omichael cognitive science laboratory · explanation. principle 3, analogy, attributet coherence to...

Explanatory Coherenceand

Belief Revision In Naive Physics

OMichael Ranney and Paul ThagardCognitive Science LaboratoryPrinceton, New Jersey 08542

4C4 Technical Report No. UPITT/LRDC/ONR/APS.17

%July, 1988

Reproduction In whole or In part Is permitted for any purpose of the United Slates Government, Approvedfor public release; distribution unlimited.

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C HPreparation of this manuscript was supported by ONR grant N00014-84-K-0223 The opinions expresseddo not necessarily reflect the position of the sponsoring agency, and no endorsement should be Inferred.I thank Lauren Resnick for stimulating discussions.

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11. rITLE (Inctuae Socuty Ctaancavonj

EXPLANATORY COHERENCE AND RELIEF REVISION IN NAIVE PHYSICS (Unclassified)

IZ. PERSONAL AUTHOR(S)Michael Ranney and Paul Thagard

I3a. TYPE OF REPORT 113tL TIME COVERED 14. DATE OF REPORT (Year. MonMf. 0ay) IS. PAGE COUNTTechnical IFROM TO 1988, July16. SUPPLEMENTARY NOTATION

'1 COSATI COOES ill SUBJECT TERMS (Continueon rewn. of nocanary J aeniooy by block number)FIELD GRU SUB-GROUP - Explanatory coherence,' Belief revisioni Naive physics,'

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19. ABSTRACT (Caomonu an rewono of nocuamy Idid 4onoly by bodck number)

Students of reasoning have long tried to understand how people revise systems of beliefs. Wemaintain that people oft en change their beliefs in ways driven by considerations of explanatorycoherence. -Zithis report,...w~escribe!rj computational model of how experimental subjectsrevise their naive beliefs about physical motion. First, we-p"en instances~in which subjectschanged their beliefs while learning elementary physics. Each of these cases involved anindividual's attempt to explain a surprising observation. Next, we show how their beliefrevisions can be modeled using ECHO, a connectionist computer program that uses constraint-satisfaction techniques to implement a theory of explanatory coherence. The resultingsimulations even captured temporal characteristics of the observed changes in beliefs. Finally,we-diseaies the model's representational sensitivity and procedural robustness.and cowiqde-.br--show' h~o ECHO can be used to generate empirical predictions about subjects' current beliefs.-

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EXPLANATORY COHERENCE ANDBELIEF REVISION IN NAIVE PHYSICS

Michael Ranney and Paul Thagard

Cognitive Science LaboratoryPrinceton University

Students of reasoning have long tried to understand how people revise systems of beliefs(see Wertheimer, 1945, for example). We will describe a computational model of how experi-mental subjects revise their naive beliefs about physical motion. We maintain that people oftenchange their beliefs in ways driven by considerations of explanatory coherence. After describ-ing instances in which subjects change their beliefs while learning elementary physics, we showhow their belief revisions can be modeled using ECHO, a connectionist computer program thatuses constraint-satisfaction techniques to implement a theory of explanatory coherence.

THE PHENOMENA: CHANGES IN SYSTEMS OF BELIEFS

Ranney (1987a) investigated belief change in naive subjects learning elementary physicsby using feedback provided on a computer display. Subjects were asked to predict the motion ofseveral projectiles and then explain these predictions. The physical contexts were quite simple,involving objects that were either thrown or released in various ways. Analyses of verbal proto-col data indicate that subjects sometimes underwent dramatic belief revisions while offeringpredictions or receivingempirical feedback. We will describe two kinds of revisions.

Pat's ChangesConsider "Pat," an individual who was asked to offer predictions about events including (a)

the motion of a heavy object dropped by a briskly walking man and (b) the motion of a heavyobject thrown obliquely upward. Using episodic memories and mental imagery, Pat initiallypredicted that the object dropped by the man would fall straight down (relative to the ground).This belief is a common finding in the naive physics literature (McCloskey, Washburn, & Felch,1983). Although she entertained the correct prediction, that the dropped object might curve for-ward due to the object's forward "force" (velocity), she preferred to stay with the straight-downbelief.

