when hyperpropositions meet

ANDRÉ FUHRMANN

WHEN HYPERPROPOSITIONS MEET ...

Received 17 August 1998

ABSTRACT. With each propositionP we associate a set of proposition (a hyperproposi-tion) which determines the order in which one may retreat from acceptingP , if one cannotfully hold on toP . We first describe the structure of hyperpropositions. Then we describetwo operations on propositions, subtraction and merge, which can be modelled in termsof hyperpropositions. Subtraction is an operation that takes away part of the content of aproposition. Merge is an operation that determines the maximal consistent content of twopropositions considered jointly. The merge operation gives rise to an inference relationwhich is, in a certain sense, optimally paraconsistent.

KEY WORDS: conjunction, consequence, contradiction, hypertheory, inconsistency, in-ference, paraconsistency, subtraction

1. INTRODUCTION

Fallbacks. When the competition started, Angela was convinced that Bra-zil would win. She firmly ruled out the possibility that any other teammight win the cup. After she had seen the first games, she had to retreatfrom her initial belief and made room for the possibility that Germanymight win against Brazil in the final round. But then France emerged asanother strong competitor. Her fallback position then was: Either Brazil orGermany or France – certainly not Croatia: that seemed just too remote apossibility.

A convenient and theoretically fruitful means of representing a belief isby way of a set of possible worlds (proposition): all those worlds at whichthe belief in question is true. Since Angela believes that Brazil will win,she believes to inhabit one of those worlds in which Brazil wins and notone of the others. These other worlds are, from her point of view, possible– are they all equally possible? It seems not. According to Angela, thatGermany might win is “more possible” than that Croatia might win.

Sometimes it is important to take notice not only of the propositionswhich are accepted as true by an agent but also of those that are next inline as candidates of belief. These are the possible fallback positions thatsomeone is prepared to take in response to new evidence overthrowingold beliefs. Borrowing from recent work by Krister Segerberg, I use the

Journal of Philosophical Logic28: 559–574, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

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termhyperpropositionfor a certain family of propositions that are, looselyspeaking, held together by the fact that they are possible fallbacks, in someorder, for a given proposition as believed by some agent. The purpose ofthis note is to probe into the structure of hyperpropositions and to describetwo simple operations on hyperpropositions.

Some History. The concept of a hyperproposition results by way of anatural generalisation from Segerberg’s (1996, 199+) notion of a hyper-theory. Hypertheories – though not under this name – are the key items inGrove’s (1988) modelling of AGM belief change. Lindström and Rabinow-icz (1991) have also taken Grove’s models as a point of departure for study-ing relational theories of belief revision. Such theories can be modelled bytaking nonnested hypertheories into consideration. The term ‘fallback’ inthe sense employed here was introduced in Lindström and Rabinowicz(1991). Grove’s models of belief change adapted Lewis’ (1973) seman-tics for counterfactual conditionals in terms of nested systems of spheres.Lewis’ spheres semantics may be may be seen as a dyadic neighbourhoodsemantics in the sense of Lemmon and Scott (1977).

The Models. Hyperpropositions are sets of propositions. But not any setof propositions is a hyperproposition. Thus, to model hyperpropositionswe need some additional structure in the universe of propositions. Let amodel

(W,P , R)

consist of

• a setW of possible worlds,• a Boolean algebraP (of propositions) on 2W (we letP also denote

thesetof propositions), and• a binary relationR between propositions, i.e.R ⊆ P 2 – thefallback

relation of the model.

Propositions are thus represented by way of sets of possible worlds.The stronger the proposition, the more possibilities it rules out whencethe smaller its associated set of possible worlds. One proposition entailsanother just in case the set that represents the former is included in the setthat represents the latter.

I have already indicated by way of an example the intended interpreta-tion of the fallback relation. A propositionB is a fallback for a propositionA, i.e.RAB, just in case one would retreat toB, if A is no longer ten-able. According to this interpretation, a model represents a sequence ofpossible beliefs. The order of the sequence is determined by judgements

WHEN HYPERPROPOSITIONS MEET ... 561

of comparative possibility: one proposition, i.e. set of worlds, being morepossible than another.

It is not controversial that we constantly engage in such comparisons.Their basis, however, is highly controversial. Some would argue (withDavid Lewis) that judgements of comparative possibility have to measureup to facts about overall similarities between possible worlds. There wouldthen be nothing essentially epistemic in the notion of a fallback relationbetween propositions. Epistemic agents either suceed or fail to retreat fromtheir beliefs in the right order. Others would hold that comparative possi-bility is more like subjective probability: it is or better be constrained byhow things are – but it is not thus determined.

