Chapter 9
Mental Models
WHAT IS A MENTAL MODEL?
In chapter 7, I examined the problem of scripts and related representational structures for events. Such representations focused on temporal and causal relationships among objects during events. Most events that I discussed (and indeed most events that have been the basis of work on scripts) are socially or culturally defined events, such as going to the doctor or taking a trip on a train. These representational structures allow people to reason about ongoing events by using knowledge about related events that have already happened.
How do people reason about novel events? Imagine a situation in which a young boy stands at the top of a hill, makes a snowball, and rolls it down the snow-covered side of the hill. A person may never have witnessed an event like this, but one can construct the event and talk about it. One can imagine that the snowball rolls down the hill and gets larger and larger as it rolls, because snow sticks to it. A mental image of this event occurring might be formed. I discussed mental imagery extensively in chapter 6, but this situation goes beyond a mere mental image; it requires reasoning about the physics of the situation to determine how the image changes over time.
Constructing this answer requires a mental model of the situation. That is, one must envision a boy on a hill and then use one’s knowledge of snow, hills, and gravity to make educated guesses about the snowball’s behavior after the boy rolls it down the hill. Mental models of this type are useful for making predictions about the behavior of objects in novel situations that are often like situations encountered before, but not so familiar that a person has an established script for dealing with them.
In addition to reasoning about novel physical systems like snowballs and hills, other instances call for similar kinds of reasoning. For example, text comprehension often involves reasoning about novel situations. As discussed in chapter 6, if one reads a story about a man jogging around a lake, and at some point the man takes off his sweatshirt, places it on the ground, and continues to run, one can recognize that the man and his shirt are at different locations. One need not have an established script for this scenario; a person can construct this situation based on knowledge of the spatial locations of the individuals involved as well as an understanding of space (e.g., that moving away from a stationary object puts distance between the person moving and the object).
Now, consider a third situation that seems unrelated to the other two at first glance. In a complicated abstract reasoning problem, one may encounter the syllogism:
If a person is asked what can be concluded from it, and if the person has taken an elementary course in logic, he or she might conclude that None of the Chemists are Actors. Even if one did not know what follows from these premises directly, one can still generate an answer by reasoning about it. On the surface, the notion of a mental model does not seem helpful in a situation that requires abstract reasoning, but one can try to instantiate this syllogism as a spatial layout of objects and then reason about the objects. For example, one may first imagine a group of actors and imagine that all of them are beekeepers. Next, one may imagine an additional group of beekeepers who are not actors (as the premise that all actors are beekeepers does not require that all beekeepers are actors). After that, one can imagine a group of chemists, none of whom are beekeepers. Having done this, one can examine the mental model, and see that none of the chemists are actors either.
Theories of mental models have been developed for all three kinds of examples described here. At the heart of all of these types of mental models are specific instantiations of situations consisting of objects and relations among them. The relations can be simple (like the simple spatial relations used in mental models of reasoning) or complex (like the mental models of physical systems). I discussed the construction of spatial models of the sort used in the text comprehension example in chapter 6 and do not discuss them further in this chapter. Instead, I examine the use of mental models in logical reasoning tasks and then turn to naive physics and mental models of physical systems.
MENTAL MODELS OF LOGICAL REASONING TASKS
The theory of mental models for reasoning has been developed by Johnson-Laird and his colleagues (Johnson-Laird, 1983; Johnson-Laird & Byrne, 1991; Johnson-Laird, Byrne, & Tabossi, 1989), in response to people’s difficulties in reasoning abstractly (as discussed in chaps. 1 and 8). For example, in chapters 1 and 8, I noted that the Wason selection task (see Figure 1.1) is difficult to solve when presented abstractly, but can be solved when put in a concrete context, particularly one with the structure of a common social rule like permission. People find some abstract logical problems easier to solve than others. For example, Johnson-Laird and Byrne (1991) summarized the results of studies that examined people’s ability to solve classical syllogisms.1 For example, people provided a correct conclusion to the syllogism:
(All the Archers are Car Owners) about 90% of the time. In contrast, the syllogism:
received the correct conclusion (Some of the Anglers are not Crooks) only about 20% of the time. If people can reason by using logical rules, it is unclear why there are such vast differences in the ability to solve these problems.
In response to such issues, Johnson-Laird and his colleagues developed a mental models approach to logical reasoning. The intuition underlying this approach is that a deductive reasoning task can be solved by creating a model of the premises and then finding a statement that describes the model. The model consists of spatial relations among objects denoting elements in the premises of an argument. If a statement describing the model that is constructed can be formulated, this statement is the conclusion of the deductive argument. A key aspect of this approach is that a set of premises may have more than one model that is consistent with it, and it is necessary to construct all possible models consistent with the set of premises. A valid conclusion to a deductive argument is one that is consistent with all the models that can be constructed on the basis of the premises.
As an illustration of how this approach works, consider the syllogism in 9.2. The model for the first premise (All the Archers are Bodybuilders) is:
The way to interpret this model is to treat each line as an individual in the model. The letters correspond to descriptions that apply to the individual. A letter in brackets means that the existence of the individual is optional. In this case, the [B] individuals leave open the possibility that there are Bodybuilders who are not Archers. The model can then be extended to the second premise (All the Bodybuilders are Car Owners) by adding:
where all the Archers who are Bodybuilders are also Car Owners. Also, all the optional Bodybuilders are also Car Owners. Finally, there are some possible Car Owners who are neither Archers nor Bodybuilders. This model leads to the obvious conclusion that All the Archers are Car Owners. For one to make this conclusion, these last Car Owners must be recognized as optional so that a conclusion like “Some Car Owners are not Archers” is not suggested.
