A Logic of Implicit and Explicit Belief

Abstract

As part of an on-going project to understand the found* tions of Knowledge Representation, we are attempting to characterize a kind of belief that forms a more appropriate basis for Knowledge Representation systems than that cap tured by the usual possible-world formalizations begun by Hintikka. In this paper, we point out deficiencies in current semantic treatments of knowledge and belief (including recent syntactic approaches) and suggest a new analysis in the form of a logic that avoids these shortcomings and is also more viable computationally. The kind of belief that underlies terms in AI such as ‘Know!edge Representation” or “knowledge base” has never been adequately characterized. r As we discuss below, the major existing formal model of belief (originated by Hintikka in [l]) requires the beliefs of an agent to be closed under logical consequence, and thus can place unrealistic computational demands on his reasoning abilitites. Here we describe and formalize a weaker sense of belief that is much more attractive computationally and forms a more plausible foundation for the service to be provided by a Knowledge Representation utility. This formalization is done in the context of a logic of belief that has a truth-based semantic theory (like the possible-world approach but unlike its recent syntactic competitors). This logic is also shown to have connections to relevance logic and, in a certain sense, to subsume it. 1. Logical Omniscience & Possible Worlds A recurring problem in the modelling of belief or knowledge is what has been called in [z] logical omniscience. In a nutshell, all formalizations of belief based on a possible-world semantics suffer from the fact that at any given point, the set of sentences considered to be believed is closed under logical consequence. It is simply built into the logic that if a is believed and a logically implies ,8, then B is believed as well. Apart from the fact that this does not allow for a resource-limited agent who might fail to draw any connection between a and fi, this has at least three other serious drawbacks from a modelling point of view: 1. Every valid sentence must be believed. 2. If two sentences are logically equivalent, then one must be believed if the other is. ‘Because what is represented in a knowledge base is typically not required to be true, to be consistent with most philosophers and computer scientists, we are calling the attitude involved here ‘belief” rather than “knowledRe”. 3. If a sentence and its negation are both believed, then so must every sentence. Any one of these might cause one to reject a possible-world formalization as unintuitive at best and completely unrealistic at worst. There is, however, a much more reasonable way of interpreting the possible-world characterization of belief. As discussed in [3], instead of taking logical omniscience as an idealization (or heuristic) in the modelling of the beliefs of an agent, we can understand it to be dealing realistically with a different though related concept, namely, what is implicit in what an agent believes. For example, if an agent imagines the world to be one where a is true and if o logically implies B, then (whether or not he realizes it) he imagines the world to be one where B also hap pens to be true. In other words, if the world the agent believes in satisfies cy, then it must also satisfy ,8. Under this interpretation, we examine not what an agent believes directly, but what the world would be like if what he believed were true. There are often very good reasons for examining the consequences of what an agent believes even if the agent himself has not yet appreciated those consequences. If the proper understanding of a possible-world semantics is that it deals not with what is believed, but what is true given what is believed, what then is an appropriate semantics for dealing with the actual beliefs of an agent? Obviously, we need a concept other than the one formalized by possible worlds. If we use the terminology that a sentence is ezplicitly believed when it is actively held to be true by an agent and implicitly believed when it follows from what is believed, then what we want is a formal logical language that includes two operators, B and L: Ba will be true when a is explicitly believed while La will be true when Q is implicit in what is believed. While a possibleworld semantics (like that of [l] or [4]) is appropriate for dealing with the latter concept, the goal of this paper is to present one for the former. 2. The Syntactic Approach When talking about what an agent actually believes, we want to be able to distinguish between believing only a and (a > 8) on the one hand, and believing a, (CY > a) and @, on the other. While the picture of the world is the same in both cases, only the second involves realizing that /3 is true. This is somewhat of a problem semantically, since the two sets of beliefs are true in nreciselv the same possible worlds and so, in some sense, semanFrom: AAAI-84 Proceedings. Copyright ©1984, AAAI (www.aaai.org). All rights reserved. tically indistinguishable. This might suggest that any realistic the syntactic and the possible-world approaches so that different semantics for belief will have to include (something isomorphic sets of sentences can represent the same beliefs without requirto) a set of sentences to distinguish between the two belief sets ing that all logically equivalent sets do so. We now show that above. The usual way to interpret a sentence like La in a stanthere is a reasonably intuitive semantics for belief that has these dard Kripke framework is to have a model structure that conproperties. tains a set of possible worlds, an accessibility relation and other things. It appears that to interpret a sentence like Ba, a model structure will have to contain an explicit set of sentences. This is 3. Situations indeed what happens in the formalizations of belief of [S] and (61 that share our goal of avoiding logical omniscience. A slightly On closer examination, the reason the possible-world ap more sophisticated