hni. In what follows, we shall call these Jennings/Schotchalgebras; we shall denote the variety of weakly additive algebras satis-fying the above inequality by the notation Vn.

The Jennings/Schotch truth condition for�, when translated into itsdual form, is as follows:

x ˆ Þ˛ ” 9y1; : : : ; ynŒRxy1; : : : ; yn ^ 8i.yi ˆ ˛/�:

3.3 A Further Generalization

Weakly additive operators can be defined from multi-place relations,in a way that generalizes both the Jennings/Schotch truth conditionfor possibility, and a construction of Jonsson and Tarski (Jonsson andTarski, 1951, Definition 3.2). Let Ek be a length n sequence of non-negative integers, X a non-empty set and R a .

P Ek C 1/-place relationon X . We define an n-place operationÞR on P .X/ as follows:

Definition 8. x 2 ÞR.A1; : : : ; An/ ”9 Ey1; : : : ; EynŒRx Ey1; : : : ; Eyn ^ 8i Œ1 � i � n) Eyi � Ai ��

3.4. DUALITY THEORY FOR WEAKLY ADDITIVE OPERATORS 39

where for each i , 1 � i � n, Eyi is a vector of length ki .

Theorem 4. For Ek a sequence of non-negative integers of length n, andR a .

P Ek C 1/-place relation on a non-empty set X , ÞR is a weaklyadditive operator of type Ek on P .X/.

Proof. Let L be P .X/. We need to prove for k D ki > 0 the inclusion

ÞRŒ EA;\

0�h<j�k.Bh [ Bj /; EC � �

[

0�j�kÞRŒ EA;Bj ; EC �;

where EA 2 Li�1, EC 2 Ln�i and B0; : : : ; Bk 2 L. Assume

x 2 ÞRŒ EA;\

0�h<j�k.Bh [ Bj /; EC �;

so that there exist Ey1; : : : ; Eyn satisfying the conditions:Rx Ey1; : : : ; Eyn ^ 8j .1 � j < i ) Eyj � Aj /

^8j .i < j � n) Eyj � Cj /;

Eyi �\

0�h<j�k.Bh [ Bj /:

Let us suppose that for all j , 0 � j � k, Eyi 6� Bj . Then for anysuch j , there is a z 2 Eyi so that z … Bj . Since j Eyi j D k, it follows(by the pigeonhole principle) that there are distinct h; j so that for somez 2 yi , z … Bh [ Bj , contrary to assumption. Hence for some j ,Eyi � Bj , proving the inclusion. The remaining conditions defining aweakly additive operator of type Ek are easily verified. �

3.4 Duality theory for weakly additive operators

The duality for weakly additive operators described in this section ispiggy-backed on Priestley’s duality for bounded distributive lattices. Webegin by giving the main results of this duality theory without proofs.For detailed discussion of the theory, the reader is referred to articles byPriestley (Priestley, 1970, 1984), Davey and Duffus (Davey and Duffus,


1982) and to the textbook by Davey and Priestley (Davey and Priestley,1990).

An ordered topological space is a topological space with a partialorder relation defined on it. A subset E of a partially ordered set is in-creasing if x 2 E, x � y imply y 2 E. A map f between partiallyordered sets is increasing if x � y implies f .x/ � f .y/, and decreas-ing if x � y implies f .y/ � f .x/. A map between two ordered spacesis said to be an order-homeomorphism if it is both a homeomorphismand an isomorphism with respect to the orderings on the spaces. APriestley space is an ordered topological space S that is compact andtotally order-disconnected, that is, for points x; y of S, if x 6� y thenthere is a clopen increasing set U such that x 2 U; y … U .

If L is a bounded distributive lattice, then the dual space of L, S.L/,is the ordered topological space in which the set of points S is the familyof all prime filters of L, ordered by containment, and the topology isdefined by taking as a sub-base the family of all sets fr 2 S W a 2 rgand fr 2 S W a … rg for a 2 L. Conversely, if S is a Priestley spacethen the dual lattice of S, L.S/, is the set of all clopen increasing sets ofS, with the lattice operations of set intersection and union.

Theorem 5. 1. If L is a bounded distributive lattice, then S.L/ is aPriestley space, and L is isomorphic to L.S.L// under the map-ping

�.a/ D fx 2 S.L/ W a 2 xgI

2. If S is a Priestley space, and L.S/ its dual lattice, S is order-homeomorphic to S.L.S// under the mapping

�.x/ D fB 2 L.S/ W x 2 Bg:

We obtain the duality theory for weakly additive operators by build-ing on Priestley’s duality theory. Let Ek be a sequence of non-negativeintegers, and

P Ek D n the sum of the entries in Ek. We define a re-lational space of type Ek to be a structure hS; Ri, where S is a Priestleyspace, andR is a .

P EkC1/-place relation on S , satisfying the followingconditions:


1. If A1; : : : ; An 2 L.S/, thenÞR.A1; : : : ; An/ is clopen;

2. If Rx Ey and x � z then Rz Ey;

3. If for all i , 1 � i � n, jyi j D ki and :Rx Ey1; : : : ; Eyn then:

.9A1; : : : ; An 2 L.S//Œ8i .1 � i � n) Eyi � Ai / ^ x … ÞR.A1; : : : ; An/�.

If R D hS; Ri is a relational space of type Ek, then the dual algebraof R is the algebra A.R/ defined on the lattice L.S/ by adding theoperation ÞR. If hL;Þi is a weakly additive algebra, where Þ is ann-place operator, and each Ey1; : : : ; Eyn is a sequence of filters in L, thenthe relation RÞ is defined by:

RÞx Ey1; : : : ; Eyn , 8Ea 2 Ln Œ8i.ai 2\Eyi /) Þ.a1; : : : ; an/ 2 x�:

If A D hL;Þi is a weakly additive algebra of type Ek, then the dualspace of A, R.A/, is defined by adding to the Priestley space of L therelation R defined by:

Rx Ey1; : : : ; Eyn, RÞx Ey1; : : : ; Eyn ^ 8i Œ1 � i � n) j Eyi j D ki �;

where n D jEkj.If L is a distributive lattice, then we employ the following termi-

nology for vectors Eu, where each ui is a subset of L. For such vec-tors Eu and Ev, where jEuj � jEvj, we write Eu � Ev if for all i where1 � i � jEuj, ui � vi (this terminology is consistent with the corre-sponding terminology introduced above for vectors of points in orderedsets). If F is a family of such vectors, and the maximum length of a vec-tor in F is k, then we define

SF to be the vector Ev of length k where

vi DSfui jEu 2 F g.

For a lattice L, x a filter in L, and a an element of L, we denote byxŒa/ the smallest filter containing x and a. The principal filter contain-ing a is denoted by Œa/.

Lemma 1. Let hL;Þi be a weakly additive algebra of type Ek, wherej Ekj D n. Then for Ea 2 Ln, x 2 S.L/,


ÞEa 2 x)

9 Ew1; : : : ; EwnŒ Rx Ew1; : : : ; Ewn ^ 8i Œ 1 � i � n ) ai 2\Ewi � �:

Proof. Let ÞEa 2 x, where x 2 S.L/. Define F to be the familyof all sequences Eu1; : : : ; Eun, where each Eui is a sequence of filters inL, so that: (1) ai 2

T Eui ; (2) For every i , 1 � i � n, there areb1; : : : ; bi�1; biC1; : : : ; bn so that for all y 2 Eui , there is an elementay 2 L such that

8z 2 Eui .z ¤ y ) ay 2 z/ ^ Þ.b1; : : : ; bi�1; ay ; biC1; : : : ; bn/ … xI

(3) RÞx Eu1; : : : ; Eun; (4) j Eui j � ki . We define an ordering on F asfollows. For Eu1; : : : ; Eun 2 F and Ev1; : : : ; Evn 2 F , Eu1; : : : ; Eun �Ev1; : : : ; Evn if and only if for all i , Eui � Evi .

First, we show that F ¤ ;. For ki > 0, set Eui D hŒai /i. ThenEu1; : : : ; Eun 2 F , since for y 2 Eui , we can set bj D 1 for j ¤ i , anday D 0, since Þ.1; : : : ; 1; 0; 1; : : : ; 1/ D 0 … x. Second, we show thatevery chain in F has an upper bound in F . Let C � F be a chain; for1 � i � n define Eui D

Sf Evi j Ev1; : : : ; Evn 2 Cg: Then it is straightfor-ward to check that Eu1; : : : ; Eun is in F , and is an upper bound for C.

By Zorn’s Lemma, F contains a maximal element Ew1; : : : ; Ewn. Tocomplete the proof of the Lemma, we need to show that all the filtersin Ew1; : : : ; Ewn are prime. Suppose that this is not so, so that for someEwi D y1; : : : ; yj ; : : : ; yl , yj is not prime; thus there are d; e 2 L whered _ e 2 yj , but d; e … yj . Let Eu; Ev be the vectors that result from Eui byreplacing yj by yj Œd / and yj Œe/ respectively. Since Ew1; : : : ; Ewn is max-imal in F , it follows that :RÞx Ew1; : : : ; Ewi�1; Eu; EwiC1; : : : ; Ewn and:RÞx Ew1; : : : ; Ewi�1; Ev; EwiC1; : : : ; Ewn. Thus there are f1; : : : ; fn 2 Lso that f1 2

T Ew1; : : : ; fi 2T Eu; : : : ; fn 2

T Ewn, Þ.f1; : : : ; fn/ … xand g1; : : : ; gn 2 L so that g1 2

T Ew1; : : : ; gi 2T Ev; : : : ; gn 2

T Ewn,and Þ.g1; : : : ; gn/ … x. Let Ez be the vector hEu; yj Œe/i. We claim thatRÞx Ew1; : : : ; Ez; : : : ; Ewn. For if not, there are elements p1; : : : ; pn 2 Lso that p1 2

T Ew1; : : : ; pi 2T Ez; : : : ; pn 2

T Ewn andÞ.p1; : : : ; pn/ … x. But since pi 2 yj Œd / and pi 2 yj Œe/, it followsthat pi 2 yj , contradicting RÞx Ew1; : : : ; Ewn.


We now show that the sequence Ew1; : : : ; Ez; : : : ; Ewn also satisfiesthe second of the three conditions defining the elements of F . Letb1; : : : ; bi�1; biC1; : : : ; bn be the sequence of elements associated withEwi by condition (2). For j ¤ i , let hj D bj ^ fj ^ gj . We claim that

the sequence h1; : : : ; hi�1; hiC1; : : : ; hn satisfies condition (2). For el-ements y in Ez other than the filters yj Œd / or yj Œe/, we can set ay tobe the element ay associated with Ew1; : : : ; Ewn by condition (2). Fory D yj Œd /, set ay D gi , and for y D yj Œe/, set ay D fi . Then itfollows from our assumptions that h1; : : : ; hi�1; hiC1; : : : ; hn and theelements ay satisfy condition (2).

Since Ew1; : : : ; Ewn � Ew1; : : : ; Ez; : : : ; Ewn, and Ew1; : : : ; Ewn is a maxi-mal element in F , Ew1; : : : ; Ez; : : : ; Ewn … F . Because Ew1; : : : ; Ez; : : : ; Ewnsatisfies conditions (1) to (3) in the definition of F , it follows that l Dj Ewi j D ki . Let a0; : : : ; al be the sequence of elements ay for y 2 Ez.Then ^

0�i<j�lai _ aj 2

\Ez;

so that

Þ.h1; : : : ; hi�1;^

0�i<j�lai _ aj ; hiC1; : : : ; hn/ 2 x:

It follows that _

0�i�lÞ.h1; : : : ; ai ; : : : ; hn/ 2 x:

But then since x is a prime filter, it follows that for some i , 0 � i �l , Þ.h1; : : : ; ai ; : : : ; hn/ 2 x, contrary to the definition of ai . Thiscontradiction completes the proof that all the filters in Ew1; : : : ; Ewn areprime.

In the definition of F , we have only required that j Eui j � ki , for 1 �i � n, while the definition of R above requires the condition j Eui j D ki .However, if RÞx Eu1; : : : ; Eui ; : : : ; Eun, then RÞx Eu1; : : : ; h Eui ; zi; : : : ; Eun,where z is any element in Eui . Consequently, we can conclude that

9 Ew1; : : : ; EwnŒ Rx Ew1; : : : ; Ewn ^ 8i Œ 1 � i � n ) ai 2\Ewi � �:

This completes the proof of the lemma. �


Theorem 6. 1. If A is a weakly additive algebra of type Ek, thenR.A/ is a relational space of type Ek andA is isomorphic toA.R.A//under the mapping

�.a/ D fx 2 R.A/ W a 2 xgI

2. If R is a relational space of type Ek, and A.R/ its dual algebra, Ris order-homeomorphic to R.A.R// under the mapping

�.x/ D fB 2 A.R/ W x 2 Bg:

Proof. (1) The main condition to be verified is that

ÞEa 2 x)

9 Ew1; : : : ; EwnŒ RÞx Ew1; : : : ; Ewn ^ 8i Œ 1 � i � n ) ai 2\Ewi � �;

for x 2 S.L/. This follows immediately from the definition of RÞ andLemma 1. The conditions defining a relational space of type Ek are theneasily verified from the definitions, using this equivalence.

Priestley’s representation theorem (theorem 5) implies that � is alattice-isomorphism, so it suffices to show that � is an isomorphism withrespect toÞ. That � is also an isomorphism with respect toÞ is exactlythe content of the above condition.

(2) By theorem 5, the map � is an order-homeomorphism from Ronto the second dual of R, so it suffices to prove that � is an isomor-phism with respect to the relationR. This follows from the definition ofÞR and the third condition defining a relational space of type Ek. �

The representation theorem we have just proved is an algebraic ver-sion of the completeness theorem described in the introductory section.In fact, that completeness theorem can be derived from the representa-tion theorem. (theorem 6).

Theorem 7. The logic Kn is complete with respect to the family of allnC 1-ary relational frames, given the Jennings/Schotch truth conditionfor the� operator.


Proof. We can construct a Boolean algebra with operator Þ by the fa-miliar Lindenbaum construction; that is to say, we construct an algebrain which the elements are classes of provably equivalent formulas. Ifwe begin from the logic Kn, then the result is an algebra A in the vari-ety Vn. By theorem 6, this algebra is isomorphic to A.R.A// under themapping

�.a/ D fx 2 R.A/ W a 2 xg:Consider the .n C 1/-ary relational frame defined on the family of allprime filters in A, with the accessibility relation R defined in the dualspace R.A//. For a propositional variable P , the set of points at whichit is true is defined to be �.ŒP �/, where ŒP � is the set of formulas equiv-alent to P inKn. It is a straightforward exercise to check that a formulais true at all points in this model if and only if it is a theorem of Kn. �

Four

A Dualization of NeighborhoodStructures

DORIAN NICHOLSON

Abstract

A dualization of neighborhood semantics is introduced for the purposeof obtaining a simplified proof that weakly aggregative modal logic iscomplete with respect to .nC 1/-ary relational frames. This new class ofquasi-semantic structures exploits the theory of transverse hypergraphs.Unlike the other proofs in the literature, the one included here does notcite chromatic, or colouring, compactness. Along the way we prove com-pleteness for a denumerable class of non-normal modal logics, whichhave deontic, as well as philosophical logical, motivations.

4.1 Introduction

The search for a completeness proof for Jennings and Schotch’s weaklyaggregative modal logic lasted for nearly twenty years (cf. Johnston(1978); Schotch and Jennings (1980b); Jennings and Schotch (1981b);Schotch and Jennings (1980e)) before its goal was attained in 1995 byApostoli and Brown in Apostoli and Brown (1995b), and also inde-pendently, algebraically, by Urquhart in Urquhart (1995). Apostoli and

47

48 CHAPTER 4. NEIGHBORHOOD STRUCTURES

Brown’s proof was subsequently simplified, in 2000, by Nicholson, Jen-nings and Sarenac in Nicholson et al. (2000a). But both proofs exploitthe compactness of colouring for hypergraphs whose edges are finitelylong. Chromatic compactness, also called colouring compactness, is theclaim that 8H , ifH is a hypergraph thenH is k-colourable .k � 1/ iffevery finite G � H is k-colourable.1 A hypergraph is a family of sets,called edges. A hypergraph H is k-colourable iff there is a partition ofits union, called its vertex set, into k pairwise disjoint, mutually exhaus-tive sets, or cells, such that no edge of H is a subset of any cell. Thesimplifying thrust of Nicholson et al. in Nicholson et al. (2000a) raisesthe question whether there is an even simpler completeness proof, onewhich avoids citing chromatic compactness. In this paper an affirma-tive answer to this question is demonstrated, by invoking the theory ofwhat are dubbed hyperframes. The theory of hyperframes implementsthe theory of transverse hypergraphs and consists essentially of a du-alization of the neighborhood semantics for modal logic explored bySegerberg in Segerberg (1971), and referred to as ‘minimal models’ byChellas in Chellas (1980).

4.2 Hyperframes

A hyperframe F is pair .U;H/ where U is a non-empty set (the uni-verse of the frame) and H is a hypergraph function from U to P P .U/.Accordingly, for each x 2 U;H.x/ is a hypergraph on U: a family ofsubsets of U, where the subsets are called the edges of the hypergraph,and the elements of the edges are called the vertices of the hypergraph.For each x in U, H.x/ is called the hypergraph on x (relative to F).A hyperframe is thus, in essence, a neighborhood frame, as the latteris defined in Segerberg (1971), for example. But a model on a hyper-frame is distinct from a model on a neighborhood frame when it comesto interpreting�.

If F D .U;H/ is a hyperframe and V W Nat ! P .U/ is a valuationfunction, then .U;H;V/ is a (hyper)model M on F. Truth at a point xin a model M D .U;H;V/, with respect to the language of a standard

1Colouring compactness is provable from the compactness of propositional logic.

4.2. HYPERFRAMES 49

ux

�ˇ

kˇk

H.x/

Figure 4.1: The truth condition for� on hyperframes.

propositional logic, is defined in the standard way for Boolean connec-tives. To interpret the unary necessity operator �, we use the notion ofthe transversal of a hypergraph.

Definition 1. If H is a hypergraph on a non-empty set U, and S is asubset of U, S is a transversal for H iff 8E 2 H; S \ E ¤ ;.

It follows from this definition that ; 2 H iff H has no transversals, andH D ; iff, vacuously, every subset of U is a transversal for H.

The truth condition for� is:

ˆMx�ˇ iff kˇkM is a transversal for H.x/:

(See Figure 1.) IntroducingÞˇ as an abbreviation for:�:ˇ, we there-fore have:

ˆMxÞˇ iff 9E 2 H.x/ W kˇkM � E

If M D .U;H;V/ is a model and 8x 2U;ˆMx˛, then ˛ is valid on M,written ‘ˆM˛’; if for every valuation function V, ˛ is valid on .U;H;V/


then ˛ is valid on F D .U;H/, indicated by ‘F ˆ ˛’. If ˛ is valid onevery member of a class C of frames, then ˛ is valid with respect to C,‘C ˆ ˛’.

The logic determined by the class C of all hyperframes is axioma-tized by Nq,2 the system defined by:

ŒRR� W ` ˛ � ˇ )` �˛ � �ˇ (4.1)

ŒPL� W `PL ˛ )` ˛ (4.2)

ŒMP � W ` ˛ � ˇ & ` ˛ )` ˇ (4.3)

ŒUS� W ` ˛ & ˇ is a substitution instance of ˛ )` ˇ (4.4)

It is easy to check that Nq is sound with respect to C; to prove com-pleteness we use a Henkin construction which capitalizes on the theoryof transverse hypergraphs.

Definition 2. Let H be a hypergraph on a set U. A set E � U is aminimal transversal for H if E is a transversal for H and 8E0 � E;E0 isnot a transversal for H. The transverse hypergraph for H, T .H/, or justTH for convenience, is the set of all minimal transversals for H.

Proposition 1. For any hypergraph H on a set U, H D ; iff TH D f;g,and ; 2 H iff TH D ;.

Proof. Œ)� Assume that H D ;. Then every subset of U is a transver-sal for H; but then if E ¤ ;, 9E0 � E such that E0 is a transversal forH. So every minimal transversal for H is empty. And, ; 2 TH be-cause ; has no proper subsets. Thus TH D f;g. Œ(� Assume now thatTH D f;g. Then 8E 2 H;; \ E ¤ ;. Therefore H D ;.Œ)� Suppose that ; 2 H. Then 8E 2 TH, E \ ; ¤ ;. Whence

TH D ;. Œ(� Lastly, suppose that TH D ;. If H D ; then by theabove reasoning ; 2 TH. So H ¤ ;. If, then, ; 62 H, 9E¤; such thatE 2 TH, which is absurd. Therefore ; 2 H. �

Definition 3. A hypergraph H is simple if 8E;E0 2 H, E 6� E0:

2The name of this system has been drawn from Jennings and Schotch (1981b).

4.2. HYPERFRAMES 51

Proposition 2. For any hypergraph H on a set U, T TH � H. If H issimple then H D T TH.

Proof. Let H be a hypergraph on U, and let E 2 T TH. Suppose thatE 62 H. Note that 8E0 2 H;E0 is a transversal for TH. Therefore8E0 2 H; 9x 2 E0 such that x 62 E, in which case 9E0 2 TH suchthat E \ E0 D ;, contrary to the assumption that E 2 T TH. ThereforeT TH � H.

Now let E 2 H, and assume that H is simple. Since E is a transversalfor TH, 9E0 � E such that E0 2 T TH. But the above reasoning showsthat T TH � H. Therefore, since H is simple, E0 D E, i.e., E 2 T TH,whence H � T TH. �

If L is a modal logic and ˛ is a sentence, the proof set for ˛ in L,j˛jL, is the set of all maximal L-consistent sets of which ˛ is a member.The canonical frame for L is the structure FL D .UL;HL/ where ULis the class of all maximal L-consistent sets of formulae, and 8x 2UL,

HL.x/ D T .fj jL W � 2 xg/:

The canonical model for L, ML, is the triple .UL;HL;VL/ where VLis defined:

8n 2 Nat; x 2 VL.n/ iff pn 2 x:

Theorem 1. Let L be any logic closed under ŒRR�; ŒPL�; ŒUS� andŒMP �. Then

8˛; x 2UL; k˛kML D j˛jL:

Proof. We show that 8x 2UL; ˛ is true at x iff ˛ 2 x. The proof is byinduction on the complexity of ˛. We omit all but the case for ˛ D �ˇ.

Suppose that x 2 k�ˇk. Then kˇk is a transversal for HL.x/, inwhich case, by the hypothesis of induction, so is jˇj. Therefore 9E �jˇj such that E 2 T .HL.x//. Whence E 2 fj j W � 2 xg (Proposition2). So let E D j j. Then j j � jˇj and� 2 x. But then ` � ˇ, andthus ` � � �ˇ (by ŒRR�). Therefore�ˇ 2 x.


Suppose now that �ˇ 2 x. Then 8E 2 HL.x/;E \ jˇj ¤ ;. I.e.,jˇj is a transversal for HL.x/. By the induction hypothesis, jˇj D kˇk.Whence x 2 k�ˇk. �

Corollary 1. The system Nq is determined by the class of all hyper-frames.

4.3 Normal Hyperframes

The logic Nq is not normal because there is at least one theorem of Nqwhose � formula is not a theorem. E.g., `Nq > while 6`Nq �>. Sincethere is a hypermodel M containing a point x such that ; 2 H.x/, it fol-lows that C 6ˆ �>, and hence 6`Nq �> (Corollary 1). By similar, dual,reasoning there is a model M containing a point x such that 8˛;ˆMxÞ˛,and thus ˆMxÞ?. Hyperframes therefore provide an opportunity for thesystematic investigation of non-normal logics, that is, logics that are notclosed under the rule:

ŒRN � W ` ˛ )` �˛ (4.5)

Definition 4. If F D .U;H/ is a hyperframe then F is normal if 8x 2U;; 62 H.x/. A model is normal if it is based on a normal hyperframe.

This is significant from a philosophical perspective for two reasons:First, non-normal logics have deontic motivations insofar as we wouldlike to not have an infinite number of obligations. A logic is normalwhen �˛ is a theorem whenever ˛ is a theorem. Thus, if � represents‘it is obligatory that’, then in any deontic logic with an infinite num-ber of theorems there is an infinite number of obligations.3 Second,if we read � as a necessity operator, then the existence of determined

3It is important to note, however, that the absence of normality does not guarantee the absenceof an infinite number of obligations. Although the absence of normality is necessary, it is notsufficient for this end. What we really need is the rule:

` ˛ ) ` :�˛: (4.6)

Jennings has suggested that what we really want is a variety of connexivist implication, which is arestriction of classical logic to contingencies.

4.3. NORMAL HYPERFRAMES 53

non-normal modal logics marks a conceptual divergence between log-ical validity and its classical, Aristotelian account. According to theclassical account, an argument is valid when it is necessary that if thepremises are true then the conclusion is true. But there are non-normallogics in which there are logically valid conditionals whose� formulaeare not theorems. This raises the philosophical question of how the-oremhood in such systems should be understood, or alternatively, thequestion of what the� operator represents.

Theorem 2. Let L be a logic which is closed under ŒRR�; ŒRN �; ŒUS�; ŒPL�,and ŒMP �. Then the canonical hyperframe for L is normal.

Proof. Let x 2 UL, and suppose that ; 2 HL.x/. Then by Proposi-tion 1, THL.x/ D ;. But HL.x/ D T .fj j W � 2 xg/; thereforeTHL.x/ D fj j W � 2 xg D ; (Proposition 2). I.e., 8 ;� 62 x. But` >, and so by ŒRN �, ` �>, in which case�> 2 x, an absurdity. �

.nC 1/-ary Relational Frames and n-Bounded Hyperframes

An .nC 1/-ary relational frame .n � 1/ is a pair .U;R/ where U is anon-empty set and R � UnC1. If V is a function from Nat to P .U/

then the triple .U;R;V/ is an .n C 1/-ary relational model based onthe frame .U;R/. Truth and validity relative to .n C 1/-ary relationalframes and models are as defined for hyperframes and models, with theexception that truth at a point x for� formulae is defined:

ˆMx�˛ iff 8hy1; :::; yni 2 R.x/; 9i 2 Œn� WˆMyi˛;

where for any positive integer n, Œn� denotes f1; 2; :::; ng, and if R �UnC1, then R.x/ D fhy1; :::; yni W hx; y1; :::; yni 2 Rg. (See Figure2.)

Definition 5. A hyperframe F D .U;H/ is n-bounded, for n � 1, if8x 2 U;8E 2 H.x/; jEj � n. A model is n-bounded if it is based onan n-bounded hyperframe.


z n = 3

z

z

z

zzz

z

z

z

x

y1

y2

y3

z1

z2

z3

w1

w2

w3

�˛

˛

˛

˛

Figure 4.2: The truth condition for� on .nC 1/-ary relational frames.

Definition 6. Let F D .U;H/ be an n-bounded normal hyperframe,and let the relation R �UnC1 be defined pointwise:

R.x/ WD fhy1; :::; ym; yi ; :::; yi„ ƒ‚ …n�m times

i W

fy1; :::; ymg 2 H.x/ & i 2 Œm�gThen the .n C 1/-ary relational transformation of F is the .n C 1/-aryrelational frame F� D .U;R/.

It is easy to see that for any model M on an n-bounded normal hy-perframe there is an equivalent .nC 1/-ary relational model. That is:

Theorem 3. For every normal n-bounded hyperframe F D .U;H/,every model M� D .U;R;V/ on the .n C 1/-ary relational transfor-

4.4. WEAKLY AGGREGATIVE MODAL LOGIC 55

mation F� D .U;R/ of F is pointwise equivalent to the hypermodelM D .U;H;V/, that is,

8x 2U;8˛; ˆMx˛,ˆM�x ˛

Proof. The (omitted) proof is by induction on the complexity of ˛. �

4.4 Weakly Aggregative Modal Logic

A consequence of the theory of n-bounded normal hyperframes is asimple proof of the completeness of the system Kn with respect to theclass of all .n C 1/-ary relational frames. The system Kn .n � 1/ isdefined:

ŒRR� W ` ˛ � ˇ )` �˛ � �ˇ (4.7)

ŒRN � W ` ˛ )` �˛ (4.8)

ŒPL� W `PL ˛ )` ˛ (4.9)

ŒMP � W ` ˛ � ˇ & ` ˛ )` ˇ (4.10)

ŒUS� W ` ˛ & ˇ is a substitution instance of ˛ )` ˇ (4.11)

ŒKn� W ` �˛1 ^ ::: ^�˛nC1 � �_

1�i<j�nC1˛i ^ j (4.12)

K1 is just the Kripke system K. For each n > 1, Kn is weaklyaggregative because it replaces the strong aggregation principle ŒK�.DŒK1�/ with the weaker ŒKn�.

It would appear that the completeness proof herein is a simplifica-tion of the other proofs in the literature, found in Apostoli and Brown(1995b) and Nicholson et al. (2000a), as both of these rely heavily onthe colouring theory of hypergraphs, reference to which is omitted inthe present proof. In particular, the previous proofs exploit colouringcompactness, the claim that if H is a hypergraph each of whose edgesis finitely long, then H is k-colourable iff every finite subgraph of His k-colourable. In contrast, the crucial lemma used here, in addition toTheorem 2, is that the canonical hyperframe for any logic that includesŒKn� .n � 1/ is n-bounded.


For convenience, we introduce the convention that if t1; t2; :::; tnC1are sets .n � 1/, then 2

nC1.ti /i2ŒnC1� denotesS1�i<j�nC1 ti \ tj ,

and if ˛1; ˛2; :::; ˛nC1 are sentences then 2nC1.˛i /i2ŒnC1� representsW

1�i<j�nC1 ˛i ^ j .

Lemma 1. 8n � 1, if H is a hypergraph such that 8E 2 H; jEj � nthen whenever t1; t2; :::; tnC1 are transversals for H, so is 2

nC1.ti /i2ŒnC1�.

Proof. Suppose that 8E 2 H, jEj � n and that 8i 2 Œn C 1�; ti is atransversal for H. Suppose further that 2

nC1.ti /i2ŒnC1� \ E D ;, forsome E 2 H. By a pigeonhole argument, 9i; j 2 ŒnC 1�.i ¤ j / suchthat ti \ tj \ E ¤ ;, which is absurd since ti \ tj � 2

nC1.ti /i2ŒnC1�. �

Lemma 2. Let H be a simple hypergraph such that 9n � 1; 9E 2H; jEj > n. Then there are n C 1 transversals for H, t1; t2; :::; tnC1,such that 2

nC1.ti /i2ŒnC1� is not a transversal for H.

Proof. Let H be a simple hypergraph and let n � 1 be an arbitraryinteger such that for some E 2 H; jEj > n. Suppose thatE D fx1; x2; :::; xi ; :::; xnC1; :::g. Then it is possible to construct nC 1transversals for H, t1; t2; :::; tnC1 such that 8i 2 ŒnC 1�, ti \E D fxig;otherwise 9E0 2 H such that E0 � E, contrary to the assumption that His simple. But then 2

nC1.ti /i2ŒnC1� \ E D ;.�

Theorem 4. 8n � 1, if L is a modal logic which is closed under ŒRR�,ŒPL�, ŒUS�, and modus ponens, then if ŒKn� 2 L, it follows that thecanonical frame for L is n-bounded.