An interesting facet of the mental models theory is that it predicts which syllogisms are likely to be easy and which are likely to be difficult. For example, because the model in 9.5 is the only one that can be constructed for the syllogism in 9.2, the syllogism should be easy. As discussed earlier, this prediction is borne out. In contrast, the syllogism in 9.3 is difficult. How can a mental model of this syllogism be constructed? The first premise of 9.3 (Some Bankers are Anglers) can be given a mental model like:
and the second premise (None of the Crooks are Bankers) can be added to form the model
This model is consistent with the conclusions that Some Anglers are not Crooks, Some Crooks are not Anglers, and None of the Crooks are Anglers. A second model can also be constructed, however. In this case, the model
for the first premise acknowledges that if Some Bankers are Anglers, there may be some Anglers who are not Bankers:
When the second premise is added, it is possible that some Anglers are Crooks:
This model rules out the conclusion that None of the Crooks are Anglers, but leaves open the conclusions that Some Anglers are not Crooks and Some Crooks are not Anglers. Finally, a third model can be constructed in which the first premise is again modeled as in 9.8, but the second premise now yields a model in which all the Crooks are Anglers:
This model rules out the possibility that Some Crooks are not Anglers and leaves only the valid conclusion that Some Anglers are not Crooks (namely those that are Bankers). This syllogism requires examination of three models to find a valid conclusion. Because three models are required, mental models theory predicts that this syllogism is more difficult than the one in 9.2, and indeed it is. In extensive tests of this theory, fewer people solve syllogisms that require three models than solve those that require two models, and fewer solve these syllogisms than solve those that require one model.
This mental models account is not limited to syllogisms. Johnson-Laird and Byrne (1991) extended their theory to deductive problems with many quantifiers in them. For example, the statement:
can be represented as
in which each line represents a location. As for syllogisms, it is possible to combine sets of sentences with multiple quantifiers like Example 9.11 into a deductive argument. Some sets of premises require examining more than one model to reach a valid conclusion, and people find it more difficult to reach valid conclusions for sets of premises that require multiple models than to find valid conclusions for sets of premises that require only one model.
To summarize, the mental models approach to reasoning posits that people approach logical reasoning problems by creating models of the premises. Descriptions of the model derived from a set of premises are the conclusions that can be drawn from the premises. In some cases, many different models are consistent with a set of premises, and one model may rule out a conclusion suggested by another. These problems are difficult because people must keep all the models in mind at once to find a description that applies to all of them. Mental models also seem compatible with content effects in deductive reasoning like those discussed in chapters 1 and 8. Because each element in mental models is assumed to be a specific object, it should be easier to create and maintain a model of a familiar situation than to create a model of an unfamiliar situation. The mental models approach has been applied to a wide variety of reasoning tasks and has been viewed as one of the most complete theories of human reasoning (Evans & Over, 1996).
What Is the Mental Models Theory?
The study of reasoning has a long history in psychology. From the time of Aristotle, authorities have been interested in how people reach the conclusions of deductive arguments. Thus, it should be no surprise that there is intense debate over whether the mental models approach to logical reasoning is a good account of the way people reason deductively. At the center of this debate is an argument about exactly what the mental models theory is. The use of diagrams (like those in the previous section) to illustrate mental models suggests that mental models are connected in some way to imagery. It is dangerous to take diagrams too seriously, however, because some process must act on the mental model to carry out the reasoning process. If a mental model was only a diagram that required someone to look at it, it would be a notational device, not the basis of a model of reasoning.
Having been acutely aware of this danger, Johnson-Laird and his colleagues implemented mental models for reasoning and for constructing spatial descriptions (as discussed in chap. 6) as computer programs. In their programs, the elements in a mental model are symbols, and the program has rules that determine how a model is constructed from premises, how a description is generated from a model, and how new models can assess whether there is a counterexample to a possible conclusion.
This description of the mental models view is significant because many attacks on mental models theory have come from people who subscribe to a natural logic theory of reasoning (e.g., Braine, Reiser, & Rumain, 1984; Rips, 1994). Briefly, natural logic theories suggest that people solving deduction problems use logical rules like those of formal logical systems, and people’s reasoning flaws arise because of limited processing capacity, flawed strategies for applying the rules, or the absence of rules needed to solve a particular problem. For example, Rips (1994) suggested that almost everyone knows the inference schema modus ponens (at least implicitly), but that few people have learned (or at least act in accordance with) the inference schema modus tollens.
A key point raised by many of these researchers is that the mental models approach to reasoning is not qualitatively different from a natural logic approach. Instead, both types of models posit some encoding of the premises of the arguments and some set of rules for examining the premises to formulate or verify a deductive argument. The models differ in their proposals about how the premises are encoded and in the types of rules (or processes) that they suggest are important for reasoning (Evans & Over, 1996; Rips, 1986, 1994).
A potential limitation of the mental models approach is that how to extend it to other forms of reasoning beyond deduction is unclear. As already discussed, the power of deductive inference lies in the fact that, because of the structure of the argument, the conclusions of a valid argument are true if the premises are true. Unfortunately, the absolute truth of premises is generally impossible to establish. Thus, pure deductive reasoning is of limited use in most ordinary circumstances. Johnson-Laird (1983) suggested that the general notion of a mental model may be appropriate in domains like language comprehension, in which the meanings of words are established in part by the context in which they appear, but these proposals for extensions of mental models are not made with the same degree of specificity as are the proposals about deductive reasoning. Thus, it may be that the mental models framework is most useful for understanding human performance on deductive reasoning problems (but see Johnson-Laird & Savary, 1996, for an extension of mental models to probabilistic reasoning).
MENTAL MODELS OF PHYSICAL SYSTEMS
Mental models for logical reasoning tasks have focused on collections of objects in particular spatial locations. The description of the model (s) generated serves as the conclusion for the reasoning problem. People also use complex mental models to describe and reason about physical systems (Gentner & Stevens, 1983). Unlike mental models for reasoning, mental models of physical systems are internal representations of external systems. The represented world for a physical mental model is a physical system in the world; the representing world is a scheme for capturing the physics of the represented world. Finally, a number of computational models of physical reasoning also have procedures for extracting the knowledge in the representation and using it to predict the future behavior of elements in the represented world (Forbus, 1984; Kuipers, 1994). For mental models of reasoning, there is generally no external represented world. Instead, the elements in these mental models are empty (i.e., nonrepresenting symbols that are used to reach a conclusion about the relationship between elements in the reasoning problem).
In this section of the chapter, I discuss psychological evidence for the nature of people’s physical knowledge. Although these data have typically not been modeled directly, they do place some constraints on what people find easy and what they find difficult. These constraints have been captured in models of physical reasoning. I then describe some general aspects of mental models of physical systems. Finally, I analyze Forbus’s (1984) qualitative process theory (QP theory) in detail as an example of how to represent qualitative information about physical systems and how to reason with this knowledge.
