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A note on non-monotonic modal logic
Artificial Intelligence 64 (1993) 183-196 Elsevier
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ARTINT 1032
A note on non-monotonic modal logic* Robert Stalnaker Department of Linguistics and Philosophy, 20D-213, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
This paper, which began as a comment on a talk by Drew McDermott, 1 was completed in the summer of 1980. It circulated in typescript, but was not published at the time. A lot has happened in the study of nonmonotonic reasoning since that time, and the paper is now quite dated, but it is being published now, just to put it on the record, since it has been widely cited (mainly as a result of citations in Robert Moore's influential work on autoepistemic logic). Other than grammatical, typographical and notational corrections, and a few cautionary comments added to endnotes, the paper is unchanged from the 1980 version. 2 I want, in this note, to raise some questions and make some suggestions about the intuitive semantic foundations of the non-monotonic modal logics that Drew McDermott and Jon Doyle have developed in [2] and [3]. In particular, I want to ask about the intended meaning and application of the non-monotonic consequence relation, and about the intended meanings of the modal operators. I will suggest some answers to the questions I will raise-answers that I think are implicit in McDermott's motivating remarks in [3]--but I will argue that these answers lead to serious problems for the
Correspondence to: R. Stalnaker, Department of Linguistics and Philosophy, 20D-213, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. E-mail: [email protected]. * Editor's Note. This article is being published long after it was written because of its importance in the history of the field. It was called to our attention by Robert Moore who wrote: "The paper was written about twelve years ago as a commentary on Drew McDermott's paper on nonmonotonic modal logics (which was ultimately published in 1982 in JACM). It was a very important influence on my development of autoepistemic logic, and has been cited by numerous other papers, but has never been published. Perhaps its most lasting contribution to the theory of nonmonotonic reasoning is the notion of a "stable set" of beliefs (that is, stable under introspection and inference), a notion which plays a key role in many nonmonotonic systems. I believe this is a very important paper, and it should be published in the A1J to get it into the archival record." 0004-3702/93/$06.00 © 1993--Elsevier Science Publishers B.V. All rights reserved
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interpretation of the theory. In conclusion I will make some brief remarks about the problem that motivated the development of this kind of logic, and suggest an alternative means of solving it. I will begin with a very general challenge to the possibility of non-monotonic logic, and a suggestion about how the general challenge might be met. A logic is non-monotonic if it has a non-monotonic consequence relation. A relation R between a set of sentences F and a sentence p is monotonic if and only if whenever F~ C_ F2, {p:R(F~, p)} C_{ p : R(F2, p)}. But how is it possible for any consequence relation to fail to be monotonic? The question is not, how can we define a non-monotonic relation between the sets of sentences and the sentences of some language. This is simple enough. The problem is to understand how any such relation could be a consequence relation. For it seems that one can give a very abstract and general definition of the notion of semantic consequence---one that should apply to any possible interpreted formal language that contains sentences that have truth values--and that any relation fitting this definition will be monotonic. Suppose we have a formalized language--any language at all--with primitive expressions divided into logical and descriptive expressions, and formation rules defining sentences, and perhaps complex expressions of other grammatical categories. A semantic analysis of such a language will be an account of the way semantic values are assigned to expressions. Semantic values will be defined relative to a model which assigns semantic values to the primitive descriptive expressions. Besides a definition of a model, the semantic analysis will include semantic rules which determine the semantic values of all complex expressions of the language as a function of the values of their parts (together, in some cases, with global features of the model, such as a domain of individuals, or a set of possible worlds). Models and semantic values can be any sorts of things, so long as the semantic value of a sentence somehow determines a truth value. Given any language and semantic theory meeting these conditions, we can define notions of logical truth and semantic consequence as follows: logical truths are sentences true in all models; a sentence p is a semantic consequence of a set F if and only if every model that makes all of the sentences in F true also makes p true. Now this characterization of the form of a semantic theory is very abstract and unconstrained. It puts no limit on what semantic values can be, or on what form the models might take, or on the way semantic rules explain the values of complex expressions as a function of the model. But no matter what weird properties the rules impose on the logical operators, no matter what strange entities are postulated by the models, the consequence relation for any semantic theory fitting this mold will obviously be monotonic. So it seems that within this orthodox approach to semantic analysis, one cannot explain the distinctive non-standard feature of non-monotonic logic in terms of non-
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standard kinds of models, semantic values or semantic rules. One must, it seems, give an alternative general account of semantic consequence. And what is wanted is not a specific technical definition for a specific non-monotonic logic, but a general account of what consequence is supposed to mean--of what concept some specific technical definition is trying to capture. And one should expect such an account to explain why it is reasonable to call the concept by the name "consequence". My point is to offer a challenge, not a refutation. It follows from what I have said that if non-monotonic modal logic is given a semantics that fits the orthodox pattern, then this semantics will determine a well-defined monotonic consequence relation, but this does not exclude the possibility that other related notions that have some claim to be called by the name "consequence" might be defined. Here is one way that this might happen: suppose the sentences of some language contain an operator that is context-dependent; suppose further that part of the context that is relevant to the truth conditions of sentences containing this operator is the fact that that sentence occurs in the course of giving an argument with certain premisses. In particular, suppose that a possibility operator " M " is interpreted relative to the fact that some set of sentences F is the set of premisses of an argument. ( " M p " might still mean "p is true in some possible world" so long as the relevant set of possible worlds is defined relative to F). In such a theory, the premiss set F would be playing two different roles, both of which are relevant to the question whether some sentence p is a consequence of some set F. First, F helps to fix the interpretation of the sentences in the argument (that is, helps determine what propositions they express); second, F determines which propositions are the premisses of the argument. One could separate, conceptually, these roles and define a well-behaved monotonic consequence relation in the standard way. But if one does not separate them, then the consequence relation one defines might be non-monotonic. Since adding a premiss may alter the interpretation of the prior premisses or the conclusion, doing this may make a valid argument invalid. Here is a trivial example that illustrates the general point. Suppose we have a language containing an indexical numerical pronoun " a " that denotes, when used in the course of giving an argument, the number of premisses of the argument. Now consider this argument: a is the number of living ex-presidents of the United States. Therefore, there is an odd number of living ex-presidents. This argument had [in 1980] a false premiss and conclusion, but it is valid according to one intelligible way of defining validity for this language. But now add a second premiss: a is the only even prime.