Several tasks later, when faced with the "upward-throw" situation, Pat noted a similaritybetween it and the "walking-drop" task - one that eventually spawned a belief revision. Whileshe offered the correct parabolic trajectory as a prediction for the upward-throw, she noted that,at the parabola's zenith, the upwardly thrown object is comparable to that just released by thewalking man. That is, at the apex of the thrown object's trajectory, it has an exactly-horizontalmotion, as does the just-dropped object. Pat then mentioned that this observation was not "con-sistent" with what she said before and, if she were to be consistent, the thrown object would"stop" its horizontal motion and "then just fall straight down" from the zenith of the parabola.This "curving-up-then-straight-down" trajectory was not consistent with her past experience offalling objects.

Ranney & Thagard Page 2

Pat then realized that her memory-driven description of the ball dropping straight downfrom the walking man involved beliefs that were incoherent with her beliefs about the parabolicmotion of thrown bodies. After a period of ignoring the incoherence, Pat stated that she had"constructed a consistent theory of how these things move." Remarkably, she went on to rejecther straight-down prediction for tho walking-drop task and accept the belief that the path wouldhave a "slight forward" arc combining the "forward force" and gravity. Eventually, Pat general-ized this notion, discriminating among the breadths of the arcs of several laterally released pro-jectiles.

Hal's ChangesA second kind of systematic belief revision occurred in subjects who offered predictions,

received feedback, and provided explanations for a set of tasks in which pendulum-bobs werereleased from their supportive strings during various points in a swing. This set of tasks wasadapted from stimuli used by Caramazza, McCloskey, & Green, (1981). Because of the similar-ity among several of the subjects, we will amalgamate them into a composite subject "Hal."

A

OTI

INS P- T FO

Figure 1. Hal's prediction (*) and four feedback paths.

Hal predicted that, at the extreme "endpoint" of a swing, a released bob will travel laterally Vand (eventually) downward. To some extent, this prediction was driven by images of children 11flying off playground swings. Via feedback, Hal learned that a bob released in this manner actu-

ally falls straight down (see Figure 1, position E). Most of the subjects observed by Ranney(1987a) were surprised by this piece of feedback, as almost 90% of the predicted trajectorieswere nonvertical. Virtually all these subjects revised some beliefs, offering explanations similarto the following prototype: t'd S

D ..ti~ Spociei

K


Unlike the bobs with the other release-positions, this bob went directly straightdown, not to the side at all. Since it had no lateral motion as it fell, this meansthat the object had no speed when it was released. Therefore, the pendulum musthave been temporarily stopped.when the bob dropped. This makes sense, sincethe pendulum was probably slowing down - and it had to stop in order to changedirectionst

In contrast to Pat's belief change, in which two incoherent predictions caused her to rejectone of the them, Hal's belief system underwent a more dramatic revision. He came to acceptboth the straight-down feedback and the notion of an instantaneous zero velocity, while rejectingboth his earlier (lateral) prediction and an impetus-driven belief regarding pendular motion.(See Halloun & Hestenes, 1985, and Ranney, 1987b, for descriptions of different sorts ofimpetus beliefs.)

EXPLANATORY COHERENCE AS A MECHANISMFOR SYSTEMATIC BELIEF REVISION

How can we account for these systematic changes in beliefs? Both cases involved asubject's attempt to adjust beliefs in order to explain a surprising observation. An adequatemodel of these phenomena must provide a mechanism by which a coherent, revised, set ofbeliefs can arise from the need for explanation.

ECHOThagard (1988a) has proposed a theory of explanatory coherence that builds on previous

ideas about the evaluation of explanatory hypotheses (Harman, 1986; Thagard, 1988b). Thetheory has been implemented in a connectionist computer program, ECHO, that uses parallelconstraint satisfaction to accept and reject hypotheses on the basis of their explanatory coher-ence. ECHO has been used to analyze a variety of scientific arguments, past and present:Lavoisier's case for his oxygen theory against the phlogiston theory, Darwin's argument for evo-lution by natural selection, controversies about continental drift (Thagard & Nowak, 1988), anddebates about why the dinosaurs became extinct. Application of ECHO to the belief revisions inPat and Hal is novel in two respects. First, we are modeling subject protocols produced duringexperiments rather than finished arguments. Second, these models are dynamic, in that ECHOchanges its coherence judgments in response to new evidence.

