Belief Change as Propositional Update

Belief Revision as Propositional Update 1 Belief Change as Propositional Update Renée Elio Francis Jeffry Pelletier University of Alberta Edmonton, Alberta T6G 2H1 Belief Revision as Propositional Update 2 Abstract In this study, we examine the problem of belief revision, defined as deciding whic h of several initially-accepted sentences to disbelieve, when new information presents a l ogical inconsistency with the initial set. In the first three experiments, the initial sentence set included a conditional sentence, a non-conditional sentence, and an inferred conclusi on drawn from the first two. The new information contradicted the inferred conclusion. Results indicated that the conditional sentences were more readily abandoned than non-c onditional sentences, even when either choice would lead to a consistent belief state, and that this preference was more pronounced when problems used natural language cover sto ries rather than symbols. The pattern of belief revision choices differed depending on whe ther the contradicted conclusion from the initial belief set had been a modus ponens or m odus tollens inference. Two additional experiments examined alternative model-theoretic definitions of minimal change to a belief state, using problems that contained multiple mo dels of the initial belief state and of the new information that provided the contradiction. The results indicated that people did not follow any of four formal definitions of minimal change on these problems. The new information and the contradiction it offered was not, for example, used to select a particular model of the initial belief state as a way of reconci ling the contradiction. The preferred revision was to retain only those initial sentences th at had the same, unambiguous truth value within and across both the initial and new info rmation sets. The study and results are presented in the context of certain logic-based for malizations of belief revision, syntactic and model-theoretic representations of belief stat es, and performance models of human deduction. Principles by which some types of sent ences might be more "entrenched" than others in the face of contradiction are also discuss ed from the perspective of induction and theory revision. Belief Revision as Propositional Update 3 Belief Change as Propositional Update Suppose you need to send an express courier package to a colleague who is away at a conference. You believe that whenever she is in New York City and the New York R angers are playing a home game, she stays at the Westin Mid-Manhattan Hotel. You also believe that she is in New York City this weekend and that the Rangers are playing this w eekend as well. You call up the Westin Mid-Manhattan Hotel and you find out that she is n't there. Something doesn't fit. What do you believe now? Well, assuming that you accep t the hotel's word that she isn't there, there are various (logically consistent) ways to recon cile the contradiction between what you used to believe and this new information. First, y ou could believe that she is in New York City and that the Rangers are indeed playing, bu t disbelieve the conditional that says whenever both of these are true, then she stays at the Westin Mid-Manhattan Hotel. Alternatively, you could continue to believe the condition al, but decide that either she isn't in New York this weekend or that the Rangers aren't pla ying a home game (or possibly both). Which do you choose as your new set of beliefs? Belief change—the process by which a rational agent makes the transition from one belief state to another—is an important component for most intelligent activity done by e pistemic agents, both human and artificial. When such agents learn new things about the world, they sometimes come to recognize that new information extends or conflicts with t heir existing belief state. In the latter case, rational reasoners would identify which of the old and new beliefs clash to create the inconsistency, decide whether in fact to accept the new information, and, if that is the choice, to eliminate certain old beliefs in favor of the new information. Alternatively, new information may not create any inconsistency with o ld information at all. In this case, the reasoner can simply add the new information to the current set of beliefs, along with whatever additional consequences this might entail. Although this is an intuitively attractive picture, the principles behind belief-state change are neither well-understood nor agreed-upon. Belief revision has been studied fro Belief Revision as Propositional Update 4 m a formal perspective in the artificial intelligence (AI) and philosophy literatures and fro m an empirical perspective in the psychology and management-science literatures. One of the practical motivations for AI's concern with belief revision, as portrayed in our openin g scenario, is the development of knowledge bases as a kind of intelligent database: one e nters information into the knowledge base and the knowledge base itself constructs and st ores the consequences of this information—a process which is non-monotonic in nature (i .e., accepted consequences of previously-believed information may be abandoned). More generally, the current belief state of any artificial agent may be contradicted either when t he world itself changes (an aspect of the so-called frame problem) or when an agent's kno wledge about a static world simply increases. Katsuno and Mendelson (1991) distinguish between these two cases, calling the former belief update and latter belief revision. Altho ugh much of the AI belief revision work focuses on formalizing competence theories of u pdate and revision, prescriptive principles for how artificial agents “should” resolve confl ict in the belief revision case—where there is a need to contract the set of accepted propo sitions in order to resolve a recognized contradiction—are far from settled. From the pers pective of human reasoning, we see an important interplay between issues of belief revisi on and deductive reasoning, particularly in terms of the kind of representational assumpti ons made about how a belief state should be modeled. But while human performance on c lassical deductive problems has been extensively studied, both Rips (1994, p. 299) and H arman (1986, p. 7) have noted the need for descriptive data and theories on how people re solve inconsistency when new information about a static world is presented. The studies we present in this article are concerned exactly with this issue. We make two simplifications in our portrayal of belief revision and the paradigm we used to investigate it. The first concerns what we refer to as "beliefs." Here, beliefs ar e sentences that people are told to accept as true, in the context of resolving some (subseq uent) contradiction arising from new information that is provided. Being told to accept so mething as true is not necessarily the same as believing it to be true. The contradictions w Belief Revision as Propositional Update 5 e introduce in our paradigm are not probes into a person's pre-existing belief system (e.g., as in social cognition investigations of attitude change; see Petty, Priester, & Wegener, 1 994) or of a person's hypotheses that are acquired over time via direct interactions with th e world. The second simplification we make is treating beliefs as propositions that are bel ieved either to be true or to be false (or, sometimes, that have a belief status of "uncertain "). This idealization characterizes the perspective of AI researchers who are interested in showing how classical deductive reasoning is related to belief revision. We will call this perspective "classical belief revision," to distinguish it from other frameworks, including one direction in formal studies of defeasible reasoning, that map statistical or probabilisti c information about a proposition into a degrees of belief in that proposition (Kyberg, 198 3, 1994; Bacchus, Grove, Halpern, and Koller, 1992; Pollock, 1990; Pearl, 1988). Both c lassical belief revision and defeasible reasoning are concerned with non-monotonicity an d it is possible to view belief revision as driving defeasible reasoning or vice versa (Gärd enfors, 1990a; Makinson & Gärdenfors, 1991). This alternative formalization of beliefs and belief change in terms of probabilisti c or statistical information have analogies in certain empirical investigations as well. A pr imary concern in the management-science literature, for example, is to understand what f actors influence a shift in the degree of belief in a particular proposition of interest. These factors include information framing (e.g., Ashton & Ashton, 1990; Shields, Solomon, & Waller, 1987) and agreement with prior beliefs and expectations (e.g., Koehler, 1993). C arlson and Dulany (1988) have proposed a model of belief revision about causal hypothes es from circumstantial evidence, in which the level of certainty in a causal hypothesis dep ends in part on the level of certainty the reasoner ascribes to circumstantial evidence supp orting it. In Thagard's (1989) computer model of explanatory coherence, propositions hav e levels of activation that roughly correspond to acceptance levels; such a model has been applied to accounts of scientific reasoning and to belief revision as evidenced in protocol s of subjects performing elementary physics (Ranney & Thagard, 1988). Belief Revision as Propositional Update 6 Notwithstanding these alternative ways to conceptualize belief states, we believe t hat the issues investigated under our simplifications are relevant to these other perspectiv es. Belief revision as a deliberate act by an agent must be driven by something, and that d riving force must include the detection of a conflict (defined logically or otherwise) withi n the belief state. The problem of explicitly "expunging" or contracting of beliefs, after ha ving noticed a conflict, has been acknowledged within some degree-of-belief frameworks (e.g., Kyberg, 1983; 1994). As soon as one attempts to define notions like "acceptance" o r "full commitment to" within a degrees-of-belief framework, for the purpose of making a decision or taking an action, then new information can introduce conflict with existing ac cepted information. Hence, the issue still remains as to which prior belief or assumption an agent continues to believe (or to increase the degree of belief in) and which the agent d ecides to abandon (or decrease the degree of belief in). 1 Belief revision has also been studied as something that does not occur when it “sh ould.” That is, there is considerable evidence indicating that people are in general very rel uctant to change their current belief sets in the face of evidence that indicates those belief s are unjustified; and that they are much more likely to reject, ignore, or reinterpret the ne w information which conflicts with their current beliefs rather than attempt to add it to th eir beliefs and make the necessary adjustments (Edwards, 1968; Einhorn & Hogarth, 197 8; Ross & Lepper, 1980; Hoenkamp, 1988; Lepper, Ross, and Lau, 1986). Although it is t rue that there are occasions in which people fail to revise their beliefs or refuse to accept new information, and there are theories offered as accounts of that reluctance, our starting point in these investigations assumes that any inertia against changing a belief set has be en overcome. Given our simplifications for the representation of belief states, the specific issue that concerns us can be easily stated. It is the question of which belief(s) out of some initi al set is (are) abandoned when new, contradictory information must be integrated. The m atters we consider in this study relate to certain formal notions that have been central to ( Belief Revision as Propositional Update 7 what we have called) the classical AI belief revision perspective. These notions are episte mic entrenchment (whether some forms or types of information are less readily abandone d to resolve contradiction) and minimal change. It is not possible to consider these ideas without considering the fundamental choice that theories make in modeling a belief state either as a set of formulae or as a set of models. The implications of choosing one framew ork or another are crucial to operationalizing ideas like epistemic entrenchment. We revie w these two alternative positions on modeling belief states, and their relation to theories o f human deduction, in the next section. On Modeling Belief States and Deduction Classical Models of Belief Revision Alchourrón, Gärdenfors, & Makinson (1985; henceforth, “AGM”) proposed a set of "rationality postulates" as a competence specification of what rational belief change sh ould be. Many of these ideas are intrinsically important to thinking about human belief re vision as we are studying it here, so we borrow some key distinctions from that literature in setting the stage for our studies. There are two predominant camps in how belief states are modeled within what w e earlier defined as the classical belief revision community: “syntactic-based theories” v. “model-based theories”. The majority of the work in either of these camps follow the idea lizations we outlined above: that beliefs are propositional in nature, that the status of a bel ief is "believed true", "believed false" or "uncertain", and that logical inconsistency is to b e avoided within the agent's chosen belief state. The difference between the syntactic and model approaches can be seen by exam ple. Consider what might be in a belief state when an agent told: All of Kim’s cars are ma de in the US; This (some particular) car is made in Germany. The syntax-based theories t ake the position that what is stored in the agent’s belief state are the two formulas mentio ned (plus whatever background information the agent already had…also stored as a set of Belief Revision as Propositional Update 8 formulas). Since beliefs are just formulas, doing a logical inference amounts to performi ng some further mental activity on these formulas. This further activity would generate a different belief state from the initial one. And so there is no guarantee that the agent will perform any logical inferencing to generate new beliefs. For instance, there is no guarante e that this agent will use the background information it may have that Germany is a differ ent country than the US and that cars made in the one are not made in the other to infer th at this car is not made in the US. Even if it does perform this inference, there is no guaran tee that it will make the further inference that the car is not owned by Kim. In this concep tion, two beliefs are different when and only when they are expressed by two syntacticall y distinct formulas. In contrast to this, the model-based theories identify a belief state with a model— an interpretation of the world which would make a group of beliefs be true. In the above e xample of Kim’s cars, a model-based theory would identify the agent’s belief state with t hose models of the world in which all of Kim’s cars are made in the US and where furthe rmore some particular car is made in Germany. Assuming the agent’s background beliefs include that Germany is a different country than the US and that cars made in the one are not made in the other, the set of background models that can accommodate such situation s in the world are merged with those describing the two stated beliefs and the output is a model (or models) in which Kim’s cars are made in the US, and this car is made in Germ any, and hence this car is not made in the US, and hence this car is not owned by Kim. Al l this sort of "inferencing" is done already in the very description of the belief state. The f act that the belief state is a model of the world described by the sentences guarantees that all logical consequences of these sentences will be represented, for otherwise it couldn't b e a model of those sentences. One common way of putting the difference is to say that the syntax-based approac h is committed only to explicit beliefs as defining a belief state, whereas a model-based a pproach is committed to defining a belief state in terms not only of explicit beliefs but als Belief Revision as Propositional Update 9 o of the implicit beliefs that are entailed by the explicit ones. Both approaches involve a c ertain amount of theoretical idealization. Under the model-based view, the very definition of an agent’s belief state already embodies finding the models that perfectly suit it, and t his in effect means that all the logical conclusions of any explicit beliefs are included. Wi thin the syntactic framework, there is an assumption that only "obvious" or "minimal" co nclusions are drawn, but how these are recognized as such goes unspecified. Secondly, it is not clear how syntactic-based theories detect arbitrary logical contradictions beyond on es that can be immediately spotted by a syntactic pattern-match, such as "p and ~p", since beliefs are represented as strings of symbols and not models of the world being described .2 A third conception of belief states—which could be seen as an intermediate stance between the syntactic and the model-based approaches— might be called a “theory-base d” theory of beliefs. Here a belief state is identified with a theory, which is taken to be a s et of sentences, as the syntactic-based theories hold. However, this set is the infinite set of all the logical consequences of the explicit beliefs. 