The Relationship Between Knowledge, Belief, and Certainty: Preliminary Report
.Joseph Y. Ha.lpern IBM Almaden Resea.rch Center San .Jose, CA 95120 [email protected] (ARPA/CSNET) [email protected] (BITNET)
Abstract: We consider the relation between lt is well known that we can capture different co knowledge and certainty, where a fact is known if it notions of knowledge by varying the nditions on is true at all worlds an agent considers possible and the acceuibility relation which defines the set of rob is certain if it holds with p ability 1. We iden worlds that an agent consid ers poRsiblc (sec [HM85) tify certainty with probabilistic belief. We show for an overview). One particular set of require that if we assume one fixed probability assignment, ments on the accessibility relation, namely tnat it then the logic KJH5, which has been identified as be uri a/, Eudidr.an, and tran�itivc (we define these c perhaps the most appropriate for belief, provides terms helow) resu lts in the logi K015 which has o a complete axiomatization for reasoning about cer been considered the logic most appr priate for be tainty. Just as an agent may believe a fact although lief [Lev81a, FH88a).
IP i!l false, he may be certain that a fact tp is true al We can also give a probabilistic interpretation to s though IP is false. However, it is ea y to see t.hat an belief. The greater the probability of'{' (according agent can have such false (probabilistic) beliefs only to an agent's subjective probability function), the rob b at a set of worlds of p a ility 0. 1£ we restrict at stronger an agent's belief is in IP· ln this paper, we tention to structures where all worlds have ositive p identify certainty - where an agent is said to be cer probabil ty, then S5 prov des a complete axiomati i i tain of tp if IP hold!! with probability 1 - with prob zation . If we consider a more general setting, where abilistic belief. We show that such an identification r there might be a different p obability assignment at is well motivated: If we have one fixed probability each world, then by pl acing appropriate cond itions assignment on the set of possible worlds, then cer on the function set .mppnrt of the probability (the tainty satisfieR precisely the axioms of KD1.5. Jn of worlds which have non -zero probability , we can ) KIH5, an agent may hold false beliefs; i.e. , he may capture man y otl1er wdl-known modal logics, such believe a fact IP that is fahe. Similarl y, an agent c as T and S-1. Finally, we onsider which axioms may be certain about a fact which is false. How characteri7.e structures satisfying llfi{{cr'·' principle. ever, we show that an agent can have such false be liefs only at worlds with probability 0; i.e. , almost surely, his (probabilistic) beliefs arc correct. If we
restrict attention to structures where all possible worlds have non-7.cro probability, then S5 gives a 1 Introduction complete axiomati7-al.ion for certainty: an agent no longer can have false beliefs.
II. great deal of interest has focussed recently on We can extend these re�wlts by considering more logics of knowledge and probability (sec, for exam general probability structures, where the agent may ple, the volumes [llal86, Var88, KL86, KL87]). Re have a different probability function at each state s searchers have used the possible-worlds approach of the world . .Ju t M different axioms for knowledge n to give semantics to knowledge hy saying an agent can be captured by placi g appropriate conditions f.now.� a fad <.p if tp is true at all the worlds the agent on the set of worlds an agent considers possible, so com1idcr!l possible. We can also give !lcmantics to different axioms for certainty can he captured by c formulas involving probability in a possible-worlds pla ing appropriate conditions on the support of a framework by s ying tp hold!! with probability a if the probability f•1nction, that is, the set of worlds cro the set or worlds when� !p is true is a set of proba� to which the probability function assigns non-7: as s bility Ci. m e u re. Indeed, we how that many other wel1-