The epistemic interpretation of the fallback relation is the more cau-tious, ontologically less committal option. This interpretation requires nomore than to acknowledge the observation that doxastic retreats proceedorderly. Whether such retreats can be grounded in certain facts about pos-sibilia is a question that need not now be decided. On the epistemic inter-pretation, each model (i.e. structure of the just described kind) representsfor each proposition a route of retreat that some epistemic agent may take,if that proposition were among his beliefs. Note that the models are notepistemic in the sense that they carry a representation of doxastic commit-ment at some time (static commitment). The models serve only to representdynamic commitment – commitments, that is, to trade in certain beliefs forcertain others.

So what is a hyperproposition? Ahyperpropositionis the closure ofsome proposition under the fallback-relation. We say that a hyperproposi-tionA∗ is generated froma propositionA just in case

A∗ = {B : RAB}.The task for the remainder of this paper is to investigate the structure ofhyperpropositions and to display some simple operations on hyperpropo-sitions.

2. THE STRUCTURE OFHYPERPROPOSITIONS

If hyperpropositions are generated by closing under the fallback relation,then the structure of hyperpropositions is determined by the conditions onfallback relations.

Reflexivity. Each proposition is a fallback for itself. Let us view this con-dition simply as a stipulation, governing a limiting case and simplifyingmatters:

RAA.(R)

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Inclusion. Fallbacks are reached by retreating from the stronger to theweaker. As one is forced to retreat from a position, more possibilities,previously ruled out, come into view. Hence, ifB is a fallback forA, thenthe set ofB should extend the range of possibilities represented byA:

if RAB, thenA ⊆ B.(I)

From Inclusion we obtain immediately anti-symmetry:

if RAB andRBA, thenA = B.

Transitivity. If B is a fallback forA andC is a fallback forB, thenC isa fallback forA in the sense thatC can be reached by a series of retreatsstarting fromA:

if RAB andRBC, thenRAC.(T)

Virtual Connectedness.This condition is difficult to justify but it makesfor a much simpler picture of hyperpropositions. Perhaps the best that canbe said for it is that it is dispensable. At the cost of complicating thedefinitions of the operations to be introduced below, one can do withoutVirtual Connectedness,

if RAB andRAC, thenRBC orRCB.(VC)

Given (I), the immediate effect of (VC) is that it makes fallbacks lin-early ordered by set-inclusion. Thus, sets of fallbacks may be pictured as“onions”, as in Figure 1.

Virtual connectedness is a particularly simple way of stating the ruleof the game. For any given belief, if that belief is to be given up, thenthere ought to be a definite answer to the question “What should I believeinstead?” There are basically two routes to implementing that rule. First,make all fallbacks comparable in the sense of (VC) and choose the clos-est, i.e. strongest fallback. Second, allow for incomparables and choosewhat they agree upon. Given (I), there will always be a common coreto all fallbacks from a given (nonempty) proposition. The first route issimple but based on a strong assumption. The second route requires noextra assumption at the cost of complicating some of the operations onhyperpropositions. We choose here the first, simple and strong option.

The Limit Assumption. This condition is known from the Lewis-sem-antics for counterfactual conditionals and we need to impose it here forreasons very similar to those advanced there. Suppose you are moved to


retreat from the proposition that Hans is tall. According to the recipe de-tailed below, you ought to retreat to the closest fallback permitting Hansnot to be tall. But tallness comes in densely ordered degrees. So insteadof a unique closest fallback at which Hans need not be tall, there seems tobe an infinite chain of closer and closer fallbacks. The limit assumptionsboldly rules out such infinite chains. The exact formulation of the limitassumption is sensitive to the presence or absence of (VC).