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Now the premisses are both true [again, in 1980], but the conclusion remains false. We have turned a valid argument into an invalid one by adding a premiss. I think something like this is going on in McDermott and Doyle's nonmonotonic modal logic. That is, I think that the explanation for the nonmonotonic character of the consequence relation is to be found in the way in which the interpretation of the modal operators is dependent on the set of sentences that constitutes the premiss set of the non-monotonic inferences. To make out this claim I will try to spell out the semantic analysis that I think is implicit in the theory. I will begin with a rough intuitive sketch of what the theory, as I understand it, is intended to represent. The premiss set (the set of proper axioms) of a non-monotonic theory represents a total set of given information--the initial beliefs of an ideally rational agent. The agent is able to make inferences--to expand its 3 set of beliefs. The crucial assumption that motivates the distinctive features of non-monotonic logic is that the agent makes inferences, not only from the given information, but also from the fact that it has the information that it has. That is, the crucial assumption is that the agent can reflect on its beliefs and draw conclusions about what it does and does not believe. If p is entailed by the given information, then the agent can infer not only p, but also that it believes that p. If q cannot be inferred from the given information, then the agent can conclude that it does not believe that q - - t h a t not-q is compatible with its beliefs. So the ideally rational agents that the theory seeks to describe have access to their own beliefs: they know, or can determine, that they believe something when they do, and that they do not when they do not. The modal operators, " M " and " L " , serve to represent information about what the agent believes. " M p " represents something like the claim that p is compatible with what the agent believes, and " L p " represents something like the claim that the agent believes that p. But we have to be careful here. Our assumption is that the agent's beliefs are changing as it reflects on them and makes inferences from them about what it believes. Does " M p " mean that p is compatible with the agent's initial beliefs, or that p will be compatible with the agent's beliefs after it is finished reflecting on them and drawing conclusions from them? Clearly it must be the latter. When an agent draws a conclusion of the form Mp, it is not concluding simply that p was compatible with its initial beliefs. It is concluding that p is compatible with its beliefs, and will remain so at least until it receives new information. If " M p " meant simply that p was compatible with the initial information, then it might be rational for an agent to conclude both not-p and Mp. The proper axioms or the premiss set, of a non-monotonic theory represents an initial belief state; the fixed points of the theory represent various possible final belief states--belief states that the agent might reach by reflecting on its beliefs and drawing conclusions from them and about them. The operators M
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and L are used to represent statements about the final belief state that will be reached. A final belief state must meet two intuitive conditions: first, it must be stable in the sense that no further conclusions could be drawn by an ideally rational agent in such a state. Second, all the agent's beliefs must be, in some sense, grounded in the initial belief state--in the given information. The first condition is relatively easy to make precise: a set of sentences F is stable, in the relevant sense, if and only if it meets the following three conditions: (1) F is deductively closed; (2) if p E F, then Lp E F; (3) if n o t - p ~ F , then Mp E F. The second intuitive condition is more difficult to turn into a formal constraint, but M c D e r m o t t and Doyle's syntactic definition of a fixed point, and their semantic definition of a non-committal model are, I take it, attempts to capture this idea (as well as the idea of stability). Fixed points will all be stable in the sense defined. In terms of this intuitive picture, what does the non-monotonic consequence relation represent? " F ~ p " represents the claim that any final belief state based on initial given information F will include the belief that p. As predicted, the premiss set F is playing two roles relevant to non-monotonic consequence: first, it determines what sentences are believed in the relevant final belief state; second, it constrains the interpretation of the sentences containing the operators M and L. It is only because it plays these two roles at once that the relation can be both a consequence relation and a relation that is nonmonotonic. Now I want to consider how this intuitive picture gets turned into a formal semantic analysis. I will assume that the language is a standard modal propositional calculus. A final belief state will be represented in the semantics for this language by a Kripke model: an ordered triple M = ( K, R, v } where K is a non-empty set of possible worlds---intuitively, the possible worlds that are compatible with the agent's final belief state; R is a binary relation on K - - t h e accessibility relation; and o is the valuation function that assigns propositions-sets of possible worlds--to the sentence letters. The intuitive role of the accessibility relation is to determine what the agent's beliefs would be if each of the various possible worlds in K were actual. The reason such a relation is necessary is that the agent has beliefs about its beliefs. To model an agent's belief that it believes that p, for example, we need not a set of possible worlds in all of which p is true, but a set of possible worlds in all of which the agent believes that p is true. But to interpret the claim that the agent believes that p in some world w E K, we need a way of determining the set of possible worlds compatible with what the agent believes in w (which may be different from the set K, which represents the possible worlds compatible with what the agent
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believes in the actual world). The accessibility relation represents this information in the following way: the possible worlds compatible with the agent's beliefs in w is the set {u: w R u } . This is the set of possible worlds accessible from w. We will want to put some constraints on this relation, but for the moment, I will defer the question of what they should be. So far, we have an ordinary monotonic modal semantics, interpreted as a logic of belief. To allow for the definition of the non-monotonic consequence relation, we need to put further constraints on the models. In terms of our intuitive picture, a model represents a final, stable belief state. But final belief states are defined relative to initial belief states--relative to the received information that initiates the reflection that leads to the final belief state. Let F be the set of sentences representing the initial information. Then an admissible F-model must be a non-committal model of F. The definition of non-committal model given in [3] is somewhat complicated and intuitively opaque, but I take it that it is intended to capture the two intuitive requirements that final belief states be stable (in the sense defined above) and that they be grounded in the given information. 4 Now we can define two consequence relations, first one that may be non-monotonic; second, a more orthodox consequence relation that separates the two roles played by the premiss set: (1) For any set of sentences F and sentence ~b, F ~ ~b iff for all noncommittal models, ( K , R, v) of F, A {v(~O): qJ E F} C_ v(~b). (2) For any set of sentences F and sentence ~b, F ~ ~b iff for all sets of sentences A and all non-committal models ( K , R, v) of zl,
N {v(~,): ~,~r} c_ v(~). Now let me raise the question that ! deferred above: what constraints should be put on the relation R? The main point I want to make is that we do not have free choice here. The intuitive assumptions that motivate the characterization of final belief states requires that R be an equivalence relation, and that the monotonic logic of belief should be S5.s The reason for this is that the formal assumption that the accessibility relation between belief worlds is an equivalence relation corresponds to the intuitive assumption that the agent has access to its own beliefs--that it knows what it believes and what it does not believe. Without this assumption, we have no reason to give fixed points and noncommittal models any special status. Whatever constraints are put on the accessibility relation, the following will be true, given McDermott's definition of non-committal model: all noncommittal models are $5 models in the sense that all theorems o f $5 are true in all possible worlds in all such models. Hence, whatever assumptions one makes about the relation R, the monotonic consequence relation ((2) above) will be
an $5 consequence relation. This means that it follows from the meanings given to M and L that the operators are $5 modal operators.
