Geometry of Psychological Space
作者: Mildred L G Shaw &am / 6893次阅读 时间: 2010年2月27日
来源: Knowledge Science Institute, University of Calgary 标签: Geometry Kelly kelly Space

Kelly's "Geometry of Psychological Space" and its Significancefor Cognitive Modeling


Mildred L G Shaw & Brian R Gaines
 Knowledge Science Institute, University of Calgary
 Calgary, Alberta, Canada T2N 1N4

 {mildred, gaines}
 (from The New Psychologist, 23-31, October, 1992) 

 Psychological Geometry
 Constructs and Concepts
 A Visual Language for the Logic
 The Repertory Grid


 Personal construct psychology is a theory of individual and group psychologicaland social processes that takes a constructivist position in modeling humanknowledge but bases this on a positivist scientific position that characterizesconceptual structures in axiomatic terms.  It provides a fundamental frameworkfor both theoretical and applied studies of knowledge acquisition andrepresentation.  This paper presents Kelly's original intuitions underlyingpersonal construct psychology and links these to its foundational role incognitive and computational knowledge representation.


 George Kelly was a clinical psychologist who lived between 1905 and 1967,published a two volume work defining personal construct psychology in 1955, andwent on to publish a large number of papers further developing the theory, manyof which have been issued in collected form (Maher, 1969).  Figure 1 attemptsto encapsulate the historic forces at work in psychology, logic, cognitivescience and artificial intelligence before and after Kelly's work.



 Figure 1 The intellectual setting of personal construct psychology  

Personal construct psychology can be seen to be an heir to Europeanlogical positivism and American pragmatism, taking an alternative path tobehaviorism that is similar in many respects to what later became termedcognitive science.  In this Kelly was preceded by Vygotsky in the USSRwhose bookThought and Languageappeared in 1934 but was suppresseduntil 1962 (Wertsch, 1985; Vygotsky, 1989).  Luria, who had been a student ofVygotsky  invited Kelly to Moscow in 1961 where he delivered a particularlyclear statement of the formal principles of personal construct psychology inthe light of reactions to his earlier book (Kelly, 1969).  Kelly was working ona second book when he died of which, unfortunately, only the preface has beenpublished (Kelly, 1970), and he also began to become involved with the computersimulation of personality (Kelly, 1963).  However, his work never became partof the mainstream cognitive science literature although it has attractedwidespread attention and application in management, education and clinicalpsychology (Shaw, 1981).

 Kelly was a keen geometer with experience in navigation and an interest inmulti-dimensional geometry.  When he came to formalize his theory he took ashis model Euclid'sElementsand axiomatized personal constructpsychology as afundamental postulatetogether with elevencorollaries, terming the primitives involveselementsandconstructs.  Kelly took far more than a vocabulary from Euclid.  TheElementswere the normative model for the science that arose out of theGreek enlightenment, and fulfilled a similar role in the second enlightenmentfor Descartes, Kant and others (Russell, 1946).  The careful definition ofterms, with attention to exact and overt presuppositions, and then thedevelopment of a rigorous deductive sequence with precise specification ofhypothesis and constructions, provided an intellectual model that we followtoday.  Indeed in the post-modern literature the modern age has beencharacterized as based on theethics of geometry(Lachterman, 1989).

 Kelly presented his theory as ageometry of psychological space(Kelly,1969), and his conceptual framework is very clear if seen in these terms.  Itmay seem strange to base cognitive science on geometry rather than logic untilone remembers that the reasoning structure of theElementswas the basisfor both Greek and modern logic, and that geometry and logic in acategory-theoretic framework are equivalent (Mac Lane, 1971).  What Kellyachieved through the use of geometry was an intensional logic, one in whichpredicates are defined in terms of their properties rather than extensionallyin terms of those entities that fall under them.  In his time there were noadequate formal foundations for intensional logic--it was not until 1963 thatHintikka (1963) published the model sets formulation that gave intensionallogic itspossible worldsformal foundations.  