LetH be a hyperproposition andB be a proposition. Let

H \\ B := {X ∈ H : X \ B 6= ∅}.Pictorially speaking,H \\ B (pronounce: “H beyondB”) is that part ofH that stretches beyong the realm ofB-worlds (cf. Figure 2). Infinitelydescending chains of falling back inH \\B are prevented by requiring thatH \\B is bounded by minimal elements underR. With (VC) in place, everysubset of a hyperproposition is linearly ordered whence a minimal element,if it exists, will be unique. So, assuming (VC), the limit assumption maybe formulated as follows. For every hyperpropositionH and propositionB, providedH \\ B 6= ∅,

H \\ B has a least element, min(H \\ B).(LA)

How plausible is the limit assumption? Take again the case of tallness.Hans, I believe, is 196 cm tall. How many centimetres does Hans measurein the closest world in which he is not tall? Silly question! There is no suchworld. Hence, it seems, there is no strongest proposition that you can fallback to from the proposition that Hans is tall. So the Limit Assumptionseems false.

There are two possible responses to this problem. First, one could try toshow that the Limit Assumption is no more than a simplifying assumptionwhich is not really needed. This is the course Lewis takes with his spheressemantics for counterfactual conditionals. Perhaps this course could alsobe taken here.

The second response is based on the observation that the problem istriggered by vague predicates. As a result of the occurance of such a pred-icate, the proposition that Hans is tall is vague. So that proposition, as wellas the proposition that Hans is not tall, should be represented as vague ina model. Whatever representation of vague propositions we choose, it willnot be a single, ordinary set of worlds. Perhaps it will be a family of suchsets or a single set with a real-valued membership relation, to name justtwo candidates. In any case, it is not so obvious that the Limit Assumptionis false for hyperpropositions generated from vague propositions, rightlyrepresented. In fact, there is reason to think that it must be true if vague

564 ANDRE FUHRMANN

Figure 1.Onions. Figure 2.H beyondB.

Figure 3.Subtraction without and. . . . Figure 4.. . . with Recovery.

Figure 5. Merge.

propositons are properly taken into consideration. For, if we are forced toretreat from the belief that Hans is tall, we do not thereby suspend all be-liefs about Hans’ height. Instead we end up with a single vague belief abouthow tall Hans is: somewhere between the Japanese prime minister and theGerman chancellor.1 This is, perhaps, the strongest fallback position thatwe are prepared to take. There is a limit – but it may be vague.

Another Definition of Hyperproposition.Hyperpropositions are not prim-itive entities in our models; they are generated from propositions by closing


under the fallback relation. General properties of hyperpropositions arethus consequences of general conditions on the fallback relation. An alter-native route would be to tackle the concept of a hyperproposition directly.The following definition is similar to the one in Segerberg (199+).2

DEFINITION. H is a hyperproposition iff

(H1)⋂H ∈ H ,

(H2) if A,B ∈ H , then eitherA ⊆ B orB ⊆ A, and(H3)

⋂(H \\ A) ∈ H \\A, if H \\A 6= ∅.

OBSERVATION. (1)If H is a hyperproposition, then there exists a propo-sitionA and a relationR ⊆ P 2 such that(i) H = A∗ = {B : RAB}, and(ii) R is a fallback relation onH .

(2) If A is a proposition andR is a fallback relation, thenA∗ = {B :RAB} is a hyperproposition.

Proof.(Ad 1) LetR be the restriction of⊆ toH , i.e. for all propositionsA andB,

RAB iff A,B ∈ H andA ⊆ B.(∗)We show that(

⋂H)∗ = H , i.e.R(

⋂H)B iff B ∈ H . The left-to-right

direction is immediate from (∗). Conversely, suppose thatB ∈ H . By(H1),

⋂H ∈ H . Since

⋂H is inclusion-minimal among the members of

H ,⋂H ⊆ B. So, by (∗), R(⋂H)B.

It remains to show that the relationR of (∗) is a fallback relation. (R),(I) and (T) are immediate from (∗) and the corresponding properties ofinclusion. For (VC) assume thatRAB andRAC. ThenB,C ∈ H whence,by (H2),B ⊆ C or C ⊆ B. So, by (∗) again,RBC or RCB. For (LA)suppose thatH \\ A 6= ∅. Then, by (H3), we have

⋂(H \\ A) ∈ H \\ A.