4 This is the approach advocated in th e original work done by AGM (1985). It too is obviously an idealization, for taken to the extreme, it would require a person's mind (or an agent's memory) to be infinite in order to hold a belief. Although theory-based theories are like syntax-based theories in containing formulas (and unlike model-based theories in this regard), they differ from syntax-based theories in obeying a principle called “The Irrelevance of Syntax”: if two formulas are lo gically equivalent, then adding one of them to a belief state will yield the same result as a dding the other, since the set of their logical consequences is the same. This principle is o beyed by both the theory-based and the model-based theories, and has been vigorously de fended (AGM, 1985; Dalal, 1986; Yates 1990; Katsuno & Mendelson, 1991) on the grou nds that all that is relevant to belief change is how the world is, or would be, if the beliefs were true. Belief Revision as Propositional Update 10 Many of the concepts and distinctions mentioned above as characterizing classical AI belief revision also apply to other belief revision frameworks. Computational framew orks in which propositions are represented as nodes in some kind of belief network (e.g., Pearl, 1988; Thagard, 1989) are syntactic, insofar as any semantic contradiction between two nodes must be reflected in the names of the links chosen to join the nodes in the netw ork. Methods proposed by Halpern (1990) and Bacchus et al. (1992) for deriving degrees of belief from statistical information are model-based approaches: the degree of belief in a sentence stems from the probability of the set of worlds in which the sentence is true. K yberg's theory of rational belief (1983;1994), in which levels of acceptance are also deriv ed from probabilities, falls into what we have called the "theory theory" category. He mo dels beliefs as a set of (first-order logic) sentences but then requires the set to obey the irr elevance of syntax principle: if two sentences have the same truth value in belief set, then their probabilities are also equivalent within that set. So we see that, although the classic al belief revision approach comes from a milieu were performance criteria are not explicit ly considered, the sorts of distinctions made within these classical belief revision framew orks can elucidate the representational assumptions of other approaches as well. Performance Theories of Human Deduction Harman (1986) has argued that the principles guiding belief revision are not the r ules of deductive logic. Certainly, any principles that can dictate which of several differe nt belief-state changes to select are outside the scope of deductive inference rules. Any ch aracterization of belief revision must first make some commitment to how a belief state is represented; as the formal theories we outlined above illustrate, making (or not making) i nferences is crucial to how the belief revision process is to be conceptualized. Certainly, t he ability to recognize inconsistency is a necessary step towards deliberate belief revision , and that step may involve some aspects of what has been studied and modeled in the lab oratory as deductive reasoning. Hence, it seems that theories about how people draw infer Belief Revision as Propositional Update 11 ences from propositional knowledge will be crucially related to the transition from one be lief state to another, if only because those inferences may define the content of the belief states themselves. Generally speaking, the theories of human deductive reasoning have split along a dimension that is similar to, but not identical with, the syntactic v. model-theoretic distinc tion in AI. On the one hand, mental-model theories of the type proposed by Johnson-Lair d and colleagues (Johnson-Laird, Byrne, & Schaeken, 1992; Johnson-Laird & Byrne, 199 1) hold that a person reasons from particular semantic interpretations (models) of sentenc es such as p→q and either p or q. 3 In this framework, a reasoner identifies or validates a particular conclusion by manipulating and comparing these models. On the other hand, pr oof-theoretic approaches (Braine & O'Brian, 1991; Rips, 1983; 1994) propose that people possess general inference rules and follow a kind of natural deduction strategy to derive conclusions from a set of premises. Like the different kinds of belief revision theories in AI, these different psychological accounts of human deduction offer distinct representatio nal assumptions about the constituent parts that are said to define a belief state. But unlik e the AI belief revision theories, psychological theories must make a commitment to a pla usible process account of how a person generates and operates upon these different repres entations. Neither mental-model nor proof-theoretic accounts of deduction were initially dev eloped for belief revision as we have portrayed it here; nor have there been, as yet, extens ions designed to accommodate aspects of this phenomenon. However, we consider some of their basic assumptions in the discussion of our tasks and results, and so here we briefl y summarize the mental-models framework proposed by Johnson-Laird and colleagues an d the proof-theoretic model proposed by Rips (1994). If we apply a state-space abstraction to mental models frameworks and to proof-th eoretic frameworks, the main distinction between proof theoretic and model-based theori es of human deduction can be summarized as differences in what defines a state and what Belief Revision as Propositional Update 12 constitutes the operators that make transitions from one state to another. In a proof-theor etic system like the one proposed by Rips (1994), a state is a partial proof and the operato rs are a set of inference rules (a subset of the classical logic inference rules). These operat ors extend a proof (and hence move the system from one state to the next) by following a natural deduction-like strategy, with heuristics that order their application within this gen eral control strategy. The goal can be viewed as a state (or a path to a state) which include s a given conclusion as an outcome of a proof (hence validating it) or includes a statemen t not already specified in the problem's premises (drawing a new conclusion). In the ment al-models theory, a state contains one or more interpretations of the sentence set, i.e., tok ens with specific truth values that correspond to some situation in the world. Operators re trieve models of sentences and move the system to new states that constitute candidate m odels of the world. More specifically, the mental models framework assumes there are pa rticular models that are initially associated with particular sentence forms (conditionals, d isjuncts, and so forth), with other models of these forms sometimes held in abeyance until there is a need to consider them. A conclusion is any truth condition that is not explicitly stated in the sentence set, but which must hold given a consistent interpretation of the sen tence set. Hence, the goal state can be seen as one in which such a truth condition is ident ified. Thus, the proof-theoretic theory of human deduction can be seen as a search for alte rnative inference rules to apply to a sentence set in order to extend a proof, whereas the m ental models theory can be seen as a search for alternative interpretations of a sentence se t, from which a novel truth condition can be identified or validated. It is important to be clear not only about the similarities but also about the differen ces between the classical, competence belief-revision theories and the psychological perf ormance theories of human deduction. What the mental-models theory shares with the for mal model-based belief revision theories is the essential idea that the states being operate d upon are models. These models capture the meaning of the connectives as a function of the possible truth values for individual atomic parts that the connectives combine. Howev Belief Revision as Propositional Update 13 er, there are three key differences between these two types of model theories. First, althou gh the irrelevance-of-syntax principle is a distinguishing feature of formal models of beli ef revision in AI, it does not distinguish between mental-models and proof-theoretic mod els of human deductive reasoning, which offer alternative accounts of the pervasive findi ng that syntactic form does influence how people reason about problems that are otherwis e logically equivalent. Second, in the mental-models theory, the models of p→q are gener ated in a serial, as-needed basis, depending on whether a conclusion is revealed or validat ed by the initial interpretation (and it is the order in which such models are generated that plays in the mental-model's account of the effect of syntactic form on deductive reasonin g). The AI model-based belief revision frameworks do not make any such process assum ptions, except in their idealization that all models are available as the belief state. Third, t he mental-models framework may be considered closer to what we have called the "theor y theory" classical belief-revision framework, than to the pure model framework, because separate models of each sentence are produced and operated upon. What a psychological proof-theoretic framework of deduction shares with its formal AI syntactic-based counte rparts is a commitment to apply deductively sound inference rules to sentences. But unlik e the syntactic-based competence theories of belief revision, psychological proof-theoreti c models of deduction do not presume that a person has a representation of every deducti ve rule of inference and they may presume there is some heuristic ordering of the availabl e rules; these differences are relevant to how a proof-theoretic perspective models the rela tive difficulties that people have with certain forms of deductive problems. Further, some of the undesirable aspects of syntactic competence models, such as uncontrolled deductiv e closure, are avoided in proof-theoretic performance models (e.g., Rips, 1994) by explici tly positing that the reasoner's current goals and subgoals directs and controls the applicat ion of inference rules. Minimal Change and Epistemic Entrenchment Belief Revision as Propositional Update 14 A basic assumption behind most AI theories of belief revision (e.g., the AGM pos tulates) and some philosophical accounts (e.g., Harman, 1986) is that an agent should mai ntain as much as possible of the earlier belief state while nonetheless accommodating the new information. But it is not completely clear what such a minimal change is. First, ther e is the problem in defining a metric for computing amounts of change. Often, this relies on counting the number of propositions whose truth value would change in one kind of re vision versus another. The revision that leaves the belief state “closest” to the original on e is to be preferred. But note that how such a definition of closeness works depends on w hether one takes a syntactic or model-based approach. As an example of the differences that can evolve, consider our earlier story about your New York-visiting colleague. Let n stand for she-is-in-New-York, r stand for Range rs-are-playing, and w stand for she-stays-at-the-Westin. Symbolically, your initial beli efs were [n & r → w, n, r], from which you deduced w. But then you found out ~w. In a model-based approach, the unique model describing your initial belief set (unique, at leas t, if we attend only to n, r, and w) is: n is true, r is true, w is true. Then you discover that t he model is incorrect because w is false. The minimal change you could make is merely t o alter w's truth value, and so your resulting belief states is: n is true, r is true, w is false. I n a syntax-based approach, you would instead keep track of the ways that as many as pos sible of the three initial sentences remain true when you add ~w to them. There are three such ways: S1= [n & r → w, n, ~r, ~w], S2=[n & r→w, ~n, r, ~w], S3=[~(n & r→w), n, r, ~w].5 Now consider what common sentences follow from each of S 1, S2, and S3, and t he answer is that the consequences of [~w, n ∨ r] will describe them. Note that this is diff erent from the version given as a model-theoretic solution. In the syntactic case, only one of n and r need remain true, whereas in the model-based belief revision version, both nee d to remain true. The notion of "epistemic entrenchment" in the belief revision literature (Gärdenfo rs, 1984, 1988; Gärdenfors & Makinson, 1988; Nebel, 1991; Willard & Yuan, 1990) has Belief Revision as Propositional Update 15 been introduced as a way to impose a preference ordering on the possible changes. Forma lly, epistemic entrenchment is a total pre-ordering relation on all the sentences of the lang uage, and this ordering obeys certain postulates within the AGM framework. Less formal ly, epistemic entrenchment can be viewed as deeming some sentences as “more useful” a nd hence more entrenched against possible abandonment than other sentences; and in cas es where there are multiple ways of minimizing a change to a new belief state, these prior ity schemes will dictate which way is chosen. Now, the general issue of whether some ty pes of knowledge (e.g., sensory observations v. reasoned conclusions) should be a priori more epistemically privileged than other types of knowledge has occupied much of philos ophy throughout its history. One particular, more modest, contrast is between what might be called statements about data v. statements about higher-order regularities. From one pe rspective, it can seem that conditional statements should enjoy a greater entrenchment in t he face of conflicting evidence because of they express either semantic constraints about t he world or express an important predictive regularity that might be the result of some lo ng-standing and reliable inductive process. As an example of this sort of perspective, one can point to scientific theorizing that is based on statistical analysis, were one rejects "out lying" data as unimportant, if other regularities characterize most of the remaining data. I n doing so, we give priority to the regularity over (some of) the data. Certain approaches to database consistency (e.g., Elmasri & Navathe, 1994, pp. 143-151) and some syntactic theories of belief revision (e.g., Willard & Yuan, 1990; Foo & Rao, 1988) advocate the e ntrenchment of the conditional form p→ q over non-conditional forms. For database cons istency, a relation like p→ q can be said to represent a semantic integrity constraint, as in "If x is y's manager, then x's salary is higher than y's salary." For classical belief revision theories, the intuition driving the idea of entrenching p→q over other types of sentences i s not because material implication per se is important, but because "lawlike relations" are often expressed in sentences of this form. For example, Foo and Rao (1988) assign the hi ghest epistemic entrenchment to physical laws, which may be especially effective in reas Belief Revision as Propositional Update 16 oning about how a dynamic world can change (e.g., the belief update, rather than revision , problem). But there is another perspective that would propose exactly the opposite intuition s about entrenchment: what should have priority are observations, data, or direct evidence . These are the types of statements which are fundamental and about which we can be mo st certain. Any kind of semantic regularities expressed in conditional form are merely hyp otheses or data-summarizing statements that should be abandoned (or at least suspected) when inferences predicted from them are not upheld by direct evidence. This sentiment fo r data priority seems entirely plausible in the context of hypothesis evaluation (e.g., Thag ard, 1989) as it did to some involved in the "logical construction of the world" (e.g., Russ ell, 1918; Wittgenstein, 1922). In sum, we note that these alternative intuitions about entrenching conditionals v. non-conditionals are more or less readily accommodated, depending on the representation of belief states. It is easy to use the form of a sentence as a trigger for entrenchment