Naive Physics
Psychological research carried out under the umbrella of naive physics has been directed at people’s understanding of the physical world. Much of this research has been descriptive and has dealt with the extent to which people have a veridical understanding of physical processes and with the depth of their understanding. This work, which has centered on finding flaws in people’s reasoning about physical systems, has shown people to have fundamental misconceptions about the way the world works and to possess only shallow knowledge about physical systems.
Some demonstrations of shortcomings in people’s physical knowledge seem to involve single phenomena for which people’s judgments of what happens in a situation do not match what actually happens. Perhaps the best-known case of an error in naive physics is the belief in curvilinear momentum (Kaiser, Proffitt, & McCloskey, 1986; McCloskey, 1983). The speed and direction of a moving object change only when a force is applied to the object, but in the real world, gravitational forces constantly pull objects toward the earth and frictional forces constantly slow them down. These unseen forces make it appear as if objects must exert a constant force to continue moving and that objects themselves have a memory of the direction in which they are traveling. In a fascinating series of tests, Kaiser, McCloskey, and Proffitt (1986) asked people what would happen in a situation like that shown in Figure 9.1, where a ball rolls into a curved tube and emerges at the end of it. The investigators asked subjects to assume that they were looking down on the tube and then asked them to predict the motion of the ball when it leaves the tube. Despite the fact that the ball has no additional forces acting on it when it leaves the tube and hence shoots out straight (the middle choice in Figure 9.1A), many people predicted that the ball would take a trajectory that (at least partially) continued the path it took inside the tube. Such a path is shown in the left-most trajectory in Figure 9.1A. This finding suggests that people believe objects maintain some memory (momentum) of their previous path. The researchers obtained this finding even when they allowed subjects to roll a ball bearing themselves. Thus, people make errors when they are performing an action, not only when they are solving problems in a pencil-and-paper task.
A similar error in physical reasoning occurs in the water-level problem (Piaget & Inhelder, 1956). In this problem, illustrated in Figure 9.1B, subjects are asked to imagine that the beaker is filled with water and they are asked to draw the surface of the water in the picture. The correct answer is that the water level is perpendicular to the pull of gravity, and the water level should be drawn horizontally (as in the middle item in Figure 9.1B). Nevertheless, 40% or more of adults given this task drew the water level incorrectly, typically as it appears in the left-most beaker in the bottom row of Figure 9.1B (McAfee & Proffitt, 1991). This phenomenon is also apparently not a function of being a pencil-and-paper presentation; it is observed with realistic pictures (McAfee & Proffitt, 1991) and also with animations of jugs pouring water (Howard, 1978).
As a final example, studies have explored people’s ability to detect the relative mass of objects that collide (e.g., Gilden & Proffitt, 1989, 1994). In one set of studies, people were shown collisions between two balls that differed in their mass. One ball was initially stationary, and when the other ball struck the stationary ball, both balls moved at some speed in some direction. People’s performance in this task was well described by simple heuristics. If one ball moved faster than the other after the collision, it was generally judged to be lighter. If one ball ricocheted backward from the collision, it was generally judged to be lighter. The incoming velocity of the colliding ball was not used to make mass judgments. These heuristics often give the correct answer, but people do make errors in their judgments of relative mass. The important finding from these studies is that people do not judge the mass of objects on the basis of accepted principles of Newtonian physics but use rules (heuristics) that may often yield an appropriate answer but may sometimes be wrong.
FIG. 9.1. Examples of errors in naive physics. A: Curvilinear momentum; B: the water-level problem.
These phenomena demonstrate that people are not always perfectly attuned to the physics underlying the action of objects in the world. Yet people are not thereby ill suited to exist in the world. Because of unseen forces like gravity and friction, the world often seems not to act in accord with Newtonian physics. People are reasonably well equipped to predict the actual motions of objects and to handle objects and liquids. Nonetheless, they do not seem to have quantitative physical models in their heads but rely on fairly simple rules that relate salient properties in the environment.
Mental Models of Physical Systems
Many studies described in the previous section focused on simple events. The tasks have a strong perceptual component, and the problem to be solved often involves information that can be extracted from a visual display of a physical system. Studies of complex physical reasoning, however, have explored how people reason with a mental model of a situation. There is no accepted definition of a mental model in this case, but generally a mental model involves understanding the causal relations among several elements in a physical system. The purpose of representing this information is to facilitate reasoning about change in the system being modeled. The knowledge of these relations allows prediction of how the system operates and allows people to reason about unexpected events (as they must do when diagnosing and repairing broken mechanical and electronic devices). Thus, mental models are larger in scope than the rules and heuristics posited to underlie performance in naive physics tasks.
FIG. 9.2. Heat exchanger. The top pipe has hot fluid flowing through it at a rate F1. The bottom pipe has cold fluid flowing through it at a rate F2. The temperature of the fluid entering the heat exchanger through the top pipe is T1, and the temperature of the fluid as it exits the heat exchanger is T2. The temperature of the fluid entering the heat exchanger through the bottom pipe is T3, and the temperature of the fluid as it exits the heat exchanger is T4.
Mental models have two primary components (de Kleer & Brown, 1983). First, there are objects in the domain. For example, Figure 9.2 depicts a heat exchanger (Williams, Hollan, & Stevens, 1983), used to cool a fluid in a machine. Pipe 1 may contain oil used to lubricate the parts of an engine. The oil heats up as it comes into contact with hot engine parts and flows through a pipe into the heat exchanger. There is also a fluid flowing through Pipe 2, and this fluid is colder than the fluid in Pipe 1. The heat exchanger puts the fluids flowing through the pipes in thermal contact so that heat can flow between them. Because heat flows from hotter to cooler areas, heat from the liquid in Pipe 1 flows out of the fluid, into the surrounding heat-conducting material, and into the cold fluid in Pipe 2. Thus, the fluid in Pipe 1 leaves the pipe cooler than it was when it entered (i.e., T2 < T1), and the fluid in Pipe 2 leaves the pipe warmer than it was when it entered (i.e., T4 > T3). The amount of heat that enters the system is dependent on the flow from the hot pipe (F1), and the amount of heat that can be taken out of the system depends on the flow from the cool pipe (F2).2
In a mental model of this heat exchanger, there are definitions for elements like the pipes (which are paths for fluid to flow). There is also general causal knowledge, like the knowledge that heat flows from hotter to cooler places. These relationships are generally assumed to be qualitative. Thus, although the model predicts that heat flows from one point to another when the first point is hotter than the second point, it does not predict precisely how much heat flows. The mental model combines the parts to form a set of relationships that govern the expected behavior of the system. This process is called model formulation. To simulate the behavior of the heat exchanger, the model must specify the relationships among the various quantities that exist in the device. For example, the greater the temperature of the fluid in the hot pipe (T1), the greater the temperature of the fluid exiting the hot pipe (T2) is likely to be. The greater the temperature of the fluid in the hot pipe (T1), the greater the temperature in the exiting cool pipe (T4) as well (provided that the entering fluid in the cool pipe was indeed cooler than the entering fluid in the hot pipe [i.e., T3 < T1]). The more hot fluid that enters the system (i.e., the higher F1), the greater the temperature of the fluid exiting both the hot pipe (T2) and the cool pipe (T4). The complete model must have relations among all the quantities that affect each other, and it must have boundary conditions on the relations (such as that increases in T1 lead to increases in T4 only when T3 < T1). One possible model for the heat exchanger is shown in Figure 9.3.
