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Generalizing Prototype Theory : A Formal Quantum Framework Aerts, Diederik, 1953-; Broekaert, Jan; Gabora, Liane; Sozzo, Sandro Mar 30, 2016

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ORIGINAL RESEARCHpublished: 30 March 2016doi: 10.3389/fpsyg.2016.00418Frontiers in Psychology | www.frontiersin.org 1 March 2016 | Volume 7 | Article 418Edited by:Kevin Bradley Clark,University of California, Los Angeles,USAReviewed by:Terrence C. Stewart,Carleton University, CanadaBruce MacLennan,University of Tennessee, USA*Correspondence:Sandro Sozzoss831@le.ac.ukSpecialty section:This article was submitted toCognition,a section of the journalFrontiers in PsychologyReceived: 13 August 2015Accepted: 09 March 2016Published: 30 March 2016Citation:Aerts D, Broekaert J, Gabora L andSozzo S (2016) Generalizing PrototypeTheory: A Formal QuantumFramework. Front. Psychol. 7:418.doi: 10.3389/fpsyg.2016.00418Generalizing Prototype Theory: AFormal Quantum FrameworkDiederik Aerts 1, Jan Broekaert 1, Liane Gabora 2 and Sandro Sozzo 3*1Center Leo Apostel for Interdisciplinary Studies, Free University of Brussels, Brussels, Belgium, 2Department of Psychology,University of British Columbia, Kelowna, BC, Canada, 3 School of Management, Institute for Quantum Social and CognitiveScience, University of Leicester, Leicester, UKTheories of natural language and concepts have been unable to model the flexibility,creativity, context-dependence, and emergence, exhibited by words, concepts andtheir combinations. The mathematical formalism of quantum theory has instead beensuccessful in capturing these phenomena such as graded membership, situationalmeaning, composition of categories, and also more complex decision making situations,which cannot be modeled in traditional probabilistic approaches. We show how a formalquantum approach to concepts and their combinations can provide a powerful extensionof prototype theory. We explain how prototypes can interfere in conceptual combinationsas a consequence of their contextual interactions, and provide an illustration of thisusing an intuitive wave-like diagram. This quantum-conceptual approach gives new lifeto original prototype theory, without however making it a privileged concept theory, aswe explain at the end of our paper.Keywords: cognition, concept theory, prototype theory, contextuality, interference, quantum modeling1. INTRODUCTIONTheories of concepts struggle to capture the creative flexibility with which concepts are used innatural language, and combined into larger complexes with emergent meaning, as well as thecontext-dependent manner in which concepts are understood (Geeraerts, 1989). In this paper, wepresent some recent advances in our quantum approach to concepts. More specifically, we followthe general lines illustrated in Gabora and Aerts (2002), Aerts and Gabora (2005a,b), and Gaboraet al. (2008), and generalize the quantum-theoretic model elaborated in Aerts (2009b) and Aertset al. (2013a).According to the “classical,” or “rule-based” view of concepts, which can be traced back toAristotle, all instances of a concept share a common set of necessary and sufficient definingproperties. Wittgenstein pointed out that: (i) in some cases it is not possible to give a set ofcharacteristics or rules defining a concept; (ii) it is often unclear whether an object is a member ofa particular category; (iii) conceptual membership of an instance strongly depends on the context.A major blow to the classical view came from Rosch’s work on color. This work showed thatcolors do not have any particular criterial attributes or definite boundaries, and instances differwith respect to how typical they are of a concept (Rosch, 1973, 1978, 1983). This led to formulationof “prototype theory,” according to which concepts are organized around family resemblances, andconsist of characteristic, rather than defining, features. These features are weighted in the definitionof the “prototype.” Rosch showed that subjects rate conceptual membership as “graded,” withdegree of membership of an instance corresponding to conceptual distance from the prototype.Moreover, the prototype appears to be particularly resistant to forgetting. Prototype theory alsoAerts et al. Generalizing Prototype Theoryhas the strength that it can be mathematically formulatedand empirically tested. By calculating the similarity betweenthe prototype of a concept and a possible instance of it,across all salient features, one arrives at a measure of the“conceptual distance” between the instance and the prototype.Another means of calculating conceptual distance comes outof “exemplar theory” (Nosofsky, 1988, 1992), according towhich a concept is represented by, not a set of defining orcharacteristic features, but a set of salient “instances” of it storedin memory. Exemplar theory has met with considerable successat predicting empirical results. Moreover, there is evidence ofpreservation of specific training exemplars in memory. Classical,prototype, and exemplar theories are sometimes referred toas “similarity based” approaches, because they assume thatcategorization relies on data-driven statistical evidence. Theyhave been contrasted with “explanation based” approaches,according to which categorization relies on a rich body ofknowledge about the world. For example, according to “theorytheory” concepts take the form of “mini-theories” (Murphy andMedin, 1985) or schemata (Rumelhart and Norman, 1988), inwhich the causal relationships among properties are identified.Although these theories do well at modeling empirical datawhen only one concept is concerned, they perform poorly atmodeling combinations of two concepts. As a consequence,cognitive psychologists are still looking for a satisfactory andgenerally accepted model of how concepts combine.The inadequacy of fuzzy set models of conceptualconjunctions (Zadeh, 1982) to resolve the “Pet-Fish problem”identified by Osherson and Smith (1981) highlighted the severityof the combination problem. People rate the item Guppy asa very typical example of the conjunction Pet-Fish, withoutrating Guppy as a typical example neither of Pet nor of Fish(“Guppy effect”) (Osherson and Smith, 1981, 1982). Studiesby Hampton on concept conjunctions (Hampton, 1988a),disjunctions (Hampton, 1988b) and negations (Hampton, 1997)confirmed that traditional fuzzy set and Boolean logical rulesare violated whenever people combine concepts, as one usuallyfinds “overextension” and “underextension” in the membershipweights of items with respect to concepts and their combinations.It has been shown that people estimate a sentence like “x is talland x is not tall” as true, in particular when x is a “borderlinecase” (“borderline contradictions”) (Bonini et al., 1999; Alxatiband Pelletier, 2011), again violating the rules of set-theoreticBoolean logic. The seriousness of the combination problem waspointed out by various scholars (Komatsu, 1992; Fodor, 1994;Kamp and Partee, 1995; Rips, 1995; Osherson and Smith, 1997).More recently, other theories of concepts have been developed,such as “Costello and Keane’s constraint theory” (Costello andKeane, 2000), “Dantzig, Raffone, and Hommel’s connectionistCONCAT model of concepts” (Van Dantzig et al., 2011),“Thagard and Stewart’s emergent binding model” (Thagardand Stewart, 2011), and “Gagne and Spalding’s morphologicalapproach” (Gagne and Spalding, 2009). However, none of thesetheories has a strong track record of modeling the emergenceand non-compositionality of concept combinations.The approach to concepts presented in this paper grewout earlier work on the application to concept theory on theaxiomatic and operational foundations of quantum theory andquantum probability (Aerts, 1986; Pitowsky, 1989; Aerts, 1999).A major theoretical insight was to shift the perspective fromviewing a concept as a “container” to viewing it as “an entityin a specific state that is changing under the influence of acontext” (Gabora and Aerts, 2002). This allowed us to providea solution to the Guppy effect and to successfully represent thedata collected on Pet, Fish and Pet-Fish by using the mathematicalformalism of quantum theory (Aerts andGabora, 2005a,b). Then,we proved that none of the above experiments in concept theorycan be represented in a single probability space satisfying theaxioms of Kolmogorov (1933). We developed a general quantumframework to represent conjunctions, disjunctions and negationsof two concepts, which has been successfully tested several times(Aerts, 2009a,b; Sozzo, 2014, 2015; Aerts et al., 2015a), andwe put forward an explanatory hypothesis for the observeddeviations from traditional logical and probabilistic structuresand for the occurrence of quantum effects in cognition (Aertset al., 2015b). We recently identified a strong and systematicnon-classical phenomenon effect, which is deeper than theones typically detected in concept combinations and directlyconnected with the mechanisms of concept formation (Aertset al., 2015c). This work is part of a growing domain of cognitivepsychology that uses the mathematical formalism of quantumtheory and quantum structures to model empirical situationswhere the application of traditional probabilistic approaches isproblematical (probability judgments errors, decision-makingerrors, violations of expected utility theory, etc.; Aerts and Aerts,1995; Aerts et al., 2000, 2013a,b, 2014, 2015; Aerts and Sozzo,2011, 2014; Busemeyer and Bruza, 2012; Haven and Khrennikov,2013; Pothos and Busemeyer, 2013; Khrennikov et al., 2014;Wang et al., 2014).This paper outlines recent progress in the development of aquantum-theoretic framework for concepts and their dynamics.Section 2 explains how the “SCoP formalism” can be interpretedas a “contextual and interfering prototype theory that is ageneralization of standard prototype theory” in which prototypesare not fixed, but change under the influence of a context, andinterfere as a consequence of their contextual interactions (seealso Gabora et al., 2008; Aerts et al., 2013a). Section 3 presentsan amended explanatory version of the quantum-mechanicalmodel in complex Hilbert space worked out in Aerts (2009b)and Aerts et al. (2013a) for the typicality of items with respectto the concepts Fruits and Vegetables, and their disjunctionFruits or Vegetables. This improved quantum model illustrateshow the prototype of Fruits (Vegetables) changes under theinfluence of the context Vegetables (Fruits) in the combinationFruits or Vegetables. The latter combination is represented usingthe quantum-mathematical notion of linear superposition ina complex Hilbert space, which entails the genuine quantumeffect of “interference.” Hence, our model shows that theprototypes of Fruits and Vegetables interfere in the disjunctionFruits or Vegetables. Sections 2, 3 also justify the fact that ourquantum-theoretic framework for concepts can be considered asa “contextual and interfering generalization of original prototypetheory.” The presence of linear superposition and interferencecould suggest that concepts combine and interact like waves do.Frontiers in Psychology | www.frontiersin.org 2 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype TheoryIn Section 4 we develop this intuition in detail and proposean intuitive wave-like illustration of the disjunction Fruits orVegetables. Finally, Section 5 discusses connections betweenthe quantum-theoretic approach to concepts presented here,and other theories of concepts. Although this