princ iples, if one has a syntactic stance; but if a reasoner works with models of the world, then this sort of entrenchment is not as easily supported (unless sentences are individually mo deled and knowledge of "form" is somehow retained). By first understanding the principl es that actually guide belief revision in people, we are in a better position to formulate wh at kinds of representations would support the execution of those principles in a cognitive system. Overview of Experiments So far, we have touched upon a number of broad theoretical issues that bear on be lief revision, at least when this is characterized as a deliberate decision to remove some pr oposition(s) that had been accepted as true, in order to resolve a contradiction noted in the belief set. Although our longer-term interest is to better understand what plausible princi ples might define epistemic entrenchment, our immediate interest in the present studies w Belief Revision as Propositional Update 17 as first to acquire some baseline data on what belief revision choices people make in relat ively content-free tasks and to tie these results to models of deduction. To do this, we con sider the simple task of choosing to abandon a conditional proposition v. a non-condition al proposition (what we will also call a "simple sentence") as a way to resolve a logical co ntradiction. This decision corresponds to the example dilemma we presented at the start o f this article. The initial belief state, defined by a conditional and a simple sentence, can b e expanded by the application of a deductive inference rule. In our paradigm, it is this res ulting inferred belief that is subsequently contradicted. Because we know that human ded uction is influenced by the particular form of the inference rule used (cf. Evans, Newstea d, & Bryne, 1993) we are secondarily interested in whether the inference rule used in defi ning the initial belief set impact a subsequent belief revision rule. While these particular experiments are not designed to discriminate between proof theoretic or mental-models th eories of deduction, such evidence is relevant to expanding either of these performance m odels of human reasoning to embrace aspects of resolving contradiction. The final two stu dies examine more directly various of the alternative model-theoretic definitions of mini mal change, and investigate whether minimal change—by any of these definitions—is a p rinciple for human belief revision. This organization notwithstanding, we note that these i ssues—the syntactic versus model-theoretic distinction, epistemic entrenchment, and min imal change—are tightly interwoven and they bear on each experiment in some way. Entrenchment of Conditionals In the first three experiments we report, we used two problem types that differed i n whether the initial belief state included a conclusion drawn by the application of a mod us ponens inference rule or by the application of a modus tollens inference rule. Modus p onens is the inference rule that from If p then q, and furthermore p, then infer q. The mod us ponens belief set consisted of a conditional, the conditional's antecedent, and the deriv Belief Revision as Propositional Update 18 ed consequent. Modus tollens is the rule that from If p then q, and furthermore ~q, infer ~ p. The initial modus tollens belief set consisted of a conditional, the negation of its conse quent, and the derived negation of the antecedent. We introduced contradiction with the i nitial belief state by providing new information—the expansion information—which cont radicted whatever the derived conclusion was. In the modus ponens case, the expansion was ~q. In the modus tollens case, the expansion was p.6 We defined belief-change problems using these well-studied problem types, both to provide a baseline for understanding the role of syntactic form in belief-change proble ms, and to make contact with existing data and theories about human performance on the se classic deductive forms in a different problem context. If a conditional enjoys some ki nd of entrenchment by virtue of its syntactic form, people should prefer a revision that ret ained the conditional but reversed the truth status of the simple sentence that permitted th e (subsequently contradicted) inferred sentence. A related question is whether this belief revision choice is made differently, depending on whether the belief state consisted of a modus ponens or modus tollens inference. From an AI model-theoretic viewpoint, modus ponens and modus tollens are just two different sides of the same coin: they differ only i n their syntactic expression. Classical AI model-theoretic approaches would consider a re vision that denied the conditional to be a more minimal change. 7 From a psychological v iewpoint, it is well documented (e.g., see the survey by Evans, Newstead, and Byrne, 199 3) that people find making a modus tollens inference more difficult than making a modus ponens inference. In this work, we did not want this feature of reasoning to come into pla y. Therefore, we provided the inferences explicitly in defining the initial belief set, and th en asked whether the deductive rule used to derive them affects the belief revision choice. The existing literature on human reasoning performance also indicates an influen ce of domain-specific content on the kinds of inferences that people are able or likely to d raw. To account for these effects, theories have proposed the use of abstract reasoning sc hemas (Cheng & Holyoak, 1989; Cheng, Holyoak, Nisbett, & Oliver, 1993) and a reasoni Belief Revision as Propositional Update 19 ng by analogy approach (Cox & Griggs, 1982). For these initial investigations of belief-re vision choices, we were not interested in investigating the direct applicability of these the oretical distinctions to the issue of belief-revision, but rather considered the general empir ical findings that people reason differently with familiar topics than they sometimes do w hen given problems involving abstract symbols and terms. If belief revision is viewed les s as a decision task driven by notions like minimal change and more as a problem of creat ing consistent explanations of past and current data, then we might expect the pattern of r evision choices to be different when the problem content is more “real-worldly” than abst ract. So, these experiments used both abstract problems (containing letters and nonsense s yllables to stand for antecedents and consequents) and equivalent versions using natural l anguage formats. Experiment 1 Method Problem Set. Table 1 gives the schematic versions of the two problem types used i n this experiment. Each problem consisted of an initial sentence set, expansion informatio n and then three alternative revision choices. The initial sentence set was labeled "the wel l-established knowledge at time 1." The expansion information was introduced with the p hrase, "By time 2, knowledge had increased to include the following." 8 Each revision al ternative was called a "theory" and consisted of statements labeled "Believe", "Disbelieve " or "Undecided About." A theory could have statements of all these types, or of just som e of these types. The task for subjects was to choose one of the alternative revision theori es as their preferred belief state change. -------------------------------Insert Table 1 about here -------------------------------- Belief Revision as Propositional Update 20 For the modus ponens and modus tollens problems, the original sentence set inclu ded a conditional of the form p→q and either the antecedent p or the negated consequent ~q, respectively. In both cases, the derived inferences were included in the initial set (q fo r modus ponens, ~p for modus tollens). The expansion information for both problems con tradicted the derived inference and this was explicitly noted to subjects in the presentatio n of the problem. Revision choices 1 and 3 offered two different logically consistent way s to reconcile this: deny the conditional (choice 3) or retain the conditional but reverse the truth status of the simple sentence that permitted the inference (choice 1). Revision choic e 2 was included to provide a choice that was non-minimal by almost any standard: it incl uded the expansion information, denied the conditional, and labeled the simple sentence t hat permitted the inference to be made as “uncertain” (signified by a ? in Table 1). Note t hat all revision alternatives indicated that the expansion information must be believed. Problems had one of two presentation forms: a symbolic form, using letters and n onsense syllables, and a science-fiction form. An "outer space exploration" cover story w as used to introduce the science-fiction forms. Here is an example of how a modus tollens problem appeared in the science fiction condition: On Monday, you know the following are true: If an ancient ruin has a protective force field, then it is inhabited by the aliens called Pylons. The tallest ancient ruin is not inhabited by Pylons. Therefore, the tallest ancient ruin does not have a protective force field. On Tuesday, you then learn: The tallest ancient ruin does have a protective force field. The Tuesday information conflicts with what was known to be true on Monda y. Which of the following do you think should be believed at this point? Belief Revision as Propositional Update 21 A corresponding symbol version of this problem was: If Lex's have a P, then they also ha ve an R. Max is a Lex that has a P. Therefore, Max has an R. The expansion information was Max does not have an R. Design. All subjects solved both modus ponens and modus tollens problem types. Presentation form (symbolic versus science-fiction) was a between-subjects factor. The s cience-fiction cover stories used several different clauses to instantiate the problems. The clauses used for each problem type are shown in Appendix A. Subjects. One-hundred twenty subjects from the University of Alberta Psychology Department subject pool participated in the study. Equal numbers of subjects were rando mly assigned to the symbol and science fiction conditions. Procedure. The modus ponens and modus tollens belief problems appeared as part of a larger set of belief revision problems. The order of revision alternatives for each pro blem was counterbalanced across subjects. Below are excerpts from the instructions, to cl arify how we presented this task to our subjects: ....The first part of the problem gives an initial set of knowledge that was true and well-established at time 1 (that is, some point in time). There were no mistakes at that time. The second part of the problem presents additional knowledge about th e world that has come to light at time 2 (some later time). This knowledge is also t rue and well-established.... The world is still the same but what has happened is th at knowledge about the world has increased....After the additional knowledge is pr esented, the problem gives two or more possible "theories" that reconcile the initi al knowledge and the additional knowledge....Your task is to consider the time 1 a nd time 2 knowledge, and then select the theory that you think is the best way to r econcile all the knowledge. Results Belief Revision as Propositional Update 22 Each subject contributed one revision-type choice for each of the two problem ty pes. This gives us frequency data for how often each revision choice was selected, as a f unction of two variables: problem form (modus ponens v. modus tollens) and presentatio n form (science-fiction v. symbolic). Table 2 presents this data as the percentages of subj ects choosing a particular revision choice. From the schematic versions of the problems in Table 1, it is clear that the three b elief revision alternatives for the modus ponens (MP) and modus tollens (MT) problems have a certain symmetry, even though the actual details of each revision are necessarily d ifferent. In Table 2's presentation of the data, we re-label these revision alternatives in a more general form that reflects this symmetry. For both problem types, revision choice 1 retains the conditional but reverses the truth status for the non-conditional that was the ot her initial belief. (For the MP problem, the expansion was p, so q was the initial non-cond itional. For the MT problem, the expansion mentioned q; so p was the initial non-conditio nal.) In revision choice 2, the conditional is disbelieved and non-conditional is uncertain. Under revision choice 3, the conditional is disbelieved and non-conditional retains whate ver truth value it had initially. -------------------------------Insert Table 2 about here -------------------------------In general, subjects preferred revisions in which the p→q rule was disbelieved (rev isions 2 and 3). Collapsing across presentation condition, the clearest difference between the MP and MT belief-change problems concerned which of these two rule-denial revisio ns subjects preferred: on MP problems, the preferred belief change saw subjects preferrin g simply to disbelieve only the rule; on MT problems, the preferred revision was to disbel ieve the rule and to regard the initial non-conditional, ~q, as uncertain. To analyze this frequency data, one could create a set of two-way tables for each l evel of each variable of interest to assess whether the distribution of frequencies is differe Belief Revision as Propositional Update 23 nt, and compute a chi-square test of independence for each subtable; however, this does n ot provide estimates of the effects of variables on each other. Loglinear models are usefu l for uncovering the relationships between a dependent variable and multiple independent variables for frequency data. A likelihood-ratio chi-square can be used to test how well a particular model's prediction of cell frequencies matches the observed cell frequencies. We can first ask whether the three revision alternatives were selected with equal pr obability, when collapsed across all conditions. The observed percentages of 22.2%, 39.9 %, and 37.9% for revision choices 1, 2, and 3, respectively, were significantly different fr om the expected percentages (χ2=13.27, df=2, p=.001). By examining the residuals, we c an identify patterns of deviation from the model. The two deviations in this case were the percentage of revision 1 and revision 2 choices. To test whether revision choice is independent of problem type and presentation m ode, we fit a model that included simple main effects for each factor, but no interaction te rms. The chi-square value indicates that such an independence model does not fit the data well (χ 2=15.33, p=.004, df=4). Models that included only one interaction term for revisi on by problem type, or only one for revision by presentation mode, were also poor fits to the observe data (χ 2=12.02 and 10.52, respectively, df's=4, p's < .05). The simplest mod el whose predicted frequencies were not significantly different from observed frequencies included both a revision by problem-type and a revision by presentation-mode interactio n term (χ 2=3.18, df=2, p=.203).9 The means in Table 2 indicate that the pattern of difference between MP and MT choices is primarily due to differences in responses on the science-fiction problems. 