142 known modal logics, such as T, 04, and S4, corre 2 Reasoning about probability spond in a natural way to conditions on the sup port. We are interested in making statements about cer tainty; that is we would like a logic that allows formulas of the form "The probability of cp is 1." This is not t e first paper to consider the rela h In order to accommodate such statements we start tionship between knowledge, belief, and certainty; with a more general logic, essentially th�t consid Gaifman [Gai86) and Frisc and Iladdawy [FH88c) h ered in [FHM88, FH88b]. In this logic, statements also consider these issues. Both of these papers of the form w(�p) � 1/2 and w(cp)< 2w(7Jr) are al focu:; on structures that satisfy Miller'� principle lowed, which can be interpreted as "the probability [Mil66 , Sky80b] (this principle is discussed in detail of cp is greater than or equal to 1/2" and "the prob l ter). Gaifman [Gai86 shows that the valid formu a ] i ability of cp s less than twice the probability of '�-'•·'· las of S5 are precisely those which hold with prob respective. I y. More generally, linear combinations of ability 1 in his logic (when restricted to structures expressions involving probability are allowed. satisfying MiUer's principle), while Frisch and Had The formal syntax of the logic is quite straight da.wy IFH8!k} argue that tl1e valid formulas of the forward. Well-formed formulas are formed by modal logic 04 are precisely those that hold with starti!lg with primitive propositions, and closing probability 1 in their logic (which is also intended off under Boolean connectives (conjunction and to capture Miller 's principle). We show that in our negation ), as well as allowing weir�ht formulas of framework, there is a precise sense in which KD45 the + ... + a.�:w(cpt) 2: b, where characterizes certainty in those structures satisfy form a1w( 143 infinite set of state., or po.ui6le worlds, ?r associates those of the form w ( r.p) = 1 (with nesting allowed); with e very state ·' E S a truth assignment 1r{s ) we abbreviate such a formula as Cetl(rp). We caU on the primitive propositions (so that ?r(s)(p) is ei this sublanguage .cc. Thus, a typical formula of ther true or false for every primitive proposition .cc is --.q A Cert(·C�rt(p) A Cert(q)). p and state ·' E S), and pr is a. discrete probability function on S (so that pr( .'!) � 0 for each s E S 3 Reasoning about knowledge and ·' = ). We can tl1 nk of as being L •ES pr( ) l i pr a n nt to the agent's subjective probability ssig me the The possible-worlds model can also be used to cap S. As usu , every worlds in al for subset A � S, we ture reasonitJg about kllowledge. We brieHy review define pr(A):::: pr(.• We have restric ted S LreA ). the necessary ideas here; the interested reader is l1ere to he cmmt.able and to be a discrete proba pr referred to [HM85) for more details. bility function for case of exposition. We discuss in The intuitive idea is that an agent knows tp if r.p is Section 6 how our results can be extended to more true in all the worlds the agent considers possible. general setting�. For now, we restrict our discussion to a situation in We can now define the satisfaction relation 1=, volving only one agent; in Section 6 we discuss how where is read is true (or (N, . 144 the binary relation K, since by restricting A.: we true, T hM been thought to be inappropriate for can capture a number of interesting properties of these applications; thus, KD45 is considered, for knowledge. Recall that a binary relation K on S example, in [Lev81aJ. KD15 is also considered to is refle:xivr. if (.•, -') E K for all ·' E S, lra.ruitive if be an appropriate logic for ch.araderi7.ing the be ( s, t) E K and (t, u) E K implies (s, v.) E K, .•ym liefs of an agent who might believe things that in mclric if ( .•, I) E K implies ( t, s) E K, Euclidean if fact turn out to be false [FH88a, Lev84h]. (.,,t) E and (s,u) E /C: implies (t,u) and K E K, We say that an axiom system A is Jound with re urial if for all s S, there is sorrte t such that E spect to a class of (knowledge or probability) struc s, t E K. Let M be the class of all knowledge ( ) tures Q if all the axioms in A are valid with respect structures. \11/e restrict M by using superscripts r, to Q and the rules of inference preserve validity; A . , e, and to denote reHcx:ive, symmetric, transi , l, l is complete with respect to a class Q if all the valid tive, Euclidean, and serial structures, respectively. formulas in Q are provable using the axioms and Thus, M'1 denotes the cla."s of all reflexive and rules of inference of el! A. transitive