The Reasonableness of Necessity

Paul Snow

P.O Box 6134Concord, NH 03303-6134 USA

paulusnix@cs.com

AbstractThe necessity calculus is a familiar adjuvant to thepossibilit y calculus and an uncertain inference tool in itsown right. Necessity orderings enjoy a syntacticalrelationship to some probabilit y orderings similar to thatdisplayed by possibilit y. In its adjuvant role, necessitymay be viewed as bringing possibilit y closer to achievingthe quasi-additi ve normative desideratum advocated by deFinetti. Nevertheless, there are occasions when one mightchoose to use possibilit y without the help of necessity, e.g.when the full range of alternative hypotheses is unknown,or to exploit possibilit y’s distinctive abilit ysimultaneously to express preference as well as credibilit yordering. Such situations arise in uncertain domains li kethe evaluation of scientific or mathematical hypotheses,where professions of belief may reflect aesthetic,utilit arian, and evidentiary considerations as much as theusual notions of credibilit y.

Introduction

The possibilit y calculus is a popular tool for uncertaintymanagement which evaluates disjunctions according tothe simple and computationally eff icient rule

Π( A ∨ B ) = max( Π( A ), Π( B ) )By convention, a tautology has a possibilit y of unity

and a contradiction has (and typicall y only knownfalsehoods have) a possibilit y of zero. In any set ofmutually exclusive and collectively exhaustive sentences,at least one of the sentences must also have a possibilit yof unity, so that the max rule yields the desired unit valuefor the tautology.

Possibilit y is easil y identified as a measure of“credibilit y” in that

if A ⇒ B, then B is ranked no less than AThis is a widely-cited criterion of intuiti ve credibilit y-li keness identified by Lukasiewicz and revived bySugeno. Probabilit y is another familiar example of aLukasiewicz-Sugeno calculus, as is the ordinary Booleanlogic. Copyright © 2002, American Association for Artificial Intelli gence (www.aaai.org). All rights reserved.

Perhaps because of its origins in the fuzzy logiccommunity (Zadeh, 1978), possibilit y has sometimes beenviewed as fundamentall y different from probabili sticapproaches to uncertainty. It has long been known,however, that there was a nexus of similarity amongpossibilit y, operations involving probabilit y intervals, andthe Dempster-Shafer calculus (Dubois and Prade andSmets, 1996 review this history), itself originall yproposed as an innovation in Bayesian technique(Dempster, 1968).

Another similarity was noted in connection withconditional logics and default reasoning (Kraus,Lehmann, and Magidor, 1990). A particular class ofprobabilit y distributions faithfull y emulates the behaviorof the default entailment connective (Snow, 1999). These“atomic bound” probabiliti es are the solutions of thesimultaneous constraint system whose typical constraint is

p( x ) > ∑ all atoms y: p( x ) > p( y ) p( y ) > 0

for example, the probabiliti es over the five exclusive andexhaustive sentences a through e:

p( a ) = 16/31, p( b ) = 8/31, p( c ) = 4/31,p( d ) = 2/31, p( e ) = 1/31

Similar emulation of the default rules is also achievedby “ linear” possibilit y distributions (Benferhat et al.1997), those where each atomic sentence has a distinctpossibilit y value, e.g.,

π( a ) = 1, π( b ) = 1/2, π( c ) = 1/4,π( d ) = 1/8, π( e ) = 1/16

For all mutually exclusive sentences A and B, atomicbound probabiliti es follow the possibili stic max rule, thatis, p(A) > p(B) just when the highest probabilit y atom inA ∨ B is in A. So, the special probabiliti es and the linearpossibiliti es are in ordinal agreement for all mutuallyexclusive sentences. Since default entailment can beunderstood as the ordering of mutually exclusivesentences, and it, too, follows a version of the max rule,either possibilit y or probabilit y works equally well foremulation.