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I should emphasize that my point concerns intuitive foundations and not technical consequences. In particular, I am not denying that one can consistently define, as M c D e r m o t t does, a notion of non-committal model in a modal semantic theory that is weaker than $5, and that the relation being a non-committal model of a certain theory will be different if the modal semantics is weaker than $5. The reason that this is possible, despite the fact that all non-committal models are $5 models is as follows: whether or not a model is a non-committal model of a theory F depends on whether or not certain other models, which need not be non-committal, exist. If one allows these other models to be non-S5 models, this will make a difference to the question whether a given $5 model is a non-committal model of F. So I am not arguing that M c D e r m o t t is mistaken when he claims, for example, that non-monotonic consequence is different if the logic is $4 from what it is if the logic is $5. My point might be put this way: the intuitive meaning of M~b in non-monotonic logic is something like this: "~b is compatible with the agent's final stable belief state." But final belief states are represented by noncommittal models, and non-committal models are always $5 models. Hence the monotonic consequence relation is the $5 consequence relation. If I am right so far, then there is a problem. For non-monotonic $5 has, as M c D e r m o t t puts it, a serious bug. The bug is that non-monotonic $5 is monotonic. The two notions of consequence which have been so carefully separated collapse together in $5. Every $5 consistent and closed extension of a set of sentences is a fixed point of that set. Every $5 model of a theory F is a non-committal model of F. So $5 will not do at all as an underlying logic for non-monotonic logic. I am suggesting that non-monotonic logic, as M c D e r m o t t develops it, faces a serious dilemma. If its underlying logic is weaker than $5, then the operators M and L cannot mean what they are supposed to mean. But if the logic is as strong as $5, then the theory cannot be used to do what it was developed to do: the interesting and distinctive feature of the logic disappears. Let me use a simple abstract example to illustrate the problem with non-monotonic $5 that McDermott points out. Suppose my initial information consists of just the single sentence p. p is the only proper axiom; initially, I believe only p. What should I conclude from reflecting on this information? First, I conclude Lp, Mp, LMp, MLp, and so forth. Also, I add all the deductive consequences of p and Lp. What about q? (Suppose q is some totally independent factual proposition.) One might think that I could conclude Mq since I can see that nothing I believe could possibly lead me to conclude that q is true. But I cannot conclude M q for the following reason: Mq is inconsistent with another possibility statement, namely M - - M q . In $5, M - - M q entails --Mq. And since what this other possibility statement claims to be possible is also compatible with p, I have as much reason to add M ~ Mq to my beliefs as I have to add Mq. I cannot consistently add both, but if my beliefs are to be
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stable, I must add one or the other. So I have a free choice. But if I choose to add M - Mq, then I will conclude - q . It is clear from this that I can conclude anything at all that is consistent with the initial beliefs. Any $5 model of { p } will be a non-committal model of {p}. McDermott's response to the collapse of non-monotonic $5 into monotonic $5 was to drop down to $4: in effect, to define non-committal models in terms of a weaker logic. This is technically possible, but it does not change the fact that the monotonic consequence relation is the $5 consequence relation: all $5 inferences remain semantically valid in the sense that they preserve truth in all possible worlds of all non-committal models, which means in all admissible interpretations of the language. The move to $4 is simply a decision to define non-committal models in terms of a semantically incomplete logic. M c D e r m o t t owes us a non-ad-hoc explanation of why one should define non-committal models in this way. 6 An alternative response to the collapse would be to keep the logic $5, but to change the definition of non-committal model (and of fixed point) so that it better represents the intuitive idea that motivated it. Recall that the intuition behind the idea of non-committal model (as I reconstructed it) had two parts. First, non-committal models represent stable belief states: belief states from which no more could be inferred simply by reflecting on them by an agent who had access to its beliefs. Second, a non-committal model of a theory F represents a belief state that is grounded in the initial information represented by F. As we saw, the first of these constraints is much clearer and was more easily turned into a formal constraint than is the second. It is the first constraint that implies that all non-committal models are $5 models, but a different definition of non-committal model---one that captured the second idea in a different way--might avoid the collapse of non-monotonic $5. It is clear from the simple abstract example that I sketched above that McDermott's definition of non-committal model fails to capture the second intuitive constraint, at least when the logic is $5. According to that definition, a model in which p and - p are both believed is a non-committal model of {p}. But there is no plausible sense in which, when p and q are totally unrelated factual propositions, a belief that - q can be said to be grounded in an initial belief that p.; there is no way that an agent, simply by reflecting on its belief that p, might find reason to conclude that q. I think that by looking more closely at McDermott's account of noncommittal models, we can see where it goes wrong. Non-committal models are supposed to be models that maximize the possibilities permitted by the initial belief state. But the problem is that this idea is interpreted too syntactically. What is maximized is the set of possibility statements that are believed to be true. But as the abstract example brings out, this does not necessarily maximize the possibilities. Compare the following two $5 models of { p}, defined relative to just the two sentence letters p and q:
Non-monotonic modal logic Model I:
Model II:
K={w)
K'={u,v}
v(p) = { w } v(q) = 0
v'(p) = {u, v} v'(q) = {u}
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There is clearly a sense in which Model II permits more possibilities than Model I, and a sense in which II admits all the possibilities of I and more. But there are possibility statements that are necessarily true in I, but not in II; one example is M - M q . That is why, if non-committal models of {p} are defined as models of {p} that maximize the possibility statements that are necessarily true, then I is as non-committal as II. Here is a more semantical definition of non-committal model that yields the conclusion that I is not a non-committal model of {p}. First, two preliminary definitions. Note that every $5 model M = ( K , v)7 determines a set of truth valuations--assignments of truth values to all the sentence letters. If w E K, then v w is a truth valuation: vw(p) = T if w E v(p), and F otherwise. Semantically non-committal models will be defined in terms of the set of truth valuations determined by the model. (1) t(M) =df {V~: W E K}. (2) A model M' is a submodel of a model M iff t(M') C t(M). (3) A model M is a semantically non-committal model of a set of sentences F iff M is a model of F that is not a proper submodel of any model of F. A non-monotonic consequence relation like McDermott's, except defined in terms of this concept of non-committal model, will be different from the $5 monotonic consequence relation, and will, in fact, be non-monotonic. It would, I think, accord better with the intuitions that provide the motivation for the definition of non-committal model, and the intuitive account of the modal operators and the non-monotonic consequence relation. But the resulting theory will still not be an appropriate theory to do the main job that non-monotonic modal logic was designed to do: the representation of default reasoning. There remains a problem that makes any non-monotonic $5 an inadequate theory for this purpose. In the concluding part of this note, I will sketch the problem of representing default reasoning, or defeasible presumptions, show why such presumptions cannot be represented in the way suggested by McDermott within any non-monotonic $5 modal logic, and finally suggest an alternative approach that uses a different modal semantic theory: a semantic theory designed for the analysis of counteffactual conditionals. The problem, as I understand it, that first motivated the development of non-monotonic logic is this: we want to be able to represent and explain what is going on when an agent adopts a policy of accepting certain propositions presumptively--that is, in the absence of evidence to the contrary. We know that not all birds can fly, but it still may be a methodologically efficient policy
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to presume that any given bird can fly until we are shown that it cannot. In McDermott and Doyle's non-monotonic logic the defeasible presumption that p is represented by the (material) conditional Mp--~ p. This representation has some initial plausibility: to accept p presumptively is to accept that unless p is known to be false, it is true. But if the logic of " M " is $5, then this representation will not work, however the non-monotonic consequence relation is defined, for the following reason: in $5, L(Mp--~p) is logically equivalent to L ( M - p - - ~ - p ) . So if the presumption that p is represented by the acceptance of (Mp --~ p), then the presumption that p is logically equivalent to the presumption that - p . But the whole idea of defeasible presumption or default assumption is to create an asymmetry between a statement and its negation: to presume that p is to say that in the absence of evidence one way or the other, one should accept that p. Obviously, one cannot coherently presume both p and its negation. So the $5 logical equivalence cited above shows that one cannot, within an $5 theory, represent defeasible presumptions in the way suggested. The problem I think, is not with $5, but with the representation of presumptions. Intuitively, presumptions are not propositions that we accept only after establishing that they are consistent with what we believe (and further, that they will remain consistent with what we believe even after reflecting on our beliefs). They are propositions we begin by accepting, and continue to accept until we are shown that we should not. The idea of presumption is to put the burden of proof on the one who denies the presumptive claim. When the defendant is presumed innocent, it is not incumbent on the defense to prove even that it is possible that the defendant is innocent. It seems to me that to presume that p is to accept that p, and not to accept some proposition weaker than p. The difference between presumptions and settled beliefs should be explained, I think, not in terms of contrasting contents of presumptions and beliefs, but in terms of different reasons that the propositions are accepted, and in terms of the policies the agent has for revising what it accepts in response to new information. Presumptions are accepted for methodological reasons, rather than because we have reason to believe they are true. In the fact of conflicting evidence, presumptive claims are first to go, and giving them up will normally be less disruptive. To characterize the status of presumptions, we need a general framework for representing the dynamics of acceptance--the policies that an agent has for changing what it accepts in response to new information, including information that conflicts with what it accepts. Such a representation could obviously involve principles of inference (that is, principles for changing beliefs, as contrasted with principles of semantic entailment that describe semantic relationships) that are non-monotonic. Adding new evidence can sometimes lead an agent to give up what was previously accepted. But it need not, I think,
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involve the troublesome modal operators representing something like an agent's beliefs about what is compatible with its beliefs that are the distinctive feature of McDermott's non-monotonic logic. To give the vague constructive suggestions I am making a slightly more concrete form, let me point to a semantic theory that I think would be a more useful framework for the representation of default reasoning, and more generally, of the dynamics of belief. Formal semantic theories of conditionals were designed primarily to represent counterfactual conditionals, and not epistemic dispositions, but I think the same abstract framework is suitable for this different task. Suppose one were to represent an agent's dispositions to change what it accepts upon acquiring the new information that p by a set of conditional propositions of the form "if p, t h e n . . . " that the agent accepts initially. 8 If a theory F represents an initial belief state, and p represents a potential piece of new information, then the belief state that the agent is disposed to move to upon acquiring the information that p may be represented by the set Fp, defined as follows:
ffp =dr {q: p > q E F } (where the conditional is expressed by the corner, " > " ) . The logical properties of the conditional (as defined in any of a number of different semantic theories of conditionals, including those of David Lewis [1], John Pollock [4] and myself and Richmond Thomason [5]) will ensure that the set Fp will have the appropriate properties. Specifically, the following claims will always hold: (1) If F is consistent and deductively closed, then so is Fp, provided p is possibly true. (If F represents a possible set of beliefs of an ideally rational agent, then so does Fp, provided that p is a coherent piece of information.) (2) p ~ Fp. (One is disposed to accept p on learning that p.) (3) If p E F, then Fp = F. (If the "new" information is something one already accepts, then one's beliefs do not change.) (4) The relation between the possible new belief states is non-monotonic in the following sense: Fp need not include F, and more generally, p ~ q does not imply that Fq C_Fp. (Learning more, or stronger, new information may dispose one to infer less, or to give up something previously accepted.) How would default reasoning be represented in such a framework? Consider the presumption that birds can fly. To presume this, the agent simply accepts all sentences of the form
Bx > Fx
(1)
(if x is a bird, x can fly). Because it is known that penguins are birds and that penguins cannot fly, the agent will also accept all sentences of the form
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(2)
Px > Bx
and Px > -Fx
.