approach can beinterpreted as a specific generalization of prototype theory, it iscompatible with insights from other theories of concepts.We stress that our investigation does not deal with theelaboration of a “specific typicality model” that representsa given set of data on the concepts Fruits, Vegetables, andtheir disjunction Fruits or Vegetables. We inquire into themathematical formalism of quantum theory as a general,unitary and coherent formalism to model natural concepts. Ourquantum-theoretic model in Section 3 has been derived from thisgeneral quantum theory, hence it satisfies specific technical andgeneral epistemological constraints of quantum theory. As such,it does not apply to any arbitrary set of experimental data. Ourformalism exactly applies to those data that exhibit a peculiardeviation from classical set-theoretic modeling; such deviationsare taken in our framework as indicative of interference andemergence. Data collected on combinations of two conceptssystematically exhibit deviations from classical set-theoreticalmodeling, and traditional probabilistic approaches have difficultycoping with this. In this sense, the success of the quantum-theoretic modeling can be interpreted as a confirmation ofthe effectiveness of quantum theory to model conceptualcombinations. We should also mention that our quantum-theoretic approach has recently produced new predictions,allowing us to identify entanglement in concept combinations(Aerts and Sozzo, 2011, 2014), and systematic deviations fromthe marginal law, deeply connected to the mechanisms ofconcept formation (Aerts et al., 2015a,c). These effects wouldnot have been identified in a more traditional investigation ofoverextension and underextension.It follows from the above analysis that our quantum-theoretic modeling rests on a “theory based approach,” as itstraightforwardly derives from quantum theory as “a theory torepresent natural concepts.” Hence, it should be distinguishedfrom an “ad-hoc modeling based approach,” only devised tofit data. One should be suspicious of models in which freeparameters are added after the fact on an ad-hoc basis to fitthe data more closely. In our opinion, the fact that our “theoryderived model” reproduces different sets of experimental data isa convincing argument to support its advantage over traditionalmodeling approaches and to extend its use to more complexcombinations of concepts.2. THE SCoP FORMALISM AS ACONTEXTUAL INTERFERING PROTOTYPETHEORYThis section summarizes the SCoP approach to concepts byproviding new insights to the research in Aerts and Gabora(2005a,b) and Gabora et al. (2008).We mentioned in Section 1 that, according to prototypetheory, concepts are associated with a set of characteristic,rather than defining, features (or properties), that are weightedin the definition of the prototype. A new item is categorizedas an instance of the concept if it is sufficiently similarto this prototype (Rosch, 1973, 1978, 1983). The originalprototype theory was subsequently put into mathematicalform as follows. The prototype consists of a set of features{a1, a2, . . . , aM}, with associated “weights” (or “applicationvalues”) {xp1, xp2, . . . , xpM}, where M is the number of featuresthat are considered. A new item k is also associated with aset {xk1, xk2, . . . , xkM}, where the number xkm refers to theapplicability of the m-th feature to the item k (for a givenstimulus). Then, the conceptual distance between the item k andthe prototype, defined asdk =√√√√ M∑m= 1(xkm − xpm)2 (1)is a measure of the similarity between item and prototype. Thesmaller the distance dk for the item k, the more representative kis of the given concept.Prototype theory was developed in response to findings thatpeople rate conceptual membership as graded (or fuzzy), withthe degree of membership of an instance corresponding to theconceptual distance from the prototype. A second fundamentalelement of prototype theory is that it can in principle beconfronted with empirical data, e.g., membership or typicalitymeasurements.A fundamental challenge to prototype theory (but also toany other theory of concepts) has become known as the “Pet-Fish problem.” The problem can be summarized as follows. Wedenote by Pet-Fish the conjunction of the concepts Pet and Fish.It has been shown that people rate Guppy neither as a typical Petnor as a typical Fish, they do rate it as a highly typical Pet-Fish(Osherson and Smith, 1981). This phenomenon of the typicalityof a conjunctive concept being greater than—or overextends—that of either of its constituent concepts has also been calledthe “Guppy effect.” Using classical logic, or even fuzzy logic,there is no specification of a prototype for Pet-Fish starting fromthe prototypes of Pet and Fish that is consistent with empiricaldata (Osherson and Smith, 1981, 1982; Zadeh, 1982). Fuzzy settheory falls short because standard connectives for conceptualconjunction involve typicality values that are less than or equal toeach of the typicality values of the conceptual components, i.e.,the typicality of an item such as Guppy is not higher for Pet-Fishthan for either Pet or Fish.Similar effects occur for membership weights of itemswith respect to concepts and their combinations. Hampton’sexperiments indicated that people estimate membership in sucha way that the membership weight of an item for the conjunction(disjunction) of two concepts, calculated as the large numberlimit of relative frequency of membership estimates, is higher(lower) than the membership weight of this item for at least oneconstituent concept (Hampton, 1988a,b). This phenomenon isreferred to as “overextension” (“underextension”). “Doubleoverextension” (“double underextension”) is also anexperimentally established phenomenon, when the membershipFrontiers in Psychology | www.frontiersin.org 3 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype Theoryweight with respect to the conjunction (disjunction) of twoconcepts is higher (lower) than the membership weights withrespect to both constituent concepts (Hampton, 1988a,b).Furthermore, conceptual negation does not satisfy the rulesof classical Boolean logic (Hampton, 1997). More, Boniniet al. (1999), and Alxatib and Pelletier (2011), identified thepresence of “borderline contradictions,” directly connectedwith overextension, namely, a sentence like “John is tall andJohn is not tall” is estimated as true by a significant numberof participants, again violating basic rules of classical Booleanlogic. More generally, for each of these experimental data, asingle classical probability framework satisfying the axiomsof Kolmogorov does not exist (Aerts, 2009a,b; Aerts et al.,2013a,b, 2015a; Sozzo, 2014, 2015). To clarify the latter sentenceno single probability space can be constructed for an itemwhose membership weight with respect to the conjunction oftwo concepts is overextended with respect to both constituentconcepts.These problems—compositionality, the graded nature oftypicality, and the probabilistic nature of membership weights—present a serious challenge to any theory of concepts.We have developed a novel theoretical model of conceptsand their combinations (Gabora and Aerts, 2002; Aerts andGabora, 2005a,b), conjunction (Aerts, 2009a; Aerts et al.,2013a, 2015a; Sozzo, 2014, 2015), disjunction (Aerts, 2009a;Aerts et al., 2013a), conjunction and negation (Aerts et al.,2015a; Sozzo, 2015). It uses the mathematical formalism ofquantum theory in Hilbert space to represent data on conceptualcombinations, which has been successfully tested several times.This quantum-conceptual approach enables us to model theabove-mentioned deviations from classicality in terms of genuinequantum phenomena (contextuality, emergence, entanglement,interference, and superposition), thus capturing fundamentalaspects of how concepts combine. More importantly, we haverecently identified stronger deviations from classicality thanoverextension and underextension, which unveil, in our opinion,deep non-classical aspects of concept formation (Aerts et al.,2015c).The approach was inspired by similarity based theories, suchas prototype theory, in several respects:(i) a fundamentally probabilistic formalism is needed torepresent concepts and their dynamics;(ii) the typicality of different items with respect to a concept iscontext-dependent;(iii) features (or properties) of a concept vary in theirapplicability.A key insight underlying our approach is considering a conceptas, not a “container of instantiations” but, rather, an “entity in aspecific state,” which changes under the influence of a context. Inour quantum-conceptual approach, a context is mathematicallymodeled as quantum physics models of a measurement on aquantum particle. The (cognitive) context changes the state of aconcept in the way a measurement in quantum theory changesthe state of a quantum particle (Aerts and Gabora, 2005a,b).For example, in our modeling of the concept Pet, we consideredthe context e expressed by Did you see the type of pet he has?This explains that he is a weird person, and found that whenparticipants in an experiment were asked to rate different items ofPet, the scores for Snake and Spiderwere very high in this context.In this approach, this is explained by introducing different statesfor the concept Pet. We call “the state of Pet when no specificcontext is present” its ground state pˆ. The context e changesthe ground state pˆ into a new state pweird person pet . Typicalityhere is an observable semantic quantity, which means that ittakes different values in different states of the concept. As aconsequence, a substantial part of the typicality variations thatare encountered in the Guppy effect are due to, e.g., changesof state of the concept Pet under the influence of a context.More specifically, the typicality variations for the conjunctionPet-Fish are in great part similar to the typicality variations forPet under the context Fish (and also for Fish under the contextPet). Not only does context play a role in shaping the typicalityvariations for Pet-Fish, but also interference between Pet and Fishcontributes, as we will analyze in detail in Section 3.In general, whenever someone is asked to estimate thetypicality of Guppy with respect to the concept Pet in the absenceof any context, it is the typicality in the ground state pˆPet thatis obtained, and whenever the typicality of Guppy is estimatedwith respect to the concept Fish in the absence of any context,it is the typicality in the ground state pˆFish that is obtained. But,whenever someone is asked to estimate the typicality of Guppywith respect to the conjunction Pet-Fish, it is the typicality ina new ground state pˆPet−Fish that is obtained. This new groundstate pˆPet−Fish is different from pˆPet as well as from pˆFish. It is closebut not equal to the changed state of the ground state pˆPet underthe context eFish, and close but not equal to the changed state ofthe ground state pˆFish under the context ePet , the difference beingdue to interference taking place between Pet and Fish when theycombine into Pet-Fish (see Section 3). The “changes of state underthe influence of a context” and corresponding typicalities behavelike the changes of state and corresponding probabilities behavein quantum theory, giving rise to a violation of correspondingfuzzy set and/or classical probability rules. This partly explainsthe high typicality of Guppy in the conjunction Pet-Fish, and itsnormal typicality in Pet and Fish, and the reason why we identifythe Guppy