The relation between atomic bound probabiliti es andlinear possibiliti es is somewhat closer than that. Aproperty of all probabilit y distributions given specialnormative emphasis by Bruno de Finetti (1937) is calledquasi-additivity,

508 FLAIRS 2002

From: FLAIRS-02 Proceedings. Copyright © 2002, AAAI (www.aaai.org). All rights reserved.

p( A ) ≥ p( B ) ⇔ p( A¬B ) ≥ p( B¬A )for arbitraril y related A and B. Since atomic boundprobabiliti es and linear possibiliti es agree for all mutuallyexclusive sentences (li ke A¬B and B¬A), we have

p( A ) ≥ p( B ) ⇔ Π( A¬B ) ≥ Π( B¬A )Conversely, using a standard possibili stic identity,

Π( A ) ≥ Π( B ) ⇔ Π( A ) ≥ Π( B¬A )we find

Π( A ) ≥ Π( B ) ⇔ p( A ) ≥ p( B¬A )again since atomic bound probabiliti es agree with thelinear possibiliti es for exclusive sentences (A and B¬A inthis case).

These results say that orderings based upon atomicbound probabiliti es and those based upon linearpossibilit y are syntactic restatements of one another. Ifone is “ reasonable” by some standard, then so is the other.The “same thought” can be expressed equally well by aprobabilit y relation or by a possibilit y relation.

Since many of the normative arguments advanced byprobabili sts are ordinal in character, ordinalreasonableness should be enough to moot attempts toportray possibilit y as normatively inferior to probabilit y(or vice versa). That linear possibiliti es also solve theequations advanced in Cox’s Theorem (Snow, 2001), aclassical pill ar of probabili st normative argumentation,should secure the point.

None of the existing results briefly reviewed aboveaddresses the status of the other calculus common in thepossibili st literature, necessity, which is defined as

N( A ) ≡ 1 - Π( ¬A )Like possibilit y, the necessity calculus has enjoyedconspicuous success in emulating other approaches touncertainty management (Dubois and Prade and Smets,1996).

In the current paper, the concern will be to examine aspecific role played by necessity in possibili stic practice,a role which might be called “ tie breaking.” That is tosay, on some occasions when

Π( A ) = Π( B )we find that the necessities of A and B differ. Forexample, if A is a tautology, and B is an uncertainsentence containing a “top atom” (one whose possibilit yis unity), then the possibiliti es tie at unity, but we findthat

N( A ) > N( B )This allows the analyst to distinguish betweenuncertainties of high possibilit y and certainties, obviouslya useful capabilit y in the management of uncertainty.

In the next section, we shall examine necessity andobserve that some aspects of its “ tie-breaking” behaviorhave normative appeal from a probabili st perspective.That said, in the subsequent sections, we consider anuncertain domain where other properties might bedesirable.

In this domain, the range of alternative hypotheses(and hence the “¬A” upon which necessity depends) is

typicall y not full y apprehended, and some subtleties ofdynamic belief revision may be especiall y germane.Possibilit y values applied within the domain may berepresenting information about preferences in addition tocredibilit y in the Lukasiewicz-Sugeno sense. Toaccomplish this dual representation, it may be best toleave possibili stic ties unbroken.

Reasonableness and Tie-breaking

Continuing in an ordinal mode of discourse, it is clear thatsince necessity’s orderings are the expression of a“ thought” in possibili stic terms

N( A ) ≥ N( B ) ⇔ Π( ¬B ) ≥ Π( ¬A )When the possibilit y is linear, then necessity inheritslinear possibilit y’s abilit y to syntacticall y restate theatomic bound probabiliti es’ orderings.

The specific relationship can be easil y worked out byan application of De Morgan’s law to the argument-sentences in the equivalences discussed in theintroduction, yielding

p( C ) ≥ p( D ) ⇔ N( C ∨ ¬D ) ≥ N( D ∨ ¬C )N( C ) ≥ N( D ) ⇔ p( C ∨ ¬D ) ≥ p( D )

So, linear necessity orderings are indeed syntacticrestatements, and full y expressive restatements, of aspecial probabili stic ordering. No special analysis isrequired to confirm the Cox-reasonableness of necessity,since in general, any function of a Cox-reasonable beliefrepresentation is itself Cox-reasonable.