Space constraints permit only a sketch of the theory of explanatory coherence and itsimplementation (see Thagard, 1988a, for greater detail). The theory is stated using seven princi-ples of explanatory coherence that can be summarized as follows. Principle I, Symmetry, statesthat coherence and incoherence are symmetric relations. Principle 2, Explanation, says thathypotheses that together explain a piece of evidence cohere with the evidence and with eachother, and that the degree of coherence decreases with the number of hypotheses used in theexplanation. Principle 3, Analogy, attributet coherence to similar hypotheses that explain simi-lar pieces of evidence. Principle 4, Data Priority, states that pieces of evidence have a degree ofcoherence in themselves, even though evidence can be rejected for theoretical reasons. Accord-ing to principle 5, Contradiction, contradictory propositions are incoherent. Principles 6 and 7claim that the explanatory coherence of a proposition or set of propositions is determined by thepairwise relations established by principles 1-5.


ECHO is a Common LISP program whose input consists of statements about the explana-* tory and contradictory relations among propositions. It creates units representing propositions

and sets up links between pairs of propositions in accord with the above principles of explana-tory coherence. If two propositions cohere because they are both arguments of a particularexplanation, then ECHO sets up an excitatory link between them. If two propositions areincoherent because they contradict each other, then ECHO sets up an inhibitory link betweenthem. In accord with the principle of data priority, propositions representing evidence receive alink from a special evidence unit. For modeling the physics students, we treat as evidence pro-positions based on either (a) the presence or absence of direct observations, (b) memories ofsuch observations, or (c) facts that are well-established for the subject, such as "gravity pullsobjects downward."

The mathematics underlying ECHO are straightforward. Following typical connectionistpractice (Rumelhart & McClelland, 1986), each unit has an activation that is updated by consid-ering the units that are linked to it. A unit's excitatory link with another unit whose activation isgreater than 0 tends to increase the first unit's activation, while an inhibitory link with the otherunit tends to decrease activation. More generally, for each unit j, the activation a. is a continu-ous function of the activation of all the units linked to it, with each unit's contribution dependingon the weight w.. of the link from unit i to unit j. The activation of a unit j can be updated fromtime t to time t+Y using the following equation.

a1(t+1) = aj(t)(1-O) + netj(max-am(t)) if net>O (I)net, (ai ()-min) otherwise (

Here 0 is a decay parameter that decrements each unit at every cycle, min is minimum activation(-), max is maximum activation (1), and net. is the net input to a unit. This is defined by:

etj = wij ai (t ) (2)

Repeated updating cycles result in some beliefs gaining acceptance (activation > 0) while otherare rejected (activation < 0). ECHO networks eventually settle into stable states in which theunits have asymptotic activations that represent their coherence with other units.

Applying ECHO To Pat's Belief RevisionWe have used ECHO to analyze the kinds of belief revision exhibited in the subjects

described above. In each case, a contradiction among the subject's beliefs appeared to serve asthe motivation for the observed changes. ECHO deals with contradictions gracefully, treatingthem as a pressures to change beliefs, but otherwise tolerating them. Pat's case involved a criti-cal incoherence between two mutually exclusive predictions: a piece of evidence that was sup-posedly observed, and a hypothesis that was not observed, yet consistent with other observationsand hypotheses.

The following is a list of Pat's initial set of propositions, as garnered from her verbal proto-col of the problem-solving session. They represent her active beliefs just after she provided herstraight-down prediction for the walking-drop task.


Evidence:E L. Carried objects fall straight-down upon release.E2. Carried objects don't fall diagonally upon release.

Negative Evidence (proposed observations that do not obtain):NEI. Carried objects fall diagonally upon release.

Common Fact:CFI. Gravity moves released objects downward.

Newtonian Hypotheses:NH1. Laterally moving objects curve downward (immediately) upon release.NH2. Released objects move forward via a forward velocity.

Alternative (non-Newtonian) Hypotheses:AHL. Horizontally moving objects fall straight-down (immediately) upon release.AH2. Released objects move forward via a forward "force."

The following are Pat's original verbalized explanations, manifested in ECHO as excitatorylinks among each of the propositions involved.

Explanations:El is explained by AHI;E2 is explained by NHI;E2 is explained by AHI;NEI is explained by CFI and AH2;NHl is explained by CFI and NH2;

The next set of relations are the inconsistencies that Pat originally mentioned. Recall thatthe contradiction that disturbed Pat was the one between El and NHl; she couldn't accept both(a) that laterally-released objects curve downward and (b) that carried objects (also beinglaterally-released) fall straight-down.