 Kelly's "fundamental postulate" for personal construct psychology was that:

 "A person's processes are psychologically channelized by the way in which heanticipates events." (Kelly, 1955, p.46)

 This was stated as a postulate to emphasize that it was presented as aconvenient viewpoint from which to understand human behavior, not imputed to anunderlying physiological or psychological reality.  The basis of Kelly'sapproach to psychotherapy was that this was also a convenient viewpoint fromwhich someone could understand, and modify, their own behavior.  He saw allpeople as "personal scientists" in anticipating the world, and attempted todevelop techniques where this anticipatory modeling activity was reflexivelyapplied to the self.  His first corollary, the construction corollary,states:

 "A person anticipates events by construing their replications." (p.50)

 This emphasis on the role in behavior of a view to the future is whatdistinguishes Kelly's approach to psychology.  He saw people as driven by theneed to cope with coming events in the world and all other aspects of behavioras deriving from this:

 "A person's processes, psychologically speaking, slip into the grooves whichare cut out by the mechanisms he adopts for realizing his objectives." (p.49)

 These grooves provide templets for construing events which he termed "personalconstructs":

 "Man looks at his world through transparent templets which he creates and thenattempts to fit over the realities of which the world is composed." (pp.8-9)

 "Constructs are used for predictions of things to come, and the world keeps onrolling on and revealing these predictions to be either correct or misleading.This fact provides the basis for the revision of constructs and, eventually, ofwhole construct systems." (p.14)

 Kelly introduces the notion of apsychological spaceas a term for aregion in which we may place and classify elements of our experience.  It isimportant to note that he did not suppose this space to pre-exist as a world ofsuch elements, but rather to come into being through a process of constructionby which we create a space in which to place elements as we come to construethem.  He sees us as creating dimensions in personal psychological space as away of providing a coordinate system for our experience, and emphasizes thatthe topology of the space comes into existence as it is divided:

 "Our psychological geometry is a geometry of dichotomies rather than thegeometry of areas envisioned by the classical logic of concepts, or thegeometry of lines envisioned by classical mathematical geometries.  Each of ourdichotomies has both a differentiating and an integrating function.  That is tosay it is the generalized form of the differentiating and integrating act bywhich man intervenes in his world.  By such an act he interposes a differencebetween incidents -- incidents that would otherwise be imperceptible to himbecause they are infinitely homogeneous.  But also, by such an intervening act,he ascribes integrity to incidents that are otherwise imperceptible becausethey are infinitesimally fragmented.  In this kind of geometrically structuredworld there are no distances.  Each axis of reference represents not a line orcontinuum, as in analytic geometry, but one, and only one, distinction.However, there are angles.  These are represented by contingencies oroverlapping frequencies of incidents.  Moreover, these angles of relationshipbetween personal constructs change with the context of incidents to which theconstructs are applied.  Thus our psychological space is a space withoutdistance, and, as in the case of non-Euclidian geometries, the relationshipsbetween directions change with the context." (Kelly, 1969)

 It is this emphasis on the space itself being created by a process of makingdistinctions rather than being defined by the elements distinguished that givespersonal construct psychology its intensional nature:

 "the construct is a basis of making a distinction...not a class of objects, oran abstraction of a class, but a dichotomous reference axis" (Kelly, 1970)

 Figure 2 shows the main features of Kelly's notion of psychological space.  Aconstruct is a dichotomous reference axis.  It defines a family of planesorthogonal to it that divide the space:

 "To catch a glimpse of psychological space we may imagine a system of planes,each with two sides or aspects, slicing through a galaxy of events" (Kelly,1970)

 However, only part of the space that is divided is used in placing elements:

 "A construct is convenient for the anticipation of a finite range of eventsonly.  A personal construct system can hardly be said to have universalutility.  Not everything that happens in the world can be projected upon allthe dichotomies that make up a person's outlook. . . The geometry of the mindis never a complete system." (Kelly, 1970)

 This division defines the dichotomous poles of the construct:

 "Each construct involves two poles, one at each end of its dichotomy.  Theelements associated at each pole are like each other with respect to theconstruct and are unlike the elements at the other pole" (Kelly, 1955)

 Even though Kelly's geometry is not metrically defined, he has no problems inusing the defined constructs to generate a metric as shown in Figure 3:

 "imagine a system of planes, each with two sides or aspects, slicing through agalaxy of events...If the set is moved into all possible positions it generatesa paracartesian hyperspace with its relatively concrete scalar axes" (Kelly,1970)

 It is possible to develop a complete theory of cognition, action, learning andintention with the geometry.  