effect as an effect at least partly due to context. Thereis also an interference effect, as we will see later.We developed this approach in a formal way, and called theunderlying mathematical structure a “State Context Property(SCoP) formalism” (Aerts and Gabora, 2005a). Let A denote aconcept. In SCoP, A is associated with a triple of sets, namelythe set 6 of states—we denote states by p, q, . . ., the set Mof contexts, we denote contexts by e, f , . . ., and the set L ofproperties—we denote properties by a, b, . . .. The “ground state”pˆ of the concept A is the state where A is not under the influenceof any particular context. Whenever A is under the influenceof a specific context e, a change of the state of A occurs. Incase A was in its ground state pˆ, the ground state changes toa state p. The difference between states pˆ and p is manifested,for example, by the typicality values of different items of theconcept, as we have seen in the case of the Guppy effect, andthe applicability values of different properties being different inthe two states pˆ and p. Hence, to complete the mathematicalFrontiers in Psychology | www.frontiersin.org 4 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype Theoryconstruction of SCoP, also two functions µ and ν are needed.The function µ : 6 × M × 6 −→ [0, 1] is defined suchthat µ(q, e, p) is the probability that state p of concept A underthe influence of context e changes to state q of concept A. Thefunction ν : 6 × L −→ [0, 1] is defined such that ν(p, a) is theweight, or normalization of applicability, of property a in statep of concept S. The function µ mainly accounts for typicalitymeasurements, the function ν mainly accounts for applicabilitymeasurements. Through these mathematical structures the SCoPformalism captures both “contextual typicality” and “contextualapplicability” (Aerts and Gabora, 2005a).A quantum representation in a complex Hilbert space of dataon Pet and Fish and different states of Pet and Fish in differentcontexts was developed (Aerts and Gabora, 2005a), as well as ofthe concept Pet-Fish (Aerts and Gabora, 2005b). Let us deepenthe connections between the quantum-theoretic approach toconcepts and prototype theory (see also Gabora et al., 2008).This approach can be interpreted in a rather straightforward wayas a generalization of prototype theory which mathematicallyintegrates context and formalizes its effects, unlike standardprototype theory. What we call the ground state of a conceptcan be seen as the prototype of this concept. The conceptualdistance of an item from the prototype can be reconstructedfrom the functions µ and ν in the SCoP formalism. Thus, aslong as individual concepts are considered and in the absence ofany context, prototype theory can be embodied into the SCoPformalism, and the prototype of a concept A can be representedas its ground state pˆA. However, any context will change thisground state into a new state. An important consequence ofthis is that when the concept is in this new state, the prototypechanges. An intuitive way of understanding this is to consider thisnew state a new “contextualized prototype.” More concretely, theconcept Pet, when combined with Fish in the conjunction Pet-Fish, has a new contextualized prototype, which could be called“Pet contextualized by Fish.” The new state can be thought ofas a “contextualized prototype.” Hence, this is a prototype-liketheory that is capable of mathematically describing the presenceand influence of context. From the point of view of conceptualdistance, this contextualized prototype will be close to, e.g.,Guppy.The interpretation of the SCoP formalism as a contextualprototype theory can be applied not just to conjunctions anddisjunctions of two concepts, but also to abstract categories suchas Fruits. It is very likely that the prototype of Fruits is close to,e.g., Apple, or Orange. But let us now consider the combinationTropical Fruits, that is, Fruits under the contextTropical. It is thenreasonable to maintain that the new contextualized prototype ofTropical Fruits is closer to, e.g., Pineapple, or Mango, than toApple, or Orange. The introduction of contextualized prototypeswithin the SCoP formalism enables us to incorporate abstractcategories as well as deviations of typicality from fuzzy setbehavior.Another interesting aspect of this approach to prototypetheory comes to light if we consider again the conceptualcombination Pet-Fish. It is reasonable that the prototypes ofPet and Fish—ground states pˆPet and pˆFish—interfere in Pet-Fishwhenever the typicality of an item, e.g., Guppy, is measuredwith respect to Pet-Fish. This sentence cannot, however, bemade more explicit in the absence of a concrete quantum-theoretic representation of typicality measurements of items withrespect to concepts and their combinations. Indeed, interferenceand superposition effects can be precisely formalized in suchquantum representation. This will be the content of Sections 3and 4.3. A HILBERT SPACE MODELING OFMEMBERSHIP MEASUREMENTSOne can gain insight into how people combine concepts bygathering data on “membership weights” and “typicalities.” Toobtain data on “typicalities,” participants are given a concept,and a list of instances or items, and asked to estimate theirtypicality on a Likert scale. In other experiments participants areasked to choose which instance they consider most typical of theconcept. Averages of these estimates or relative frequencies of thepicked items give rise to values representing the typicalities ofthe respective items. A membership weight is obtained by askingparticipants to estimate the membership of specific items withrespect to a concept. This estimation can be quantified using the7-point Likert scale and then converted into a relative frequency,and then into a probability called the “membership weight.”Hampton used membership weights instead of typicalities(Hampton, 1988b), because all you can do with typicalities isfuzzy set type calculations: the minimum rule of fuzzy sets forconjunction or the maximum rule for disjunction. This approachhas many serious shortcomings; indeed the Pet-Fish problemcould not be addressed by it. More serious failures are revealedby membership weight data. Hampton measured “membershipweights” and “degrees of non-membership or membership,”making these two measurements in one experiment. Morespecifically, Hampton’s experiment generates magnitude data,measuring the “degree of membership or non-membership”using a 7-point Likert scale providing −1, −2, −3 for degreesof non-membership, 1, 2, 3 for degrees of membership and, 0for borderline cases. From the same experiment membershipweight data are obtained, with 8 possible triplets [±,±,±] peritem. Each triplet indicating with a + whether the participantconsidered item k to be a member of the first category (A), thesecond category (B) and the third disjunction category (A orB),and with a − respectively otherwise. In the present Hilbert spacemodel we use the “degree of membership or non-membership”values obtained by Hampton, add +3 to them to make them allnon-negative, sum them, and divide each one by this sum. Sincethere are 24 items in total, in this way we get a set of 24 values inthe interval [0,1], that sum up to 1. We will use these values as asubstitute for membership collapse probabilities.Let us first explain how we arrive at the membership collapseprobabilities as a consequence of a measurement, and why wecan use the above-mentioned calculated values of “degree ofmembership or non-membership” as substitutes. Suppose thatinstead of using the data obtained by Hampton, we performedthe following experiment. For each pair of concepts and theircombination we ask the participant to select one and only oneFrontiers in Psychology | www.frontiersin.org 5 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype Theoryitem that they consider the best choice for membership. Thenwe calculate for each of the 24 items the relative frequency ofits appearance. These relative frequencies are 24 values in theinterval [0,1] summing up to 1, and their limits for increasingnumbers of participants represent the probabilities for eachitem to be chosen as the best member. These probabilities arewhat in a quantum model are called the “membership collapseprobabilities.” Of course, the above described experiment todetermine the membership collapse probabilities has not beenperformed. However, the values calculated from Hampton’smeasurement of “degree of membership or non-membership,”after renormalization as explained above, are expected tocorrelate with what these membership collapse probabilitieswould be if they were measured. This is why we use themas substitutes for the membership collapse probabilities in ourquantum model. As we will see when we construct the quantummodel, the exact values of the substitutes for the membershipcollapse probabilities are not critical. Thus, if we can model thesubstitutes for the membership collapse probabilities calculatedfrom Hampton’s data, we can also model the actual membershipcollapse probabilities (the data we would have if the experimenthad been done).So, we repeat, in Table 1, Hampton’s experimental data(Hampton, 1988b) have been converted into relative frequencies.The “degrees of non-membership and degrees of membership”give rise to µk(X) and now stand for the probability of conceptsFruits (X = A), Vegetables (X = B) and Fruits or Vegetables(X = “A or B”) to collapse to the item k, and thus add up to1, that is,24∑k= 1µk(A) =24∑k= 1µk(B) =24∑k= 1µk(A or B) = 1 (2)for the 24 items. The quantum model for concepts and theirdisjunction in complex Hilbert space is developed by buildingappropriate state vectors and projection operators for a givenontology of 24 items of two more abstract “container” concepts.In our model, the Hilbert space is a complex n-dimensionalCn, in which state vectors are n-dimensional complex numberedvectors. We use the “bra-ket” notation – respectively 〈·| and |·〉—for vector states (see the Appendix for further explanation). Thecomplex conjugate transpose of the |·〉 ket-vector (nx1 dim.) isthe 〈·| bra-vector (1xn dim.). Projectors and operators are thencombined as matrices |·〉〈·|, while scalars are obtained by innerproduct 〈·|·〉. We represent the measurement, consisting in thequestion “Is item k a good example of concept X?,” by means ofan orthogonal projection operatorMk. Each self-adjoint operatorin the Hilbert spaceH has a spectral decomposition on {Mk|k =1, . . . , 24}, where each Mk is the projector corresponding toitem k from the list of 24 items in Table 1. A priori we setno restrictions to the dimension of the complex Hilbert space,and thus neither to the projection space of the operators Mk.Each separate concept Fruits and Vegetables is now representedby its proper state vector |A〉 and |B〉 respectively, while theirdisjunction Fruits or Vegetables is realized by their equiponderoussuperposition 1√2(|A〉 + |B〉). It is precisely this feature of themodel—its ability to represent combined concepts as superposedstates—that provides the interferential composition of whatcould not be classically composed using sets.Following the standard rule of average outcome values ofquantum theory, the probabilities, µk(A), µk(B) and µk(A or B)are given by:µk(A) = 〈A|Mk|A〉 (3)µk(B) = 〈B|Mk|B〉 (4)µk(A or B) =〈A| + 〈B|√2Mk|A〉 + |B〉√2(5)After a straightforward calculation, the membership probabilityexpression µk(A or B) becomes:µk(A or B) =12(〈A|Mk|A〉 + 〈A|Mk|B〉 + 〈B|Mk|A〉+ 〈B|Mk|B〉)= 12(µk(A)+ µk(B))+ℜ〈A|Mk|B〉 (6)where ℜ takes the real part of 〈A|Mk|B〉. This expressionshows the contribution of the interference term ℜ〈A|Mk|B〉in µk(A or B) with respect to the “classical average” term12(µk(A)+ µk(B)). It consists of the real part of the complexprobability amplitude of the k-th item in Vegetables (concept |B〉)to be the one in Fruits (concept |A〉).The quantum concept model imposes the orthogonality of thestate vectors corresponding to different concepts. Therefore, wehave for the states of Fruits and Vegetables,〈A|B〉 = 0. (7)Each different item of the projector Mk also provides anorthogonal projection space. Since the conceptual disjunctionFruits or Vegetables spans a subspace of 2 dimensions in thecomplex Hilbert space (along the rays of |A〉 and |B〉), we set forththe possibility for a complex 2-dimensional subspace for eachitem. This brings the dimension of the complex Hilbert space to48. However, we will choose the unit vectors of these subspacesin such a way as to eliminate redundant dimensions wheneverpossible. Each category vector is built on orthogonal unit vectors,defined by the projection operators Mk. i.e., we define |ek〉 theunit vector on Mk|A〉, and define |fk〉 the unit vector on Mk|B〉.Thus, each item is now represented by a vector spanned by|ek〉 and |fk〉. Due the orthogonality of the projectors Mk, wehave〈ek|fl〉 = δklckeiγk (8)where the Kronecker δkl = 1 for same indices and zero otherwise,i.e., different item states are orthogonal as well. And ck expressesthe angle between the two unit vectors |ek〉 and |fk〉 of each2-dimensional subspace of item k. Notice that should some ckbe 1, then the required dimension of the complex Hilbert spacediminishes by 1, since the vectors |ek〉 and |fk〉 then coincide—a property that we will use to minimize the size of the requiredFrontiers in Psychology | www.frontiersin.org 6 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype TheoryTABLE 1 | Membership collapse probability values µk (X) of 24 items for the categories Fruits, Vegetables, and Fruits or Vegetables (Hampton, 1988b).A = FRUITS B = VEGETABLESk Item µk (A) µk (B) µk (A or B) λk λ-rank ǫk φk1 Almond 0.0359 0.0133 0.0269 0.0217 16 +1 84.0◦2 Acorn 0.0425 0.0108 0.0249 0.0214 17 −1 −94.5◦3 Peanut 0.0372 0.0220 0.0269 0.0285 10 −1 −95.4◦4 Olive 0.0586 0.0269 0.0415 0.0397 9 +1 91.9◦5 Coconut 0.0755 0.0125 0.0604 0.0260 12 +1 57.7◦6 Raisin 0.1026 0.0170 0.0555 0.0415 7 +1 95.9◦7 Elderberry 0.1138 0.0170 0.0480 0.0404 8 −1 −113.3◦8 Apple 0.1184 0.0155 0.0688 0.0428 5 +1 87.6◦9 Mustard 0.0149 0.0250 0.0146 0.0186 19 −1 −105.9◦10 Wheat 0.0136 0.0255 0.0165 0.0184 20 +1 99.3◦11 Root Ginger 0.0157 0.0323 0.0385 0.0172 22 +1 49.9◦12 Chili Pepper 0.0167 0.0446 0.0323 0.0272 11 −1 −86.4◦13 Garlic 0.0100 0.0301 0.0293 0.0146 23 −1 −57.6◦14 Mushroom 0.0140 0.0545 0.0604 0.0087 24 +1 18.5◦15 Watercress 0.0112 0.0658 0.0482 0.0253 13 −1 −69.1◦16 Lentils 0.0095 0.0713 0.0338 0.0252 14 +1 104.7◦17 Green Pepper 0.0324 0.0788 0.0506 0.0503 4 −1 −95.7◦18 Yam 0.0533 0.0724 0.0541 0.0615 3 +1 98.1◦19 Tomato 0.0881 0.0679 0.0688 0.0768 1 +1 98.5◦20 Pumpkin 0.0797 0.0713 0.0579 0.0733 2 −1 −103.5◦21 Broccoli 0.0143 0.1284 0.0642 0.0423 6 −1 −99.5◦22 Rice 0.0140 0.0412 0.0248 0.0238 15 −1 −96.7◦23 Parsley 0.0155 0.0266 0.0308 0.0178 21 −1 −61.1◦24 Black Pepper 0.0127 0.0294 0.0222 0.01929 18 +1 86.7◦Notice also the membership collapse probabilities for Mustard and Pumpkin still show the mark of double underextension of the disjunction. Membership collapse probability data withδµ ≈ 10−4 entail phase data δφ ≈ 2 · 10−1 and lambda data δλ ≈ 4 · 10−4.Hilbert space. Should ck be different from 1, then |ek〉 and |fk〉span a subspace of 2 dimensions. The state vectors |A〉 and |B〉of the concepts can then be expressed as a superposition of thevectors |ek〉 and |fk〉 for the items:|A〉 =24∑k= 1akeiαk |ek〉, |B〉 =24∑k= 1bkeiβk |fk〉 (9)where ak, bk, αk, βk ∈ R.We can express their inner product as follows:〈A|B〉 =(24∑k= 1ake−iαk〈ek|)(24∑l= 1bleiβl |fl〉)=24∑k= 1akbkckei(βk−αk+γk) =24∑k= 1akbkckeiφkwhere we have defined phase φk as φk: = βk − αk + γk in the laststep. The membership probabilities given in Equations (3 and 4)and the interference terms in Equation (6) can be expanded onthe projection spaces of the items:µk(A) =(24∑l= 1ale−iαl 〈el|)(akeiαk |ek〉) = a2k (10)µk(B) =(24∑l= 1ble−iβl〈fl|)(bkeiβk |fk〉) = b2k (11)〈A|Mk|B〉 =(24∑l= 1ale−iαl 〈el|)Mk|(24∑m= 1bmeiβm |fm〉)= akbkei(βk−αk)〈ek|fk〉 = akbkckeiφk (12)Notice that the phase of the k-th component of the conceptualdisjunction is not at play in the interference term 〈A|Mk|B〉(Equation 6). Taking the real part of the interference term inEquation (12), we can rewrite the membership probability of thedisjunction in Equation (6) as follows:µk(A or B) =12(µk(A)+ µk(B))+ ck√µk(A)µk(B) cosφk (13)Rearranging this equation we now choose φk must satisfyFrontiers in Psychology | www.frontiersin.org 7 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype Theorycosφk =µk(A or B)− 12 (µk(A)+ µk(B))ck√µk(A)µk(B)(14)Since all the membership probabilities on the right side arefixed, the only remaining free parameters are the coefficientsck. These parameters must now be tuned in order to satisfythe orthogonality of |A〉 and |B〉. Using the expansion on theunit vector sets {|ek〉}, {|fk〉} we obtain for their orthogonality(Equation 7):24∑k= 1ck√µ(A)kµ(B)k cosφk = 0, (15)24∑k= 1ck√µ(A)kµ(B)k sinφk = 0. (16)The “cosine sum” (Equation 15) is automatically satisfied due tothe definition of cosφk and the normalization of membershipprobabilities (Equation 2). This can be seen by substitutingthe expression of cosφk in Equation (14) and then applyingthe normalization condition of the membership probabilities(Equation 2). The “sine sum” equation still needs to be satisfied.With the defining relation (Equation 14) of φk, and sinφk =ǫk√1− cos2 φk, where ǫk = ± provides the sign, this becomes124∑k= 1ǫk√c2kµk(A)µk(B)− (µk(A or B)−12(µk(A)+ µk(B)))2= 0. (17)In order to satisfy this equation a simple algorithm was devised(Aerts, 2009a). For convenience of notation we denote the squareroot expression, with ck = 1, by a separate symbol:λk: =√µk(A)µk(B)− (µk(A or B)−12(µk(A)+ µk(B)))2.(18)First, we order the values λk from large to small and then assigna sign ǫk to each of them in such a way that each next partialsum (increasing index) remains smallest. The λ-ranking withcorresponding values have been tabulated in Table 1. We assignindex m to the item with the largest λ-value. In the present case,the item Tomato has the largest value, 0.07679.We now adopt an optimized complex Hilbert space for ourmodel in which ck = 1 (k 6= m), which reduces the spaceto 25 complex dimensions. We again note that all items exceptTomato receive a 1-dimensional complex subspace, whileTomatois represented by a 2-dimensional subspace. The “sine sum”equation in Equation (17) can be written as24∑k= 1,k 6=mǫkλk + ǫm√c2mµm(A)µm(B)− (µm(A or B)− 12 (µm(A)+ µm(B)))2= 0.(19)1The cosine value only defines the phase up to its absolute value |φk|. Thus, thesign of the sine value is undefined. If ǫk = −1, then φk = −|φk|.Next, we define the partial sum of the λk according a scheme ofsigns ǫk such that from large to small the next ǫkλk is added tomake the sum smaller but not negative.Sj =j∑size ordered λiǫiλi (20)Sj+1 = Sj − λj+1 and ǫj+1 = −1, if Sj − λj+1 ≥ 0 (21)= Sj + λj+1 and ǫj+1 = +1, if Sj − λj+1 < 0 (22)The first summand is thus λm, with ǫm = +1. Finally thisprocedure leads toS24 =24∑k= 1ǫkλk ≥ 0In the Fruits and Vegetables example with membershipprobability data in Table 1, this procedure gives:S24 = 0.0154 (23)In general the “sine sum” equation then becomesS24 − λm +√c2mµm(A)µm(B)− (µm(A or B)− 12 (µm(A)+ µm(B)))2= 0. (24)From which we can fix cm, the remaining ck not equal to 1:cm =√(S24 − λm)2 + (µm(A or B)− 12 (µm(A)+ µm(B)))2µm(A)µm(B)(25)In the present example we obtain the value cm = 0.8032. Wethus have fixed the inner product—or “angle”—of the vectors|em〉 and |fm〉, and can now write an explicit representation inthe canonical 25 dimensional complex Hilbert space C25. We cantakeMk(H) to be rays of dimension 1 for k 6= m, andMm(H) tobe a 2-dimensional plane spanned by the vectors |em〉 and |fm〉.We let the space C25 be spanned on the canonical base {1i},i ∈ [1 . . . 25]. All items k 6= m are represented by the respective1i. While for k = m we express the projections of |A〉 and |B〉 byMm(H) accordingly on 1m and 125ameiαm |em〉 = a˜meiαm1 1m + a˜25eiαm2 125 (26)bmeiβm |fm〉 = b˜meiβm1 1m + b˜25eiβm2 125 (27)with a˜m, b˜m, a˜25, b˜25 ∈ R to be specified. The parameters inthese expressions should satisfy the inner product (Equation 8)for k, l = mambm〈em|fm〉 = a˜mb˜me−i(αm1−βm1 ) + a˜25b˜25e−i(αm2−βm2 ),(28)= ambmcmei(γm−αm+βm) (29)and the probability weights for k = ma2m = a˜2m + a˜225, (30)b2m = b˜2m + b˜225. (31)Frontiers in Psychology | www.frontiersin.org 8 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype TheoryFinally, the representation of all vectors of the items can nowbe rendered explicit by simply choosing αk = γk = 0, andthus βk = φk, ∀k. A further simplification for Tomato is doneby setting a˜25 = 0, which also allows free choice of βm2 = 0.Then a˜m = am and b˜m = bmcm, and b˜25 = bm√(1− c2m).We have rendered explicit these membership probabilities andphases in Table 1. Thus we can write the vectors |A〉 and |B〉in C25 Hilbert space corresponding to the categories Fruits andVegetables respectively.|A〉 = (0.1895, 0.2062, 0.1929, 0.2421, 0.2748, 0.3203, 0.3373,0.3441, 0.1221, 0.1166, 0.1253, 0.1292, 0.1000, 0.1183,0.1058, 0.0975, 0.1800, 0.2309, 0.2968, 0.2823, 0.1196,0.1183, 0.1245, 0.1127, 0.0000) (32)|B〉 = (0.1153ei84.0◦ , 0.1039e−i94.5◦ , 0.1483e−i95.4◦ , 0.1640ei91.9◦ ,0.1118ei57.7◦, 0.1304ei95.9◦, 0.1304e−i113.3◦, 0.1245ei87.6◦,0.1581e−i105.9◦, 0.1597ei99.3◦, 0.1797ei49.9◦, 0.2112e−i86.4◦,0.1735e−i57.6◦, 0.2335ei18.5◦, 0.2565e−i69.1◦, 0.2670ei104.7◦,0.2807e−i95.7◦, 0.2691ei98.0◦, 0.2606ei96.8◦, 0.2670e−i103.5◦,0.3583e−i99.5◦, 0.2030e−i96.7◦, 0.1631e−i61.1◦, 0.1715ei86.7◦,0.1552). (33)This completes the quantum model for the membershipprobability of items with respect to Fruits, Vegetables andFruits or Vegetables. It captures the enigmatic aspects ofconceptual overextension and underextension identified inHampton (1988b), explaining them in terms of genuine quantumphenomena.Recalling the terminology adopted in Section 2, the unitvectors |A〉 and |B〉 in Equations (32) and (33) represent theground states of the concepts Fruits and Vegetables, respectively.Equivalently, these unit vectors represent the prototypes of theconcepts Fruits and Vegetables in prototype theory. The unitvector 1√2(|A〉 + |B〉) instead represents the “contextualizedprototype” obtained by combining the prototypes of Fruitsand Vegetables in the disjunction Fruits or Vegetables. If onenow looks at Equation (6), one sees that the prototypes Fruitsand Vegetables interfere in the disjunction Fruits or Vegetables,and the term ℜ〈A|Mk|B〉 in Equation (6) specifies how muchinterference is present when the membership probability of k ismeasured.4. AN ILLUSTRATION OF INTERFERINGPROTOTYPESIn this section we provide an illustration of contextual interferingprototypes. It is not a complete mathematical representation aspresented in Section 3 but, rather, an illustration that can helpthe reader with a non-technical background to have an intuitivepicture of what a contextual prototype is and how contextualprototypes interfere. Consider the concepts Fruits,Vegetables andtheir disjunction Fruits or Vegetables. The contextual prototype ofFruits can be represented by the x-axis of a plane surrounded by acloud containing items, features, etc.