A C T A U N I V E R S I T A T I S L O D Z I E N S I S ____ FOLIA OECONOMICA 152, 2000
G r z e g o r z M a l i n o w s k i *
L O G IC A L M A N Y -V A L U E D N E S S V E R S U S P R O B A B IL IT Y
Abstract. The aim of the papers is to present and discuss the most direct issues on relation between logical many-valuedness and logical probability i.e. probability related to propositions. Having introduced the reader into the realm of many-valued logics, we outline two faces of the problem. One is that logical values must not be identified with the probability values, the other concerns the so-called subjective probability which, as shown by Giles, may be interpreted within the infinite-valued logic o f Lukasiewicz. ’ T he m athem atical probability calculus in its simplest form resembles m any- -valued logic. Therefore, the question o f a connection between probability and m any-valuedness em erges qu ite naturally. T h e aim o f the p ap e r is to present and discuss the m ost direct issues on relation between logical m any-valuedness and logical probability i.e. the probability related to propositions. Section 1 is a sh o rt in tro d u ctio n in to the realm o f logical m any-valuedness. Section 2 is devoted to the three-valued Lukasiew icz logic, which serves as a p re p a ra to ry exam ple for the sequel. In Section 3 we present the argu m en ts show ing th a t logical values o f any m any-valued logic m u st no t be identified w ith the probability values. Section 4 provides an overview o f an ingenious construction by Giles showing the way in which the so-called subjective probability m ay be interpreted w ithin the infinite-valued logic o f Lukasiew icz.
I. PRINCIPAL MOTIVATIONS FOR MANY-VALUED LOGIC
J he assu m p tio n stating th a t to every p ro p o sitio n it m ay be ascribed exactly one o f the tw o logical values, truth o r falsity, called the principle o f bivalence, constitutes the basis o f classical logic. It determ ines b o th the subject m a tte r and the scope o f applicability o f th e classical logic an d it found its expression th ro u g h the tw o h o n o u re d logic laws: law o f the excluded middle,
( E M ) p \ —\p an d the principle o f contradiction,
(CP) —| ( р л- 1 p).
G iven the classical u n d ersta n d in g o f the logical connectives, E M and CP m ay be read as stating th a t o f the tw o p rop o sitio n s p and - 1 p: a t least one is tru e and a t least one is false, respectively.
T h e m o st n a tu ra l and straig h tfo rw ard step beyond the tw o-valued logic is th e in tro d u ctio n o f m o re logical values, rejecting sim ultaneously the principle o f bivalence. T h e ro o ts o f m any-valued logics can be traced back to A risto tle (4th century BC) w ho considered f ut ure contingents sentences like
“T h ere will be a sea-battle to m o rro w ” .
T h e P hilosopher from S tagira em phasizes the fact th a t such sentences describing accidental events are neither actually tru e n o r actually false. C on sequently, he suggests th a t there is a th ird logical statu s o f prop o sitio n s.
T h e p rehistory o f m any-valuedness falls on th e M iddle Ages an d first serious attem p ts to create three-valued logical co n stru ctio n s ap p eared at the tu rn o f the X lX th century. T he final thoroughly successful form u latio n o f the three-valued logic was m ad e by L u k a s i e w i c z in 1913, see L u k a s i e w i c z (1920). Independently P o s t (1920) introduced a family o f finite-valued logics. Finally, tw o years later Lukasiew icz constructed logics having infinitely m any logical values, see Section 4. N ow adays, the area o f m any-valued logic is an a u to n o m o u s field o f investigation, see e.g. M a l i n o w s k i (1993).
2. THREE-VALUED LOGIC OF LUKASIEWICZ
T h e actual in tro d u ctio n o f a third logical value next to tru th and falsity, was preceded by th o ro u g h philosophical studies o f the problem s o f induction and the theory o f probability. Lukasiewicz, a fierce follower o f indeterm inism , finally introduced the third logical value to be assigned to n on -d eterm ined pro positions; specifically, to p ro p o sitio n s describing casual fu tu re events, i.e. fut ure contingents. L u k a s i e w i c z (1920) refers to A ristotle fut ure contingents and analyses the sentence: “ I shall be in W arsaw at n o o n on 21 D ecem ber o f the next year” . H e argues th a t at the m om ent o f the utterance this sentence is n either tru e n o r false, since otherw ise w ould get fatalist conclusions ab o u t necessity o r im possibility o f the contin g en t fu tu re events.