Proof. Assume that L is a modal logic which satisfies the antecedentconditions, including that ŒKn� 2 L for some n � 1. We show that8x 2 UL, 8E 2 HL.x/, jEj � n. Suppose not. Let x 2 ULbe such that E 2 HL.x/ and jEj > n. From Lemma 2 it followsthat there are n C 1 transversals for HL.x/, t1; t2; :::; tnC1, such that2nC1.ti /i2ŒnC1� is not a transversal for HL.x/. But since ti is a transver-sal for HL.x/ .i 2 Œn C 1�/, 9Ei � ti such that Ei 2 T .HL.x//, i.e.,from Proposition 2, Ei 2 fj j W � 2 xg. So let Ei D j i j .i 2 ŒnC1�/,

4.4. WEAKLY AGGREGATIVE MODAL LOGIC 57

where � i 2 x. Since ŒKn� 2 L, and L is closed under uniform sub-stitution, � 1 ^ ::: ^ � nC1 � � 2

nC1. i /i2ŒnC1� 2 x. Therefore� 2nC1. i /i2ŒnC1� 2 x, and thus ˆML

x � 2nC1. i /i2ŒnC1� (Theorem 1), in

which case k 2nC1. i /i2ŒnC1�kML is a transversal for HL.x/. But:

k 2

nC 1. i /i2ŒnC1�kML D 2

nC 1.k ikML/i2ŒnC1�

D 2

nC 1.j i j/i2ŒnC1�

D 2

nC 1.Ei /i2ŒnC1�

And 2nC1.Ei /i2ŒnC1� � 2

nC1.ti /i2ŒnC1�. Therefore 2nC1.Ei /i2ŒnC1� is

not a transversal for HL.x/, which is absurd. �

Since 8n � 1, Kn is sound with respect to the class of all normaln-bounded hyperframes (see Lemma 1), given Theorems 2 and 4 wehave:

Corollary 2. 8n � 1, the modal system Kn is determined by the classof all normal n-bounded hyperframes.

In closing we have:

Theorem 5. 8n � 1; the system Kn is complete with respect to theclass of all .nC 1/-ary relational frames.

Proof. Assume that ˛ is valid with respect to the class of all .n C 1/-ary relational frames. Then ˛ is valid on the .n C 1/-ary relationaltransformation of the canonical hyperframe FKn for Kn (Theorems 2,4). Therefore, by Theorem 3, ˛ is valid on the canonical model MKn

for Kn, whence `Kn ˛. �

Five

Polyadic Modal Logics and TheirMonadic Fragments

KAM SING LEUNG AND R.E. JENNINGS

Abstract

It is well known that the smallest normal modal logic, called K, is deter-mined by the class of binary relational frames. The collection of normalmodal logics can be expanded by considering polyadic modal languages,each possessed of an n-ary modal connective and interpreted in (n+1)-aryrelational frames. We survey a number of such logics and their classes offrames, and give metatheorems of soundness, completeness, and defin-ability. Finally a series of monadic logics are obtained by diagonalizingthe n-ary modal connectives. These monadic fragments support aggrega-tion principles which are weaker than that of K, thus making them moresuitable than K for such applications as deontic reasoning.

5.1 Introduction

The originating pulse of virtually all of the work presented in this vol-ume lay in the inexpressibility, within the received Kripkean modalsystems, of one or two deontically fundamental distinctions. The firstwas the distinction between deontic conflict, the common enough moral

59

60 CHAPTER 5. POLYADIC MODAL LOGICS

pickle in which, through incompetence or inattention, or circumstance,we find ourselves forced to neglect one moral duty to fulfill another. Thetopic has confronted moral systematizers for at least two hundred years.John Stuart Mill evidently supposed that if obligations are objectivelygrounded, any apparent such conflict can be no more than the startingpoint of a moral calculation that would ultimately dissolve it. Any like-minded philosopher might find little to complain of in the deontic read-ing of the modal� of some of the weaker Kripkean systems as It oughtto be the case that, where a claim of deontic conflict�˛^�:˛ (a con-junction of contradictory obligations) is equivalent to a claim of a singleobligatory contradiction �?. Most philosophers are convinced of theprinciple embodied in the Kantian slogan ”Ought implies can”, repre-sentable in the weak form ŒCon� :�? and some are no doubt happyto learn that as a matter of logic, that fundamental principle, of itself,rules out moral conflict, being equivalent to ŒD� �p ! :�:p. If itis already a matter of logic, then it would seem that the question as towhether obligations are objectively grounded need not be answered. Forsome, no doubt, that resolution is a vindication of the Kripkean analysisof deontic necessity. For such people, it is an early discovery of deonticlogic that the principle (call it “Kant”) that ought implies can is equiv-alent to the principle that there are no moral conflicts, and allows thelatter to be elevated to the status of “The Deontic Law”.

But for deontic logicians, there can be no such repose; nor for moralphilosophers who accept the Kantian proscription of unfulfillable singleobligations, but who recognize the presence of pairs or triples or largerfamilies of obligations that force us to a kind of moral triage. For suchtheorists even the question as to what principles of deontic reasoning canbe relied upon to constrain deontic conclusions remains itself a deonticmatter. Logic does not relieve us of the responsibility to sort out suchmesses; the existence of such messes forces us to choose among logics.For such theorists, the distinctness of the two cases is itself a matter oflogic. Accordingly, no system that does not preserve it is adequate.

For the healthy skeptic, every principle of logic can be interrogatedas to its suitableness for a particular application. For the deontic case,this applies to all of the principles that the Kripke systems force upon us.


The principle ŒK��p^�q ! �.p^q/ is certainly suspect, but so arethe principles of normality and monotonicity. The former is embodiedin the rule of necessitation ŒRN� ` ˛ H) ` �˛, which makes everytheorem an obligation. There would seem to be wanting in the moraltradition, that no obligation is unshirkable. Indeed, when we reflect thatlanguage is an evolved biological characteristic of humans, and morallanguage in particular is likely the product of selective social pressures,the exclusion of logically warranted sentences has some point. It is pre-cisely the contingency of social arrangements that presses us to preservethe language by which we promote some and discourage others.

Again, the principle ŒRM� ` ˛ ! ˇ H) ` �˛ ! �ˇ (the ruleof monotonicity), which takes all logically provable consequences ofobligatory acts as themselves obligations, can be ruled out on the relatedgrounds that theorems are among the logically provable consequencesof every obligatory act. Thus the adoption of a rule excluding theoremsfrom obligation, would seem to force some restriction on [RM] or adoptas the underlying propositional system, one weaker that the classicalPL.

Now all of logic is, within varying limits, negotiable. Systems oflogic are artifacts, and can be fashioned variously to serve varying goals,howsoever local. Nevertheless, there remains such a thing as pure re-search, and all of current deontic logical research must yet come underthat rubric. So there is some point to exploring systems and classes ofsystems that have no other desirable feature than that they admit or denyparticular distinctions, whatever other, independent distinctions they ne-glect. We might explore such a system for no purpose other than to sat-isfy our curiosity or to expand our understanding. In doing so, we mightdo well to proceed gradually from the familiar to less familiar, retaining,for the sake of illumination, principles which we expect ultimately to re-ject. We gain another benefit from this Fabian approach, for we learnabout relationships and dependencies among principles that are maskedin the more particular setting, but become evident in the more general.

The simple expedient of generalizing the semantic analysis of the�to n-ary relational frames, substituting the binary truth-condition

M; x ˆ �˛ ” .8y;Rxy H) M; y ˆ �˛/


with the general

M; x ˆ �˛ ”.8y1; : : : ; yn; Rxy1 � � �yn H) 9i.1 � i � n/;M; yi ˆ ˛/

preserves both [RM] and [RN] as rules, but fails to validate [K]. Instead,it validate the considerably weaker

ŒKn� �p1 ^ � � � ^�pnC1 ! �nC1_

iD1

nC1_

jDiC1.pi ^ pj /:

In consequence, it distinguishes [D], which corresponds to no first-orderrestriction on n-ary R, and [Con], which corresponds to n-ary seriality:

8x; 9y1; : : : ; yn W Rxy1 � � �yn:As we demonstrate below, the matter is not so simple as that. In the

first place, although we do not discuss the matter here, a generalizationto a set of n binary relations would serve to make at least some of thedistinctions that a single binary relation conceals. And n-ary relationalstructures introduce their own conflations. Nor is the language of unary�. the most natural or most expressive modal language for a study ofn-ary relational structures. So we begin with a discussion of the cor-respondences between the language of an (n-1)-ary modal connectiveand the language of a single n-ary relation. Again, once we have seenthe greater generality, it becomes evident that such a language generatesmore than one monadic fragment. In this introductory essay, however,we consider only the diagonal fragments of n-ary systems.

5.2 Polyadic modal languages and multi-ary relationalframes

In this section, we introduce a series of modal languages, each of whichextends the language of propositional logic with an n-ary modal oper-ator. The frames and models for interpreting these languages are thendescribed.

5.2. POLYADIC MODAL LANGUAGES AND MULTI-ARYRELATIONAL FRAMES 63

Definition 1. The n-adic modal language, denoted Ln, has thefollowing symbols: denumerably many propositional variables(p1, p2, p3, . . . ), connectives (?, :, _, and�n), and punctuations(the left and right parentheses ( and ), and the comma ,). The set ofLn-formulas is specified by the following rule in BNF:

˛ WWD pj?j:˛j.˛1 _ ˛2/j�n.˛1; : : : ; ˛n/

where p is any propositional variable.

Note that ? is a nullary connective while :, _, and �n are unary,binary, and n-ary connectives, respectively. Other connectives such as>, ^,!,$, andÞn (the dual of�n) are introduced as shorthand:

> abbreviates :?.

.˛ ^ ˇ/ abbreviates :.:˛ _ :ˇ/.

.˛ ! ˇ/ abbreviates .:˛ _ ˇ/.

.˛ $ ˇ/ abbreviates ..˛ ! ˇ/ ^ .ˇ ! ˛//.

Þn.˛1; : : : ; ˛n/ abbreviates :�n.:˛1; : : : ;:˛n/.We mention here some of the meta-logical conventions adopted in

this paper. Outermost parentheses are omitted when writing formulas._ and ^ bind more strongly than ! and $. Throughout this paper,we use lower case letters from the Greek alphabet ˛, ˇ, , . . . to denoteformulas, and upper case letters � ,�,†, . . . , to denote sets of formulas.We write 8, 9, :, &, H) , and ” (or iff) for “for every”, “thereexists”, “it is not the case that”, “and”, “(if . . . ) then”, and “if and onlyif”, respectively. Other meta-logical symbols will be introduced whenneeded.

Definition 2. An n-ary relational frame F is a duple hU;Ri, whereU , the universe of the frame, is a non-empty set of points, and R


an n-ary relation on U . Given an n-ary relational frameF D hU;Ri, an n-ary relational model M is a duple hF; V i (or atriple hU;R; V i) where V is a valuation, i.e. a function assigningeach propositional variable a set of points in U .

Definition 3. An Ln-formula ˛ is said to be true at a point x in an.nC 1/-ary relational model M D hU;R; V i (notation:M; x ˆ ˛) according to the following set of inductive conditions(where M; x 6ˆ ˛ means that ˛ is false at x in M):

For every propositional variable pi , M; x ˆ pi iffx 2 V.pi /.M; x 6ˆ ?.

M; x ˆ :˛ iff M; x 6ˆ ˛.

M; x ˆ ˛ _ ˇ iff M; x ˆ ˛ or M; x ˆ ˇ.

M; x ˆ �n.˛1; : : : ; ˛n/ iff 8y1; : : : ; yn; Rxy1 � � �yn H)9yi .1 � i � n/ WM; yi ˆ ˛i .

Given the above definition ofÞn, we have the following condition:

M; x ˆ Þn.˛1; : : : ; ˛n/ iff 9y1; : : : ; yn W Rxy1 � � � yn &8yi .1 �i � n/;M; yi ˆ ˛i .

Truth conditions for the other defined connectives are straightforwardand are omitted here.

Definition 4. An Ln-formula ˛ is said to hold in a model M

(notation: M ˆ ˛) if for every x in M, we have M; x ˆ ˛.Moreover ˛ is said to be valid on a frame F (notation: F ˆ ˛) if itholds in every model on F. If it is valid on every frame in a class C

5.3. NORMAL POLYADIC MODAL LOGICS 65

of frames, we say that it is valid on C (notation: C ˆ ˛). If ˛ isvalid on the class of all frames, we simply say that ˛ is valid(notation: ˆ ˛).

The presentation of formulas of an n-adic modal language and prop-erties of an n-ary relational frame will be made simpler by adopting thefollowing shorthand.

Notation 1. 1. We often drop the subscripts of �n and Þn sincetheir arities are obvious from the number of propositions that arewithin their scope.

2. Ep is an n-termed sequence p1; p2; : : : ; pn, and Epi is an n-termedsequence pi;1; pi;2; : : : ; pi;n. Similarly for Ex and Ex1 etc.

3. ?k is a k-termed sequence of ?’s, and similarly for >k .

5.3 Normal polyadic modal logics

A formal system S (in an object language) is essentially a proof-theoreticentity: it consists of a decidable set of formulas (called its axioms) and afinite set of inferential rules, each of which specifies what formula (con-clusion) can be derived from other formulas (hypotheses). A formula ˛is said to be provable from a set † of formulas (called assumptions) inS if there is a finite sequence of formulas, with the last member being ˛,and each member of the sequence being an axiom, or an assumption, orthe conclusion of a rule of inference applied to some previous formulasin the sequence. We call such a sequence a proof of ˛ from † in S, andrecord its existence by writing † `S ˛ or simply † ` ˛ if the systemis clear in the context. If ˛ is provable from the empty set of formulas,we call it a theorem of S and write `S ˛ or simply ` ˛ if the systemis obvious. S is said to be inconsistent if the false is a theorem of it(`S ?), consistent if not (6`S ?).

A logic, in comparison with a system, is usually characterized as aset of formulas (in an object language) that is closed under certain rules


of inference. Given our definition of formal systems above, the set oftheorems of a formal system constitutes a logic, but the converse is notgenerally true: there are logics which cannot be axiomatized as formalsystems. Since we deal with axiomatizable logics only in this paper, wewill no longer make the distinction between a system and a logic, anduse the terms interchangeably from now on.

Definition 5. An n-adic modal logic (in the modal language Ln)is a consistent system that has the set of all the tautologies ofpropositional logic (or a suitable subset of it) included in its set ofaxioms, and the following rules in its set of inferential rules:

ŒMP�˛; ˛ ! ˇ

˛

ŒUS�` ˛

` ˛Œpi=ˇ�where ˛Œpi=ˇ� is the formula that results from substituting ˇ forevery occurrence of pi in ˛.

Note that an n-adic modal logic (or simply an n-adic logic) is anextension of propositional logic in the sense that its set of theoremsincludes all the tautologies of propositional logic and is closed underthe rules of modus ponens and uniform substitution. We express thesame thing by saying that an n-adic logic includes PL. (Here PL refersto the set of tautologies together with the rules ŒMP� and ŒUS�.)

Since the “early” days of modern modal logic, logicians have workedon monadic systems characterized as “normal”. Normality in this con-text means that the systems or logics have the following rules and ax-ioms in addition to PL.

ŒRM�` ˛ ! ˇ

` �˛ ! �ˇŒRN�

` ˛` �˛

ŒK� �p ^�q ! � .p ^ q/


The smallest normal (monadic) system is called K (after Kripke). Vari-ous logics are obtained by adding axioms to K.

KCon W ŒCon� :�?KD W ŒD� �p ! ÞpKT W ŒT� �p ! p

KB W ŒB� p ! �Þ pK4 W Œ4� �p ! ��pK5 W Œ5� Þp ! �Þ p

In what follows, we generalize normal monadic logics to normal n-adic logics where n is a positive integer.

Definition 6. An n-adic logic is called normal if it has in additionto PL the following schemas of rules of inference and axioms(where 1 � i � n, and ˇ, q, and pi ^ q occur in the i th argumentplace of� as ˛i and pi do).

ŒRMin�

` ˛i ! ˇ

` �.˛1; : : : ; ˛i ; : : : ; ˛n/! �.˛1; : : : ; ˇ; : : : ; ˛n/ŒRNin�

` ˛i` �.˛1; : : : ; ˛i ; : : : ; ˛n/

ŒGin� �.p1; : : : ; pi ; : : : ; pn/ ^�.p1; : : : ; q; : : : ; pn/!�.p1; : : : ; pi ^ q; : : : ; pn/

We mention here an alternative way to characterize normal logicswhich makes use of PL, ŒRNin�, and the following schema of axiomsŒGŠin�.

ŒGŠin� �.p1; : : : ; pi ! q; : : : ; pn/! .�.p1; : : : ; pi ; : : : ; pn/!�.p1; : : : ; q; : : : ; pn//

Note that there are n instances of each of ŒRMin�, ŒRNin�, ŒG

in�, and ŒGŠin�.

We shall refer to any one of them, or all of them, simply as ŒRMn�,ŒRNn�, ŒGn�, and ŒGŠn�, respectively.


We call the smallest normal n-adic logic Gn. Note that G1 is just K.The name “Gn” comes from D.K. Johnston, who names the system afterGoldblatt for his introducing what amounts to G2 in an unpublishedpaper “Temporal Betweenness” (see ?). Other names have also beenused in the literature: EŒn� (E for entailment) in ?, K� where � is amodal similarity type in Blackbutn et al. (2001).

Various normal n-adic logics can be obtained by adding to Gn ax-ioms which are generalizations of their monadic counterparts.

Definition 7. The following normal n-adic systems extend Gn withthe indicated axioms. (In what follows, )

GnConn W ŒConn� :�?n

GnDn W ŒDn� � Ep !n_

iD1Þ .>i�1; pi ;>n�i /

GnTn W ŒTn� � Ep !n_

iD1pi

GnBn W ŒBn�n

iD1pi !

n

iD1�.?i�1;Þ Ep;?n�i /

Gn4n W Œ4n� � Ep !n

iD1�.?i�1;� Ep;?n�i /

Gn5n W Œ5n� Þ Ep !n

iD1�.?i�1;Þ Ep;?n�i /

Each of the axioms and rules of inference mentioned previously hasa dual form, which is logically equivalent to it (in any logic which pro-vides propositional logic). We list the dual forms here for future refer-


ence.

ŒRMÞin�` ˛i ! ˇ

` Þ.˛1; : : : ; ˛i ; : : : ; ˛n/! Þ.˛1; : : : ; ˇ; : : : ; ˛n/ŒRNÞin�

` ˛i` :Þ .˛1; : : : ;:˛i ; : : : ; ˛n/

ŒGÞin� Þ.p1; : : : ; pi _ q; : : : ; pn/!Þ.p1; : : : ; pi ; : : : ; pn/ _Þ.p1; : : : ; q; : : : ; pn/

ŒGÞin� :Þ .p1; : : : ; pi ; : : : ; pn/! .Þ.p1; : : : ; q; : : : ; pn/!Þ.p1; : : : ;:pi ^ q; : : : ; pn//

ŒConÞn� Þ>n

ŒDÞn�n

iD1�.?i�1; pi ;?n�i /! Þ Ep

ŒTÞn�n

iD1pi ! Þ Ep

ŒBÞn�n_

iD1Þ.>i�1;� Ep;>n�i /!

n_

iD1pi

Œ4Þn�n_

iD1Þ.>i�1;Þ Ep;>n�i /! Þ Ep

Œ5Þn�n_

iD1Þ.>i�1;� Ep;>n�i /! � Ep

To conclude this section, we list some of the theorems of normal n-adic logics below. (To highlight the axioms and rules used, we assumethe base logic is just an n-adic logic.)

Proposition 1. The following are provable in any n-adic logic (i.e. logicin Ln that provides PL), given the specified rules and axioms.


1. In the presence of ŒRMn� and ŒRNn�:

ŒBn�$� n_

iD1

n

jD1pi;j ! �.Þ Ep1; : : : ;Þ Epn/

�

2. In the presence of ŒRMn�:

ŒBn�! ŒGn�

3. In the presence of ŒTn� and Œ4n�:

� Ep $ �.?i�1;� Ep;?n�i /where 1 � i � n.

4. In the presence of ŒTn� and Œ5n�:

� Ep $ �.?i�1;Þ Ep;?n�i /where 1 � i � n.

5. In the presence of ŒRMn� and ŒGn�:

ŒConn�! ŒDn�

6. In the presence of ŒRNn�:

ŒDn�! ŒConn�

We leave the proof of the above proposition to the reader. Its importis as follows: (1) provides an alternative formulation of our ŒBn� insystems that have ŒRMn� and ŒRNn�; (2) is a generalization to the n-ary � of the result reported in ? for the unary �; (3) and (4) show thatŒ4n� and Œ5n� are in some sense reduction axioms (in systems that haveŒTn�); (5) and (6) show that ŒConn� and ŒDn� are indeed equivalent innormal n-adic systems. Observe that what has been called ŒDn� here isweaker than the following formula.

ŒDŠn� � Ep ! Þ Ep

5.4. FRAME DEFINABILITY 71

We remark here that both ŒD1� and ŒDŠ1� are the same formula, viz.ŒD��p ! Þp. So ŒDn� and ŒDŠn� represent two different ways to gen-eralize ŒD�. In normal monadic logics, ŒD� and ŒCon� are inter-derivable.Bearing this in mind, our ŒDn� appears to be a better way to generalizeŒD� than ŒDŠn� does. (This is also reflected in the correspondence resultin the next section: while ŒDn� and ŒConn� correspond to seriality, ŒDŠn�is not even first-order definable.)

5.4 Frame definability

Proposition 2. The following modal formulas correspond to the indi-cated first-order conditions:

ŒConn� W .8x/.9 Ey/Rx Ey (Seriality)

ŒDn� W .8x/.9 Ey/Rx Ey (Seriality)

ŒTn� W .8x/Rxx � � � x (Reflexivity)

ŒBn� W .8x/.8Ey/.Rx Ey ! .8yi 2 Ey/Ryix � � � x/ (Symmetry)

Œ4n� W .8x/.8Ey/.Rx Ey ! .8yi 2 Ey/.8Ez/.Ryi Ez ! RxEz//(Transitivity)

Œ5n� W .8x/.8Ey/.Rx Ey ! .8yi 2 Ey/.8Ez/.RxEz ! Ryi Ez//(Euclideanness)

Proof. For correspondence between any one of the modal formulas listedabove (say ˛) and the respective first-order condition (say �), we provethat a frame F validates ˛ if and only if it validates �. The if directionis trivial and is omitted here. For the only if direction, we show that if F

does not validate �, then there is a model on F in which ˛ fails at somepoint in the model.

For ŒConn�, assume that F D hU;Ri is not serial. Then there existsan x inU which is not related to any n-tuple of points inU . Consider anarbitrary model M on F. By the truth condition for �, any �-formulais true at x in M. So M; x 6ˆ :�?n.

For ŒDn�, assume, as in the case of ŒConn�, that F D hU;Ri is notserial, and so some x in U is not related to any n-tuple of points inU . Consider an arbitrary M D hF; V i. Trivially M; x ˆ � Ep, but


M; x 6ˆ Þ.>i�1; pi ;>n�i / for any i such that 1 � i � n. So M is acounter-model of ŒDn�.

For ŒTn�, assume that F D hU;Ri is not reflexive, i.e. there is an xin U such that Rxx � � � x does not hold. Consider a model M D hF; V iwith V.pi / D U � fxg for every i between 1 and n. It is clear thatM; x ˆ � Ep, since for any Ez such that RxEz, at least one of Ez, say zj , isnot x, and so M; zj ˆ pj . On the other hand, M; x 6ˆ p1 and similarlyfor p2; : : : ; pn. So M; x 6ˆ ŒTn�.

For ŒBn�, assume F D hU;Ri is not symmetric, i.e. there exist anx 2 U and a Ey 2 U n such that Rx Ey and for some yi 2 Ey, it is false thatRyix � � � x. Consider a model M D hF; V i with V.p1/ D fxg, . . . , andV.pn/ D fxg. Clearly, M; x ˆ p1 ^ � � � ^ pn. But M; yi 6ˆ Þ Ep, for ifM; yi ˆ Þ Ep then there exists Ez such that Ryi Ez and M; zj ˆ pj for allj from 1 to n, which implies that zj D x and so Ryix � � � x, contraryto assumption. So M; x 6ˆ �.?i�1;Þ Ep;?n�i /. It thus follows thatM; x 6ˆ ŒBn�.

For Œ4n�, assume F D hU;Ri is not transitive, i.e. there exist x 2 U ,Ey 2 U n, and Ez 2 U n such that Rx Ey, Ryi Ez (for some yi 2 Ey), and:RxEz. Consider a model M D hF; V i with V.p1/ D U � fz1g, . . . ,and V.pn/ D U � fzng. We argue that M; x ˆ � Ep as follows: foran arbitrary Ew 2 U n such that Rx Ew and for some wj 2 Ew, we havewj 6D zj and so M; wj ˆ pj . However, M; x 6ˆ �.?i�1;� Ep;?n�i /since M; yi 6ˆ � Ep (the reason being that Ryi Ez, M; z1 6ˆ p1, . . . , andM; zn 6ˆ pn). In other words, M; x 6ˆ Œ4n�.

For Œ5n�, assume F D hU;Ri is not euclidean, i.e. there existx 2 U , Ey 2 U n, and Ez 2 U n such that Rx Ey, RxEz, and :Ryi Ez, forsome yi 2 Ey. Consider a model M D hF; V i with V.p1/ D fz1g,. . . , and V.pn/ D fzng. Clearly M; x ˆ Þ Ep. However M; x 6ˆ�.?i�1;Þ Ep;?n�i / or equivalently M; x ˆ Þ.>i�1;�: Ep;?n�i /,since M; yi ˆ �: Ep, for which we argue as follows: assume arbitraryEw 2 U n such that Ryi Ew, then wj 6D zj (for some wj 2 Ew) and soM; wj 6ˆ pj , i.e. M; wj ˆ :pj . In other words, M; x 6ˆ Œ5n�. �

Theorem 2. The classes of .nC1/-ary relational frames for the follow-

5.5. SOUNDNESS AND COMPLETENESS 73

ing normal n-adic logics are as indicated.

GnConn W Serial framesGnDn W Serial framesGnTn W Reflexive framesGnBn W Symmetric framesGn4n W Transitive framesGn5n W Euclidean frames

Proof. The theorem follows directly from Proposition 2. �

5.5 Soundness and completeness

A consequence of Theorem 2 is the soundness of those logics with re-spect to their classes of frames. In this section, we demonstrate that theyare also complete.

Our strategy of proving the completeness of a normal n-adic logic Lwith respect to a class C of .nC 1/-ary relational frames is to show thatevery set of Ln-formulas consistent in L has a model on a frame in C.In fact, for any normal modal logic, there exists a model that satisfiesany consistent set of formulas. (We call this model the canonical modelof the logic, and the corresponding frame its canonical frame.) Giventhis result, all that remains to show the completeness of L with respectto the class C of frames is to show that the canonical frame of L belongsto C.

In the following, we first define the canonical model of a normal n-adic logic. Before showing that the canonical model is indeed a modelfor any consistent set of formulas, we prove an existence lemma and atruth lemma pertaining to such a logic and its canonical model. (Theproof for what we call the existence lemma here is based on ?. ? hasanother proof. For a more recent version, readers can check Blackbutnet al. (2001).)

Definition 8. [Canonical frames and models] Let L be a normaln-adic logic (in modal language Ln). The L-canonical model,denoted ML, is a triple hUL; RL; VLi where:


UL is the set of all L-maximal consistent set of Ln-formulas.

For every x, y1, . . . , and yn, RLxyi � � � yn iff the followingcondition holds:

�.˛1; : : : ; ˛n/ 2 x H) 9˛i .1 � i � n/ W ˛i 2 yi :

For every x, x 2 VL.pi / iff pi 2 x.

We call the duple hUL; RLi the canonical frame of L.

Lemma 1 (Existence Lemma for normal n-adic logics). Let ML DhUL; RL; VLi be the canonical model of a normal n-adic logic L. Forany point x 2 UL and Ln-formulas ˛1; : : : ; ˛n, if :�.˛1; : : : ; ˛n/ 2 x,then there exist y1; : : : ; yn 2 UL such that :˛1 2 y1, . . . , and :˛n 2yn, and RLxy1 � � � yn.

Proof. Assume:�.˛1; : : : ; ˛n/ 2 x. We show, by induction, that thereexist y1; : : : ; yn 2 UL such that each yi (1 � i � n) satisfies both ofthe following requirements.

(E1) :˛i 2 yi :(E2) For any formulas 1; : : : ; i�1, ˇ, if : 1 2 y1, . . . , : i�1 2

yi�1, and�. 1; : : : ; i�1; ˇ; ˛iC1; : : : ; ˛n/ 2 x, then ˇ 2 yi .For the existence of y1, we first show that y01 defined by letting

y01 D f:˛1g [ fˇj�.ˇ; ˛2; : : : ; ˛n/ 2 xg

is L-consistent. Assume, for reductio, y01 is not L-consistent. Then, forsome ˇ1; : : : ; ˇm 2 fˇj�.ˇ; ˛2; : : : ; ˛n/ 2 xg, the following hold.

fˇ1; : : : ; ˇm;:˛1g `L ?`L ˇ1 ^ � � � ^ ˇm ! ˛1`L �.ˇ1 ^ � � � ^ ˇm ! ˛1; ˛2; : : : ; ˛n/ .ŒRNn�/`L �.ˇ1 ^ � � � ^ ˇm; ˛2; : : : ; ˛n/! �.˛1; ˛2; : : : ; ˛n/ .ŒGn�/`L

VmjD1�. j ; ˛2; : : : ; ˛n/! �.˛1; ˛2; : : : ; ˛n/ .ŒGn�/


Since�. j ; ˛2; : : : ; ˛n/ 2 x for every j , and x is maximal L-consistent,we have �.˛1; ˛2; : : : ; ˛n/ 2 x. But this is impossible, for by assump-tion :�.˛1; ˛2; : : : ; ˛n/ 2 x. Thus, by reductio, y01 is L-consistent andso has a maximal L-consistent extension y1 (by Lindenbaum’s Lemma).It is straightforward to see that y1 satisfies both requirements (E1) and(E2) (for i D 1).