(3)
(1), (2), and (3) are consistent, and because the hypothetical syllogism is invalid for the conditional, Px > Fx
(4)
will not be a consequence of (1) and (2). The universal closures of the three statements do together imply that there are no penguins, but to presume that all birds can fly is to presume that there are no penguins. What makes (1), as contrasted with (2) and (3), a mere presumption is not its logical form or content, but its status: what happens to it on learning new information. Presumptions tend to be given up before settled beliefs. So, for example, if we discover a penguin and so come to accept a sentence of the form P x , consistency requires that we give up one of (1), (2), or (3). That (1) is clearly the one to go is an indication that it is merely a presumption. While the status of (1)--the fact that it has low priority relative to (2) and (3)--is not indicated by the content of (1), or by the fact that it is accepted, it is indicated by the total set of accepted propositions. One indication of the priority of (1) over (2) is the fact that ( ( P x A B x ) > - - F x ) is accepted. The framework is capable of representing a lot of information about epistemic priorities. One can represent in a natural way not only generalizations with simple exceptions, but also exceptions to exceptions, such as in an example discussed by McDermott [3, p. 42]. All snakes are non-poisonous (All S are non-P), except those in South America (except A), except Andean snakes living above 5,000 feet (except F), except those kept in greenhouses (except G). These presumptive beliefs might be represented by acceptance of the conditional sentences of the following forms: Sx > ~ Px (Sx A Ax) > Px ( S x ^ A x ^ Fx) > - - P x ( S x A A x A F x ^ G x ) > Px.
These are undeveloped suggestions. A lot of work would have to be done to show that conditional semantics provides a suitable framework for representing the kind of reasoning that is the subject of McDermott and Doyle's investigation. But it is a rich and flexible framework with considerable intuitive content. I recommend it to the developers of non-monotonic logic.
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Endnotes 1. The paper and comment were presented at a conference on artificial intelligence and philosophy sponsored by the Sloan Foundation and the Center for Advanced Study in the Behavioral Sciences in March, 1980. 2. I want to thank Drew McDermott, Jon Doyle, Robert Moore and Rich Thomason for comments, discussion, advice and encouragement. 3. Since we are talking about artificial intelligence, I think it is fair to avoid the he/she problem by calling my ideally rational agent "it". 4. Here is a sequence of definitions leading to a definition of "non-committal model of F " . It is slightly simpler than, but equivalent to, McDermott's. (a) For any model M = ( K , R, o ), we define the necessary possibles of M, n(M), as follows: n(M) = {M~b: v(M~b) = K}. (b) For any model M and set of sentences F, we define the expansions of (M, F ) as the set of all models of F U n(M). (c) A sentence M~b is a metapossible of (M, F ) iff for some expansion M' = ( K ' , R', v ' ) of (M, F ) , v'(th) n K' is nonempty. (d) A model M is a non-committal model of F iff the set of metapossibles of (M, F ) coincides with n(M). 5. Strictly, I think, the appropriate logic of belief should be a weakened S5--the logic K45, which is $5 without the axiom L~b---> ~b, and without the assumption that all beliefs are true. In the semantics for K45, R must be transitive and euclidean. A relation is euclidean iff any two things that stand in the relation to the same thing stand in the relation to each other. In K45, the principle L(L~b--> 4~) is valid, but not the principle Lib--> ~b. Rational agents must believe their beliefs are true, but their beliefs need not be true. We can ignore the difference between $5 and K45 because we define K to be the set of possible worlds compatible with the agent's beliefs. Relative to this set, the relation must be an equivalence relation. That is, if R is euclidean and the range of R is restricted to the set {x • Rax}, where a is the actual world, then R becomes an equivalence relation. Intuitively, the point is this: our models, as defined in the text, represent belief states (the way the agent believes the world to be) but contain no representation of the way the world actually is. If the agent has any false beliefs, then the actual world will not be in the set K. If we were to add a world to represent the actual world--a world in which the agent has the not necessarily correct beliefs represented by K - - t h e n we would need a weaker belief logic. But if we keep the assumption that the total set K represents just the worlds compatible world the agent's beliefs, then we can use the stronger belief logic with equivalent results. [Added in 1992] As Robert Moore's later work in autoepistemic logic made clear, the difference between the logics of knowledge and belief--between $5 and K45--is more significant and relevant than this note suggested. The main argument of this paper against
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M c D e r m o t t ' s syntactic definition of non-committal model works as well for K45 as for $5. 6. [ A d d e d in 1992] As r e m a r k e d in note 5 above, since it is belief rather than knowledge that is represented by " L " , strictly speaking, our assumptions imply that the consequence relation is a K45 consequence relation, rather than an $5 relation. But it is true in K45 as well as $5 that M - M p entails - M p , so this problem infects a theory based on K45 as well as a theory based on $5. 7. Since we are here talking just about $5 models we can drop reference to the relation R, since we may assume that the relation is universal. 8. It must be emphasized that the ordinary counterfactual conditionals one accepts do not represent such epistemic dispositions. My acceptance of the conditional, "if Oswald had not shot Kennedy, Kennedy would be alive t o d a y " does not indicate that I am disposed to accept that K e n n e d y is alive today upon learning that Oswald did not shoot him. Obviously, if I learned that I would give up the counterfactual. I am suggesting only that conditionals with the same logical properties as counterfactuals can represent epistemic dispositions.
References [1] D. Lewis, Counterfactuals (Oxford University Press, Oxford, 1973). [2] D. McDermott and J. Doyle, Non-monotonic logic I, Artif. Intell. 13 (1980) 41-72. [3] D. McDermott, Non-monotonic logic II: non-monotonic modal theories, J. ACM 29 (1982) 33-57. [4] J. Pollock, Subjunctive Reasoning (Reidel, Dordrecht, Netherlands, 1976). [5] R. Stalnaker and R. Thomason, A semantic analysis of conditional logic, Theoria 36 (1970) 23-42.
