Logic and Aggregation

BRYSON BROWN and PETER SCHOTCH

LOGIC AND AGGREGATION

ABSTRACT. Paraconsistent logic is an area of philosophical logic that has yet to findacceptance from a wider audience. The area remains, in a word, disreputable. In this essay,we try to reassure potential consumers that it is not necessary to become a radical in orderto use paraconsistent logic. According to the radicals, the problem is the absurd classi-cal account of contradiction: Classically inconsistent sets explode only because bourgeoisclassical semantics holds, in the face of overwhelming evidence to the contrary, that bothA

and∼A cannot simultaneously be true! We suggest (more modestly) that there is, at leastsometimes, something else worth preserving, even in an inconsistent, unsatisfiable premiseset. In this paper we present, in a new guise, a very general version of this “preservationist”approach to paraconsistency.

KEY WORDS: adjunction, aggregation, colouring, graph, hypergraph, implication, incon-sistency, paraconsistent logic

INTRODUCTION

Paraconsistent logic1 is an area of philosophical logic that has yet to findacceptance from a wider audience, by which we mean a wider audienceof philosophical logicians, let alone others. The area remains, in a word,disreputable. For some this is a positive attraction, but for most, it hasproven a real deterrent, even to those in need of a paraconsistent accountof reasoning. In this brief essay, we try to reassure potential consumersthat, despite first appearances, it is not necessary to become a radical inorder to use paraconsistent logic.2

One central use of a logical consequence relation – and one close to ourhearts, if not to everyone’s – is as arationally minimal closure operationon sets of sentences representing possible cognitive commitments. Usedthis way, a logical consequence relation should produce a semanticallyor syntactically conservative extension of the original set: What followslogically from a set of sentences will be all and only the sentences bearingthis conservative relation to the set.

The standard forms of this conservatism require (semantically) that thetruth of the premise set’s members guarantee the truth of all the membersof the set’s closure under the consequence relation, or (syntactically) thatonly those sentences that can be proven from the premise set be includedin its closure. These accounts have many virtues. For one thing, they en-

Journal of Philosophical Logic28: 265–287, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

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266 BRYSON BROWN AND PETER SCHOTCH

sure that whatever we add to a set when we close it under the syntacticconsequence relation will be consistent with anything the original set wasconsistent with;0 ∪ A 0 ⊥ → C`(0) ∪ A 0 ⊥, and, mutatis mutandis,the same applies to the semantic relation: Anything that can be satisfiedsimultaneously with the satisfaction of the original set can be satisfiedsimultaneously with the satisfaction of the original set’s closure under�.Thus guaranteeing preservation of truth or provability ensures consistencypreservation in a very strong sense, just as a truly conservative extensionof our commitments should: We beg no questions when we reason deduc-tively, i.e. we rule out nothing that our commitments don’t already rule out.

The same point can also be made in this way: So long as we reasonin this conservative way, we can reason freely from whateversubsetof 0we like; what we get will be consistent with0, assuming0 was consistentin the first place. This is why deductive inference is so safe; we can’t beled to any conclusions at odds with other commitments we have, unlessour commitments are already at odds with themselves. As a result, detach-ment is perfectly safe in deductive reasoning (without any need for “totalevidence” requirements), but dangerous in any less conservative form ofreasoning. And of course, monotonicity is fine wherever detachment issafe. If every consistent extension of0 will be a consistent extension of0’s closure under deduction, then any consequence of0 can safely beadded to any extension of0: It will not render the extension inconsistent(though of course the extension may already be inconsistent).

MOTIVATION

Anybody who teaches introductory logic knows that it is not hard to be-come disenchanted with the classical account of inconsistency. So do thosewho have grappled with the possibility of inconsistent sets of obligations orbeliefs. Our commitments are not always consistent, and sometimes eventhe idealization that they are is untenable. Nevertheless, our commitmentsshould never be regarded as trivial – trivial commitments can play no usefulrole in explanation, whatever our psychology of commitments and theirrelations to action may be.

But rather than work from these examples, let us reconsider a positionin philosophical logic that dates from the beginning of the century andwhich, for a time, was quite influential (though controversial). Starting in1918 or so, C. I. Lewis launched an attack on thePrincipia account of ‘im-plication’. In a nutshell, Lewis held that the ‘⊃’ of Russell and Whiteheadwas too frail a reed to support Russell’s reading of it as ‘implies’. Therewere several examples of ‘bad’ theorems that were intended to bolster

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LOGIC AND AGGREGATION 267

this view: the most prominent were the so-calledparadoxesof materialimplication.

Everybody has their favorite3, but the one that Lewis highlighted mostoften wasex falso quodlibet: a false proposition implies any propositionwhatever. This was his reading of the classical tautology:

∼A ⊃ (A ⊃ B).Lewis thought that any account of ‘implies’ which had that as a law

was wrong. His reasoning was that the intuitive semantics of implicationrequires it to be the converse of deducibility; i.e., that “A impliesB” istrue if and only ifB is deducible fromA. Further,ex falso quodlibetisnot true of deducibility. In response to these intuitions, Lewis introduceda new connective, strict implication, which does support the deducibilitysemantics (at least as far as he was concerned). Having done that, he dis-covered that in his preferred system4 there remained a modal version ofthe offending principle according to which anecessarilyfalse proposition(strictly) implies any proposition. This discovery did not signal a return tothe drawing board,5 however, since Lewis was able to demonstrate (to hisown satisfaction and that of many others besides) that this modal versionis supported by the deducibility semantics.

Lewis’ status as a pioneer of modal logic is not in jeopardy, but theconnection between his motivation and his logic has been viewed with ajaundiced eye, even by his sympathizers.6 This is because logicians of his‘school,’7 the ‘laws of thought’ school, had no very clear grip on the idea ofa rule of inference – which notion they mostly took to be unformalizeable.8

In other words they didn’t grasp the significance of “Frege’s sign of asser-tion,”9 viz. `. One way to attack the Lewis program is to observe thatimplication is a relation, not an operator.10 Having said that, we can nextobserve that the relation which holds betweenA andB exactly whenB isdeducible fromA is just`, so we don’t need any funny operators.

At this point the enemies of modal logic are satisfied that we haveshown the utter futility and wrong-headedness of the enterprise and wantto put this whole sordid chapter in the history of logic behind them. Thoseof us with less interest in the religious issues however, might want to pausefor a moment to see if anything remains of the Lewis motivation once thereal nature of the program has been made clear. This becomes the question:Is the classical , afflicted withex falso quodlibet?