58% of the science-fiction condition subjects chose to disbelieve p—>q on modus ponens beli ef states, while only 29% did so in the modus tollens case. The most frequently-chosen re vision (54%) for a science fiction MT belief-revision was a non-minimal change: disbelie Belief Revision as Propositional Update 24 ving p—> q and changing q's initial truth status from false to uncertain. Only 29% of the subjects choose this revision on the modus ponens belief state. Experiment 2 In Experiment 1, subjects may have been evaluating merely whether each revision opt ion was logically consistent, independently of what the initial sentence set and expansion information was. Only two of the revisions alternatives offered minimal-changes to the in itial sentence set, and this might have accounted for the close pattern of responses betwee n symbolic-form MT and MP problems. Asking subjects to generate, rather than select, a revision would most directly address this possibility, but for these studies, we decided to r etain the selection paradigm and to increase the alternatives. For Experiment 2, we includ ed an extra non-minimal change revision and a revision in which the sentences were logic ally inconsistent. Method Problem Set and Design. Table 3 presents the response alternatives for the modus ponens and modus tollens problems used in Experiment 2. The first three response choice s were the same as those used in Experiment 1. The fourth choice denies both the rule and changes the original truth status of the initial non-conditional sentence. This is a non-min imal change and results in an inconsistent set of sentences as well. The fifth revision choi ce labels both the conditional and the non-conditional from the initial belief set as uncerta in. These changes too are non-minimal, but the final belief set is logically consistent. -------------------------------Insert Table 3 about here -------------------------------- Belief Revision as Propositional Update 25 Subjects and Procedure. Forty-three subjects participated as part of a course requir ement for an introductory psychology course. All subjects solved both MP and MT probl ems, as part of a larger set of belief-revision problems. Only symbolic forms of the proble ms were used in this follow-up experiment. The instructions were the same as those used for Experiment 1. Results The percentage of subjects choosing each revision choice are also given in Table 3. There is some consistency in the patterns of responses across both Experiments 1 and 2 . The frequency of revisions in which the initial non-conditional's truth value was change d (revision choice 1) was still relatively low (about 25%) on both problem types, as we ha d found in Experiment 1. About 33% of the subjects opted simply to disbelieve the condit ional (revision 3) on the MP problem (as they had in Experiment 1). However, on the MT problem, changing both the conditional and the initial simple sentence to uncertain (revis ion 5) accounted for most of the choices. A simple chi-square computed on the revisionchoice by problem-type frequency table confirmed there was a different pattern of revisio n choices for these modus ponens and modus tollens problems ( χ2=15.33, df=4, p=.004) Experiment 3 In the first experiments, we explicitly included the derived consequences in the m odus ponens and modus tollens problems. In Experiment 3, we tested whether or not this inclusion of consequences as explicit elements of the initial belief set (versus allowing th e subjects to draw their own conclusions) would affect revision choice. Consider, for exa mple, problem type 1 in Table 4. This problem's initial belief set supports a simple modus ponens inference from a conditional m & d → g and the simple sentences m and d to gen erate the conclusion g. As in the previous experiments, there were two logically consisten t ways to reconcile the ~g expansion information: deny the conditional, or deny one or m ore of the simple sentences that comprise the conditional's antecedent. The two revision c Belief Revision as Propositional Update 26 hoices reflect these two choices. Alternative 1 disbelieves the conditional and retains beli ef in the simple sentences; alternative 2 retains belief in the conditional and calls into que stion one or both of the simple sentences. ------------------------Insert Table 4 here ------------------------ Whether or not the initial sentence set includes derived consequences can have m ore profound implications when the initial belief set supports a chain of inferences. Consi der problem type 2 in Table 4, in which the initial belief state is {c→h, h→m, c} and the expansion information is {~h}. One conclusion supported in the initial belief set is h. And this is in conflict with the expansion information. There are two ways to resolve this conf lict: deny the conditional c→h , arriving at the final belief set of {c, h→m, ~h}. Or deny c and retain the conditional c→h, to obtain the revised belief set {c→h, h→m, ~c,~h}. Not e that m cannot be inferred from either of these two revised belief states, but note also tha t it was a consequence of the initial belief set. Should we continue to believe in m? We ca n do that only if we believed in m in the first place, that is, if we drew m as a logical cons equence of the first set of sentences. Otherwise, its status would be uncertain—neither bel ieved nor disbelieved. Belief revision alternatives were provided for both these possibiliti es and this was investigated both in the case where logical consequences of beliefs were e xplicitly included in the initial belief set (as in Experiments 1 and 2) and also without exp licit inclusion. A second factor we considered in this follow-up was whether the conditional sent ences in the initial belief set were propositional sentences or were universally-quantified s entences. The belief revision problem hinges on the reconciliation of conflicting informat ion, but how that reconciliation proceeds may depend on whether it contradicts what is be lieved about a class (hence, is a factor relevant to predicate logic), versus what is believed Belief Revision as Propositional Update 27 about an individual (and hence is a feature of propositional logic). Therefore, we manipu lated whether the initial belief set was specified by universally quantified sentences or pr opositional sentences for each of the problems studied in Experiment 3. Method Problem Set and Design. The schematic versions of the two problem types given i n Table 4 were used to create 8 different problems. Two factors were crossed for both pro blem types 1 and 2. The first factor was whether the minimal logical consequences of the initial sentence set were explicitly given as part of the initial belief set. In Table 4, the bra cketed simple sentences were explicitly listed as part of the initial belief set in the conseq uences-given condition or were omitted in the no-consequences given condition. The second factor, sentence-form, was whether the belief set was based only on pr opositional sentences, or concerned sentences about universally-quantified arguments. Th us, one propositional form of a conditional was If Carol is in Chicago, then she stays at th e Hilton Hotel, while the universally-quantified form was Whenever any manager from y our company is in Chicago, s/he stays at the Hilton Hotel. The associated simple sentence s in each case referenced a particular individual. For the propositional example, the sente nce instantiating the antecedent was You know that Carol is in Chicago. For the universal ly-quantified condition, it was You know that Carol, one of the company managers, is in Chicago. For problem type 1, the revision choices were either to disbelieve the conditional ( revision alternative 1) or to disbelieve one or both of the initial simple sentences (revision alternative 2). The same distinction holds for problem type 2, which had four revision alt ernatives: alternatives 1 and 3 involved denying the conditional c→h, while revision choi ces 2 and 4 retained the conditional and instead changed c to ~c. The other key distinctio n in problem type 2's revision alternatives concerned the status of m, which was the chain Belief Revision as Propositional Update 28 ed inference that the initial belief set supports. Revision choices 1 and 2 labeled m as unc ertain; revision alternatives choices 3 and 4 retained m as a belief. All of the problems were presented in natural language formats. The following tex t illustrates how Problem Type 1 appeared in the consequences given—propositional con dition: Suppose you are reviewing the procedures for the Photography Club at a nearby uni versity, and you know that the following principle holds: If the Photography Club receives funding from student fees and it also charges membership dues, then it admits non-student members. You further know that the Photography Club does receive funding from student fees . It also charges membership dues. So you conclude it admits non-student members. You ask the Photography Club for a copy of its by-laws and you discover The Photography Club does not admit non-student members—all members must be registered students. Subjects and Procedure. Thirty-five University of Alberta students served as subje cts, to fulfill a course requirement for experiment participation. Problems were presented in booklet form, which included other belief-revision problems as fillers. All subjects sol ved all four versions of both problem types 1 and 2: no consequence—propositional, cons equences given—propositional, no consequences—quantified, consequences given—qua ntified. There were six pseudo-random orders for the problems within the booklet; within each order, the four versions of any given problem were separated by at least two other pr oblems of a different type. The order of response alternatives for each problem was also r andomized. Results Belief Revision as Propositional Update 29 For problem type 1, revision choice 1 (disbelieving the conditional; see Table 4) a ccounted for 82% of the revision choices. This is consistent with the pattern of choices in Experiment 1 results on science-fiction problems and this preference to disbelieve the con ditional was not affected by whether or not the modus ponens inference was explicitly list ed in the initial sentence set nor by the use of propositional v. universally-quantified sente nces. In terms of the first factor, we note that people generally find modus ponens an eas y inference to make, and these results confirm that the general preference to disbelieve th e conditional does not rest on whether the contradicted inference is explicitly provided. Concerning propositional v. universally-quantified sentences, we observe that it is difficu lt to construct if p —>q sentences that are not, somehow, interpretable as universally-qua ntified over time. Thus, even sentences like If Carol is in Chicago, then Carol is at the Hil ton, may be interpreted as For all times when Carol is in Chicago, ..... There seems to be little in the line of systematic, empirical study of the effect of propositional v. single quan tifier v. multiple quantifier logic upon people's reasoning (although both Rips, 1994, Cha pts. 6 and 7, and Johnson-Laird & Byrne, 1991, Chapts. 6 and 7, address this issue in thei r respective computational frameworks). Nonetheless, it seems clearly to be an important issue for studies that place an emphasis upon recognition of contradictions, since the imp act of contradictory information upon "rules" is different in these different realms. There was also no impact of either the consequences-given or the sentence-form f actor on the patterns of revision choices for Problem Type 2, in which the initial belief set contained an intermediate conclusion h and then a chained conclusion m, that depended on h, and where expansion information contradicted h. The percentage of revision choice 1 (denying the conditional c—> h) accounted for 52% of the choices; choice 2 (denying t he non-conditional sentence c) accounted for 29% of the choices. In both these cases, the status m, the chained inference that depended on h, was labeled uncertain. Revision altern atives 3 and 4, which were analogous to alternatives 1 and 2 except that they retained beli ef in m, accounted for 14% and 5%, respectively, of the remaining choices. The preferen Belief Revision as Propositional Update 30 ce to change m's truth status from true to uncertain rather than retain it as true is interestin g: it is an additional change to the initial belief state beyond what is necessary to resolve t he contradiction. Perhaps people's revision strategy is guided more by the recognition tha t a belief depends on another than upon minimizing the number of truth values that chang e from one state to the next. Discussion In Experiments 1-3, we aimed to identify what kinds of revision choices subjects would make in symbolic and non-symbolic types of problems, with the former providing some kind of baseline for whether a conditional statement enjoys some level of entrench ment merely as a function of its syntactic form. Our second concern was to assess whethe r belief revision choices were affected by the composition of an initial belief set, i.e., whe ther it was defined through the use of the conditional in a modus ponens or modus tollens inference. This offers us a bridge between belief revision (as a task of making a deliberat e change in what is to be "believed" in the face of contradictory information) and the data and theories on deductive reasoning. There was no evidence that people preferred to entrench the conditional on these tasks. In the choices we gave subjects, there was one way to continue to believe the condi tional and two ways to disbelieve it. If people were equally likely to retain the conditional as they were to abandon it, we might expect 50% of the choices falling into the keep-theconditional revision, with the two ways to disbelieve it each garnering 25% of the choices . On the symbolic problems in Experiments 1 and 2, the frequency of retaining the conditi onal after the expansion information was only about 25% on both modus ponens and mod us tollens problems; it was even lower on the natural language problems. Although subjects' preference was to abandon belief in the conditional, the way i n which this occurred on modus ponens and modus tollens problems was slightly differen t. On modus ponens problems, subjects disbelieved the conditional but continued to belie ve the non-conditional sentence as it was specified in the initial belief set. On modus tolle Belief Revision as Propositional Update 31 ns problems, subjects tended towards more "uncertainty" in the new belief state: either de nying the conditional and deciding the non-conditional was uncertain (Experiment 1), or l abeling both as uncertain when that was an option (Experiment 2). These tendencies on m odus tollens problems could be interpreted as conservative revision decisions, since neith er the initial conditional nor the initial non-conditional sentence is explicitly denied; on th e other hand, they correspond to maximal changes because the truth values of both initial beliefs are altered. We leave further discussion of entrenchment issues to the General Dis cussion. It is natural at this point to consider the relationship between this belief-change tas k and standard deduction, and to ask whether this task and its results can be understood as a deduction task in some other guise. There are two reasons we think it is not. First we c onsider the task demands and results for the modus ponens and modus tollens belief revis ion problems, and then briefly outline results we have obtained on belief expansion probl ems that did not involve a contradiction. The task demands of the modus ponens and modus tollens belief-revision problems We can neutrally rephrase the modus ponens belief-change problem that subjects f aced as "Make sense of [p→q, p, q] + [~q]," where the first sentence set represents the ini tial belief set and the second signifies the expansion information. Since subjects had to ac cept the expansion information, what we call the modus ponens problem thus becomes " Make sense of [p→q, p, ~q], such that ~q is retained." Similarly, the modus tollens proble m is "Make sense of [p→q, ~q, p], such that p is retained." Because these two problems a re semantically equivalent, the forms in the set of propositions to be considered are the sa me and the models of these sentence sets are the same. The difference lies only in the nat ure of the derivation in the initial sentence set, and the corresponding constraint on what must be retained after the revision. What we have called the modus ponens belief revision problem could be construe d as an modus tollens deduction problem, if subjects consider only the conditional in com Belief Revision as Propositional Update 32 bination with the expansion information: "Given [p →q] + [~q], what can I derive?" The i nvited modus tollens inference is ~p. If they derived this, they could at least consider reta ining the conditional and changing p to ~p in their belief-state change. The trouble that m odus tollens inferences present for people could in this way explain the observed prevalen ce of disbelieving the conditional on modus ponens belief revision problems. Applying this same perspective on the task to the modus tollens problem, we woul d see the modus tollens belief revision problem becoming an modus ponens deduction pr oblem, if only the conditional and the expansion information are considered: "Given [p→ q] + [p], what can I derive?" People have little difficulty with modus ponens and under th is analysis, it would be an "easy inference" to conclude q, and so be led to reverse the trut h status of ~q as the belief change. But the majority of subjects did not do this—on these problems as well, they disbelieved the conditional. Therefore, it does not seem that our ge neral pattern of disbelieving the conditional in belief revision can be reduced to, and acco unted for by, the nature of the difficulties in making certain types of standard deductive in ferences. It is possible that subjects did not accept the modus tollens belief set as consistent in the first place. (People have difficulty both in generating modus tollens inferences and in validating them when they are provided—cf. Evans, Newstead, & Byrne (1993), p.36). So perhaps this could be used to account for why