Since⋂(H \\ A) is minimal under inclusion inH \\ A ⊆ H and sinceR

is just⊆ restricted toH ,⋂(H \\ A) is minimal underR, as required by

(LA).(Ad 2) Make the assumptions. For (H1) note first that (R),RAA, gives

us (a)A ∈ A∗. Moreover, from (I) we have that ifRAB, thenA ⊆ B, forall B ∈ A∗. Thus,

⋂A∗ = A. Putting (a) and (b) together it follows that⋂

A∗ ∈ A∗.For (H2) suppose thatB,C ∈ A∗, i.e.RAB andRAC. Then, by (VC),

RBC orRCB whence, by (I),B ⊆ C orC ⊆ B.For (H3) suppose thatA∗ \\ B 6= ∅. Then, by (LA),A∗ \\ B has an

R-min memberM. Since by (I),R reflects inclusion among the membersof a hypertheory,M is⊆-min inA∗ \\ B. SinceA∗ \\ B is a nested familyof sets,M =⋂(A∗ \\ B) as required. 2

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3. HYPERPROPOSITIONMEETS PROPOSITION: SUBTRACTION

Wittgenstein wrote (in thePhilosophical Investigations, §621):Let us not forget this: when I raise my arm, my arm goes up. And the problem arises: whatis left over if I subtract the fact that my arm goes up from the fact that I raise my arm?3

Of course, in this passage Wittgenstein draws attention to a problem aboutintention. But his formulation also points towards another problem, thistime of philosophical logic: what is it, tosubtractone fact from another?We have a number of theories of implication (or consequence or entail-ment) that tell us what it is toextractone fact from another. But subtractionlooks like a hitherto much neglected logical operation.

Jaeger (1973) was the first to notice that §641 of thePhilosophical In-vestigationsposed a problem for the philosophical logician. Hudson (1975)responded with the thesis thatA − B ought to be the weakest proposi-tion such that, when conjoined withB, it entailsA. According to Hudsonthen, subtraction is just the converse of extraction alias material implica-tion. This is appealingly simple but deeply unsatisfactory for a number ofreasons recounted in Humberstone (1981) and Fuhrmann (1997a).4

In a way, the neglect of logical subtraction is quite startling. For the easewith which we read and understand the passage quoted from Wittgensteinsuggests that logical subtraction is nothing unfamiliar. Tosubtract oneproposition,B, from another,A, one removes the contents ofB from thecontents ofA. Let us writeA−B for the result of subtractingB fromA. Inone case the result of subtracting is immediately obvious: if the contents ofB is not part of the contents ofA, then nothing needs to be removed fromA whenceA− B ought to be the same asA. But what about the principalcase whereB is part ofA?

Apart from manifest occurrences as in §641 cited above, there are cer-tain families of locutions in natural language which suggest subcutaneouspresences of subtraction:

(1) A lawlike statement is a law, except it need not be true.(2) Everyone wears a hat, except Bob.(3) All birds fly, except penguins.(4) A hypothesis is a belief that is not fully accepted.(5) A gratin is a quiche that is not baked in a shell.(6) No one will succeed, except possibly Jane.

While some of these sentences admit of a coherent first-order para-phrase (such as∀x(Bird(x) ∧ ¬Peng(x) → Fly(x)) others do not. Forexample (1) cannot be analysed in any obvious way as having the logicalform of a conjunction, as in


(1′) A lawlike statement is a lawand it need not be true.

Such a conjunctive rephrasal issues in a necessary falsehood. Yet it is(possibly) true. So some alternative analysis is called for. Perhaps thesesentences are of a subtractive form: a lawlike statement is a lawminusit istrue? All birds flyminuspenguins fly?

Defining Subtraction. Let us explore the prospects of treating the abovesentences as involving a subtraction operation. This requires that the con-tents of sentences have apart-whole structuresuch that parts of the con-tents of a sentence can be removed, “subtracted”, and the remainder canstand on its own feet. Once the parts of a proposition have been identified,we can ask how parts can be removed from whole proposition.

First Proposal: X is a part of a propositionA if and only if A ⊆ X.This is not promising for he present purpose becauseX may not be aproposition. It would be difficult to guarantee that the result of removing anon-propositional part from a proposition results in a proposition.

Second Proposal: X is a part of a propositionA if and only if A ⊆X andX is a proposition. Since propositions are closed under Booleanoperations,AwithoutX is likely to result in a proposition. But the proposalsins against the spirit of the models presented here.