To demonstrate the existence of the other members of the series, viz.,y2; : : : ; yn, assume that we already have y1; : : : ; yk 2 UL which satisfy(E1) and (E2) in place (where k < n). As in the case of y1, we definean initial set y0

kC1 that can be shown to have a maximal L-consistentextension ykC1 satisfying both (E1) and (E2). So let

y0kC1 D f:˛kC1g [ fˇj9 1; : : : ; k W : 1 2 y1; : : : ;: k 2 yk &

�. 1; : : : ; k; ˇ; ˛kC2; : : : ; ˛n/ 2 xg:To show that y0

kC1 is L-consistent, we assume otherwise. Then, forsome ˇ1; : : : ; ˇm 2 y0kC1 � f:˛kC1g, the following hold.

fˇ1; : : : ; ˇm;:˛kC1g `L ?`L ˇ1 ^ � � � ^ ˇm ! ˛kC1

For each j (1 � j � m), there exist : j:1 2 y1; : : : ;: j:k 2 yk suchthat

�. j:1; : : : ; j:k; j ; ˛kC2; : : : ; ˛n/ 2 x:Then by ŒRMn� and ŒGn� we get

�.m_

jD1 j:1; : : : ;

m_

jD1 j:k;

m

jD1j ; ˛kC2; : : : ; ˛n/ 2 x:

Since ˇ1^ � � � ^ˇm ! ˛kC1 2 x, we also have the following by ŒRNn�

�.m_

jD1 j:1; : : : ;

m_

jD1 j:k;

m

jD1j ! ˛kC1; ˛kC2; : : : ; ˛n/ 2 x:

Thus by ŒGn� we have

�.m_

jD1 j:1; : : : ;

m_

jD1 j:k; ˛kC1; ˛kC2; : : : ; ˛n/ 2 x:


Note that :WmjD1 j:1 2 y1, since : j:1 2 y1 for all j (1 � j � m),

and y1 is maximal L-consistent. Similarly, :WmjD2 j:2 2 y2, . . . , and

:WmjD1 j:k 2 yk . But

WmjD1 j:k 2 yk , since yk complies with our

requirement (E2). Hence we derive a contradiction. By reductio y0kC1

is L-consistent, and so has a maximal L-consistent extension ykC1. It isstraightforward to check that ykC1 satisfies requirements (E1) and (E2)(for i D k C 1).

We have now demonstrated the existence of y1; : : : ; yn 2 UL allof which satisfy requirements (E1) and (E2). It remains to show thatRLxy1 � � �yn. Assume that for any ˇ1; : : : ; ˇn, �.ˇ1; : : : ; ˇn/ 2 x,ˇ1 … y1, . . . , ˇn�1 … yn�1. Then :ˇ1 2 y1, . . . , :ˇn�1 2 yn�1.Since yn satisfies (E2), we have ˇn 2 yn. Thus RLxy1 � � �yn accord-ing to the definition of RL. This completes our proof of the ExistenceLemma. �

Lemma 2 (Truth lemma for n-adic logics). Let L be a normal n-adiclogic. For any Ln-formula

ML; x ˆ ˛ ” ˛ 2 x:Proof. The proof is by induction on ˛. In the following we show themodal case only. Let ˛ be�.˛1; : : : ; ˛n/, and show that for an arbitraryx 2 UL,

ML; x ˆ �.˛1; : : : ; ˛n/ ” �.˛1; : : : ; ˛n/ 2 xby assuming the inductive hypothesis that the theorem holds for ˛1, . . . ,and ˛n.

For the direction H), assume that�.˛1; : : : ; ˛n/ … x, i.e.:�.˛1; : : : ; ˛n/ 2 x. Then, by the existence lemma, there exist y1; : : : ; yn 2UL such that :˛1 2 y1, . . . , and :˛n 2 yn, and RLxy1 � � �yn. Thenfor each i such that 1 � i � n, ˛i … yi and by the inductive hypothesisML; yi 6ˆ ˛i . Thus ML; x 6ˆ �.˛1; : : : ; ˛n/, as desired.

For the direction(H, assume that�.˛1; : : : ; ˛n/ 2 x. To show thatML; x ˆ �.˛1; : : : ; ˛n/, we consider arbitrary y1; : : : ; yn 2 U suchthat RLxy1 � � �yn. Then by the definition of RL, ˛i 2 yi for some iwhere 1 � i � n. It follows from the inductive hypothesis that M; yi ˆ˛i , whence we conclude that ML; x ˆ �.˛1; : : : ; ˛n/. �


Corollary 1. Let L be a normal n-adic logic. Then any L-consistent setof formulas is satisfiable in the canonical model of L.

Proof. Let† be an L-consistent set of formulas. By Lindenbaum Lemma,† can be extended to a maximal L-consistent set x of formulas. But ev-ery formula in† is true at x in ML, the canonical model of L, accordingto the truth lemma. In other words, † is satisfiable in ML. �

Theorem 3. The following normal n-adic logics are complete with re-spect to the indicated classes of .nC 1/-ary relational frames:

Gn W All framesGnConn W Serial framesGnDn W Serial framesGnTn W Reflexive framesGnBn W Symmetric framesGn4n W Transitive framesGn5n W Euclidean frames

Proof. For the completeness of a logic L with respect to a class C offrames, it suffices to show that every L-consistent set of formulas is sat-isfiable in a model on a frame in class C. In the case of a normal modallogic L, we need only show that its canonical model belongs to C, forevery L-consistent set of formulas is satisfiable in the canonical modelaccording to the above corollary. (In the following, we use ML and RLfor the canonical model and relation of the modal logic in context.)

For Gn. It suffices to note that the the canonical model of Gn is an.nC 1/-ary relational frame.

For GnConn. Let x in ML be arbitrary. Since ŒConn� is in x, wehave by the fundamental theorem ML; x ˆ Þ>n. Hence the canonicalrelation is serial.

For GnDn. Let x in ML be arbitrary. Since both ŒDn� and �>n arein x, we haveÞ>n in x as well. Hence the canonical relation is serial.

For GnTn. To show that the canonical relation is reflexive, we as-sume�.˛1; : : : ; ˛n/ 2 x, where x in ML, and formulas ˛1, . . . , and ˛nare arbitrary. Since ŒTn� is in x, we have ˛1 _ � � � _ ˛n in x. Then at


least one of ˛1, . . . , ˛n is in x. Hence RLxx � � � x by the definition ofthe canonical relation.

For GnBn. We assume RLxy1 � � � yi � � �yn where x, y1; : : : ; yn inML are arbitrary, and show that RLyix � � � x. This will follow if wecan show that if ˛1 … x, . . . , and ˛n … x then �.˛1; : : : ; ˛n/ … yi .So assume ˛1 … x, . . . , and ˛n … x. Then :˛1 2 x, . . . , and:˛n 2 x. Then :˛1 ^ � � � ^ :˛n 2 x. Since ŒBn� 2 x, we have�.?i�1;Þ.:˛1; : : : ;:˛n/;?n�i / 2 x. Given the assumption thatRLxy1 � � �yn, we concludeÞ.:˛1; : : : ;:˛n/ 2 yi , i.e. :�.˛1; : : : ; ˛n/ 2yi . But this just means that�.˛1; : : : ; ˛n/ … yi , which is what we want.

For Gn4n. We assume RLxy1 � � � yi � � �yn and RLyiz1 � � � zn wherex, y1; : : : ; yn, z1; : : : ; zn are arbitrary points in ML. To show thatRLxz1 � � � zn, we argue that for any formulas ˛1; : : : ; ˛n, if�.˛1; : : : ; ˛n/ 2 x, then j 2 zj , for some j where 1 � j � n. Soassume�.˛1; : : : ; ˛n/ 2 x. Since Œ4n� 2 x, we have�.?i�1;�.˛1; : : : ; ˛n/;?n�i / 2 x. Then �.˛1; : : : ; ˛n/ 2 yi , giventhat RLxy1 � � �yn. Then, as desired, we have j 2 zj for some j suchthat 1 � j � n, given that RLyiz1 � � � zn.

For Gn5n. We assume thatRLxy1 � � �yi � � �yn andRLxz1 � � � zn wherex, y1; : : : ; yn, z1; : : : ; zn are arbitrary. To show thatRLyiz1 � � � zn, we argue that for any formulas ˛1; : : : ; ˛n, if j … zj ,where j ranges from 1 to n, then �.˛1; : : : ; ˛n/ … yi . So assume forall j from 1 to n, j … zj . Since RLxz1 � � � zn, we have�.˛1; : : : ; ˛n/ … x, i.e. :�.˛1; : : : ; ˛n/ 2 x, i.e. Þ.:˛1; : : : ;:˛n/ 2x. Then �.?i�1;Þ.:˛1; : : : ;:˛n/;?n�i / 2 x, given that Œ5n� 2 x.Then Þ.:˛1; : : : ;:˛n/ 2 yi , given that RLxy1 � � �yn. In other words,�.˛1; : : : ; ˛n/ … yi , as desired. �

5.6 Diagonal fragments of normal polyadic modal logics

We introduce the unary operator� and its dualÞ into the n-adic modallanguage Ln by the following identities:

�˛ D �n.˛; : : : ; ˛/ Þ˛ D Þn.˛; : : : ; ˛/

5.6. DIAGONAL FRAGMENTS OF NORMAL POLYADIC MODALLOGICS 79

These unary operators can be described as “diagonalization” of theirn-ary counterparts: consider the set of n-tuples of formulas as an n-dimensional matrix, and the set of tuples whose coordinates are thesame formula as a diagonal across such a matrix. The truth conditionsfor the diagonal operators are as follows:

M; x ˆ �˛ iff 8y1; : : : ; yn; Rxy1 � � �yn H) 9yi .1 � i � n/ WM; yi ˆ ˛.

M; x ˆ Þ˛ iff 9y1; : : : ; yn W Rxy1 � � �yn & 8yi .1 � i � n/;M; yi ˆ ˛.

Let Ln � � be the set of Ln-formulas that can be expressed byusing only the unary � and truth-functional connectives. We call theset of theorems of Gn that can be so expressed, i.e. the intersection ofGn and Ln � �, the diagonal fragment of Gn. This fragment has beenshown to be axiomatized by the following system Kn. For details, see?, ?, and ?.

Definition 9. Kn has PL, ŒRM�, ŒRN�, and the following axiom.

ŒKn� �p1 ^ � � � ^�pnC1 ! �nC1_

iD1

nC1_

jDiC1.pi ^ pj /

Note that K1 is just K. Kn can be described as a “weakly aggregativemodal logic” since its aggregate principle ŒKn� is a weakening of thefollowing principle of complete aggregation, which is a theorem of K.

�˛1 ^ � � � ^�˛n ! �.˛1 ^ � � � ^ ˛n/

Kn can be extended by adding the familiar formulas ŒK�, ŒCon�, ŒT�,ŒB�, Œ4�, and Œ5�, and the resulting systems are called, respectively, KnK,KnCon, KnT, KnB, Kn4, and Kn5. Note that KnK is just K1 or K, sinceŒKn� is derivable from ŒK�. In the following, we list correspondence anddetermination results for these formulas and logics.


Theorem 4. The following modal formulas correspond to the indicatedfirst-order conditions.

ŒK� W .8x/.8Ey/.Rx Ey ! .9yi 2 Ey/Rxyi � � �yi /(Quasi-binarity)

ŒCon� W .8x/.9 Ey/Rx Ey (Seriality)

ŒT� W .8x/Rxx � � � x (Reflexivity)

ŒB� W .8x/.8Ey/.Rx Ey ! .9yi 2 Ey/Ryix � � � x/ (Symmetry�)

Œ4� W .8x/.8Ey; Ez1; : : : ; Ezn/.Rx Ey ^Ry1Ez1 ^ � � � ^RynEzn !.9 Ew � Ez1 [ � � � [ Ezn/Rx Ew/ (Transitivity�)

Œ5� W .8x/.8Ey; Ez/.Rx Ey ^RxEz ! .9yi 2 Ey/Ryi Ez/(Euclideanness�)

Note: In the condition of transitivity, 9 Ew � Ez1 [ � � � [ Ezn means thefollowing: there exists a Ew such that every wk 2 Ew belongs to the set ofzi;j ’s where 1 � i; j � n.

Observe that the frame properties corresponding to ŒB�, Œ4�, and Œ5�are weaker than those corresponding to ŒBn�, Œ4n�, and Œ5n�, which isexpected since the former formulas are derivable from the latter ones.We distinguish the weaker properties from the stronger ones by the sign�.

Theorem 5. The following diagonal logics are determined by the indi-cated classes of .nC 1/-ary relational frames.

Kn W All framesKnK W Quasi-binary framesKnCon W Serial framesKnT W Reflexive framesKnB W Symmetric� framesKn4 W Transitive� framesKn5 W Euclidean� frames

Six

Preserving What?

GILLMAN PAYETTE AND PETER SCHOTCH

Abstract

In this essay Gillman Payette and Peter Schotch present an account of thekey notions of level and forcing in much greater generality than has beenmanaged in any of the early publications. In terms of this level of gen-erality the hoary notion that correct inference is truth-preserving is care-fully examined and found wanting. The authors suggest that consistencypreservation is a far more natural approach, and one that can, further-more, characterize an inference relation. But an examination of the usualaccount of consistency reveals problems that, in general, can be correctedby means of an auxiliary notion of inference (forcing) which relies upona kind of generalization of consistency, called level. Preservation of thelatter is shown to be another of the properties which characterize a logicand forcing is shown to preserve it. The essay ends with a sketch of a re-sult which locates forcing among all possible level-preserving inferencerelations.

6.1 Introduction

The (classical) semantic paradigm for correct inference is often giventhe name ‘truth-preservation.’ This is typically spelled out to the awe-struck students in some such way as:

81

82 CHAPTER 6. PRESERVING WHAT?

An inference from a set of premises, � , to a conclusion, ˛,is correct, say valid, if and only if whenever all the membersof � are true, then so is ˛.

This understanding of the slogan may be tried, but is it actually true?There is a problem: the way that ‘truth’ is used in connection with thepremises is distinct from the way that it is used with the conclusion. Inother words, this could be no better than a quick and dirty gloss. Thechief virtue of the formulation is that of seeming correct to the naiveand untutored.

But what of the sophisticates? They might well ask for the precisesense in which truth is supposed to be preserved in this way of unpack-ing. On the right hand side of the ‘whenever’ we are talking about thetruth of a single formula while on the left hand side we are talking aboutthe truth of a bunch of single formulas. Is it the truth of the whole gangwhich is ‘preserved?’

Of course it is open to the dyed-in-the-wool classicalist to replyscornfully that we need only replace the set on the left with the conjunc-tion of its members. In this way is truth preserved from single formulato single formula as homogenously as anyone could wish.

It is open, but not particularly inviting. In the first place, this strategyforces us to restrict the underlying language to one with conjunction andconditional connectives—that must operate in something like the usual(which is to say classical) way. There are enough who would chafeunder this restriction that a sensitive theorist would hesitate to imposeit.

Apart from the objection, we are inclined to think of this business ofcoding up the valid inferences in terms of their ‘corresponding condi-tionals’1 as an accident of the classical way of thinking, and that it is nopart of the definition of a correct account of inference. We also noticethat on the proposal, we are restricted to finite sets.

Setting aside this unpalatable proposal then, we ask how is this no-tion of truth-preservation supposed to work? Since there is no gang on

1We think this usage was coined first by Quine.


the right we seem to be talking about a different kind of truth, individ-ual truth maybe, from the kind we are talking about on the left—masstruth perhaps. Looked at in that somewhat jaundiced way, there isn’tany preserving going on at all, but rather a sort of transmuting.

The classical paradigm really ought to be given by the slogan ‘truthtransmutation.’ In passing from the gaggle of premises to the conclu-sion, gaggle-truth is transmuted into single formula truth. It may bemore correct to say that, but it makes the whole paradigm somewhatless forceful or even less appealing.

What we need in order to rescue the very idea of preservationismis to talk entirely about sets. So we shall have to replace the arbitraryconclusion ˛ with the entire set of conclusions which might correctly bedrawn from � . We even have an attractive name for that set—the theorygenerated by � or the deductive closure of � . In formal terms this is

C`.�/ D f˛j� ` ˛g

Now that we have sets, can we say what it is that gets preserved—can we characterize classical inference, for instance, as that relationbetween sets of formulas and their closures such that the property ˆ ispreserved?

We can see that gaggle-truth would seem to work here in the sensethat whenever � is gaggle-true so must be C`.�/, for ` the classicalnotion of inference at least. We are unable to rid ourselves, however,of the notion that gaggle-truth is somewhat lacking from an intuitiveperspective. Put simply, our notion of truth is carried by a predicatewhich applies to sentences, or formulas if we are in that mood. Theseare objects which might indeed belong to sets but they aren’t themselvessets. So however we construe the idea of a true set of sentences (orformulas) that construal will involve a stipulation or more charitably, anew definition.

Generations of logic students may have been browbeaten into ac-cepting: ‘A set of sentences is true if and only if each member of theset is true.’ But, it is a stipulation, and is no part of the definition of‘true.’ It doesn’t take very much imagination to think that somebodymight actually balk at the stipulation. Somebody who is attracted to the


idea of coherence for instance, might well want to say that truth must bedefined for (certain kinds of) sets first and that the sentential notion isderived from the set notion and not conversely. All of which is simplyto say that a stipulation as to how we should understand the phrase ‘trueset of sentences’ is unlikely to be beyond the bounds of controversy.2

It may gladden our hearts to hear then, that there is another prop-erty, perhaps a more natural one, which will do what we want. Thatalternative property is consistency.

6.2 Making A Few Things Precise

By a logic X , over a language L we understand the set of pairs h�; ˛isuch that � is a set of formulas from the language L and ˛ is a formulafrom that same language, and � `X ˛. In the sequel we frequentlyavoid mention of the language which underlies a given logic, when noconfusion will thereby be engendered.

This set of pairs is also referred to as the provability or inferencerelation of X . In saying this we expose our extensional viewpoint ac-cording to which there is nothing to a logic over and above its inferencerelation. This has the immediate consequence that we shall take twologics X and Y which have the same inference relation, to be the samelogic.

When X is a logic, we refer to the X-deductive closure of the set �by means of ‘CX .�/’

Unless the contrary is specified, Every logic mentioned below willbe compact, which is say that whenever � ` ˛ it follows that there mustbe some finite subset, say �, of � , which proves ˛.

In mentioning consistency, we have in mind some previously givennotion of inference, say `X . Each inference relation spawns a notion ofconsistency according to the formula

2It may be helpful here to consider an analogy between sentences and numbers, taken to beurelementen. We can define the idea of a prime number easily enough but be puzzled about howto define a prime set of numbers. Somebody might be moved to offer: ‘Why not simply define aprime set of numbers to be a set of prime numbers?’ The answer is likely to be: ‘Why bother?’Indeed the whole idea of a prime set of numbers seems bizarre and unhelpful. We can easilyimagine circumstances in which we would require a set of prime numbers but the reverse is truewhen it comes to a prime set.

6.2. MAKING A FEW THINGS PRECISE 85

� is consistent, in or relative to a logic X (alternatively, �is X-consistent) if and only if there is at least one formula ˛such that � °X ˛.

To say this in terms of provability rather than non-provability wemight issue the definition:

� is inconsistent in a logic X if and only if CX .�/ D S,where S is the set of all formulas of the underlying languageof X .

Where X is a logic, the associated consistency predicate (ofsets of formulas) for X , is indicated by CONX .

We were interested in how an inference relation might be character-ized in terms of preserving some property of sets. We have singled outconsistency as a natural property of sets and having done that we can seethat preservation of consistency comes very naturally indeed. The timehas come to say a little more exactly what we mean by ‘characterized.’In order to do this we shall be making reference to the following threestructural rules of inference.

[R] ˛ 2 � H) � ` ˛[Cut] �; ˛ ` ˇ & � ` ˛ H) � ` ˇ[Mon] � ` ˛ H) � [� ` ˛

Unless there is a specific disavowal, every inference relation we con-sider will be assumed to admit these three rules. It should be noted thaton account of [Mon], if the empty set ¿ is inconsistent in X , then theinference relation for that logic contains every pair h�; ˛i. In such acase we say that X is the trivial logic over its underlying language. Weshall take the logics we mention from now on to be non-trivial, barringa disclaimer to the contrary.

Let us say that an inference relation, say ’`X ’, preserves consistencyif and only if:


If � is X-consistent (in the sense of the previous definition),then so is CX .�/.

It is easy to see that every inference relation with [Cut] and [Mon]must preserve consistency since if the closure of a set � proves someformula, ˛, then by compactness some finite sequence of [Cut] opera-tions will lead to the conclusion that � proves ˛. It may be that we endup showing that some subset of � proves ˛, which is why we require[Mon] in this case.

We say that X preserves consistency in the strong sense when thecondition given above as necessary is also sufficient.

It is similarly easy to see that since every set is contained in its de-ductive closure by [R], and since inconsistency is preserved by super-sets, given [Mon], every inference relation satisfying the three structuralrules preserves consistency in the strong sense.

This is all very well, but we haven’t really gotten to anything thatwould single out an inference relation from among a throng of such, allof which preserve consistency. In order to do that it will be necessary totalk about a logic X preserving the consistency predicate of a logic Y ,in the strong sense.

A moment’s thought will show us that when the preservation is mu-tual, whenX and Y preserve each other’s consistency predicates, (whichimplies that they share a common underlying language) then they mustagree on which sets are consistent and which are inconsistent.

For consider, if CONX .�/ and Y preserves the X consistency pred-icate then CONX .CY .�//. Suppose that � is not Y -consistent, thenCY .�/ D S. By [R] CX .CY .�// D CX .S/ D S which is to say thatCY .�/ is not X -consistent, a contradiction. Similarly for the argumentthat � is consistent in Y and X preserves the Y consistency predicate.

When two logics agree in this way, i.e. agree on the consistentand inconsistent sets, we shall say that they are at evens. Another mo-ment’s thought reveals that two logics which are at evens will preserveeach other’s consistency predicates. Assume X and Y are at evens andCONX .�/ but CONX .CY .�//, where the overline indicates predicatenegation. Then CONY .CY .�// because X and Y agree on inconsistent

6.2. MAKING A FEW THINGS PRECISE 87

sets. Hence by idempotentcy of CY � is not consistent in Y , a contra-diction. Thus we have:

Proposition 1. Any two logicsX and Y over a language L are at evensif and only if X and Y preserve each other’s consistency predicates.

This is nearly enough to guarantee that X and Y are the same logic.All we need is a kind of generalized negation principle:

Definition 1. A logic X is said to have denial provided that forevery formula ˛, there is some formula ˇ such that CONX .f˛; ˇg/.

In such a case we shall say that ˛ and ˇ deny each other (inX , whichqualification we normally omit when it is clear from the context). Wewill assume that any logic we mention has denial.

Clearly if a logic has classical-like negation rules then it has denial,since the negation of a formula will always be inconsistent with theformula. Of course classically, there are countably many other formulaswhich are inconsistent with any given formula—namely all those whichare self-inconsistent. The generalized notion doesn’t require that therebe distinct3 denials for each formula, only that there be some or otherformula which is not consistent with the given formula.

Evidently, if two logics are at evens, then if one has denial, so doesthe other. In fact something stronger holds, namely:

Proposition 2. If two logics X and Y are at evens, and X has denialthen, for every formula ˛ there is some formula ˇ for which bothCONX .f˛; ˇg/ and CONY .f˛; ˇg/.

Finally, we shall require of our logics that they satisfy the principlethat denial commutes with provability in the right way:

[Den] � `X ˛ ” CONX .� [ fˇg/ where ˇ denies ˛.

Now we are ready to state our result:

3Distinct up to logical equivalence, it goes without saying.


Theorem 1 (Generalized Consistency Theorem). Two logics X and Yare at evens if and only if, X and Y are the same logic.

Proof. For this argument we split the equivalence into its necessary andsufficient halves.

( H) / Assume CX .�/ D CY .�/ for every set �—which is to say thatX = Y . To say that CONX .�/ is to say that the X-closure of �is S. But then so must be the Y -closure of � . Similarly, to saythat CONX .�/ is to say that there is some ˛ which is not in theX -closure of � , but then neither can ˛ be in the Y -closure of � ,hence CONY .�/. So X and Y are at evens.

((H) Suppose then that X and Y are at evens. Let � be a consistentset, which means by the assumption, that it is consistent in bothlogics. Assume for reductio that � `X ˛ and � °Y ˛, and let ˇdeny ˛. Thus,by [Den] CONX .� [ fˇg/ and CONY .� [ fˇg/, acontradiction.

�

6.3 What’s Wrong With This Picture?

To answer the question in the section heading, there really isn’t any-thing wrong with an approach which characterizes inference in terms ofpreserving consistency. It’s consistency itself, or at least many accountsof it, which casts a shadow over our everyday logical doings.

The way we have set things up, a set � of formulas is either consis-tent in a logic X , or it isn’t. But it doesn’t take much thought to see thatsuch an all-or-nothing approach tramples some intuitive distinctions. Inparticular, we may find the reason for the inconsistency to be of interest.

In the logic X , for example, there may be a single formula ı whichis, so to speak, inconsistent by itself. In other words CONX .fıg/. For-mulas of this dire sort are sometimes described as being self-inconsistent(in X) or absurd in X . By [Mon] any set of formulas which contains aself-inconsistent formula is bound to be inconsistent.

6.4. SPEAK OF THE LEVEL 89

Thinking of the possible existence of absurd formulas leads us tosharpen our previous notion of denial:

Definition 2. We shall say that X has non-trivial denial if andonly if for every non-absurd formula ˛ there is at least onenon-absurd formula ˇ which is not X -consistent with ˛.

We are now struck by the contrast betweenX -inconsistent sets whichcontain X -absurdities and those which do not. Isn’t there an importantdistinction between these two cases? If we think of consistency as adesirable property which we are willing to trouble ourselves to achieve,then the trouble will be light indeed if all we need do is reject absurdi-ties. On the other hand, an entire lifetime of angst may await those whowish to render consistent their beliefs or their obligations.4 There is agreat deal more that one could say on this topic and some of the currentauthors have said much of it. For now, we shall take it that the need fora distinction has been established and our job is to construct an accountof consistency which allows it.

We have in mind building upon what we have already discoveredinstead of pursuing a slash-and-burn policy. This means, among otherthings, that the predecessor account should appear as a special (or lim-iting) case of the new proposal. The intuitive distinction bruited above,is clearly a distinction between different kinds of inconsistency, or per-haps different degrees. We might think of one kind being worse than theother, which leads to a rather natural way of classifying inconsistency.

6.4 Speak of the Level

The account of inconsistency which we propose is a generalization ofthe one first suggested in the 20th Century, in the work of Jennings andSchotch5, namely the idea of a level (of incoherence, or inconsistency).

4In saying this, we assume that nobody is obliged to bring about anything impossible and thatwhatever is self-inconsistent cannot truly be a belief.

5See especially Schotch and Jennings (1980c, 1989).


The basic idea is that we can provide an intuitive measure of how incon-sistent a set is by seeing how finely it must be divided before all of thedivisions are consistent. What in the earlier account is stated in termsof classical provability, we now state in terms of arbitrary inference re-lations that satisfy the minimal conditions given in the earlier section.

It all begins with the notion of a certain kind of indexed collection ofsets being a logical cover for a set � of formulas, in the logic X . Firstwe need a special kind of indexed family of sets (of formulas).

Definition 3. A.�/ D ˚a0; a1; : : : ; a�

is an indexed set startingwith �, provided a0 D � and all the indices 0 : : : � are drawnfrom some index set I .

Definition 4. Let F be an indexed set starting with ¿. F is said tobe a logical cover of the set †, relative to the logic X , indicated byCOVX .F; †/, provided:

For every element a of the indexed family,CONX .a/ and

† � Si2ICX .ai /

So an X-logical cover for � is an indexed family of sets startingwith the empty set, such that there are enough logical resources in thecover to prove, in the logic X , each member of � . Evidently, given therule [R], f¿; �g will always be a logical cover of � if the latter set isX -consistent though it won’t, in general, be the least.

If F† is a logical cover for the set†, the cardinality jI j � 1 where Iis the index set for F†, is referred to as the width of the cover, indicatedby w.F†/.

In the special circumstance that all the members of a logical coverof † are disjoint, the cover is said to partition †.6

6This should be contrasted with a covering family being a partition. We can recover the latter

6.4. SPEAK OF THE LEVEL 91

And finally we introduce the notion at which we have hinted sincethe start of this section.

Definition 5. The level (relative to the logic X) of the set � offormulas of the underlying language of X , indicated by `X .�/ isdefined:

`X .�/ D8<:

minw.F/

ŒCOVX .F; �/� if this limit exists

1 otherwise

In other words: the X-level (of incoherence or inconsistency) of aset † in a logic X is the width of the narrowest X-logical cover of †, ifthere is such a thing, and if there isn’t, the level is set to the symbol1.

One might think that there will fail to be a narrowest logical coverwhen there is more than one—when several are tied with the least width,but this is a misreading of the definition. There might indeed be severaldistinct logical covers but there can only be one least width (which theyall share). The uniqueness referred to in the definition attaches to thewidth, not to the cover, so to speak.

The only circumstance in which there might fail to be a narrowestlogical cover, is one in which † has no logical covers at all. In thiscircumstance † must contain what we earlier called an absurd formula.

This notion satisfies both the requirement that it distinguishes be-tween inconsistent sets which contain absurd formulas, and those whichdon’t, and the requirement that the predecessor notion of consistencyrelative to X appears as a special case. For it is clear that if � is anX -consistent set of formulas which does not consist entirely of X theo-rems, then a narrowest logical cover of � is f¿; a1g where � � CX .a1/and CONX .a1/. So at least part of the earlier notion of CONX .�/ is cap-tured by `X .�/ D 1.

notion from this one by intersecting the covered set with each of the disjoint sets in the logicalcover.


There is even an interesting insight which comes out of this newidea. For there are two levels of X-consistency, 0 and 1. In our earliernaive approach, we thought of consistency as an entirely monolithicaffair, but once we give the matter some thought we see that the emptyset does indeed occupy a unique position in the panoply ofX -consistentsets. If we know only that � and† are both X -consistent, nothing at allfollows about theX-consistency of �[†. But we may rest assured thatboth †[¿ and � [¿ are X -consistent. And the same goes for the X-consequences of the empty set, namely X-theorems. By our definition,`X .�/ D 0 if � is empty or any set of X-theorems, and of course suchsets are consistent with any X-consistent set of formulas. It is temptingto call these level 0 sets hyperconsistent.

6.5 Level Preservation

So now that we have the concept of anX-level, should we be concernedabout preserving such a thing? Perhaps there is no need for such con-cern, since it is at least possible that the logic X preserves its own level,isn’t it? Well, in a word, no. It is not in general true that X preserveslevel beyond, of course the levels 0 and 1 of X-consistency. All thelogics we consider not only do that, but are characterized by doing that.

Suppose the set � contains not only the formula ˛ but also a denial ˇof ˛, although it does not contain any X -absurdities. Now, since X hasthe rule [R], � must X-prove both ˛ and ˇ. If X permits the arbitraryconjunction of conclusions, then � will prove an X -absurd formula,namely ˛ ^ˇ. Hence by the meaning of closure, that absurdity belongsto the X-closure of � , which must thus have X -level1. This amountsto a massive failure to preserve level.

The obvious question to raise is this: given that `X is a generaliza-tion of CONX , is it the case that level characterizes logics in the sameway that preserving consistency (in the strong sense) does? If not, thenit seems that the generalization is not perhaps as central a notion as theroot idea upon which it generalizes. Fortunately for supporters of thegeneral notion, we may prove the following generalization of the Gen-eralized Consistency Theorem.

6.5. LEVEL PRESERVATION 93

Theorem 2 (Level Characterization Theorem). Suppose that X and Yare inference relations over the same language L and let `X and `Y

be the level functions associated with the respective inference relations.ThenŒ`X .�/ D `Y .�/ for every set � of formulas of L� ” X D Y .

Proof. The proof depends upon the Generalized Consistency Theorem.