there was high percentage of "everythi ng but the expansion information is uncertain" revisions on modus tollens problems in Ex periment 2. However, this does not account for why, on these modus tollens problems, su bjects would not simply focus on both the conditional and the expansion information, and then draw an modus ponens inference—that would lead to changing the truth status of th e initial simple sentence, as opposed to what they in fact did. Deductive reasoning and belief-state expansions Belief Revision as Propositional Update 33 The second reason we believe these tasks are not reducible to equivalent deductiv e reasoning problem stems from results we obtained on other belief-state expansion probl ems, in which the expansion information did not contradict the initial belief set (Elio & P elletier, 1994). These problems used two different but logically equivalent forms of a bic onditional: p if and only if q and ( (p & q) ∨ (~p & ~q) ). The expansion information was sometimes p and at other times ~p. Unlike the belief revision problems, these problems h ave a deductively "correct" answer: given p↔q (in either form) as an initial belief, with t he sentence p as the expansion, it logically follows that q should be asserted and made pa rt of the belief state. (And if ~q is the expansion, then ~p should be believed). If we view the biconditional-plus-expansion information problems as biconditional modus ponens (o r biconditional modus tollens) problems, then we would expect that subjects presented wi th our biconditional and disjunctive belief expansion problems should behave like the sub jects given biconditional and disjunctive deductive problems in other studies. Yet we fou nd that subjects asserted q on the p↔q form of our biconditionals much less frequently ( about 72%) than typically reported for these problems presented as standard deduction tas ks (e.g., 98% accuracy in Johnson-Laird et al., 1992). And fully 56% of subjects given th e biconditional in disjunctive form followed by the belief expansion p did not augment th eir belief set with q, when the problem was presented with a science-fiction cover story. I nstead, they decided q was uncertain and that the biconditional itself was uncertain or unb elievable. In sum, we believe that the task of belief revision, even in the relatively constraine d way we have defined it here, does not simply unpack into deductive reasoning, particul arly when natural-language formats are used for the problem. That is, subjects may not in tegrate information arriving across time (e.g., learning “later” that p holds true) into a beli ef set in the same way as information known to be true at the same time (“From If p is no w true, then q is also true, and furthermore p is now true, what follows?”). It may be that t he belief revision task invites the reasoner to make certain assumptions about evidence th Belief Revision as Propositional Update 34 at is not explicitly included in the initial or subsequent information; it may also be that co uching the task as changes in beliefs invites a more conservative strategy than what chara cterizes people's choices on formal logic problems. On models of belief states and deduction The experiments we designed do not speak to whether belief states are best model ed as sets of sentences or sets of models. However, we can observe the following. First, A I competence models are typically not concerned with human performance, yet they some times appeal to human rationality to justify their particular perspective. For example, a sy ntax-based competence model proponent may point to the fact that a model-based perspe ctive involves an infinite number of models, when taken to the extreme; and because that is so clearly beyond the capability of human cognition, such modeling cannot be appropri ate. A model-theoretic proponent might say that it is only via models of the actual world t hat the meaning of the sentences has any reality. Even acknowledging that competence m odels are not likely to be interested in belief revision decisions on modus ponens and mod us tollens based belief states; we can nonetheless say that a model-theoretic competence f ramework could never model any of these kinds of differences, since modus ponens and modus tollens are indistinguishable from the perspective of formal model theories. Furthe r, our finding that people seem to prefer to abandon the conditional is problematic for mo del-theoretic frameworks, unless they retain some mapping between each sentence and th e model which that sentence generates. But there are difficulties for a syntactic-based per spective. It is unclear that the syntactic form of sentences per se should be a primary tag for guiding belief revision decisions. Indeed, our finding that people were more willing t o abandon the conditional on natural language problems than on symbolic problems sugg ests that there are other, non-syntactic considerations at play that may serve as pragmatic belief revision principles. We return to this issue in the General Discussion. Belief Revision as Propositional Update 35 The belief revision results we obtained do not speak directly to performance theor ies of human deduction, but there are some important observations we can make here as well. First, the Johnson-Laird mental models framework could possibly accommodate the general preference to deny the conditional, by the preference ordering it puts on models t hat different types of sentences generate. The mental model of p→q is "[p q]..." where [p q] represents the initial explicit model, in which both p and q are true, and the ellipsis “... ” represents that there are additional models of this sentence (corresponding to possible m odels in which p is not true; Johnson-Laird, Byrne, and Schaeken, 1992). For our modus ponens problem, the initial sentence set is p→q, p, and ∴ q. Let C indicate models of the conditional, and S to indicate models of simple sentences in the initial belief set. Hence, t he initial modus ponens model set would be C: [p q]..., S: [p], S:[q], respectively. Note th at the models for the simple sentences are consistent with what the mental models theory proposes as the initial explicit model for the conditional. The modus ponens expansion in formation is ~q and we denote its model as E:[~q]. Suppose a subject compares the expan sion model E:[~q], which must be retained in any revision, to each of the models from the initial set. The expansion model would eliminate the model S:[q], be silent on the model S:[p], and eliminate the model C:[p q] of the conditional. By this process, the preferred re vision choice should be to deny this model of the conditional and the retain the non-condi tional sentence p. In fact, this choice accounted for 75% of the modus ponens revisions in Experiment 1 and about 60% in Experiment 2. By the same general reasoning, the menta l-models approach would find itself predicting a preponderance of conditional denials for modus tollens problems. While we did find this is true in general, there would have to be some further account for people's greater tendency to decide the conditionals are uncertai n (rather than false) on modus tollens problems than on modus ponens problems. From a proof-theoretic perspective, Rips (1994, pp. 58-62) directly considers the problem of belief revision as the issue of which of several premises to abandon in the fac e of contradiction, acknowledging that deduction rules cannot alone "solve" the belief rev Belief Revision as Propositional Update 36 ision problem. He discusses a multi-layer approach, in which the principles governing be lief revision decisions are themselves "logic-based processing rules" that co-exist with th e deduction rules that he proposes as components of reasoning and problem-solving. Thu s, a proof-theoretic approach might be extended to deal with our belief revision results by having an explicit higher-level rule that, when contradiction is recognized, indicates the action of disbelieving a conditional form when it is one of the premises. But even withou t an appeal to this approach, it is possible to consider a proof-theoretic account of our res ults, as we did for the mental-models perspective, using Rips' (1994) framework. Recall a gain the above perspective that portrayed the modus ponens belief-revision problem as bo iling down to " Given [p—>q, p] + [~q] and the constraint that ~q must be retained as a b elief, what can you prove?" One can imagine that a subject formulates two competing set s of premises. One set is [p—>q, ~q]. There is no direct modus tollens rule in Rips' theor y (the modus tollens inference is accomplished through the application of two other infer ence rules), thus accounting for the notion that modus tollens proof for ~p is difficult and may halt. On the other hand, there is a readily available inference rule ("and introduction" ) that can apply to the other combination of premises [p, ~q] to yield [~p and q]. From thi s perspective, subjects might reach a state that they can more easily recognize as valid an d that may be why they prefer a revision in which these sentences are retained and the co nditional is disbelieved. On the modus tollens problem, we can characterize the belief re vision dilemma as "Given [p—>q, ~q] + [p] and the constraint that p must be retained, w hat can you prove?" The modus ponens rule is readily available according to Rips' theory, and so the premise combination [p—>q, p] easily yields q. Just as easily, the other comb ination of premises [~q, p] yields [p and ~q]. The greater tendency to prefer revisions that label the conditional (and the non-conditional) "uncertain" in the modus tollens belief re vision case relative to the modus ponens belief-revision case may reflect subjects' ability t o prove something from both combinations of premises (as we have stated them) and thei Belief Revision as Propositional Update 37 r appreciation that they have no reason to prefer the premises of one proof over the other in these simple problems. Our goal in considering how two contrasting perspectives of deductive reasoning might accommodate our results was not to support one over the other; neither was it our motivating intent. The accounts we sketched above are offered as speculations on how ea ch perspective might be extended into the realm of belief revision, given their representati on and processing assumptions about deductive reasoning. Such extensions are an import ant component for an integrated theory of reasoning and required much more consideratio n than we have briefly allowed here. Models and Minimal Change As we noted earlier, one of the desiderata of the classical AI belief revision perspe ctive is that an agent should make a minimal change to its initial belief set, when resolvin g any conflict that results from new information. Within a syntactic approach, the definiti on of change is computed from the number of formulas retained from one belief state to a nother; there are not many different ways to compute this number, since the formulas are fixed. The primary issue is whether or not the set of formula is closed, i.e., includes all co nsequences of the initially-specified set of sentences. When the set of formulae is not clo sed, making a formula become part of the explicit belief set is regarded as more of a chan ge than having it be in the implicit beliefs. Within a model-theoretic approach, it turns out there is more than one way to com pute what a minimal change might be, even for the simplest problems. In this section, we present the gist of some alternative computational definitions of minimal change. None of these approaches were devised as psychological models of how humans might manipulat e alternative models in the face of conflicting information. And while the ways the algorit hms that compute minimal change might not be psychologically plausible, the final chang Belief Revision as Propositional Update 38 e that each one deems minimal often corresponds to an intuitively reasonable way of inte grating both the old and new belief information. We provide simple algorithmic interpret ations of each of these minimal change definitions in Table 5 and highlight the functional effects of computing minimal change according to one algorithm or another. A straightforward way to quantify the degree of change is to count the number of propositions whose truth values change if one model (e.g., expansion information) is inte grated with another model (e.g., the initial belief set). The tricky part comes when there is more than one model of the initial belief set, or of the expansion information, or both. Cl early, there will be more than one possible interpretation for a sentence set whenever ther e is an explicit uncertainty. By explicit uncertainty, we mean a belief sentence that directl y mentions that the truth status of some proposition is either true or false. Hence, in the se ntence set (p, q ∨ ~q), q is explicitly uncertain, so there are two models of this sentence se t: [p ~q], [p q]. Suppose, however, that the initial sentence set were "Either p and q are tru e at the same time, or they are false at the same time" and that the expansion information is "p is false, q is true, and furthermore r is true." The initial belief state has two models, [ p q], [~p ~q], and both p and q are explicitly uncertain. The proposition r was not in eithe r of the initial models of the world. But clearly, its truth status (along with every other po ssible sentence) in the initial belief set was, in hindsight, uncertain. This is what we call i mplicit uncertainty, and all the algorithms in Table 5 construct different models of the init ial belief set to accommodate the implicit uncertainty about r just as if it were explicitly u ncertain in the first place. Thus, the computations for minimal change for this problem w ould begin with these models of the initial belief set [pq~r], [pqr], [~p~q~r], and [~p~q~r] . As we shall see in the first example below, this same approach of creating extra models also applies when a sentence that is present in the initial belief set is not mentioned the ex pansion information. One approach to determining a minimal change is to chose a model of the expansi on sentences that is the minimal distance from some model of the initial belief set. Suppo Belief Revision as Propositional Update 39 se an initial belief is "Either p, q, r, s are all true at the same time, or they are all false at t he same time." So there are two different models of this initial belief: [p q r s] and [~p ~q ~r ~s]. Expansion information such as "p is true, s is false, and r is false" contradicts this i nitial belief state and furthermore does not mention anything about q. There are then two models of the expansion, one in which q is true [p q ~r ~s] and one in which it is false [p ~q ~r ~s]. The latter model of the expansion is "close" to the second model (disjunct) of t he initial belief set and is indeed "closer" than either expansion model is to the first model of the initial belief set. By this reasoning, a new belief state that represents a minimal cha nge on the initial state is [p ~q ~r ~s]). This is the gist of the minimal change approach pr oposed by Dalal (1988) and summarized as Algorithm D in Table 5. More formally, Dala l's revision of a belief set by an expansion sentence is a set of minimal models where (a) e ach member of this set satisfies the expansion information, and (b) there is no other mode l of the initial belief set that also satisfies the expansion information and differs from any model of initial belief set by fewer atoms than the set of minimal models. The revision re sults in the set of all these minimal models. Thus, Dalal's algorithm settles on one model of the expansion information, if possible, and in doing so, can be viewed as retroactively settling on one particular model of the initial belief set. An alternative intuition would hold that: only informative (non-tautological) initia l beliefs can be used to choose among multiple interpretations of the expansion informati on, if they exist. This is one way to interpret an algorithm proposed by Weber (1986). Si mply put, Weber's algorithm first identifies the initially-believed sentences that must take on whatever truth values are specified for them in the expansion. For the same example i n the preceding paragraph, this set would contain the sentences p, r, and s, because they e ach have a specific value they are required to take, according to the new information. The se sentences are then eliminated from the initial belief set to identify what (if any) inform ative sentences propositions might be retained from the initial belief set. Subtracting p, r, and s from the initial belief set {[p q r s], [~p ~q ~r ~s]}leaves [q ∨ ~q], which is a tautol Belief Revision as Propositional Update 40 ogy, and