knowledge structures, M denotes the It turns out that there is a close connection be class of Euclidean, serial, and transitive structures, tween condition!! placed on K and the axiom�. In and so on. particular, T corresponds to K being reflexive, 4 to Consider the following collection of axioms: K being transitive, 5 to /C being Euclidean, and D to K being serial. To make this precise, we define P All instances of axioms of proposlllonallogic an axiom system A to be normal if it consists of K (/(cpA K(cp :::> 1/;)) =>[('if; the axioms P, K, rules of inference R.l, R2, and some subset (possibly empty) of the axioms T, 4, T K 'P :::> 'P 5, and D. The class of structures corre.!ponding to 4 Kcp :::> KKcp A is that class that results by restricting to the 5 -.Kcp:::} 1(---,J(cp relations corresponding to the axioms as discussed M..tr D --.Kfa.lu above. For example, is the class correspond ing to KD45 and M' is the class corresponding to 'We usc A to denote the class of structures cor and rules of inference: T. M responding to the normal axiom system A. We then get the following well-known result (whose proof Rl From cp and cp :::} 1/; infer 1/; can be found in (Che80, HM85J): R.2 From 145 Theorem 3.2: TJ A i� a normal axiom �1J6tem and Theorem 4.4: sse j$ a 4ound and Ct)mplete ax cp i8 con.•i,tr:nt with A, ·then cp i.J ,,ati.Jfiahle in a iomatization with re.•ped to N1• finite l:nowled,qe htructure in M A. It is well-known that using the axioms of KD45, The second result relates S5 provability to KD45 we can prove that any formula in _cK is equivalent provability. The re1mlt is undoubtedly known to formula to a with no nesting of /Cs. {This is proved experts, although a proof does not seem to appear by using the equivalences K(cp /1. '!/!):: (Kcp /1. T<'!/1), in the literature. Kl Theorem KD4,Sl i! a 1ound and complete 4.1: 5 Generalized probability structures axiomatization for the language .cc with re.Jpr:ct to No. There are situations for which the probability struc tures discusse in Section 2 may not be general Corollary 4.2: If cp i1 a formula in r,K, then cp i.• d enough to capture what is going on. In particular, S!i provahfr. iff .No I= Cert(cpc). since there is only one probability function in the Corollary 4.2 is closely related to Theorem 5 of picture, we cannot capture situations where there i [Gai86]; we discuss the precise relationship in Sec is some uncertainty about the probab lity function . tion 7. For example, consider an agent tossing a coin, Note that KD15 allows the agent to have false which he k nows to be either a fair coin (so that beliefs; -.cp /1. K cp is consistent with KD15. By in the probability of both heads and tails is l/2) or terpreting l( as certainty (by translating a formula a biMecl coin (s0 that the probability of heads is, cp to cpc), we get some added insight into the prob say, 1/3, while the probability of tails is 2/J). This ability of having false beliefs. Given a probabiHty suggests that we allow one possible world where the structure N = (S,w,pr), let FB consist of those probability function assigns probability l/2 to the states .v E S where the agent has some false belief�, event heads (i.e., to the set of possible worlds where i.e., those states ,, where for some formula cp we the coin lands heads) and another possible world have ( N, ·•) I= -.cp 1\ Cert( cp) . Then it is easy to sec where the probability function assigns probability that FB is a set of measure 0. l/3 to the event heads. We might even consider a situation where the agent does not know his own Proposition = 0. a 4.3: pr(FB) prob bility function (this is analogous to situations regarding the modelling knowledge, where we Proposition 4.3 shows that if there arc no states of have false want to allow an who does not know what he of measure 0, then the agent will not agent beliefs. This suggests that S5 will form a com know�), and thus considers a number of worlds pos plete axiomatization in this case. To make this sible where he has different probability functions. precise, let N1 consist of those probability struc These scenarios lead us to a more general ap tures where all states have positive measure (thus proach: associating a (posRibly different) probabil N = (S, 1r, pr) E N1 iff pr(s) > 0 for all .9 E S). ity function with each posRible world. We capture 146 this intuition by means of generalized pro6a6iliiy is always serial: there must be at least one state t �tructure.,. A generali1:ed probability structure N such that P R( s )(t) > 0, since if we sum P R( .