Once the model has been formulated, the behavior of the system can be simulated. For example, with the set of relations in Figure 9.3, one can ask what happens if the flow in Pipe 2 (F2) is increased. Under normal circumstances, this change should lead to a decrease in temperature in the exiting fluid in the hot pipe (T2) and a decrease in temperature of the exiting fluid in the cold pipe (T4). If these changes are not observed, in view of the conditions on the rules, one can speculate that the temperature of the fluid in the cold pipe (T3) is not less than the temperature of the fluid in the hot pipe (T1). The effects of a variety of changes in temperatures and flow rates can be assessed by using these simple rules.
FIG. 9.3. A simple qualitative model of the heat exchanger in Figure 9.2.
In support of this way of thinking about mental models, Williams, Hollan, and Stevens (1983) asked people to reason about a heat exchanger like that in Figure 9.2. As people reasoned about various changes in quantities, they were asked to think aloud. Consistent with the mental model depicted in Figure 9.3, people tended to reason by using simple qualitative relationships among components, and their errors in reasoning could be traced to the absence of rules governing the relationships among quantities in the model. White and Frederiksen (1990) suggested that such simple models are useful for automated teaching systems that help people learn to reason about simple physical systems. In their view, many people who reason about physical systems think in terms of zero-order models (i.e., whether a particular quantity is present or absent at a particular location) and first-order models (i.e., the direction of influence of a change in a quantity would be in the system). In this parlance, the simple model in Figure 9.3 is a first-order model because it describes the effects of changes in a quantity. White and Frederiksen further suggested that quantitative models are typically not used. That is, people are unable to reason about the specific values of quantities and the specific degrees of change of quantities in physical systems.
Having a mental model of a physical system allows people to reason effectively about a variety of situations without having to represent all possible interactions among components in advance. The ability to “run” a mental model allows people to specify a set of relationships, and to determine the actual behavior of a system only when it is important to know how the system works (Schwartz & Black, 1996). The model limits reasoning in two important ways. First, the model is only as good as the assumed reladonships among components; if there are flaws in these relationships, they show up as errors in reasoning. Second, mental models are assumed to omit many details; if someone has only a first-order model, he or she cannot reason about indirect influences of one quantity on another. Because mental models are qualitative, one can reason only about the direction of change, not about its precise degree. Of course, it is not surprising that reasoning is qualitative: Quantitative reasoning about physical systems requires enormous computational power.
Qualitative reasoning generally permits accurate reasoning with minimal cost, but it can pose a problem when two quantities both directly influence a third quantity in different directions. For example, the pressure of a gas in an enclosed area increases with the temperature of the gas and decreases with its volume (Boyle’s law). In a qualitative system, it is impossible to reason about the pressure of a gas in a situation in which the temperature and the volume increase simultaneously.
D. Gentner and D. R. Gentner (1983) provided some evidence of the power of models to mislead. They suggested that people can conceptualize electricity either as analogous to the flow of water through pipes or as analogous to a crowd of people in a constricted hallway. These models can be helpful in many cases. For example, voltage can be conceptualized as the pressure of water flowing through a pipe or as the speed of people walking through a hallway. A resistor in an electrical circuit can be thought of as a constriction in a pipe or a gate in a hallway, either of which limits the flow (of water or people) through the path. Gentner and Gentner pointed out, however, that batteries are easier to conceptualize in the water model than in the crowd model. In the water model, a battery is simply a pump. In contrast, there is no particularly good way of thinking about a battery in the crowd model, except perhaps as a waiting room full of people who want to leave.
The effect of this difference is that people find it easier to reason about configurations of batteries if they use the water model than if they use the crowd model. In the water model, two batteries connected in serial (as in Figure 9.4) can be conceptualized as two pumps, each of which increases the pressure of the water. In this way, there is twice the pressure in the pipe, just as connecting two batteries of the same voltage in serial yields twice the voltage of a single battery. Connecting two batteries in parallel is like having two pumps in parallel. Both push water to the same pressure, and the total pressure in the pipe is the pressure generated by each pipe alone (there is just more available water). Likewise, two batteries of the same voltage connected in parallel yield the same voltage as one battery. In contrast, picturing two waiting rooms connected either in serial or in parallel does not provide much insight into the action of batteries. Consistent with this analysis, people asked to reason about the voltage in circuits with serial and parallel batteries were more accurate when they held the water model than when they held the crowd model.
FIG. 9.4. Simple circuits consisting of batteries and resistors.
Errors in people’s mental models have been demonstrated to lead directly to errors in real-world decisions. Kempton (1986) interviewed people about how their home heating systems worked. The proper model of a thermostat for a heater (which is held by many people) is that it is set to the desired temperature, and the heat stays on until the room reaches the desired temperature, at which time the heat shuts off. However, a significant number of people have a model of the thermostat that assumes it is like the accelerator of a car. That is, the higher it is set, the more heat that comes out. People with this model believe that setting the thermostat to a high temperature causes the heat in the house to rise faster than setting it to a lower temperature (that is still greater than the temperature in the room). By studying home heating records, Kempton was able to isolate families that appeared to act on the basis of both these models. In some homes, the thermostat was set twice a day (in the morning and at night). In other homes, the thermostat was constantly adjusted up and down during the day. This latter strategy wastes energy and is expensive, but it is consistent with the mistaken accelerator model of the thermostat.