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ARTINT 1032
A note on non-monotonic modal logic* Robert Stalnaker Department of Linguistics and Philosophy, 20D-213, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
This paper, which began as a comment on a talk by Drew McDermott, 1 was completed in the summer of 1980. It circulated in typescript, but was not published at the time. A lot has happened in the study of nonmonotonic reasoning since that time, and the paper is now quite dated, but it is being published now, just to put it on the record, since it has been widely cited (mainly as a result of citations in Robert Moore's influential work on autoepistemic logic). Other than grammatical, typographical and notational corrections, and a few cautionary comments added to endnotes, the paper is unchanged from the 1980 version. 2 I want, in this note, to raise some questions and make some suggestions about the intuitive semantic foundations of the non-monotonic modal logics that Drew McDermott and Jon Doyle have developed in [2] and [3]. In particular, I want to ask about the intended meaning and application of the non-monotonic consequence relation, and about the intended meanings of the modal operators. I will suggest some answers to the questions I will raise-answers that I think are implicit in McDermott's motivating remarks in [3]--but I will argue that these answers lead to serious problems for the
Correspondence to: R. Stalnaker, Department of Linguistics and Philosophy, 20D-213, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. E-mail: [email protected]. * Editor's Note. This article is being published long after it was written because of its importance in the history of the field. It was called to our attention by Robert Moore who wrote: "The paper was written about twelve years ago as a commentary on Drew McDermott's paper on nonmonotonic modal logics (which was ultimately published in 1982 in JACM). It was a very important influence on my development of autoepistemic logic, and has been cited by numerous other papers, but has never been published. Perhaps its most lasting contribution to the theory of nonmonotonic reasoning is the notion of a "stable set" of beliefs (that is, stable under introspection and inference), a notion which plays a key role in many nonmonotonic systems. I believe this is a very important paper, and it should be published in the A1J to get it into the archival record." 0004-3702/93/$06.00 © 1993--Elsevier Science Publishers B.V. All rights reserved
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interpretation of the theory. In conclusion I will make some brief remarks about the problem that motivated the development of this kind of logic, and suggest an alternative means of solving it. I will begin with a very general challenge to the possibility of non-monotonic logic, and a suggestion about how the general challenge might be met. A logic is non-monotonic if it has a non-monotonic consequence relation. A relation R between a set of sentences F and a sentence p is monotonic if and only if whenever F~ C_ F2, {p:R(F~, p)} C_{ p : R(F2, p)}. But how is it possible for any consequence relation to fail to be monotonic? The question is not, how can we define a non-monotonic relation between the sets of sentences and the sentences of some language. This is simple enough. The problem is to understand how any such relation could be a consequence relation. For it seems that one can give a very abstract and general definition of the notion of semantic consequence---one that should apply to any possible interpreted formal language that contains sentences that have truth values--and that any relation fitting this definition will be monotonic. Suppose we have a formalized language--any language at all--with primitive expressions divided into logical and descriptive expressions, and formation rules defining sentences, and perhaps complex expressions of other grammatical categories. A semantic analysis of such a language will be an account of the way semantic values are assigned to expressions. Semantic values will be defined relative to a model which assigns semantic values to the primitive descriptive expressions. Besides a definition of a model, the semantic analysis will include semantic rules which determine the semantic values of all complex expressions of the language as a function of the values of their parts (together, in some cases, with global features of the model, such as a domain of individuals, or a set of possible worlds). Models and semantic values can be any sorts of things, so long as the semantic value of a sentence somehow determines a truth value. Given any language and semantic theory meeting these conditions, we can define notions of logical truth and semantic consequence as follows: logical truths are sentences true in all models; a sentence p is a semantic consequence of a set F if and only if every model that makes all of the sentences in F true also makes p true. Now this characterization of the form of a semantic theory is very abstract and unconstrained. It puts no limit on what semantic values can be, or on what form the models might take, or on the way semantic rules explain the values of complex expressions as a function of the model. But no matter what weird properties the rules impose on the logical operators, no matter what strange entities are postulated by the models, the consequence relation for any semantic theory fitting this mold will obviously be monotonic. So it seems that within this orthodox approach to semantic analysis, one cannot explain the distinctive non-standard feature of non-monotonic logic in terms of non-
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standard kinds of models, semantic values or semantic rules. One must, it seems, give an alternative general account of semantic consequence. And what is wanted is not a specific technical definition for a specific non-monotonic logic, but a general account of what consequence is supposed to mean--of what concept some specific technical definition is trying to capture. And one should expect such an account to explain why it is reasonable to call the concept by the name "consequence". My point is to offer a challenge, not a refutation. It follows from what I have said that if non-monotonic modal logic is given a semantics that fits the orthodox pattern, then this semantics will determine a well-defined monotonic consequence relation, but this does not exclude the possibility that other related notions that have some claim to be called by the name "consequence" might be defined. Here is one way that this might happen: suppose the sentences of some language contain an operator that is context-dependent; suppose further that part of the context that is relevant to the truth conditions of sentences containing this operator is the fact that that sentence occurs in the course of giving an argument with certain premisses. In particular, suppose that a possibility operator " M " is interpreted relative to the fact that some set of sentences F is the set of premisses of an argument. ( " M p " might still mean "p is true in some possible world" so long as the relevant set of possible worlds is defined relative to F). In such a theory, the premiss set F would be playing two different roles, both of which are relevant to the question whether some sentence p is a consequence of some set F. First, F helps to fix the interpretation of the sentences in the argument (that is, helps determine what propositions they express); second, F determines which propositions are the premisses of the argument. One could separate, conceptually, these roles and define a well-behaved monotonic consequence relation in the standard way. But if one does not separate them, then the consequence relation one defines might be non-monotonic. Since adding a premiss may alter the interpretation of the prior premisses or the conclusion, doing this may make a valid argument invalid. Here is a trivial example that illustrates the general point. Suppose we have a language containing an indexical numerical pronoun " a " that denotes, when used in the course of giving an argument, the number of premisses of the argument. Now consider this argument: a is the number of living ex-presidents of the United States. Therefore, there is an odd number of living ex-presidents. This argument had [in 1980] a false premiss and conclusion, but it is valid according to one intelligible way of defining validity for this language. But now add a second premiss: a is the only even prime.