( H) ) Assume that `X .�/ D `Y .�/ for every set � of formulas of L.Then by definition the two agree on which sets have level 1 andlevel 0. But this is to say that X and Y agree on which sets areconsistent. But by the Generalized Consistency Theorem, any twosuch logics (logics which we say are at evens) must be identical.

((H) Assume that X D Y and suppose that for some arbitrary set �of formulas of L `X .�/ D �, for � some cardinal. Then, bythe definition, there is a narrowest X -logical cover F� such thatw.F�/ D �. Since X D Y , it must be the case by the GeneralizedConsistency Theorem that the two logics agree on consistency (inthe strong sense). Further, by definition each ai 2 F� is suchthat CONX .ai /. But then, since X and Y are at evens,CONY .ai /.Thus, F� must be a Y -logical cover of � of width �. Moreover,this must be the narrowest such logical cover or else by parity ofreasoning, there would be an X -logical cover of cardinality lessthan � contrary to hypothesis. Since � was arbitrary it followsthat `X and `Y must agree on all sets of formulas of the languageL.

�

This suggests that level is worth preserving, that it is a sort of naturallogical kind, but doesn’t show how the preservation may be carried out.It is time to repair that lack.

Perhaps the most straightforward route to preserving X-level is todefine a new inference relation based on X . Evidently the definitionin question must also connect somehow with the notion of X-level andthus ultimately to CONX (from now on we shall mostly drop reference


to the background logic, like X , when no confusion will result). Theprocess might have been informed by the ancient joke:

Question: How do you get down from an elephant?

Answer: You don’t get down from an elephant, you get downfrom a duck.

except in our case the question and answer would go:

Question: How do you reason from inconsistent sets?

Answer: You don’t reason from inconsistent sets, since everyformula follows in that case, you reason from consis-tent subsets.

In other words, an inconsistent set is one for which the distinctionbetween what follows and what doesn’t has collapsed. This lack ofmeaningful contrast means that it no longer makes sense to talk aboutinferring conclusions from such a set. In order to regain the distinctionwe are going to have to drop back to the level of consistency and theonly way to do that, is to look at consistent subsets of the original set.

Absent the notion of level, there are different ways to do this. Theone suggested in Quine and Ullian (1970) to deal with inconsistent setsof beliefs, involves two stages: At the first stage we discover the small-est subset of the inconsistent set which still exhibits the inconsistency.At the second stage we discard the member of the inconsistent subsetwith the least evidence, and repeat as necessary until the set is consis-tent. Having thus cleansed the belief set, we may now draw conclusionsas we did before.7

We don’t say that this process can’t work. We do say that it doesn’tseem to work in every case. There is a clear difficulty here when thetwo conditions on rational belief: Consistency (which we might call theexternal condition) and evidential support, (the internal condition) pullus in different directions.

7This procedure was intended to apply to classical inference, but the method obviously gen-eralizes to cover cases in which the base inference isX .

6.5. LEVEL PRESERVATION 95

In the lottery paradox, for instance, we seem to have good evidencefor each one of the lottery beliefs (ticket 1 won’t win, ticket 2 won’twin,. . . , ticket n won’t win.) and we can make the evidence as strongas we like by making the lottery ever larger. Now conjoin the beliefsand we get ‘No ticket will win.’ which contradicts fairness. We couldget consistency by throwing out the belief that the lottery is fair, butthat would be cheating. The problem is that each of the lottery beliefshas exactly the same support as the others. They stand or fall as one, itwould seem. If we let them all fall, then the rationality,or at least thenon-irrationality of buying a lottery ticket would seem to follow. Butisn’t it true that it isn’t rational, according to the accepted canons atleast, to buy a lottery ticket?8

Leaving aside the possibly controversial issue of the lottery paradox,take any situation in which we are unable to find a rationale for discard-ing one member of an inconsistent subset rather than another. Quineseems to suggest that in this situation, the counsel of prudence is to waituntil we do find some way to distinguish among the problematic beliefs.Those with less patience seem to regard random discarding until at lastwe get to consistency, to be the path of wisdom.9 We are inclined toreply to Quine that patience, for all that it is a virtue, is sometimes alsoa luxury we cannot afford or even a self-indulgence that we do well todeny ourselves.

To the others we say consistency is not a virtue which trumps ev-erything else. Suppose we might achieve consistency by throwing awayone of ˛ or ˇ though we have no reason to prefer one over the other.Flipping a coin is a method for determining which goes to the wall, butwe have no way of knowing if we have determined the correct one. Wehave left ourselves open to having rejected a truth and accepted a false-hood. ‘Yes, but at least we now have consistency!’ won’t comfort usmuch if the consequences of picking the wrong thing to throw away areunpleasant enough.

Let us take up level once more.10 In saying the level of the set † is8Not for nothing have lotteries long been known as ‘a tax on fools.’9This seems to be the route advocated by some of those in computing science who have

devised so-called truth-maintenance systems.10We realize that the Quinean suggestion is not the only one, though it might be the most


k, we are saying two things. First that there is a way to divide the logicalresources of † into k distinct subsets each of which is consistent. Fromnow on we shall refer to these consistent subsets as cells. Second, thatany way of thus dividing † must have at least k cells. Here we havegot to the level of consistency not once, but k times. Not only mightwe wonder which of the k cells is the ‘real’ one, the one which bestrepresents ‘the way things really are,’ but there may be lots and lots ofdistinct ways to form the k cells. Which of the possibly many waysshould we privilege?

At this point, we cannot answer these questions, which means thatwe must treat the cells on an equal footing along with the various waysof producing them.11 In saying this, we say that we shall count as aconsequence of † in the derived inference relation, whatever formulafollows (in the ‘underlying’ logic, say X) from at least one cell in everyway of dividing † into k cells.

When the underlying logic is X , the derived inference relation iscalled X -level forcing, which relation in indicated by

� X . We can

give the precise definition as:

Definition 6.�� X ˛ if and only if, for every division of � into `X .�/ cells,

for at least one of the cells �;� `X ˛

It is easy to see that:

Proposition 3. If �� X ˛ then `X .�/ D `X .� [ f˛g/

Proof. Suppose the condition obtains and let `X .�/ D k. It followsfrom the definition that every division of � into k cells, results in atleast one cell that X-proves ˛. But then we could add ˛ to the cellin question without losing the cell property since X is a logic whichpreserves consistency. In such a case, after adding ˛ we would have a

well-known, to deal with inconsistent sets of formulas. We do not, however, intend this essay as asurvey of all of so-called paraconsistent logic.

11Which is not to say that there is no way. Elsewhere one might find suggestions which narrowthe range of ways of dividing up our initial set. In this connection see the discussion of A-forcingin chapter 9.

6.6. YES, BUT IS IT INFERENCE? 97

division of � [ f˛g into k cells. Moreover there couldn’t be a divisionof � [f˛g into fewer than k cells without there being a similar divisionof � which would contradict the hypothesis. �

It obviously follows directly from this that:

Corollary 1.� X preserves X-level, in the sense that

`X .�/ D `X .Œ X.�//

6.6 Yes, But Is It Inference?

To be perfectly honest, or at least honest enough for practical purposes,all we have shown is that X-level forcing is a relation that preservesX -level. There is a gulf between this, and the assertion that

� X is

an inference relation which preserves X-level. The obvious problemfor anybody wishing to assert such a thing resides in the fact that wehaven’t, for all our efforts at precision, actually said which relationscount as inference relations. What we have said, is that we assume thatthe inference relations we mention admit certain rules. Shall we takethe collection of these rules to be constitutive of inference?

We shall not, because some, at least, of the rules which the underly-ing logic admits simply don’t make sense for the derived relation. Thisshould not come as a surprise. It is our palpable annoyance with the un-derlying logic which leads us to propose

� X . How silly then to require

that the derived logic inherit everything from the underlying logic, sincethat would make the derived logic another source of irritation rather thanthe balm for which we hope.

Although it is easy to check that� X inherits from its underlying

inference relation X both [R] and [Cut], we can see that it fails to admitthe rule [M] of monotonicity, which we would do better to label therule of unrestricted monotonicity from now on. But this is one of thosecases in which the rule ought not to apply to the derived relation. If weare allowed to dilute premise sets in an arbitrary way, there is nothingto prevent us from raising the X -level of such sets. But raising the levelgives us, in general, (logically) weaker cells in each logical cover. Whatused to X-follow from at least one cell of every such cover might no


longer do so, as we are cut off from vital logical resources by the finerdivision.12

This is not to say that no form of monotonicity makes sense for thederived relation. Quite the contrary, in fact, what most emphaticallydoes make sense is that X-level forcing consequence must survive anydilution which preserves the level of the premise set. Such a restrictedversion of monotonicity manifestly is a rule for X-level forcing, as istrivial to verify.

Along with level-preserving dilution, there are certain consequenceswhich must survive any dilution at all, whether or not theX-level of thepremise set increases. These are the consequences which dilution can-not affect, and we can say precisely which they are: TheX -consequencesof the empty set and of any unit set will remain X-consequences of atleast one cell of every logical cover of any set which contains any ofthese privileged sets. In earlier work these sets were called singular.

The other properties which we have been mentioning for the under-lying logics are non-triviality and having denial. It should be clear thatwhen the underlying logic is non-trivial so will be its derived forcingrelation. In fact, keeping to our original definition of consistency, inpassing from an underlying logic X to its derived

� X , many of the

X -inconsistentsets fail to be� X -inconsistent, which is after all, the

whole point of the derived relation.Which brings us to denial. If the underlying logic has denial, nothing

follows about the derived forcing relation, which is not necessarily abad thing. This is because in the underlying logic, inconsistent setsare (typically) relatively easy to come by, but in the derived logic, theonly inconsistent sets have inconsistent unit subsets, or what we calledX -absurd formulas. Having denial doesn’t imply having absurdities. Sothe derived logic will have denial only if the underlying logic has absurdformulas, but in no case will the derived logic have non-trivial denial.

And since ˇ denies ˛ in the derived logic if and only if one or both

12Here is a concrete example where the underlying logic is classical. The premise setf˛; ˛ � ˇg has the classical-level forcing consequence ˇ since it has level 1. If we add the for-mula :ˇ the resulting set has level 2, and now there is a logical cover: f¿; f˛;:ˇg ; f˛ � ˇggno cell of which classically proves ˇ . Thus ˇ is not a classical-level forcing consequence of thediluted set.

6.7. OTHER LEVEL-PRESERVING RELATIONS 99

of the two are absurd in the underlying logic, the principle [Den] musthold of the derived relation, but it is much less interesting there than itis in the underlying logic.

So for the derived relation, we would seem to be on solid groundwhen we require [R], [Cut], and the restricted version of monotonicity.Those we might well regard as the hallmarks of inference, or at least ofderived inference. And let us not forget that the derived relation agreesexactly with the underlying relation X on the consequences of the X -consistent sets. So for this reason alone, we ought to admit the forcingrelation into the fold.

Perhaps we should put it this way: Anybody who thinks that X isfine and dandy except for its failure to be properly sensitive to the va-rieties of inconsistent sets, must think that X -level forcing is an ade-quate account of inference. This is because when premise sets are X -consistent, X-level forcing just is X . And while it surely isn’t X for(some) X -inconsistent sets, in those cases X isn’t an inference relation.X has abdicated, throwing up its hands and retiring from the inferentialstruggle, offering the hopeful reasoner nothing beyond a contemptuous‘Whatever!’

6.7 Forcing In Comparison With Other Level-preservingRelations

Finally, we consider the place of the X -level forcing relation comparedwith other possible relations which preserve X-level. We cannot claimuniqueness here, for there may be plenty of relations, even inferencerelations, which preserve X -level. What we can claim however, is in-clusiveness, in a sense to be made precise.

That precision will require another property13 of the underlying logicX .

13If we regard deductive systems as categories, then to call a logic productival is simply to saythat the category (logic)X has products. The first condition amounts to the assertion of canonicalprojections while the second amounts to the universal mapping property of products.


Definition 7. A logic X will be said to be productival if and only iffor every finite set � there is some formula � such that

� `X for every 2 � , and

� `X �

Evidently being productival is another of those properties more hon-ored at the level of underlying logics. If a productival logic X hasdenial, then X-level forcing will certainly not be productival. But ofcourse at the underlying level, products are useful. For instance:

Theorem 3. If X is productival then for any pair �; ˛ with � a finiteset of formulas and ˛ a formula:If Y preserves X-level and admits level-preserving monotonicity, then� `Y ˛ H) �

� X ˛

Proof. We shall content ourselves with a sketch only—a fuller treatmentcan be found in ‘Level Compactness’ (Payette and d’Entremont, 2006).Assume for indirect proof that � `Y ˛ and that � fails to X-level force˛. From the latter we know that � has finite level, say k, and that thereis a logical cover of � of width k such that none of the k cells X-proves˛. Where ˇ (non-trivially) denies ˛ add ˇ to each cell and then formeach of the k products of the cells. The set of these products must haveX -level k but the Y closure of the set must have X-level k C 1. So Yfails to preserve X-level, contrary to hypothesis. �

The restriction to finite premise sets will chafe us only until we seeits removal in the more general result referenced above.

So while there may be many inference relations which preserve X-level, X -level forcing is the largest of them.

Seven

Preserving Logical Structure

GILLMAN PAYETTE

Abstract

In this paper Gillman Payette looks at various structural properties of theunderlying logic X , and ascertains if these properties will hold of theforcing relation based on X . The structural properties are those that donot deal with particular connectives directly. These properties include thestructural rules of inference, compactness, and compositionality amongothers. The presentation of the logic X is carried out in the style of al-gebraic logic; thus, a description of the resulting “forcing algebras” isgiven.

7.1 Introduction

Faced with the possibility of logical pluralism, if not the reality of it,one should follow a methodology of research which is sensitive to theplurality of logics. What has not been dealt with so sensitively is thequestion of how best to deal with inconsistent sets if we don’t want totrivialize inference in every one of these cases.

In classical and intuitionistic logic the sets fP ^ :P g and fP;:P ghave the same deductive closure: Everything! In Latin the rule of infer-

101

102 CHAPTER 7. PRESERVING LOGICAL STRUCTURE

ence is phrased as ex falso quodlibet, which may be translated as ‘fromthe false whatever.’ This is the notion of inconsistency first presentedby Post and used throughout this volume and elsewhere. For sets offormulas this general definition of inconsistency has it that a set is in-consistent just when everything follows from it.1 It should not escapeour keen attention that the definition is mute on the subject of falsity ingeneral, and that there is likewise no mention in particular of the nega-tion connective.

The paraconsistent 3-valued logic of Priest, the Logic of Paradox(LP), uses the same language as classical logic, and has all of the sametheorems.2 Thus, it has .P ^ :P / � Q as a theorem so, in a sense,it rings true that from the necessarily false, i.e., the absurd, everythingfollows in LP. However, the two sets above do not have trivial closuresin LP. So, this logic can, in its way, deal with inconsistency. But, falsityand truth apply to sentences. Only by type raising do they apply tosets. What has happened, historically speaking, is a running together ofabsurdity, inconsistency and falsity. LP still runs together absurdity andinconsistency; it merely renders the two, inert.

Although the sets above have the same closure in classical logic (andLP respectively) there is an obvious difference in the composition of thetwo: one contains what is often taken to be an absurd formula, and theother does not. One might ask ‘Is there a logic that can distinguish be-tween these two kinds of set?’ But, this question may be wrongheaded.Inconsistency, as well as consistency, is a relative notion; it is relativeto a particular logic. Therefore, when a logic like LP says that fP;:P gdoes not explode it says that this set is not LP-inconsistent. Lackinganother notion of inconsistency, we see that some paraconsistent logicsmerely regard what are inconsistent sets in other logics as consistent.

This shifts the ground by proposing what was called in Chapter 2 areplacement of the original theory (logic), but it leaves the originatingproblem untouched. Think of it this way: Axiomatic set theory does notfix naive set theory—the latter remains paradoxical. Neither does LP fixthe classical account of inconsistency—we are still unable, classically,

1Such sets are often said to explode (inferentially).2See Avron (1994).


to distinguish inferentially between classically inconsistent sets whichcontain an absurd formula and those which do not.

In opting for a replacement strategy, what happens is that the rulesof the game have changed to solve the problem. Of course this maybe entirely respectable—it may even be the only way to proceed. Wechange the rules all of the time. But, when the rules have changed inlogic it seems that the meanings of the connectives involved change aswell. And that may be a problem.

Consider a set that we regard as consistent. In that case the logicdoes not permit us to treat that set as classically consistent. We mustcontinue according to the rules unless we are constantly changing themeanings of the connectives to suit our desires. But that seems wrong.

A different solution is possible. There are many logics which sufferunder the iron yoke of ex falso. Thus, it would make sense to develop,rather than a new logic to deal with classical account of inconsistency,a method of determining, for a whole class of logics which have thisproblem, a notion of derived consequence that enables the distinctionwe want without having to construct a whole new logic in each case.We seek a method which we can apply to a logic X described quitegenerally so long as we can representX’s notion of inconsistency as theexplosive Postian kind. This may be considered a method of Paracon-sistentizing the logic.3 The notion of Forcing a la Schotch and Jennings(1989), which is the subject of this volume, is what shall be used, butmassaged into the framework of algebraic logic.

Just what constitutes a good way of constructing such a derived in-ference relation is obviously crucial. The Quinean idea of minimal mu-tilation is what we take to be an obvious criterion for ‘goodness.’ Themore of a logic that can be kept the better. What is of interest here is howthe ‘structural properties’ of an underlying logic X may be transferredto the forcing relation

� X . We call properties ‘structural’ when they

don’t have a direct relation to particular connectives. These are prop-erties like compactness, preservation of consequence by substitutions,and the structural rules of inference. The compositionality of a logic,

3This term was used first by Alexandre Costa-Leite; however, the method that appears here isdifferent from his approach.


i.e., the meaning of whole is given by the meaning of the parts, is alsoa structural property since it does not deal with particular connectives.I will also make a few comments on how the existence of certain kindsof formulas, like denials,4 may affect these structural properties.

The present study makes up part of the justification of the forcing re-lation as an acceptable notion of paraconsistent consequence. The pointis to see that much of the general structure of a logic X can be pre-served by the forcing relation. The other part involves the idea of levelpreservation which is taken up in detail in Chapter 6 of this Volume.We also show where the forcing relation sits amongst other similar re-lations on X . The final discussion concerns first-order classical logic asan instance of many of the properties mentioned above.

7.2 Definitions

This study is diverse and long. Therefore I will present many definitionsto give some background to what follows. In what follows there is a aminor departure from previous work this will be pointed out as it isnecessary. The reason for the departure is the focus on the semanticversion of a logic, rather than the syntactic. First we will begin with alogic.

Logics

Definition 1. A logic X is given by a tuple hSX ;`X ;MX ;ˆX isuch that

1. SX is the set of formulas.

2. `X is the provability relation, thus proof theoreticconsequence. This may not exist for some logics.

3. MX is the class of models. Whatever it may be.

4. ˆX is the satisfaction relation between models and formulas.

4in the sense of that term introduced in Chapter 6 of this volume

7.2. DEFINITIONS 105

Note that ˆ is not semantic consequence. Semantic consequence, orentailment, is defined in the usual way using ˆ, i.e., � � ˛ iff for allM 2 M;M ˆ � H) M ˆ ˛. I will follow the convention and usethe � symbol for both relations. Also, when M � ' for every M 2 Mwe write � ' which means ' is a truth of the logic. Relative to eachlogic there is a consistency predicate CONX which holds � when either� � ˛ for every ˛, or, there are no models of � .

An important part of the algebraic presentation of logic is the mean-ing function mng. The meaning function will, in a sense, be definitiveof the logic since the meaning algebra of a logic is what is studied inalgebraic logic. We will present a candidate for the meaning function ofthe derived forcing relation, and show some properties which it inheritsfrom the meaning function for the original logic. So in what follows alogic is given with a meaning function right from the start.

Definition 2. A meaning function for a logic X is a mapmng WMX � SX ! H where H is some class. Thus it takesformula-model pairs to something. A function mng is a meaningfunction for a logic X–subscripts omitted–just when.8M 2M ;' 2 S/Œmng.M; / D mng.M; '/ H) M ˆ' ” M ˆ �.

There could be many meaning functions for a logic but, there arecanonical examples. The composition of the set H is left undefinedin general; it will vary from logic to logic as the composition of Mdoes. In the case of first-order logics the meaning of a formula, relativeto a model, is given as the set of variable assignments that satisfy thatformula in that model. For the usual 2 valued propositional logic themodels are just the truth value assignments and the meaning functionsare their extensions to all of the formulas.

A meaning function will only be of use if its image forms an alge-bra. An algebra is a setA (often called the underlying set of the algebra)along with a set of functions from An to A for various natural numbersn (including zero). A logic is said to ‘have connectives’ when for anyn-tuple of formulas 1; :::; n and any n-ary connective4 in the set Cn


of connectives of the language of X , the formula 4. 1; :::; n/ is alsopart of the language. The set of formulas then can be seen as an algebra,and it is called the ‘formula algebra’ denoted by AX . Recursively gen-erated sets of formulas will produce formula algebras. In that case, theunderlying set is just the set of all formulas generated, and the functionsare the connectives.

A function (or map) h W hA;�Ai �! hB;�Bi is a homomorphismwhen h.a �A b/ D h.a/ �B h.b/, and if h W A �! A, then h is anendomorphism. Homomorphisms are structure preserving maps. Whenthe meaning function of a logic is a homomorphism for each model, thelogic is called compositional. We can give a nice mathematical defini-tion of compositionality as follows. The subscripts for the logic X areomitted where confusion is unlikely.

Definition 3. A logic X is compositional just when the meaningsof compound formulas depends on the meanings of the componentformulas. More precisely, let 1; :::; n and '1; :::; 'n be n-tuplesof formulas from S, and4 an n-ary connective in the language ofX , then given a model M 2M if for each 1 � i � nmngM. i / D mngM.'i / then

mngM.4.'1; :::; 'n// D mngM.4. 1; :::; n//

When a logic’s meaning function is a homomorphism for each modelwe can generate a meaning algebra for a logic. That is what it meansfor the image of a meaning function to form an algebra.

There are two interesting sets that deserve mention.

Definition 4. Let K �M and † � S.

1. The ‘theory of K’ is T h.K/ D f 2 S WM ˆ 8M 2 Kg, i.e. the set of formulas every model in K models.


2. The ‘models of †’ is the classMod.†/ D fM 2M WM ˆ 8 2 †g. Which is theclass of models which model all the formulas in †.

The aim of this essay is to discuss the properties which carry overto the forcing relation. In furtherance of this aim we introduce someimportant properties of consequence relations. The substitution prop-erty comes in two flavors viz. weak and strong. A substitution s isa morphism from the set of atomic formulas to the set of all formulaswhich can be extended uniquely to a homomorphic endomorphism ofthe formula algebra. So a substitution s on a propositional languagemaps each atom to a formula, not necessarily atomic. Then one canreconstruct formulas with the substituted atoms.

Definition 5. The substitution properties for a logic X are:

1. (Weak) Ifˆ thenˆ .P=s.P // where .P=s.P // is theuniform substitution of s.P / for P in , with P an atom ands a substitution.

2. (Strong) If � � , then �s � s where �s is f's W ' 2 �gand 's is '.P=s.P // for each atom mentioned in '.

A logic which obeys the substitution properties preserves conse-quences under uniform substitutions of the atoms in the formulas. Thestrong property implies the weak property by letting � D ¿. The weakproperty asserts that theorem-hood or truth-hood is preserved by uni-form substitutions.

There are logics that have what are called ‘structural rules’ of in-ference. Such logics are not called ‘structural logics’; however, everystructural logic must have these rules. A common presentation of thestructural rules is (I omit the subscripts on the `):

� [R] for reflexivity, if 2 � then � ` .


� [M] for monotonicity, if � ` and � � � then � ` and

� [Cut-1] if � ` and �; ` ' then � ` '.

The first two rules are fine; however, there are many formulations of[Cut]. The formulation we shall use is the following.

[Cut]: Let � ` ı for all ı 2 �0 � � and � ` then�;� ��0 ` . Where � ��0 is set of members of � notin �0.

This version of the rule is slightly more general than the one above. Itis also more general than the following version:

[Cut-2]� ` ı for all ı 2 � and � ` ˛ then � ` ˛.

But the previous version is a bit too strong for our purposes in this paperthus we use [Cut]. There is a very interesting fact to be notice about thelast formulation of the rule.

Proposition 1. [R] and [Cut] imply [M].

Thus, only in certain situations will the three rules really ‘comeapart’; forcing is, as will be shown, one of those situations.

Definition 6. A logic X is structural if and only if it obeys both thestrong substitution property (see below) and the structural rules ofinference (with [Cut-2] in place of [Cut]).

Covers, Models and Levels

Essential to forcing are the notions of cover and level.A cover C inX , or anX-cover, is a tuple of sets of formulas indexed

by some ordinal � C 1 � ! C 1: C Dh�i W i 2 � C 1 & �0 D ¿i such that for each i 2 � C 1, CONX .�i /

where CONX is the consistency predicate for X . The � from the indexordinal of C is referred to as the width of the cover C and denoted w.C/.Each �i in C, indicated by a pardonable abuse of notation as ‘�i 2


C’, is called a cell. The collection of X-covers will be referred to asM�. This notion of cover departs from the definitions in the literature,e.g., Schotch and Jennings (1989); Brown and Schotch (1999) and thepresent volume, but it will help avoid technical problems later. Since acover is defined as an ordered tuple of length less than or equal to !5

the condition for two covers C;C� to be equal is: w.C/ D w.C�/ D �

and 8i � � �i D ��i , where �i 2 C and ��i 2 C�. So covers mayhave repetitions of cells, or two covers with different orderings of thesame sets are not equal.

Definition 7. C is a cover of � , that is C �� /, just when

1. �0 D ¿2. CONX .�i /, 8i 2 � C 13. For each 2 � there is a �i 2 C such that �i � and,

4. if the constant symbol a or free variable x of the language ofX is mentioned in �i for some i 2 � C 1 then it is mentionedin � .

For a symbol t to be mentioned in a set means that for some formula inthe set t appears in that formula.

It is easy enough to see that a cover covers a single formula whenthere is a cell in the cover that entails that formula. In such case it iswritten C �� . The condition on terms and constant symbols will beof use when we discuss first order logic. As the definition stands it isonly a small divergence from the one found in earlier literature.

This notion of cover can be interpreted as something like a model;it is a kind of model for certain inconsistent sets relative to the logicX , though not all X -inconsistent sets will have these kinds of models.To see that the C’s function as models in this way suppose C �� .Thus, there is some cell �i in C that X-entails , and CONX .�i / by

5! is the first countable ordinal, or the order type of the natural numbers.


the definition of cover. We call a formula absurd when its unit set isinconsistent. An absurd formula does not have any models, nor does ithave any covers. Consider an absurdity ˛. If ˛ had a cover, then therewould be a C and �i 2 C such that �i � ˛, but that would require thatCONX .�i /, because X has [Cut], which contradicts the second condi-tion in the definition cover. Thus, sets like f˛g have no covers and theconverse of this also holds. Note the important assumption that the logicX is not trivial, i.e., the empty set’s closure is not everything.

Proposition 2. � does not have a cover iff � contains an absurdity.

A related fact is that the minimal cover for a set is a cover justwhen there are covers for a set. The minimal cover of � is Cm� Dh¿; f g W 2 �i. Cm� is minimal in two senses. The cover Cm� is thefinest partition one can make of the set � , and it is ‘contained’ in anyother cover of the set. It is contained in the sense that, for any cover C

of a set � , if Cm� �� ˛ then C �� ˛. This cover will be of use whendiscussing infinite levels.

A level function relative to X , `X , is defined using the notion ofcover. The codomain, or range, of the level function will depend on thesize of S. We restrict the cardinality of S to the countable cardinal forthis essay; thus, jSj D !.

Definition 8. The X -level (i.e. level relative to the logic X) of theset � of formulas of the underlying language of X , indicated by`X .�/ is defined:

`X .�/ D8<:

minw.C/

ŒCOVX .C; �/� if this limit exists

1 otherwise

Thus, the level of a set is determined by the minimum width of thecollection of covers of a set � . This is of course a function, and underthe assumptions made in this paper its domain is ! [f!g[ f1g. Levelactually induces a kind of measure on the set � . Level measures how


inconsistent a set is. One must use more classically consistent sets tocover fP ^Q;:P ^R;:R ^ :Qg than to cover fP;:P;Q;Rg. Andone cannot cover fP ^ :P g; it has level1! This is how we can finallysee the difference between inconsistency and absurdity. An inconsistentset may have a level, but absurdity does not.

Also note that if there are theorems, then any set of theorems haslevel 0. Since the cover C D h¿i is a cover for a set of theorems,w.C/ D 0 because C is indexed over the ordinal 1=0+1=� C 1. Withlevel we can define the forcing inference relation on P .S/ � S.

Definition 9. [Forcing]6 �� X ” .8C/Œ.C ��

� & w.C/ D `X .�// H) C �� .

This is read ‘� level forces , relative to X ’. A set will force aconclusion, , just when every cover of � of width ‘level of �’ is acover of . This relation in some cases will collapse into just whatfollows from the unit sets, but not always, and, considering the size ofP .S/, one may say only half of the time.

What is needed to fill out the algebraic approach to forcing is ameaning function. However, meaning functions are semantic entities;they take models as one input and formulas as the other and spit outsomething. If the derived meaning function for forcing is to depend onthe underlying logic it will, ideally, depend on the models of the un-derlying logic. The models for the forcing relation are, as mentionedabove, covers. Thus, what must be constructed are covers, not out ofsets, but, rather, out of models. This can be accomplished by the fol-lowing construction.

Definition 10. A semantic cover F of a set � , written F �� , is atuple of models from M , hMi W i 2 �i, � an ordinal less than! C 1, such that for any 2 � there is an Mi 2 F such thatMi � .


Thus, semantic covers are just tuples of models. Again, the � inthe definition above will be referred to as the width of the cover, andcovering a single formula is to cover its unit set. In this case the widthof the cover is just the cardinality of the cover. The level of a set offormulas is defined thus,

`X .�/ D

8ˆ<ˆ:

0 if � � CX .¿/minw.F/

ŒF �� if this limit exists

1 otherwise

where CX is the consequence closure operator for X . It is the set of allconsequences of a set of formulas relative to the logic X . Of course aforcing relation can then be defined relative to this notion of cover in ananalogous way.

Definition 11. �Œ� ˛ if and only if for all F �� such thatw.F/ D `X .�/, F �� ˛.

But, do these two notions of cover give rise to the same forcing re-lation, extensionally speaking? The sort answer is yes.

Lemma 1. If C is a (syntactic) cover of � then there is a semantic coverof � of the same width, and vice versa.

Proof. Suppose that C �� , and w.C/ D �. Then for any 2 �there is �i 2 C such that �i � . Thus if M 2 Mod.�i / thenM � . Each Mod.�i / ¤ ¿ by definition of cover, so let FC DhMi W some Mi 2Mod.�i / i ¤ 0i. So the width of F is that of C,and every member of � is modeled by some member of F. The oppositedirection follows by making up CF of ¿ and �i D T h.Mi /, Mi 2 F.�

And what about the different types of level?