The answer to this question is clearly yes. Consider the classical infer-ence{A,∼A} ` B. Here we may infer an arbitrary conclusion when allwe know is that one of the premises is false. But surely this is wrong, saysLewis. The contrast is with{A∧∼A} ` B. Here we know we have a falsepremise (in fact, a necessarily false premise), but we also know which one

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it is, so this inference is correct because it is, in effect, a way of telling us toget rid of the bugger. Put this way, there really does seem to be somethingto complain about. For we can all think of contexts in which we don’t wantto abandon the distinction between what follows and what doesn’t, just be-cause we are stuck with bad data from which to reason and we cannot pointthe finger of blame at one particular datum (or set of data). If wecoulddothat, then of course we would get rid of it (or them as the case may be).Alternatively, we see an intuitive difference between these two inferences,a difference which the classical account of deduction fails to respect.

This is one way to motivate a paraconsistent account of deduction, andalthough it has been carefully chosen, let us not forget that it was there tochoose. Paraconsistent logics remedy this classical malaise in one of twoways. The radicals (who are, we regret to report, in the majority amongstparaconsistent logicians elsewhere in the world) modify classical logic byrendering sets that are inconsistent and unsatisfiable consistent and sat-isfiable. We suggest (more modestly) that there is, at least sometimes,something else worth preserving, even in an inconsistent, unsatisfiablepremise set. In this paper we present, in a new guise, a very general versionof the second of these approaches to paraconsistency. We shall carry outthis program under the followingconservative principle:

Preserve as much as possible of classical logic. In particular, when the tacit assumption ofthe classical theory of inference holds, viz. that the premise set is consistent, preserve allof classical logic.

If we needed something to put on aT -shirt, perhaps “Fix only what’sbroken” would do. The idea behind this stricture is clear: Many of thosewho stand in need of our product are practitioners of classical logic. Forall (or at least most) of these, the idea of “giving up” classical methodswill not be attractive. So what we must do is provide a theory which workslike the classical theory, when the classical theory works, and keeps onworking when the classical theory gives up. More precisely, our theoryshould enable a clear distinction to be made between the two cases ofinconsistency which we earlier flagged.

ALTERNATIVES

The radicals have a different slant on this issue. That classical logic has togo is a given, since it stands exposed as an instrument of exploitation andincorrect thinking. Many radical logicians come to the paraconsistent en-terprise with some already prepared non-classical framework within whichthey prefer to work. Often enough to be mildly ironic, their prior program

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is an ‘updated’ version of the original Lewis one, viz. an alternative to theclassical account of ‘implication.’

To the radicals, our conservative principle is so much classical clap-trap and our informing distinction a deluded half-measure. Why settle formaking such a feeble distinction and leave the real problem untouched?The problem is the absurd classical account of contradiction. Classicallyinconsistent sets explode only because bourgeois classical semantics holds,in the face of overwhelming evidence to the contrary, that bothA and∼Acannot simultaneously be true! Once we free ourselves from this falseclassical doctrine, we shall be able to reason from inconsistent sets ofpremises, in just the same way that we reason from consistent ones. Inthis brave new world, we will not be able to deduce an arbitrary sentencefrom {A,∼A} because there will be ways of making both premises trueand the conclusion (for at least some conclusions) false. The same, it goeswithout saying, will hold for the single premiseA ∧ ∼A. So Lewis wasjust wrong about this, as he was just wrong about the correct theory ofimplication.

If one’s aim is to attack the classical picture of reasoning root andbranch, then such a program might well be just the ticket. Lacking thatgoal however, the ticket in question involves a destination that most wouldprefer to avoid. Market research clearly shows that, with the exceptionof the radicals, and certain others for whom the science of logic is anoccult branch of knowledge, the notion that there are true contradictionsis repulsive. The conservative principle is intended to guard us against anysimilar excess.

PRESERVING CONSISTENCY GENERALIZED

We are closer to our brethren on the radical left than it might at firstappear. Like them, we are concerned with a sort of loophole in the clas-sical conception of inference. According to this conception, an inferenceis semantically correct if and only if it is impossible for the premises tobe true and the conclusion false. It seems a cheat then (to introductorystudents at least) for this relation to hold because (and only because) it isimpossible that the premises be truesimplicater. The radical way to closethis loophole, is to fix it so that every premise set, no matter how contradic-tory, can be made true11. Our more modest proposal will be to amend thecriterion so that it reads “except for the case in which the premises cannotbe simultaneously true, in which case. . .”.

To see how to fill in the ellipsis let us return to the Lewis distinction.What is it about{A,∼A} that makes us think it should be distinct from

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{A ∧ ∼A}? The (semantic) answer that occurs naturally, if not most nat-urally, is that while there is no way of making either set true, there is away to partition the first so that for each cell of the partition, there is a wayof making that cell true. For the second set however, no similar strategycan work. It is not hard to parlay this intuition into a way of measuring theamount or degree of inconsistency which a set of sentences suffers. We canavail ourselves of the fact that classical logic is model complete and dropsatisfiability talk altogether, for now, in favour of talk of consistency.

The central idea here is a generalization of consistency called alevelor degree of incoherence. The original definition of a set’s level was theminimum number of members in a covering of the set such that everymember of the covering set is consistent. If there is no covering set suchthat every member is consistent, the level∞ was assigned. More recently,we define the predicate CON(0, ξ) to hold if and only if there is a familyof setsA = ∅, a1, . . . , aξ , ξ ≤ ω such that each member of the familyis (classically) consistent and for each memberA of 0 there is someakin A such thatak ` A.12 Obviously, such a family of sets can be said toconsistently cover0 in almost the same sense that a partition of0 does.We can now define our measure as the functioni:

i(0) ={

Minξ [CON(0, ξ)] if this limit exists,∞ otherwise.

The function i is named to suggest (degree of)incoherence. It is aspecial case of something called alevel function, about which we shallhave a bit more to say in the sequel. For now we should note that wecan distinguish two ‘good’ degrees, 0 and 1.i(0) = 1 means that0 isclassically consistent, whilei(0) = 0 means that0 consists entirely ofclassical tautologies. It seems natural to say that sets of the latter kindare more consistent (less incoherent) than those of the former because theunion of a set of tautologies with any consistent set must be consistent.This is not true of an arbitrary consistent set.

Being able to measure degrees of incoherence is the key to ‘fixing’ theclassical account of inference. We are already used to thinking about infer-ence inpreservationistterms; in terms, that is, of certain properties of setswhich must be preserved or transmitted by the inference relation. In fact,we started down our current path by noticing that the ‘truth-preservation’definition of validity had an annoying loophole. In order to achieve a greaterprecision in this, we shall talk about preservation in terms of the deductiveclosure operation:C`(0) = {A|0 ` A}. It is clear that preserves degrees0 and 1, in the sense that if0 has either of these two degrees of incoher-ence, then so doesC`(0). But, we can now see, in order to maintain the

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Lewis distinction, and jettisonex falso quodlibet, we want our inferencerelation to preserve more than just this.