In discussing the reasonableness of necessity in thespecific application of tie-breaking in concert withpossibilit y, it is convenient to combine the two calculiinto a single ordering,

A ≥* B ⇔ Π( A ) ≥ Π( B ) andA >* B ⇔ Π( A ) > Π( B ) or Π( ¬B ) > Π( ¬A )

where the symbol “>*” means “ is ranked ahead of,” andthe meanings of “≥* ” and “=*” are parallel.

A simple way to reali ze this ordering is to create acomposite function,

f( A ) = [ Π( A ) + N( A ) ] / 2This composition is possible since the necessity of allsentences whose possibilit y is less than unity is zero, awell -known feature of the relationship between necessityand possibilit y. Since the function is invertible, noinformation about the original possibilit y or necessityvalues is lost in the composition.

f( A ) < 1/2 ⇒ Π( A ) = 2f( A ); N( A ) = 0otherwise Π( A ) = 1; N( A ) = 2[ f( A ) - 1/2 ]

So conceived, it is easy to very that f() is aLukasiewicz-Sugeno credibilit y function. The f() orderingis also reasonable in the two senses discussed, ordinall yrestating and Cox-reasonable.

The ordering based on f() is transiti ve, and so linear or“one dimensional.” This should not be interpreted as arefutation of the common characterization (e.g., Dubois

FLAIRS 2002 509

and Prade, 1994) of combining necessity and possibilit yas offering “ two dimensions” of information aboutuncertainties. Distinct information is indeed offered bythe two contributors to f(), different “ thoughts” if youwill . The two strands are linearly consonant, however,which is a different matter.

It is interesting from a probabili st-normative point ofview to consider which possibili stic ties are resolved bynecessity. The tie-breaking f() ordering is a kind of quasi-additi ve extension of the strict orderings in the underlyingpossibilit y distribution. That is,

f( A ) > f( B ) ⇒ Π( A¬B ) > Π( B¬A )To see this, it is obvious that all the strict inequaliti es inpossibilit y are echoed in f(), and

Π( A ) > Π( B ) ⇒ Π( A¬B ) > Π( B )since order is determined by the max rule (something in Aand not in B must beat B). But B¬A implies B, so byLukasiewicz-Sugeno and transiti vity,

Π( A ) > Π( B ) ⇒ Π( A¬B ) > Π( B¬A )For the ties which are broken by necessity,

Π( ¬B ) > Π( ¬A ) ⇔Π( A¬B ∨ ¬A¬B ) > Π( B¬A ∨ ¬A¬B )

and by similar reasoning to the previous case based on themax rule, Lukasiewicz-Sugeno, and transiti vity,

Π( ¬B ) > Π( ¬A ) ⇒ Π( A¬B ) > Π( B¬A )Of course, necessity breaks only some of the ties in the

underlying possibilit y ordering. In general, a completetransiti ve ordering based on max cannot be quasi-additi vefor all sentence pairs. For example, consider the ordering

A =* B >* C >* DIf quasi-additi vity obtained in a max calculus, then wewould have the cycle

A ∨ C =* B ∨ C (by max) >* B ∨ D (by quasi-additi vity)=* A ∨ C (by max)

Nevertheless, to the extent that necessity breakspossibili stic ties, it does so in a way which is consonantwith a probabili st’s intuition about good credal ordering.

The Polya Domain

There is an interesting domain of uncertain reasoning inwhich necessity may experience diff iculties, yetpossibilit y itself may be an especiall y attractive reasoningtool.

A pioneer in the exploration of the domain is themathematician George Polya (1954). It concernsreasoning about the possible truth of mathematicalhypotheses, particularly as one’s opinion in the mattermight be affected by the discovery of analogous, implied,or otherwise logicall y or intuiti vely related facts. In thecourse of his studies, Polya noted that the domain wassimilar to other inferential arenas, notably thedevelopment of scientific theories and historical questionssuch as determination of guilt i n criminal investigations.

Perhaps of most immediate relevance to necessity isthat in this domain, “not X” is often unavailable for credalevaluation with any useful specificity. For example, it iseasy enough to think about the import of f inding thedefendant’s DNA at the scene of a crime in relation to thehypothesis that she is guilt y. But in relation to “ the”contrary hypothesis? Would that be “not guilt y, butsomeone with innocent access to the scene,” or “notguilt y, and contamination occurred,” or something else?