A t the early stage Lukasiew icz interp reted the th ird logical value as “ possibility” o r “ indeterm inacy” . F ollow ing intu itions o f these co ncepts, he extended the classical in terp re tatio n o f negation and im plicatio n in the follow ing ta b le s1:
X — X — 0 1/2 1
0 1 0 1 1 1
1/2 1/2 1/2 1/2 1 1
1 0 1 0 1/2 1
T h e o th e r connectives o f disjunction, co n ju n ctio n an d equivalence were introdu ced th ro u g h the sequence o f the follow ing definitions:
a v /? = ( « - > /? ) - > / ? ,
а л /? = —i ( —i a v —i /?),
<x = /J = ( а - * 0 ) л ( ß - * o t ) . T h eir tables are as follows:
V 0 1/2 1 л 0 1/2 1 = 0 1/2 1
0 0 1/2 1 0 0 0 0 0 1 1/2 0
1/2 1/2 1/2 1 1/2 0 1/2 1/2 1/2 1/2 1 1/2
1 1 1 1 1 0 1/2 1 1 0 1/2 1
A v a lu a tio n o f fo rm u las in the th ree-v alu ed logic is an y fu n c tio n v: F o r —* {0, 1/2, 1} com patible with the above tables. A tautology is a form ula which under any valu atio n v takes on the designated value 1.
T h e set L 3 o f tautologies o f three-valued logic o f L ukasiew icz differs from T A U T . So, for instance, neither the law o f th e excluded m idd le, n o r the principle o f co n trad ictio n is in i 3. T o see this, it suffices to assign 1/2 for p: any such valuation also associates 1/2 w ith E M and CP. T he th o ro u g h -g o in g re fu tatio n o f these tw o laws was in ten ded , in L ukasiew icz’s opinion, to codify the principles o f indeterm inism .
T o close up it w ould be in o rd e r to add th a t m o st o f the three-valued p ro p o sitio n al logics are com patible with the Lukasiew icz logic having the sam e ch a rac te rizatio n (the table) o f d isju nctio n an d, m o st often, also the tab le o f negation.
1 The truth-tables of binary connectives • are viewed as follows: the value of a is placed in the first vertical line, the value of ß in the first horizontal line and the value of a «/j at the intersection of the two lines.
3. NON-CLASSICAL LOGICAL VALUES AND PROBABILITY OF PRPOSITIONS
It is interesting to note th a t still before the construction o f his three-valued logic, L ukasiew icz classified as undefinite the p ro p o sitio n s with free nom inal variables an d assigned to them fractional “ logical” values ind icating the p ro p o rtio n s between the n u m b er o f actual variable values verifying a p ro p ositio n and the n um ber o f all possible values o f th a t variable. C learly, only finite dom ain s were adm itted and values were relative: Lukasiew icz values depend o n the set o f individuals actually evaluated. So, for exam ple, the value o f the p ro p o sitio n ‘jc2 — 1 = 0’ am o u n ts to 1/2 in the set { — 1, 0} and to 2/3 in the set { — 1, 0, 1}.
T h e m ath em atical prob ab ility calculus in its sim plest form resem bles m any-valued logic. T herefore, the question o f a conn ection betw een p ro b ability and m any-valuedness em erges q u ite n aturally. L u k a s i e w i c z (1913) invented a th eo ry o f logical p robability . T h e differen tiating feature o f thu s com prehended probability is th a t it refers to p ro p o sitio n s and no t to events. T h e co n tin u ato rs o f L ukasiew icz’s co ncep tio n, R eich en bach and Z aw irski am o n g them , exerted m u ch effort to create a m any -v alu ed logic w ithin w hich logical p ro b ab ility could find a satisfactory in te rp re ta tio n , see e.g. Z a w i r s k i (1934a), (1934b), R e i c h e n b a c h (1935). T h e R eichcn- bach-Z aw irski conception is based on the assu m ption th a t th ere is a fu n ction Pr ranging over the set o f p ro positions o f a given st andard p ro p o sitio - nal language, with values from the real interval [0, 1]. T h e p o stu lates for Pr are:
P I. 0 < P r ( p ) < l , P2. Pr(j) v - i p ) = l,
P3. Pr(p v q) = Pr(p) + Pr(q) if p an d q a re m u tu a lly exclusive (Pr(p л q) = 0,
P4. Pr(p)Pr(q) when p and q are logically equivalent.
F ro m P 1 -P 4 it is possible to infer o th er expected pro perties o f Pr. If then we identify the logical value v(p) with the m easure o f prob ab ility Pr{p) th en for Pr(p) = 1/2 from the properties m ention ed we w ould get th a t
1/2 v 1/2 = Pr(p v —i p) = 1 and 1/2 v 1/2 = Pr(p v p) = Pr(p) — 1/2.
C onsequ ently, logical p ro b ab ility m u st n o t be identified with logical values o f any o rd in ary extensional m any-valued logic.
4. INFINITE-VALUED LUKASIEWICZ LOGIC AND SUBJECTIVE PROBABILITY
In 1922 Lukasiew icz generalizes his logical co n stru ctio n and defines the fam ily o f fin ite n-valued logics h av in g as th e ir values th e sets {0, 1} and tw o infinite-valued logic: X0- and K r valued. T h e first is based on the set o f all ratio n al n u m bers o f th e interval [0, 1] and the second the whole real interval [0, 1].