In fact, another principle occurs to us at this point:don’t make thingsworse. With this in mind, let us call an inference relationI , Hippocratic(primum non nocere), provided:i(0) = i(CI (0)). Clearly,` fails to beHippocratic. In fact it converts every set of degree 2 or higher (e.g.{A,∼A}has degree 2), into a set of degree∞! We can set this to rights, in what isobviously a ‘classical’ way by means of a construction which is suggestedby the definition ofi.

We shall call the revised notion of (classical) inferenceforcing, andrefer to it by the symbol[`. The construction goes like this: Supposei(0) = k; and letcon(0) be the class of all families of setsA, of thesort required by the definition ofi, of ‘width’ k. We can think of eachAas a sort of decomposition of0 into subsets each of which is ‘appropriate’for using` (although such an oversimplification is not strictly speakingcorrect).

0 [À⇔ ∀A ∈ con(0) : ∃aj ∈ A : aj ` A.Here is an Hippocratic inference relation. One can see, in more general

terms, that this construction must preserve degree of incoherence wheneverthe underlying inference relation preserves level 0 and 1. We also claimthat this way of doing things accords with our conservative principle, forconsider the following consequences of the definition of[`:

∅ ` A⇒ ∅ [À,{A} ` B ⇒ {A} [`B.

In the interests of a more general view, we shall call the empty setand all unit setssingular. Then we can put this ‘bridge’ principle: Theconstruction preserves all inferences from singular sets. Notice that thisprinciple requires that we not only agree with` on the laws of thought, butalso on all inferences from singletons. One might be inclined to argue herethat [` agrees exactly with on the meaning of the logical words. Thiswould be a pretty non-Quinean thing to say, however, so perhaps we’d bebetter off to steer clear of controversy.

Before we go on, we will spend a little time clarifying the relationbetween this proposal and our earlier characterization of classical logicin terms of preservation.0 [`α will hold if and only if, for any degree-preserving extension of0, A, i(0 ∪ A,α) = i(0 ∪ A). This is easily seengiven the definition of[` above. For anyA covering0, since` preservesconsistency, any[`-consequence of0,α, is covered by at least one elementof A, ai , such thatai ` α. As a result, ifA could be extended (by consis-tently adding elements to some of its members) to anA′ covering0 ∪ A,

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the same goes trivially for someA covering0, α. Just as preserves theconsistency of all consistent extensions of0, [` preserves the degree of alldegree-preserving extensions of0:

i(0) = i(0 ∪ A)→ i(0 ∪ A) = i(C[`(0) ∪A).Level-preservation –i(0) = i(C[`(0)) – is just a special case of this.

But just as in the classical case, level-preservation alone is not enough. Pre-serving level, or consistency, is compatible with eliminating some level-or consistency-preserving extensions – something a properly conservativeclosure operator will not do.

There is one special case here that deserves mention. In classical logicwe do not normally distinguish between the consistency of tautologoussets and that of contingent sets. Thus classical logics ignore the degreeincrease that occurs when we extend a set of level 0 to a set of level 1, andthe fact that our principle does not suggests[` may differ from classicallogic on the consequences of tautologous sets. Standard logic requires thatwe preserve all consistency-preserving extensions, i.e. all extensions thateither preserve level 0, or that increase it to level 1, while forcing requiresonly that we preserve all extensions that preserve level 0. In fact, however(as the definition of[` above already makes clear) there is no differencebetween[` and` over the consequences of tautologous sets. We definelevel 0 thus:i(0) = 0 iff ∀A i(0 ∪A) = i(A). Sincei(C[`(0)) = i(0),

i(0) = 0→ i(A) = i(0 ∪ A) = i(C[`(0) ∪ A).If 0 has degree 0, closing0 under [` will preserve not only degree-preserving increases, but also the degree-increaseof any degree-increasingextension; in particular, closing0 under[`will preserve the consistency ofall consistent extensions of0, whether the result has degree 0 or degree 1.

FORCING

Natural Deduction Rules

Those whose specialty is not formal logic will be most concerned with theoperator rules. Since certain of the classical rules are central to its non-Hippocratic nature, these must undergo a sea change. Perhaps most promi-nent is the rule ofconjunction introduction, AKA adjunction. In its classi-cal guise this rule has the form:0 ` A and0 ` B ⇒ 0 ` A∧B. Clearly itwon’t do, since by its means we can move from a set of degree less than∞to a set of degree∞. Nor are we by any means the first to suggest that al-lowing the deduction of arbitrary conjunctions of conclusions correctly ar-rived at might well give us something untoward.13 Our contribution in this

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area isnot the suggestion that we give up adjunction.14 It is the suggestionthat the classical adjunction rule is not stated with proper generality.

We replace adjunction with a degree-determined aggregation principle,2/n+1-intro (originally called “cockroach intro”:

∧∨-intro, because of the

way it combined∧ and∨).15 When the degree of0 is n, this principle sayswe’re entitled to infer the disjunction of pairwise conjunctions amongstanyn+ 1 members of the set.

2/n+ 1: 0 [À1& · · ·&0 [Ài(0)+1 ⇒ 0 [` ∨ (Ai ∧ Aj)(1≤ i 6= j ≤ i(0)+ 1).

The soundness of this is obvious. We want to ensure that closing underour new consequence relation won’t add anything to the set that wouldincrease the degree of any degree-preserving extension of the set: If addingsome set of sentences to our original commitment set0 would not haveraised0’s degree, then neither will adding it to the closure of our commit-ment set under the new inference relation. But given any covering familyof sets for0, A = ∅, a1, . . . , an, at least 2 out of everyn + 1 mem-bers of0 must be consequences of the same member of the coveringset. So every consistent covering family for0 is already also a consis-tent covering for0 ∪ 2/n + 1(γ1 . . . γn+1), whereγ1 . . . γn+1 ∈ 0, since2/n+1(γ1 . . . γn+1)must be a consequence of whichever element ofA hastwo of ourn + 1 members of0 as consequences. It follows that 2/n + 1preserves all degree-preserving extensions – ifA covers0∪A, A will alsocover0, 2/n + 1(γ1 . . . γn+1) ∪ A. Of course, when0 has degree 1 (i.e.the degree of ‘ordinary’ consistent sets), 2/n + 1-intro collapses into theclassical adjunction rule.

It is somewhat less obvious that this is complete, i.e. that all aggregationforced by such a division of a premise set is captured by the 2/n+ 1 rule.But it is, and the fact that it is has some real mathematical interest, as weshall see.

Similarly, the ordinary rule ofconditional elimination(or, as it is im-properly called,modus ponens) is no respecter of degrees. Consider, forexample, the premise set{A,A→ (B ∧∼B)}, which has degree 2. Goodold MP used here will generate a closure which has degree∞. But, again,when the premise set has degree no higher than 1, we can cross as many ofthese conditional bridges as we come to.