The potentially misleading effects of people’s models on their reasoning are as true for scientists as for novices reasoning about electricity. Hutchins (1983, 1995) described the anthropological study of navigation among Micronesian navigators. These navigators are able to sail outrigger canoes hundreds of miles between islands without any instruments used by Western navigators (e.g., logbooks and compasses). They accomplish this feat by a number of means, among them by using stars as guidance for directions and by using landmarks (real and fictitious islands) as a measure of how far they are from their destination. The landmarks are envisioned as moving along the boat, which is conceptualized as being stationary. The landmark islands are seen as being ahead of the boat at the start of the voyage and moving slowly down its length and behind it as the journey is completed.
Early investigators studying Micronesian navigation practices had difficulty understanding how the islands figured into the navigational system, because they assumed that the navigators used the islands to create a bird’s-eye view of the journey, in which the boat is conceptualized as moving through space. This assumption is deeply ingrained in Western navigation, in which the course of a vessel is plotted on charts that show an overhead perspective. In this system, landmarks and the boat itself are treated as objects viewed from above, and the function of landmarks is to locate the boat in the plane of the map. The location of the moving boat in relation to a fixed frame of reference is determined by finding the angle between the boat and at least two (usually three) landmarks whose location relative to the frame of reference is known.
It is difficult to see the function of the landmark islands in the Micronesian system from this perspective because the landmark islands seem not to be used to fix the boat’s location in space. Instead, the movement of the islands relative to the boat functions as a measure of how much of the journey has been completed. Only when the Micronesian navigational system is conceptualized independently of the Western system do the practices make sense (see Hutchins, 1995, for an elegant description of this navigational system). Hutchins used this example as a warning of the problems that even trained scientists can encounter when they are forced to reason about situations that diverge from their own mental models.
A second issue of importance involves the completeness of a mental model. In the model of the heat exchanger, there was a series of relations among quantities but very little other supporting knowledge. The model had no other information about thermodynamics and so it could not be used to reason about properties of the materials in the heat exchanger. With this model, there is no way of assessing the impact of a change in the type of fluid used in the cooling pipe of the exchanger or of a change in the materials used to build the pipes. Such information could be important to someone looking to improve the performance of the machine, but extensive knowledge including quantitative information is required to answer these questions. Clearly, people in the business of building heat exchangers would know this information and might even incorporate it into their mental models. It is an open question whether people think that their models are incomplete without extensive knowledge or whether they are happy with even a minimal set of relations among the objects in a model.
A quick check of one’s own intuitions suggests that an extensive mental model is not always necessary. I have a vague idea of how the engine in my car works: Pistons compress gas and fuel in a chamber; a spark plug creates a controlled explosion that causes gas to expand and drives the piston outward. The pistons are arranged so that the upward motion of some pistons leads to the compression motion of others. There is quite a bit missing in my model. I am not sure I understand the impact of pressing the gas pedal. I am sure I do not understand how the valves in the engine allow fuel to enter the chamber or gas to escape. Yet, despite this marked absence of knowledge, I am happy to drive my car, to fill it with gasoline, and to have the oil and other fluids checked regularly.
FIG. 9.5. Examples of some children’s conceptions of the Earth.
Such general intuitions have been examined in an extensive investigation of children’s understanding of the solar system undertaken by Vosniadou and Brewer (1992; Samarapungavan, Vosniadou, & Brewer, 1996). In these studies, researchers asked children to discuss the Earth and the solar system. The results showed that children often have some information about the Earth and the solar system (the Earth is round, it spins, and it goes around the sun), but they have not received extensive training in the causal mechanisms involved. As a result, there are significant gaps in children’s mental models of the solar system, and they try to fill these when pressed by using knowledge from their own experience. For example, most children say that the Earth is round, but when asked for more information, some children suggest that it is round like a pancake, with the people living on the flat side (see Figure 9.5). This answer conforms to the observation that the ground is flat. Others know that the Earth is round like a ball, but they assume that people live inside the ball and that the stars are on the top side of the ball. Still others know that the Earth is a ball and that people live outside it (although it takes time for some to realize that people live all over the surface, not just on the “top”). Eventually, children are exposed to more detailed models of the Earth until they reach the currently accepted scientific model. Children also display misconceptions about what causes the sun to move across the sky and what causes the seasons to change; these also seem to arise from a lack of causal knowledge. The studies demonstrated that children do not have a complete mental model of the solar system. Instead, aspects of the world always lie beyond what they know. Nonetheless, children (like adults) are content to have partial knowledge and may not even notice the limits of what they know until pushed to answer questions that extend beyond these limits. Thus, it appears that mental models are not assessed for their completeness. Instead, people are willing to reason with whatever knowledge they have.
To summarize, mental models of physical systems consist of representations of objects. The objects can be given functional descriptions that outline how they work in a broad mechanism. The model of a whole mechanism is formulated by combining the objects and specifying the relationships that hold among the objects. These relationships are typically qualitative. They specify broad correlations among quantities (e.g., whether the quantities have a positive or negative relationship). Quantitative relationships do not appear to be part of people’s everyday mental models. The choice of a mental model influences what kinds of problems are easy or hard to reason about. Finally, mental models are not complete specifications of a system. Rather, they contain enough information to carry out whatever tasks a person typically faces. It is easy to construct situations that go beyond the information people have about a situation and to demonstrate the brittleness of their knowledge about a domain.
Qualitative Process Theory
In the preceding section, I described the general characteristics of a mental model and focused primarily on people’s ability to generate and reason about physical situations. In artificial intelligence (AI), there have been many attempts to build systems to represent and reason about physical systems (Bobrow, 1984; Kuipers, 1994). Rather than summarizing the diversity of approaches to qualitative physical reasoning in this section, I go into one such approach in some detail: qualitative process theory (or QP theory: Falkenhainer & Forbus, 1991; Forbus, 1984). This system contains all the basic elements of a system for generating computational mental models and so provides a good example of the issues already raised in this chapter. The system has been incorporated into detailed models of complex physical systems and so is known to work. QP theory was motivated by the desire to understand how people reason about physical systems and hence is meant as a cognitive model as well as a computational tool. In the following subsections, I first discuss the general notion of a process and a process history. Then, I describe the way that the theory represents objects and quantities. Finally, I describe how models are created and how QP theory represents physical processes.
