Betting on Fuzzy and Many-valued Propositions

Transcription

1 Betting on Fuzzy and Many-valued Propositions Peter Milne 1 Introduction In a 1968 article, Probability Measures of Fuzzy Events, Lotfi Zadeh proposed accounts of absolute and conditional probability for fuzzy sets(zadeh, 1968). Where P is an ordinary( classical ) probability measure defined on a σ-fieldofborelsubsetsofaspace X,and µ A isafuzzymembershipfunctiondefinedon X,i.e.afunctiontakingvaluesintheinterval [0,1],the probabilityofthefuzzyset Aisgivenby P(A) = µ A (x)dp. X Thethingtonoticeaboutthisexpressionisthat,inaway,there snothing fuzzy aboutit.tobewelldefined,wemustassumethatthe levelsets {x X : µ A (x) α}, α [0,1], are P-measurable. These are ordinary, crisp, subsets of X. And then P(A)isjusttheexpectationoftherandomvariable µ A. Thisisentirely classical.ofcourse,youmayinterpret µ A asafuzzymembershipfunction butreallywehave,ifyou llpardonthepun,inlargemeasurelostsightof the fuzziness. Soyoumightask: isthistheonlywaytodefinefuzzyprobabilities? Theanswer,Ishallargue,isyes. Defining conditional probability Zadeh offered P(A B) = P(AB), when P(B) > 0, P(B)

2 146 Peter Milne where One might wonder: x X µ AB (x) = µ A (x) µ B (x). is this the only way to define conditional probabilities? Theanswer,Ishallsuggest,isno,itisnottheonlywaybutitistheonly sensible way. Zadeh assigns probabilities to sets. What I offer here, using Dutch Book Arguments, is a vindication of Zadeh s specifications when probability is assigned to propositions rather than sets.(but translation between proposition talk and set and event talk is straightforward. It s just that proposition talk fits better with betting talk.) 2 Bets and many-valued logics Iapply thedutchbookmethod,asjeffpariscallsit(paris,2001),to fuzzy and many-valued logics that meet a simple linearity condition. I shall call such logics additive. Additivity Foranyvaluation vandforanysentences Aand B v(a B) + v(a B) = v(a) + v(b) where and the conjunction and disjunction of the logic in question. Additivity is common: the Gödel, Łukasiewicz, and product fuzzy logics are all additive, as are Gödel and Łukasiewicz n-valued logics. InordertoemployDutchBookarguments,weneedabettingscheme suitably sensitive to truth-values intermediate between the extreme values 0and 1.Settingouttheclassicalcasetherightwaymakesonegeneralization obvious. Rather than betting odds, which are algebraically less tractable, we use, as is standard, a normalized betting scheme with fair betting quotients. Classically, with a bet on A at betting quotient p and stake S: thebettorgains (1 p)sif A; thebettorloses psifnot-a. Taking 1fortruth, 0forfalsity,and v(a)tobethetruth-valueof A,wecan summarise this scheme like this: thepay-offtothebettoris (v(a) p)s.

3 Betting on Fuzzy and Many-valued Propositions 147 Andnowweseehowtoextendbetstothemanyvaluedcase:weadoptthe sameschemebutallow v(a)tohavemorethantwovalues.thesloganis: thepay-offisthelargerthemoretrue Ais. 1 Using this betting scheme, we obtain Dutch Book arguments for certain seemingly familiar principles of probability, seemingly familiar in that formally they recapitulate classical principles. 0 Pr(A) 1; Pr(A) = 1when = A; Pr(A) = 0when A =; Pr(A B) + Pr(A B) = Pr(A) + Pr(B). Here and are the conjunction and disjunction, respectively, of an additive fuzzy or many-valued logic. Other principles that may or may not be independent, depending on the logic: Pr(A) + Pr( A) = 1when v( A) = 1 v(a); Pr(A) xwhen,underallvaluations, v(a) x; Pr(A) xwhen,underallvaluations, v(a) x; Pr(A) Pr(B)when A = B. I ll show how two of the arguments go as there s an interesting connection with the standard Dutch Book arguments used in the classical, two-valued case. Welet xrangeoverthepossibletruth-values(whichalllieintheinterval [0,1]).Clearly,forgiven p,wecanchooseavalueforthestake Sthatmakes G x = (x p)s negative,forallvaluesof xintheinterval [0,1],if,andonlyif, pislessthan 0 or greater than 1. Hence 0 Pr(A) 1. 1 Thesuggestedpay-offschemeis,ofcourse,onlythemoststraightforwardwaytoimplement the slogan. One could distort truth values: take a strictly increasing function f : [0, 1] 2 [0, 1]with f(0) = 0, f(1) = 1,andtakepay-offstobegivenby (f(v(a)) p)s. Analogously,Zadehcouldhavetaken X f(µa(x))dptodefinedistortedprobabilities. Andthepointisthatsuch probabilities aredistortedforwhen fisnottheidentity functionitmaybethat P(A) < ceventhough µ A(x) > c,forall x X.