—all the contextual elementsconnected with the prototype of Fruits. Similarly, the contextualprototype of Vegetables can be represented by the y-axis of thesame plane surrounded by a cloud containing items, features,etc.—all the contextual elements connected with the prototypeof Vegetables. How can we represent the contextual prototype ofthe disjunction Fruits or Vegetables? Although as we have seenit cannot be represented in traditional fuzzy set theory, it can berepresented in terms of waves, with peaks and troughs. Indeed,waves can be summed up in such a way that peaks and troughs ofthe combined wave reproduce overextension and underextensionof the data. In other words, waves provide an intuitive geometricillustration of the interference taking place when contextualprototypes are combined in concept combination as discussedin Section 3. For example, let us demonstrate the interference ofthe item Almond when its membership probability with respectto the disjunction Fruits or Vegetables is calculated based onits membership probabilities for Fruits and for Vegetables. Themembership probabilities for the categories Fruits,Vegetables andFruits or Vegetables have been calculated from the Hampton’sdata and are reported in Table 1.The idea of an illustration would be to show that in additionto “fuzziness” (as modeled using a fuzzy set-theoretic approach)there is a “wave structure.” How can we graphically representthis “wave structure” of a concept? We start from the standardinterference formula of quantum theory, which is the following.For an arbitrary item k we haveµk(A or B) =12(µk(A)+ µk(B))+ ck√µk(A)µk(B) cosφk.(34)Now, we haveφk = βk − αk + γk (35)where αk is the phase angle connected with µk(A), βk the phaseangle connected to µk(B), and γk the phase angle connected to〈A|Mk|B〉. This has not yet been emphasized but if one analysesthe rest of the construction in Hilbert space, it is possible to seethat one can always choose γk = 0, which means that, with thischoice, φk becomes the difference in phases βk and αk.This is all we need to represent the “wave” nature of a conceptin a manner analogous to that of quantum theory. Indeed, it isthe “phase difference” between the waves—their phases being αkand βk respectively – that we attach to µ(A)k and µ(B)k. Theydetermine, together with the membership probabilities µ(A)kand µ(B)k the interference that gives rise to the measured datafor µ(A or B)k.The choice of the ck is such that only for the biggest value ofλk, which in this case of Tomato, the ck is chosen different from1. The only choice different from 1, for Tomato, still does notinfluence the fact that φk is the difference between βk and αk,when we decide to choose γk = 0. Let us consider for examplethe first item Almond of the list of 24 in Table 1. We haveµ(A)1 = 0.0359 (36)µ(B)1 = 0.0133 (37)µ(A or B)1 = 0.0269 (38)Frontiers in Psychology | www.frontiersin.org 9 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype TheoryThese are the data measured by Hampton, and also what existsfor the concepts Fruits, Vegetables and their combination Fruitsor Vegetables with respect to membership probability of the itemAlmond in the realm where fuzzy set probability appears. Theseare the values that do not fit into a model in this realm, and forwhich a wave-like realm underneath is necessary. Calculating theangle φ1 we getφ1 = 84.0◦ (39)(see Table 1). This angle is the result of a wave being presentunderneath the fuzzy, probability realm for µ(A)1 and µ(B)1,such that both waves give rise to a difference in phase—wherethe crests of one wave meet the troughs of the other—which isequal to β1 − α1, and is the value of φ1. This can be representedgraphically by attaching a wave pattern toµ(A)1 and another onetoµ(B)1, such that both have a phase difference of 84.0◦—see alsoFigure 1.Let us apply quantum theory to each of the items apart.Each item k has a Schrödinger wave function vibrating in theneighborhood of A, another one vibrating in the neighborhoodof B and a third vibrating in the neighborhood of “A or B,” andthey are related by superposition. We have:ψAk =√µk(A)eiαk (40)ψBk =√µk(B)eiβk (41)ψAorBk =√µk(A or B)eiδk (42)In each case, this gives us the membership probabilities. Squaring(multiplying by its complex conjugate), we have〈ψAk |ψAk 〉 = (ψAk )∗(ψAk ) =(√µk(A)eiαk)∗ (√µk(A)eiαk)=(√µk(A)e−iαk) (√µk(A)eiαk)=µk(A)ei(α−α)=µk(A) (43)〈ψBk |ψBk 〉 = (ψBk )∗(ψBk ) =(√µk(B)eiβk)∗ (√µk(B)eiβk)=(√µk(B)e−iβk) (√µk(B)eiβk)=µk(B)ei(β−β) = µk(B) (44)〈ψAorBk |ψAorBk 〉 = (ψAorBk )∗(ψAorBk )=(√µk(A or B)eiδk)∗ (√µk(A or B)eiδk)=(√µk(A or B)e−iδk) (√µk(A or B)eiδk)= µk(A or B)ei(δ−δ)= µk(A or B) (45)If we write the quantum superposition equation for each item weget1√2(ψAk + ψBk ) = ψAorBk (46)⇔ 1√2(√µ(A)keiαk +√µ(B)keiβk)=√µ(A or B)keiδk (47)where 1√2is a normalization factor. It is the squaring (i.e.,multiplying each with its complex conjugate) that gives rise tothe interference equation. Let us do this explicitly to see it.FIGURE 1 | Interference of items Almond, Acorn and Coconut in theconcept Fruits or Vegetables. Elementary oscillatory waves√µk (A) cos(x)and√µk (B) cos(x + φk ) are associated to the components of each given itemin Fruits and Vegetables respectively. The weight amplitude of the item in thedisjunction Fruits or Vegetables emerges at the origin of(√µk (A) cos(x)+√µk (B) cos(x + φk ))/√2.First we multiply the left hand side with its complex conjugate.We do the multiplication explicitly writing each step of it,to see well how the interference formula appears. Hence, wehave(1√2(√µk(A)eiαk +√µk(B)eiβk))∗ ( 1√2(√µk(A)eiαk+√µk(B)eiβk))Frontiers in Psychology | www.frontiersin.org 10 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype Theory=(1√2(√µk(A)e−iαk +√µk(B)e−iβk))( 1√2(√µk(A)eiαk+√µk(B)eiβk))= 12(√µk(A)e−iαk+√µk(B)e−iβk) (√µk(A)eiαk+√µk(B)eiβk)= 12(√µk(A)e−iαk ·√µk(A)eiαk +√µk(A)e−iαk ·√µk(B)eiβk+√µk(B)e−iβk ·√µk(A)eiαk +√µk(B)e−iβk ·√µk(B)eiβk)= 12(µk(A)ei(αk−αk) +√µk(A)µk(B)ei(βk−αk)+√µk(A)µk(B)e−i(βk−αk) + µk(B)ei(βk−βk))we use now that ei(αk−αk) = e0 = 1, ei(βk−βk) = e0 = 1,ei(βk−αk) = cos(βk − αk) + i sin(βk − αk) and e−i(βk−αk) =cos(βk − αk)− i sin(βk − αk), to get to the following= 12(µk(A)+√µk(A)µk(B)(cos(βk − αk)+ i sin(βk − αk))+√µk(A)µk(B)(cos(βk − αk)− i sin(βk − αk))+ µk(B))see that the term in i sin(βk − αk) cancels, to get= 12(µk(A)+ 2√µk(A)µk(B) cos(βk − αk)+ µk(B))= 12(µk(A)+ µk(B))+√µk(A)µk(B) cos(βk − αk) (48)Let is multiply now the right hand sight of Equation (46) with itscomplex conjugate. This gives=(√µk(A or B)eiδk)∗ (√µk(A or B)eiδk)= µk(A or B) (49)Hence, we get, as a consequence of squaring (Equation 46),exactly our interference formula12(µk(A)+ µk(B))+√µk(A)µk(B) cos(βk − αk) = µk(A or B)(50)Note that the difference in phase βk − αk between the wavesconnected with the item k and A and the item k and B is whatgenerates the interference. The new wave connected to the item kand A or B, of which the phase is δ is not influenced by it, is theamplitude of this new wave which is affected. This is the reasonthat interference is visible in the realm where the fuzzy natureappears, while it is provoked by the realm where the waves occur.We put forward this “wave nature” aspect of concepts notjust as an illustration, but to help the reader understand themanner in which such an underlying wave structure increasessubstantially the possible ways in which concepts can interact,as compared to the interaction possibilities in a modeling withfuzzy set structures. Of course the notion of a “wave” only addsclarification if we can imagine it to exist in some space-like realm.This is the case for the type of waves we all know from ourdaily physical environment, such as water waves, sound waves orlight waves. The quantum waves of physical quantum particlescan also be made visible in general by looking at probabilisticdetection patterns of these quantum particles on a physicalscreen, and noting the typical interference patterns when thewaves interact and the particles are detected on the screen. Onemight believe that an analogous situation is not possible forconcepts, because intuitively concepts, unlike quantum particles,do not exist “inside” space. If we look at things is an operationalway, however, an analysis can be put forward for the quantummodel of the combination of the two concepts, and the gradedstructure of collapse probability weights of the 24 items, whichdoes illustrate the presence of an interference pattern, and as aconsequence reveals the underlying wave structure of conceptsand their interactions. Let us explain how we can proceed toaccomplish such an analysis.We start by considering Figure 2. We see there the 24 differentitems of Table 1 represented by numbered spots in a plane wherea graded pattern, starting with the lightest region around thespot number 8, which is Apple, systematically becomes darker.Different numbers of items are situated in spots in regions ofdifferent darkness, for example, number 16, Lentils, is situatedin a spot in the darkest region. Let us explain how the figureis constructed. The “intensity of light” of a specific regioncorresponds to the “weights of the items” with respect to theconcept Fruits in Table 1. Looking at Table 1, it is indeed Apple,which has the highest weight, equal to 0.1184, and hence isrepresented by spot number 8 on Figure 2, in the lightest region.Next comes Elderberry with weight equal to 0.1134, representedby spot number 7 on Figure 2, on the border of the lightestand second lightest region. Next comes Raisin, with weight equalto 0.1026, represented by spot number 6 on Figure 2, on theborder of the third and the fourth lightest region. Next comesTomato, with weight equal to 0.0881, represented by spot number19 on Figure 2, in the seventh lightest region, etc. last is Lentils,with weight equal to 0.0095, represented by spot number 16 onFigure 2, in the one to darkest region. Hence Figure 2 contains arepresentation of the values of the collapse probability weights ofthe 24 items with respect to the concept Fruits. There is howevermore; we can, for example, wonder what the reason is to choosea representation in a plane? To explain this, turn to Figure 3. Letus first note with respect to the two figures, although it might notseem the case at first sight, all the numbered regions are locatedat exactly the same spots in both Figures 2, 3, with respect to thetwo orthogonal axes that coordinate the plane. What is differentin both figures are the graded structures of lighter to darkerregions, while they are centered around the spot number 8,representing the itemApple, in Figure 2 they are centered aroundthe spot number 21, representing the item Broccoli, in Figure 3.And, effectively, Figure 3 represents analogous to Figure 2 ofthe same 24 items, their collapse probability weights, but thistime with respect to the concept Vegetables. This explains whyin Figure 3 the lightest region is the one centered around spotnumber 21, representing Broccoli, while the lightest region inFigure 2 is the one centered around spot number 8, representingApple. Indeed, Broccoli is the most characteristic vegetable of theconsidered items, while Apple is the most characteristic fruit, if“characteristic” is measured by the size of the respective collapseFrontiers in Psychology | www.frontiersin.org 11 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype TheoryFIGURE 2 | The probabilities