 Figure 2 The geometry of psychological space 


 Figure 3 Scales in psychological space 

 Constructsand Concepts

 One obvious question about Kelly's use of the term "construct" was how itdiffers from the more conventional term "concept."  He discusses this in thefollowing terms:

 "We use the termconstructin a manner which is somewhat parallel to thecommon usage of `concept.'  However, if one attempts to translate ourconstructinto the more familiar term, `concept,' he may find someconfusion.  We have included, as indeed some recent users of the term `concept'have done, the more concretistic concepts which nineteenth-centurypsychologists would have insisted on calling `percepts.'  The notion of`percept' has always carried the idea of its being a personal act--in thatsense, ourconstructis in the tradition of `percepts.'  But we also seeourconstructas involving abstraction--in that sense ourconstructbears a resemblance to the traditional usage of `concept.' ...Now when we assume that the construct is basically dichotomous, that itincludes percepts, and that it is a better term for our purposes than the term`concept,' we are not quarreling with those who would use it otherwise.  Withinsome systems of logic the notion of contrast as something distinct fromirrelevancy is not part of the assumptive structure.  We, on the other hand aresimply assuming that this is the way people do, in fact, think." (Kelly, 1955,p.70)

 The dichotomous aspect of constructs is the most significant aspect of thedifference between Kelly's constructs and current usage of the term, `concept.'Hisdichotomy corollarystates this:

 "A person's construction system is composed of a finite number of dichotomousconstructs." (p.59)

 and it is a consequence of the two-sided nature of a distinction represented inthe geometry.  The range of convenience captures the notion of relevancy andthe distinction within it then generates a natural opposition.  That peopletend to conceptualize the world in terms of restricted sorts that are thendichotomized is a phenomenon identified in antiquity (Lloyd, 1966) and commonacross many cultures (Maybury-Lewis & Almagor, 1989).  

 Since the term "concept" is used in psychology and in knowledge representationin somewhat different ways, it is useful to define it clearly in these contextsfor the purposes of this paper.  A psychological concept is defined to be thatmental entityimputedto a distinction making agent as enabling it tomake a particular distinction.  Note that concepts are separated both from thedistinctions they support and the entities they distinguish, and are notreified but seen as imputed to the agent.  They are themselves distinctionsmade by an observer--possibly, a reflective observer.  Concepts are statevariables we impute to a knowledgeable agent.  This definition also correspondsto Anglin's:

 "a concept is all of the knowledge possessed by an individual about a categoryof objects or events"--"Concepts mediate categorization but concepts are notthe resultant categories." (Anglin, 1977)

 For personal construct psychology, the geometry and the logic, distinctions arethe only primitives.  One might ask "what is distinguished?", but the answerwill be "distinctions."  This corresponds to deriving the two phenomena inKelly'sconstruction corollaryfrom a single primitive, that wedistinguish events in the undifferentiated stream of circumstance, and then wefurther distinguish among the distinguished events by construing.

 AVisual Language for the Logic

 Kelly's `construct' in psychological space is conveniently represented by apair of disjoint concepts corresponding to the construct poles, both subsumedby a third corresponding to the range of convenience as shown in Figure 4.This diagram is already a simple semantic network in the style of KL-ONE(Brachman & Schmolze, 1985) or KRS (Gaines, 1991), but it has well-definedlogical semantics as defined above, and also strong psychological foundationsin personal construct psychology.



 Figure 4 A visual language for the logic 


 Figure 5 Logic of subsumed foci of convenience  

There is an analogy between the visual language and the representationof chemical structures as atoms and bonds.  Distinctions are the atomicprimitives in personal construct psychology, and further constructions may beseen as complex `molecules' formed by distinctions joined through subsumptionand disjoint `bonds.'  For example, Figure 5 shows how the geometry and thelogical visual language may both be used to represent different foci ofconvenience and contexts:

 "A construct's range of convenience comprises all those things to which theuser would find its application useful.  A construct's focus of conveniencecomprises those particular things to which the user would find its applicationmaximally useful.  The somewhat more restricted than the range ofconvenience...somewhat more extensive than the focus of convenience" (Kelly,1955)

 The distinctions, that cut out regions for `person' and `wine' each involve anadditional dimension and are not represented in a two-dimensional diagram.This is a problem for the geometry that is overcome in the visual language forthe logic.