4 148 Peter Milne Sofarsogood,buthere sthecutebit: G x = xg 1 + (1 x)g 0, so G x isnegativeforallvaluesof x [0,1]if,andonlyif, G 1 and G 0 areboth negative. From the classical case, we know that the necessary and sufficient conditionforthelatteristhat plieoutsidetheinterval [0,1].Itsufficesto lookattheclassicalextremestofixwhatholdsgoodforalltruth-valuesin the interval [0, 1]. Next we consider four bets: 1.abeton A,atbettingquotient pwithstake S 1 ; 2.abeton B,atbettingquotient qwithstake S 2 ; 3.abeton A B,atbettingquotient rwithstake S 3 ; 4.abeton A B,atbettingquotient swithstake S 4. Weassumethatforallallowedvaluesof v(a)and v(b), v(a B) + v(a B) = v(a) + v(b) and v(a B) min{v(a),v(b)}. Then,where x, y,and zarethetruth-valuesof A, Band A Brespectively, the pay-off is G x,y = (x p)s 1 + (y q)s 2 + (z r)s 3 + ((x + y z) s)s 4. Thiscanberewrittenas G x,y = zg 1,1 + (x z)g 1,0 + (y z)g 0,1 + (1 x y + z)g 0,0. Theco-efficientsareallnon-negativeandcannotallbezero.Thus G x,y is negative,forallallowable x, y,and z,justincase G 1,1, G 1,0, G 0,1,and G 0,0 areallnegative.fromthestandarddutchbookargumentforthetwovalued,classicalcase,weknowthistobepossibleif,andonlyif, p+q r+s. Hence Pr(A B) + Pr(A B) = Pr(A) + Pr(B). 3 The classical expectation thesis for finitely-many-valued Łukasiewicz logics AsaninitialvindicationofZadeh saccount,wefindthatinthecontext of a finitely-many-valued Łukasiewicz logic, all probabilities are classical expectations. That is, the probability of a many-valued proposition is the

5 Betting on Fuzzy and Many-valued Propositions 149 expectation of its truth-value and that a proposition has a particular truthvalue is expressible using a two-valued proposition. So in this setting, in analogy with Zadeh s assignment of absolute probabilities to fuzzy sets, all probabilities are expectations defined over a classical domain. In all Łukasiewicz logics, conjunction and disjuction are evaluated by the functions max{0,x + y 1}and min{1,x + y},respectively. Employing Łukasiewicz negation and one or more of Łukasiewicz conjunction, disjunction, and implication, one can define a sequence of n + 1 formulasofasinglevariable, J n,0 (p),j n,1 (p),...,j n,n (p),whichhavethis property(rosser& Turquette, 1945): in the semantic framework of (n + 1)- valuedłukasiewiczlogicitisthecasethatforeveryformula A,forall k, 0 k n,andforeveryvaluation v, v(j n,k (A)) = 1,if v(a) = k n ; v(j n,k (A)) = 0,if v(a) k n. In the semantic framework of (n + 1)-valued Łukasiewicz logic, for all sentences A, = J n,0 (A) Ł J n,1 (A) Ł Ł J n,n (A) and J n,i (A) Ł J n,j (A) =, 0 i < j n. (*) From the probability axioms, we have, for all sentences A, that Pr(J n,i (A)) = 1. 0 i n Thepropositionsoftheform J n,i (A)aretwo-valued,so, (n + 1)-valued Łukasiewiczlogicreducingtoclassicallogiconthevalues 0and 1,thelogic of these propositions is classical. Thus, when restricted to these propositions and their logical compounds, the probability axioms give us a classical, finitely additive, probability distribution. What we show next is that this classical probability distribution determines the probabilities of all propositions in the language. Theorem 2(Classical Expectation Thesis). In the framework of (n + 1)- valued Łukasiewicz logic, Pr(A) = 1 n 0 i n ipr(j n,i (A)). Proof.From(*)andthetwo-valuednessofthe J n,i (A) swehave A = = (A Ł J n,0 (A)) Ł (A Ł J n,1 (A)) Ł Ł (A Ł J n,n (A)).

