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The Relationship Between Knowledge, Belief, and Certainty: Preliminary Report

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The Relationship Between Knowledge, Belief, and Certainty: Preliminary Report 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(<p!) We a,!, ... , a_�:, b are arbitrary integers and 'I'll· .•, cpk ing Miller's principle. remark that Morgan has are arbitrary formulas. We call the resulting lan­ also considered the relationship between axioms for gu ge A formula � ch as w(cp) is, probability and axioms for more standard modal � [/'. u 2: 1/2 stnctly speaking, a.n abbreviation of the for­ logics [Mor82a, Mor82b] , but his focus is on con­ t:.P mula 2w(cp);::: while w(cp) 2w(.P)is an abbre­ ditional probabilities and the resulls have a much 1, < viation for • w( !p) 2: 2w( . We also use a num­ different flavor from ours. ( 1/J)) ber of other obvious abbreviations without further comment, such as w(cp) ::=; b for -w(cp) -band -> The est of this paper is organized as follows. In b) (w( r w(fP) = b Cor (w(cp)2: 1\ cp) :$b). the next section, we present the formal model for Just as in [F1188b], we allow arbitrary nesting of reasoning about probability (which is a slight vari­ probability formulas, so that w(w('P) 2: I/2) < 1/3 ant of the model discussed in [FilM88, FH88bJ). In is a legal formula of t:.P .1 Such higher-order prob­ Section 3 we review tl1e formal semantics for rea­ ability statements will be one of our main interests soning about knowledge, stating a number of re­ here. They are not as unmotivated as they might sults that are needed in the sequel. In Section 4 we first appear. Suppose we take cp to be the statement show that KD45, the logic of belief, is a complete "it will rain tomorrow," and we have just heard the axiomatization for reasoning about certainty (with weatherman say that it is likely to rain tomorrow. respect to the probability structmes introd uced in Thus, according to the weatherman, w(cp) � 1/2 Section 2), and that if we restrict attention to struc­ holds. However, suppose we have found this weath­ tures where all worlds have non-zero probability, erman to be quite unreliable in the past, so Lhat tlwn S5 is a r.omplete axiomatization. In Section 5 his predictions turn out to be wrong far more often we consider generalized probability structures, ami than they are right. Thus, we might place proba­ �how how dilferenl conditions on u.e support of the bility less than 1/3 on his statement, which leads us probabili ty measure correspond to dilferent axiom­ exa tly to the formula 1/3. (See � w(w(<p) 2: l/2) < atizations. In particular, we show that many of the [Ga186, Sky80b] for further discussion of higher­ classical modal logics can be captured by placing order probabilities.) t�e appropriate conditions on the probability struc­ We usc a possible-worlds approach to give se­ tures. While these rcsu Its are all quite straightfor­ mantics to the formul s in t:.P. (This essen­ ward, they do sl1ow an interesting and not alto­ a is tially same approach as that taken by Nilsson gether obvious connedion between certainty and the knowledge. In Section 6 we briefly discuss some (Nil86].) We take a proha.hility !!ructure N to be a tuple ( 1r, pr , where S is a finite or countably extensions to our results.
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