Lemma 2. For all � � S, `X .�/ D `X .�/.

7.3. SUBSTITUTIONS 113

Proof. Suppose that `X .�/ D 0. Obviously `X .�/ D 0, and viceversa. If `X .�/ D � , then let C �� and w.C/ D �. If the levels weredifferent, then we could generate a semantic cover, or regular cover,with the Mod and T h operators respectively to derive a contradiction.If `X .�/ D 1 Then there are no covers of � of either kind, because� must contain an absurdity. This is the case with either definition ofcover. �

Then we can prove the result that we want.

Theorem 1. For any � and ˛, �� ˛ ” �Œ� ˛.

Proof. Suppose �Œ6 ˛. Then suppose C 6�� ˛. Construct a semanticcover out of C as in lemma 1. By lemma 2 this cover has the samewidth, which is the `X .�/, as C and it is a semantic cover of � . So�Œ6� ˛. For the other direction, the dual argument as in lemma 1 is usedand the definition of the T h operator. �

With the two notions amounting to the same thing from two differentdirections I will use whichever notion is appropriate for the context.

7.3 Substitutions

Recall definition 5. There are two versions of a logic being syntacticallysubstitutional. One is strong the other weak. Given that X is structural,the forcing relation is weakly substitutional (proof omitted), and onlyobeys a qualified version of the strong substitutional property.7

I shall prove the theorem that forcing has a restricted form of strongsubstitution. Recall that the logics considered are ones that obey thethree structural rules mentioned in the last section. In what follows Iwill use the syntactic version of forcing since I am discussing a syntacticnotion of substitution.

7An interesting side note to this point is that forcing will also have the interpolation propertyif the underlying logic does.


Definition 12. A ‘Partition Cover’ of � is a cover C such thatw.C/ � � D `X .�/ and

SC D � . Further, if �i and �k

.i; k � �/ are distinct members of C then �i \�k D ¿.

This definition says that C is a partition of the set � . It is the kindof partition in which each cell is a consistent set relative to X . Sucha thing is not unique in general, but there is always the possibility ofhaving such a partition if the set’s level is not1.

Lemma 3 (Partition Cover Lemma). If `.�/ D � 2 !, then, given acover C of � , there is a partition cover C0 which one can construct fromC.

Proof. Assume that `.�/ D �. Then let C be a cover whose widthis � . We shall construct the partition cover out of this one. First, let�0i D CX .�i / \ � for all 1 � i � � . Note that i does not start at 0.Once the construction has finished we add¿ to the rest of the sets to getthe C0 wanted. Let �ij D �0i \�0j where i ¤ j . Further, define

��k Dk�1[

mD1.�0k � .�0k \�0m//

for k � � . Since C is well ordered by its width ordinal one can speak ofthe ‘first �i such that...’ . In a sense the ��’s check to see if they aredisjoint with all of the sets ‘before’ them.

Claim: These ��k

’s are all disjoint, and � D Sk��

�k

. Supposei ¤ j , and 2 ��i \ ��j . Without loss of generality assume thati < j , since it is one or the other. Then, since 2 ��i , it must be that 62 �0

kfor all k < i , but 2 �0i . So, 2 �0i \�0j . However, by the

construction of ��j it is impossible that 2 ��i , since i < j .It is clear that

Sk��

�k� � so assume that 2 � . Then for

some �i 2 C, 2 CX .�i /, but that means 2 �0i . We may assumethat i is the smallest such index, since there must be one. If 62 ��ithen there would have to be a smaller j such that 2 �0j which is notpossible. Thus, 2 ��i �

Sk��

�k� �; therefore, � D S

k�� k

.

7.3. SUBSTITUTIONS 115

Let C� D h¿; ��i W 1 � i � �i. Then C� is clearly a partition cover of� . �

The next lemma allows us to restrict what covers we must considerof the collection of covers of a set � . Notice that although this lemmais proved for the forcing case it will work in the X� case since in thelatter case all covers are considered. Only covers whose width is thelevel of the set � are considered in forcing. Thus the obvious extensionwill suffice for that case.

Lemma 4. For `X .�/ > 0, �� ” 8 C a partition cover of �

of width `X .�/, C �� .

Proof. ( H) ) If �Œ� then all covers of � of appropriate width willcover . Partitions are covers of � , so they must cover .

((H) By contrapositive. If �Œ 6 then there is a cover C of � , ofwidth `X .�/ which does not cover . Let C� be a partition cover of� associated with C as in the partition cover lemma, lemma 3. Noticethat ��i � CX .�i / for �i 2 C. This is because ��i � �0i , and �0i DCX .�i / \ � . Thus, CX .��i / � CX .�i / by assumptions on X whichtransfer to CX . So, 62 CX .�i / for all �i 2 C, and 62 CX .��i / foreach ��i 2 C�. Thus, C� 6�� . �

The case where `X .�/ D 0 and � is non-empty will not have anypartition covers of width 0 since the only cover of width 0 is h¿i. How-ever, this poses no problem because if a set of theorems forces some-thing then that something is also a theorem. But forcing is weakly sub-stitutional.

We can see that strong substitution does not hold for forcing by con-sidering the following counterexample. Let � D fP;:Qg. Let thesubstitution s be the function which takes P 7! P and Q 7! P . Aswe shall see in the next section forcing is just like the underlying logicX when the sets are consistent; thus, �Œ� P ^ :Q. Notice though�s D fP;:P g and s.P ^ :Q/ D P ^ :P . But then �S Œ6� P ^ :Psince forcing is level preserving (see the next section theorem 6). Al-though the underlying logic may be substitutional, the forcing relationis not.


Having said this if we restrict the class of substitutions, we can de-rive a substitution theorem for substitutions from that class.

Theorem 2. Suppose that s is a substitution from the set of atoms tothe set of all formulas such that s 2 ˚g 2 SA W `X .�g/ D `X .�/

, i.e.,

all of the substitutions which are level preserving. If X is such that forany pair h�; 'i and any substitution s0, � � ' H) �s

0 � 's0; then�Œ� ' H) �sŒ� 's .

Proof. By contraposition. Assume that for some level preserving sub-stitution s, �sŒ6� s . X has been assumed to obey the substitutionproperty which is equivalent to: if �s 6� 's then � 6� '. Let C be apartition of �s such that C 6�� , which exists by definition.

Then for each �i 2 C, �i 6� s . Consider the sets s�1Œ�i �: theinverse image of the �i ’s from C. If s is applied to these sets the �iis returned, since each member of the inverse image gets mapped tosome element of �i , and since �i � �s , there must be somethingwhich was mapped to each element of�i . Thus, .s�1Œ�i �/s D �i . Let��i D s�1Œ�i � \ � .

Claim: ��i 6� . Suppose that ��i � . Then, by the assumptionthat X has the substitution property, .��i /

s � s . But �i D .��i /s . So�i � s , but it does not. Thus, ��i 6� . This implies CONX .�

�i / for

all �i 2 C.Claim: � � S

��i . Each element of the �i ’s is an image of someelement of � , so all of � is recovered in the inverse images of the �i ’s.We don’t get more than � since we restrict the inverse images to � in theconstruction of the ��i ’s. Let C� D h¿; ��i W �i 2 Ci. This is clearlya cover of � , and not a cover of . But there are only as many ��i ’sas there are �i ’s; thus, C� has the same width as C. But the width of C

is the level of �s , i.e., is equal to `X .�/. So �Œ6� which is what iswanted. �

Thus not all is lost, but there must be some room given for failuresto balance out the gains.

7.4. ALGEBRAIC CONCERNS 117

7.4 Algebraic Concerns

There are two types of algebras which are of particular interest to thealgebraic logician. These algebras arise from a logic through the appli-cation of the meaning function. The first type of algebra is the meaningalgebra, and the second is the Lindenbaum-Tarski algebra. The mean-ing algebra is constructed out of the meanings of the formulas of thelogic. Lindenbaum-Tarski algebras are made up of equivalence classesof formulas. Both are necessary to study the properties of the logic inquestion. However, forcing has many of the necessary properties thatmake it susceptible to algebraic methods, but it fails in a crucial place.To begin this study some definitions are needed.

Definition 13. [Algebra, subuniverse] An algebra A D hA;�Ai isa set A with a collection of (finitary) operators�A D

nf ki W i 2 �; k 2 �

o. The � is referred to as the ‘type’ of

the algebra and A as the ‘universe’ or ‘carrier’ of the algebra.A subuniverse B of an algebra A is a subset B of A such that B isclosed under the operations in �A.

The important definitions for algebraic logic are as follows:

Definition 14. A Lindenbaum-Tarski (LT) algebra is given withrespect to a class of models K. The LT-algebra is a isomorphic8

copy of copy of the quotient algebra of the formula algebra AXmodulo the equivalence relation �K: �K ' iffmngM. / D mngM.'/ for all M 2 K.

The class of LT-algebras is denoted AlgLT .L/.

Definition 15. The meaning algebra relative to a model M isdenoted mng.M/. It is the set fmngM.'/ W ' 2 Sg. The type ofmng.M/ is the interpretation of the connectives of the language


in the models of X . The class of meaning algebras for L isdenoted as Algm.L/

With these definitions in place it is time to construct meaning func-tions for forcing.

Meaning Functions and X�

Meaning functions have less to do with the consequence relation andmore with the relation between the class of models and formulas. Forthe rest of this section we will assume that M is a set. The idea nowis to, somehow, define a meaning function for X� from the meaningfunction for the logic X . The one presented may not be the only oneavailable, but it does do enough of the work we would want it to. In thissection I will use the semantic version of cover.

Recall that the “models” ofX� are (semantic) covers so the meaningfunction will have to be a function which takes cover-formula pairs tosome class. Let F D hMi W i 2 �i 2M� and ' 2 S. Then,

mng�F.'/ D hmngMi.'/ WMi 2 Fi

So for two instances of mng�F to be equal, at and ' say, means thatfor each Mi 2 F, mngMi

.'/ D mngMi. /.The meaning function

that has been defined maps M� � S toS�2.!C1�¿/.H/

� , i.e., eachmng�F.'/ 2 .H/� where � is the non-zero width of F.

Before proceeding it must be proved that this mng� is a meaningfunction. The reader should recall definition 2.

Theorem 3. mng� is a meaning function for X�. That is, for ; ' 2 S

and any cover F, ifmng�F. / D mng�F.'/ then F �� ” F �� '.

Proof. Let mng�F. / D mng�F.'/, for some ; ', and F. SupposeF �� '. By definition there is some Mi 2 F such that Mi � '.By hypothesis, mngMi

. / D mngMi.'/. Since mng is a meaning

function for X , Mj � ” Mj � ' for all Mj 2 F, i.e., itdoes so for Mi . Thus, F �� . And similarly from the assumption thatC �� . �

7.4. ALGEBRAIC CONCERNS 119

Thus mng� is a meaning function. What can be said about compo-sitionality?

Theorem 4. If a logic X is compositional, then the derived logic X�

will also be compositional relative to the meaning function mng�.

Proof. Suppose thatX is compositional. Then suppose there is an n-aryconnective4, and some n-tuples of formulas as in definition 3 such thatgiven a cover F 2 M �, for each 1 � i � n, mng�F. i / D mng�F.'i /.These assumptions imply that given a Mi 2 C and j and 'j one has

mngMi. j / D mngMi

.'j /

This implies that one hasmngMi

. 1/ D mngMi.'1/; :::; mngMi

. n/ D mngMi.'n/. By hy-

pothesis mng is a compositional meaning function so it follows that:

mngMi.4. 1; :::; n// D mngMi

.4.'1; :::; 'n//

For each Mi . So,mng�F.4. 1; :::; n// D mng�F.4.'1; :::; 'n//, whichis what is wanted. �

Algebras

What can this information do for this study? It is clear that a mean-ing algebra can be formed relative to any semantic cover F. But onecan also form the quotient algebras relative to covers as well. So, onecan generate both LT- and meaning algebras. Indeed, it is possible, butwhat can be said about the resulting algebras given knowledge of theunderlying algebras?

Actually, more than one may expect. In the case of meaning alge-bras, which is the more complex case, the meaning algebra relative toa cover is a subsuniverse of a product of meaning algebras. Recall thatmng�F.˛/ is a tuple of meanings of the underlying logic. That means itis a member of the direct product of the meaning algebras in F. That is,

mng�F.˛/ 2Y

Mi2Fmng.Mi /


However, it is not clear thatmng�.F/ is the product. Think of the prod-uct displayed as an array with the formulas heading the columns, andthe models from F heading the rows. mng�.F/ is made up of just thecolumns, no other combinations of elements from the meaning algebrasappear.

˛ ˇ ı : : :

M1 mngM1.˛/ mngM1

.ˇ/ mngM1. / mngM1

.ı/ : : :

M2 mngM2.˛/ mngM2

.ˇ/ mngM2. / mngM2

.ı/ : : :

M3 mngM3.˛/ mngM3

.ˇ/ mngM3. / mngM3

.ı/ : : ::::

::::::

::::::

:::

But, because of compositionality, these columns are closed under theoperations of the product algebra. Thus, mng�.F/ is a subset of theproduct which is closed under the operations on the product, i.e., a sub-suniverse. Of course, if one knows more about the class of meaningalgebras ofX then one can say more about the meaning algebras ofX�.For instance it is well known that the class of meaning algebras of clas-sical propositional logic is a variety. This means that it is closed underproducts and subalgebras, among other operations on the class. Thus,for such logics, the forcing meaning algebra is a member of meaningalgebras for the original logic.

Consider the LT-algebras. These are made of quotients of the for-mula algebra. If the definition of an LT-algebra is scrutinized some-thing very interesting pops out. For the forcing LT-algebras take a classof covers K. Then consider the equivalence relation relative to K, �K .

�KD fŒ˛� W ˛ 2 Sg

but ˇ 2 Œ˛� ” ˇ �K ˛, and that is just to say that for all F 2 K,mng�F.˛/ D mng�F.ˇ/. Breaking this definition down further we cansee that what really matters is that the meanings relative to the modelsin F are equal.

TakeK a class fromM �, andKm D fM WM 2 Fg for each F 2 K,then ˛ �K ˇ ” ˛ �Km ˇ. In this way every forcing LT-algebracorresponds to an LT-algebra of X . But does it go in the other way too?

7.5. DENIALS 121

Theorem 5. The Lindebaum-Tarski algebras for forcing and the logicX are the same.

Proof. The comments above imply that the forcing LT-algebras are asubclass of the LT-algebras for X . Take a set of models of X say U .For each M 2 U generate the unit cover hMi. Then the collection ofthese unit covers forms a class of covers. The LT-algebra correspondingto that class of covers is of course a forcing LT-algebra. �

7.5 Denials

A denial of a formula ' is a formula such that CONX .'; / (the pairset is inconsistent). According to this definition of denial, a logic whichhas only ?, with its usual meaning, as the only denial will be said tohave denials. Formulas like ? are called absurd formulas. A logic Xhas non-trivial denial when there are denials of formulas other than ab-surdities. A formula is contingent just when it is neither absurd nor atheorem (or logical truth). Suppose that ' is a contingent formula, andthere is another contingent formula such that the pair set is inconsis-tent. These will be called ‘contingent denials’. The next definition ismore complex.

Definition 16. Let ˛ be a contingent formula of a logic X and ˇbe a denial of ˛. ˇ, is a Negation-Denial of ˛ (ND) if and only if,

1. ˇ is contingent,

2. CONX .˛; ˇ/ and,

3. if CONX .˛; ı/ then ı `X ˇ.

We have chosen the term ‘ND’ rather than ‘negation’ since it may bethe case that a logic has a connective in its language which is commonlycalled negation; but, there may be a derived connective which satisfies


the definition of ND.9 A derived connective is a formula schema thatdoes the work of a connective. A formula does the work of anotherwhen the inferential relationships of the two are the same. For instancein a classical system with only ^ and : the derived connective :.:A^:B/ does the work of _. The formula :.:A ^ :B/ obeys the sametruth conditions as A _ B and it will obey the same introduction andelimination rules in some calculi.10

Notice that ND is not a symmetric relationship. For instance thenegation of intuitionistic logic (IL) satisfies this formulation, but A isnot an ND of :A in IL. That is because although A � ::A the impli-cation does not go in the other direction.

In the sequel we make the following assumption about the logic X :Suppose that CONX .�; ˇ/. Then there is a denial ı of ˇ such that � `Xı. If one further assumes that ND is a symmetric relationship then thelogic X will have the following property:

[Den*]8�; ˛; ˇŒˇND˛ H) .� `X ˛ ” CONX .�; ˇ//�

Note that ND is used to represent the relation of negation-denial.

Proposition 3. Given that a logic X has the following properties:

1. If CONX .�; ˇ/ then there is ı such that CONX .ı; ˇ/ and � `X ı.

2. X has symmetric ND’s.

Then the logic satisfies [Den*].

Proof. Suppose that ND is symmetric for X , and the other assumptions1. and 2. hold of X . Then assume ˇND˛. Suppose that � `X ˛.Then, of course, CONX .�; ˇ/ since ˇ is a denial of ˛. Now supposethat CONX .�; ˇ/. Then there is some denial of ˇ, , such that � `X

by assumption. Since ND is symmetric, ˛NDˇ. By definition of anND it follows that `X ˛, and by [Cut] � `X ˛. �

9Consistency preservation can characterize a logic under certain assumptions on denials. Fordetails see Payette and Schotch (2006).

10See the Fitch style system in Schotch (2004).

7.6. FORCING AND PRESERVATIONISM 123

7.6 Forcing and Preservationism

The theme of this Volume is to investigate inference relations accord-ing to what property or properties they preserve. The obvious questionthat one might ask is what does this new forcing relation preserve? Asmentioned elsewhere in this Volume forcing preserves level. To see thisnote that the level function is monotonic, i.e., if � � � then `X .�/ �`X .�/. This follows from CONX being downward monotonic–I omitthe proof.

Theorem 6. If `X .�/ D � then `X .CŒ .�// D � .

Proof. Again the level 1 case is trivial. Suppose `X .�/ D � ¤ 1.Then let C be a cover of � such that w.C/ D `X .�/. Suppose that 2 CŒ .�/. Then, by definition, C �� for all C, of appropriatewidth, which cover � . However, was arbitrary, as was C. Thus, onecan see that C is a cover of CŒ .�/, and has width � . Thus `X .CX .�//has level at most � , but `X is monotonic, and � � CŒ .�/; therefore,`X .CŒ .�// D �. �

The relation of forcing as a paraconsistent inference relation is ‘nice’in the sense that one does not lose out on consequences that �� mayhave to give up. If it is the case that a set of premises is X -consistentthen the forcing closure is the closure of the set relative to the logic X .

Theorem 7. If `X .�/ D � , � 2 f0; 1g then CX .�/ D CŒ .�/.

Proof. Assume that ˛ 2 CŒ .�/. Then for any partition C there is�i 2 C such that �i � ˛. Then, �i � � , and, by monotonicity of X ,it follows that ˛ 2 CX .�/.

Now suppose that `X .�/ D 0, then h¿i is a cover of � . In fact itis the only cover, since it is the only cover of width 0. So CŒ .�/ DCX .�/. Now assume that `1.�/ D 1. One may restrict the investigationof the covers of � to the partition covers by lemma 4. Thus, all one mustconsider is the cover C D h¿; �i. The result follows. �

So for consistent sets the relation collapses into the underlying logic.


What about other levels beyond the finite into the transfinite? Thelevel function can assign the level ! to a set under the assumptionswhich have been made. And of course the forcing closure will preservethat level, but can we say anything about what that closure will be?Recall that the language which we are working with is denumerable atmost.

Proposition 4. If `X .�/ D ! then CŒ .�/ DS 2� CX .f g/.

Proof. Assume that `X .�/ D !. Then the largest the cardinality of �can be is !. Thus, the cover Cm� is a cover of width !, the level of � .Clearly Cm� � CŒ .�/. Let ˛ 2 CŒ .�/. Then for every cover C ofwidth the level of � , C �� ˛. Thus, ˛ 2 Cm� . �

Until now, the only assumptions that have been used are that theconsequence relations, and/or operators, obey [M], [R] and [Cut]. Noconcerns about compactness or connectives other than that the logic Xhas connectives and is compositional have been addressed. Now it istime to ask these difficult questions. Namely, what structural rules arepreserved in the “move up” and what can be said about compact logics,and the effects of connectives on forcing?

7.7 Structural Rules and Compactness

Continuing with the study of the properties inherited from the underly-ing logic one might ask if the forcing relation has [R], [M] and [Cut].That forcing has [R] is perfectly obvious, but that it has [Cut] is not.Before [Cut] is discussed we must discuss [M]. There is a problemwith [M]. If �

� ˛ then it is possible that the level will rise if one

adds a set to it. For instance, if the underlying logic is classical thenfA;Bg will force A ^ B but, fA;B;:Bg will not force A ^ B , sincef¿; fBg ; fA;:Bgg is a partition of width 2, the level of fA;B;:Bg.But that cover will not cover A ^ B . This is the downfall of forcing.It is not monotonic, but it is not anti-monotonic. With a caveat it ismonotonic. That restriction is that the set being added must not makethings worse. That means that when one adds a set of premises, the newset must not be more inconsistent. The level of the superset must not

7.7. STRUCTURAL RULES AND COMPACTNESS 125

change, i.e., `X .�/ D `X .� [�/. The relation �� does not suffer fromthis; it has all three structural rules. For forcing one needs somethingdifferent. The new version of [M], [M�]11 is: If `X .�/ D `X .� [�/,and �

� ˛ then � [�� ˛. However, this is not something assumed

about forcing; it follows from the definition.

Proposition 5. The forcing relation� , has [M�], given that X has

[M].

Proof. Assume `X .�/ D `X .� [ �/, and �� ˛. Then, let C ��

� [�, and w.C/ D `X .� [�/. Then C �� so by definition C �� ˛,since w.C/ D `X .�/. Hence � [ �� ˛, since C was an arbitrarycover of the proper width. �

Given [M�], one can see that the version of [Cut] given in the sec-tion above is not satisfied in all cases. However, the formulation of[Cut] as: If � ` ı for all ı 2 � and � ` ˛, then � ` ˛ is also notsatisfied. Also mentioned in the section above, the version of [Cut] justmentioned gives the correlation between the consequence operator andthe consequence relation. Consider the following counter example. Let� D fA;Bg and � D fA;B;C;:C;Dg. Then although any cover of� of width 2 is a cover of �, it may not cover � properly. The set �forces A ^ B , but � does not. Thus [Cut] cannot be satisfied, at leaston the formulation that has been chosen. But not all is lost, just mostof it. A different, but not finitist, version can be satisfied. In fact thisprinciple has been used implicitly many times over in the proofs above.It is as follows: If � ` ı for all ı 2 � and �;� ` ˛ then � ` ˛. Thiswill be known as [Cut�].

Proposition 6. If the relation � of X has [R], [M] and [Cut] then�

has [Cut�]. That is if �� ı for all ı 2 � and �;�

� ˛ then �

� ˛.

Proof. Assume �� ı for all ı 2 �, and �;�

� ˛. Then assume that

C �� , and w.C/ D `X .�/. � � CŒ .�/ so C �� by definitionbecause C �� . Since `X .�/ D `X .CŒ .�//, proposition 6, it mustbe that `X .� [ �/ D `X .�/. Thus C is a cover of � [ � of the rightwidth to be a cover of ˛ by definition of forcing. Since C is an arbitrarycover of � of the right width, �

� ˛. �

11This is also known asX-level preserving monotonicity in Payette and d’Entremont (2006).


It would be at best misleading and at worst cowardly to omit men-tion of some rather odd results that flow from the definition of forcing.Forcing can take a logicX that does not have [Cut],12 and give it [Cut�].Because of this peculiarity, for the logic and its derived forcing relationto agree on consistent sets the logic must obey the structural rules. How-ever, the same cannot be said about the rule [R]. The case of [M�]issimilar to that of [Cut�]. The restricted version of monotonicity mayhold even though X does not have [M]. But we place these problemsout of bounds because the logics we are considering are ‘structural’ asin definition 6.

The preservation of compactness is not from the logic to forcing,but, rather, from the logic to the level function. The forcing relation iscompact given the compactness of X , albeit in an uninteresting way. If�� then take a partition cover of � , C. By definition of forcing

there is a �i 2 C such that �i � . By the compactness of X we get a�0i �! �i such that �0i � . Here �! is the finite subset of relation.However, �0i �! � so we satisfy compactness. Level compactness ismuch more interesting.

For level compactness we need a theorem first. These theorems orig-inally appeared in Payette and d’Entremont (2006), and they will alsobe proved here. We assume that the consequence relation of the logicXis compact. That means that the CONX predicate is also compact.

Theorem 8. The following are equivalent when `X .�/ < ! and CONXis compact:(The subscripts will be omitted.)

1. `.�/ � n”8� 0 � � that are finite, `.� 0/ � n.

2. If `.�/ D n then 9� 0 � � which is finite, such that `.� 0/ D n.

3. If there is a finite subset �� of � such that for any other finite� 0 � � , `.� 0/ � `.��/ then `.��/ D `.�/.

Proof. We proceed by showing the equivalence in a triangle. All n;m; ketc. are elements of !.

12Although, strictly speaking, some might wish to argue that there is no such thing as a logicwhich fails to have [Cut].

7.7. STRUCTURAL RULES AND COMPACTNESS 127

1) 2 Assume 1 and assume for reductio that `.�/ D n and that thereis no finite subset of � which has level n. We know by monotonicityof level that all of the subsets of � must have level less than that of � ,so there is an upper bound. This upper bound will also apply to finitesets. Call this upper bound m. This m is strictly less than n becauseotherwise there would be a finite subset of level n which there isn’t.With 1 it follows that `.�/ � m < n, which is a contradiction.2 ) 3 Assume 2 and the existence of a �� as in the antecedent of3. There must be, by 2, a finite � 0 � � with `.� 0/ D `.�/ but then,`.�/ D `.� 0/ � `.��/. By monotonicity of ` so `.��/ � `.�/ hence,`.��/ D `.�/.3) 1 Assume 3. The only if direction of 1 follows from monotonicityof ` thus, assume that for every finite � 0 � � `.� 0/ � n. Letm D maxfkj`.� 0/ D k & � 0 � � f initeg This must exist since thereis an upper bound, viz. n. Let �� D � 0 such that `.� 0/ D m. Then theconditions for 3 are satisfied thus, `.��/ D `.�/ D m � n. �

With this theorem we can prove:

Theorem 9 (Finite Level Compactness). If � is a set of formulas with`.�/ < ! and CONX is compact then: If `.�/ D n then for some finitesubset � 0 of � , `.� 0/ D n.

Proof. We will proceed by induction on the level of � and use theorem8. For `.�/ D 0 then � D ¿ or � � CX .¿/, so ¿ � � and isfinite with `.¿/ D 0, and by the downward monotonicity of CONX anynonempty subset of a nonempty � will have level 0. The result for thebasis step follows. Assume: if `.�/ D k then, For some � 0 �! � ,`.� 0/ D k for k � n. Assume that `.�/ D n C 1. We will show that3 holds to get the result for this stage. If there is a finite set then we aredone, so assume that there is no finite set with level n C 1. All finitesubsets must have level (strictly) less than n C 1 by monotonicity oflevel. Since there is an upper bound on the levels of the finite sets, callitm,m � n < nC 1 and let �� be a finite subset with levelm. Then byinduction hypothesis using 3 above we get `.��/ D `.�/ D m < nC1but that is a contradiction. Therefore for all n 2 !, if `.�/ D n then,for some finite subset � 0 of � , `.� 0/ D n. �


Notice the crucial restriction to finite levels in these two theorems.The level of a set is bounded by its cardinality. If a set has a level thatis not 1 then Cm� is a cover. But, w.Cm� / D j�j. If � had a levelgreater than its cardinality, Cm� could not be a cover. So `X .�/ � j�j.If `X .�/ D ! for example then there is no finite subset that could havelevel ! since the set can have at most a finite level. What this means isthat any set with level ! will have a subset of every finite level.

To conclude the general look at properties that are preserved in themove to forcing, consider a particular example of a logic that meetsthese properties: classical first-order logic.

7.8 First-Order Logic and Forcing

The properties studied thus far can be used to judge the ‘goodness’ ofthis method of paraconsistentizing a logic. However, they are not justidle properties; they have applications that imply more reasons to thinkthat forcing is a good method.

We will show this in two tiers. First we will give the general casethen a specific case: first-order logic. What we want to show is thatforcing under certain conditions will be the largest relation that can bedefined that has many of the aforementioned properties. In particularthe key properties will be the restricted versions of the structural rulesand level preservation. To accomplish this we must introduce some newobjects.

Definition 17. �C is a maximal level preserving extension (MLPE)of � if and only if:

1. � � �C,

2. `.�C/ D `.�/, and

3. for any formula , if `.�C [ f g/ D `.�C/ then 2 �C.

It follows, with the help of theorem 9 above, that any set with finite

7.8. FIRST-ORDER LOGIC AND FORCING 129

level can be extended to an MLPE . We can show this by considering thefollowing result about partial orders.

Definition 18. A partially ordered set hS;�i is a set S with arelation � such that for a; b; c 2 S

1. a � a (Reflexivity)

2. If a � a and b � c then a � c (Transitivity)

3. If a � b and b � a then a D b (Anti-symmetry)

A monotone operator on a partial order, hS;�i, say, is an operator,ˆ W S �! S such that if a � b then ˆ.a/ � ˆ.b/. Given a set T � Sone says that T has an upper bound if, there is an a 2 S such that forall t 2 T , t � a. A least upper bound for T is an upper bound a ofT , such that for any other upper bound b, a � b. An element m 2 Sis maximal just when, for any a 2 S , if m � a then m D a. A fixedpoint of the operator ˆ is an a 2 S such that ˆ.a/ D a. A ‘chain’ in Sis a set C � S such that for each a; b 2 C , either a � b or b � a. Asmallest member of S is a b 2 S such that, for any a 2 S , b � a. Withthis terminology in mind the following can be shown.

Theorem 10. Fitting (1986, pp. 78-9) Let hS;�i be a partial ordersuch that S has a smallest member, every non-empty set of membersof S which has an upper bound has a least upper bound, and ˆ isa monotone operator on S . Suppose every chain in S has an upperbound. Then, if a 2 S is such that a � ˆ.a/ then there is a maximalfixed point of ˆ, call it b, such that a � b.

Proof. Suppose that hS;�i is a partial order which satisfies the condi-tions of the theorem. Letˆ W S �! S be a monotone operator. Supposethat a 2 S and a � ˆ.a/. Define E Dfx 2 S W x � ˆ.x/g. Clearly a 2 E. If one restricts � to E thereis another partial order. Suppose that C is a chain in E. C will alsobe a chain in S thus it has a upper bound, and since it is a nonempty


set in S it has a least upper bound call itWC . Thus for all x 2 C ,

x � WC so ˆ.x/ � ˆ.WC/ since ˆ is monotone. Since x � ˆ.x/because C � E thus for all x 2 C x � ˆ.

WC/. So ˆ.