• )(I) is a tuple ( 8, 11", PH), where 8 is a finite or count over all states t we get l. We can impose other ahly in fi n ite set of state�, 1r ( ·' ) is a truth assignment restrictions on the support relation, just as we did to the pri m iti ve propositions for each state 3 E 8, fo r the accessibility relation X..::; we then get anal and P R( ., ) is a probability [unc tion on S for each ogous classes of generali1:ed probability structures state 3 E S. General ized probabili ty structures can N• , N� 11 , and i'!O on. Just at'! in the r:ase of knowl he viewed as a generalization of knowledge struc edge, given a normal axiom system A, we can talk tures. Tnstead of just having a set of states that about the class of generalized probabili ty structures an agen t considers possible from each state ,, , each N11 corresponding to A. world that tl1e agent r:onsidcrs possible is assign ed a probahility (where the worlds that the agent docs Theon!m 5.2: If A i� a normal axiom �y.•tem that not consid r possible arc assigned probability 0) e . include� T or D, then Ac i!! ,, ound and complete that the We remark Kripl:e Jtruclure! for knowl with re,pect to NA (for the la n,quage cc ). edge and probability of [FII88b] are in fact a gcn erali�>:ation of generalized probability struc:tures (in This result shows that most of the standard log that they allow many agents anrl include modal op ics of knowledge can be interpreted as logics of cer e ators no ledge . r fo r k w ) tainty. We give semantics to probability formulas just as .. bdorc, except that when eval uati ng the trnth of a 6 Extensions weight formula in the state s, we use the probability fu nction PR(.,). Thus, we get As we mentioned above, we can easily ex tend our stru ct u res to allow for many agents. Suppose we (N, s) F a,w(t,et)+ . . . at 111('fk)� b (l) have many agents, each with his own subjective iff at PR(.•)(S'�' 1)+-..a.�: l'R(�)(S�", ) _ � b (2) probability fu nction. In the case of probabili ty structnres, this would arn onn t to considering struc Note that the prohability structures of Section 2 tures of the . . . ) where is can be viewed as a special case of generalized prob. form (S, 1r, pr" ,pr,. , pr; agent i's probability function , while i n the case of ability structures, where PR(.,) = pr for all states generalized probability structures, we would have s E S. structures of the form ( S, 1r, P R1 , •••, P R,. ), where When rea.'lonin g about certainty, it is clear that, PR;(.• ) is agent i's probability function in stale ·' · in some sense, all that is relevan t arc the stales We would then ex tend the l anguage to allow for with non-zero measure. Given a generalized prob rnulaR of the fo rm Cer(j (r.p): agent i is certain that ability structure N = (8, 11', PR), let the .! upport 0; i.e., ( s , t) E .'iuppN if the probabil iom systems can he extended in tl1e obvious way to ity function in state s assigns positive probability allow reason ing about many agents (d. [1JM85])_ to state t. Tt is easy to r:heck from the definitions A II our results then go tlJTough with essentially no that (N, .! ) I= Cerl(t,e) iff (N, t) f= 147 llOP.t, in which the expert's probability fu nction to us here can be expressed can be time-dependent. We can easily deal with this in our framework by adding temporal opera tors, and a temporal accessibility relation. where w1 and w2 can be viewed as the probability Frisch and Haddawy [FH88c] present a structure fu nctions of two agent� (we present possible inter along the same lines as those of Gaifman, except pretations for these agents below) and I is an in that they actually allow the agent to have different terval, which for the purposes of this discussion we probability functions at each state. However [Had], can take to be a dosed interval [a, b) where a and they view their structure as only appropriate for b arc rational endpoints with 0 � a � b � 1. Intu giving semantics to fo rmulas with depth of nesting itively this says that the conditional probability of at most two (and thus inappropriate for a formula cp with respect to Wt , given that the probability of of the form Cert(Cert(Certp))}. In order to deal If! with respect to w2 is a , is a. with deeper nesting, they require a whole sequence There arc a number of possible interpretations of probability functions. This makes their approach of w1 and w2• One is to view w1 as referring to for nested formulas quite different from ours and rational degree.