Contradictions:El versus NHI;E2 versus NEI;NHI versus AHI;

When Pat was later asked to offer a prediction for the upward-throw task, she added thefollowing beliefs:

New Evidence:E3. Upwardly thrown objects curve up-and-down.E4. Upwardly thrown objects do not curve up and fall straight-down.

New Negative Evidence:NE2. Upwardly thrown objects curve up, then fall straight-down.


Finally, Pat verbalized the following explanations and contradictions. Note that the expla-

nation of NE2 is essentially a (higher-order) explanation of a hypothesis by other hypotheses:

New Explanadons:E3 is explained by NHI1;E4 is explained by NHI;NE2 is explained by NHI and AHI;

New Contradicions:E3 versus AH1;E4 versus NE2;E4 versus AHI;

I

CF

""Il -NE "-NrE2 E4

E2 E3

I "H

Figure 2. Pat's explanatory coherence network.

Figure 2 displays the network ECHO forms from the above explanations and contradic-tions, with solid lines representing symmetrical excitatory links and dashed lines representingsymmetrical inhibitory links. We suggest that the figure displays the essential structural aspectsof Pat's working memory during the belief change in question. The graph shows that predictionNHI is well-supported by evidence E2, E3, and E4, as well as by fact CF1 and hypothesis NH2.Prediction El, being a "remembered" observation, has a direct source of activation via principle(4) yet is supported only by the Aristotelian hypothesis AHI.

In order to approximate Pat's belief change, ECHO should exhibit both an initial accep-tance of El, followed by its rejection in favor of NHL. As Figure 3 illustrates, these characteris-tics are indeed captured by ECHO. The activation (from -1 to +1 on the y-axis) of each node isplotted against time (from 0 to 200 cycles of activation updating). With each node initially setto zero activation, the system relaxes into more and more coherent states, such that El's trajec-tory follows the desired nonmonotonic path -- rising sharply, then falling into the rejected region


- as NHI advances and AHI declines. The other propositions are similarly accepted or rejected(or held in limbo, as is AH2), depending upon their local coherence relationships within theoverall constraint-satisfaction system. Note that the model also simulates the temporal aspect ofPat's reasoning, as the "new" propositions, E3, E4, and NE2, as well as their associated explana-tions and contradictions, are introduced after a brief lag (after 15 cycles). The final, most stableconfiguration of beliefs happens to be one that roughly corresponds to Newtonian motion. (Ofcourse, if Pat happened to recall other evidence that supported her alternative hypotheses, thisneed not have been the case.)

£2 cf3

N~l 11E

ziIZIZFigure 3. The activation trajectories of Pat's beliefs.

A Dynamic Simulation Of Hal's Belief RevisionA simulation of Hal's belief changes involves a more intensive temporal analysis. Recall

that Hal's revision was due to an empirically-driven contradiction, in contrast to Pat's morememory-driven contradiction. Here are Hal's essential original beliefs (i.e., his beliefs prior toreceiving any trajectory feedback about pendulum-bobs that are released during a swing). Keepin mind that Hal is a composite subject: these are beliefs that were characteristic of many of thesubjects who underwent essentially the same belief revision.

Evidence:El. Kids can fly off the end of a playground swing.ES. A pendulum reverses directions at the endpoints.

Common Facts:CFI. Gravity pulls obiects downward.CF2. A swing is a pendulum.

Classical Physical (Newtonian) Hypotheses:CPI. At the endpoints, a pendulum is at rest.CP2. A laterally-released object moves over and down.CP3. The slower a pendulum-bob's speed at release, the smaller the curved trajectory.


Alternative (nco Newtonian) Hypotheses:AHI. At the er 'points, a pendulum-bob continues its preceding lateral motion.

Predictions:PI. At the endpoint, a released bob will move over and down.P2. At the endpoint, a released bob will fall straight-down.

Both El and E5 are remembered observations. E2, E3, and E4 were intentionally left outfor now, since these pieces of evidence will be sequentially added as feedback, as describedlater. The following explanations and contradictions were common to protocols reflecting Hal'sbelief revision. Note that here the critical incoherence (which feedback eventually resolves) isbetween P1 and P2, two mutually-exclusive predictions with different levels of support andcompetition.