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Now the premisses are both true [again, in 1980], but the conclusion remains false. We have turned a valid argument into an invalid one by adding a premiss. I think something like this is going on in McDermott and Doyle's nonmonotonic modal logic. That is, I think that the explanation for the nonmonotonic character of the consequence relation is to be found in the way in which the interpretation of the modal operators is dependent on the set of sentences that constitutes the premiss set of the non-monotonic inferences. To make out this claim I will try to spell out the semantic analysis that I think is implicit in the theory. I will begin with a rough intuitive sketch of what the theory, as I understand it, is intended to represent. The premiss set (the set of proper axioms) of a non-monotonic theory represents a total set of given information--the initial beliefs of an ideally rational agent. The agent is able to make inferences--to expand its 3 set of beliefs. The crucial assumption that motivates the distinctive features of non-monotonic logic is that the agent makes inferences, not only from the given information, but also from the fact that it has the information that it has. That is, the crucial assumption is that the agent can reflect on its beliefs and draw conclusions about what it does and does not believe. If p is entailed by the given information, then the agent can infer not only p, but also that it believes that p. If q cannot be inferred from the given information, then the agent can conclude that it does not believe that q - - t h a t not-q is compatible with its beliefs. So the ideally rational agents that the theory seeks to describe have access to their own beliefs: they know, or can determine, that they believe something when they do, and that they do not when they do not. The modal operators, " M " and " L " , serve to represent information about what the agent believes. " M p " represents something like the claim that p is compatible with what the agent believes, and " L p " represents something like the claim that the agent believes that p. But we have to be careful here. Our assumption is that the agent's beliefs are changing as it reflects on them and makes inferences from them about what it believes. Does " M p " mean that p is compatible with the agent's initial beliefs, or that p will be compatible with the agent's beliefs after it is finished reflecting on them and drawing conclusions from them? Clearly it must be the latter. When an agent draws a conclusion of the form Mp, it is not concluding simply that p was compatible with its initial beliefs. It is concluding that p is compatible with its beliefs, and will remain so at least until it receives new information. If " M p " meant simply that p was compatible with the initial information, then it might be rational for an agent to conclude both not-p and Mp. The proper axioms or the premiss set, of a non-monotonic theory represents an initial belief state; the fixed points of the theory represent various possible final belief states--belief states that the agent might reach by reflecting on its beliefs and drawing conclusions from them and about them. The operators M
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and L are used to represent statements about the final belief state that will be reached. A final belief state must meet two intuitive conditions: first, it must be stable in the sense that no further conclusions could be drawn by an ideally rational agent in such a state. Second, all the agent's beliefs must be, in some sense, grounded in the initial belief state--in the given information. The first condition is relatively easy to make precise: a set of sentences F is stable, in the relevant sense, if and only if it meets the following three conditions: (1) F is deductively closed; (2) if p E F, then Lp E F; (3) if n o t - p ~ F , then Mp E F. The second intuitive condition is more difficult to turn into a formal constraint, but M c D e r m o t t and Doyle's syntactic definition of a fixed point, and their semantic definition of a non-committal model are, I take it, attempts to capture this idea (as well as the idea of stability). Fixed points will all be stable in the sense defined. In terms of this intuitive picture, what does the non-monotonic consequence relation represent? " F ~ p " represents the claim that any final belief state based on initial given information F will include the belief that p. As predicted, the premiss set F is playing two roles relevant to non-monotonic consequence: first, it determines what sentences are believed in the relevant final belief state; second, it constrains the interpretation of the sentences containing the operators M and L. It is only because it plays these two roles at once that the relation can be both a consequence relation and a relation that is nonmonotonic. Now I want to consider how this intuitive picture gets turned into a formal semantic analysis. I will assume that the language is a standard modal propositional calculus. A final belief state will be represented in the semantics for this language by a Kripke model: an ordered triple M = ( K, R, v } where K is a non-empty set of possible worlds---intuitively, the possible worlds that are compatible with the agent's final belief state; R is a binary relation on K - - t h e accessibility relation; and o is the valuation function that assigns propositions-sets of possible worlds--to the sentence letters. The intuitive role of the accessibility relation is to determine what the agent's beliefs would be if each of the various possible worlds in K were actual. The reason such a relation is necessary is that the agent has beliefs about its beliefs. To model an agent's belief that it believes that p, for example, we need not a set of possible worlds in all of which p is true, but a set of possible worlds in all of which the agent believes that p is true. But to interpret the claim that the agent believes that p in some world w E K, we need a way of determining the set of possible worlds compatible with what the agent believes in w (which may be different from the set K, which represents the possible worlds compatible with what the agent
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believes in the actual world). The accessibility relation represents this information in the following way: the possible worlds compatible with the agent's beliefs in w is the set {u: w R u } . This is the set of possible worlds accessible from w. We will want to put some constraints on this relation, but for the moment, I will defer the question of what they should be. So far, we have an ordinary monotonic modal semantics, interpreted as a logic of belief. To allow for the definition of the non-monotonic consequence relation, we need to put further constraints on the models. In terms of our intuitive picture, a model represents a final, stable belief state. But final belief states are defined relative to initial belief states--relative to the received information that initiates the reflection that leads to the final belief state. Let F be the set of sentences representing the initial information. Then an admissible F-model must be a non-committal model of F. The definition of non-committal model given in [3] is somewhat complicated and intuitively opaque, but I take it that it is intended to capture the two intuitive requirements that final belief states be stable (in the sense defined above) and that they be grounded in the given information. 4 Now we can define two consequence relations, first one that may be non-monotonic; second, a more orthodox consequence relation that separates the two roles played by the premiss set: (1) For any set of sentences F and sentence ~b, F ~ ~b iff for all noncommittal models, ( K , R, v) of F, A {v(~O): qJ E F} C_ v(~b). (2) For any set of sentences F and sentence ~b, F ~ ~b iff for all sets of sentences A and all non-committal models ( K , R, v) of zl,
N {v(~,): ~,~r} c_ v(~). Now let me raise the question that ! deferred above: what constraints should be put on the relation R? The main point I want to make is that we do not have free choice here. The intuitive assumptions that motivate the characterization of final belief states requires that R be an equivalence relation, and that the monotonic logic of belief should be S5.s The reason for this is that the formal assumption that the accessibility relation between belief worlds is an equivalence relation corresponds to the intuitive assumption that the agent has access to its own beliefs--that it knows what it believes and what it does not believe. Without this assumption, we have no reason to give fixed points and noncommittal models any special status. Whatever constraints are put on the accessibility relation, the following will be true, given McDermott's definition of non-committal model: all noncommittal models are $5 models in the sense that all theorems o f $5 are true in all possible worlds in all such models. Hence, whatever assumptions one makes about the relation R, the monotonic consequence relation ((2) above) will be
an $5 consequence relation. This means that it follows from the meanings given to M and L that the operators are $5 modal operators.