An Overview of QP Theory. A central goal when reasoning about a physical situation (or for that matter, a social situation) is to understand change. A physical system that does not change is not very interesting. For example, if I put a plate on a table and walk away, I do not need to spend much cognitive effort to reason about what happens to the plate. I assume that if I come back some time later, the plate will still be on the table, and only if the plate is not there (and something has changed) is there any need to reason about the situation. Although the idea that change is important may seem trivial, it is the focus of the frame problem, one of the most vexing problems in AI and cognitive science (Ford & Hayes, 1991; Ford & Pylyshyn, 1996; Hayes, 1985). Part of the problem’s difficulty lies in its apparent simplicity. The frame problem can be illustrated with a simple scenario.
Imagine being in a room, sitting on a chair, with a table in front of you. On the table are 500 different objects, including a revolver, a flashlight, an oil lamp that is lit, a pitcher of water, and a ball of string. You move the ball of string to the edge of the table, and it drops to the floor.
(9.13)
What has changed in the room? Intuitively, this question seems straightforward. Most people would say that the ball of string has fallen to the floor and is no longer on the table. Nothing else seems to have happened.
Does one manage to restrict attention to the ball of string because it was mentioned in the passage describing the situation? Imagine that situation 9.13 is changed just slightly:
Imagine being in a room, sitting on a chair, with a table in front of you. On the table are 500 different objects, including a revolver, a flashlight, an oil lamp that is lit, a pitcher of water, and a ball of string. You accidentally tip the oil lamp, and it falls over.
(9.14)
In 9.14, the situation seems more dire. There is the possibility of a fire, and some objects on the table may burn. The ball of string is quite likely to burn; the revolver may get hot, but is unlikely to burn. Something else about one’s knowledge allows a determination of what changes to consider when reasoning about a physical system. The goal of a system that reasons about physical systems is to determine what changes and what does not.
Despite its apparent simplicity, the frame problem is difficult to solve, because it is analogous to the holism problem already discussed. To determine what changes in a situation, one may need to check each fact to see whether it has an influence on the current situation. Each object in the table in these examples is a potential source of change (not to mention every other object in the fictitious world I have created). For a system to solve the frame problem, there must be a mechanism that limits the elements that are considered when reasoning about change. Of course, checking only a subset of information may lead to errors, but the job of a reasoning system is to find procedures that are reasonably accurate and still to reach conclusions in an acceptable time.
Qualitative reasoning systems like QP theory solve the frame problem by reasoning locally. This local reasoning is done by representing objects (and agents) in the world as histories (Hayes, 1985).3 A history is a spatially bounded object that exists in time (and makes the history four-dimensional: three spatial dimensions and one temporal dimension). In the example of the plate on the table, the history for the plate on the table is bounded spatially by the three-dimensional space filled by the plate and temporally by the time between when it is left on the table and when it is picked up again. This representation can solve the frame problem, because only other objects whose histories intersect with the history of the plate need to be reasoned about. The table that supports the plate intersects the spatial boundary beneath the plate and intersects the time at which the plate rests on the table, and the histories of the table and the plate are intertwined. When the state of the table changes, one must reason about the plate because it has an intersecting history. One need not reason about other objects whose histories do not intersect with those of the table and plate. In Scenario 9.13, when the ball of string falls to the floor, it does not intersect with the histories of any of the objects on the table, and there is no need to consider those objects. In contrast, when the oil lamp falls over, because the oil and flame may spread to other objects on the table and thereby intersect with the histories of all of those objects, they must be reasoned about. Of course, there must be a mechanism that reasons about the behavior of objects that tip over, but that is the goal of physical reasoning systems like QP theory. That is, there must be a detailed specification of how a history is constructed and how the objects that make up a process interact.
Objects and Quantities. QP theory permits representations of physical systems and reasoning about them by using qualitative representational machinery. There are no statements of exact quantities. Instead, quantitative relations are stated qualitatively. QP theory uses structured representations in which attributes and relations describe objects, quantities, causal relationships among objects, and (qualitative) numerical relationships between quantities.
FIG. 9.6. A: Container being filled with water. B: Situation in which water flow is about to occur.
A key aspect of a physical system to be defined is the set of quantities in it; understanding the behavior of a system often involves reasoning about quantities. One may want to reason about the amount of water in Container 1 in Figure 9.6A. QP theory has methods for representing quantities in qualitative terms. First, something is defined as a quantity type. There are many different types of quantities in physical systems; an amount is a type of quantity. This type of quantity is interesting, because it can take on only values greater than or equal to zero (i.e., never less than zero). Electrical charge is another type of quantity. Unlike amount, electrical charge can be positive or negative. Elements in a physical system whose quantities are going to be reasoned about must be defined as being describable by the appropriate quantity type. For example, it must be stated explicitly that water has an amount. Because it seems obvious that water has an amount, it may seem like overkill to define this property explicitly. Without defining quantities and objects at this level of detail, however, a reasoner lacks sufficient information to analyze this system later.
Quantities have four parts. The first two are sign and magnitude. Different quantity types permit different ranges of signs. The sign of a quantity of liquid is always positive, because there is never less than nothing of a physical substance, but the sign of the electrical charge of a substance can be positive or negative. Magnitudes of quantities specify the degree of the quantity. The magnitude of the quantity of liquid is the amount of liquid in the container. The magnitude of an electrical charge is the degree of charge. The second two components of a quantity are the sign and magnitude of the derivative of the quantity. This part of the representation specifies the direction and size of the change in the quantity over time. For example, a positive derivative of an amount of water means that the amount is increasing over time, and a negative derivative means that the amount is decreasing over time. The magnitude of the derivative specifies the rate at which the level of the water is changing.
Quantities are typically not described by using exact values. Rather, there is a quantity space, which defines landmark values of a quantity for a given situation. The landmark values are important for reasoning about a particular physical system. In Figure 9.6A, Empty and Full form landmark values of the level of water in the container. The quantity space for a level of a liquid is bounded by zero at one end and positive infinity (∞) at the other. The zero point is important in quantity spaces, because it is the point at which the sign of a quantity changes from negative to positive. The landmark values are important, because they are typically the points at which processes acting on the system become active or are disabled. A model for the container in Figure 9.6A being filled with water may describe the level of water rising as water is added. This process remains active until the water level reaches the Full level, at which time a new process becomes active. In this process, the water level remains the same, but the added amount of water spills over the side of the container.