Explanations:El is explained by AH and CF2;E5 is explained by CPI;P1 is explained by AHI and CP2;P2 is explained by CPI and CP3;P2 is explained by CPI and CFI;CP2 is explained by CFI;

Contradictions:El versus P2; PI versus P2; CPI versus AH1;

Figure 4 shows that, when ECHO is loaded in such a fashion at time to, the system reachesa stable state by t (after 150 processing cycles). Among other dynamic relationships, thesegraphs show that P (the curving-down-at-endpoint prediction) is believed, while its antagonist,P2 (the straight-down-at-endpoint prediction), is disbelieved -- as indicated by its negativeactivation. Thus, t, represents the state of Hal's belief system prior to any feedback.

At t2 , t3 , and t4 (of Figure 4), evidence about other pendular-release positions is acquiredin the forn of direct observations (i.e., feedback) E2, E3, and E4. These "within-swing" pathsare readily explained by (and hence support) propositions CFI, CP2, and CP3:

New Evidence:E2. A bob released on a downswing curves down after its release.E3. A bob released from midswing curves out (a lot) after its release.E4. A bob released on an upswing curves up-and-down after its release.

New Explanations:E2 is explained by CFI, CP2, and CP3;E3 is explained by CFI, CP2, and CP3;E4 is explained by CF1, CP2, and CP3;

The system then settles into state t (after 400 total cycles). Figure 4 shows that, except forthe generalization expressed in CP3 (relating release-velocity to the breadth of curves), little haschanged from state tl; P1 is still believed and P2 is not. Figure 5 shows Hal's belief systemfrom t5 onward, including all excitatory and inhibitory links.


All

c1E2

t 2

EL t3 I tP2 l t

t 6 -- t 7

Figure 4. The activation trajectories of Hal's beliefs.

CEI CP2 E 4 E

Figure 5. Hal's explanatory coherence network.


As described earlier, it is at time t that the dramatic belief revision begins, driven by thesurprising feedback that, contrary to P1, the endpoint release yields a straight-down pat, (aspredicted by the disbelieved P2). This feedback is simulated in ECHO by making P2 a datanode, thus providing it with a direct source of activation (like El-E5, CFl, and CF2 which alsohave data priority.) As Figure 4 indicates, this single change has five dramatic consequencesbetween t6 and Hal's ultimate state (after 850 total cycles), t7: (a) P2 gains acceptance, flippingfrom a negative to a positive activation-state, while (b) the antagonistic PI is rejected. (c) CPI,the notion of instantaneous zero velocity, achieves acceptance, while (d) its non-Newtonianantagonist, AH1, is rejected. (e) Even El, a fallacious piece of "evidence" (i.e., that kids can flyoff the end of swings) loses support. These changes essentially reflect the belief revisions ver-balized by subjects like Hal.

ASSESSING THE MODEL

While these simulations provide general correspondence with Pat's and Hal's changes inbelief, there are several methodological questions to consider. We must ask how sensitiveECHO is to (a) the particular representation of an individual's beliefs and (b) the particularparameters involved in activation-passing.

How arbitrary are the representations that are put to ECHO? Although Pat's beliefs weregarnered directly from audio-taped protocols, there is no fool-proof algorithm for translatingutterances into propositions, so analysis has some latitude. Similarly, although we tried toinclude only relations that were explicitly used in Pat's explanations, this part of the analysisalso involves some subjectivity. It is particularly difficult for the coder to refrain from adding anobvious node or a link even though the particular subject didn't vocalize that obvious belief orrelationship. (For instance, the authors found it difficult to not add an inhibitory link betweenPat's AH2 and NH2.) Constructing Hal's belief system allowed for more latitude than Pat's,since he is a composite. Still, care was taken to create the network first -- before tinkering withthe processing parameters -- so that we would be less likely to "kludge" the representation.

There is also another kind of representational question: What does one of these networksactually represent? Generally, we conceive of the networks as models of the current contents ofworking memory. Note, however, that by "current" we also mean "contextual," because sub-jects can hold a belief in one context that they disbelieve in another. For instance, in an abstractcontext, most subjects explicitly held CPI, that there is no speed at a pendulum's endpoints --even those, like Hal, who would reject it (in favor of AHI) during the context of the pendular-release tasks.