 Multiple constructs in psychological space correspond to multiple axes ofreference, and the plans representing their distinctions and ranges ofconvenience intersect to define regions of the space corresponding toconjunction, composition and multiple inheritance in the logic as shown inFigure 6.

 Kelly introduces anticipation within the geometry by attaching actions toregions of psychological space:

 "A young girl anticipates marriage...The predicted husband does not exist forher in the flesh, but simply as the intersect of a limited number of conceptualdimensions.  One day a young man plumps himself down on this waitingintersect...she marries him" (Kelly, 1955, p.121)

 Kelly's theory of anticipation is based on attaching significance to suchrecognizable intersections:

 "What one predicts is not a fully fleshed-out event, but simply the commonintersect of a set of properties" (Kelly, 1955)



 Figure 6 Logic of overlapping distinctions and their intersections  

The logic remains intensional because there is no implication thatelements have already been construed within the intersections.  Kelly oftenexplaineda construct in terms of how elements were placed, but he doesnotdefinea construct as necessarily having elements placed along theaxis.  Elements are placed in the framework of existing constructs, or theconstruct system is adjusted to account for a new element.  For example, in themarriage anticipation example given above, the predicted husband `exists' onlyas the "intersect of a limited number of conceptual dimensions."  Theattachment of an anticipation to this intersect corresponds to a commitment toplace an element that falls in this intersect in the region defined by the poleof some other construct also.  In logic this is amaterial implication rather than an entailment in that it is not necessitated by the way in whichthe distinctions are defined but is instead an auxiliary commitment orrule.

 Rules allow a cognitive system to be anticipatory in containing structureswhich from one set of distinctions made about an event will imply that othersshould be made leading to prediction or action.  Rules play a similar role incomputational systems in generating recommendations for decision or action.Overtly modeling the conceptual system of an expert as such a structure is abasis for emulating the expert's performance in a knowledge-based system.

 As shown in Figure 7, Kelly's model of anticipation is represented in thevisual language by an additional primitive, a rounded corner rectangle,representing material implication or a rule.  This is not so readilyrepresented as a static image in the geometry because it is a dynamicprinciple, "if you place an element in this intersection then move it to thissub-intersection."  A rule corresponds to anattractorin dynamics, thatan element placed in a region of space is unstable and falls into the basin ofan attractor (Abraham & Shaw, 1984).  This can be represented in thegeometry but the diagrams for activities of any reasonable complexity becomevery difficult to visualize and understand.  It does, however, provide a nicelink to the corresponding phenomena in the geometrical dynamics of neuralnetworks underlying the phenomenon (Domany, Hemmen & Schulten, 1991).

 The abstraction of the geometry in the visual language of the logic correspondsto taking multiple cross sections of the geometrical representation fromdifferent perspectives and presenting them together.  It is way of dealing withthe visualization of a high-dimensional space.



 Figure 7 Logic of anticipation through rules  

 TheRepertory Grid

 Kelly (1955) introduces the "role repertory grid" as a means for investigatinga person's conceptual structure relevant to inter-personal relations by havingthem classify a set of people significant to them in terms of elicited personalconstructs.  Figure 8 shows the general form of a repertory grid from expertsystem development in the literature concerning contact lens prescription, andits relation to the conceptual structures already discussed.  The elements arecases or patients, shown as their initials, and the constructs are the way theexpert distinguishes his different cases.  



 Figure 8 The repertory grid as a matrix of concepts, individuals andconstraints 

If one takes a particular concept somewhere in the lattice, and a set ofelements asserted to fall under that concept, then each distinction that may bemade about elements falling under that concept forms the rows of a matrix, theelements form the columns, and the constraints applying to a particularindividual relative to a particular distinction form the values in thematrix.