Suppose Angela believes that reading skills are always situated in theleft hemisphere of the brain – call this beliefp. This entailsq, the proposi-tion that reading skills are situated somewhere in the human body. Angelacertainly believesq and if it turned out thatp was false, i.e. that read-ing skills can be encoded somewhere else but in left brain hemisphere,she would still hold on toq as a fairly ultimate fallback position. Butsandwiched betweenp and q are infinitely many other propositions –propositions entailed byp and entailingq. There is for example the propo-sition r that reading skills are located either in the left hemisphere of thebrain or in the right heel. I submit that Angela does not considerr to be apossible fallback fromp. In that sense,q but notr is a part of what Angelabelieves when she believes thatp; cf. Figure 6. Thus we have as a

Third Proposal: X is a part of a propositionA if and only if X is afallback forA (whence, by Inclusion,A ⊆ X). In adopting this proposalhere I am, of course, not claiming that this is the right conception of theparts of a proposition in any absolute sense. The proposal is made with aview towards the definition of subtraction below. The proposal is right tothe extent that it gives rise to a plausible account of subtraction.

Having thus identified the parts of a proposition for the purpose of sub-traction, let us next ask what it means to subtract part of a proposition fromthe whole of which it is a part. The answer, recorded in the next definition

568 ANDRE FUHRMANN

Figure 6.

is this: The result of subtractingB from A is the “biggest” part ofA notentailingB.

DEFINITION. Subtraction without Recovery (cf. Figure 3).

A− B ={

min(A∗ \\ B), if A∗ \\ B 6= ∅;A otherwise.

As a consequence of this definition, all models satisfy the following con-ditions:

(S1) IfB 6= W , thenA− B 6⊆ B;subtractionssucceed, i.e.A − B does not entailB, unlessB is nec-essary.

(S2) A ⊆ A− B;subtraction is aweakening-operation.

(S3) IfA 6⊆ B, thenA− B ⊆ A;if A does not entailB, then subtraction isvacuous, i.e. A − B isincluded inA – indeed, in that case we haveA = A− B.

Limiting Cases. It may be instructive to see how the definition works byconsidering some limiting cases.

CaseW −A. SinceW ∗ = {W } it follows thatW −A = W . One cannotsubtract from a necessity.

CaseA − W . Since there are no fallbacks beyondW , A∗ \\ W = ∅.So the otherwise-condition rules thatA −W = A. A necessity cannot besubtracted.

Note that it is not generally the case that ifA∗ \\ B is empty, thenB = W . It might just be thatB is “inaccessible” with respect toA∗. Per-haps there is no fallback fromA that would admitB as a possibility. (LetB, e.g., stand for the proposition that the moon is made of green cheese.)


Inaccessible propositions could be ruled out by topping each hyperpropo-sition withW as an ultimate fallback. In that case it would be true thatA∗ \\ B = ∅ just in caseB = W .

Case∅ − A. Then∅∗ = {X ∈ ∅∗: X \ A 6= ∅}. This is not reallya limiting case. A contradiction can be associated with any sequence offallbacks and, hence,∅∗ may be any perfectly well-behaved hyperpropo-sition. The only thing we know for sure is that(∅ − A) ∩ A is nonemptyprovidedA 6= W . This is, of course, as it should be. To recover from acontradiction by subtracting some contingent propositionA should resultin a proposition consistent with not-A.

CaseA − ∅. ThenA∗ \\ ∅ = A∗ \ ∅. If A is consistent, then nothingis subtracted by subtracting a contradiction; hence,A − ∅ = A. If A isinconsistent, i.e.A = ∅, then∅ − ∅ is the the strongest fallback in∅∗, i.e.the strongest proposition that results from an unconstrained recovery fromcontradiction. (In general,A−A is the “closest”, i.e. strongest fallback inA∗ beyondA.)

Recovery. Note that the following condition, Recovery, is not a conse-quence of the above definition.

(S4) (A− B) ∩ B ⊆ A;A is recovered by intersectingA− B with B.

Recovery is, of course, a condition much discussed in connection with for-mal theories of belief change. If one wanted it to be satisfied, the followingdefinition could be adopted.

DEFINITION. Subtraction with Recovery (cf. Figure 4).

A− B ={A ∪ (min(A∗ \\ B) ∩ B), if min(A∗ \\ B) 6= ∅;A otherwise.

This will validate Recovery as well as the other conditions mentionedabove.

4. HYPERPROPOSITIONMEETS HYPERPROPOSITION: MERGE

Conjunction is a fairly reckless operation. It is safe only as long as whatis being conjoined is consistent. But suppose you need to fuse incosistentpieces of information into a consistent whole. Unless one of the two propo-sitions to be joined clearly overrules the pretensions of the other one, eachof the two has to give way a little – just as much as is necessary to achievea consistent blend of the two.