The case of the negation rules – classically both are forms ofreductioad absurdum– is a bit stranger. Here we have the rules as is, without, thatis, any restriction on degrees, but they are trivial. It certainly is true that:0 ∪ {A} [`⊥ ⇒ 0 [`∼A, but in the absence of adjunction in its simpleminded classical form, we shall not be able to discharge the hypothesisunless0 contains⊥, or A is ⊥. In fact, another way of saying that an

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inference relation respects the Lewis distinction is to say that if0 hasdegree∞, then it must contain a singular subset with that degree (whereonly sets of degree∞ explode). This could lead officials of the logicalconsumers protection bureau to complain that our logic does not meetadvertised specs, sincereductio is very closely aligned with the classicalapproach. In answer we say that our approach is consistent with the truemeaning ofreductio, which is not faithfully represented in the above rulebut rather in:

a. i(0) > 0⇒ (i(0 ∪ {A}) > i(0)⇒ 0 [`∼A).b. i({A}) = ∞⇒ 0 [`∼A.

And, of course, these are both correct rules for[`.In summary, what we must give up on this conservative approach are

only certain kinds of inferences from classically inconsistent sets. Wheninconsistency has not reared its ugly head we give up nothing! This is acrucially important point. Since classical logic simply collapses in the faceof inconsistent premise sets, it would be better to say that there are actualgains rather than that we have no losses.

These considerations about rules will leave many consumers with ques-tions about what it would be like to use such a logic in a natural deductioncontext. Giving up even a few of the classical rules may seem a heavy bur-den to those with a large investment in classical logic. However, things arereally not all that bad. Before we turn to more fundamental concerns, let’sconsider what a familiar natural deduction system might look like whenforced to respect degrees. Fitch systems give a particularly clear presenta-tion of the relation between classical logic and the degree-preserving logicbased on it. We adopt 2/n+ 1-intro for aggregation, replacing∧-intro:

2/n+ 1 intro:∣∣∣∣∣∣∣∣∣A0...

An2/n+ 1(A0 . . . An)

The second and third rules,Classical-reiterationandClassical-preser-vationmakes the role of the classical base logic clear:∣∣∣∣∣∣∣∣∣∣

A0

B

L

∣∣∣∣∣∣∣A0...

B

C-Reit.

C-pres.

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The only differences between the degree-preserving logic and classicallogic are the weakening of adjunction to 2/n + 1-introduction, and therestriction onL-subderivations that allows them to import only a singleformula. Within theL-derivations, the entire apparatus of classical logicis available, and outside them, the only issue is aggregation, and the onlyrule is 2/n+ 1 intro.

Structural Rules

Those whose study is the theory of deductive inference are liable to befar more concerned over the fate of the structural rules under this newscheme than they are over that of the operator rules. Here we have somereassurance to offer, and some explaining to do. First the reassurance: wedo indeed have CUT, which in a pseudo-natural deduction context lookslike

[CUT] 0,A [`B and0 [À⇒ 0 [`Band we also have the (true16) rule of reiteration:

[R] A ∈ 0 implies0 [À.What we don’t have is the rule of monotonicity or dilution; at least we

don’t have it in its classical form:0 ` A ⇒ 0 ∪1 ` A. The reason thisfails is that the result of dilution might be a set of higher degree whichwould require more ‘splitting up’ of the premise set, and so an inferencemight be lost. Of the versions that wedohave (the correct versions, we aretempted to say) one mentions degrees:

[MON] 0 [À andi(0 ∪1) = i(0)⇒ 0 ∪1 [Àand the other mentions what we earlier called singular sets (in the currentsetting, unit sets and the empty set):

[MON∗] 0 [À and SING(0)⇒ 0 ∪1 [À.So this logic of ours might be labeled non-monotonic.17 This makes

for an interesting comparison with what is usually called non-monotoniclogic in the AI literature. In that setting, one often specifies bits of ‘defaultreasoning’ which take the form of allowing certain conclusions to be drawnso long as the default assumptions hold. If new premises arrive (somehow)which contradict (or ‘override’ as some say) the assumptions, then theconclusions drawn ‘by default’ must be withdrawn. So in the AI approachwe are told which inferences are at risk, but in our limited form of the

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principle of monotonicity, we are told rather which inferences arenot atrisk.

In fact, detachment is safe for forcing in the same way that it is safein standard deductive logic: Any sentence that is a degree-preserving con-sequence of a subset of0 (where the degree in question is0’s degree,not that of the subset) can be added to0 without risk of increasing0’sdegree of incoherence. MON follows immediately from this. Furthermore,we can have the standard form of monotonicity offered by classical logic,0 ` A⇒ 0∪1` A, backif we want it. All we need to do is begin with thedegree of0, and apply the principle of aggregation that comes with that de-gree to all extensions of0 regardless of whether they are degree-preservingor not. Then any non-degree-preserving extension trivializes0 just as anynon-consistency-preserving extension of0 is trivialized by classical logic.So classical monotonicity is easy to obtain. What is more, thinking aboutthese strictly monotonic logic turned out to make completeness easier toprove. But monotonicity comes at the usual price: An extension that makesthings worse, by increasing0’s degree, will trivialize0’s consequence set.

Of course, this proposed improvement on classical logic has not re-ceived universal acclamation. Some have pointed out in particular the inel-egance of the new aggregation rule, which introduces a complex involvingtwo connectives rather than the usual single instance of one connective.This objection, however, is easily answered, after a short detour into math-ematics.

GRAPH THEORY

The first interesting connection between these logics and mathematics waspointed out by D. Wagner, then a student of R. Jennings. In thinking aboutthe semantics of forcing, Schotch and Jennings invented the notion of ann-trace: An n-trace over a set0 is a set of sets such that at least one ofits members is a subset of some cell in anyn-partition of0. When0 is aset of sentences, then-traces can be (right-)formulatedby conjoining themembers of each element of then-trace, and then taking the disjunction ofall these conjunctions. If everyn-partition of0 includes a superset of someelement of our trace, then the formulated trace can be added to0 (and toany level-preserving extension of0) without raising0’s level.

A non-n-colourable hypergraph is (with some here unnecessary detailsomitted) a set of sets (callededges) H = {h0, . . . , hn} such that on anyn-colouring18 of

⋃H there is at least one monochrome edge, i.e. an edge

all of whose members are assigned the same colour. Linking this to thevocabulary introduced above, a set of sets,T , is ann-trace over a set0,

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iff every n-colouring of0 makes some element ofT monochrome. Thiswill be the case if and only ifT contains19 a non-n-colourable hypergraph,T ′ such that

⋃T ′ ⊆ 0 – this is the crucial link between the semantics of

forcing and hypergraph colourings.The operation on hypergraphs that corresponds to 2/n + 1 is the con-

struction of new hypergraphs fromn + 1 input hypergraphs by takingthe union of pairwise unions of edges of each pair of then + 1 inputhypergraphs:⊗

n

(G0 . . . Gn)

= {gk ∪ gm|gk ∈ Gi, gm ∈ Gj, 0≤ i 6= j ≤ n}.