In a quantity space, it is possible to know of the existence of a system’s particular landmark quantities without knowing their relative ordering. For example, a system can know that a liquid freezes at some point and that a liquid boils at some point without knowing which temperature is higher. Although it seems implausible that someone would not know the relative temperatures associated with freezing or boiling, it is possible in complex situations to be unsure of the relative ordering of landmark values.
Time is a key quantity that is treated qualitatively. Although assumed to be continuous, time is represented in discrete steps. It is typically stated in terms of intervals and instants, in which the intervals of interest are the ones bounded by important changes in quantities in the world. Figure 9.6A depicts a container being filled with water at a constant rate. An important time interval for this system is the time interval before the filling of the container. A second important interval begins at the instant when the filling starts and ends at the instant when the water reaches the maximum (full) level of the container. A third important interval begins at the instant when water has reached the maximum level of the container. The intervals are defined on the basis of the value of some other quantity. The exact values of the quantity (the amount of water in the container) are not important, but landmark values like empty and full are (see also Kuipers, 1994).
Relationships Among Quantities. In QP theory, the relationship between a pair of quantities can be established in a qualitative way. The sign of the relationship between variables is more important than the magnitude of the relation. In addition, the type of influence of one quantity on another is represented explicitly. QP theory distinguishes between direct and indirect influences. Direct influences represent situations in which one element in the world is directly applied to another to cause a change in the element. Figure 9.6B shows two containers connected by a pipe. The flow rate from the water source (Container 1) causes the amount of water in the container to decrease. Likewise, this flow rate causes the amount of water in the destination container (Container 2) to increase. These relationships can be written as:
Direct influences can be thought of as specifying specific additive effects in an equation that defines the change in a quantity. For example, the influences in 9.15 specify:
where Δ amount-of (?x) specifies the change in the amount of a quantity and A (?x) specifies the magnitude of the amount of a quantity. These equations have left room for other factors that may also directly influence the amount of water in a container.
In addition to direct influences, there are also indirect influences. An indirect influence occurs in a situation in which a change in one quantity is associated with a change in a second, but the first element cannot be applied directly to the second. Indirect influences are represented by using qualitative proportionalities (denoted αQ, in which positive relationships are denoted αQ+, and negative relationships are denoted αQ-). This relation suggests that as one quantity changes, a second quantity changes as well; it does not imply that the first quantity was applied directly to the second. For example, one may be interested in the relationship between the flow rate into the destination container in Figure 9.6B and the level of the water in the container. This relationship is stated:
that is, a positive flow rate is associated with an increase in the water level and a negative flow rate with a decrease in the water level. This relationship is indirect; the flow rate changes the amount of water in the container, and the amount of water changes the level. The specific value of the level of the container is determined by factors like the circumference of the container (if it is round). Thus, one can only say that some function relates flow rate to water level such that:
where Δ level (?x) is the change in the level of the fluid in some container and f(?x) is some monotonically increasing function that includes the flow rate in it. The qualitative proportionality is qualified by the idea that everything else in the system has been held constant. If many quantities in a system are changing at once, the relationship specified by a qualitative proportionality may not hold; other arguments to the function in Equation 9.18 may have a different influence on the output of the function.
Model Fragments. Systems are represented by using model fragments (Falkenhainer & Forbus, 1991).4 A model fragment consists of four components: individuals, operating conditions, assumptions, and relations.
The individuals component of a model fragment represents the objects and the relations between them describing the physical system of interest. The process of water flow depicted in Figure 9.6B shows a pair of containers connected by a pipe. There is water in each of the containers. The objects involved in the water flow are the containers (source and destination), the pipe between them (p1), the water in Container 1 (water source), and the water in Container 2 (water destination). When defining the objects, the relevant quantities must be defined as well. The relevant quantities are the amount of water and the level of the water in Containers 1 and 2. In addition, the pipe must be defined as a path that permits the flow of a fluid.
The operating conditions are a set of conditions that must hold for this model fragment to be useful for reasoning about the current situation. For example, in a model fragment describing water flow, it is important that the pipe (p1) be open so that water can flow through it. If the model fragment describes a situation of water flow, the level of water in the source container must be higher than the level of water in the destination container. If these conditions are not met, the water flow model does not hold. As another example, to construct a model of the heat exchanger pictured in Figure 9.2, one must assume as one of the operating conditions that the fluid in the cold pipe is in fact colder than the fluid in the hot pipe.
The assumptions part of a model fragment describes the assumptions made by the model. A few different kinds of assumptions are important for model fragments. The first provides an ontology for thinking about the process of interest. When reasoning about fluid flow, it is important to know whether the system is trying to reason about fluids or about the behavior of individual molecules, because the explanatory constructs needed to reason about fluid flow are different from those needed to reason about molecular interactions. A second important assumption involves the grain size of reasoning. In Figure 9.6B, one is explicitly interested in fluid flow between two containers. To reason about a large hydraulic system, however, one may use a system of containers connected by a pipe as a subcomponent of the system and assume that its behavior is known. In general, when reasoning about a system, it is important to know the grain size of the elements that are basic components of a system. A final assumption worth making explicit is the approximation for reasoning in a particular case. For example, a physics student may reason about fluid flow by assuming a frictionless environment. This assumption must be made explicitly because it may lead to reasoning errors.5
Finally, the relations in the model fragment specify the mathematical relations among quantities on the basis of the set of individuals and the assumptions, if the operating conditions in the model are true. In a qualitative simulation, the relations consist of the direct and indirect influences. In a numerical simulation, these relations are precise equations specifying the correlations between variables.
Dynamic Aspects of a Model. After a model is described, its behavior can be simulated in a series of episodes and events. Episodes are situations with a temporal extent; events are instantaneous situations. Typically, episodes describe time intervals in which a process takes place; events describe instants in time that switch from one process to another. For example, earlier I described important time intervals associated with filling the container in Figure 9.6A with water. These episodes (and bounding events) are the history describing the behavior of this system. This history begins with the episode in which the container is empty; the episode continues until the event of beginning to fill the container with water. At this time, the episode of filling the container begins. There is no time between the end of each episode and the beginning of the next event or between the end of an event and the beginning of a subsequent episode. Once the container has begun to be filled, the episode of filling the container begins immediately. The episode of filling the container continues until the event of reaching the full container. At that point, a new episode, consisting of stopping the flow of water or of continuing the flow of water and having water spill over the edge of the container and onto the floor, begins.