570 ANDRE FUHRMANN

The emerging picture is this: If the information from two propositionsis to be combined, our strongest and therefore first choice would be toproceed by ordinary conjunction. But if the two propositions contradicteach other, certain falsity would result. We should therefore retreat fromeach of the two propositions just as much as is needed to combine themconsistently. In other words, the task is to identify the strongest compatiblefallbacks admitting the respective other proposition and to conjoin theseconsistent fallbacks rather than the original inconsistent propositions.

It seems that William James thought of such a procedure when hewrote:

The process [by which any individual settles into new opinions] is always the same. Theindividual has a stock of old opinions already, but he meets a new experience that putsthem to a strain. [. . .] The result is an inward trouble to which his mind till then hadbeen a stranger, and from which he seeks to escape by modifying his previous mass ofopinions. He saves as much of it as he can, for in this matter of belief we are all extremeconservatives. So he tries to change first this opinion, and then that (for they resist changevery variously), until at last some new idea comes up which he can graft upon the ancientstock with a minimum of disturbance of the latter, some idea that mediates between thestock and the new experience and runs them into one most felicitously and expediently.

The new idea is then adopted as the true one. It preserves the older stock of truths witha minimum of modification, stretching them just enough to make them admit the novelty,but conceiving that in ways as familiar as the case leaves possible. [...] New truth is alwaysa go-between, a smoother-over of transitions. It marries old opinions to new facts so asever to show a minimum of jolt, a maximum of continuity. (Lectures on Prgmatism, 1907,pp. 59ff)

Let us try to capture what James describes with the resources of thefallback models. Suppose thatA andB are incompatible propositions, i.e.A ∩ B = ∅. ThenA andB aremergedby taking what the following twopropositions agree upon:

(a) the strongest fallback forA that admitsB as a possibility (= the leastelement ofA∗ that overlapsB), and

(b) the strongest fallback forB that admitsA as a possibility (= the leastelement ofB∗ that overlapsA).

In other words, the merge ofA andB,A◦B, is the intersection of min(A∗\\B) with min(B∗ \\ A).

This describes the principal case where bothM = A∗ \\ B andN =B∗\\A are nonempty. There are three more cases to consider: (a)M but notN is empty, (b)N but notM is empty, and (c) bothM andN are empty.These cases obtain wheneverA or B or both are either contradictory orinaccessible (in the sense explained above). Rather than working out thesecases by hand, we let subtraction take its course by defining:5


DEFINITION [Cf. Figure 5].

A ◦ B = (A− B) ∩ (B − A).

Since there is in general no guarantee that eitherA or B fully survivesthe merge of the two, merge is an example of what Hansson (1996) hascalled a ‘non-prioritized revision operation’. Some of the key properties ofthe merge operation are listed below.

(M1) A ◦ B = B ◦A.(M2) A ◦ A = A.(M3) If A ∩ B 6= ∅, thenA ◦ B = A ∩ B.(M4) A ∩ B ⊆ A ◦ B.(M5) A ◦ ∅ ⊆ A.(M6) If A 6= ∅, thenA ◦W = A.(M7) If A andB are contingent, thenA ◦ B is contingent.

Some brief explanations are in order. (It might be helpful to consultFigures 3 and 5.) The condition (M1) requires merge to be commutative:order makes no difference. According to (M2), merge is idempotent. Thisis so because only one fallback inA∗ reaches beyondA, viz. A itself.Condition (M3) requires that ifA andB are consistent, then merge is con-junction. In that caseA is the strongest fallback admittingB in A∗ andBis the strongest fallback admittingA in B∗. In general, however, merge is,according to (M4), weaker than conjunction. IfA andB are inconsistent,thenA∩B = ∅; otherwiseA∩B = A ◦B. The postulates (M5) and (M6)take care of special cases: merging with the impossible (∅) and mergingwith the necessary (W ). The result of merging a propositionA with theimpossible always entailsA. For,A−W = A andA∩∅− A ⊆ A. If A isconsistent, then merging with the necessary proposition has no effect. Thisis so becauseW − A = W whenceA ◦W is whateverA − ∅ happens tobe, i.e.A if A is consistent. Finally, (M7) gives expression to the fact thatmerge blends any two contingent propositions to a consistent whole.