µ(A)k of a person choosing the item k as a “good example” of Fruits are fitted into a two-dimensional quantum wavefunction ψA(x, y). The numbers are placed at the locations of the different items with respect to the Gaussian probability distribution |ψA (x, y)|2. This can be seen as alight source shining through a hole centered on the origin, and regions where the different items are located. The brightness of the light source in a specific regioncorresponds to the probability that this item will be chosen as a “good example” of Fruits.FIGURE 3 | The probabilities µ(B)k of a person choosing the item k as a “good example” of Vegetables are fitted into a two-dimensional quantumwave function ψB(x, y). The numbers are placed at the locations of the different items with respect to the probability distribution |ψB (x, y)|2. As in Figure 2, it can beseen as a light source shining through a hole centered on point 21, where Broccoli is located. The brightness of the light source in a specific region corresponds to theprobability that this item will be chosen as a “good example” of Vegetables.probability, i.e., the probability to choose this item in the courseof the study. What might not seem obvious is that in a planeit is always possible to find 24 locations for the 24 items suchthat a graded structure with center Apple and a second gradedstructure with center Broccoli can be defined, fitting exactly alsothe other items in their correct value of “graded light to dark,”corresponding to the collapse probability weights in Table 1.Such a situation is what we show in Figures 2, 3. It can beprovenmathematically that a solution always exists, although nota unique one, which means that Figures 2, 3 show one of thesesolutions.We have chosen on purpose the graded structure form light todark to be colored yellow, because we can interpret Figures 2, 3such that an interesting analogy arises between our study ofthe 24 items and two concepts Fruits and Vegetables, andthe well-known double slit experiment with light in quantumFrontiers in Psychology | www.frontiersin.org 12 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype Theorymechanics. It is this analogy that will also directly illustrate the“wave nature” of concepts. Suppose we consider a plane figuringin the experiment as a detection screen, and put counters forquantum light particles, i.e., photons, at the numbered spotson the plane. Then we send light through a first slit, whichwe call the Fruits slit, which is placed in front of the screen.The slit is placed such that the counters in the spots detectnumbers of photons with fractions to the total number ofphotons send equal the collapse probability weights of the itemsrepresented by the respective spots with respect to the conceptFruits. The light received on the screen would then look likewhat is shown in Figure 2. Similarly, with counters placed inthe same spots, we send light through a second slit, whichwe call the Vegetable slit. Now the counters detect numbers ofphotons with fractions to the total number of photons equal tothe collapse probability weights of the same items with respect tothe concept Vegetables. The light received on the screen wouldthen look like what is shown in Figure 3. We can obtain the samefigures directly for our psychological study, consisting of eachparticipant choosing amongst the 24 items the one that he orshe finds most characteristic of Fruits andVegetables respectively.The relative frequencies of the first choice gives rise to the imagein Figure 2, while the relative frequencies of the second choicegives rise to the image in Figure 3, if, for example, we wouldmarkeach chosen item by a fixed number of yellow light pixels on acomputer screen.Before we combine the two slits to give rise to interference,let us specify the mathematics of the quantum mechanicalformalism that underlies the two Figures. The situationcan be represented quantum mechanically by complexvalued Schrödinger wave functions of two real variablesψA(x, y), ψB(x, y). For the light and the two slits, this situationis the “interaction of a photon with the two slits.” For thehuman participants in the concepts study, this situation is the“interaction with the two concepts of the mind of a participant.”We choose for ψA(x, y) and ψB(x, y) quantum wave packets,such that the radial part for both wave packets is a Gaussian intwo dimensions. Considering Figures 2, 3, we choose the top ofthe first Gaussian in the origin where spot number 8 is located,and the top of the second Gaussian in the point (a, b), wherespot number 21 is located. HenceψA(x, y) =√DAe−(x24σ2Ax+ y24σ2Ay)eiSA(x,y)ψB(x, y) =√DBe−((x−a)24σ2Bx+ (y−b)24σ2By)eiSB(x,y) (51)The phase parts of the wave packets eiSA(x,y) and eiSB(x,y) aredetermined by two phase fields SA(x, y) and SB(x, y) which willaccount for the interference and hence carry the wave nature.Of course, these phase parts vanish when we multiply each wavepacket with its complex conjugate to find the connection with thecollapse probabilities. Hence,|ψA(x, y)|2=DAe−(x22σ2Ax+ y22σ2Ay)|ψB(x, y)|2=DBe−((x−a)22σ2Bx+ (y−b)22σ2By)(52)are the Gaussians to be seen in Figures 2, 3, respectively. Let usdenote by 1k a small surface specifying the spot correspondingto the item number k in the plane of the two figures. We thencalculate the collapse probabilities of this item k with respectto the concepts Fruits and Vegetables in a standard quantummechanical way as followsµk(A) =∫1k|ψA(x, y)|2dxdy =∫1kDAe−(x22σ2Ax+ y22σ2Ay)dxdy(53)µk(B) =∫1k|ψB(x, y)|2dxdy =∫1kDBe−(x22σ2Bx+ y22σ2By)dxdy(54)We can prove that the parameters of the Gaussians,DA, σAx, σAy,DB, σBx, σBy can be determined in such away that the above equations come true, and for the images ofFigures 2, 3, exactly as we have done—using an approximationfor the integrals, which we explain later.If we open both slits it will be the normalized superpositionof the two wave packets that quantummechanically describes thenew situationψAorB(x, y) =1√2(ψA(x, y)+ ψB(x, y)) (55)We haveµk(A or B) =∫1kψAorB(x, y)∗ψAorB(x, y)dxdy= 12(∫1kψA(x, y)∗ψA(x, y)dxdy+∫1kψB(x, y)∗ψB(x, y)dxdy)+∫1kℜ(ψA(x, y)∗ψB(x, y))dxdy= 12(µk(A)+ µk(B))+∫1kℜ(ψA(x, y)∗ψB(x, y))dxdy (56)Let us calculate∫1kℜ(ψA(x, y)∗ψB(x, y))dxdy. We have∫1kℜ(ψA(x, y)∗ψB(x, y))dxdy=∫1k√DAe−(x24σ2Ax+ y24σ2Ay)√DBe−((x−a)24σ2Bx+ (y−b)24σ2By)ℜ(e−iSA(x,y)eiSB(x,y))dxdy=∫1k√DADBe−(x24σ2Ax+ (x−a)24σ2Bx+ y24σ2Ay+ (y−b)24σ2By)ℜ(ei(SB(x,y)−SA(x,y)))dxdyFrontiers in Psychology | www.frontiersin.org 13 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype Theory=∫1k√DADBe−( x24σ2Ax+ (x−a)24σ2Bx+ y24σ2Ay+ (y−b)24σ2By)cos(SB(x, y)− SA(x, y))dxdy (57)We can hence rewrite (Equation 56) in the following way∫1kf (x, y) cos θ(x, y)dxdy = fk (58)wheref (x, y) =√DADBe−(x24σ2Ax+ (x−a)24σ2Bx+ y24σ2Ay+ (y−b)24σ2By)(59)is a known Gaussian-like function, remember that we havedetermined DA, DB, σAx, σAy, σBx, σBy and a and b in choosinga solution to be seen in Figures 3, 4, andfk = µk(A or B)−12(µk(A)+ µk(B)) (60)are constants for each k determined by the data, and we haveintroducedθ(x, y) = SB(x, y)− SA(x, y) (61)the field of phase differences of the two quantum wave packets.This field of phases differences will determine the interferencepattern and it is the solution of the 24 nonlinear Equations in(58). This set of 24 equations cannot be solved exactly, but evena general numerical solution is not straightforwardly at reachwithin actual optimization programs. We have introduces twosteps of idealization to find a solution. First, we have looked fora solution where θ(x, y) is a large enough, polynomial in x and y,more specifically consisting of 24 independent sub-polynomialsthat are independentθ(x, y) = F1 + F2x+ F3y+ F4x2 + F5xy+ F6y2 + F7x3+ F8x2y+ F9xy2 + F10y3 + F11x4 + F12x3y+ F13x2y2 + F14xy3 + F15y4 + F16x5 + F17x4y+ F18x3y2 + F19x2y3 + F20xy4 + F21y5 + F22x6+ F23x5y+ F24x4y2 (62)Secondly, we suppose that 1k = 1 is a sufficiently smallsquare surface such that a good approximation of the integralin Equation (58)—and it is also the approximation we have usedfor the integrals (Equations 53 and 54)—is given by 1 times thevalue of the function under the integral in the center of 1. Thistransforms the set of 24 nonlinear (Equation 58) into a set of 24linear equations1f (xk, yk)θ(xk, yk) = fk (63)We have solved them for the points (xk, yk) where the 24 itemsare located in Figures 2, 3, for 1 = 0.01, which gives us θ(x, y),and hence also the expression for |ψAorB(x, y)|2 containing theexpected interference term, and we have|ψAorB(x, y)|2 =12(|ψA(x, y)|2 + |ψB(x, y)|2)+|ψA(x, y)ψB(x, y)| cos θ(x, y) (64)FIGURE 4 | The probabilities µ(A or B)k of a person choosing the item k as a “good example” of Fruits or Vegetables are fitted into thetwo-dimensional quantum wave function 1√2(ψA(x, y)+ ψB(x, y)), which is the normalized superposition of the wave functions in Figures 2, 3. Thenumbers are placed at the locations of the different exemplars with respect to the probability distribution |ψA (x, y)+ ψB (x, y)|2 = 12 (|ψA (x, y)|2 +|ψB (x, y)|2 )+|ψA (x, y)ψB (x, y)| cos θ (x, y), where θ (x, y) is the quantum phase difference at (x, y). The values of θ (x, y) are given in Table 1 for the locations of the different items. Theinterference pattern is clearly visible.Frontiers in Psychology | www.frontiersin.org 14 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype TheoryTABLE 2 | The parameters of the interference pattern solution illustrated in Figure 4.Parameters of the solutionk item (x, y) coordinates of items Sub-polynomial Coefficients Fk Gaussian Parameters1 Almond (−7.2826, 3.24347) 1 87.6039 DA 1.184122 Acorn (−7.3316, 2.3116) x 2792.02 σAx 5.653903 Peanut (−5.2957, 4.56032) y 8425.01 σAy 3.803604 Olive (−4.3776, 3.41765) x2 19.36 DB 1.284215 Coconut (−5.0322, 1.24573) xy −2139.87 σBx 8.208236 Raisin (−2.7149, 0.896651) y2 −7322.26 σBy 2.415787 Elderberry (−1.420, 0.487598) x3 −39.28118 Apple (0, 0) x2y −55.52639 Mustard (1.7978, 7.64549) xy2 586.67410 Wheat (2.4786, 7.73915) y3 2205.8111 Root Ginger (2.8164, 7.41004) x4 −2.2286812 Chili Pepper (3.9933, 7.03549) x3y 4.1940813 Garlic (4.7681, 7.81803) x2y2 13.357914 Mushroom (5.6281, 6.89107) xy3 −72.23315 Watercress (7.233, 6.67322) y4 −275.83416 Lentils (8.1373, 6.56281) x5 0.42673117 Green Pepper (3.8337, 5.55379) x4y 1.5876418 Yam (1.5305, 4.69497) x3y2 0.58253619 Tomato (2.4348, 2.42612) x2y3 −1.1316720 Pumpkin (3.9873, 2.06652) xy4 3.4400821 Broccoli (10, 4) y5 12.258422 Rice (11.6771, 0.392458) x6 −0.0094313223 Parsley (11.3949, −0.268463) x5y −0.053588124 Black Pepper (11.9389, −0.107151) x4y2 −0.200688The first column lists the different items, and the second column the coordinates of their locations in Figures 2–4. The third column contains the orthogonal set of sub-polynomials usedas first approximation for the phase field θ (x, y), and the fourth column their values. The fifth and sixth columns contain the Gaussian parameters and their values of the solution.In Figure 4 we have graphically represented this probabilitydensity |ψAorB(x, y)|2. The interference pattern shown inFigure 4 is very similar to well-known interference patterns oflight passing through an elastic material under stress. In ourcase, it is the interference pattern corresponding to “Fruits orVegetables” as a contextual, interfering prototype. The numericalvalues of the solutions represented in Figures 2–4 are in Table 2.We have thus completed our illustration of contextualinterfering prototypes. It is, however, important to rememberthat this