Model fragments are used as the basis of simulations of the behavior of a physical system. In an AI model, it is possible to create a complete simulation of the behavior of the system; this simulation determines all possible states of the system and all possible events that lead from one episode to another. Creating a complete description of the behavior of a system is called envisioning. This process can be useful for reasoning, but because it requires too much computation, it is not a good candidate for a cognitive process. Instead, cognitive models of qualitative reasoning focus on simulating the behavior of a system (de Kleer & Brown, 1983). As discussed previously, simulation involves determining the behavior of a system on the basis of its description. In QP theory, the behavior of a system can be described as a history guided by the operating conditions and quantity conditions in the model fragment. In general, the critical events in a process history occur when changes in the world influence the elements in an operating condition and allow a process to become active or when quantities that appear in the operating conditions of a process go from being equal to unequal (or from unequal to equal). The search for changes in equality relations among quantities in a model is called limit analysis.
An example of a history of a process is the situation in Figure 9.6B, in which the source and destination containers have different water levels and a pipe connecting the containers has a valve on it. Initially, the valve is closed, and so there is an episode in which the valve is closed and no water flows. When the valve is opened (an event) and there is more water in the source container than in the destination container, a new episode begins in which water flows from the source to the destination. Because of this event, the operating conditions for the model fragment associated with water flow are now relevant; there is an open path through which the fluid can flow. This flow causes the amount of water in the source to decrease and the amount of water in the destination to increase. The change in the amount of water produces a change in the level of water in each container as well. At some point, the water level in each container is equal (an event), and the flow stops. The situation now reaches an equilibrium in which there is no additional water flow. This equilibrium reflects a case in which an operating condition that was true (i.e., that the water levels were unequal) became untrue. In general, processes become active or cease to be active when operating conditions that were untrue become true or when operating conditions that were true cease to be true. The important transitions in this process occurred when the pipe became an open path and when the water levels in the source and destination containers became equal.6
In the interest of clarity, this discussion has glossed over some details. For example, the model of a process must have relations describing that the water is actually contained in the containers. Creating and reasoning with these relations requires domain-specific knowledge about the action of water in containers. The importance of domain-specific knowledge should not be surprising: People have little understanding of the physics of the world in novel domains. Only after experience in a new domain does one have a sense of how objects act. Qualitative physical models similarly require domain-specific information to operate.
QP theory represents physical situations using model fragments. The model fragments describe the objects in the system and relevant relations among them. The model fragments also specify the assumptions made by the model. These assumptions are important to track, because errors in reasoning may occur when a faulty set of assumptions is adopted. Another key component of the model fragment is the set of operating conditions, which specify the circumstances under which the model can be used. Among these operating conditions are preconditions specifying when a particular process is active. Finally, a model fragment contains a record of the direct and indirect influences on quantities that allow the behavior of a system to be simulated. These simulations track changes in quantities, changes that may eventually lead other processes to become active (by satisfying their operating conditions) or inactive (by disabling their operating conditions). QP theory permits reasoning about complex physical systems by using only qualitative information about time and quantities. In this way, the theory can be used both as an effective computer model of the behavior of physical systems as well as a model of the way people reason about physical systems.
CONTRASTING TYPES OF MENTAL MODELS
In this chapter, I have reviewed two different types of systems known as mental models. Although these systems differ in significant ways, both are aimed at the problem of reasoning about complex problems by using specific elements. Both mental models have mechanisms for representing and reasoning about novel situations (in contrast to the scripts discussed in chap. 7, which focused on representations of familiar situations). In the case of mental models of reasoning, the specific elements are tokens created for the purpose of solving a problem. In the case of mental models of physical systems, the elements are representations of objects, quantities, and processes.
Keane, Byrne, and Gentner (1997) characterized the difference between these types of models as reflecting a different emphasis on working memory and long-term memory. They suggested that mental models of reasoning are designed to explain how people solve difficult, novel problems. Thus, the primary limitation on processing is the working memory available as the problem is solved, because multiple models may need to be held in mind at the same time. In contrast, mental models of physical systems focus on explaining the behavior of physical systems. These mental models are knowledge intensive: Reasoners must have extensive knowledge about the domain to form a model of a novel system in the domain. This need for background knowledge is a reflection of people’s inability to reason about the dynamics of systems in unfamiliar domains. Thus, these models emphasize the way that people store and use models of physical systems that they already know something about. Keane et al. suggested that, although research on these two models has been independent, a rapprochement of the lines of research may lead to insights about how working memory and long-term memory come together in the solution of complex problems.
1As a brief review, a classical syllogism has two premises. The first premise relates two terms, A and B, and the second premise relates B to a C term. The conclusion must find a valid relation between A and C. The four relations in syllogisms are All X are Y, No X are Y, Some X are Y, and Some Xare not Y. There are 64 ways that these four relations and the arguments (in either order) can be assigned to the two premises. Typically, a conclusion is also given, and the task is to determine whether the conclusion follows from the premises. The conclusions can be constructed from the four relations with either assignment of the arguments A and C, and there are 512 possible syllogisms containing two premises and a conclusion. If the task is set up so that there are just the 64 combinations of two premises, 27 have valid conclusions. For the others, there are multiple possible conclusions. For example, for the syllogism:
there is no valid conclusion. One can apparently conclude that Some Artists are Cooks, but it is also possible that some Barbers are not Artists and that only Barbers that are not Artists are Cooks. Thus, it is not possible to draw a conclusion from this form.
2The diagram in Figure 9.2 is schematic. Actual heat exchangers try to maximize the surface area in the region of thermal contact to maximize the amount of heat flowing from one fluid to the other.
3This is by no means the only proposed solution to the frame problem (see Ford & Hayes, 1991; Ford & Pylyshyn, 1996).
4The notion of a model fragment generalizes the concepts of individual view and process, which were originally part of QP theory (Forbus, 1984).
5See Falkenhainer and Forbus (1991) for an extended discussion of assumptions.
6The process of generating the behavioral states of a system on the basis of the process description (i.e., envisioning) in QP theory has been implemented in the program GIZMO and its successor QPE (Forbus, 1990). A discussion of the mechanisms involved in this work is beyond the scope of an introductory chapter like this, but the topic is well worth exploring for those who want more details on the processes involved in using QP theory to reason about a physical system.