by Weber's algorithm, leaves no (informative) proposition. (Had there been som e other sentence which was in both of the initial models, it would have then been assigne d to the revised belief state). The algorithm then conjoins these two components: the truth values of p, r, and s as determined by the expansion information [p ~r ~s] and whatever c an be retained with certainty from the initial belief set, which here is the empty model [ ]. Whereas Dalal's revision for this problem would be [p ~r ~s ~q], Weber's minimal revisio n would be [p ~r ~s], with q implicitly uncertain by virtue of its absence from the model. A simple algorithm that corresponds to this approach is given as Algorithm W in Table 5. ------------------------Insert Table 5 about here ------------------------Borgida (1985) proposes an algorithm that is similar to Dalal's, but produces what might be considered a more conservative belief-state change. Essentially, each expansion model is compared to each initial belief-set model: the expansion model that produces a minimal change for a particular initial-belief interpretation is remembered. All these expa nsions that are minimal with respect to some model of the initial belief set are then used t o define the new belief set. An algorithm that captures this approach is given as Algorith m B in Table 5. Consider a case where there is more than one interpretation of the initial belief set. If [p q ~s] is the initial belief set, and [~p ~q r ~s] and [~p ~q ~r s] are two mod els of the expansion information, then two models of the initial belief set are considered: t he first contains r and second contains ~r. Both interpretations of the expansion informati on define a minimal change with one of the interpretations of the initial belief set (the firs t expansion disjunct with the first interpretation of the belief set, and the second expansio n disjunct with the second interpretation of the belief set). Thus, both [~p~q r ~s] and [~p ~q ~r s] are on the stack after step B1.2. Since neither of these is minimal with respect to the other, the final belief set consists of guaranteed truth values for those propositions on which the interpretations agree and uncertain truth values for propositions on which they Belief Revision as Propositional Update 41 disagree, yielding a final belief state of [ ~p~q {r~s ∨ ~rs}]. Algorithm B differs from Al gorithm D in that each model of the initial belief set identifies, in Algorithm B, what mod el of the expansion information would result in a minimal change (by number of propositi ons changed). Once one of the expansion models is identified as minimal with respect to a particular model of the initial belief set, there is no further check of whether one change is more or less minimal than some other combination of initial-belief interpretation and e xpansion-interpretation (as Algorithm D does on step D2). This can be viewed as a more conservative belief-state change, because there isn't the possibility of settling on one parti cular model of the initial belief state. Satoh (1988) proposed belief revision operator that is a less-restricted version of Borgida’s revision operator, when applied to the propositional case. The feature that mak es it less restricted is illustrated in Algorithm S, which is identical to Algorithm B, except that step B1.2 in Algorithm B occurs outside the first control loop as step S2 in Algorith m S. Functionally, this difference means that there is no pruning of non-minimal changes with respect to a particular belief-set model (as on Step 1.2 in Algorithm B). Instead, the entire set is saved until step S2, which removes any change that subsumes another change . After S2, all changes that remain are minimal. Step S3 then finds a model of the expansi on that is consistent with the minimal set of necessary changes. Put more intuitively, this algorithm crosses all interpretations of the initial belief set and all interpretations of the e xpansion set to create the model set from which a minimal change is computed. The funct ional effect is that, when there is just one model of the initial belief set, that model may " choose" the closest interpretation of the expansion information; when there is just a single version of the expansion information, that model may "choose" among alternative model s of the initial information. Only the latter may occur under the Borgida algorithm. We are not interested so much in the means by which these alternative model-base d revision frameworks define minimal change, as we are in the way they capture alternati ve intuitions about manipulating multiple models. In Algorithm D, the way that minimal Belief Revision as Propositional Update 42 change is computed can have the effect of "selecting" one of multiple interpretations of th e initial belief set. The effect of Algorithm B is to retain multiple models in the new belie f set when there are multiple models of the expansion information. Algorithm S will com pute a new belief state with multiple models, when multiple models exist in both the initi al and expansion information; but it can use a single model of either to produce a single model of the new belief set. Finally, Algorithm W uses the expansion information to defi ne what can be believed with certainty; other belief-set sentences not mentioned in the ex pansion information may decide between multiple interpretations of the expansion inform ation, but only if their truth value was known with certainty in the first place (i.e., was tru e in every model or false in every model of the initial belief state). There are plausible elements in each of these approaches for principles that might dictate how people deal with multiple interpretations of information when resolving inco nsistencies. Our interest was whether which, if any of them, corresponded to how people integrate multiple models in a belief revision task. As the reader might surmise, for any p articular problem, some or all of the methods could yield the same final belief set. It is po ssible, however, to define a set of problems for which a pattern of responses would distin guish among these alternative approaches. We developed such a problem set to obtain dat a on whether people follow a minimal change principle, as defined by any of these approa ches. The revision problems were very simple: there were either one or two models of the initial belief set and either one or two models of the expansion information. The problem sets were designed to distinguish among the four model-based minimal change framewor ks described above. Experiment 4 Method Problem Set. Table 6 gives the problem set used for Experiments 4 and 5. The firs Belief Revision as Propositional Update 43 t five problems in this table were used only in Experiment 4; problem 6 was added for Ex periment 5. For economy of space, we write sentence letters adjacent to one another to m ean ‘and’. Thus, the problem 1 notation (pqrs) ∨ (~p~q~r~s) means “Either p, q, r, and s a re each true at the same time or else they are each false at the same time.” -----------------------Insert Table 6 about here -----------------------The subscripts for the revision choices in Table 6 correspond to the particular mod el-theoretic definition of minimal change: D for Algorithm D, W for Algorithm W, and s o forth. Experiment 4 offered subjects two revision choices for Problems 1-5 (of Table 6) ; these each corresponded to one or more of the four definitions of minimal change we ou tlined in the previous section. It can be seen that each of the four algorithms selects a diff erent set of answers across these five problems: Algorithm D selects answers <1,1,1,1,1> for its five answers; Algorithm B selects answers <2,2,2,1,1>; Algorithm S selects answe rs <2,2,1,1,2>; and Algorithm W selects answers <2,2,2,2,2>. Design. Problem type was a within-subjects factor; all subjects solved all five pro blems. As in Experiment 1, presentation form (symbolic v. science-fiction stories) was m anipulated as a between-subjects factor. Appendix B shows how the initial-belief sentenc es and the expansion sentences were phrased in the symbolic condition; the revision alter natives were phrased in a similar manner. Different letters were used in each of the probl ems that the subjects actually solved. The five different science-fiction cover stories were paired with the problems in six different ways. Subjects and Procedure. The same 120 subjects who participated in Experiment 1 provided the data presented here as Experiment 4. Sixty subjects were assigned to the sy mbolic condition and sixty were assigned to the science-fiction condition. Equal numbers of subjects received the six different assignments of science-fiction cover stories to probl ems. No science-fiction cover story appeared more than once in any subject's problem bo Belief Revision as Propositional Update 44 oklet. Other details about the procedure and instructions were as described for Experimen t 1. Results Unlike the modus ponens and modus tollens belief revision problems, there was n o significant effect for the symbolic versus science-fiction manipulation on these problem s. Table 6 presents the percentage of subjects choosing each possible revision choice, coll apsed across presentation condition. The only planned comparisons concerning these data were within-problem differences, i.e., whether one revision choice was preferred signific antly more often than another. Within each problem, there is a clear preference for one re vision over the other: subjects chose revisions that most closely matched the form of the e xpansion information. We also tabulated the number of subjects whose response pattern a cross problems matched the particular pattern associated with each revision algorithm des cribed in Table 5. Virtually no subjects matched a particular response pattern for all five problems. Experiment 5 A concern about these data is that subjects were not following any particular mod el of change at all, but simply using the expansion sentence to define the new belief set. T his could mean that they viewed the problem as an update, rather than a revision, problem (i.e., the world has moved to a new state defined by the expansion and there is no reason to maintain anything from the initial belief state), or it could mean that they were simply not engaged in the task. Since the same subjects generated distinct problem-specific patte rns of responses in Experiment 1, we do not believe the latter possibility holds. In Experiment 5, we included two additional response alternatives for each proble m in order to test whether subjects continued just to adopt the expansion information (whi ch might be the simplest interpretation of the results). Revision choice 3 was a non-mini mal change model that was consistent with some interpretation of the expansion informati Belief Revision as Propositional Update 45 on. Revision choice 4 included only those sentences whose truth values were not contradi cted within the expansion information or between some model of the initial sentences and the expansion. Basically, revision choice 4 offered the minimal number of sentences that could be known with certainty and made all other conflicts between truth values become "uncertain." We also added Problem 6, which was isomorphic in form to Problem 5, except tha t the initial belief set consisted of all negated sentences rather than of all positive sentence s. If subjects have a bias for models that consist primarily of non-negated sentences, then they should prefer such "positive" models regardless of whether they are minimal change models. Problems 5 and 6 differed only in whether the sentences in the initial set were all true or all false. Note the symmetry between revision choices 1 and 3 for these problems: the revision [~pqr], with one negated sentence, is a minimal change model for Problem 5 but a non-minimal change model for Problem 6. Conversely, [p~q~r] is the minimal chan ge model for Problem 6 and a non-minimal change model for Problem 5. If subjects are b iased towards revisions that maximize non-negated sentences, then there should be an int eraction between the form of the initial belief set and the revision selected. Finally, we str essed in the instructions that both the initial and subsequent information should be consid ered before determining what should or should not be believed, just in case subjects belie ved that the expansion information should replace the initial belief set. Method Forty-three subjects solved problems 1-6 from Table 6 in random order. Since Ex periment 4 had shown no effect for symbolic v. science-fiction presentation, the problems were presented in symbolic form only and the response alternatives appeared in different random orders for each subject. Results and Discussion Belief Revision as Propositional Update 46 The percentages of subjects choosing each revision choice in Experiment 5 are gi ven in Table 6. As in Experiment 4, Experiment 5's subjects did not consistently obey any particular pattern of minimal change. First, it is striking that revision choice 1 was never the most preferred revision—it is the syntactically simplest way of specifying a model th at accommodates the expansion sentence and corresponds to Algorithm D, which has an i ntuitively simple notion of minimal change. The second feature of the results concerns th e relative percentages of revision 2 (in which the new belief state is simply the adoption o f the new information) and revision 4. While revision choice 2 was the clear preference i n Experiment 4, it was no longer the clear favorite here. Generally speaking, if subjects w ere given the option of tagging certain sentences as "uncertain" (revision 4), they gravitat ed to this choice over a revision that more precisely (and more accurately) specifies the u ncertainty as multiple models (revision 2). One conjecture is that subjects elect to use revi sion 4 as short-hand way of expressing the uncertainty entailed in having multiple models of the world. That is, they may see "p and q are both uncertain" as equivalent to (p~q) ∨ (~pq), although, of course, it is not. It is unclear whether subjects appreciate the 'loss of in formation' inherent in such a specification. Problems 5 and 6 were of particular interest, because they differed only in whethe r the initial belief set consisted of positive or negated sentences; the expansion informatio n was the same. The set of revision alternatives is also identical. As with the other proble ms, the most preferred revision choice (about 40%) was to declare all sentences uncertain , when their truth value differed in two different models of the expansion information (an d they did not merely adopt the complex specification of the expansion information as a b elief state, as they had in Experiment 4). However, if we restrict our attention just to the percentage of revision 1 and revision 3 choices in these problems, we see that about the s ame number of subjects (20%) chose the revision ~pqr when it served as minimal change revision 1 for problem 5 and also when it was the non-minimal revision 2 for Problem 6. Conversely, only 7% of the subjects chose p~q~r when it was the non-minimal revision Belief Revision as Propositional Update 47 1 for Problem 5, but also only 7% chose it when it was (the minimal change) revision 3 f or Problem 5. A simple chi-square computed on the response-choice by problem type (Pr oblem 5 v. Problem 6) frequency table was marginally significant (χ2 =7.52, df=3, p =.05 7). These results suggest that there may be a bias against revisions that have more negate d beliefs than non-negated beliefs in them. There is some suggestion of this in problem 2 as well, in which 35% of the subjects choose a non-minimal change revision (revision 3) than either of the two minimal change revisions (revisions 1 and 2). Such a finding itself is certainly consistent with body of evidence indicating that reasoning about negated sent ences pose more difficulties for subjects (see, e.g., Evans, Nestead, Byrne, 1993, on "neg ated conclusions"); hence, people may prefer to entertain models of situations that contai n fewer negations, when possible. This possibility of a bias against models with negations needs further, systematic study. In sum, Experiments 4 and 5 suggest that subjects are n ot following any single model-based minimal change metric and do not integrate the expa nsion information whole-heartedly. Despite the availability of choices that could be select ed via simple matching procedure between disjuncts appearing in the initial and new info rmation (revision 1 across all problems), our subjects seem to prefer belief states that con sist of single models and models with non-negated beliefs, when possible. General Discussion We can summarize the main findings from this study as follows. First, to