• of beliP-j of an agent and w2 to that of Gaifman. refer to propen1itie8 or objective probabililie1 (see Another way we can extend our structures is by [Sky80a, Sky80b, Hal89] fo r fu rther discussion of dropping the assumptions that the set of possible these issues). Another viewpoint is taken by Gaif worlds is countable and that the probability func· man [Gai86] , where as we mentioned in the previous tion is discrete. We briefly discuss how to do so section, w1 is taken to represent the expert and 'U.I2 here. the agent about whom we are reasoning. A third Ir we drop the as sumption that the probabil possibility mentioned by Skyrms [Sky80aJ is that ity function is discrete, we have to explicitly de we can identify �v 1 and w2 as degrees of belief of an scribe with each probability function its domain, agent who docs not necessarily know his own mind. the set of sets to which the function assigns a prob This last interpretation is easily captured within a ability. These sets are called the m�a.,urable .t el,,, generalized probability structure. We fo cus on that We then have to slightly redefine the semantics of interpretation for now, and then relate our results Cert(cp) to take into account the possibility that to those of Gaifman and Frisch and Iladdawy. the set might not be measurable. If N is a S"' Since we assume that w1 and w2 in Miller's probability structure, we define (N, .'! ) p Cert(cp) principle now represent the same probability func if there is some measurable �et A such that A � tion, we replace both by w. This still does not S"' and = 1. This essentially amounts to 11(A) does not correspond to a formula in t:P, since We considering the inner mea8ure induced by 11. (see do not allow conditional probabilities. But since [FHM88, FH88b] for more details). It is easy to w( cplt/1)= w ( cp 1\ 1/1)/ w ( 1/J ), Miller's principle can be check that this definition agrees with our old defi rewritten as nition if S"' is measurable. We make similar modi fications if N is a generalized probability structure. aw(w(cp)E I)�w(cpl\(w(cp) E I)) � bw(w(cp)E J ), (*) In this case, we also redefine the support relation where we take to be the interval [a, b] and so that ( ... t), E SuppN iff t E n{A:PR(s)(A)=l} A. I w(cp) E to be an abbreviation of This is Again, this definition agrees with our old definition I a � w( cp) � b. of support if all sets are measurable. We leave it to (an abbreviation of) a formula in £P .4 the reader to check that, with these modifications, Miller's principle (the axiom ( *)} for the full Ian· all our proofs go through with essentially no change. guage £P is not sound with respect to any of the These modifications also enable us to deal with the classes of structures we have considered so far. This case that the set of possible worlds is uncountable. is perhaps not surprising, since information about We leave details to the reader. support is not sufficient to capture an axiom that talks about arbitrary probabilities, rather than jttst certainty. In [FH88b], probability structures satis 7 Miller's principle fying a condition called uniformity are considered; Gaifman [Gai86] and Fr isch and Iladdawy [FH88c] �Note that our requirement that I be an interval with are mainly interested in structures that embody rational endpoints is necessary in order to m.-.ke this " for mul" in c;P. also remark We th11tr"ther than expressing the Miller's principle [Mil66, Sky80a, Sky80b]. In one conditional probability "" term divided by another, we [Sky80a, Sky80b] , a number of variants of Miller's h�.ve cle�tted the denomine.tor to avoid having to deol wi th denomin11lor principle are presented. The one of most interest the problems that a.rise when the is 0. 148 these arise naturally in distributed systems appli We now want to show that when we restrict to cations. In the notation of this paper, a general reasoning about certainty, KD45 provides a com ized probability structure N = (S, 1r, P R) is uni plete axiomatization for uniform structures. Let form if for all s, t E S, if (.