representation is at the subtle level of an illustration,while the real working representation of contextual interferingprototypes needs the complete quantum-mechanical formalism.It can be considered as a pre-representation, exactly as thewave-like representations by de Broglie and Schrödinger in theearly days of quantum physics can be considered as useful pre-quantum representations that capture something of the waveaspects of microscopic particles.5. DISCUSSIONIn this paper we showed that a generalization of prototype theorycan address the “Pet-Fish problem” and related combinationissues. This was done by formalizing the effect of the cognitivecontext on the state of a concept using a SCoP formalism (Gaboraand Aerts, 2002; Aerts and Gabora, 2005a,b; Gabora et al.,2008). We also developed a quantum-theoretic model in complexHilbert space to show that, in this contextualized prototypetheory, prototypes can interfere when concepts combine, asevidenced by data where typicality measurements are performed.This could then lead one to think that the general quantumapproach to concepts only presupposes a (contextual) prototypetheory. We now explain why this inference is not true.Let us make more explicit the relationship between ourquantum-conceptual approach and other concept theories, suchas prototype theory, exemplar theory and theory theory. A deeperanalysis shows that our approach is more than a contextualgeneralization of prototype theory. Roughly speaking, othertheories make assumptions about the principles guiding theformation and intuitive representation of a concept in thehuman mind. Thus, prototype theory assumes that a conceptis determined by a set of characteristic rather than definingfeatures, the human mind has a privileged prototype for eachconcept, and typicality of a concrete item is determined by itssimilarity with the prototype (Rosch, 1973, 1978, 1983). Exemplartheory assumes instead that a concept is not determined by aFrontiers in Psychology | www.frontiersin.org 15 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype Theoryset of defining or characteristic features but, rather, by a set ofsalient instances of it stored in memory (Nosofsky, 1988, 1992).Theory theory assumes that concepts are determined by “mini-theories” or schemata, identifying the causal relationships amongproperties (Murphy and Medin, 1985; Rumelhart and Norman,1988). These theories have all mainly been preoccupied withthe question of “what predominantly determines a concept.” Weagree on the relevance of this question, though it is not themain issue focused on there. Transposed to our approach, thesetheories mainly investigate “what predominantly determines thestate of a concept.” Conversely, the main preoccupation ofour approach has been to propose a theory with the followingfeatures:(i) a well-defined ontology, i.e., a concept is in our approachan entity capable of different modes of being with respectto how it influences measurable semantic quantities such astypicality, membership weight and membership probability,and these modes are called “states”;(ii) the capacity to produce theoretical models fitting data onthese measurable semantic quantities.We seek to achieve (i) and (ii) independent of the question thatis the focus of other theories of concepts. More concretely, and inaccordance with the results of investigations into the question of“what predominantly determines a concept,” as far as prototypetheory, exemplar theory and theory theory are concerned, webelieve that all approaches are partially valid. The state of aconcept, i.e., its capability of influencing the values of measurablesemantic quantities, such as typicality and membership weight,is influenced by the set of its characteristic features, but alsoby salient exemplars in memory, and in a considerable numberof cases—where more causal aspects are at play—mini-theoriesmight be appropriate to express this state. It is important that“a conceptual state is defined and gives rise, together with thecontext, to the values of the measurable semantic quantities.” Thefact that the specification of these values can be only probabilisticis a confirmation that potentiality and uncertainty occur evenif the state is completely known, hence quantum structures areintrinsically needed.It follows from the above that resorting and giving newlife to prototype theory does not necessarily entail thatcontextual prototype theory is the only possible theory ofconcepts for what concerns the question of “what predominantlydetermines a concept.” However, we choose to identify ourgeneral approach as a “generalized contextual interferingprototype theory,” because the “ground state” of a concept isa fundamental notion of the theory, and this ground stateis what corresponds to the prototype. There is not a similaraffinity with exemplar theory and theory theory. However, theconceptual state and its interaction with the cognitive contextcan potentially capture the other conceptual aspects, exemplarsand schemata, which are instead predominant in alternativeconcept theories. In this respect, an interesting analogy mustbe emphasized. The quantum-theoretic approach only aimsat modeling concepts and their combinations in a unitaryand coherent mathematical formalism. We do not pretend togive a universal definition of what a concept is and how itforms. Using a known analogy in mathematics, we can say thatthe quantum-theoretic model is to a concept as a traditionalKolmogorov model is to a probability. A Kolmogorovian modelspecifies how a probability can be mathematically formalizedindependent of the definition of probability that is chosen(favorable over possible cases, large number limit of frequencies,subjective, etc.). Analogously, the quantum-theoretic frameworkfor concepts enables mathematical modeling of conceptualentities independent of the definition that is adopted in a specificconcept theory (prototype, exemplar, theory, etc.).We conclude with an epistemological consideration. Thequantum-theoretic framework presented here constitutes a steptoward the elaboration of a general theory for the representationof any conceptual entity. Hence, it is not just a “cognitive modelfor typicality, membership weight or membership probability.”Rather, we are investigating whether “quantum theory, in itsHilbert space formulation, is an appropriate theory to modelhuman cognition.” To understand what we mean by this let usconsider an example taken from everyday life. As an exampleof a theory, we could introduce the theory of “how to makegood clothes.” A tailor needs to learn how to make goodclothes for different types of people, men, women, children,old people, etc. Each cloth is a model in itself. Then, one canalso consider intermediate situations where one has models ofseries of clothes. A specific body will not fit in any clothes: youneed to adjust the parameters (length, size, etc.) to reach thedesired fit. We think that a theory should be able to reproducedifferent experimental results by suitably adjusting the involvedparameters, exactly as a theory of clothing. This is different froma set of models, even if the set can cope with a wide range ofdata.There is a tendency, mainly in empirically-based disciplines,to be critical with respect to a theory that can cope with allpossible situations it applies to. This is because the theorycontains too many parameters, which may lead one to thinkthat “any type of data can be modeled by allowing all theseparameters to have different values.” We agree that, in casewe have to do with an “ad-hoc model,” i.e., a model speciallymade for the circumstance of the situation it models, thissuspicion is grounded. Adding parameters to such an ad-hocmodel, or stretching the already contained parameters to othervalues, does not give rise to what we call a theory. On theother hand, a theory needs to be well defined, its rules, theallowed procedures, its theoretical, mathematical, and internallogical structure, “independent” of the structure of the modelsdescribing specific situations that can be coped with by thetheory. Hence also the theory needs to contain a well defineddescription of “how to produce models for specific situations.”Coming back to the theory of clothing, if a tailor knows the theoryof clothing, obviously he or she canmake a cloth for every humanbody, because the theory of clothing, although its structure isdefined independently of a specific cloth, contains a prescriptionof how to apply it to any possible specific cloth. In this respect,we think that one should carefully distinguish between a modelthat is derived by a general theory, as the one presented in thispaper, and a model specifically designed to test a number ofexperimental situations.Frontiers in Psychology | www.frontiersin.org 16 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype TheoryThis brings us to the important question of the “predictivepower” of existing quantum-theoretic models. Models derivedfrom a theory will generally need more data from a biggerset of experiments to become predictive for the outcomes ofother not yet performed experiments than this is the case formodels that are more ad-hoc. The reason is that in principlesuch models—think of the analogy we present with the theory ofclothing above—must be able to faithfully represent the data ofall possible experiments that can be performed on the conceptualentity in the same state. A tailor knowing the theory of clothingcan in principle make clothes for all human bodies but hencealso predicts outcomes of not performed experiments, e.g., themeasure of a specific part of the cloth, if enough data of aset of experiments are available to the tailor, e.g., data thatdetermine the possible types of clothes still fitting these dataand as a consequence also determine the measure of this partof the clothe. In general in quantum cognition, the scarcity ofdata is preventing models from having systematic and substantialpredictive power. One can wonder, if predictive power is notyet predominantly available in the majority of existing quantum-theoretic models, why so much attention and value is actuallyattributed to them? Answering this question allows us to clarifyan aspect of quantum cognition that is not obvious and evenmakes it special in a specific way, at least provisionally untilmore data is available. The success of quantum cognition is dueto it “being able to convincingly model data that theoreticallycan be proven to be impossible to model with any model thatrelies on classical fuzzy set theory and/or classical Kolmogorovianprobability theory.” Hence, a different criterion than predictivepower is provisionally used to identify the success of quantumcognition. Of course, as soon as more data are collected, themodels will also be able to be tested for their predictive power.Recent work in quantum cognition is starting to reach thelevel of being predictive, for example study of order effects(Wang et al., 2014), and an elaboration and refinement ofthe model presented in this article (Aerts et al., 2015a,c). Thelatter model simultaneously investigates the “conjuntion” andthe “negation” of concepts, starting from data collected on suchconceptual combinations. To explain the exact nature and alsoaccurateness of the predictive power we gained in the modelin Aerts et al. (2015a,c), consider the following mathematicalexpressionIABA′B′ = 1−µ(A andB)−µ(A andB′)−µ(A′ andB)−µ(A′ andB′)(65)where A and B are the concepts Fruits and Vegetables,respectively, while A′ and B′ are their