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I should emphasize that my point concerns intuitive foundations and not technical consequences. In particular, I am not denying that one can consistently define, as M c D e r m o t t does, a notion of non-committal model in a modal semantic theory that is weaker than $5, and that the relation being a non-committal model of a certain theory will be different if the modal semantics is weaker than $5. The reason that this is possible, despite the fact that all non-committal models are $5 models is as follows: whether or not a model is a non-committal model of a theory F depends on whether or not certain other models, which need not be non-committal, exist. If one allows these other models to be non-S5 models, this will make a difference to the question whether a given $5 model is a non-committal model of F. So I am not arguing that M c D e r m o t t is mistaken when he claims, for example, that non-monotonic consequence is different if the logic is $4 from what it is if the logic is $5. My point might be put this way: the intuitive meaning of M~b in non-monotonic logic is something like this: "~b is compatible with the agent's final stable belief state." But final belief states are represented by noncommittal models, and non-committal models are always $5 models. Hence the monotonic consequence relation is the $5 consequence relation. If I am right so far, then there is a problem. For non-monotonic $5 has, as M c D e r m o t t puts it, a serious bug. The bug is that non-monotonic $5 is monotonic. The two notions of consequence which have been so carefully separated collapse together in $5. Every $5 consistent and closed extension of a set of sentences is a fixed point of that set. Every $5 model of a theory F is a non-committal model of F. So $5 will not do at all as an underlying logic for non-monotonic logic. I am suggesting that non-monotonic logic, as M c D e r m o t t develops it, faces a serious dilemma. If its underlying logic is weaker than $5, then the operators M and L cannot mean what they are supposed to mean. But if the logic is as strong as $5, then the theory cannot be used to do what it was developed to do: the interesting and distinctive feature of the logic disappears. Let me use a simple abstract example to illustrate the problem with non-monotonic $5 that McDermott points out. Suppose my initial information consists of just the single sentence p. p is the only proper axiom; initially, I believe only p. What should I conclude from reflecting on this information? First, I conclude Lp, Mp, LMp, MLp, and so forth. Also, I add all the deductive consequences of p and Lp. What about q? (Suppose q is some totally independent factual proposition.) One might think that I could conclude Mq since I can see that nothing I believe could possibly lead me to conclude that q is true. But I cannot conclude M q for the following reason: Mq is inconsistent with another possibility statement, namely M - - M q . In $5, M - - M q entails --Mq. And since what this other possibility statement claims to be possible is also compatible with p, I have as much reason to add M ~ Mq to my beliefs as I have to add Mq. I cannot consistently add both, but if my beliefs are to be
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stable, I must add one or the other. So I have a free choice. But if I choose to add M - Mq, then I will conclude - q . It is clear from this that I can conclude anything at all that is consistent with the initial beliefs. Any $5 model of { p } will be a non-committal model of {p}. McDermott's response to the collapse of non-monotonic $5 into monotonic $5 was to drop down to $4: in effect, to define non-committal models in terms of a weaker logic. This is technically possible, but it does not change the fact that the monotonic consequence relation is the $5 consequence relation: all $5 inferences remain semantically valid in the sense that they preserve truth in all possible worlds of all non-committal models, which means in all admissible interpretations of the language. The move to $4 is simply a decision to define non-committal models in terms of a semantically incomplete logic. M c D e r m o t t owes us a non-ad-hoc explanation of why one should define non-committal models in this way. 6 An alternative response to the collapse would be to keep the logic $5, but to change the definition of non-committal model (and of fixed point) so that it better represents the intuitive idea that motivated it. Recall that the intuition behind the idea of non-committal model (as I reconstructed it) had two parts. First, non-committal models represent stable belief states: belief states from which no more could be inferred simply by reflecting on them by an agent who had access to its beliefs. Second, a non-committal model of a theory F represents a belief state that is grounded in the initial information represented by F. As we saw, the first of these constraints is much clearer and was more easily turned into a formal constraint than is the second. It is the first constraint that implies that all non-committal models are $5 models, but a different definition of non-committal model---one that captured the second idea in a different way--might avoid the collapse of non-monotonic $5. It is clear from the simple abstract example that I sketched above that McDermott's definition of non-committal model fails to capture the second intuitive constraint, at least when the logic is $5. According to that definition, a model in which p and - p are both believed is a non-committal model of {p}. But there is no plausible sense in which, when p and q are totally unrelated factual propositions, a belief that - q can be said to be grounded in an initial belief that p.; there is no way that an agent, simply by reflecting on its belief that p, might find reason to conclude that q. I think that by looking more closely at McDermott's account of noncommittal models, we can see where it goes wrong. Non-committal models are supposed to be models that maximize the possibilities permitted by the initial belief state. But the problem is that this idea is interpreted too syntactically. What is maximized is the set of possibility statements that are believed to be true. But as the abstract example brings out, this does not necessarily maximize the possibilities. Compare the following two $5 models of { p}, defined relative to just the two sentence letters p and q:
Non-monotonic modal logic Model I:
Model II:
K={w)
K'={u,v}
v(p) = { w } v(q) = 0
v'(p) = {u, v} v'(q) = {u}
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There is clearly a sense in which Model II permits more possibilities than Model I, and a sense in which II admits all the possibilities of I and more. But there are possibility statements that are necessarily true in I, but not in II; one example is M - M q . That is why, if non-committal models of {p} are defined as models of {p} that maximize the possibility statements that are necessarily true, then I is as non-committal as II. Here is a more semantical definition of non-committal model that yields the conclusion that I is not a non-committal model of {p}. First, two preliminary definitions. Note that every $5 model M = ( K , v)7 determines a set of truth valuations--assignments of truth values to all the sentence letters. If w E K, then v w is a truth valuation: vw(p) = T if w E v(p), and F otherwise. Semantically non-committal models will be defined in terms of the set of truth valuations determined by the model. (1) t(M) =df {V~: W E K}. (2) A model M' is a submodel of a model M iff t(M') C t(M). (3) A model M is a semantically non-committal model of a set of sentences F iff M is a model of F that is not a proper submodel of any model of F. A non-monotonic consequence relation like McDermott's, except defined in terms of this concept of non-committal model, will be different from the $5 monotonic consequence relation, and will, in fact, be non-monotonic. It would, I think, accord better with the intuitions that provide the motivation for the definition of non-committal model, and the intuitive account of the modal operators and the non-monotonic consequence relation. But the resulting theory will still not be an appropriate theory to do the main job that non-monotonic modal logic was designed to do: the representation of default reasoning. There remains a problem that makes any non-monotonic $5 an inadequate theory for this purpose. In the concluding part of this note, I will sketch the problem of representing default reasoning, or defeasible presumptions, show why such presumptions cannot be represented in the way suggested by McDermott within any non-monotonic $5 modal logic, and finally suggest an alternative approach that uses a different modal semantic theory: a semantic theory designed for the analysis of counteffactual conditionals. The problem, as I understand it, that first motivated the development of non-monotonic logic is this: we want to be able to represent and explain what is going on when an agent adopts a policy of accepting certain propositions presumptively--that is, in the absence of evidence to the contrary. We know that not all birds can fly, but it still may be a methodologically efficient policy
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to presume that any given bird can fly until we are shown that it cannot. In McDermott and Doyle's non-monotonic logic the defeasible presumption that p is represented by the (material) conditional Mp--~ p. This representation has some initial plausibility: to accept p presumptively is to accept that unless p is known to be false, it is true. But if the logic of " M " is $5, then this representation will not work, however the non-monotonic consequence relation is defined, for the following reason: in $5, L(Mp--~p) is logically equivalent to L ( M - p - - ~ - p ) . So if the presumption that p is represented by the acceptance of (Mp --~ p), then the presumption that p is logically equivalent to the presumption that - p . But the whole idea of defeasible presumption or default assumption is to create an asymmetry between a statement and its negation: to presume that p is to say that in the absence of evidence one way or the other, one should accept that p. Obviously, one cannot coherently presume both p and its negation. So the $5 logical equivalence cited above shows that one cannot, within an $5 theory, represent defeasible presumptions in the way suggested. The problem I think, is not with $5, but with the representation of presumptions. Intuitively, presumptions are not propositions that we accept only after establishing that they are consistent with what we believe (and further, that they will remain consistent with what we believe even after reflecting on our beliefs). They are propositions we begin by accepting, and continue to accept until we are shown that we should not. The idea of presumption is to put the burden of proof on the one who denies the presumptive claim. When the defendant is presumed innocent, it is not incumbent on the defense to prove even that it is possible that the defendant is innocent. It seems to me that to presume that p is to accept that p, and not to accept some proposition weaker than p. The difference between presumptions and settled beliefs should be explained, I think, not in terms of contrasting contents of presumptions and beliefs, but in terms of different reasons that the propositions are accepted, and in terms of the policies the agent has for revising what it accepts in response to new information. Presumptions are accepted for methodological reasons, rather than because we have reason to believe they are true. In the fact of conflicting evidence, presumptive claims are first to go, and giving them up will normally be less disruptive. To characterize the status of presumptions, we need a general framework for representing the dynamics of acceptance--the policies that an agent has for changing what it accepts in response to new information, including information that conflicts with what it accepts. Such a representation could obviously involve principles of inference (that is, principles for changing beliefs, as contrasted with principles of semantic entailment that describe semantic relationships) that are non-monotonic. Adding new evidence can sometimes lead an agent to give up what was previously accepted. But it need not, I think,
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involve the troublesome modal operators representing something like an agent's beliefs about what is compatible with its beliefs that are the distinctive feature of McDermott's non-monotonic logic. To give the vague constructive suggestions I am making a slightly more concrete form, let me point to a semantic theory that I think would be a more useful framework for the representation of default reasoning, and more generally, of the dynamics of belief. Formal semantic theories of conditionals were designed primarily to represent counterfactual conditionals, and not epistemic dispositions, but I think the same abstract framework is suitable for this different task. Suppose one were to represent an agent's dispositions to change what it accepts upon acquiring the new information that p by a set of conditional propositions of the form "if p, t h e n . . . " that the agent accepts initially. 8 If a theory F represents an initial belief state, and p represents a potential piece of new information, then the belief state that the agent is disposed to move to upon acquiring the information that p may be represented by the set Fp, defined as follows:
ffp =dr {q: p > q E F } (where the conditional is expressed by the corner, " > " ) . The logical properties of the conditional (as defined in any of a number of different semantic theories of conditionals, including those of David Lewis [1], John Pollock [4] and myself and Richmond Thomason [5]) will ensure that the set Fp will have the appropriate properties. Specifically, the following claims will always hold: (1) If F is consistent and deductively closed, then so is Fp, provided p is possibly true. (If F represents a possible set of beliefs of an ideally rational agent, then so does Fp, provided that p is a coherent piece of information.) (2) p ~ Fp. (One is disposed to accept p on learning that p.) (3) If p E F, then Fp = F. (If the "new" information is something one already accepts, then one's beliefs do not change.) (4) The relation between the possible new belief states is non-monotonic in the following sense: Fp need not include F, and more generally, p ~ q does not imply that Fq C_Fp. (Learning more, or stronger, new information may dispose one to infer less, or to give up something previously accepted.) How would default reasoning be represented in such a framework? Consider the presumption that birds can fly. To presume this, the agent simply accepts all sentences of the form
Bx > Fx
(1)
(if x is a bird, x can fly). Because it is known that penguins are birds and that penguins cannot fly, the agent will also accept all sentences of the form
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(2)
Px > Bx
and Px > -Fx
.