 The first two constructs are the possible recommendations for contact lensprescription, and the third refers to the reduced or normal tear production ofthe patient.. It can be seen how it relates to the concepts of acontactlens patient, who is apatient, who is aperson.  There areother types of patient such as aspectacle patientsome of whom may bebifocal patients.  



 Figure 9 A repertory grid about contact lens presciptions 

Figure 9 shows the full grid in its conventional form, as it wouldnormally appear in the literature.  

 In clinical psychology, repertory grids have been used extensively withoutcomputer support, and techniques for manual elicitation are well-documented(Fransella & Bannister, 1977).  However, repertory grid methodologies aredifficult to undertake manually as they require feedback and management fromthe elicitor while at the same time attempting to avoid inter-personalinteractions that would distort the elicitee's conceptual structures.  Hencethe advent of the personal computer in the mid-1970s and its evolution into thegraphic workstations of the 1980s has been very significant for the practicalapplication of the approach (Shaw, 1980, 1981; Mancuso & Shaw, 1988).Figures 10 and 11 show screens from a graphical repertory grid system (Gaines& Shaw, 1990) at work on a computer, using the contact lens prescriptionexample.  

 The screen in Figure 10 shows a list of the constructs so far elicited, andvarious options available to the user to select her next task.  The adviceboxes can be turned off for an experienced user, or the next most sensible taskchosen by clicking on theAdvicebutton.  Each button shown in the toppart of the screen gives the user a different option.

 Constructs are elicited in this system by the Macintoshclick and drag facility.  A construct axis can be generated from a triad or a pair ofelements, from matched elements or just adding a new construct.  The poles arelabeled, and the element labels dragged on to the appropriate place on theaxis.  Element matches and construct matches at any level are shown by clickingon the bar at the middle right top of the screen, and can be edited, or newelements or constructs added as the user wishes.

 Figure 11 shows a principal components analysis of the contact lens data.  Itis reminiscent of Kelly's model of psychological space, and that may be onereason for the popularity of such analyses.  Each construct is represented by adimension on the two axes explaining the most variance in the construct system.The elements are located in this space.  We can see the two main constructclusters, one beingastigmatic -- not astigmatic, hard -- not hard, andnot soft -- soft.  These later turn out to be therulesforprescribing lenses, the others specify the exceptions.  An alternative form isthe FOCUS analysis which shows the beginnings of the development of the logicalhierarchical structure in space.  



 Figure 10 The RepGrid Screen showing existing constructs and availableoptions 


 Figure 11 The RepGrid Screen showing a principal components analysis  


 Personal construct psychology is a theory of individual and group psychologicaland social processes that has been used extensively in knowledge acquisitionresearch to model the cognitive processes of human experts.  The psychology hasthe advantage of taking a constructivist position appropriate to the modelingof specialist human knowledge but basing this on a positivist scientificposition that characterizes human conceptual structures in axiomatic terms thattranslate directly to computational form.

 This paper has given an overview of personal construct psychology and itsexpression as an intensional logic describing the cognitive processes ofanticipatory agents.  A theoretical framework has been developed and shown toprovide logical foundations for personal construct psychology and computationalknowledge representation schema.  The framework is generated from the singleprimitive of "making a distinction."  It has been used to provide cognitive andlogical foundations for existing knowledge acquisition tools and techniques,and for the design of integrated knowledge acquisition systems.

  • Kelly's geometry of psychological space gives personal constructpsychology strong theoretical foundations.
  • Developments in intensional logic since his death allow his geometry tobe expressed as a logical system.
  • Developments in computing allow his vision of the computer simulation ofhuman personality as an expression, and a test, of his theory be realized.
  • Personal construct psychology provides foundations for cognitive scienceand artificial intelligence that are consistent with current positions in thesedisciplines, but supplies an integrative framework that is currently lacking.
  • Tools based on personal construct psychology are already having animpacton the development of computer-based knowledge systems.
  • The theory itself will have a significant impact on computer support ofhuman knowledge processes.
  • Computer tools and simulations developed for industrial objectives offerexciting potential in other applications of personal constructpsychology--clinical, educational, etc.
  • Twenty five years after George Kelly's death, personal constructpsychology looks more fresh, original and applicable than ever.  The world isbeginning to catch up with his ideas.
  • All the pieces are in place for a systematic development theconstructivist position as a foundation for a theory of anticipatory systemsencompassing human psychology, society, & technologies.
In conclusion,we suggest that personal construct psychology provides a very attractivefoundation for knowledge acquisition methodologies and tools that have tobridge between human cognitive processes and computational knowledgerepresentation.  The many tools so far developed based on personal constructpsychology encourage this conclusion.  However, what has so far been developedtaps only a small part of the potential and we may expect to see major newdevelopments in the future.  In particular, personal construct psychologyoffers the opportunity to integrate many different knowledge acquisition andknowledge representation approaches within a single principled framework.