5. HYPERPROPOSITIONS, MERGE AND PARACONSISTENCY

When extracting information from a collection of premisses, one usuallytakes the collection in a conjunctive sense: one assumes that all the pre-misses are true and sees what follows. However, in the limiting case thatthe premissescannotbe true together, this procedure produces a result thatis not even of limited but of no value at all. For, byex falso quodlibet,

572 ANDRE FUHRMANN

anything follows from contradictory premisses. This observation has ledto research in inference relations that would yield useful results even whenapplied to inconsistent premisses. A logic codifying such an inferencerelation is calledparaconsistent.

Approaches to paraconsistency can be grouped together in various fam-ilies. The great majority of paraconsistent logics emerge as attempts totackle directly the problem of nontrivial inference from inconsistent pre-misses. They hold on to the conjunctive understanding of how premissesare collected for the purpose of inferring. They deny, however, that incon-sistent premisses cannot be true together. As a result we are presented withtheories of inference that depart to a rather large extent from a classicalunderstanding of the logical notions involved. This creates a dilemma forthe paraconsistent logician. On the one hand, one aims at a logic that isstrong enough to produce the body of theory we are accustomed to (e.g.in mathematics). On the other hand, the departure from a classical un-derstanding of logical notions like negation should be well-motivated andaccompanied by a clear semantic understanding of the syntactic machin-ery. Unfortunately these two aims have a tendency to pull into oppositedirections. Logics that are good on the first count (like da Costa’s C-systems) fail completely on the second and logics that are carried by astrong semantic motivation (like Priest’s Logic of Paradox), are arguablytoo weak to serve as a grounding for, say, arithmetic.

Classicists with some sympathy for the concerns that underly the para-consistent enterprise have tried a different route. They have denied thatinference from inconsistent premisses should proceed by taking their con-junction. Instead, inconsistent premisses should be fenced off from eachother, they should be insulated or compartmentalised, each considered inisolation from the rest. This approach has been recommended by Jaskowskiand, more informally, by David Lewis. The main problem with non-adjunc-tive approaches to paraconsistent inference is that it impoverishes the no-tion of inference. The resulting logics leave no room for multiple premissesinference; they essentially reduce inference to reasoning from single pre-misses.

The first, direct approach to paraconsistency allows inconsistent pre-misses to interact and to combine their resources for the purpose of in-ference – at the cost that inference then goes non-classical. The second,non-adjunctive approach keeps to the familiar notion of inference but can-not then allow premisses to join forces and to deliver their full informa-tional potential.

The merge operation points to a third approach. This approach starts byrejecting the assumption – underlying the previous two approaches – that


premisses are always combined conjunctively. Conjunction is the preferredmode of premise combination whenever the circumstances are favourable.But when the information supplied is inconsistent, one should try to makethe best of it. The merge operation is an attempt to spell out what optimalpremise combination should be like.

Classical inference from merged premisses (merge inference) is para-consistent. It falls short of classical inference from conjoined premissesonly where it has to, viz. when the premisses are inconsistent. Whenevermerge goes sub-conjunctive, it does so in an optimizing way by combiningthe strongest possible fallbacks of all premisses involved. Merge inferencethus extracts as much information as possible under the circumstances. Thetool of extraction is the familiar notion of classical consequence. Mergeinference illustrates how the paraconsistent end can be served withoutengaging in barocque deviance.

ACKNOWLEDGEMENTS

I wish to thank audiences at the LOGICA ‘98 meeting in Liblice (Bo-hemia) and at the PUC (Rio de Janeiro), in particular Oswaldo Chateau-briand. I am also grateful to a referee for the journal for helping me toimprove an earlier version of this paper. The work reported here was sup-ported by a Heisenberg Fellowship of the Deutsche Forschungsgemein-schaft and a travel grant of CAPES (Brazil) and DAAD (Germany).

NOTES

1 ... at the time this paper was written.2 Segerberg has the condition that hyperpropositions are nonempty instead of our con-

dition (H1). Nonemptyness follows from our (H1).3 “[W]as ist das, was übrigbleibt, wenn ich von der Tatsache, daß ich meinen Arm hebe,

die Tatsache abziehe, daß mein Arm sich hebt?”4 Humberstone (1981, unpublished) left the subtraction problem unresolved. A version

of the account outlined here appeared first in Fuhrmann (1997a, Ch. 3) and Fuhrmann(1997b).

5 This definition differs slightly from the one offered in Fuhrmann (1997).

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Department of PhilosophyUniversity of Konstanz78 434 Konstanz, GermanyE-mail: [email protected]

when hyperpropositions meet

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