6 150 Peter Milne From our probability axioms it follows that logically equivalent propositions must receive the same probability, so Pr(A) = Pr(A Ł J n,i (A)). ( ) 0 i n Weconsidertwobets,oneon A Ł J n,k (A)atbettingquotient pand stake S 1,theotheron J n,k (A)atbettingquotient qwithstake S 2. The pay-offs are: G = k n G k n = ( ) k n p S 1 + ((1 q)s 2 ) when Ahastruth-value k n, = ps 1 qs 2 when Ahastruth-valueotherthan k n. Setting ( S 2 ) = k n S 1givesapay-off,independentofthetruth-valueof A,of qk n p S 1,whichcanbemadenegativebychoiceof S 1 provided p qk n.ontheotherhand,forarbitrary S 1and S 2,when p = qk n thetwo pay-offs are [ ] k G = k = (1 q) n n S 1 + S 2 when Ahastruth-value k n, and [ ] k G k = q n n S 1 + S 2 when Ahastruth-valueotherthan k n. These cannot both be negative. Hence Substituting in( ), we obtain: Pr(A Ł J n,k (A)) = k n Pr(J n,k(a)). Pr(A) = 1 n 0 i n ipr(j n,i (A)). Two comments Firstly, having been obtained by an independent Dutch Book argument, the Classical Expectation Thesis may seem to be an additional principle. In fact itisnot;itisderivablefromouraxiomsforprobability.toshowthiswehave to introduce a propositional constant, introduced into Łukasiewicz logic by Słupecki in order to obtain expressive completeness(słupecki, 1936).

7 Betting on Fuzzy and Many-valued Propositions 151 Inthesemanticsof (n + 1)-valuedŁukasiewiczlogic,inwhichallformulasareassignedvaluesintheset { 0, 1 n, 2 n 1 n,..., n,1},thepropositional constant t has this interpretation: underallvaluations v, v(t) = n 1 n. Let t 1 bethe (n 2)-fold Ł -conjunctionof twithitself. For 1 < k n, let t k bethe (k 1)-fold Ł -disjunctionof t 1 withitself. v(t 1 ) = 1 n and v(t k ) = k n.sincewehave t k Ł t 1 =, 1 k < n, and = t n, from our probability axioms we obtain: Pr(t k ) = k Pr(t 1 ), 1 k n, and Pr(t n ) = 1, hence Pr(t k ) = k n, 1 k n. Usingthe t i swecanderivetheclassicalexpectationthesis.(i llskipthe details here.) Secondly, the Dutch Book argument for the Classical Expectation Thesis goes through with any notion of conjunction for which v(a&b) = v(a) when v(b) = 1and v(a&b) = 0when v(b) = 0. Also,the J n,i (A) s being truth-functional, the Classical Expectation Thesis holds good of every proposition in the semantic framework, not just those expressible using the Łukasiewicz connectives. 4 The extension to infinitely many truth-values(a sketch) Foranyrationalnumber xintheinterval [0,1],thereisaformula φ(p)ofa single propositional-variable p, constructed using Łukasiewicz negation and any one or more of Łukasiewicz conjunction, disjunction, or implication, suchthat,underanyvaluationtakingvaluesin [0,1], v (φ(a/p)) = 0if v(a) x and v (φ(a/p)) > 0 otherwise(mcnaughton, 1951). EmployingtheGödelnegation, 2 then,wehave, 2 TheGödelnegationis,tobesure,notusuallytakentobepartofthevocabulary of the Łukasiewicz logics. Semantically, however, it can be defined in the Łukasiewicz fuzzy/many-valued frameworks as the external negation that maps 0 to 1 and all other valuesto 0.