WC/ is

an upper bound for C in S so becauseWC is a least upper boundW

C � ˆ.WC/. Thus

WC 2 E. Hence al chains in E have upper

bounds in E. Then apply Zorn’s lemma to get a maximal element ex-tending any element of E, including a. Suppose that m is a maximalelement ofE. Then by definition ofE,m � ˆ.m/ and by monotonicityof ˆ, ˆ.m/ � ˆ.ˆ.m// so ˆ.m/ 2 E. Since m is maximal in E, i.e.there can be no proper, or non-equal, extensions of m in E, ˆ.m/ D m(m is a fixed point of ˆ). Take any fixed point, say g, of ˆ in S theng D ˆ.g/ so g � ˆ.g/ thus, g 2 E. Since m is maximal in E nofixed point in S can properly extendm so no maximal element of S canextend m because maximal points must be fixed points of ˆ. Thereforem is maximal in S . Which is what is wanted. �

In the logics we are considering, all of the suppositions of the the-orems are satisfied, save the condition on chains. For a compact logicthe chain condition follows immediately. However, this theorem appliesvery easily to some propositional logics. But for first-order logics it isnot so clear; a maximally consistent extension may not correspond toa model in some cases. We may need something more, but that wouldtake us off track. What we want to show is that a partial order can bemade out of sets so that the maximum items will be the MLPE ’s.

Consider a set � with a finite level relative to the compact logic X .Define the set L� as set of all level preserving supersets of �. That isthe set

L� D f� [� W `X .� [�/ D `X .�/gProposition 7. hL�;�i is a partial order which satisfies the conditionsin theorem 10.

Proof. Any non-empty set D of L� that has an upper bound � 0 will besuch that `X .� [ � 0/ D `X .�/, and, since

SD � � 0, and level is

monotonic D has a least upper bound,SD.

That unions of chains have upper bounds follows from the FiniteLevel Compactness theorem, since, if the union of the chain C ,

SC ,


has the property that `X .SC[�/ > `X .�/, then there is a finite subset

of this set that has `X .SC [�/, or there is some finite subset that has

a level greater than `X .�/ if the level ofSC [ � is infinite. Either

way that finite set would be contained in some set in the chain, but thatis impossible. The monotone operator is the forcing closure operator.This operator is not monotone in general, but, since it has [M�], it ismonotone on L�. �

We then have the result that we want:

Lemma 5. Let � have level m 2 ! then it can be extended to a �C.

Proof. For any � � S it is the case that � � CŒ .�/. Apply proposi-tion 7. It is clear that the maximal sets in this are the upper bounds weget from this theorem. Every upper bound must be in theL� so they arelevel preserving supersets, which takes care of the first and second con-ditions. The third condition holds because of the maximality of thesesets in L�: if something could be added to the set that maintained thelevel, the set would not be maximal in L�. The result holds. �

Notice that if the level of � is ! then the extensions will be every-thing except those formulas which have level1.

We can then define the relation which will be the largest on a com-pact logic, even for sets of infinite levels. Notice that the followingholds for MLPE ’s: �C

� ” 2 �C. This follows at once

since forcing obeys [R], and forcing is level preserving. The largestrelation, regardless of underlying logic X , can be defined as follows:

� `MLPE ' , ' 2\

�C2MLPE

�C

where MLPE stands for the class of maximal level preserving extensionsof � .13 We abbreviate the right hand side as

T�C.

13The general case will have to obey some other condition if the logic is first order namely theone like in the completeness theorem of Henkin. That is one only considers formulas which occurin the original language of � not the extensions.


Proposition 8. Given a compact logic X and a relation Y which obeys[Cut�], [R], [M�] and preservesX -level we have that for all� CY .�/ �T�C.

Proof. Assume for reductio that ' 2 CY .�/, but ' 62 T�C. Thenthere is an MLPE of � , call it �C' , such that ' 62 �C' . By definition`.�C' [ f'g/ > `.�C' / D `.�/. And since � � �C' , by hypothesis onY , it follows that CY .�/ � CY .�C' / by [M�]. Thus ' 2 CY .�C' /, andso `.CY .�C' // > `.�C' /. Therefore, Y does not preserve X-level. Or,Y does not obey [M�]; either option is contrary to hypothesis. �

The next question to raise is whether a sufficient and/or necessarycondition for forcing to agree with this ‘largest’ relation

T�C, can be

established. It can indeed be shown that there is a sufficient condition,but to establish that we need to discuss the object language of the logicX in question. The next theorem holds for logics said to be productivalor said to have products. The definition of such a logic is as follows.

Definition 19. A logic X is productival iff, given any finite set �there is a formula ' such that ' `X for each 2 � , and� `X '.

This notion of a product is like that of a denial. A product will besome derived connective which will allow one to satisfy the conditionsabove. It is worth noticing that if a logic has binary products, then it hasall finite products. This is a well known theorem from category theory.

Somebody might be tempted to say ‘Oh I see, products are just akind of conjunction!’ but that would be to put the matter backwards. Inparticular, a logic may have products without its underlying languagehaving any explicit connectives at all. Or if it has connectives it mightnot have an explicit conjunction. Classical logic with _ and : as theonly primitive connectives serves as an example. The product of ˛ and ˇis :.:˛_:ˇ/. The further assumption needed about logics is that theyhave symmetric negation denials. The reader should recall the definition16. Then we have,


Theorem 11 (Maximal Forcibility Theorem). LetX be a compact, pro-ductival logic, which has symmetric negation denials, so obeys [Den*],and `X .�/ 2 !. Then, �

� if and only if, for each MLPE �C of � ,

�C� .

Proof. ())(I will omit the subscript X’s.) Suppose �

� . Then let �C be an

MLPE of � . So, `.�/ D `.�C/ by definition. Suppose that C covers�C and has width `.�/. Then C is also a cover of � because � � �C.But, the width of C is also the level of � so, C is a cover of � of theappropriate width for there to be a �i 2 C such that �i � by thedefinition of forcing. Since C was arbitrary �C

� .

(() By contrapositive. Assume �Œ6� and `.�/ D m. Then, thereis an C which is a partition cover of � of width m where 8 �i 2 C,�i 6� . Thus, by [Den*] CON.�i [ f 0g/ for each i , 0 a ND of . For each �i ¤ �j , �i \ �j D ¿. And CON.�i [ �j /. Bycompactness of CON, for each pair of cells there are finite sets �0i and�0j , contained in �i ; �j respectively, which are inconsistent with oneanother. Thus, form their respective products and get 'ij and 'j i . So itfollows that CON.f'ij g [ f'j ig/.

There are only finitely many of these 'ij ’s for each i ; thus, there isa product for each i , call it 'i , such that 'i � 'ij for each j ¤ i . The'i ’s are clearly consistent, by definition of products, with the �i ’s, andany two distinct 'i ’s are inconsistent. Use the fact that CON.�i [f 0g/to form the product of f'i ; 0g to get '�i for each i , which will also beconsistent with each �i . Form the cover:

C0 D h¿; �i [ f'�i g W �i 2 C; & 1 � i � miC0 is a cover of � [ f 0g of width m. Let �� D � [ ˚'�i W 1 � i � m

By monotonicity of level it follows that `.��/ D `.� [ f'�i W 1 � i �mg/ D `.�/. This new set �� will have a MLPE �C. �C will have levelm by definition, and, since each '�i entails 0, and must be contained ina different cell, cannot be added to �C without increasing its level.But that is to say: �CŒ 6� with � � �C. �

What this theorem implies is that for logics satisfying the specifiedproperties what is forced by a given set is all that ends up in every MLPE


of the set. Thus forcing agrees withT�C, and

T�C is the biggest

relation there is that fits our description.The reason for the assumptions of symmetric ND’s and products is to

guarantee that the extensions will be able to exclude a given formula—the one which is not to follow from the set. If given the set fA;:Ag,this set does not force :B , but without the ability to make the formulas,:A ^ B and A ^ B , one may be able to add :B to the set withoutraising the level. The symmetric ND’s are needed so that entailment andconsistency will commute in the right way, i.e., so that non-provabilitywill imply that the set is consistent with an ND. This will of course failfor intuitionistic logic. The general problem is that, depending on thelogic, the maximal extension may include things that do not follow fromthe sets. In IL there are formulas that are consistent with a set, make itinto every maximal extension, but are never proved by the original set.These restraints, if absolutely necessary, greatly limit the number oflogics for which forcing is the largest relation, but it does not push outall but classical logic. The Łukasiewicz many-valued Łn logics all haveconjunctions, and have a type of bar negation which act a symmetricND.

Next comes a particular example of these properties. Like the max-imal Henkin-extensions of first-order logic used to prove completenesswe need a more complex extension for forcing.

Definition 20. An �-MLPE �C of a set � is such that: it is anMLPE and if �C

� .9x/ then �C

� .a/. That is every

existential claim is witnessed.

Theorem 12. If � is the relation of CFL, then for all �; ' � S, withfinite level, �

� ' ” �C

� ' for all �-MLPE ’s �C such that

� � �C.

To prove this we need another extension lemma, one that says thateach � can be extended to an �-MLPE . Here the reader should alsorecall definition 7.

Lemma 6. Each � with CFL-level n 2 ! can be extended to an �-MLPE .


Proof. Let `CFL.�/ D n 2 !. Let 1; :::; k; ::: such that k 2 ! be anenumeration of the formulas of the language. Form sets †n for n 2 !by:

†0 D �

†n D

8ˆ<ˆ:

†n�1 [ f ng if `.†n�1 [ f ng/ D `.�/†n�1 [ f ng [ f'.a/g if `.†n�1 [ f ng/ D `.�/

and n D .9x/'†n�1 otherwise

Where a is not mentioned in †n�1 [ f ngLet �C D S

n2! †n. Claim: this set is an �-level preserving max-imal extension of � . By the recursive construction and compactness`.�C/ D `.�/.

First it must be established that for each clause of the construction`.†n/ D `.†n�1/. For the first clause it is simple, by monotonicity oflevel. Similarly for the third clause. The second clause is more compli-cated: Assume the result holds.

There are three cases if `.�C/ ¤ `.�/.1. `.�C/ D12. `.�C/ � !3. ! > `.�C/ > `.�/

If 1. then an absurd formula was added which cannot be covered, butthat is impossible since that would have occurred at some stage k andwould raise the level from the previous stage. If 2. then for any n 2 !,there is a finite subset of �C of level n. If there was a finite upperbound on the levels of the finite subsets of �C then the compactnesstheorem, theorem 9, says the whole set would have that finite level,which is contrary to assumption. So choose n D `.�/ C 1, the finitesubset of level n will be contained in some †k so `.†k/ > `.�/ whichis impossible. Finally, if 3. there was a finite set which was the culprit,and it would be contained in some †n, which is also impossible. Thus`.�C/ D `.�/.

Claim: �C is an MLPE . Suppose `.�C[f'g/ D `.�C/, then eithera) ' 2 � or, if not, b) ' D n for some n 2 !. If a) then ' 2 �C


a fortiori. If b) then ' was considered for membership at stage n, andsince adding it to �C does not change its level, adding ' to †n�1 doesnot change the level of †n�1 since `.†n/ D `.�/ D `.�C/. Thus, 'was added at stage n. Therefore, ' 2 �C.

Claim: �C is �-complete. Suppose that �C� .9x/'. Then

`.�C [ f.9x/'g/ D `.�C/ so, by the previous result, .9x/' 2 �C.And for some k 2 ! .9x/' D k , so k was added at stage k, butthen so was '.a/ for some constant a, by the construction. Therefore�C

� '.a/ a fortiori. This completes the proof. �

Lemma 7. The second clause of the construction preserves level.

Proof. This uses theorem 9 essentially. Let†n D †n�1[f ng[f'.a/gwith n D .9x/' as is stipulated. By assumption `.†n�1 [ f ng/ D`.†n�1/ D k 2 !. For any †0 �! †n�1 [ f.9x/'g, `.†0/ � k bymonotonicity of `. Note that for any †0 �! †n�1 [ f.9x/'g the levelof †0 [ f.9x/'g must also be � k.

Claim: For all †0 �! †n�1 [ f.9x/'g, `.†0 [ f'.a/g/ � k. Sup-pose not, then there is some †0 � †n�1 [ f.9x/'g, such that for eachcover C D h¿; �1; : : : �ki with w.C/ D k, we haveCONCFL.�i ; '.a// for each 1 � i � k. If there wasn’t, then by com-pactness the result would follow since each finite subset would havelevel � k. However, CFL obeys [Den*]; thus, one has �i � :'.a/for 1 � i � k. Note that a is not mentioned in any �i , since C is anelementary cover, and a is not mentioned in †n�1 [ f.9x/'g by hy-pothesis, so �i � .8x/:'.x/ by the rules for classical logic. But thatmeans, �i � :.9x/'.x/ also by classical logic.

But, some�i must entail .9x/' because C is a cover of†0[f.9x/'g.So, for some�i , CONCFL.�i /, which is contrary to the hypothesis thatC is a cover. Therefore, each finite subset of †n�1 [ f.9x/'g [ f'.a/ghas level � k, and by compactness, i.e., theorem 9, the whole set musthave level � k. However, `.†n�1/ D k, and †n�1 � †n�1 [f.9x/'g [ f'.a/g so by monotonicity of level, `.†n/ D `.†n�1/.Which is what is wanted. �

Using these results the proof of theorem 12 follows easily.

7.9. CONCLUSION 137

Proof of theorem 12. Assume that � has finite level. Then suppose that�� . Then every �-MLPE will force it, since it must contain .

Suppose that �Œ 6� , then proceed as in theorem 11 by conjoining theproper finite sets with the negation of . Then by lemma 6 extend thisnew set to an �-MLPE which will not force since it would raise itslevel to do so. �

This concludes the study of forcing as an approach to paraconsis-tent logic that derives a paraconsistent relation from an inconsistencyintolerant logic X .14

7.9 Conclusion

This chapter’s purpose is to display many properties that can be pre-served by the move to the forcing version of some underlying logic(which is assumed to have the properties). Even when the full version ofthe property fails (which is guaranteed since the whole point of a forc-ing relation is to fix perceived problems with the underlying relation)there are usually restricted versions of the properties which hold forforcing. In fact the restricted versions can be seen to make more sensefor a paraconsistent relation, which is certainly the case for monotonic-ity and substitutionality (in the strong sense). The hope is that theseconsiderations may persuade the reader that this method is, from a tech-nical standpoint, the right way to go. What is important to recognize isthat the method does not completely change the rules of the game. Itrecognizes a failing, and then offers a solution. It does not change themeaning of logical constants; it merely does not permit one to use thoseconstants in certain contexts: namely inconsistent contexts. In all of theconsistent contexts one may proceed as one always does using the logicone wants.

What happens in the inconsistent contexts is that something has goneawry. But, we want to keep as much around as possible of the originalway of doing things, for which we presumably have strong motivation.

14Please note that Theorem 8, Theorem 9, Proposition 8 and Theorem 11 were originallypublished in Payette and d’Entremont (2006).


Forcing allows us to keep within sight of the familiar even while stray-ing into unexplored logical territory.

Eight

Representation of Forcing

DORIAN NICHOLSON AND BRYSON BROWN

Abstract

This essay shows how the forcing relation with underlying logic Xmight be represented in a way which more closely resembles an axiomaticapproach. Following the initial result we take up the case of representingthe forcing relation in which the underlying logic X allows sets on thethe right of `X . This requires us to redefine the notion of X -level forcingto take into account the ‘handedness’ of sets. We must also expand thedefinition of X-level to take this difference into account.

8.1 Introduction

Once it is realized that X-level forcing1 inherits only some of the prin-ciples of the underlying logicX and in particular that an abridgement ofthe structural rule [Mon] of monotonicity is required,2 the question ofalternative presentations of the relation naturally arises. The goal is togive something like an axiomatization of the relation, and once that has

1See ‘On Preserving’ in this volume for the definitive account.2In fact many, otherwise well-intentioned folk, have gone so far as to equate monotonicity

with deduction, in the sense that if a relation is not monotonic, then it can’t be deductive inference.

139

140 CHAPTER 8. REPRESENTATION OF FORCING

been accomplished, we can, those of us who were worried, heave a sighof relief. The project is not so urgent as the corresponding problem inso-called n-ary modal logic, but that shouldn’t be regarded as an excusefor not undertaking it.

8.2 Background

We consider logics X which satisfy the usual structural rules:

[R] ˛ 2 � H) � `X ˛[Cut] �; ˛ `X ˇ & � `X ˛ H) � `X ˇ[Mon] � `X ˛ H) � [� `X ˛

For these logics, we understand the notion of consistency in the man-ner of Post.

� is consistent, in or relative to a logic X (alternatively, �is X -consistent) if and only if there is at least one formula ˛such that � °X ˛.


Having a notion of consistency allows us to define a certain relationbetween pairs of formulas (of the underlying language of the logicX—after this we won’t bother to keep mumbling that particular mantra)which we term non-trivial denial.

X has non-trivial denial if and only if for every non-absurdformula ˛ there is at least one non-absurd formula ˇ whichis not X -consistent with ˛. Two formulas which are relatedin this way are said to deny each other. We also say each isa denial of the other.

There are, in addition, two properties which we may require of alogic X :

8.2. BACKGROUND 141

We shall say of the logic X that it is productival (or that it hasproducts), which is to say two things:

(Q1) For every finite set † there is a formula

Q.†/ such thatQ

.†/ `X �i for every �i 2 † and

(Q2) † `X

Q.†/

We say of the logic X that it is coproductival (or that it has co-products) when it satisfies these two conditions

(`1) For every finite set of formulas † there is a formula

`.†/

such thatfor every � 2 †, � `X

`.†/

(`2) If, for any formula ˇ, � 2 † H) � `X ˇ then`

.†/ `x ˇ

When the logic X enjoys all of these, it will be said to have a full setof properties.

Given a set � , of formulas drawn from the language of the logic X ,we say that an indexed family of sets (starting with ¿), is an X-logicalcover of � provided:

For every element a of the indexed family, CONX .a/

and

� � Si2ICX .ai /

whereCX .ai / is the (X ) logical closure of ai which is to say fˇjai `X ˇg.When F is an (X ) logical cover for � we write COVX .F; �/. Mentionof the particular logic X is often suppressed when no confusion willbe thereby engendered. We shall refer to the size of the index set ofthe indexed family in this definition as the width of the logical cover,indicated by w.F/.

And now for the central idea:The level (relative to the logic X) of the set � of formulas, indicated

by `X .�/ is defined:


`X .�/ D8<:

minw.F/


1 otherwise


Finally we define the relation of X-level forcing which, as its namesuggests, is derived from some underlying logic X . The relation is de-fined:

�� X ˛ if and only if, for every division of � into `X .�/ cells,


8.3 Representation

In order to represent, as we shall say, the relation� X of X -level forc-

ing, we shall provide another relation to be called simply X-forcing,indicated by

� `X and eventually show that the two are the same re-lation. In doing this we axiomatize, in a certain sense, the relation ofX -level forcing for a variety of logics X .

X -forcing

We describe�`X by saying first that it connects to X by means of the

singular sets of formulas in this manner:

[Sing] If � is singular then

��`X ˛ ” � `X ˛

where a set is singular if it cannot be broken up by logical covers. Inthis case the empty set is singular, along with all unit sets.

As a result of this bridge rule, we should notice the consequence:

[�] f g �`X f˛g & f g �`X fˇg H) f g �`X � f˛:ˇg

8.3. REPRESENTATION 143

Next we stipulate that X-forcing satisfies [Ref] and [Cut] but only arestricted form of monotonicity (or dilution) which has two parts:

[M1] If ��`X ˛ then � [ ��`X ˛

provided `X .�/ D `X .� [ ��/[M2] If �

�`X ˛ then� [ ��`X ˛ if � is singular.

We shall have need of bunches of products as well as coproducts ofsuch bunches. To maximize the ease of displaying such conglomera-tions of formulas we define:

Definition 1. �ij for � n ˚ i ; j

Definition 2. PAIRS.�/ for the set of pairs of elements of �’

For example, if our set is fx; y; zg,then PAIRS.fx; y; zg/ D ffx; yg ; fy; zg ; fx; zgg

Definition 3. PPAIRS.�/ for the set of products of pairs ofelements of �

For example PPAIRS.fx; y; zg/ D f… fx; yg ;… fy; zg ;… fx; zggWe the aid of this notation, we can now state two rules of inference

for�`X .

[RPig] ��`X ˛1 & : : : & �

�`X ˛k H)��`X

` fPPAIRS.f˛1; : : : ˛kg/g[LPig] �ij ;…

˚ i ; j

�`X ˛ for every˚ i ; j

2 PAIRS.�/ H)�;`

PPAIRS.�/�`X ˛

Where ‘Pig’ suggests ‘Pigeonhole.’


The assumption of the two rules is that the cardinality of � exceeds theX -level of � .

The soundness of [RPig] is no serious issue since it is simply a wayof stating the pigeonhole principle. The soundness of [LPig] requires abit more than just seeing however. The requirement is provided by thefollowing:

Lemma 1. Let ��ij D �ij [˚…. i ; j /

. Then `.��ij / D `.�/, when

i ; j 2 � and there is some X -cover of � which is a cover of theirproduct.

Proof. Suppose that F is an X -cover of � and w.F/ D `.�/ and F

is a cover of …˚ i ; j

for some i ; j 2 � . Suppose further that

`.�/ D k. Now let F D f�0; : : : �kg, and for some �i 2 F, �i `X�. i ; j / as in the hypothesis. Let ��i D �i [

˚…˚ i ; j

, and form

F� D ˚�0; : : : ; �

�i ; : : : ; �k

. This is clearly an X-cover of ��ij ; thus,

`.��ij / � `.�/.Suppose for reductio that `.��ij / < `.�/. By definition there is

F0 that covers ��ij with width w.F0/ D `.��ij / < `.�/. Let F0 Df�0; : : : ; �mg, and m < k.

We want to show that F0 is a cover of � . There is some �i 2 F0 thatproves �. i ; j / since it is a cover of ��ij . By definition of product, theproduct must prove both i and j . And there must be a cell that provesthe product so by [Cut] on the underlying logic we have that that cell,�i , must prove both i and j . But � � ��ij D

˚ i ; j

. So that makes

F0 a cover of � . However, that means `.�/ � w.F0/ < `.�/, and thatis a contradiction. Therefore `.��ij / D `.�/. �

Lemma 2. [LPig] holds for� X

�ij ;Q˚

i ; j � X ˛ for every

˚ i ; j

2 PAIRS.�/ H)�`

PPAIRS.�/� X ˛

Proof. Arguing indirectly, we show is that there is some �ij ; �. i ; j /that does not X -level force ˛, given that the consequent of [LPig] fails.We do this by showing that there is a cover, of appropriate width, of�ij [

˚�. i ; j /

that does not cover ˛.

8.3. REPRESENTATION 145

By assumption, the antecedent of the conclusion does not X-levelforce ˛, so there must be a cover of � , of width `.�/, that does notcover ˛. Since the cardinality of � exceeds the X -level of � , theremust be a cell in this cover with at least two elements by pigeonholereasoning. Take the �ij [

˚�˚ i ; j

relative to those two elements

and that set will not X -level-force ˛. What needs to be shown to giveus that conclusion is that the level of this set is the same as the level of� . That way the cover that we construct has the right width to act as anX -level forcing cover for �ij [

˚�. i ; j /

. This follows at once from

the previous lemma.�

The Representation Theorem

With all this under our belt, we are finally ready to prove the Represen-tation Theorem for X-level forcing. We assume that X enjoys a full setof properties.

Theorem 1 (Representation Theorem for X-level Forcing). For everyset � of formulas and every formula ˛

�� X ˛ ” �

�`X ˛Proof. The proof is by induction on the cardinality of the premise set� . The basis of the induction is the case in which the cardinality inquestion is less than or equal to the X -level of the premise set. In thiscase, since there will be at least one logical cover of the premise set inwhich each cell contains no more than one formula, there must be anX -inference of the form f g � X ˛ and thus the result follows at onceby [Sing].

In the induction step, the hypothesis of induction is that the cardi-nality of the premise set is n+1, and that the result holds for all premisesets of cardinality up to and including n. To avoid loss of generality wemust take n to be greater than the X-level of � .

Assume �� X ˛. It is easy to see that �ij ;…

˚ i ; j

� X ˛

for every pair˚ i ; j

in PAIRS.�/. Since the cardinality of each of

the premise sets is less than the cardinality of � , by the hypothesis ofinduction we may replace

� X by

�`X in each of the inferences. But


by now invoking the rule [LPig] we get �;` fPPAIRS.�/g �`X ˛. But

��`X

` fPPAIRS.�/g by [RPig] so the result follows at once by [Cut].�

8.4 More Definitions and motivation

It would be an exaggeration, although a pardonable one, to say that themost general way to do proof theory is to take an inference relation tobe a relation between sets of formulas.3 This is different from studyingthe relation between a set of premises and its closure, which is simplygathering together all of the single formulas which follow from the thepremises. In this new approach we, in effect, draw a distinction betweentwo kinds of sets (of formulas).

We could label the two kinds, premise sets, and conclusion sets,though that terminology invites a confusion between the notion of de-ductive closure which is after all a set of conclusions, and the newspecies which we now consider. For the most part, we call the setsthat appear on the left side of the provability symbol, l-sets while thoseon the right are r-sets.

Perhaps the easiest way to think of what is going on here is as a kindof dualism, that r-sets and l-sets are dual to one another in a sense thatwe shall try to illuminate as we go. As a starting point, we can thinkof sets on the left as candidates for assertion, and sets on the right ascandidates for denial. The consequence relation holds when assertionof all members of the left set is incompatible (according to some logicalstandard or measure) with denial of all members of the right set.

First we shall formulate our basic structural rules, generalizations ofthe rules which we have already seen, which apply to all the logics wediscuss, unless we issue an explicit disclaimer.

[R] � \ � ¤ ¿ H) � ` �[Cut] �; ˛ ` �& � ` ˛;� H) � ` �

3Those who favor Gentzen-style proof theory, also known as sequent calculus, would arguethat sequents are more general than sets since the latter, in effect, build in a number of structuralrules rather than making them explicit.

8.4. MORE DEFINITIONS AND MOTIVATION 147

[Mon] � ` � H) � [† ` � [ƒ

In previous work we took the notion of consistency as central. Alas,this is peculiar to l-sets, which is to say that it doesn’t translate to r-sets.Of course, this is what we ought to expect. What we shall require is amore general notion, which we call triviality which comprehends bothsides of the provability relation. Here it is for l-sets:

A set � is said to be (X) l-trivial, indicated by TRIVXL.�/

if and only if � `X ¿.

Notice that, since we are assuming that `X satisfies [Mon], thisamounts to saying that � proves every set, which is simply a way ofsaying that there is no distinction between what � proves inX and whatit doesn’t. Whatever notion of X-provability has interested us thus farcan interest us no longer, when it comes to � .

For r-sets, we must take the dual formulation.

A set � is said to be (X ) r-trivial, indicated by TRIVXR.�/

if and only if ¿ `X �.

l-triviality is of course merely inconsistency in sheep’s clothing whiler-triviality seems much more radical or even gratuitous. An r-trivial unitset, say f˛g, which we would normally write ˛, is what others wouldcall a theorem of the logic in question. So far from being trivial, theseare the whole point of a logic, aren’t they?

In a word, no. Logic is about inference, and the r-trivial sets tramplethe distinction, dual to the one offended by inconsistency, between whatproves some set and what doesn’t. Once again, given [Mon], every setproves an r-trivial one.

In general terms we might say this in defence of our derogatory ter-minology: In so far as an inference relation draws a distinction betweenthose kinds of reasoning which are correct and those which aren’t, bothkinds of triviality are well-named. In both cases there is no longer adistinction. In the l-case we have lost the distinction between what a setproves and what it fails to prove and in the r-case the dual distinctionbetween what it would take to prove a set and what it would take not to


prove it. There is no reason to reject one sort of trampling while em-bracing its dual. And while we are at it, let us not forget that ‘tautology’is indeed a term of derogation, often prefixed by some such qualifier as‘mere.’

First, we define an ambidextrous predicate:

Definition 4. Relative to a logic X , a set † is said to beX-consistent indicated by CONX .†/ if and only if TRIVR.†/ if †is on the right, and TRIVL.†/ if † is on the left.

Here the overline indicates predicate negation.Having increased the scope of the consistency predicate to cover

both kinds of sets, we carry over our earlier definitions of logical coverand level which have become similarly ambidextrous.

Level applies to both flavors of set, but rather than use explicit no-tation to display which kind of level is at issue, we shall let the contextdetermine whether we are dealing with left-level or right. It ought togo without saying that one and the same set will not in general have thesame level on both sides of the inference relation.4

Having level functions for both left and right handed sets, permitsus to define two derived inference relations. The first is just X-levelforcing, while the second, more exotic one we shall dub inverse X-levelforcing. Once the second relation is defined, the aptness of the namewe have chosen will become apparent. Note that while we speak ofdivisions rather than covers of � and � here, the effect is the same.

X -level forcing:


for at least one of the cells a; a `X ˛inverse X-level forcing:

˛� X � if and only if, for every division of � into `X .�/ cells,

for at least one of the cells a; ˛ `X a4A significant exception is ¿ which has the same level on the left as it does on the right,

namely 0.

8.4. MORE DEFINITIONS AND MOTIVATION 149

Although the inverse case looks a bit unusual, both these relationspreserve level, though we need to specify what that means in the inversecase. Here we must define, for every conclusion-set †, the inverse X-theory of †, indicated by C�1X .†/ as

Definition 5. C�1X .†/ D f� j� `X †g

To say that X -level forcing preserves level is to say that

`X .�/ D `X .CX .�//

and to say that inverse X-level forcing preserves level is to say thesame thing except for inverse closure, i.e.

`X .�/ D `X .C�1X .�//

And this in turn means that neither left or right-handed sets can betrivial unless the set in question contains an absurd formula.

Just as we early showed how to represent X -level forcing, we cannow do the same for the inverse relation.

Inverse X-Forcing

It is relatively easy to see that what we could do by way of represent-ing the relation of X -level forcing, we can also do with the relation ofinverse X -level forcing, by ‘dualizing.’

This certainly applies to the syntactical presentation of the relationwe call inverse X-forcing. We say, as before, that it connects to X bymeans of the singular sets of formulas in this manner:

[Sing] If � is singular then

˛�`X � ” ˛ `X �

where a set is singular if it cannot be broken up by logical covers.Once again the empty set is singular, along with all unit sets.


Next we stipulate that inverseX -forcing satisfies [Ref] and [Cut] butthese rules should be stated:

[Ref] ˛ 2 � H) ˛�`X �

[Cut] .˛�`X �; ˇ & ˇ

�`X �/ H) ˛�`X �

While the restricted form of monotonicity (or dilution) still has twoparts, namely the duals of the previous ones.

[M1] If ˛�`X � then ˛

�`X � [ ��provided `X .�/ D `X .� [ ��/

[M2] If ˛�`X � then

˛�`X � [ �� if � is singular.

Using the notation we introduced for X-forcing, we can state thedual pigeonhole principles which hold for the inverse relation.

[LPig] ˛1�`X � & : : : & ˛k

�`X � H)… fCOPPAIRS.f˛1; : : : ˛kg/g

�`X �

[RPig] ˛�`X �ij ;

`˚ i ; j

for every

˚ i ; j

2 PAIRS.�/ H)˛�`X �;… fCOPPAIRS.�/g

The assumption of the two rules is that the cardinality of � exceeds theX -level of � .