•, t) E SuppN, then .N..... if be the class of uniform structures. PR(s) = PR(t). As we now show, uniform struc tures do capture Miller's principle. Theorem 7.2 : If to be a pair F = (S, PR), where Sis a set of states Corollary 7.3: KD.{� i� a !otmd and complete and P R( s) is a discrete probability function on S with re.•peci to .Nnif for the la nguage £C . for each � E S. Thus, a frame is a (generalized) probability structure without the truth assignment Thus KD45c is a sound and complete axiom 11. A probability structure (S1, 1f1, PR') is baud on atization for the class of structures characterized frame (S, PR) if S = S' and PR = PR'. Unifor by Miller's principle, given our interpretation of mity and all the conditions on support that we have Miller's principle with w1 and w2 identified. Com considered can be viewed as conditions on frames, bining Corollary with Theorem we get rather than conditions on structures, since they do 7.3 3.3, not depend on the truth assignment at all. Thus, Corollary 7.4: If 'f! i11 a form ula in £K, then 'f! i! fo r example, we can define a frame { S, P R) to be provable in S5 iff .N... i/ f: Cert('f!c). uniform if for all s, t E S, if ( .1 , t) E SuppF, then R( II = R( No te a frame Fill uniform iff p ) r t). some As we mentioned above, Gaifman does not iden probability structure based on F is uniform iff ev tify w1 and w2 ; rather, he associates Wt with the ery probability struc ur based on F is t e uniform. expert'::� probabilities and w2 with the agent's prob say a formula is valid in frame written We 'f! F, abilities. In addition, Gaifman actually considers a if for every probability structure F I= '{!, N I= 'f! stronger fo rm of Miller's principle, namely N based on F. The following theorem shows that Miller's principle characterizes uniform frames. Theorem 7.1 : Th e following two condition/! are where t/1 is a conjunction of formulas of the form 1').6 Ile equivalent: w2 ( 'l/;1 E sh ows that structures satisfying this stronger principle can be characterized in a way 1. F ill a uniform frame. rather similar to Theorr.m 7.l. We provide a few of the dr.tails here. Suppose we are given a gener £. Every inlltance of Miller '_, principle (i.e., the a\izr.d probability structme N = (S, 1r, PR1, PR2) axiom (*)) ill valid in F.5 fo r two agents. Recall that Gaifman assumes that m li 5 We rem3.rk that usi�g fro.rnes to characterize axio s is the agent's probabi ty assignment PR2 is inde c modal a well-known te h niqu e in logic [Gol87, HC84]. Con pendent of the state, thus there is a fixed proba oider, for example, the axtom T for knowledge. Although we bility fu ndion such that I' R2( ·' = for all have noted that pr ) pr it i• •ound fo r reflexive knowledge structures E ( i.e., knowledge structures where the 1C rdo.tion is ren�xive) states � 8. Given a slate ·' E 8, let C(.•) 8.n d, together with P, K, Rl, and R2 provides !l co mplete be the equi,,aJr.nce cla�s of states in 5 consist axiornalizo.tion such structures, it is not hard to construct for ing of all states ·' ' such that /'R1(.•') = I'Rl (-•). a. non-reflexive st ructure where every instance of T is va,lid Thus, C( ·• ) is the sr:t of states where the expert (see [HM8.1] ). On the other hB.nd, T dMs charoderi7.ereflex. ive knotl>ledge fm me.• ( where a knowledge frame is just 11 pnir has the same probability fu nction that he dor:s in (8, K), and It reflexivekn owlerlgefrB me io a knowledge frame s. Now consider the set of states Sgood where the where the relation /( is reflexive), in that every instn.nce of expert believes that with pmhability I, his prob T is val id in 11 knowledge frame F" iffF io reflexive. Simi ability distribu tion is the right one, i. ., = lar rem• rks hold fo r nil the other axioms we con sid ered for e Sgood knowledge. However, al though :Mi ller's principle does cha.r {.