The completeness proof that was originally arrived at for the logicturns out to be (in essence) a proof that closing the set of singleton loopsunder this operation together with operations that trivially preserve non-n-colourability (adding further edges, or removing vertices from edges)is sufficient to produce all the non-n-colourable hypergraphs. Since thenon-n-colourable graphs are just the subset of the non-n-colourable hyper-graphs whose edges are pairs, this is (a fortiori) a method for constructingthe non-n-colourable graphs.20

This leads to an important open question in graph theory: Is there ahumanly surveyable proof of the 4-colour theorem? The 4-colour theoremsays that no non-4-colourable graph is planar; we know, further, that anygraph edge-contractible to the complete graph on 5 vertices,K5, is non-planar. So a proof that any non-4-colourable graph is edge contractible toK5 is a proof of the 4-colour theorem. But the construction of the non-4-colourable graphs by means of our

⊗-operator begins withK5, and then

usesK5 as a sort of template to construct further non-4-colourable hy-pergraphs. So a proof that this construction preserves edge-contractibilityto K5 would be a proof of the 4-colour theorem. In fact, a proof thatconstruction based onKn preserves edge-contractibility toKn would bea proof of a well-known generalization of the 4-colour hypothesis knownas Hadwigger’s conjecture. So far (unsurprisingly) a proof has eluded us,but we haven’t yet lost all hope.

Forcing generalized

An alternative approach to expressing the degree-preserving logic based onL can avoid any worries about odd connective rules. It uses a new concept,closely related to the concept of singular sets: We declare that for any

LOGID421.tex; 28/05/1999; 10:12; p.13


singular subset of0, γ , {γ } is anaggregateset, and then declare that ifα0 . . . αn are aggregate sets, so is

⊗(α0 . . . αn), where⊗

(α0 . . . αn) = {a ∪ b|a ∈ αi, b ∈ αj , 0≤ i 6= j ≤ n}.That is,

⊗(α0 . . . αn) is the set of all pairwise unions amongst elements

of α0 . . . αn.The rule that we require to capture the resulting version of a degree-

respecting logic is a modification of the second rule above – we apply theprocedure above for generating aggregate sets, together with the followingrule: ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

Where{{A10, . . . , A

1i }, {A2

0, . . . , A2j },

{Am0 , . . . , Amk }} is an aggregate set.A1

0...A1i

A20

...A2j

...Am0...Amk

C

∣∣∣∣∣∣∣∣∣∣

A10

...A1i

...B

C

∣∣∣∣∣∣∣∣∣∣

A20

...A2j

...B

...

C

∣∣∣∣∣∣∣∣∣∣

Am0...Amk...B

B C-Pres2.

LOGID421.tex; 28/05/1999; 10:12; p.14


B is derivable from0 in the [`-sense iffB is derivable from0 in the`-sense from every member of some aggregate set:

0 [`B iff ∃A ∈ Agg | ∀a ∈ A, a ` B.As a result, in this version of things there is nothing to add to clas-

sical logic beyond the definition of aggregate set, which begins with thesets of individual singular sets as defined above as a base, and applies anaggregative principle to them. In classical logic any set of subsets of thepremise set is an aggregate set(S ⊆ P (0)→ AGG(S)); this is the resultof beginning with singleton sets of individual formulae as the singular setsand applying the trivial classical aggregation principle. But if we insteaddefine the aggregate sets as the closure of the sets of singleton subsets of0 under

⊗, we get a degree-respecting version of the original logic which

builds all the necessary aggregation into the definition of the aggregatesets.

ALTERNATIVE FORMULATIONS

Suppose we decide to use a different “template” for aggregation. RatherthanKn+1 we could use some other hypergraph of the same chromaticindex(n+ 1); the result, we have recently realized, would again be soundand complete for degreen.

The way in which aggregation proceeds using another template is this:Given a template hypergraph of chromatic indexn + 1 and orderm, wewould take as input an orderedm-tuple of aggregate sets, and arrangethem as the vertices of the template; call this the template hypergraph.The resulting aggregate set would be the set of allp-wise unions amongstthe edges of thep vertex sets on each edge of the template hypergraph.

A proof of soundness: If every input hypergraph is non-n-colourable,then anyn-colouring of all their vertices makes at least one edge of everyinput hypergraph monochrome. But the template is also non-n-colourable,which means thatn-colouring its vertices (now occupied by the input hy-pergraphs) will also produce a monochrome “edge”. Assign to each inputhypergraph the colour of some monochromatic edge, so that for everycolouring of the vertices of the input hypergraphs, we get a correspondingcolouring of the template hypergraph.

The hypergraph that results from applying the template to these inputhypergraphs has as its edges thep-wise unions of the edges of thep inputhypergraphs appearing on each edge of the template. Since some edgeof the template is monochrome, the colour of the template’s hypergraphvertices is the colour of their monochromatic edges, and the union of those

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edges is an edge in the resulting hypergraph, the resulting hypergraph musthave a monochrome edge too. 2

The result, as we prove in the appendix, is an alternative formulation ofthe logic we get from theKn aggregation rule, wheren is the chromaticindex of the new template. So the range of aggregation principles ini-tially explored by Schotch and Jennings covers the entire range of weaklyaggregative logics.

CONCLUSIONS

The sketch just given is enough to give the flavor of the theory beingproposed, though it does not touch on many matters of importance tologicians. Rather than repair these deficiencies, we shall consider the ad-judication of the conservative approach on a somewhat larger scale. Fora theory to succeed, it must have more than niceness (which we have inspades); it must offer generality and fruitfulness as well. Whatever thebenefits, so far uncanvassed, under the latter heading, the prospects underthe former might be thought a bit thin. After all, the main thrust of ourapproach has been to effect a patch on the classical account of deduction.This is rather misleading however.

Here we will drop some of our camouflage. So far we have pretendedthat our only objective is to reform classical logic. And of course, by far thelargest number of potential customers are presently consumers of classicallogic – it would be folly not to target that market. But there is absolutelynothing about the formulation just given of aggregational logic that ties itto any particular base logic. If the original logicL is classical propositionallogic and the singular subsets are the singleton subsets of0, the result isSchotch and Jennings’ original forcing relation. However, if we are in-terested in a more general approach – in particular, if we want to avoidmaking any assumptions about the language of the base logic, and insteadfocus purely on the general issue of aggregation – the second proposal for aFitch-style natural deduction system makes clear exactly what is involved.

We propose a wide range ofweakly aggregative logicsbased on thisgeneralization of consistency and satisfiability. These logics improve thestability of any consistency/satisfiability-based logic, preventing the trivi-alization of inconsistent or unsatisfiable premise sets which do not containa singular inconsistent/unsatisfiable subset. Furthermore, the presentationof the weakly aggregative logics just given allows generalizations alonggraph theoretical lines to be presented very simply as proposals regardingwhat sets are aggregate.