resolve t he inconsistency that new information creates with an existing belief set that consists of si mple sentences (p, q) and conditional sentences (p→q), the preferred revision was to disb elieve the conditional rather than alter the truth status of one of the initial simple sentence s. This preference was even stronger on problems using science-fiction or familiar topic c over stories than it was using symbolic formulas. Second, there were some differences in revision choices depending on whether the initial belief set was constructed by using a m odus tollens or modus ponens inference. Subjects more often changed the truth status of t Belief Revision as Propositional Update 48 he initial simple sentence (and the conditional, when there was that option) to "uncertain" on the modus tollens problems than they did on the modus ponens problems. Third, we o bserved that the patterns of revision choices on the simple problems we investigated does not depend on whether or not the (modus ponens) inference was explicitly listed in the ini tial belief set or whether subjects were left to perform the inference themselves. Fourth, w e note that the patterns of revision did not change when the initial belief state was constru cted from purely propositional reasoning or used universally-quantified inferences. Fifth, we discovered that when an implied conclusion of the initial belief set itself gives rise to yet another conclusion, and when the first of these conclusions is contradicted by the exp ansion information, then the status of the second conclusion is regarded as "uncertain." Fi nally, we investigated alternative model-theoretic definitions of minimal change. We fou nd that subjects did not adhere to any of these particular prescriptions, some of which (e.g ., Algorithm D) can be construed as a fairly straightforward matching strategy between a model in the initial information and a model of the expansion information. Even when th e initial belief state had only one model, subjects did not use it to chose among alternative models of (uncertain) expansion information; and even when there was only a single mo del of expansion information, subjects did not use this to chose among alternative models of an (uncertain) initial belief state. A disjunct of multiple models can specify how the tr uth value of one sentence co-varies with another's; subjects did not prefer such multiplemodel specifications of a belief state as a way to represent uncertainty. They instead singl e-model revisions that retained only sentences that had an unambiguous truth values acro ss the initial and expansion information, and labeled all other sentences as uncertain (eve n though this results in a loss of information). There is a possibility as well that people pr efer revisions that contain positive rather than negated sentences; this requires further stu dy. In the remainder of this section, we consider these results for notions of epistemic ent renchment and minimal change. Belief Revision as Propositional Update 49 On Epistemic Entrenchment The rationale behind a notion like epistemic entrenchment is that, practically, an a gent may need to choose among alternative ways to change its beliefs, and intuitively, the re will be better reasons to chose one kind of change over another. These better reasons ar e realized as a preference to retain or discard some types of knowledge over another; the i ssue is what those epistemically-based principles of entrenchment are or ought to be. As we noted in the introduction, some theorists have argued that conditional statements like p→ q may warrant, a priori, a higher degree of entrenchment than some other sentence ty pes, not because there is something to be preferred about material implications, but becau se that form often signals "law-like" or predictive relations that have explanatory power. And law-like relations, because of their explanatory power, should be retained over other types of knowledge when computing a new belief state. We did not find evidence for this kind of entrenchment as a descriptive principle of human belief revision in the tasks we studied. In general, the frequency of continuing t o believe the conditional was lower than what might be expected by chance, and lower sti ll on natural language problems. Finding that belief-revision choices changed when the pr oblems involved non-abstract topics is not surprising, for there are many results in the de ductive problem solving literature indicating that real-world scenarios influence deductiv e inferences, serving either to elicit, according to some theories, general pragmatic reason ing schemas (e.g., Cheng & Holyoak, 1989) or, according to other interpretations, specifi c analogous cases (Cox & Griggs, 1982). On the other hand, there was no domain-specifi c knowledge subjects could bring to bear about a science-fiction world. Indeed, the clause s used to make science-fiction sentences are not unlike those used by Cheng and Nisbett ( 1993) as "arbitrary" stimuli to investigate causal interpretations of conditionals. Nonethel ess it is clear that subjects revised and expanded non-symbolic belief sets differently than they did symbolic belief sets. Belief Revision as Propositional Update 50 Subjects may have interpreted the science-fiction conditional relations as predictiv e, or possibly causal, relations. The instructions that set up the science-fiction problems e njoined subject to imagine that information about an alien world was being relayed from an scientific investigative team. This may have prompted a theory-formation perspective, based on the assumption that even alien worlds are governed by regularities. The generati on of, and belief in, these regularities depends on observations. The initial belief set had s uch a regularity in it (the conditional), plus a “direct observation” sentence. When the exp ansion information indicated that the inference from these two was contradicted, the "den ial" of the conditional is one way of asserting that the regularity it expresses, as specified, does not hold, in this particular case. Cheng and Nisbett (1993) found that a causal interp retation of if p, then q invokes assumptions of contingency, namely that the probability of q's occurrence is greater in the presence of p than in the absence of p. Subjects may have viewed the (contradictory) expansion information in the modus ponens and modus tollens problems as calling this contingency into question. Such a perspective only makes sense when the problems are not manipulations of arbitrary symbols, and is consistent with our finding a higher rate of rule denials on non-abstract problems than on symbolic problems . When simple statements of p and q are viewed as observations about some world, p→q can be interpreted as a theory, or summarizing statement, about how the truth value s of these observations are related. This is, essentially, a model-theoretic viewpoint: an e xpression such as p→q is shorthand for how the truth values of p and q occur in the world . Taking this understanding of conditionals, the preference of our subjects to deny the con ditional as a way of resolving contradiction can be interpreted as a preference to retain th e truth value of "data" (the non-conditional sentences) and deny the particular interdepen dence that is asserted to hold between them. This seems straightforwardly rational from a n empiricist viewpoint: the "regularities" are nothing more than a way of summarizing th e data. So, for a through-and-through empiricist, it is not even consistent to uphold a "law Belief Revision as Propositional Update 51 " in the face of recalcitrant data. Such a perspective puts a different light on the observati on that people did not make the "easy" modus ponens inference from the expansion infor mation combined with a modus tollens belief set: to have opted for this revision would ha ve required changing the truth values of observational data. While doing so may be a plau sible alternative when problems involve meaningless symbols, it may not seem rational al ternative when working with information that is interpretable as observational data. The idea that data enjoys a priority over regularities has been offered as a belief re vision principle in other frameworks (Thagard, 1989; Harman, 1986) particularly when re gularities are (merely) hypotheses under consideration to explain or systematize observed facts. There is a natural role, then, for induction mechanisms in specifying the process o f belief revision, once the conditional "regularity" is chosen by the agent as suspect. We n ote that the classical AI belief revision community presents the belief revision problem as denying previously believed sentences, including conditionals. But replacing p—>q with (p & r )→ q or (p & ~s) —> q are equally good ways to deny p—>q. In such a case, the conditional regularity can either be "patched" or demoted to the status of default rule ("M ost of the time, p → q, except when r holds"). In our view, this method of denying a cond itional as belief-revision choice seems to be preferable to merely lowering a degree of bel ief in the conditional, for the latter leaves the agent is no wiser about when to apply such a rule, only wiser that it should be less confident about the rule. This approach is being p ursued in some classical approaches to belief revision (e.g., Ghose, Hadjinian, Sattar, Yo u, and Goebel, 1993) and in explanation-based learning approaches to theory revision in t he machine learning community, where the inability of a domain theory to explain some data causes changes to the domain theory rules (Ourston & Mooney, 1990; Richards & M ooney, 1995). While some aspects of the belief revision process can be viewed as inductive proc esses searching for a better account of some data, we note that the such a perspective itsel f does provide principles for guiding such a process when there are alternative ways to re Belief Revision as Propositional Update 52 concile a contradiction. Specifically, we don't always believe the data at the expense of a regularity or contingency that we currently believe holds in the world. As we noted in th e introduction, there are intuitions opposite to those that would deny or change a regularit y to accommodate data: Kyberg's (1983) belief framework includes a place for both meas urement error and the knowledge that some types of observations are more prone to error than others. Thagard (1989) offers an explanatory coherence metric by which data can be discounted, if they cohere with hypotheses which themselves are poor accounts of a large r data set. Carlson & Dulany's (1988) model of reasoning with circumstantial evidence in cludes parameters for degrees of subjective belief in the evidence. So the broader questio ns for epistemic entrenchment might be to ask what kinds of data and what kinds of regul arities are more differentially entrenched than others. On our simple belief revision tasks, we found some baseline results that suggest a tendency to abandon the conditional. But it has long been recognized by researchers in bo th linguistics and human deduction that the if p then q form is used to express a broad ran ge of different types of information, e.g., scientific laws, statistical relationships, causal re lations, promises, and intentions ( "If it doesn't rain tomorrow, we will play golf"). More recent studies using both the selection paradigm described here as well as one in which s ubjects gave degrees-of-belief ratings to the initial belief sentences (Elio, 1996) have indi cated that different types of knowledge expressed in this common syntactic form—causal relations distinguished by different enabling and disabling conditions, promises, and defi nitions—are differentially entrenched, when given new and contradictory evidence. For u nderstanding the pragmatic principles of belief revision, these meta-knowledge distinctio ns may be important in formulating entrenchment preferences in the face of contradiction . On Multiple Models and Minimal Change Belief Revision as Propositional Update 53 One clear result we obtained is that people retain uncertainty in their revised belie f states—they did not use single models of the new information to chose among alternativ e interpretations of the initial information, or conversely, in the tasks we gave them (e.g., they did not follow Algorithm D). Further, they tended to select revisions that include mo re uncertainty than is logically defensible, opting for "p is uncertain and so is q" as often or more frequently than "p is true and q is false, or else p is false and q is true." It seems c lear that people recognize that the former is less informative than the latter about possible combinations of p and q's truth values, but our subjects chose it anyway. One way to vie w the results we obtained is to say that many of our subjects preferred revisions which we re not minimal with respect to what was changed, but were instead minimal with respect t o what they believed to hold true without doubt when both the initial and expansion infor mation were considered jointly. It certainly seems more difficult to work with a "world" s pecification like {[~p~q r s] or [p ~q ~r ~s]} than it is with one that says "q is false and I' m not sure about anything else," even though (from a logical point of view) the former sp ecification contains much more information than the latter. What we learned from our initial investigations on minimal change problems may have less to do with the metrics of minimal change and more to do with issues of how pe ople manipulate multiple models of the world. Rips' (1989) work on the knights-and-knav es problem also highlights the difficulty that people have in exploring and keeping track o f multiple models. In that task, the supposition that one character is a liar defines one mod el, being a truth-teller defines another model, and each of these might in turn branch into other models. Even working such a problem out on paper presented difficulties for subjec ts, Rips reported. Yet in real life, we can certainly reason about vastly different hypotheti cal worlds that could be viewed as being equivalent to disjunctions of complex sentence s ets. Unlike the arbitrary problems give to our subjects or even the knights and knaves pro blems, alternative hypothetical worlds about real-world topics may have some "explanato ry glue" that holds together the particular contingencies, and no others, among the truth v Belief Revision as Propositional Update 54 alues of the independent beliefs. The question is whether for more real world situations, are people better able to retain and integrate the interdependencies among truth values in multiple models? Alternative Representations of Belief States Representing a belief state as a set of sentences or even as a set of models is a sim plification. We believe that number of important issues arise from this simple conceptuali zation and this study offers data on some of those issues. We noted alternative approache s to modeling belief states in the introduction, specifically those that use probabilistic info rmation and degrees of belief. But there are two other perspectives that have long been co nsidered from a philosophical viewpoint: the foundationalist view and the coherentist vie w. The foundationalist view (Swain, 1979; Alston, 1993; Moser, 1985, 1989) distinguishe s between beliefs that are accepted without justification and those that depend on the prio r acceptance of others. Such a distinction is used in truth-maintenance systems (e.g., Doyl e, 1979; deKleer, 1986) for keeping track of dependencies among beliefs and to prefer the retraction of the latter ("assumptions") over the former ("premises") when contradictions are caused by new information. Pollock's (1987) defeasible reasoning theory defines a wi der class of distinctions (e.g., "warrants" and "undercutters") and such distinctions can als o be used to define normative foundationalist models of belief revision. The coherentist v iew (BonJour, 1985; Quine & Ullian, 1978; Harman, 1986) does not consider some belief s as more fundamental than others, but rather emphasizes the extent to which an entire set of beliefs "coheres". One set of beliefs can be preferable to another if it has a higher cohe rence, however defined. Thagard's (1989) theory of explanatory coherence is an instance of this perspective and operational definitions of coherence can, in such a framework, be a means of implementing belief revision principles (Thagard, 1992). Pollock (1979) gives a whole range of epistemological theories that span the spectrum between foundationalist and coherentist. Belief Revision as Propositional Update 55 It is widely believed (e.g., Harman, 1986; Gärdenfors, 1990b; Doyle, 1992; Nebel , 1992) that the original AGM account of belief revision, as