• : f'Rt(-•)(G(s)) = 1}. Gaifman shows that the ac terize uniform frames, it i s nnt the case that Miller'• pri n strongr.r form of Miller's principle is equivalent to ciple together with I he other a> 149 equivalence class constraint, and that the equiva ity relationship, we can capture properties of cer lence class constraint holds in all frames where the tainty by placin g conditions on the support. In stronger fo rm of Miller's prin ciple i11 valid. More particular, if we aasum e one fi xed prohability fu nc over, Gaifrnan Rl10ws that if cp is provable in S5, tion, we !lhowed that KD45 provides a complete tl1cn cp hold!! with probability 1 ( with rcRpcc:t to axiomatization. Moreover, the set of worlds where pr) in all !ltructures !latisfying the equivalence class an agent has false beliefs has probability 0. Inter constraint. Note th at by Corollary 7.4, th e analo estingly, KD45 also provides a complete axiomali gous re!!ult holds for uniform structures. zation with respect to st ructures sati!!fying Miller's principle. A!! mentioned in the previous section, in [FII88c], where they only discus!! ptobabi1ity formulas of Many researchers have rejected S5 as an appro depth 2, Frisch and I l addawy do not assume that priate axiomatization for an agent's beliefs since l ow se beliefs. 1' n2 is t. he same for all agents. Since they ai!!O they want to a l an agent to have fal want to consider structu res where Miller's princi Instead , they consider KTH5, which allow!! false be ple holds ( th ey do not consider the stronger ver liefll. Our results suggest that there is a reason sion of Miller's principle), they assume that the able interpretation for belief that is charac::terized equivalence class constraint holds for every prob by the KD45 axioms, but still make!! rather strong ability function rn2(.,), i.e., they assume that assumptions about the corredne!ls of an agent's be liefs. l' R2(·1 )(89,.,,1) :::: 1 for all 10tates s. However, in order to deal with more deeply nested formulas, These resul ts show how the tools of modal logic Frisch and llarldawy plan to have a sequence of can he brought to bear on reason ing about prob probability fu nctions. Thus, in their structure!! ability. We believe that further work along these they will ha1•e fu nction11 P R1, I' R2, P R.3 , . . . , when� ]jncs should yield fu rther insights into probabilit.y, l'R;(s) gives a probability function at each state belief, and knowledge . .�. They do not provide interpretations fo r these AcknowlcdgrrwntR: Tll is paper wa11 inspirer! by probabilit)' fu nctions, but they want to con sider discussion!! with Peter Haddaw )'. M oshe Vardi structures where Miller's principle holds between made a number of U!leful comrnentR on an earli�r all consecutive pairs o[ probahility functions given draft of the paper. by l'R; and J'R; +1 • Thus, they plan to assume that the equivalence class constraint holds between consecu tive pairs of probability functions. Miller's References principle is easily seen to be sound in structure!! Che80 B. F. ChellM, Modal Logir, Cambridge satisfyin g this constraint. Rather than having a ! ] University Pres!!, 1980. diffe rent modal operator for each of t hese probabil ity assignmen ts, they only hM·e one modal operator [FH88a] R. Fagin and J. Y. Halpern , Helid, r n A rli· C crt. They usc I. he r Il;'s to give semantics to more awareness, and limited eason i g, deeply ne!il.ed occurrences of Cert. More fo rmally fir:ial lnteUig ence 34, 1988, pp. 39 --76. to interpret an oc (using our notation), in order [F'fl88b] R. Fa gin and J. Y. Halpern, Rea.�on ing currence of the modal operator C r.r t app earing at ab ou t knowledge and probability: pre depth i at state ., , they use the probability fu nction liminary report, Proceedin,qlf oj ihe Sec /'rob;(.,), where l'rob1 ( ., ) = /'HJ(.,) and for i> 1, ond Conferenre on Th eoretical A"'pect., we define !'rob; ( ·' ) as the mean value of I' R; ( ·' ') of Rr.a.10nin,q ahout K nowlr.dgc ( M. Y. over all stales ·' ', where the wt�ighting is done wilh Vard i, eel.), Morgan Kan hnann, 1988, resped to l'rob;_1 • Thus, if S' i!! a subset of 8, tl1cn pp. 277 2!);1. /'rob; (.�)(8') is L•'ES l'U;(.,)(.,') · l'rob;-1(·�')(5") [FII88c] A. M. Frisch and P. l laddawy, Prob Iliad]. hey show that under this interpretation T ability as a modal operator, Procecd for Gc l, the axioms of KJHC are sound , but not r in_qlf of lh� Fo urth A A A I Uncertainty in necessarily com plcte. Artificial ln tellig rncc Worbhop, 1988, pp. 109 118. 8 Conclusions [FI IM88J R. Fagin , .J . Y. Halpern, and N. Megiddo, A logic for rea11oning about probabil.ities, We have exami11cd tl1c relationship between knowl Proc ..'lrd TF;F;E Symp. on Logic in Com edge, belief, and probability. We showed that, just puter Science, 1988, pp. 277- 291. 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