We use the expression COPPAIRS.†/, by analogy with PPAIRS, torepresent the set of coproducts of pairs of elements of †.

We can argue for the soundness of the Pig rules and for the rep-resentation of inverse X-level forcing by inverse X-forcing, by simplyreusing the original arguments for the representation ofX -level forcing,in dual form.

We now consider the proper way to characterize the set-set versionof forcing.

8.5. ��`X � 151

8.5 ��`X �

We start by describing the relation�`X , called X-forcing, based upon

the underlying (multiple conclusion) logic X . Our aim is to show thatthis relation is the same as the multiple conclusion X-level forcing thatwe describe below. It will turn out that there are two ways to accomplishthis, as we have been calling it, representation result, and that the twotheorems are equivalent.

We shall stipulate that X satisfies the structural rules already men-tioned and is productival.

We shall also, as we expect from the earlier account, want X to becoproductival which we may now define more elegantly than we didearlier.

(`

1) For every finite set �, there is a formula`� such that

For every ı 2 �, ı `X`� and

(`

2)`� `X �

We describe�`X by saying first that it connects to X by means of

the singular sets of formulas as we have done earlier.

[Sing] If � and � are singular then

��`X � ” � `X �

The notion of singularity need suffer no change from the earlierconcept so that the empty set is singular, along with all unit sets.

As a result of this bridge rule, we should notice the consequence:

[Q

] f g �`X f˛g & f g �`X fˇg H) f g �`X � f˛; ˇgNext we stipulate that X-forcing satisfies the multiple conclusion

form of [Ref] and [Cut] but again, only a restricted form of monotonicity(or dilution) which has two parts:

[M1] If ��`X � then � [ ��`X � [��

provided `X .�/ D `X .� [ ��/ and `X .�/ D `X .� [��/


[M2] If ��`X � then

� [ ��`X � if � is singular and��`X � [�� if � is singular.

Finally, we must expand our collection of pigeonhole principles.

[RPigL] ��`X ˛1; � & : : : & �

�`X ˛k; � H)��`X

`PPAIRS.f˛1; : : : ˛kg/;�

[LPigL] �ij ;Q˚

i ; j �`X ˛;� for every

˚ i ; j

2 PAIRS.�/ H)�;`

PPAIRS.�/�`X ˛;�

[RPIGR] ��`X �ij ;

`˚ıi ; ıj

for every

˚ıi ; ıj

2 PAIRS.�/ H)��`X �;

QCOPPAIRS.�/

[LPIGR] �; ı1�`X � or : : : or �; ık

�`X � H)�;Q

COPPAIRS.fı1; : : : ; ıkg/�`X �

In the statement of these principles, we assume that the cardinalityof the sets � and � are strictly greater than their corresponding levels.

8.6 �� X �

We shall say that �� X � provided that for every pair F� , F� of

logical covers of the premise set and the conclusion set respectively,that there is a kind of ‘onto’ relation between them of this form:

.9a 2 F� & existsb 2 F� W a `X b/In other words, some cell in each logical cover of the premise setproves (in X ) some cell in each logical cover of the conclusion set.

8.7 The Representation of Forcing

We shall consider two theorems which characterize, or represent therelation

�`X .

Theorem 2. (First Representation Theorem) For every pair � ,� of setsof formulas��`X � ” �

� X �

8.7. THE REPRESENTATION OF FORCING 153

Theorem 3. (Second Representation Theorem) For every pair � , � ofsets of formulas��`X � ” for some formula ˛, �

�`X ˛ & ˛�`X �

Lemma 3. (Equivalence of the two representations) For every pair � ,� of sets of formulas�� X � ” for some formula ˛; �

�`X ˛ & ˛�`X �

Proof. The right to left direction of this equivalence is a simple (in ef-fect) soundness proof which we can omit with a clear conscience. Thehard direction, from left to right, we prove by double induction on thecardinalities of � and �.

In the basis case, the cardinalities of the two sets are less than orequal, respectively to the X-level of � and �. In such a case the onlyway that �

� X � can hold is if there is an X proof `X ı between

some pair of formulas, say 2 � and ı 2 �. By singularity, it mustfollow that �

� X ˛ and that ˛

� X �, for some formula ˛.

As the hypothesis of induction, assume that the result holds for car-dinalities of � , � up to n.

In the first part of the induction step, we assume that j�j is greaterthan `X .�/ (else the result would hold trivially by singularity), and that�� X �. Now it follows that there must be pairs i , j of distinct

members of � such that the X -level of the set � ij D �ij [�˚ i ; j

is

equal to theX-level of � and that � ij� X �. As before we indicate by

�ij the set � with the pair i ; j removed. Moreoverˇ� ij

ˇD n � 1, so

that the hypothesis of induction gives us the existence of some formula,let us call it ˛ij such that � ij

�`X ˛ij and ˛ij�`X �

If, for each one of these pairs i ; j , we gather together the corre-sponding results gained from the hypothesis of induction, we shall have,after using a [PIG] rule and the fact that X has coproducts:

�;` fPPAIRS.�/g �`X

`˚˛ij.i; j / 2 PAIRS.�/

A use of the other [PIG] together with [Cut] results in�� `X

`˚˛ij.i; j / 2 PAIRS.�/ But since each element of the co-

productX -proves�, so does the coproduct which serves as the formulaX -forced by � which X-forces �. This ends the first induction step.


The second induction step is, once again, the dual of the first inwhich we replace pairs ıi ; ıj from delta with their coproducts, whichwe must be able to do for at least some pairs, with increasing the levelof �. The we simply use the same kinds of moves (except dualized) toget the existence of some formula X-forced by � which X-forces �.

�

But this serves as a guideline for the proof of the first representationtheorem as follows:

Suppose �� X �. Then as we showed in the equivalence lemma,

��`X ˛ and ˛

�`X �.

But then by the allowable uses of monotonicity, ��`X ˛;� and

�; ˛�`X �

But then, by [Cut] ��`X �.

Nine

Forcing and Practical Inference

PETER SCHOTCH

Abstract

In this essay we consider two variations on the theme of forcing. In one ofthese we restrict the number of possible logical covers of a given premiseset by specifying certain subsets, which we call clumps and requiring thatany cell which contains a member of a clump, must also contain the othermembers. The other variation allows for redefining the notion of logicalcover in order to require that the cells not only be logically consistent butalso practically consistent. In other words in this mode we require of anylogical cover that it recognize our theoretical (and perhaps also practical)commitments, by defining the cells to be those subsets (if any) which donot prove (in the underlying logic) any denial of those commitments.

9.1 Introduction

What we might call the pure or perhaps general theory of the forcingrelation is all very well in a theoretical setting. That shouldn’t come asmuch of a surprise since our treatment of forcing tends to be abstractand general. On the other hand, sometimes we want to study inferencein something more closely resembling ordinary life, more ordinary thanmathematical life at least. In these cases the general theory is too spare.

155

156 CHAPTER 9. FORCING AND PRACTICAL INFERENCE

In fact we shall distinguish two ways in which one might wish to comedown from the Olympian heights of generality to no greater altitudethan a foothill might boast.

First a sketch of the general theory:1

We consider logics X which satisfy the usual structural rules:

[R] ˛ 2 � H) � `X ˛

[Cut] �; ˛ `X ˇ & � `X ˛ H) � `X ˇ

[Mon] � `X ˛ H) � [� `X ˛

For these logics, we understand the notion of consistency in the man-ner of Post:

� is consistent, in or relative to a logic X (alternatively, �is X -consistent) if and only if there is at least one formula ˛such that � °X ˛.


Any formula the unit set of which is not consistent, is saidto be an absurd formula.

Having a notion of consistency, we go on to define a certain relationbetween pairs of formulas of the underlying language of the logic X(mention of which is often suppressed when it is unlikely to lead toconfusion) which we term non-trivial denial.

X has non-trivial denial if and only if for every non-absurdformula ˛ there is at least one non-absurd formula ˇ whichis not X -consistent with ˛. Two formulas which are relatedin this way are said to deny each other. We also say each isa denial of the other.

1The definitive account is presented in the essay ‘Preserve What?’ in this Volume.

9.2. DIRTY HANDS 157

Given a set of formulas � , we say that an indexed family of sets(starting with ¿) is a logical cover of � provided:

For every element a of the indexed family, CONX .a/

and

† � Si2ICX .ai /

whereCX .ai / is the (X ) logical closure of ai which is to say fˇjai `X ˇg.When F is an (X) logical cover for � we write COVX .F; �/. We shallrefer to the size of the index set of the indexed family in this definitionas the width of the logical cover, indicated by w.F/.

And now for the central idea:The level (relative to the logic X) of the set � of formulas, indicated

by `X .�/ is defined:

`X .�/ D8<:

minw.F/


1 otherwise


Finally we define the relation of X-level forcing which, as its namesuggests, is derived from some underlying logic X . The relation is de-fined:



9.2 Dirty Hands

Now this is all very well in general, for that most abstract plane of log-ical existence on which epistemology intrudes not at all upon how wereason. However this is seldom a good representation of real inferen-tial life. When we face the world of applications, there may well benon-logical considerations to weigh in the balance.


In real life, we have commitments. Some we clasp to our bosomsafter striving to acquire them, while others fall upon us unbidden. Atthis point, some of our opponents may become existentialist enough todeclare that they themselves renounce all commitments and refuse to bebound by anything but their own mighty wills. Such as these have putthemselves not only beyond polite society, but also beyond science.

If the mission statement of science includes the aim of framing boldhypotheses and then suffering these to be tested, science involves com-mitments. We see this directly we begin to think about how hypothesistesting works. It is clear that, whether we construct a new experiment,or merely use data previously gathered, we have to assume that the lawsof science, except of course for one that we may be testing, hold. Itsimply makes no sense to talk about testing all the laws of science atonce. Testing them against what? We are committed, in this endeavor,to the correctness of whatever bits of science are not being tested.

This is merely one example, though it is a particularly compellingone. But what does it mean to say that we are committed to this or thatprinciple? This much at least: For the duration of our commitment, weshall brook no denial of the principle in question. In other words, weshall treat any such denial as a sort of practical contradiction.2

Apart from worrying about commitments there are clearly cases whenmerely having consistent cells is not enough even if we expand our hori-zons to include practical consistency. Cases that is, in which we wantto insist on the integrity of certain of our data. We may be prepared tocarve up our premise set into subsets, but we wish to place limits uponhow much carving can be done.

9.3 †-Forcing

Whatever else we may mean by commitment, intolerance of any de-nial of the commitment must be high on the list. There may come aday in which an argument is so convincing that it releases us from oneor more of our commitments, but when that happens, it goes without

2Such was a popular device in Buddhist logic where a paradigm of absurdity was the sentence‘Lotus in the air’ which we might take as an Eastern version of ‘Pie in the sky.’

9.3. †-FORCING 159

saying we are no longer committed to whatever it was that previouslyclaimed our allegiance. So we aren’t saying that commitment is forever,even commitment worthy of the name. Commitments come and go—anever shifting pattern as we fail to renew our subscription to one theoryand subscribe instead to another.

While our commitment is in force however, while we are still boundby it, we shall not suffer any denial to be concluded except in the specialcircumstances noted below. This is the grand picture at any rate, and likeall such, the devil is in the details. One thing to give us pause here isthe recognition that our logical commitments are in no way inferior toour other commitments3 which places definite limits upon how muchrefusing we can do.

Think of it this way: We want to treat the denials of commitments,we refer to the set of these by †, as absurd—in some wider sense thanlogically absurd perhaps, but absurd nonetheless. Since we shall bedealing with these commitments in the manner of forcing, there willbe some underlying logic X in terms of which initial inferences takeplace, Given what we have said so far, no member of † can be a(n X )consequence of a singular set and still be blocked. Earlier we definedas singular, the sets that suffer no diminution in any logical cover. Inother words, our ability to block these bad conclusions has the samelimit as our ability in ordinary forcing to block the inference to logicalabsurdities.

But might we not change this? It all depends upon how we choose toregard the notion of singularity for sets. For the most part, we have beenthinking of the singular sets as defined as we did just above. If we stickwith that line, then the analogy between practical absurdity and logicalabsurdity continues to hold. On the other hand, we have the optionof regarding singularity as a primitive predicate of sets (of formulas)which might be the victim of whatever we want to stipulate in the wayof exceptions to the way we have understood the idea previously.4

We might, that is, choose to excommunicate all those sets which

3Though we wouldn’t follow Quine in giving them a more central place in our ‘web of belief.’4In effect, this was the approach taken when the topic was introduced in Schotch and Jennings

(1989).


were previously singular but consist of unit sets of members of †. Tocarry out this program of banishment we would also have to limit our-selves to those commitments the denials of which are notX -consequencesof the empty set. We shall examine the upshot of this alternative sternapproach below, but we should note here the existence of opposition toit. Some of our friends have suggested that striking off some sets previ-ously regarded as singular, is contrary to the spirit of preservationism.

The most strongly held intuition behind the notion of forcing is that,in the face of problematic inferences in the underlying logic, we fallback to looking at inference from subsets which are problem free. Un-der any such regime, those inferences in the underlying logic from setswhich cannot be partitioned, must be reckoned correct in the derivedinference. To deny this is to break the connection between the derivedlogic and the underlying logic, a connection which forms a large partof the the justification for preservationism as a revision rather than areplacement theory.5 That is one of the things that separates preserva-tionism from alternative approaches to paraconsistency.

We reply that any general scheme for setting aside those X-licensedinferences, even though they must survive partition, would indeed beanathema. But, we are not suggesting anything so general as all that.What we must do is balance our distaste for jettisoning inferences whichseem otherwise unassailable, with our distaste for practical absurdities.In the sequel we shall present both approaches side by side.

Imagine a case in which there is a physical theory which provides uswith plenty of well-confirmed predictions and also, unfortunately, otherpredictions which are simply absurd, in practical terms at least. Lets say,for instance, that the theory predicts that some physical constant e hasan infinite value. What shall we do? A philosopher will immediatelysay that we have disconfirmed the theory and that we must get rid of itat once. A natural scientist might well have an entirely different slanton things.

A scientist is apt to regard a theory as rather like a salami. Late atnight while working in the lab, a scientist overtaken by hunger might goto the fridge and take out a salami to make herself a snack. Alas, there is

5For a discussion of these terms, consult the first chapter of this Volume.

9.3. †-FORCING 161

some mould on one end of the salami. Does she throw it out? Probablynot. After all when faced with a salami which has mould on one end, nolaw compels us to eat the mouldy part. Cut a few slices from the goodend! And the same goes for our theory. Don’t use the mouldy part ofthat either.

In fact, why not cut away the mouldy part of the theory entirelyand, in order to make it sound better, we could refer to this operationas ‘renormalization.’ Of course it would be better if we didn’t haveto resort to this sort of thing, but the world being the kind of place itis, we often do. Show her a physical theory which does all that herrenormalized theory does, without invoking that somewhat disreputableoperation, and our scientist will embrace it with open arms. Later, atsome learned conference or other, she and her colleagues might have agood laugh at the bad old days of renormalization, and proclaim that theyounger folk in the profession have it much easier than they did.

But right now there is no such theory and the lab is waiting for ourexperiments and the journals our articles. There are promotions to getand prestigious fellowships. There are even prizes to win and often re-spectable amounts of cash are involved. Little wonder then that nobodyis prepared to twiddle their scientific thumbs while awaiting a more wor-thy theory.

This makes philosophers sound a lot purer in spirit than scientists,but it really isn’t so. Consider the ever fascinating realm of moral phi-losophy. Here we have a rather similar problem. We have two grandapproaches to moral theorizing which we might term axiological anddeontological. Each of them ‘sounds right’ in the sense of agreeing withmany of our central moral intuitions. But not only are they not compat-ible, each is refutable by appeal to a central intuition highlighted in theother.6 In other words, we cannot abandon axiology for deontology orvice versa in order to gain a more secure foundation for morals.

So it seems in moral philosophy as in physics, we make do. We usethe non-mouldy part of the theory and pretend as well as we can thatthe mouldy part doesn’t really exist. If we generalize our notion of X -

6Thus deontological theories are vulnerable to ‘lifeboat’ counterexamples while axiologicaltheories are vulnerable to ‘scapegoat’ counterexamples.


consistency, we can make our logic conform to this behavior. We canagree that it is a stop-gap measure, one to be used as a last resort, butevidently a last resort is needed.

The central idea is that we get all the mould into one set called †.Less colorfully,† contains the (X) denials of each of our commitmentsso that every member of† is a sentence which we would like to excludefrom the output of our theory.

A set � is (X) †-consistent indicated by CON†X , providedthat� `X ˛ H) ˛ 62 †.7

Evidently, †-consistency implies the usual sort.

From this, the definition of †-logical cover and †-level is obvious.In those definitions we merely replace CONX with CON†X . The revisednotions will be denoted by COV†X and `X† respectively, while the accountof forcing which uses † will be represented by

� †X . It should be clear

that there will be, in general, far fewer †-logical covers for a given set� than there are logical covers.

Not all of the old rules will hold if we modify singularity by re-moving members of †; there is bound to be some cost associated withbanishing certain contingent sentences. The earlier form of [Ref] had itthat:

� \� ¤ ¿�� X �

when sets are allowed on the right, and

˛ 2 � H) �� X ˛

when they aren’t.This will no longer hold when either � \ † ¤ ¿ or � \ † ¤ ¿.

What we require is a more general form of [Ref] viz.:

7In the more general calculus where sets are allowed on the right, this definition can be statedmore simply using � °X †.

9.3. †-FORCING 163

[Ref†]� \� ¤ ¿& .� [�/ \† D ¿

�� †X �

or

[Ref†] .˛ 2 � & ˛ 62 †/ H) �� †X ˛

We get back The usual form of [Ref] by replacing † with ¿. In factthis move collapses †-forcing into vanilla forcing. We also get back[Ref] if we don’t tinker with the description of singularity.

Tweaking the notion of singularity also requires other changes. Wecannot inherit quite as many inferences from the underlying logic X aswe did. It remains true that anyX-theorem will also be an X-†-forcingtheorem, since the members of † are contingent by stipulation, but itwill no longer be the case that every X-consequence of a unit premiseset will be preserved. We must drop all those unit sets the members ofwhich belong to †. Of course this means that such sets cannot figure inthe generalization to X-forcing of the rule [Mon] of monotonicity.†-forcing is a true generalization of forcing and not one of the those

approaches which use something akin to counter-axioms. Neither dowe simply add the denials of all members of† to every premise set andproceed from that point by ordinary forcing.

The difference is simply that on the counter-axiom approach, or themodified forcing method, the denials of members of † are, one and all,consequences of every premise set. This is not true for†-forcing since,for example, each ‘bad-guy’ in † is contingent, none will be †-forcedby the empty set. Neither will any of the denials, presumably ‘good-guys’, although not so good that we want to be able to infer them froman arbitrary premise set.

Having said that, we must acknowledge that while †-forcing mightprovide the basis for some operation like renormalization in physics, itcan’t be the whole story. This is because we don’t simply block theunpalatable in the latter case, we also replace it with the palatable. Soif the raw theory tells us that the value of e is infinite, we not onlysnip that out, we replace it with what we take the actual value to be.One can imagine different ways this might happen, including adopting


extra ‘non-logical’ axioms, but more work would need to be done beforesomething convincing could be offered.

9.4 A-Forcing

This is another of the variations on the notion of forcing which reducesthe number of logical covers to be considered in calculating whetheror not certain conclusions follow from a given premise set. In manyapplications, unadorned forcing seems to do a bit too much violence inbreaking up premise sets (and conclusion sets too, perhaps). If we thinkof forcing as a candidate for an inference relation to use in reasoningfrom sets of beliefs, this quickly becomes evident. Representing thebelief set of an individual i by Bi , it might be argued that in most casesBi has a level greater than 1. It might even be argued that Bx must havea level of at least 2 if i is rational.

The conflict between this motivation and ordinary forcing or †-forcing, is that we usually identify within some Bi natural clumps ofbeliefs. Thus, while the whole set of i’s beliefs may have level 2 ormore, we expect and allow i to draw inferences from certain consistentsubsets of Bi . In other words, we refuse to allow logical covers to prej-udice the integrity of any clump of beliefs. In effect, we rule out allthose logical covers which do not respect clumps.

For example, if Simon is unable to start his car one morning, he maycome to believe, as the result of inference, that his distributor is at fault.In general, we would be willing to allow such reasoning as correct eventhough `.BS / D 2 and there exist covering families of the appropriatewidth which would prevent the inference:

BS� X Simon’s distributor is at fault.

We would say, in this situation, that the set of Simon’s beliefs aboutcars and about his car in particular (or at least a respectable subset ofthese) is immune. The question of which subsets ofBi are to be countedas immune is not to be conclusively settled on logical grounds. Thisdetermination must depend upon many factors, not least the identityand circumstances of i .

9.4. A-FORCING 165

For ease of exposition and to avoid unnecessary complications, westipulate that the immune subsets of any premise set have level 1.

Our formalization of the intuitive idea of immunity involves a func-tion A, defined on sets of sentences and having as value for a given set� , a set of subsets of � i.e.

For all sets � W A.�/ � P .�/, where P .�/ is the set of all subsets of�

with A.�/ being interpreted informally as the set of all immunesubsets of � . In distinction to the definition of †-forcing, we needmake no change in the definition of consistency, but rather to thedefinition of logical cover. To this end we introduce the notion of alogical cover relative to both the underlying logic X and the functionA, indicated by COVAX .F; �/:

COVAX .F; �/ ” ŒCOVX .F; �/& .8a 2 A.�/9xi 2 F/a � xi �

Thus we allow only those logical covers which do not break up theimmune subsets of � . It seems most natural in this setting to considersome single conclusion presentation of the inference relation since wedo not usually think of conclusion sets as forming clumps. In eithercase, however, the definition of the relation we shall call “

� AX” will be

obvious.There are, of course, many conditions which might be placed upon

A-functions, each giving rise to a distinctive sort of A-forcing. Someare of great interest in the studies of modality mentioned earlier in thisvolume, while others recommend themselves mainly to the inferentiallyminded. To recover ordinary forcing, we simply set A.�/ empty, i.e.

�� ˛.�

� �/, �

� A ˛.�

� A �/& A.�/ D ¿.

The other end of this spectrum is not obtained by setting A.�/ D �

since this might violate our restriction on the level of immune subsets.The right condition is instead:

COVX .A.�/; �/


(which obviously reduces to A.�/ D � when `.�/ D 1.) This turnsout to be one of the modally interesting relations particularly useful indiscussions of deontic logic.

Under A-forcing we cannot break up immune sets, which will havea nice consequence that we can state if the underlying logic X is pro-ductival.

[A]

�� AX ˛1 & : : : & �

� AX ˛k & 9a 2 A.�/ W f˛1; : : : ; ˛kg � a

�� AX ….˛1; : : : ; ˛k/

where ….˛1; : : : ; ˛k/ is the product (in X) of the ˛’s.

9.5 Wrap-up

What seems most interesting about forcing and its near relatives is thatthey actually have room to make some accommodation with the actualworld. Think about making changes in classical logic or intuitionism tothe same end. It isn’t even clear what changes those might be. Perhapsmany-valued logic, as many of its champions have averred, is one ofthese attempts to meet the real world if not halfway, then at least part ofthe way.

However, forgetting for a moment the deafening howls of protestat such a statement, it is undeniably difficult to see many-valued logic,unless we are talking about a completely Boolean many-valued logic,as any species of classical logic. That is rather the point of classicallogic, isn’t it? It is as big as any logic can get over that language, havinga Post number of 0.8 So we can’t add anything to classical logic, butneither can we take anything away without the residue being resolutelynon-classical.

On the other hand. both †-forcing and A-forcing are recognizableas kinds of forcing. Both are defined in terms of level and preserve thatproperty and, at bottom, that is a near definition of what we might callthe general notion of forcing.

8The Post number of a logicX is the number of non-trivial distinct proper extensions ofX .

Ten

Ambiguity Games and PreservingAmbiguity Measures

BRYSON BROWN

Abstract

Brown (1999) applied preservationist ideas to generate consequence re-lations first exploited by relevance and dialetheic logicians. The centrallesson of the paper was that a systematic application of ambiguity canproduce consistent images of inconsistent premise sets, allowing us tosystematically constrain the consequences that can be inferred from them.Here we present several different ways to apply ambiguity and the preser-vation of ambiguity measures to obtain paraconsistent logics. The firstuses ambiguity to project consistent images of inconsistent premise sets.The second shifts the focus from the syntactic to the semantic, allowingsome atoms to receive ambiguous assignments, i.e. assignments that dif-fer from instance to instance. Finally, the third extends the second with asimple game that captures the consequence relation of first degree entail-ment, dealing symmetrically with triviality on the right as well as the left.The chapter ends with the observation that preservation of a consequencerelation under a range of images offers a new way of applying the generaltheme of preservationism to produce new (and sometimes better behaved)consequence relations.

167

168 CHAPTER 10. PRESERVING AMBIGUITY MEASURES

10.1 Ambiguity

Brown (1999) began a line of work that has applied preservationist ideasto generate consequence relations first exploited by relevance and di-aletheic logicians. The starting point of this approach was the realiza-tion that, by treating certain sets of atomic sentences as ambiguous, wecan produce consistent images of inconsistent premise sets. In this chap-ter we present several different approaches to using ambiguity and thepreservation of ambiguity measures to arrive at some familiar conse-quence relations.

The first of these uses ambiguity to project consistent images of in-consistent premise sets. The second shifts the focus from the syntactic tothe semantic, allowing some atoms to receive ambiguous assignments,i.e. assignments that differ from instance to instance. The third extendsthe second with a simple game that captures the consequence relationof first degree entailment. Along the way, dualization of these tricks isapplied to cope with triviality on the right as well as the left.

We begin with the original, syntactic story of premise sets and theirconsistent images. A set of formulae, � 0, is a consistent image of �based on A (which we write ConIm.� 0; �; A/) iff A is a set of atoms, � 0

is consistent, and � 0 results from the substitution, for each occurrenceof each member ˛ of A in � , of one of a pair of new atoms, f and ˛t .

A first, crude measure of how far � departs from consistency isthe cardinality of the smallest set of atoms A such that for some � 0,ConIm.� 0; �; A/. But this measure seems to assume that treating anatom as ambiguous carries a fixed, equal cost. We can obtain a morereasonable measure by assuming only that treating any atom as am-biguous carries a non-zero cost. This makes the set of least sets each ofwhich is sufficient for projecting a consistent image of a premise set, �a good measure of �’s inconsistency. So we define the ambiguity set of�: Amb.�/ D

fAj9� 0 W ConIm.� 0; �; A/ ^ 8A0 W A0 � A;:9� 0 W ConIm.� 0; �; A0/g

This is the set of smallest sets, A1 : : : An where for each Ai there is

10.2. THE LOGIC OF PARADOX 169

some �’ such that ConIm� 0; �; Ai .Extensions that require adding more elements to some of the mini-

mal projection bases are unacceptable. However, some acceptable ex-tensions of � will rule out some of these ‘minimal’ projection basesby, in effect, insisting on one or (in general) a proper subset of thebases. For example we might extend f˛, ˛ ! ˇ;:ˇg, whose Ambset is ff˛g,fˇgg by adding ˇ to it; the ambiguity set of f˛, ˛ ! ˇ;:ˇ,ˇg is just ffˇgg, since treating ‘ˇ’ as ambiguous is both necessary andsufficient to produce a consistent image of the original set. So we willregard � as an acceptable extension of � so long as some elements ofAmb(�/ are also in Amb(� [�/:

Accept(�,�/, � � � & Amb.� [�/ � Amb.�/.Many consequence relations, including the classical consequence re-

lation, can be defined as preserving the acceptability of all acceptableextensions; in this case, our new consequence relation is:� Àmb ˛, 8�: Accept(�,�/! Accept(f˛g, � [�/It follows that the ambiguity set of the closure of � under Àmb,

Amb../CAmb.�;Àmb// D Amb.�/. By including the necessary liter-als in � we can acceptably extend � to eliminate all but an arbitrarymember of Amb(�/. But any Àmb consequence of � must be an ac-ceptable extension of the result. It follows that every minimally costlyway of resolving �’s inconsistency remains available when we close �under this consequence relation.

10.2 The Logic of Paradox

Graham Priest’s logic of paradox (LP) is intended to be a dialetheiclogic; it is often presented as a simple illustration of such a logic. Priest’soriginal semantics for LP used Kleene’s strong 3-valued matrices, treat-ing the non-classical value, which is a fixed-point for negation, as des-ignated. But in general, the same consequence relation can be givenvery different interpretations; in particular, the preservation of ambigu-ity consequence relation above is identical to the consequence relationof LP.

This was originally shown by an induction on the formulae of a


propositional language. The idea was to establish a one-to-one relationbetween LP valuations and valuations determined by sets of valuationsall of which treat certain atoms ambiguously. To distinguish the newvalues from the old, we replace T and F in the LP valuations with 1 and0. The pairs of valuations agree on the atoms assigned true and false,mapping T into 1 and F into 0. But the atoms assigned both by the LPvaluation are treated ambiguously. Each instance of each such atom isreplaced with one or another of a pair of new atoms, one of which re-ceives the value 1, and the other 0. We quantify across all the resultingprojected images, and assign a formula the value 1 if the formula re-ceives the value 1 in some such image. Every sentence assigned eitherT or B by the LP valuation thus receives the value 1, while the rest re-ceive the value 0. The resulting ambiguity-generated 1/0 valuations arethe Scott-valuations1 that determine the LP consequence relation.

However, we can give a simpler proof by showing how ‘truth-tables’for the two logics match up. There are two ways in which an atom canacquire the value 1 in the Scott valuations produced above. It can beassigned the value 1 at the outset, or it can be one of the atoms treatedambiguously (obviously, if it’s one of those, some image of � will sub-stitute the replacement atom assigned 1 for it, and so it receives thevalue 1 in this second way). Let’s distinguish these two values as 1and 10. The difference between them, of course, is just that formulaeassigned 1 receive that value in every projected image, while formulaeassigned 10 receive 1 in some image and 0 in some image. Now con-sider how these values combine with : and ^. If a sentence is assigned1, then its negation is assigned 0 in every image, and so gets the value 0in the Scott valuation. But if a sentence is assigned 10, then its negationis also assigned 1 in some image, and so gets the value 1 in the Scottvaluation, and the value 10 in our new truth tables. Of course 0 worksjust as the classical 0 in the tables, so we get, as our new truth table for‘:’:

1See Scott (1974)

10.2. THE LOGIC OF PARADOX 171

ˆ :�1 010 10

0 1

This, of course, is isomorphic to the strong 3-valued Kleene nega-tion.

Now we turn to conjunction. Consider �, a formula of form � ^ .Once again, we consider the three values, 1, 1’ and 0. If both � and get the value 1, so does �; � gets the value 1’ if one gets the value1’ while the other gets the value 1, or both get the value 1’. Finally, ifeither or both get the value 0, so does �, of course. So our truth table is

^ 1 10 01 1 10 010 10 10 00 0 0 0

Again, the table for ^ is isomorphic to the table for the strong 3-valued Kleene conjunction. The other connectives can be defined in theusual way. It follows that our ambiguity-based consequence relation isidentical to LP’s.