T h e functions co rresponding to the connectives are defined in all these system s, including infinite-valued, by the follow ing form ulas:
(i) —i x = 1 - X,
x - + y = m in (l, 1 - x + y),
(ii) x v y = ( x - + y ) - * y = m a x ( x , y), x л у — —i ( —i jc v —iy ) = m in (x, y), х = у = ( х - * у ) л ( у - * х ) = l - \ x - y \ .
T h e in tro d u ctio n o f new m any-valued logics was n o t su p p o rted by any sep a rate arg u m e n tatio n - Lukasiew icz did n o t give new reason s fo r the choice o f m ore logical values. It w ould be, how ever, easy to see, th a t these g eneralizations were correct: for n = 3 one gets exactly the m a trix o f his
1920’ three-valued logic.
T he researches o f Giles in the early 1970’s directed tow ards finding a logic a p p ro p ria te fo r the fo rm alization o f physical theories, q u an tu m m echanics including, resulted in a very convincing in terp re tatio n o f th e N0-valued L ukasiew icz logic, see G i l e s (1974). T h e m ain p o in t o f G iles’ ap p ro ach consists in the so-called dispersive physical in terp re tatio n o f stan d ard logical language: each prime proposition in a physical theory is associated throug h the rules o f interpretation with a certain experimental procedúre which ends in one o f the tw o possible outcom es, “ yes” and “ n o ” . T h e tangible m ean in g o f a pro p o sitio n is related to the observers and expressed in term s o f subjective probability. In the case o f prim e propositions it is determ ined from the values o f prob ab ility o f success ascribed by observers in respective experim ent, w hereas in the case o f com pound propositio ns it is determ ined from the rules o f o bligation form ulated in the dialogue logic (see L o r e n z (1961). T h e fo rm alizatio n starts w ith an assu m p tio n th a t
(*) all prime propositions are definite f o r all speakers ( observers) taken into consideration
and th a t speakers are com m itted to pay certain sum o f m o ney for every single assertio n o f a prim e pro p o sitio n , w hen the experim ent associated with it results in “ n o ” . T h e m eaning o f com pound p ro p o sitio n s is th en ap po in ted
by the rules o f d eb ate o f tw o participants: a given person an d th eir p a rtn e r w ho can be a fate as well. T h e rules o f ob ligation generate a gam e, which starts with an utteran ce o f a com pound p ro po sition. F o r the stan d ard connectives they are the following:
A ssertion O bligation (C om m itm ent)
p v q u n d erta k in g to assert either p o r q a t o n e’s ow n choice p л <7 undertaking to assert either p o r q at the o pp on ent’s choice p —*q agreem ent to assert q if the o p p o n en t will assert p
—i p agreem ent to pay SI to o p p o n en t if they will assert p.
Giles tran slates subjective prob ab ility into “ risk values” : assigning to prim e prop o sitio n s risk values is a valuation. S ubsequently, he em ploys results o f gam e theory and show s th a t each valu atio n o f prim e p ro p o sitio n s h as an unique extension o n to the whole language g u aranteeing b o th p artic ip a n ts no increase in the risk value o f the initial po sitio n (a fo rm u la whose u tteran ce starts the game). T hus, G iles establishes th a t for every fo rm u la and each p artic ip a n t an optimal strategy exists.
T h e risk value function < > is defined for any form ulas a, ß th ro u g h the schemes:
< a-+ /?> = m ax {0, <0> - <a>, <a v /?> = m in {<a>, </?>},
<a л /Í = m ax {<a>, </»>}, < —ia> = 1 — <a>.
T h e form ulae co rresp onding to p ropositio ns, the u tteran ce o f w hich m ay lead only to no t losing final positions, are referred to as tautologies. N ow , using o f the equality
pr(a) = m in { l, 1 - < « > }
one m ay change risk value associations with the subjective p ro b a b ility valuations:
p r ( a —►/?) = m i n { l , 1 - pr (a) + pr (ß)}, p r ( a v /J) = m ax {p r(a), pr(ß)},
p r ( a л ß) = m in {p r(a), pr(ß)}, p r ( - ia ) = 1 - p r ( a).
A m o m e n t’s reflection show s th a t pr if a v a lu a tio n o f N 0-valued Lukasiew icz logic an d , therefore, the set o f tautolog ies o f G iles’ dialogue logic coincides with the set o f tautologies o f Lukasiewicz logic. In probabilistic term s the prop erty o f being the tau to lo g y is the p rop erty o f th o se fo rm u las w hose prob ab ility am m o u n ts to 1 independently o f th e values assigned to prim e p ropositions as its com ponents.
5. CONCLUSION
T h e considerations in Section 3 show th a t logical p ro b ab ility i.e. the p ro b ab ility associated to p ropo sitions m ust n o t be identified w ith logical values o f any ordin ary extensional m any-valued logic. O n the o th er h an d , the results by Giles open new possibilities. T hey show th a t the so-called subjective prob ab ility (o f a speaker) associated with events an d verified by elem entary experim ents found a satisfactory in terp re tatio n as logical value o f the infinite-valued Lukasiew icz logic.
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