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So, while it is true that our conservatism restricted what changes wecould make in , the means by which our changes were effected are bothgeneral and flexible. The functioni, which assigns degrees of incoherence,is got by a method which generalizes quite easily to produce what we haveelsewhere calledlevel functions. What we require in the general case istwo predicates of sets8 and9 (which we assume to be non-empty) suchthat8 – the general counterpart of consistency – and9 – the ‘singularity’predicate – are assumed to be downward monotonic (preserved by sub-sets), while neither is assumed to be preserved by supersets. Once we gothrough this generalization, we can show that the corresponding general-ization of the construction which fixes a given inference relation will beHippocratic, provided that the starting relation preserves levels 0 and 1.21

We can also see that every fixed relation will have the structural rule[R],provided that all unit sets are singular, and each will be compatible with[CUT], in the sense that the result of adding[CUT] to any Hippocraticrelation will also be Hippocratic. Finally it can be shown that the result ofadding the classical version of dilution will, under quite general conditions,produce a relation which is not Hippocratic.22

This might seem like a catalog of arcana rather than evidence of gen-erality, but it is through playing with the predicates8 and9 that onecan obtain those variations on the forcing relation which have had themost appeal to those interested in applications. One gets a more pointedform of the consistency predicate, for example, by letting it be defined by:8(0)⇔ 0 0 B for everyB ∈ 6, rather than0 0 ⊥. Here6 is some non-empty set (the set of appalling and repulsive sentences which one must barfrom conclusionhood no matter what). Once adjusted by the correspondingconstruction, we get from this a species of inference appropriate to thecase when we like some but not all of the consequences of a theory, butcannot find (at least right away) some ‘nice’ modification of our theory orinference which will block the nasties.

Similarly we might have an application in which the construction usingi looses too many ‘good’ inferences because premise sets get ‘broken up’too finely. In much of our reasoning about physical theory, for example, weshall expect certain natural ‘clumps’ of principles always to be availabletogether no matter how high the degree of our premise set turns out tobe. All we require in order to guarantee this, is to require the clumps inquestion to be singular.23

Finally we consider the question of fruitfulness. It is too soon to getthe judgement of history on this score, but the cupboard is far from bare.One of the first places we should expect a theory of inference to showsome application muscle is in the field of logic. This has certainly been the

LOGID421.tex; 28/05/1999; 10:12; p.17


case with our account, principally in the area of modal logic. Standard, oras some call it,normal modal logic is axiomatized by the addition of thefollowing to the classical rules:

[R ρ] 0 ` A⇒ ρ[0] ` ρA,whereρ[0] refers to the set{ρB | B ∈ 0}.

If we replace the on the left side of this rule by[` we get a modallogic which resembles the normal variety except for the principle of com-plete aggregation[K] ρA∧ρB → ρ(A∧B). [K] is replaced (as the readerno doubt will anticipate) with[Kn] ρA1 ∧ · · · ∧ ρAn+1→ ρ(

∨(Ai ∧Aj)

(1 ≤ i 6= j ≤ i(0) + 1)), wheren is the degree of incoherence ofthe set of formulaeA such that�w ρA. The result is a generalizationof standard modal logic including a large family of modal logics withincreasingly weak principles of modal aggregation, whose base logic lacksmodal aggregation altogether. These modal logics make an interesting andworthy study in their own right, especially in applications where we need amodality that tolerates inconsistency in the same way that[` does. Studiesby the authors of the logic of cognitive commitments and of systems ofrules have made use of them.24 They have also turned up in the study ofthe existing systemsS1and the systemE of H. B. Smith.

One of the most interesting applications of these logics is in the studyof various forms of cognitive commitment. To represent the commitmentsinvolved in accepting a set of sentences we must close the set under someform of logical implication. But classical logic is a poor candidate for thisjob. Few, if any, of us are up to its standards, since any inconsistency inwhat we have accepted will trivialize our commitments. Worse, even if werecognize the inconsistency and try to put it right, we are not always cleverenough to see immediately how to do this. We may find the claims that arejointly inconsistent individually indispensable in various applications. So,while classical logic tells us our commitments are trivial, we may find itpreferable to try to muddle through, at least temporarily, without fixing theinconsistency. Classical accounts of commitment cannot help us here.

APPENDIX

We present here a procedure for constructing the non-n-colourable hypergraphs based onnon-normal modal logic. A hypergraphG is usually treated as an ordered triple,〈V,E,F 〉.V is the set ofG’s vertices, E the set ofG’s edges, andF ∈ E → 2I is G’s incidencefunction. A propern-colouring(or ann-colouring, for short) ofG is a functionC ∈ V →{1, . . . , n} such that

(∀e, i)F(e) * C−1(i),

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i.e., ann-colouring of the vertices ofG that leaves no edgemonochrome. We sayGis (weakly) n-colourable iff there is a propern-colouring of G, and defineχ(G), G’schromatic level, as the leastn such thatG is n-colourable, and∞ if there is no suchn.So the non-(k − 1)-colourable hypergraphs are all the hypergraphs that have no proper(k − 1)-colouring, i.e. the hypergraphsG such thatχ(G) ≥ k.

Whether or not a hypergraph can be properlyn-coloured depends only on theF -imageof E. So we will treat hypergraphs in a somewhat simplified way, as subset of 2V . Also, bythe compactness of chromatic level, itself a straightforward consequence of the compact-ness of first-order predicate logic, we will assume that all the hypergraphs we’re dealingwith are finite.

I. Initial Stages

Let Hk be the set of (isomorphism classes of) hypergraphs of chromatic level at leastk.Then forX a hypergraph of orderp and chromatic indexk, and hypergraphsG1 . . . Gp,let X(G1 . . . Gp) be the result of combiningG1 . . . Gp according to the template thatXprovides, that is, treatG1 . . . Gp as the vertices ofX, and subsets of{G1 . . . Gp} asX’sedges. Then make a hypergraph by taking the union of allm-wise unions of all membersof eachGi belonging to a given edge of thisX. Thus

X(G1 . . . Gp) = {g1 ∪ g2 ∪ . . . gm | ∃i1, i2, . . . im1≤ i1 < i2 < · · · < im ≤ k,g1 ∈ Gi1, g2 ∈ Gi2, . . .gm ∈ Gim ∧ {Gi1, Gi2, . . . Gim} ∈ X(G1 . . . Gp)}.

(i) For hypergraphsG,G′, we will write

G B G′ iff ∀g ∈ G, ∃g′ ∈ G′ | g′ ⊆ g.Let Ak be the smallest set such that:

1. All singleton loops are inAk.2. G ∈ Ak andG B G′ ⇒ G′ ∈ Ak.3. G1 . . . Gp ∈ Ak ⇒ X(G1 . . . Gp) ∈ Ak .

II. Main Theorem:Hk = Ak

LEMMA 0. We can useX to obtainKk, the complete graph onk vertices.