Another way to present this preservationist reading of the LP con-sequence relation focuses on the semantic side, rather than on imagesof the premise set. Wildcard valuations allow inconsistent sets of sen-tences to be ‘satisfied,’ by treating a set of ’wild-card’ atoms in a waythat allows ambiguity.

Let L be a propositional language with ˛0, ˛1, ˛2,. . . the atoms ofL, and �1,. . .�n,. . . the formulae of L. A wildcard valuation begins byselecting a set of atoms, W , to be the wild cards. We assign values tothe sentences of L first by assigning of 0 or 1 uniformly (that is, in theusual way) to each atom not in W .

Call this assignmentAAt�W . Next, we assign 0 or 1 to each instanceof an atom inW in each formula of L.2 We call the resulting assignment

2This allows each instance of a wild-card atom in each sentence of L to receive either value


WAAt�W . From here, we assign 0 or 1 to each complex formula, usingthe usual truth functional interpretation of the connectives.

The result is a wildcard valuation, W VAt�W . Let VAt�W be the setof all such valuations based on a given AAt�W .3 We quantify acrossVAt�W to obtain a more stable valuation based on all the wildcard val-uations for each wildcard set W: Let VAt � W be the valuation de-termined by this quantification over the members of VAt�W . ThenVAt�W 2 L! f1; 0g, where

VAt�W .S/ D 1 if 9V 2 VAt�W W V.S/ D 1.

VAt�W .S/ D 0 else.

Finally, we define our consequence relation in the usual way:

� `W ˛ , 8VAt�W Œ.8 2 �;VAt�W . / D 1/) VAt�W .˛/ D1�.

10.3 Restoring Symmetry

LP is inelegant for the same reasons that singleton forcing is inelegant:It copes with inconsistency on the left, but leaves the dual triviality onthe right untouched. In LP, classical contradictions on the left don’ttrivialize, but classical tautologies on the right do: Any such tautologyfollows from every premise set. Further, in its multiple-conclusion formLP eliminates the triviality of inconsistent premise sets, but trivializesall conclusion sets whose closure under disjunction includes a tautol-ogy. First degree entailment (FDE) is a closely related logic that treatsinconsistency on the left and its dual on the right symmetrically, elimi-nating the trivialization found in classical logic altogether. (?) presentsan ambiguity-based account capturing the consequence relation of firstdegree entailment.

freely.3So VAt�; is just the singleton set of one classical valuation on L.

10.3. RESTORING SYMMETRY 173

This treatment of FDE requires careful development of the symme-tries of the consequence relation. The first step towards re-establishingthe symmetries of classical logic in our ambiguity semantics for LP isto dualize the property to be preserved. In the first step we used ambi-guity to project consistent images of premise set. Now we will use am-biguity to project consistently deniable images of conclusion sets. LetAmb*(�/ be the set of minimal sets of sentence letters whose ambiguityis sufficient to project a consistently deniable image of �. We requirethat any sentence from which � follows be an acceptable extension ofevery acceptable extension of�, where acceptability is consistent deni-ability:

� is an Amb*(�/-preserving extension of �, Amb*(� [ �/ �Amb*(�/

Which we write, Accept*(� ,�/. So a set � is acceptable as an ex-tension of a commitment to denying � if and only if extending � with� doesn’t make things worse, i.e. does not require any more ambiguityto produce a consistently-deniable image than merely denying � does.

This leads to a right-to-left consequence relation in which individualsentences are placed on the left and sets of sentences on the right:

Àmb� �, 8�: Accept*(� ,�/ H) Accept*(�[f g,�/

In English, a set � follows from a formula if and only if is anacceptable extension of every acceptable extension of �, a set we arecommitted to denying.

These two asymmetrical consequence relations can be combined toform a symmetrical one: Simply treat sets on the left as closed underconjunction and sets on the right as closed under disjunction, and de-mand that both these consequence relations apply:

� `Sym � , 9ı 2 Cl(�, _/: � Àmb ı & 9 2 Cl(� , ^/: Àmb� �.

Or, by linking the set-set relation to an underlying formula-formula


relation, we can put it this way instead:

� `Sym �, 9ı 2 C.�, _/, 9 2 Cl(� , ^/: `Sym ı.

Where `Sym ı, f g `Amb ı & `Amb�fıg.

The upshot of this maneuver is a logic sometimes called K*. Theconsequence relations of FDE and K* agree except when classicallytrivial sets appear on both the left and the right. In those cases the trivi-ality of the set on the other side ensures that the property we’re preserv-ing on each side is trivially preserved. So K* trivializes when classicallytrivial sets appear on both the left and the right. FDE demands a littlemore subtlety.

The trick is to produce consistent images of premise sets and non-trivial images of conclusion sets simultaneously, while requiring that thesets of sentence letters used to project these images be disjoint4. Then� `FDE � iff every such consistent image of � can be consistentlyextended by some member of each each non-trivial image of � basedon a disjoint set of sentence letters, or (now equivalently): � `FDE � iffevery such non-trivial image of the conclusion set can be extended bysome element of each non-contradictory image of the premise set whilepreserving its consistent deniability.

10.4 Another Approach to FDE

First degree entailment is, of course, the base of relevance logics, whichadd to it different accounts of the relevant conditional. The standardsemantics for FDE due to Dunn adds two new truth values called ‘both’and ‘neither,’ to the two familiar ones. In this section we present a newpreservationist reading of the job these two values do. On this reading,they express two different treatments of ambiguity in a way that can becaptured in a simple game.

4In effect, ambiguity allows us to capture the results of using ‘both’ and ‘neither’ as (respec-tively) designated and non-designated fixed points for negation, while insisting that the two sets ofambiguously-treated letters be disjoint ensures that we never treat the same sentence letter in boththese ways.

10.4. ANOTHER APPROACH TO FDE 175

The value ‘both’ is a designated fixed-point for negation; similarly,the value neither is a non-designated fixed-point for negation. The con-sequence relation for FDE preserves Both-or-True from left to right,or (what is the same, in a multiple-conclusion setting) Neither-or-Falsefrom right to left.

Consider the following recursive definition of a Dunn-valuation, de-fined on a classical sentential language. We use lower-case Greek lettersas variables ranging over the formulae of the language. The definitionapplies a simple way of identifying Dunn’s four values with the sub-sets of the set of traditional truth values, fT,Fg: Both= fT,Fg, True=fTg,False=fFg and Neither=;. An assignment to the atoms, Vat 2 At !fBoth,True,False,Neitherg is extended to the rest of the language ac-cording to the following rules:

1. F 2 V(:A) iff T 2 V(A)

2. T 2 V(:A) iff F 2 V(A)

3. F 2 V(A ^ B) iff F 2 V(A) or F 2 V(B)

4. T 2 V(A ^ B) iff T 2 V(A) and T 2 V(B)

5. F 2 V(A _ B) iff F 2 V(A) and F 2 V(B)

6. T 2 V(A _ B) iff T 2 V(A) or T 2 V(B)

Note that T 2 V(A) entails that A receives a designated value (eitherBoth or True), and the consequence relation preserves this from left toright, while preserving (in a symmetrical, set-set consequence presenta-tion) Neither or False from right to left.

We will prove that our new, symmetrical ambiguity logic is FDE bycomparing truth tables, as we did for LP. The truth tables we use emergefrom the game mentioned at the beginning of this chapter. The resultshows that the work of a Dunn valuation can be done by a game that hasnothing to do with peculiar truth-values, because we can arrange theresults of the game in tables isomorphic to Dunn’s 4-valued tables forFDE. The game has two players named ‘Verum’ and ‘Falsum.’ It beginswith a formula and a partial classical valuation, which matches that to


the atoms assigned T or F in the corresponding Dunn valuation. Atomsthat have not received a classical value are divided between the play-ers. Verum receives all the atoms assigned the value Both in the Dunnvaluation, while Falsum receives all those assigned the value Neither.

The game is over a selected formula of the language. For Verum, theobject of the game is to assign classical values to the instances of heratoms in such a way that the whole formula receives the value T. ForFalsum, the object is to assign classical values to all the instances of hisatoms in such a way that the whole formula receives the value F. Eachplayer is free to assign either classical value to each distinct instance ofany atom they’ve been assigned.

This game is either a won game for Verum or a won game for Fal-sum. If it’s a won game for Verum, then the Dunn valuation assignseither true or both to the formula. If it’s a won game for Falsum, theDunn valuation assigns either false or neither to the formula. Moreover,the Dunn valuation assigns the value true to the formula if and onlyif the game is won for Verum even if she and Falsum exchange theirassigned letters instances; similarly, it assigns the value false to the for-mula if and only if the game is won for Falsum even if he and Verumexchange their assigned letters. The preservationist consequence rela-tion based on this game preserves won games for Verum from left toright, or, equivalently, won games for Falsum from right to left. That is,if every game on the left is won for Verum, some game on the right isalso won for Verum.

We represent each game as a triple, < �; Vc;D >, where � is theformula, Vc is the partial classical assignment and D 2 (At – Atc/ !fVerum,Falsumg assigns the atoms not assigned a value by Vc to eitherVerum or Falsum. A game can be won in two ways: In the first case, Vctogether with the division of the remaining atoms fixes which of Verumand Falsum wins regardless of which set of atoms is given to Verum andwhich is given to Falsum; in the second, the win depends also on whichof the two sets of atoms is assigned to which player.

Proof: By induction on the number of connectives in �.

1. Base: For the atoms it’s trivial. The atomic game is won for Verum(Falsum) in the first way if and only if the partial classical valua-

10.4. ANOTHER APPROACH TO FDE 177

tion assigns it the value T (F). It’s won for Verum (Falsum) in thesecond way if the atom is assigned to Verum (Falsum).

2. Induction hypothesis: This hold for formulae with up to n connec-tives.

3. Induction Step: Let � be a formulae with n+1 connectives. Wewill deal with : and ^ here; the rest of the connectives can bedefined in terms of these in the usual way.

4. � has the form: : We have four cases to consider. First, supposethat the game is won for Verum in the first way. Then the gameresulting from exchange of letters between Verum and Falsum isalso won for Verum. This implies that Falsum can force the valueT on by some assignment of T and F to his letters. But thisis exactly what Falsum must do to ensure � receives the value F.So � is won in the first way for Falsum. Second, suppose thatthe game is won for Verum in the second way. Then the gameresulting from exchange of letters between Verum and Falsum iswon for Falsum. That is, the letters assigned to Verum wouldallow Verum to force the value F on . But this is exactly whatVerum must do to assign the value T to �. So � is also a wongame for Verum. The third and fourth cases parallel these two,with Verum and Falsum trading places, so if is won for Falsumin the first way, � is won for Verum in the first way, and if iswon for Falsum in the second way, � is won for Falsum in thesecond way as well.

5. � has the form ^ �: Here we have eight cases to consider, sinceboth and � can be won in either way for either player. However,the cases go through very straightforwardly. First, suppose both and � are won for Verum in the first way. Then Verum can forcethe value T on � whether she plays her own atoms or Falsum’satoms—that is, the A game is won for Verum in the first way aswell. But if either or both of and � is won for Verum in the sec-ond way, then Falsum wins the � game on an exchange of letters,and � is won for Verum in the second way. If either or both of


and � is won for Falsum in the first way, then � is won for Falsumin the first way, while if both and � are won for Falsum in thesecond way, � is won for Falsum in the second way. �

We can summarize the results of this theorem in a table:

� :�won 1 for Verum won 1 for Falsumwon 2 for Verum won 2 for Verumwon 2 for Falsum won 2 for Falsumwon 1 for Falsum won 1 for Verum

^ won 1 for Verum won 2 for Verumwon 1 for Verum won 1 for Verum won 2 for Verumwon 2 for Verum won 2 for Verum won 2 for Verumwon 2 for Falsum won 2 for Falsum won 2 for Falsumwon 1 for Falsum won 1 for Falsum won 1 for Falsum

^ won 2 for Falsum won 1 for Falsumwon 1 for Verum won 2 for Falsum won 1 for Falsumwon 2 for Verum won 2 for Falsum won 1 for Falsumwon 2 for Falsum won 2 for Falsum won 1 for Falsumwon 1 for Falsum won 1 for Falsum won 1 for Falsum

These tables are isomorphic to Dunn’s four-valued tables for FDE:Won1, for Verum and Falsum respectively, corresponds to true and tofalse, while Won2 similarly corresponds to both and to none. So ourlogical game between Verum and Falsum does the work of a Dunn val-uation.

This trick provides yet another way of applying ambiguity to replacestrange truth values—the rules of the game allow Verum and Falsumto treat the atoms assigned to them ambiguously, as they attempt toproduce an assignment that makes the target formula true or false, but

10.5. AMBIGUITY AND QUANTIFICATION 179

each player uses the leeway that ambiguity grants her in a particularway.

10.5 Ambiguity and quantification

Recall the wild card atoms considered above as a treatment of LP. Ourlittle game shows that we can also replace the values both and none withwild cards. For each Dunn valuation, we consider the correspondingwild card valuations. These are wild-card valuations such that:

1. Atoms to which the Dunn valuation assigned T and F are assignedthe values 1 and 0 respectively.

2. The rest of the atoms are treated as wild cards, divided betweenthe atoms assigned both by the Dunn valuation and those assignednone.

3. If some assignment of T, F to instances of the atoms assigned Bothby the Dunn valuation forces the value T on the formula, whatevervalues are assigned to the instances of atoms assigned none by theDunn valuation, we also assign 1 to the formula.

4. Otherwise we assign 0 to the formula.

The resulting 1,0 valuations each correspond to a Dunn valuation, as-signing 1 to every formula assigned a designated value by the Dunnvaluation and 0 to every formula assigned an undesignated valuation.So the consequence relation determined by our game also results fromconsidering all the possible assignments to instances of the wild cardatoms, and looking for whether a ‘win’ for Verum, an ambiguous as-signment to the atoms assigned Both by the Dunn valuation that forcestruth on the formula, exists among them.

10.6 Echoes of supervaluation

Our methods here bear a resemblance to Bas van Fraassen’s supervalu-ations. A supervaluation also begins as a partial classical valuation, ex-tends that valuation by assigning values to the ‘gaps’ in every way, and


then quantifies across all the resulting extended valuations, conserva-tively assigning T (F) to a formula � only if all the extended valuationsassign T (or F) to �.

Our ambiguity logics retain both the starting point, a partial classi-cal valuation, and the use of quantification across a range of extendedvaluations to compensate for the arbitrariness of the extensions of thatpartial valuation. These devices allow us to achieve our results withoutadding strange truth values to the mix, just as they do for van Fraassen.By adding the further element of ambiguity, we have been able to cap-ture paraconsistent consequence relations that resist the trivialization ofinconsistent sets on the left and of sets which are not consistently deni-able on the right.

10.7 Final Remarks on Preservation

There is another way to express what is preserved by these logics, whichopens up a broader understanding of preservation. This approach fo-cuses on preserving the consequence relation itself. We can say that theclassical consequence relation is preserved here, under a range of min-imally ambiguous, consistent (or consistently deniable) images of ourpremises and conclusions:� `FDE � iff every image of the premise and conclusion sets, I.�/,

I�.�/ obtained by treating disjoint sets of sentence letters as ambigu-ous is such that I.�/ ` I�.�/.

This suggests a new preservationist strategy for producing new con-sequence relations from old. We can say that the new consequence re-lation holds when and only when the old relation holds in all of a rangeof cases anchored to the original premise and conclusion sets. Thisstrategy eliminates or reduces trivialization by ensuring that the rangeof cases considered includes some non-trivial ones, even when the caseforming our ‘anchor’ is trivial.

This idea can also be applied to the weakly aggregative forcing rela-tion; the ambiguity-based treatment of forcing allows ambiguity at thelevel of formulas, but not within formulas, and uses a maximum num-ber of distinct ‘colors’ corresponding to the levels of the premise and

10.7. FINAL REMARKS ON PRESERVATION 181

conclusion sets. The familiar forcing consequence relation can then bepresented not as preserving levels of incoherence but as preserving theclassical consequence relation itself, under quantification across a rangeof recolorings of the elements of premise and conclusion sets.

Nomenclature

1; 0; 10 The three values of LP for Scott-valuations, page 170

A A set of atoms, page 168

AAt�W The pre-wildcard assignment, page 171

H The domain of a meaning function, page 105

I.�/ A consistent image of � , page 180

L� The set of sets which preserve level by union, page 131

M � The class of covers, page 109

MX The class of models of an algebraic logic, page 104

Mod.†/ The class of models of the set †, page 107

RK3 The canonical frame relation for K3, page 8

Rxyz The objects x; y and z stand in the relation R, page 4

TH.x/ the set of all minimal transversals for H.x/, page 50

T h.K/ The theory of the class of models K, page 106

VAt�W The set of wildcard valuations based on AAt�W , page 172

Won1 The game-theoretic truth value for true and false, page 178

Won2 The game-theoretic truth value that corresponds to the truth val-ues ‘both’ and ‘none’, page 178

183

184 NOMENCLATURE

X; Y Variables for logics, page 84

¿ The empty set, page 85

Amb.�/ The Ambiguity set of � , page 168

C Variable for covers, page 108

Cm� The minimal cover of � , page 110

ConIm.� 0; �; A/ The set � 0 is the consistent image of � with respectto the set of atoms A, page 168

L The language of a logic, page 84

F A variable for a logical cover of the set kind, page 90

�;† sets of formulas, page 6

� �� Set difference, page 108

�C An MLPE , page 131

M;R Models of a logic, page 107

MK3 The canonical model of K3, page 8

�-MLPE An �-consistent like MLPE , page 134

` The syntactic or proves relation, page 5

˛; ˇ; ; ' Formula variables, page 4

� The (n-1)-ary modal operator, page 5

CONX Consistency predicate for the logic X , page 85

[ Set union, page 85� The syntactic level-forcing relation, page 11

`X The level function of the logic X , page 110

`X The level function relative to a logicX where the covers are sets,page 91

NOMENCLATURE 185

�! The finite subset of relation, page 126

2nC1.˛i /i2ŒnC1�

W1�i<j�nC1 ˛i ^ j for sentences ˛1; ˛2; :::; ˛nC1,

page 56

2nC1.ti /i2ŒnC1�

S1�i<j�nC1 ti \ tj for sets t1; t2; :::; tnC1, page 56

� The material conditional., page 5

H) The meta-language conditional, page 11

1 The special value of level functions for absurd formulas, page 91

� The ‘less than or equal to’ relation on the integers, or a generalrelation of partial order, page 129

L a modal logic, page 51

Nq The set of axioms for hyperframes, page 50

H hypergraph function, page 48

U universe of a hyperframe, page 48

M a hypermodel, page 48

F Varible for a hyperframe, page 48

ˆX The satisfaction relation of an algebraic logic, page 104

� The necessity operator from modal logic, page 5

: The negation operator, page 6

ha; bi The ordered pair of a and b, page 4

f˛1; : : : ; ˛ng The set of ˛1; : : : ; ˛n, page 20

! The classical conditional (alternative), page 169

� Semantic consequence, page 105

B3 The K3 provability relation, page 7

186 NOMENCLATURE

`W The ‘wildcard’ consequence relation, page 172

`Amb The ambiguity set consequence relation, page 169

_ The ‘or’ operator, page 122

^ The ‘and’ operator, page 6

WAAt�W The wild card assignment, page 172

W VAt�W The wildcard valuation, page 172

f W A! B In general, f is a map from A to B , page 105

mngX The meaning function of a logic X , page 105

w.C/ The width of the cover C, page 108

[Cut-1] A version of [Cut], page 108

[Cut] The structural rule of inference of called [Cut], page 85

[Den*] The relation of Symmetric Negation Denial, page 122

[Den] The property that denial commutes with provability, page 87

[K3] The aggregation rule for the modal logic K3, page 6

[K] The rules of modal aggregation, page 5

[M] The rule of monotonicity (alternative), page 108

[Mon] The structural rule of inference: monotonicity, page 85

[N] The rule of necessitation, page 6

[RM] The rule of distribution of the� operator over the� connective,page 5

[R] The structural rule of inference of reflexivity, page 85

MLPE Maximal level preserving extension, page 131

[M�] The rule of level preserving monotonicity for forcing, page 125

NOMENCLATURE 187

[Cut�] The special version of [Cut] for forcing, page 125

Accept(�,�/ � is an acceptable extension of � , page 169

Cn The set of connectives of a logic, page 105

FDE First degree entailment, page 172

IL Intuitionistic Logic, page 122

LP Priest’s Logic of Paradox, page 102

ND Negation Denial, page 121

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Index

X-level forcing; axiomatization, 139urelementen, 84‘A and not-A’ capable of being true, 29

absurd, 88, 91, 110; (ity), 102; formula, 92; formulas, 10, 20, 91

acceptable extension, 169, 173aggregation principle; for ternary modal logic,

6alternative inferential lifestyle, 25ambiguity measures, 168ambiguity set, 168ambiguous; assigments, 168; sets of atomic

formulas, 168anti-symmetry, 129Apostoli, Peter, 47arithmetic, 24atomic formula, 168, 170Avron, Avrin, 102

beliefs; active, 22; attempt to deny incon-sistency of, 25; drawing infer-ences from, 22; real vs. appar-ent, 25

Boolean Algebras, 5Brown, Bryson, 13, 47, 109

calculi, 122cardinality, 110, 168category theory, 132chain; in a partially ordered set, 129Chellas, B., 6Chellas, Brian, 48chromatic compactness, 48classical consistency; as an ideal, 22coloring; k-colourable, 48; compactness for,

48

compactness, 84complete modal aggregation, 5compositional; definition of, 106compositionality, 103conclusion sets, 172–174, 181conjunction, 171; of conclusions, 92connective, 105; derived, 121connectives, 171, 172, 177consistency, 84; definition of, 85; definition-

preservation of, 85, 86; inade-quacy of classical account, 15; predicate, 85; preservation inthe strong sense, 86; preserva-tion of, 85, 88

consistent, 114consistent deniability, 174consistently deniable, 173constant symbol, 109contradictions; implicit vs explicit, 22corresponding conditionals, 82cost; of treating atoms as ambiguous, 168Costa-Leite, Alexandre, 103cover, 108; intrepreted as a model, 109

d’Entremont, Blaine, 13, 125, 126Dalhousie University, 3deducibility; semantics of implication, 18deductive closure, 83denial, 87; contingent, 121; negation-, 121

; non-trivial, 89, 121; symmet-ric negation-, 122

dialetheic, 168dialethism, 29dialethists; and truth-preservation, 12dilution, see monotonicitydisjoint, 174doctrine of original sin, 25

193

194 INDEX

dualization, 168Dunn, M., 174Dunn, Michael, 175

endomorphism, 106, 107entailment, 105ex falso quodlibet, 19, 102explicit contradiction, 23; as a single sen-

tence, 23; believing an, 23

Falsum, 175fixed point; for negation, 175fixed-level forcing relations, 10forcing, 103, 107, 115, 180 ; X-level, 98

; A-forcing, 96 ; closure rela-tive to X , 123; collaspes intothe underlying logic, 123; def-inition of, 96; fixed-level ver-sion of, 12; is not monotonic,124; level-forcing, 100; preserveslevel, 12; relation (not Cohen’sversion), 11

formalism, 17formula, 107; absurd, 102, 121; algebra of. . . (s),

107; complex, 172; contingent,121; denial of a, 121; schema,122

frame relation, 7; binary, 4; diagonal frag-ment, 5; n-ary, 4; relaxing con-dition on, 4; ternary, 4

function, 106

gaggle-truth; vs single formula truth, 83Galois Connected, 7game, 174, 175general moral principle, 26Goethe, 5

Halifax, 7homomorphic, see homomorphismhomomorphism, 106; meaning functions are,

106Hume, 21, 22Hume’s Labyrinth, 21hyperconsistent, 92hyperframe; n-bounded, 53; canonical frame,

51; simple, 50hyperframes, 48, 53; is axiomatized by, 50hypergraph; function, 48hypergraphs, 48; transverse, 48

hypermodel, 48

implicit contradiction ; as mutually incon-sistent sentences, 23

incoherence, see inconsistencyinconsistency, 89, 102; need to distinguish

two kinds, 20inconsistent, 94; definition of, 85; premise

sets, see premise sets; relativeto forcing, 98

inconsistent belief sets; two approaches, 22individual quantifiers; a needed revision of

classical logic, 16inference, 97; from inconsistent sets, 94; struc-

tural rules of, 107inference relation, 84; paraconsistent, 9intuitionists, 16

Jonsson, B., 33Jonsson, B, 5Jonsson, B., 5, 31Jennings and Schotch, 5Jennings, Ray, 3–109; Jennings’ crucial in-

sight, 7

K3 modal logic, 7, 9 ; canonical model, 8; determined by the class of allternary frames, 9; K3-canonicalrelation, 8; maximal K3-consistentset, 8

Kleene, S., 169, 171Kripke, S., 5

left-handed sets, 20Leung, Kam Sing, 13level, 89, 108; -forcing, 111; as a measure

of inconsistency, 110 ; defini-tion of, 91; is compact, 130; isdownward monotonic, 123; mono-tonicity of, 127; of a set, 111; of a set of formulas, 11; of aset of theorems, 111 ; of con-sistency, 92; of inconsistency,91; preservation of, 92; trans-finite, 124

level preservation; fixes truth-preservation,12

Lewis, C.I., 13, 16–18, 20literal(s), see asoatomic formula169Logic of Paradox, 102, 169

195

logic ; 3-valued, 102 ; algebraic, 103, 105; ambiguity, 180; at evens, 87; Classical, 101

any logic is a sublogic of,16

as a recent invention, 17as an entrenched theory, 17

; compact, 124; compositional,106 ; definition of, 84 ; exten-sion view of, 84; first-order, 105,130; inconsistency intolerant, 137; intuitionisitic, 134; Intuition-istic

as the most successful replace-ment theory, 17; intuitionistic,101, 122; modal, see modal logic; non-classical, 24, 25; non-monotonic,25; non-normal, 52; paracon-sistent, see paraconsistent logic,123 ; propositional, 16 ; struc-tural, 108 ; substitutional, 113; trivial/non-trivial, 85

logical cover, 90; cells of a, 96; conditionson equality, 109; narrowest, 91; partition, 90, 114; relative toa logic X , 90 ; width of, 91,108

logical pluralism, 101logical truths; their privileged status?, 19lottery paradox, 95

many-valued logics, 16map, 106material implication, 16maximal level preserving extension, 128meaning function, 105; definition of, 105metalanguage; concept not accepted by Lewis,

19; metalinguistic level, 19metaphysics, 21Minas, J.S., 14minimal models, 48minimal transversal, 50modal aggregation, 6modal logic, 51 ; axiomatized by the rule

[N], 6; n-ary, 7; as disreputable,18 ; frame conditions, 4 ; nor-mal, 3, 4 ; rule of monotonic-ity, 5; rule of normality or ne-cessitation, 5 ; the logic K, 7; weakly aggregative, 55

modal semantics; and paraconsistency, 7Modus Morons; rule of, 24monotone operator, 129monotonicity, 108; level preserving, 125; level-

preserving, 98, 100; unrestricted,97

moral argument, 26moral dilemma, 26, 28more humanistic logic, 21Morgan, C., 4morphism, see also map and function

necessity ; characterized in terms of infer-ence, 7

neighborhood semantics, 48Nicholson, Dorian, 13, 32Nicholson, Dorion, 48non-adjunctive; slighting term for ‘preser-

vationist’, 10non-negative; slighting term for ‘dialethic’,

10not knowing whatX is; Quinean sense, 19

obligations, 89 ; closed under logical con-sequence?, 26; conflict of, 26,28

ordinal, 108, 111ordinary doxastic agents, 21ordinary moral experience, 26

paraconsistent, 102, 104paraconsistent logic, 3, 9–10, 96; dialethic

approach to, 10; first heard of,6

paradoxes of material implication, 18Partee, B., 6partially ordered set, 129partially paraconsistent inference relation,

9Payette, Gillman, 13, 125, 126pigeonhole reasoning, 6Pittsburgh, 3possible worlds, 4Post, Emil; and the completeness of classi-

cal logic, 16premise set; consistent image, 168premise sets, 168 ; some cannot be parti-

tioned into consistent subsets,10; syntatic story of, 168; triv-iality of, 172

196 INDEX

preservationism, 7, 83; origins, 3preservationist, 171, 174, 176, 180Priest, Graham, 10, 102Priest,Graham, 169Priestley,H., 31prime number, 84probability calculus, 25productival, 100, 132projection bases, 169provability relation, see inference relation

Quine, W.V.O., 18, 19, 82, 95

rationality; our ordinary understanding, 22reductio; sometimes wrongly invoked, 20reasoning from inconsistent data, 6reflexivity, 107nC 1-ary relational frames, 53relevance of logic to ethics, 27right-handed sets, 20Russell, B., 18

satisfaction relation, 104Schotch, Peter, 3–122Scott, D., 6, 170Scott-valuations, 170Segerberg, Krister, 48self-contradictory premise, see absurd for-

mulassemantics; consequence with respect to, 105Simon Fraser University, 3singular set of premises, 98Society for Exact Philosophy, 3, 6strict implication, 16, 18, 19; is just prov-

ability, 19; Quine’s criticism im-proved, 19

structural; rules of inference, 85substitution; level preserving, 116substitution property, 107; weak and strong

versions of, 107supervaluations, 179symbolic value1; as value of level func-

tion, 11

Tarski, A., 5, 31, 33tautology, 172 ; as dual-self-contradiction,

20The Lewis position ; applied to belief, 24

; Quinean criticisms, 18; recast

as a complaint about provabil-ity, 19

theorem, 107, 121theory, 106; as set of formulas, 83true contradictions; counterintuitive, 29; hall-

mark of dialethism, 10true sets; must be stipulated, 84truth ; coherence account, 83 ; defined for

single sentences, 84; on a polyadicframe, 53; predicate of sentences,83

truth value assignment; partial, 179truth values; ambiguity replaces strange, 178truth-condition for necessity; in ternary frames,

4truth-maintenance systems, 95truth-preservation, 82; and para-consistency,

11; defined, 81; transmuting ratherthan preserving truth, 83; twokinds of truth used, 82

truth-table, 170truth-value; classical, 176; designated, 169,

179; Dunn’s 4-truth values, 174; non-classical, 169

two strategies for fixing a logic, 15

uniform substitution, 107Urquhart, Alasdair, 13Urquhart, Alisdair, 47

valid argument, 25van Frassen, B., 179variable; free, 109Venn diagrams, 24Verum, 175

wild card atoms, 179wildcard valuations, 171Wolf, R., 6

Contributors

Bryson Brown is Professor of Philosophy at the University of Leth-bridge.

Raymond Jennings is Professor of Philosophy at Simon Fraser Univer-sity.

Kam Sing Leung is a recent graduate of the doctoral program of thePhilosophy Department of Simon Fraser University.

Dorian Nicholson is a recent graduate of the doctoral program of thePhilosophy Department of Simon Fraser University.

Gillman Payette is in the doctoral program of the University of Calgaryand the recipient of the Governor General’s gold medal for his graduatework at Dalhousie University.

Peter Schotch is Professor of Philosophy at Dalhousie University.

Alasdair Urquhart a Professor of Philosophy and of Computing Scienceat the University of Toronto.

Logica Paraconsistente

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