well as model-based versions of it, are coherentist in nature. Harman (1986) and Gärdenfors go so far as to say that a f oundationalist approach to belief revision (as advocated, e.g., by Doyle, 1979; Fuhrmann, 1991; Nebel 1991) is at odds with observed psychological behavior, particularly concern ing people's ability to recall the initial justifications for their current beliefs. More marsha ling of this and other experimental evidence (including the type we have reported in this a rticle) could be a reasonable first step towards an experimentally-justified account of how human belief structures are organized; and with this is perhaps an account of how belief structures of non-human agents could best be constructed. Finally, we note that it remains a difficult matter to examine "real beliefs" and the ir revision in the laboratory (as opposed to the task of choosing among sentences to be ac cepted as true); the paradigm of direct experimentation with some micro-world, which ha s been used to study theory development, is one direction that can prove fruitful (e.g., Ra nney & Thagard, 1988). However, conceptualizing a belief state merely as a set of beliefs can still afford, we think, some insight into the pragmatic considerations people make in resolving contradiction. Future work There are many issues raised in these investigations that warrant further study; we have touched upon some of them throughout our discussions. The possibility of bias agai nst changing negated beliefs to non-negated ones, or in preferring revisions with non-neg ated sentences, needs systematic study. We used a selection paradigm throughout this stu dy and it is important to establish whether similar results hold when subjects generate thei r new belief state. A more difficult issue is whether there are different patterns of belief re vision depending on whether the belief set is one a person induces themselves or whether it is given to them. In the former case, one can speculate that a person has expended some Belief Revision as Propositional Update 56 cognitive effort to derive a belief, and a by-product of that effort may create the kind of c oherentist structure that is more resistant to the abandonment of some beliefs in the face o f contradictory information. This kind of perspective can be applied to an early study by Wason (1977) on self-contradiction. He found that subjects given the selection task were quite reluctant to change their conclusions about how to validate a rule, even when they were shown that such conclusions were contradicted by the facts of the task. Yet on the di fferent sort of task, he found that subjects can recognize and correct invalid inferences ab out the form of a rule they are actively trying to identify from a data set, when the data set leads them to valid inferences that contradict the invalid ones they make. Whether reco gnizing contradiction depends on the demands a task makes of a reasoner might elucidate something about how premises are formulated and about how inferences are validated; in the belief revision scenarios we used in this study, the contradiction occurs not because o f the reasoner's inferencing process, but because additional information about the world i ndicates that one of initially accepted premises must be suspect. The recognition and res olution of contradiction is important to general theories of human reasoning that employ deduction, induction, and belief revision. How general performance models of deductive and inductive reasoning can embrace belief revision decisions is an important open issue. Belief Revision as Propositional Update 57 References Alchourrón, C., P. Gärdenfors, D. Makinson (1985). On the logic of theory change: Partia l meet contraction and revision functions. Journal of Symbolic Logic, 50, 510-530. Alston, W. (1993). 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Portions of this work appeared in the Proceedings of the 16th Conference of the Cognitive Science Society, Atlanta, 1994. We thank Sarah Ho ffman and Sioban Neary for their assistance in conducting these experiments, the Depart ment of Psychology at the University of Alberta for the use of their subject pool, and Dr. Terry Taerem for discussions on statistical analyses. We are especially grateful to Aditya Ghose, Randy Goebel, Philip Johnson-Laird, Gil Harman, and an anonymous reviewer, w hose comments on an earlier version of this manuscript greatly improved the presentation of this work. Correspondence concerning this manuscript can be sent to Renée Elio, Dep artment of Computing Science, University of Alberta, Edmonton, Alberta, Canada, T6G 2H1 or via email at ree@cs.ualberta.ca. Belief Revision as Propositional Update 64 Footnotes 1 We note, however, that not all proponents of probabilistic frameworks concur th at 'acceptance' is a required notion. Cheeseman (1988) and Doyle (1989), for example, ar gue that “acceptance” is really a mixture of two distinct components: the theory of degree of belief together with a theory of action. The latter theory uses degrees of belief plus a t heory of utility to produce a notion of "deciding to act in a particular circumstance." Jeffr ey (1965) also proposes a framework that avoids an acceptance-based account of belief. 2 Most syntax-based approaches put into their definitions of belief revision that th e set of all logical consequences is computed for the original belief state in order to deter mine the contradictions. But only changes to this original "base" belief set are considered in constructing the new belief state. One intuition behind this variety of belief revision ca n be that certain beliefs (the ones in the "base") are more fundamental than other beliefs, and any change in belief states should be made to the implicit beliefs first and only to the base if absolutely required. This view has relations to the foundationalist conception of be lief states that we return to in our general discussion. 3 Some works, e.g., Fagin, Ullman, and Vardi (1986), use the term "theory" to inc lude both what we call a syntax-based approach and what we call a theory-based approac h. When they want to distinguish the two, they call the latter a "closed theory." 4 We aim to carefully distinguish our remarks about model-theoretic competence frameworks, as proposed by what we have been calling the classical AI belief revision co mmunity, from remarks concerning model-theoretic performance frameworks of human d eduction, such as the mental-models theory. It is proper to talk of "models" in the context of either framework. Context will normally convey which framework we intend, but we u se the term "formal AI models" or "mental models" when it is necessary. 5 The explicit inclusion of ~r in S1 and ~n in S2 is, by some accounts, an extra inf erence step beyond what is necessary to incorporate ~w, since they could be considered a Belief Revision as Propositional Update 65 s implicit beliefs rather than explicit beliefs; this could be accommodated simply by drop ping r and n from S1 and S2, respectively. 6 The actual problems used in these first experiments were really quantified versi ons of modus ponens and modus tollens. Our modus ponens problem type is more accurat ely paraphrased as: from For any x, if p holds of x, then q holds of x, and furthermore p h olds of a we can infer q holds of a. Similar remarks can be made for our modus tollens. 7 The reason is this. In a model approach, the initial belief state is the model [p is true, q is true]. When this is revised with ~q, thereby forcing the change from q's being tr ue to q's being false in the model, we are left with the model [p is true, q is false]. Such a model has zero changes, other than the one forced by the expansion information; and in t his model p→q is false. In order to make this conditional be true, a change to the model t hat was not otherwise forced by the revision information would be required, to make p be false. (Similar remarks hold for the modus tollens case). Thus model theories of belief re vision will deny the conditional in such problems. 8 We used the term "knowledge" rather than "belief" in instructions to subjects, be cause we wanted them to accord full acceptance to them prior to considering how they mi ght resolve subsequent contradiction. The use of "knowledge" here, as something that cou ld subsequently change in truth value, is nonstandard from a philosophical perspective, al though common in the AI community. Subsequent studies in which we called the initial b elief set as "things believed to be true" have not impacted the type of results we report her e. 9 The loglinear model for this data is ln(Fijk) = µ + λr i + λpres j + λprob k + λ r ip resj + λ r iprobk + λ pres jprobk, where Fijk is the observed frequency in the cell, λri is th e effect of the ith response alternative, λpres j is the effect of the jth presentation-form cat egory, λprobk is the effect of the kth problem type category, and the remaining terms are two-way interactions among these. The equivalent "logit" model, in which response is id entified as the dependent variable, has terms for response, response by presentation mode Belief Revision as Propositional Update 66 , and response by problem type; it yields identical chi-square values. Loglinear and logit procedures from SPSS version 5.0 were used for these analyses. Simple chi-squares comp uted on several two-dimensional frequency tables are consistent with the loglinear analys es and the conclusions presented in the text. The effect of symbol v. science-fiction prese ntation approached significance on both MP and on MT problems, when simple chi-squar es were computed for separate two-dimensional frequency tables (χ 2=5.65 and 4.87, p = . 059 and .087; df=2 in both cases). Belief Revision as Propositional Update 67 Table 1 Definitions of Initial Belief States and Revision Alternatives for Experiment 1's Problem Set Problem Type Revision Alternatives Modus Ponens Initial SS: Expansion: p—>q, p, q 1. p—>q, ~p, ~q ~q 2. ~(p—>q), ~q ?p 3. ~(p—>q) p ~q Modus Tollens Initial SS:p —>q, ~p, ~q 1. p—>q, p, q Expansion: 2. ~(p—>q), p, ?q p 3. ~(p—>q), p, ~q Note: SS means sentence set. Expansion means the expansion information. ? means uncertain. Belief Revision as Propositional Update 68 Table 2 Percentage of subjects choosing each revision alternative, Experiment 1 Problem Type Modus Ponens Symbol SciFi Mean Modus Tollens Symbol SciFi Revision Alternative 1. disbelieve initial non-conditional .25 .14 .20 .33 .17 .25 about non-conditional .38 .29 .34 .38 .54 .46 3. disbelieve conditional .37 .58 .48 .28 .29 .29 2. disbelieve conditional, uncertain Mean Belief Revision as Propositional Update 69 Table 3 Percentage of subjects choosing each response alternatives, Experiment 2 Problem Modus Ponens Modus Tollens Revision Choice 1. disbelieve non-conditional .23 .26 2. disbelieve conditional; non-conditional uncertain .12 .16 3. disbelieve conditional .35 .12 4. disbelieve both conditional and non-conditional .14 .02 5 both conditional and non-conditional uncertain .16 .44 Belief Revision as Propositional Update 70 Table 4 Experiment 3 problem types Problem 1 Initial Sentence Set m & d —> g, m, d [Therefore, g] Expansion ~g Revision Alternatives 1. ~[m & d —> g], m, d 2. m & d —> g, (~m & d) or (m & ~d) or (~m & ~d) Problem 2 Initial Sentence Set c —> h, h —> m, c. [Therefore, h and m] Expansion Revision Alternatives ~h 1. h —> m, ~[c —> h], c, ?m 2. h —> m, ~c, ?m 3. h —> m, ~[c —> h], c, m 4. h —> m, m c —>h, c —> h, ~c, Note: Bracketed consequences appeared in the initial sentence set for "consequences giv en" condition and were omitted in the "no consequences given" condition. All response c hoices included the expansion sentence as part of the revision description. See text for per centages of subjects choosing each option. Belief Revision as Propositional Update 71 Table 5 Algorithms for Minimal Change Algorithm D D1 For each model of the expansion information do D1.1 For each model of the initial belief set do Find and save the differences. D1.2 From the set of differences, identify the smallest change. Put this smallest change and the expansion model responsible for it on the candidate stack. D2. From the candidate stack, chose as the new belief state the expansion model that is responsible for the smallest of all the minimal changes saved from D1.2. If there is more than one, use their disjunction Algorithm W W1 For each model of the belief set do W1.1 For each model of expansion do Find and save the propositions that must change W1.2 Retain just the minimal set of propositions that must change for this pairing of an belief set model and an expansion model W2 Take the union of all proposition sets identified in 1.2 and remove them from the initial belief set W3. Identify the set of remaining KB propositions with known (certain) truth values. If this set is empty, then the new belief set is the expansion information. Otherwise, the new belief set is the conjunction of the old KB propositions with the expansion information Table 5 continued next page Belief Revision as Propositional Update 72 Table 5 continued Algorithm B B1. For each model of the initial belief set do B1.1 For each model of the expansion do Find the differences and save them B1.2 From the set of differences, identify the minimal change and put the expansion model responsible for it on the candidate stack B2. Combine all models of expansion information on the candidate stack to determine the new belief state. Algorithm S S1 For each model of the initial belief set S1.1 For each model of the expansion, stack the differences between them. S2. From the set of differences, eliminate non-minimal changes S3. Combine all models of expansion information on the candidate stack to determine the new belief state. Belief Revision as Propositional Update 73 Table 6 Problems and percentage of subjects choosing each revision alternative, Experiments 4 and 5 Experiment Problem 1 Initial: Expansion: Revision Alternative (pqrs ) or (~p~q~r~s) 1. p ~q ~r ~s p~r~s 2. p ~r ~s ?q B, S, W 4 D 3. p q ~r ~s . 06 .07 .94 .58 .05 4. p ?q ?r ?s 2 .30 Initial: (pqrs ) or (~p~q~r~s) 1. p ~q ~r ~s D Expansion: (~p~qrs) or (p~q~r~s) 2. (~p~qrs) or (p~q~r~s) B, S, W 3. ~p ~q r s .11 .07 .89 .21 .35 4. ~q ?p ?r ?s 3 Initial: Expansion: .37 pq~s 1. ~p ~q r ~s D, S, W .22 .20 (~p~q) & [(r~s) or (~rs)] 2. (~p~q) & [(r~s) or (~rs)] B .78 .43 3. ~p ~q ~r s .0 4. ~p ~q ?r ?s 4 Initial: Expansion: 5 .37 pq 1. p or q, not both ~p or ~q or (~p~q) 2. ~p or ~q or (~p~q) W 3. ~p ~q 4. ?p ?q D, B, S .12.07 .88.30 .12 .51 (Table 6 continued next page) Belief Revision as Propositional Update 74 Table 6 continued Experiment Problem 5 Initial: Expansion: 6 Initial: Expansion: Revision Alternative 4 pqr 1. ~p q r (~pqr) or (p~q~r) 2. (~p q r) or (p ~q ~r) S, W D, B 5 .10 .21 .90 .26 3. p ~q ~r .07 4. ?p ?q ?r .46 ~p~q~r 1. p ~q ~r (~pqr) or (p~q~r) 2. (~p q r) or (p ~q ~r) .07 D, B S,W .30 3. ~p q r .23 4. ?p ?q ?r .40 Note: Initial means initial sentence set. Expansion means expansion information. Belief Revision as Propositional Update 75 Appendix A Clauses used for Science-Fiction Stimuli , Experiment 1 Subjects received one of the three possible science-fiction versions of the modus ponens and modus tollens rules, given below. Each version was used equally often across subjects. Modus Ponens Rules If a Partiplod hibernates during the day, then it is a meat eater. If a cave has a Pheek in it, then that cave has underground water. If a ping burrows underground, then it has a hard protective shell. Modus Tollens Rules If Gargons live on the planet's moon, then Gargons favor interplanetary cooperation. If an ancient ruin has a force field surrounding it, then it is inhabited by aliens called Pylons. If a Gael has cambrian ears (sensitive to high-frequency sounds), then that Gael also has tentacles. Belief Revision as Propositional Update 76 Appendix B Phrasing of Problems in the Symbolic Condition for Experiments 4 and 5 Initial Belief Set Expansion Problem 1 Either A, B, C, and D are all true, or none of them are true. A is true. C is true. D is false. 2 Either A, B, C, and D are all true, or none of them are true. B is false. Exactly one of these is true, but no one knows for sure which one: • A is true, and C and D are both false. • A is false, and C and D are both true. 3 A is true. B is true. D is true. is true, but not both of them. A is false. B is false. Either C is true or D 4 A is true. B is true. possibly both of them are. At least one of A and B is false, and 5 A is true. B is true. C is true. Either A is false and Band C are both true, or A is true and B and C are both false. No one knows for sure which it is. 6 A is false. B is false. C is false. Either A is true and B and C are both false, or A is false and B and C are both true. No one knows for sure which it is.