Although Polya developed his analyses alongprobabili stic lines, total probabilit y (that p(X) + p(¬X) =1) plays very littl e role in his work apart from algebraicformaliti es. In fact, his probabili sm is at least skepticalabout any important use for additi vity within the domain,itself suggestive of a role for a non-additi ve calculus li kepossibilit y here.

Naturall y, his viewpoint is subject to criti cism fromBayesians (notably de Finetti, 1949 in response to ajournal exposition of part of Polya’s ideas). Nevertheless,Polya pursued his work with a high level of normativesophistication (including some rebuttal to de Finetti,politely unreferenced, in the 1954 book).

This lack of access to a usefull y specific “¬X” wouldjustify some thought about a divorce of probabilit y fromnecessity, at least in this domain. In itself, though, thenegation problem would not forestall what might becalled “contingent necessity,” the complement of thepossibilit y of all known alternatives. Much of the earlierdiscussion of reasonableness and perhaps theconsiderations of quasi-additi ve extension might bemarshaled on its behalf.

Necessity has another diff iculty, however. While it isnot specific to the Polya domain, the diff iculty acquiressome urgency here and is worthy of mention in any case.

Much of the belief revision which occurs in Polya’sapproach consists of the elimination of knownalternatives. For example, it is easy to imagine that aDNA test might eliminate one or more suspects, and thiselimination would have some significance for theassessment of guilt of the remaining suspects.

The diff iculty may be ill ustrated by recalli ng the fourexclusive hypotheses whose possibiliti es are in the order

A =* B >* C >* DObviously, we have

B ∨ C =* B ∨ Dwhich tie cannot be broken because the li veliness of Aforces a tie in necessity. If , however, we were then tolearn that A is untrue, but the possibilit y ordering of theremaining hypotheses is unchanged, then we arrive at

B ∨ C >* B ∨ Din the new f() ordering, since A no longer defeats thequasi-additi ve conclusion based on C’ s advantage over D.

At which point, we have an impasse in intuitions. Theidea that the elimination of an alternative can change thecredal order among the surviving assertions is acceptableto some. It is, for example, the nub of the famous “Peter,Paul, and Mary” case (Dubois, Prade, and Smets, 1996).

510 FLAIRS 2002

A point made in advancing that case, however, isprecisely that probabili st intuition differs. That is, barringsomething special about how A was eliminated, its demiseshould leave the survivors in the same order as before intypical probabili st accounts of belief change.

This divergence of opinion will not be settled here. It isoffered as a factor to consider, and as a neededcounterbalance to the earlier recitation of probabili st-reasonableness results, all of which relate to what mightbe called the static cogency of the f() ordering.

A final point about necessity in the Polya domain is nota matter of reasonableness or normative dispute, butconcerns rather a unique property of the possibilit ycalculus which it does not share with necessity, nor withthe f() composition. The property will be the subject ofthe next section; its motivation belongs here with Polya.

A distinctive feature of Polya’s work is an ambivalenceabout the goals of inference. Obviously, themathematician, scientist, and jurist are all concerned withthe discovery of truth. But that is not the whole of theirjobs.

Polya also wishes to counsel mathematicians aboutwhat problems are worthwhile to work on. Discoveringthe truth of some implication of a conjecture not onlyencourages belief in its truth, but also establi shes that theconjecture has consequences, that it might explain thoseconsequences, that it is interesting.

For their part, jurists self-consciously adopt rules ofevidence which incorporate notions of fairness as well asprobative value. A revealing hearsay may be ruled out ofcourt not because it is uninformative, but because toadmit it is to compromise a defendant’s right to cross-examine witnesses against him.

Scientists sometimes engage in especiall y nuancedinferential episodes. Theories may be judged on theirtractabilit y and elegance along with their fidelit y to theexperimental record. Beauty may also be a factor inscientific thought (McAlli ster, 1998), both as a value inits own right, and as a heuristic guide to truth.