(3)
(1), (2), and (3) are consistent, and because the hypothetical syllogism is invalid for the conditional, Px > Fx
(4)
will not be a consequence of (1) and (2). The universal closures of the three statements do together imply that there are no penguins, but to presume that all birds can fly is to presume that there are no penguins. What makes (1), as contrasted with (2) and (3), a mere presumption is not its logical form or content, but its status: what happens to it on learning new information. Presumptions tend to be given up before settled beliefs. So, for example, if we discover a penguin and so come to accept a sentence of the form P x , consistency requires that we give up one of (1), (2), or (3). That (1) is clearly the one to go is an indication that it is merely a presumption. While the status of (1)--the fact that it has low priority relative to (2) and (3)--is not indicated by the content of (1), or by the fact that it is accepted, it is indicated by the total set of accepted propositions. One indication of the priority of (1) over (2) is the fact that ( ( P x A B x ) > - - F x ) is accepted. The framework is capable of representing a lot of information about epistemic priorities. One can represent in a natural way not only generalizations with simple exceptions, but also exceptions to exceptions, such as in an example discussed by McDermott [3, p. 42]. All snakes are non-poisonous (All S are non-P), except those in South America (except A), except Andean snakes living above 5,000 feet (except F), except those kept in greenhouses (except G). These presumptive beliefs might be represented by acceptance of the conditional sentences of the following forms: Sx > ~ Px (Sx A Ax) > Px ( S x ^ A x ^ Fx) > - - P x ( S x A A x A F x ^ G x ) > Px.
These are undeveloped suggestions. A lot of work would have to be done to show that conditional semantics provides a suitable framework for representing the kind of reasoning that is the subject of McDermott and Doyle's investigation. But it is a rich and flexible framework with considerable intuitive content. I recommend it to the developers of non-monotonic logic.
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Endnotes 1. The paper and comment were presented at a conference on artificial intelligence and philosophy sponsored by the Sloan Foundation and the Center for Advanced Study in the Behavioral Sciences in March, 1980. 2. I want to thank Drew McDermott, Jon Doyle, Robert Moore and Rich Thomason for comments, discussion, advice and encouragement. 3. Since we are talking about artificial intelligence, I think it is fair to avoid the he/she problem by calling my ideally rational agent "it". 4. Here is a sequence of definitions leading to a definition of "non-committal model of F " . It is slightly simpler than, but equivalent to, McDermott's. (a) For any model M = ( K , R, o ), we define the necessary possibles of M, n(M), as follows: n(M) = {M~b: v(M~b) = K}. (b) For any model M and set of sentences F, we define the expansions of (M, F ) as the set of all models of F U n(M). (c) A sentence M~b is a metapossible of (M, F ) iff for some expansion M' = ( K ' , R', v ' ) of (M, F ) , v'(th) n K' is nonempty. (d) A model M is a non-committal model of F iff the set of metapossibles of (M, F ) coincides with n(M). 5. Strictly, I think, the appropriate logic of belief should be a weakened S5--the logic K45, which is $5 without the axiom L~b---> ~b, and without the assumption that all beliefs are true. In the semantics for K45, R must be transitive and euclidean. A relation is euclidean iff any two things that stand in the relation to the same thing stand in the relation to each other. In K45, the principle L(L~b--> 4~) is valid, but not the principle Lib--> ~b. Rational agents must believe their beliefs are true, but their beliefs need not be true. We can ignore the difference between $5 and K45 because we define K to be the set of possible worlds compatible with the agent's beliefs. Relative to this set, the relation must be an equivalence relation. That is, if R is euclidean and the range of R is restricted to the set {x • Rax}, where a is the actual world, then R becomes an equivalence relation. Intuitively, the point is this: our models, as defined in the text, represent belief states (the way the agent believes the world to be) but contain no representation of the way the world actually is. If the agent has any false beliefs, then the actual world will not be in the set K. If we were to add a world to represent the actual world--a world in which the agent has the not necessarily correct beliefs represented by K - - t h e n we would need a weaker belief logic. But if we keep the assumption that the total set K represents just the worlds compatible world the agent's beliefs, then we can use the stronger belief logic with equivalent results. [Added in 1992] As Robert Moore's later work in autoepistemic logic made clear, the difference between the logics of knowledge and belief--between $5 and K45--is more significant and relevant than this note suggested. The main argument of this paper against
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M c D e r m o t t ' s syntactic definition of non-committal model works as well for K45 as for $5. 6. [ A d d e d in 1992] As r e m a r k e d in note 5 above, since it is belief rather than knowledge that is represented by " L " , strictly speaking, our assumptions imply that the consequence relation is a K45 consequence relation, rather than an $5 relation. But it is true in K45 as well as $5 that M - M p entails - M p , so this problem infects a theory based on K45 as well as a theory based on $5. 7. Since we are here talking just about $5 models we can drop reference to the relation R, since we may assume that the relation is universal. 8. It must be emphasized that the ordinary counterfactual conditionals one accepts do not represent such epistemic dispositions. My acceptance of the conditional, "if Oswald had not shot Kennedy, Kennedy would be alive t o d a y " does not indicate that I am disposed to accept that K e n n e d y is alive today upon learning that Oswald did not shoot him. Obviously, if I learned that I would give up the counterfactual. I am suggesting only that conditionals with the same logical properties as counterfactuals can represent epistemic dispositions.
References [1] D. Lewis, Counterfactuals (Oxford University Press, Oxford, 1973). [2] D. McDermott and J. Doyle, Non-monotonic logic I, Artif. Intell. 13 (1980) 41-72. [3] D. McDermott, Non-monotonic logic II: non-monotonic modal theories, J. ACM 29 (1982) 33-57. [4] J. Pollock, Subjunctive Reasoning (Reidel, Dordrecht, Netherlands, 1976). [5] R. Stalnaker and R. Thomason, A semantic analysis of conditional logic, Theoria 36 (1970) 23-42.