8 152 Peter Milne foreachinterval [0,x]with xrational,aformula J [0,x] (A)thattakes thevalue 1underanyvaluation vforwhich v(a) xandotherwise takesthevalue 0; foreachhalf-openinterval (x,y]withrationalendpoints xand y, x < y,aformula J (x,y] (A)thattakesthevalue 1underavaluation vwhen v(a) (x,y]andotherwisetakesthevalue 0. Givenastrictlyincreasing,finitesequence x 0,x 1,...,x n 1 ofrational numbersintheopeninterval (0,1),considerthefamilyof n + 1bets: abeton Aatbettingquotient qwithstake S; abeton J [0,x1 ](A)atbettingquotient p 1 withstake S 1 ; abeton J (xi 1,x i ](A)atbettingquotient p i withstake S i, 1 < i < n; abeton J (xi,1](a)atbettingquotient p n withstake S n. 2 i n x i 1 Pr(J (xi 1,x i ](A)) Pr(A) x 1 Pr(J [0,x1 ](A)) + 2 i n x i Pr(J (xi 1,x i ](A)), where x n = 1.Sobytakingfinerandfinerpartitionswecanmoreclosely approximate the probability of A from above and below. This may not quite dotofix Pr(A)exactly. Forthatwemayalsoneedtheprobabilitiesofat most a countable infinity of(two-valued) statements of the form v(a) x where xisanirrationalnumber. 3 Withtheseinhand,wethenfindthat Pr(A) = 1 0 xdf A (x), where F A istheordinary, classical distributionfunctiondeterminedby theprobabilitiesofthe J [0,x] (A) s, J (x,y] (A) sandhowevermany v(a) x s with x irrational we have used. By introducing a countably infinite family of logical constants, we can derive this classical representation from the previously given principles of probability together with the principle 3 RecallZadeh sassumptionregardingthe P-measureabilityof levelsets.

9 Betting on Fuzzy and Many-valued Propositions 153 foranyproposition Alogicallyconstrainedtotakeonlythevalues 0 and 1andforrationalvaluesof xintheinterval [0,1], Pr(t x A) = xpr(a), where t x takesthevalue xunderallvaluations v. The really neat feature of infinitely many-valued Łukasiewicz logics is that this principle is derivable from the basic principles 0 Pr(A) 1; Pr(A) = 1when = A; Pr(A) = 0when A =; Pr(A Ł B) + Pr(A Ł B) = Pr(A) + Pr(B). 5 Conditionalprobabilities Intheclassicalsetting,abeton Aconditionalon Bisabetthatgoesahead if,andonlyif, Bistrueandisthenwonorlostaccordingastowhether Aistrueornot. Thepay-offsforsuchaconditionalbetwithstake Sat betting quotient p are: thebettorgains (1 p)sif Aand B; thebettorloses psifnot-aand B; thebettorneithergainsnorlosesifnot-b. We can summarise this betting scheme like this: v(b)(v(a) p)s. Andso,aswithordinarybets,wenowknowonewaytoextendthe scheme for conditional bets on classical, two-valued propositions to manyvalued propositions. A straightforward Dutch Book argument, which again piggy-backs on theproofinthetwo-valuedcase,thentellsusthat where Pr(A B) = Pr(A B) Pr(B) v(a B) = v(a) v(b). Allowingforthechangeofsetting,justwhatZadehsaid.

10 154 Peter Milne Youcan,ifyouaresominded,generalizetheclassicalschemeusingany many-valued or fuzzy conjunction that is classical at the extremes : (v(a B) v(b)p)s. ADutchBookargument inallessentials,thesamedutchbookargument will then deliver: Pr(A B) = Pr(A B) Pr(B). However, Pr( B) satisfies the axioms for an absolute probability measure onlywhentheproductconjunction, isused. 4 Peter Milne Department of Philosophy, University of Stirling Stirling FK4 9LA, United Kingdom peter.milne@stir.ac.uk References McNaughton, R.(1951). A theorem about infinite valued sentential logic. Journal ofsymboliclogic,16,1 13. Paris,J.(2001).Anoteonthedutchbookmethod.InG.DeCooman,T.Fine, & T. Seidenfeld(Eds.), ISIPTA 01, Proceedings of the second international symposium on imprecise probabilities and their applications, Ithaca, NY, USA(pp ). Maastricht: Shaker Publishing.(A slightly revised version is available on-line at jeff/papers/15.ps) Rosser, J.,& Turquette, A.(1945). Axiom schemes for m valued propositional calculi. Journal of Symbolic Logic, 10, Słupecki, J.(1936). Der volle dreiwertige Aussagenkalkül. Comptes rendues des séancesdelasociétédessciencesetlettresdevarsovie,29,9 11. Zadeh, L. A.(1968). Probability measures of fuzzy events. Journal of Mathematical Analysis and Applications, 23, Beyondtheclassical,two-valuedcase,productconjunctionrequiresthattherebean infinity of truth-values.