The conclusions scientists arrive at may also becomplicated, as evidenced by the survival in practice ofNewtonian mechanics and the simultaneous “acceptance”of incompatible theories (wave and particle models oflight as the occasion demands, or the unresolveddiscrepancies between general relativity and quantummechanics). This complexity is unsurprising in anenterprise which aspires to approach the truth, rather than(solely) to attain it.

Throughout the domain, then, a “good” conclusion isnot necessaril y determined by the truth of the matter.Merit is more a matter of preference (interestingness,usefulness, fairness, ...), unrebutted by the evidence ratherthan proven by it.

If this characterization of expert goal-setting practice isaccurate, then there would seem to be a role for aninferential calculus which served both preferential andmore ordinary credibilit y inference in the style ofLukasiewicz-Sugeno.

A Distinctive Feature of the PossibilityCalculus

The value of the Lukasiewicz-Sugeno insight is that itcaptures some of the intuiti ve force of what people meanwhen they speak of credibilit y, while at the same timepreserving a high degree of generalit y. Suppose one setout to look at preferential reasoning with a similar goal.

If we have a domain of mutually exclusive rewards, wesee immediately that preferential reasoning is unli keLukasiewicz-Sugeno. Offered the choice between

a commitment to be paid $5a commitment to be paid $5 or else $1

it is the stronger, rather than the weaker, commitmentwhich is preferred. If the issue were credibilit y (perhapsan onlooker wondering how much money will changehands), then Lukasiewicz-Sugeno favors the weakersentence (weakly).

But preference does not always follow logical strength,since between

a commitment to be paid $1a commitment to be paid $5 or else $1

things are more complicated. Realisticall y, it woulddepend on what you thought about the prospects foractuall y getting the $5 if you opted for the disjunctivecommitment.

Nevertheless, this homely example suggests onecandidate for a general description of a preferenceordering among sentences describing outcomes:

if A ⇒ B, then B >* A implies B¬A >* Aor equivalently,

if A ⇒ B, then A ≥* B¬A implies A ≥* BAs with Lukasiewicz-Sugeno, we might expect more froma practical calculus, but it is plausible that we would notbe content with less.

Although derived from an elementary observationabout preference, the relationship also echoes somethingof what is found in orderings of evidentiary support usingordinary conditional probabiliti es. It is easil y verified that

if A ⇒ B, then p( e | B ) > p( e | A ) impliesp( e | B¬A ) > p( e | A )

It is also interesting that even though the criterion is notoffered as a description of credibilit y, nevertheless, itstates an ordinal property of the ordinary Boolean logic,just as Lukasiewicz-Sugeno does.

In comparing the two kinds of ordering criteria

Lukasiewicz-Sugeno: if A ⇒ B, then B ≥* Apreference-support: if A ⇒ B, then A ≥* B¬A

implies A ≥* Bit is straightforward that a max rule for disjunctionssatisfies both criteria. It is only a bit more work to showthat max is the only rule which is a function of its

FLAIRS 2002 511

disjuncts and which relates stronger and weaker sentencesas required in a transiti ve ordering.

Sketch proof. If A ⇒ B, then B ≥* A byLukasiewicz-Sugeno. If B >* A, then B¬A >* Aimmediately from preference-support. If B =* A andB¬A >* A, then since B ≥* B¬A by Lukasiewicz-Sugeno, by transiti vity B >* A, contrary to thehypothesis.

In finite domains, transiti vity is easil y shown to imposethe max rule for implications on all comparisons ofdisjunctions.

Sketch proof. If a is the top atom in A, and b is thetop atom in B, then a =* A and b =* B by the maxrule for implications. Whatever order obtainsbetween a and b will obtain between A and B: A = a?? b = B.Possibilit y, the max calculus, stands alone as the only

calculus which is both Lukasiewicz-Sugeno andpreference-support. It is easil y confirmed that the f()composite ordering is not preference-support, and thus thecombination of calculi does not have the distinction thatpossibilit y alone has.

Of course, any mechanism for tie-breaking would forcethe new combined calculus to be either Lukasiewicz-Sugeno alone or preference-support alone (or perhapsneither). The ties are how the possibilit y calculusmanages to walk the tightrope between the all -but-confli cting criteria.