This is a straightforward consequence ofX having chromatic indexk. By definition aproperk-colouring leaves no edge ofX monochrome, so we can contractX to k verticeswithout reducing any edge to a singleton loop by identifying all vertices assigned the samecolour by any properk-colouring. The result will be a hypergraphXk of chromatic index atleastn (homomorphic contraction can only increase a hypergraph’s chromatic index, sincecolouring a contracted hypergraph is equivalent to considering only the colourings of theoriginal which assign the same colours to the identified vertices).

But the only hypergraphsXk with k vertices, no singleton edges and chromatic index ofat leastk includeKk , with, perhaps, some extra edges containing more than two vertices.25

And any suchXk is such thatXk B Kk. So we can obtainKk by applyingX to k verticesarranged according to the required pattern (i.e. repeating vertices in the input vector forX

LOGID421.tex; 28/05/1999; 10:12; p.19


to identify the right vertices inX, so that applyingX producesXk) and then applyingB(if necessary).

LEMMA 1. Ak ⊆ Hk .Proof.By induction on the generation ofG. AssumeG ∈ Ak .(i) Base:G is a non-(k − 1)-colourable hypergraph of order less thanp. But then

trivially G ∈ Hk(ii) G is obtained from an earlier stage byB. Then by the hypothesis of induction,

∃G′ | G′ B G andχ(G) ≥ k.ButB preserves chromatic level – if there is nok−1 or less colouring ofG′, andG′ B G,then there is nok − 1 or less colouring ofG. ThereforeG ∈ Hk , as required.

(iii) G is obtained from an earlier stage byX. Then by the hypothesis of induction,

∃G1 . . . Gp | G = X(G1 . . . Gp) and∀Gi, χ(Gi) ≥ k.But by a simple colouring argument,

χ(G1) . . . χ(Gk) ≥ k⇒ χ(X(G1 . . . Gp)) ≥ k.Ex hypothesi, there is no properk − 1 colouring of anyGi , 0 ≤ i ≤ p. Note that wecan represent somek − 1 colourings ofX(G1 . . . Gp) by the colours of the vertices inthe monochrome edges of theGi produced by ak − 1 colouring of the vertices of theGi . SinceX has no properk − 1 colouring, there must be an edge inX(G1 . . . Gp) that ismonochrome on any suchk−1 colouring induced by ak−1 colouring of the vertices of theGi . Let that edge be{Gi . . .Gm}. Now recall howX(G1 . . . Gp) is formed; it will includethe union of those monochrome edges of{Gi . . .Gm} as an edge, and so it will includea monochrome edge on anyk − 1 colouring of its vertices (which are just the vertices oftheGi ).

ThereforeG ∈ Hk as required.

LEMMA 2. LetAk(H) be the result of adding a given hypergraph,H , to the base of ourconstruction. Then

G ∈ Ak(J )⇒ G ∪1 ∈ Ak(J ∪1).

Proof.SupposeG ∈ Ak(J ). We’ll prove thatG∪1 ∈ Ak(J ∪1) by induction on thestage of construction ofG.

(i) Base.G = J orG is an element of the base ofAk . ThenG∪1 ∈ Ak(J ∪1) eithertrivially or by a single application of 2.

(ii) G results from an earlier stage by 2. Then∃G′ | G′ B G, andG′ ∪1 ∈ Ak(J ∪1)by our hypothesis of induction. But it’s easy to see that

G′ B G⇒ G′ ∪1 B G ∪1.Therefore by 2,G ∪1 ∈ Ak(J ∪1) as required.

(iii) G results from earlier stages by 3. Then by our hypothesis of induction,

∃G1 . . . Gp | X(G1 . . . Gp) = G, and ∀i, 1≤ i ≤ p,Gi ∪1 ∈ Ak(J ∪1).

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But then

X(G1 ∪1 . . .Gp ∪1) ∈ Ak(J ∪1).

And

X(G1 ∪1 . . .Gp ∪1) B X(G1 . . . Gp) ∪1.

So by 3 and 2,G ∪1 ∈ Ak(J ∪1) as required.

COROLLARY 3. G ∈ Ak(J1) andG ∈ Ak(J2) ⇒ G ∈ Ak(J1 ∪ J2). By Lemma2,G ∈ Ak(J1) yieldsG ∪ J2 ∈ Ak(J1 ∪ J2). But again by Lemma2, G ∈ Ak(J2) yieldsG ∈ Ak(G ∪ J2). HenceG ∈ Ak(J1 ∪ J2).

LEMMA 4. LetG be a hypergraph withv0 ∈ V (G) andv1, . . . , vi /∈ V (G), and letL be{{u1 . . . v1, . . . vj . . . um} | {u1 . . . v0 . . . um} ∈ G}. Then ifG ∈ Ak, L ∈ Ak(F), where

F = {{v1, . . . vj }}.

Proof.By induction on the stage of construction ofG.(i) Base.G is a singleton loop, or another hypergraph in the base ofAk . Trivial, by the

construction of the base ofAk(F).(ii) G results from an earlier stage by 2. Then∃G′ | G′ B G, andG′ is obtained at an

earlier stage. Therefore, by the induction hypothesis,

L′ = {{u1 . . . v1, . . . vj . . . um} | {u1 . . . v0 . . . um} ∈ G′} ∈ Ak(F).

But then ifG′ B G, L′ B L. Hence by 2,L ∈ Ak(F) as required.(iii) G results from an earlier stage by 3. Then∃G1 . . . Gp | X(G1 . . . Gp) = G and

G1 . . . Gp are obtained at earlier stages. But ifX(G1 . . . Gp) = G, thenX(L1 . . . Lp) =L whereLi = {{u1 . . . v1, . . . vj , . . . um} | {u1 . . . v0 . . . um} ∈ Gi}, and by our hy-pothesis of induction∀i, 1 ≤ i ≤ k, Li ∈ Ak(F). Therefore by 3,L ∈ Ak(F) asrequired.

COROLLARY 5. Let L be a hypergraph,v1, . . . , vj ∈ V (L), and letG be the homo-morphic image ofL obtained by identifyingv1 through vj with a new vertex,v0. LetF = {{v1, . . . , vj }} as above. ThenG ∈ Ak ⇒ L ∈ Ak(F).

Proof.SupposeG ∈ Ak . By Lemma 4,L′ ∈ Ak(F), whereL′ = {{u1 . . . v1, . . . , vj ,

. . . um} | {u1 . . . v0 . . . um} ∈ G}. But every element (edge) ofL′ has a subset inL. ThusL′ B L, and so by 2,L ∈ Ak(F) as required.