approach is that of [7] where the semantic preach to belief or knowledge leads to logical omniscience is that structure contains only an initial set of sentences (representing beliefs are characterized completely by a set of possible worlds a base set of beliefs) and a set of logically sound deductive rules (namely, those that are accessible from a given possible world). for obtaining new derived beliefs. Logical omniscience is avoided Intuitively, these possible worlds are to be thought of as the full there by allowing the deductive rules to be logically incomplete. range of what the agent thinks the world might be like. If he With or without deductive rules, I will refer to this approach to only believes that p is true, the set of worlds will be all those modelling belief as the .yntactic approach since syntactic entities where p is true: some, for example, where q is true, others, where have to be included within the semantic structures. q is false. However, because sentences which are tautologies will Apart from this perhaps ill-advised mixture of syntax and also be true in all these possible worlds, the agent is thought of semantics, the syntactic approach suffers from a serious defect as believing them just as if they were among his active beliefs. that is the opposite of the problem with possible worlds. A In terms of the possible worlds, there is no way to distinguish p possible-world semantics is, in some sense, too coarse-grained to from these tautologies. model belief in that it cannot distinguish belief sets that logically One way to avoid all these tautologies is to to make this noimply the same set of sentences. The syntactic approach, on tion of what an agent thinks the world is like be more relevant the ocher hand, is too fine-grained in that it considers any two to what he actually believes. This can be done by replacing the sets of sentences as distinct semantic entities and, consequently, possible worlds by a different kind of semantic entity that does different belief sets. not necessarily deal with the truth of all sentences. In particTo see why this a problem, consider, for example, the disjuncular, sentences not relevant to what an agent actually believes tion of LY and 8. There is no reason to suppose that (including some tautologies) need not get a truth value at all. B(a v ,9) E B(/3V a) Following [8] (but not too closely), we will call this sort of partial possible world a Gtuation. bughly speaking, a situation may would be valid given a syntactic understanding of B since (@VP) support the truth of some sentences and the falsity of others, may be in the belief set while (/? V a) may not.2 The trouble but may fail to deal with other sentences at all. with this is that if we consider intuitively what For example, consider the situation of me sitting at my ter“It is believed that either o or /I is true.’ minal at work. We might say that this situation supports the is saying, the order seems to be completely irrelevant. It is fact that I’m at work, that somebody is at my terminal, that almost an accident of lexical notation that we had to choose one there is either a terminal or a book at my desk, and so on. On of the disjuncts to go first. Yet, the syntactic approach makes the other hand, it does not support the contention that my wife the left to right order of disjuncts semanticallysignificant in that is at home, that she is not out shopping, or even that she is at we can believe one ordering but fail to believe the other. home or not at home. Although the latter is certainly true, me sitting at my terminal does not deal with it one way or another. The obvious counter to this is that the logic of the syntactic approach has to be embellished to avoid these spurious syntactic One way of thinking about situations is as generalizations of distinctions. For example, we might insist as part of the semanpossible worlds where not every sentence in a language is retics that to be well-formed, any belief set containing (ckVb) must quired to have a truth value. Conversely, we can think of posalso contain (/? V cr) (or, for Konolige, the obvious deduction sible worlds as those limiting cases of situations where every rule must be present). The trouble with this kind of constraint sentence does have a truth value. Indeed, the concept of a posis that it is semantically unmotivated. For example, should we sible world being compatible with a situation is intuitively clear: also insist that any set containing 11~ must also contain cr? every sentence whose truth is supported by the situation should Should every belief set containing a and b also contain (a ha)? come out true in that possible world and every sentence whose Should every belief set contain the ‘Lobviousn tautologies such falsity is supported should come out false. Again drawing from as (a > a)? Where do we stop ? Clearly, it would be preferable (81, we will also allow for incoherent situations with which no to have a semantics where restrictions such as these follow from possible world is compatible. These are situations that (at least the meaning of Ba and not the other way around. In other seem to) support both the truth and falsity of some sentence. words, we want a semantics (like that of possible worlds) that From the point of view of modelling belief, these are very useful is based on some concept of truth rather than on a collection since they will allow an agent to have an incoherent picture of the world. of ad hoc restrictions to sets of sentences. Ideally, moreover, the granularity of the semantics should lie somewhere between The “trick”, then, that underlies the logic of belief to follow is to identify explicit belief with a Bet o{aituationa rather than 21n Konolige’s #y&em, one disjunction may be deducible while the other possible worlds. Before examining the formal details, there is mav not. one point to make. Traditional lonics of knowledge and belief

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@inproceedings{Levesque1984ALO, title={A Logic of Implicit and Explicit Belief}, author={Hector J. Levesque}, booktitle={AAAI}, year={1984} }