Although it is not the purpose of this short paper tooffer alternatives to necessity, it is interesting to note inpassing that if one wished to break possibili stic ties in thepreference-support direction, one would expect thatsomething very much li ke “quasi-additi vity” would be aplausible ingredient. The notion that two sentences A andB would be compared based on how they differ (that is,how A¬B stands with respect to B¬A) is a widely-sharedintuition about how preferences work.

In any case, possibilit y’s twin aspect combiningcredibilit y reasoning and preference-support reasoningmay offer a promising vehicle for excursions into Polya’srealm of incompletely formulated alternatives andambiguous inferential goals. It may even provide a usefulalternate formulation of Polya’s already hardly-additi ve“probabili stic” account of the territory.

Conclusions

The normative case in favor of necessity, both in its ownright and as a tie-breaking mechanism for possibilit y, isconsiderable. That a portion of that case arises fromoutside the fuzzy-possibilit y community is especiall ynoteworthy, and this portion forms the bulk of the casepresented here.

The paper is also frank about a principled disagreementamong scholars concerning an important feature of beliefrevision which the combination of necessity andpossibilit y fall s on one side of, while possibilit y alone can

be placed (in at least some interpretations) on the otherside.

Possibilit y and only possibilit y, unassisted by anymechanism for tie-breaking, gives rise to an uncertaintyordering which combines credibilit y and preferenceelements. Thus, unadorned possibilit y seems especiall ywell -suited for exploring an interesting and importantdomain which engaged George Polya.

That domain’s hostilit y to evaluation based oncomplementation further diminishes the appeal ofnecessity there, echoing Polya’s own conclusion about theusefulness of some aspects of conventional probabilit y.

The possibilit y calculus’ ties are what supports theabilit y to mirror the ambiguity of the usual notions ofmerit within the domain. On at least some occasions,then, the use of possibilit y without necessity has someattraction.

References

Benferhat, S., Dubois, D., and Prade, H. 1997,Possibili stic and Standard Probabili stic Semantics ofConditional Knowledge, in Proceedings of the AAAI, 70-75, Menlo Park, CA: AAA I Press.de Finetti, B. 1937, La Prévision, ses Lois Logiques, sesSources Subjectives, Annales de l’I nstitut Henri Poincaré;English translation by H.E. Kyburg, Jr. in Studies inSubjective Probabilit y, H.E. Kyburg and H. Smokler eds.1964, New York: John Wiley.de Finetti, B. 1951, La ‘Logica del Plausibile’ secondo laConcezione di Polya, in Atti della XLII Riunione dellaSocieta Italiana per il Progresso delle Scienze, 227-236.Dubois, D., Prade, H., and Smets, P. 1996, RepresentingPartial Ignorance, IEEE Transactions on System, Man,and Cybernetics 26:361-377.Dubois, D. and Prade, H. 1994, Fuzzy Sets - a ConvenientFiction for Modeling Vagueness and Possibilit y, IEEETransactions on Fuzzy Systems 2:16-21.Dempster, A.P. 1968, A Generali zation of BayesianInference, Journal of the Royal Statistical Society B 30:205-247 (with discussion).Kraus, S., Lehmann, D., and Magidor, M. 1990,Nonmonotonic Logic, Preferential Models andCumulative Logics, Artifi cial Intelli gence 44:167-207.McAlli ster, J.W. 1998, Is Beauty a Sign of Truth inScientific Theories?, American Scientist 86:174-183.Snow, P. 1999, Diverse Confidence Levels in aProbabili stic Semantics for Conditional Logics, Artifi cialIntelli gence 113:269-279.Snow, P. 2001, The Reasonableness of Possibilit y fromthe Perspective of Cox, Computational Intelli gence17:178-192.Polya, G. 1954, Mathematics and Plausible Reasoning,Princeton, NJ: Princeton University Press (two volumes).Zadeh, L.A. 1978, Fuzzy Sets as a Basis for a Theory ofPossibilit y, Fuzzy Sets and Systems 1:3-28.

512 FLAIRS 2002