 Abraham, R.H. & Shaw, C.D. (1984)Dynamics: The Geometry ofBehavior.  Santa Cruz, California: Aerial Press.

 Anglin, J.M. (1977)Word, Object, and Conceptual Development.  Norton:New York.

 Brachman, R.J. & Schmolze, J. (1985). An overview of the KL-ONE knowledgerepresentation system.Cognitive Science, 9(2) 171-216 (April-June).

 Domany, E., Hemmen, J.L. & Schulten, K., Eds. (1991).Models of NeuralNetworks. Berlin: Springer.

 Fransella, F. & Bannister, D., Eds. (1977)A Manual for Repertory GridTechnique. London: Academic Press.

 Gaines, B.R. (1991) Empirical investigation of knowledge representationservers: design issues and applications experience with KRS.AAAI SpringSymposium: Implemented Knowledge Representation and Reasoning Systems. pp.87-101. Stanford (March)--alsoSIGART Bulletin 2(3) 45-56.

 Gaines, B.R. & Shaw, M.L.G. (1990) RepGrid ManualCentre for PersonComputer Studies,3019 Underhill Drive NW, Calgary, AB, Canada T2N 4E4.

 Hintikka, J. (1963). The Modes of Modality.Acta PhilosophicaFennica,16, 65-81.

 Kelly, G.A. (1955).The Psychology of Personal Constructs. New York:Norton.

 Kelly, G.A. (1963) Aldous: the personable computer. Tomkins, S.S. &Messick, S. Eds.Computer Simulation of Personality. pp.221-229. NewYork: Wiley.

 Kelly, G.A. (1969) A mathematical approach to psychology. Maher, B. Ed.Clinical Psychology and Personality: The Selected Papers of GeorgeKelly. pp.94-113. New York: Wiley.

 Kelly, G.A. (1970) A brief introduction to personal construct theory.Bannister, D. Ed.Perspectives in Personal Construct Theory. pp.1-29.London: Academic Press.

 Lachterman, D.R. (1989).The Ethics of Geometry. New York: Routledge.

 Lloyd, G.E.R. (1966)Polarity and Analogy.  Cambridge: CambridgeUniversity Press.

 Mac Lane, S. (1971)Categories for the Working Mathematician.  New York:Springer-Verlag

 Maher, B., Ed. (1969).Clinical Psychology and Personality: The SelectedPapers of George Kelly. New York: Wiley.

 Mancuso, J.C. & Shaw, M.L.G., Eds. (1988)Cognition and Personalstructure: Computer Access and Analysis. New York: Praeger.

 Maybury-Lewis, D. & Almagor, U. (1989). (Eds)The Attraction ofOpposites.Ann Arbor: University of Michigan Press.

 Russell. B. (1946).History of Western Philosophy.  London: Allen &Unwin.

 Shaw, M.L.G. (1980).On Becoming A Personal Scientist: Interactive ComputerElicitation of Personal Models Of The World. London: Academic Press.

 Shaw, M.L.G., Ed. (1981).Recent Advances in Personal ConstructTechnology. London: Academic Press.

 Vygotsky, L.S. (1989).Thought and Language. Cambridge, Massachusetts:MIT Press.

 Wertsch, J.V., Ed. (1985).Culture, Communication and Cognition: VygotskianPerspectives. Cambridge: Cambridge University Press.

TAG: Geometry Kelly kelly Space
«A bibliography on G.A.Kelly’s Psychology of Personal Constructs G.A. Kelly 乔治·凯利
《G.A. Kelly 乔治·凯利》
乔治·凯利 George Alexander Kelly简介»