THEOREM.Hk ⊆ Ak .Proof.Suppose thatG ∈ Hk . We will proveG ∈ Ak by induction on|V (G)|.(i) Base. For|V (G)| < k the result is trivial by definition of our base forAk.(ii) Induction step. Assume|V (G)| ≥ k, andv∗ /∈ V (G). ConsiderGi...k = {{u1 . . . v

∗. . . um} | {u1 . . . vi or . . .orvk . . . um} ∈ G}. |V (Gi...k)| = |V (G)| − n, wheren ≥ 1, soby the hypothesis of induction,Gi...k ∈ Ak . Therefore by Corollary 5,G ∈ Ak(Fi...k),whereFi...k = {{vi, . . . , vk}} for all vi 6= · · · 6= vk ∈ G. But then by repeated application

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of Corollary 3 usingk vertices ofG arranged according to the pattern required to obtainKkfromX, (u1 . . . uk) (see Lemma 0),G ∈ Ak(X(u1 . . . uk)). But by 3,X(u1 . . . uk) ∈ Ak .ThereforeG ∈ Ak.26

NOTES

1 The term is due to F. Miro Quesada, who used it in 1976 to refer to an inconsistencytolerating group of logics including those of Newton da Costa’s (one of the pioneer of thesubject). One of us (Schotch) first heard it in the presentation of a paper by Wolf and daCosta at the Pittsburgh meeting of the Society for Exact Philosophy in 1978.

2 See P. K. Schotch and R. E. Jennings, “On detonating,” in Priest, Routley and Nor-man, eds.,Paraconsistent Logic: Essays on the Inconsistent(München: Philosophia Ver-lag, 1989), “Inference and necessity,”J. Philos. Logic, and P. Apostoli and B. Brown,(“A solution to the completeness problem for non-normal modal logics”, (forthcomingin J. Symbolic Logic)), “A solution to the completeness problem for weakly aggregaturemodal logic”,J. Symbolic Logic60 (3) (September 1995) 832–842.

3 A personal favorite is:(A ⊃ B) ∨ (B ⊃ C). It is mildly ironic that this is required tobe a tautology in order for the Lewis modal logicS1 to have a semantics (or at least thesemantics that it in fact has).

4 The logicS2. The logicS1has only that version of this ‘paradox’ in which the mainconnective is material rather than strict implication.

5 The drawing board in question was getting rather worn out by this time anyway. Lewis’first attempts to formalize strict implication had collapsed into classical logic and once thatproblem had been solved he discovered that his logic had the ‘wrong’ form of transitivityprinciple.

6 Both W. T. Parry and R. Barcan-Marcus.7 In this group belong not only Russell but also Quine. Strange bedfellows indeed.8 In this connection, Judith Pelham’s recent paper “Russell’s early philosophy of logic”

in Russell and Analytic Philosophy, forthcoming University of Toronto Press, is mostilluminating.

9 The phrase is Wittgenstein’s. He gives an account of going to visit Russell over aperiod of several days before the first war, with the express purpose of puzzling out whatthis “sign” meant. The two of them finally decided that it was meaningless. This may beshocking but it isn’t surprising given what Russell has to say in thePrinciplesconcerningassertion.

10 Quine uses this mode of attack (among many others of course) but he does not followthrough as we do in the next sentence.

11 In the usual sense that a set of sentences is true provided all of its member sentencesare true. One might take the position here (starting a competing radical school) that thisbetrays deeply embedded classical thinking. We really ought to construct an alternativeto the classical theory of true sets, such that individual sentences inherit their truth fromthe fact that they are members of true sets rather than the contrary way of doing things. Itmight even be the case, on this alternative radical proposal, that one need not embrace truecontradictions. To the best of my knowledge, this new school is without members.

12 In other places this sort of predicate is defined in terms of partitions or covering fami-lies. This works, and is to some extent more intuitive, but it fails to generalize to the entire

LOGID421.tex; 28/05/1999; 10:12; p.22


class of level functions. In addition, it fails to account for those0 such that CON(0,0)which must be dealt with by a ‘convention’.

13 Such, for example, is Henry Kyburg’s diagnosis of what goes wrong in the lotteryparadox. See “Conjunctivitis,” in Marshall Swain, ed.,Induction, Acceptance and RationalBelief, 55–82 (D. Reidel, Dordrecht,1970).

14 It is, to my mind, rather unfortunate that paraconsistent logic in our style has come tobe called “the non-adjunctive approach”.

15 See Schotch and Jennings, “On detonating,” 1989.16 In most introductory logic books what iscalled the rule of reiteration, is an amalgam

of this rule and the rule of monotonicity or dilution.17 R. Sylvan has remarked in a survey of non-monotonic logics that we can comfortably

take “non-monotonic” as a synonym for “non-deductive”. If forcing really is to be regardedas non-monotonic then, we trust we have made this view rather more uncomfortable.

18 An n-colouring may be defined as a functionC∈⋃H→{1, . . . , n}.19 This sense of containment is straightforward:T containsT ′ iff every member ofT ′ is

a subset of some member ofT .20 See Brown, and Apostoli, “A solution to the completeness problem for weakly ag-

gregative modal logic,”J. Symbolic Logic60(3) (September 1995), 832–842.21 There is a bit more at stake here than just Hippocraticality; we might wish to concen-

trate on properties8 which satisfy certain ‘nice’ conditions e.g. compactness. It can alsobe shown that this is ‘transmitted’ by the construction in the sense that if8 is compact,then its corresponding level function islevel-compact. By this is meant that if (0) = n

then`(1) = n for1 some finite subset of0 (where` is a level function). This result is dueto Blaine d’Entremont,Inference and Level(M. A. Thesis, Dalhousie University, 1982).

22 This is quite important for the adjudication of a suggestion of D. Lewis in “Logic forequivocators”,Nous16, 431–441. He suggests in that essay that his approach is akin toours, but it is easy to see that the logic he proposes has the classical dilution rule, meets theconditions referred to, and hence fails to be Hippocratic.

23 In “On detonating” this species of forcing is calledA-forcing, and its presentationthere is rather different. This is largely because the reformulation of level functions interms of the predicate9 had not then appeared.

24 See B. Brown, “A paraconsistent approach to old quantum theory”,PSA 1992, Vol. 2(Philosophy of Science Association, 1993), and D. Braybrooke, B. Brown, and P. K. Schotch,Logic on the Track of Social Change(Oxford University Press, 1995).

25 To see this is pretty easy. Withk vertices andk − 1 colours we can assign differentcolours to all but two vertices inXk. So if any two vertices fail to be alone on an edgetogether, it will be possible to produce a properk−1 colouring ofXk , simply by assigningthose two vertices the same colour and every other vertex a unique colour.

26 The present form of this proof derives from a long correspondence between one of theauthors (Brown), who first discovered it, and P. Apostoli, whose extensive contributionsare here gratefully acknowledged.

University of Lethbridge,Department of Philosophy,4401 University Drive,Lethbridge AB T1K 3M4, Canada

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Logic and Aggregation