A general problem with connectionist cognitive models is that they usually have numerousnumerical parameters that can be manipulated to produce desired results. Does our simulationdepend on fine parameter tuning? The most important parameters in ECHO include the weightvalue of excitatory links, the negative weight value of inhibitory links, the weight value of the(data priority) links between evidence and the special evidence unit, and the decay of each unitat each cycle. The simulations of both Pat and Hal used the same parameter settings, andyielded the desired trajectories over similar ranges for each parameter. These common parame-ter ranges were: .015 to .05 for excitatory weights, -.05 to -.065 for inhibitory weights, .035 to.075 for data-priority weights, and .01 to .065 for the decay rate.

The simulations might have employed even more parameters. For instance, we treated unitsrepresenting direct observations, memories, and facts all as evidence, with each linked to the


special evidence unit by the same weight. But one can argue for varying these weights for dif-ferent kinds of evidence, increasing them for current observations and decreasing them for fuzzymemories. Not all evidence has the same epistemic status. In particular, when Hal is directlypresented with a phenomenon on the computer screen in front of him, this becomes a verysalient piece of evidence. Accordingly, one might argue that the unit representing the surprisingobservation that the pendulum bob falls straight down at the end of its swing should be a multi-ple of the data priority of remembered evidence.

FUTURE RESEARCH

We have been modeling previously performed experiments, but ECHO can also be used tomake predictions about the beliefs of subjects. Our simulation of Hal predicted that he wouldcome to doubt the belief that kids can fly off the end of a playground swing, but very few sub-jects explicitly re-evaluated this belief. ECHO predicts that the subjects may have experiencedthis belief change even if they did not mention it, and this prediction can be tested in new experi-ments by asking subjects to state their confidence in belief El following the relevant feedback.Additional experimental tests of the extent to which ECHO models human performance can bedone in situations where people face difficult inference problems involving judgments of expla-natory coherence. We conjecture that problems that are relatively hard for people, as measuredperhaps by the length of time to generate answers, will also be relatively hard for ECHO, asmeasured by the number of cycles it takes the system to reach a stable state. Legal reasoning, inwhich jurors attempt to construct a coherent account of the evidence (Pennington & Hastie,1987), appears to be a particularly promising domain for future empirical tests of the ECHOmodel.

ACKNOWLEDGMENTS

This research was supported by a grant from the James S. McDonnell Foundation toPrinceton University, by grant N-00014-85-K0337 from the Office of Naval Research, awardedto Lauren B. Resnick and Stellan Ohlsson, and by the Learning Research and DevelopmentCenter of the University of Pittsburgh. We are also indebted to Gilbert Harman, Stephen Han-son, and members of a discussion group on explanatory coherence.

REFERENCES

Caramnazza, A., McCloskey, M., Green, B. (1981). Naive beliefs in "sophisticated" subjects:misconceptions about trajectories of objects. Cognition, 9, 117-123.

Halloun, 1. & Hestenes, D. (1985). Common sense concepts about motion. American Journal ofPhysics, 53, 1056-1065.

Harman, G. (1986). Change in View. Cambridge, MA: MIT Press.

McCloskey, M., Washburn, A., & Felch, L. (1983). Intuitive physics: The straight-down beliefand its origin. Journal of Experimental Psychology: Learning, Memory, and Cognition, 9,636-649.

Pennington, N. & Hastie, R. (1987). Explanation-based decision making. Proceedings of theNinth Annual Conference of the Cognitive Science Society, 682-690.


Ranney, M. (1987a) Changing Naive Conceptions of Motion. Doctoral dissertation, Universityof Pittsburgh, Learning Research and Development Center.

Ranney, M. (1987b, April). Restructuring Conceptions of Motion in Physics-Naive Students.Paper presented at the annual meeting of the American Educational Research Association,Washington, DC.

Rumelhart D., & McClelland, J. (Eds.). (1986). Parallel Distributed Processing. (Vols. 1 & 2).Cambridge, MA: MIT Press.

Thagard, P. (1988a). Explanatory coherence. Princeton University Cognitive Science Labora-tory Technical Report. Princeton, NJ.

Thagard, P. (1988b). Computational Philosophy of Science. Cambridge, MA: MITPress/Bradford Books.

Thagard, P., & Nowak, G. (1988). The explanatory coherence of continental drift. Manuscriptsubmitted for publication.

Wertheimer, M. (1945). Productive Thinking. New York: Harper.

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