negations. Thus, “A andB′” means Fruits and not Vegetables, while “A′ and B” meansnot Fruits and Vegetables and “A′ and B′” means not Fruits andnot Vegetables. In Aerts et al. (2015a,c) we published the datafor the outcomes of experiments that test the membership of thesame 24 items which we considered in the present article, but thistime not only for the conjunction of A and B, but also for theconjunctions “A and B′,” “A′ and B,” and “A′ and B′.” Supposethat the data follow a classical probabilistic structure, then IABA′B′has to be theoretically equal to zero for each considered item, andthis purely follows from a general “law of probability calculus”related to the so called “de Morgan laws” of classical probability.This means that, under the hypothesis of a classical probabilisticstructure, if we measure the relative frequencies of “A and B,”“A and B′” and “A′ and B,” and hence determine experimentallythe values of µ(A and B), µ(A and B′) and µ(A′ and B),a “prediction” for µ(A′ and B′) can be made theoretically,namely,µ(A′ and B′) = 1−µ(A and B)−µ(A and B′)−µ(A′ and B) (66)for each considered item. Let is explain what are our findingsin Aerts et al. (2015a,c) that make it possible for us to speakof some specific type of predictability for the more elaboratedand refined model we developed for the combination of conceptsand their negations. In Aerts et al. (2015a,c) we have collecteddata not only for the pair of concepts Fruits and Vegetablesand the 24 items treated also in the present article, but forthree more pairs of concepts, and for each of them again 24items. Due to the already identified non classical nature ofoverextension of the conjunction we expected that IABA′B′ wouldnot be equal to zero, and that indeed showed to be the case.However, we detected a high level of systematics of the valueof IABA′B′ fluctuating around an average of −0.81. A statisticalanalysis showed the different values for individual items to bepossible to be explained as fluctuations around this average(see Tables 1–4 in Aerts et al., 2015a). Next to the detailedstatistical analysis to be found in Aerts et al. (2015a) we also putforward a theoretical explanation of this value. The elaboratedand refined model for concept combinations developed in Aertset al. (2015a) introduces within the model the combination ofa pure quantum model and a classical model. It can be shownthat for a pure quantum model the value of IABA′B′ would be−1. We also find that the quantum effects are dominant ascompared to the classical effects in case concepts are combined,which explains why our refined model gives rise to a value ofIABA′B′ in between the classical one, which is 0, and the purequantum one, which is−1, but closer to the quantum one, hence−0.81. This finding can be turned into a predictive one in thefollowing way. Suppose we measure µ(A and B), µ(A and B′)and µ(A′ and B) for two arbitrary concepts and an item. Ourmodel allows us to put forward the following prediction forµ(A′ and B′)µ(A′ and B′) = 1.81−µ(A and B)−µ(A and B′)−µ(A′ and B)(67)By comparing Equations (66) and (67), we get that thequantum-theoretic model in Aerts et al. (2015a) providesa “different prediction” from a classical probabilistic modelsatisfying the axioms of Kolmogorov, and experimentsconfirm the validity of the former over the latter. Weadd that the quantum model has different predictionsfrom a classical model also for the values of otherfunctions than IABA′B′ , and these predictions are “parameterindependent,” in the sense that they do not depend onthe values of free parameters that may accommodate thedata.The results above can be considered as a strongconfirmation that quantum-theoretic models of conceptFrontiers in Psychology | www.frontiersin.org 17 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype Theorycombinations provide predictions that deviate, insome situations, from the predictions of classicalKolmogorovian models, which is confirmed by experimentaldata.AUTHOR CONTRIBUTIONSAll authors listed, have made substantial, direct and intellectualcontribution to the work, and approved it for publication.REFERENCESAerts, D. (1986). A possible explanation for the probabilities of quantummechanics. J. Math. Phys. 27, 202–210. doi: 10.1063/1.527362Aerts, D. (1999). Foundations of quantum physics: a general realisticand operational approach. Int. J. Theor. Phys. 38, 289–358. doi:10.1023/A:1026605829007Aerts, D. (2009a). Quantum structure in cognition. J. Math. Psychol. 53, 314–348.doi: 10.1016/j.jmp.2009.04.005Aerts, D. (2009b). Quantum particles as conceptual entities: a possible explanatoryframework for quantum theory. Found. Sci. 14, 361–411. doi: 10.1007/s10699-009-9166-yAerts, D., and Aerts, S. (1995). 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Cognition 64,189–206. doi: 10.1016/S0010-0277(97)00025-5Pitowsky, I. (1989).Quantum Probability, Quantum Logic. Lecture Notes in Physics.Vol. 321. Berlin: Springer.Pothos, E. M., and Busemeyer, J. R. (2013). Can quantum probability providea new direction for cognitive modeling? Behav. Brain Sci. 36, 255–274. doi:10.1017/S0140525X12001525Rips, L. J. (1995). The current status of research on concept combination. MindLang. 10, 72–104. doi: 10.1111/j.1468-0017.1995.tb00006.xFrontiers in Psychology | www.frontiersin.org 18 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype TheoryRosch, E. (1973). Natural categories. Cogn. Psychol. 4, 328–350. doi: 10.1016/0010-0285(73)90017-0Rosch, E. (1978). “Principles of categorization,” in Cognition and Categorization,eds E. Rosch and B. Lloyd (Hillsdale, NJ: Lawrence Erlbaum), 133–179.Rosch, E. (1983). “Prototype classification and logical classification: thetwo systems,” in New Trends in Conceptual Representation: Challengesto Piaget Theory?, ed E. Scholnick (Hillsdale, NJ: Lawrence Erlbaum),133–159.Rumelhart, D. E., and Norman, D. A. (1988). “Representation in memory,”in Stevens Handbook of Experimental Psychology, eds R. C. Atkinson, R. J.Hernsein, G. Lindzey, and R. L. Duncan (New York, NY: John Wiley & Sons),511–587.Sozzo, S. (2014). A quantum probability explanation in Fock space for borderlinecontradictions. J. Math. Psychol. 58, 1–12. doi: 10.1016/j.jmp.2013.11.001Sozzo, S. (2015). Conjunction and negation of natural concepts: a quantum-theoretic modeling. J. Math. Psychol. 66, 83–102. doi: 10.1016/j.jmp.2015.01.005Thagard, P., and Stewart, T. C. (2011). The AHA! experience: creativity throughemergent binding in neural networks. Cogn. Sci. 35, 1–33. doi: 10.1111/j.1551-6709.2010.01142.xVan Dantzig, S., Raffone, A., and Hommel, B. (2011). Acquiring contextualizedconcepts: a connectionist approach. Cogn. Sci. 35, 1162–1189. doi:10.1111/j.1551-6709.2011.01178.xWang, Z., Solloway, T., Shiffrin, R. M., and Busemeyer, J. R. (2014). Context effectsproduced by question orders reveal quantum nature of human judgments. Proc.Natl. Acad. Sci. U.S.A. 111, 9431–9436. doi: 10.1073/pnas.1407756111Zadeh, L. (1982). A note on prototype theory and fuzzy sets.Cognition 12, 291–297.doi: 10.1016/0010-0277(82)90036-1Conflict of Interest Statement: The authors declare that the research wasconducted in the absence of any commercial or financial relationships that couldbe construed as a potential conflict of interest.Copyright © 2016 Aerts, Broekaert, Gabora and Sozzo. This is an open-access articledistributed under the terms of the Creative Commons Attribution License (CC BY).The use, distribution or reproduction in other forums is permitted, provided theoriginal author(s) or licensor are credited and that the original publication in thisjournal is cited, in accordance with accepted academic practice. No use, distributionor reproduction is permitted which does not comply with these terms.Frontiers in Psychology | www.frontiersin.org 19 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype TheoryAPPENDIXA. Quantum Mathematics for ConceptualModelingWe illustrate in this section how the mathematical formalismof quantum theory can be applied to model situations outsidethe microscopic quantum world, more specifically, in therepresentation of concepts and their combinations. We will limittechnicalities to the essential.When the quantum mechanical formalism is applied formodeling purposes, each considered entity—in our case aconcept—is associated with a complex Hilbert spaceH, that is, avector space over the field C of complex numbers, equipped withan inner product 〈·|·〉 that maps two vectors 〈A| and |B〉 onto acomplex number 〈A|B〉. We denote vectors by using the bra-ketnotation introduced by Paul Adrien Dirac, one of the pioneers ofquantum theory. Vectors can be “kets,” denoted by |A〉, |B〉, or“bras,” denoted by 〈A|, 〈B|. The inner product between the ketvectors |A〉 and |B〉, or the bra-vectors 〈A| and 〈B|, is realized byjuxtaposing the bra vector 〈A| and the ket vector |B〉, and 〈A|B〉 isalso called a “bra-ket,” and it satisfies the following properties:(i) 〈A|A〉 ≥ 0;(ii) 〈A|B〉 = 〈B|A〉∗, where 〈B|A〉∗ is the complex conjugate of〈A|B〉;(iii) 〈A|(z|B〉 + t|C〉) = z〈A|B〉 + t〈A|C〉, for z, t ∈ C, where thesum vector z|B〉 + t|C〉 is called a “superposition” of vectors|B〉 and |C〉 in the quantum jargon.From (ii) and (iii) follows that inner product 〈·|·〉 is linear in theket and anti-linear in the bra, i.e., (z〈A| + t〈B|)|C〉 = z∗〈A|C〉 +t∗〈B|C〉.The “absolute value” of a complex number is defined as thesquare root of the product of this complex number times itscomplex conjugate, that is, |z| = √z∗z. Moreover, a complexnumber z can either be decomposed into its cartesian form z =x + iy, or into its polar form z = |z|eiθ = |z|(cos θ + i sin θ).As a consequence, we have |〈A|B〉| = √〈A|B〉〈B|A〉. We definethe “length” of a ket (bra) vector |A〉 (〈A|) as |||A〉|| = ||〈A||| =√〈A|A〉. A vector of unitary length is called a “unit vector.” Wesay that the ket vectors |A〉 and |B〉 are “orthogonal” and write|A〉 ⊥ |B〉 if 〈A|B〉 = 0.We have now introduced the necessary mathematicsto state the first modeling rule of quantum theory, asfollows.A.1. First Quantum Modeling RuleA state A of an entity—in our case a concept—modeled byquantum theory is represented by a ket vector |A〉 with length1, that is 〈A|A〉 = 1.An orthogonal projectionM is a linear operator on the Hilbertspace, that is, a mapping M : H → H, |A〉 7→ M|A〉 whichis Hermitian and idempotent. The latter means that, for every|A〉, |B〉 ∈ H and z, t ∈ C, we have:(i) M(z|A〉 + t|B〉) = zM|A〉 + tM|B〉 (linearity);(ii) 〈A|M|B〉 = 〈B|M|A〉∗ (hermiticity);(iii) M ·M = M (idempotency).The identity operator 1 maps each vector onto itself and isa trivial orthogonal projection. We say that two orthogonalprojections Mk and Ml are orthogonal operators if each vectorcontained in Mk(H) is orthogonal to each vector contained inMl(H), and we write Mk ⊥ Ml, in this case. The orthogonalityof the projection operators Mk and Ml can also be expressed byMkMl = 0, where 0 is the null operator. A set of orthogonalprojection operators {Mk |k = 1, . . . , n} is called a “spectralfamily” if all projectors are mutually orthogonal, that is,Mk ⊥ Mlfor k 6= l, and their sum is the identity, that is,∑nk= 1Mk = 1.The above definitions give us the necessary mathematics tostate the second modeling rule of quantum theory, as follows.A.2. Second Quantum Modeling RuleA measurable quantity Q of an entity—in our case a concept—modeled by quantum theory, and having a set of possible realvalues {q1, . . . , qn} is represented by a spectral family {Mk |k =1, . . . , n} in the following way. If the entity—in our case aconcept—is in a state represented by the vector |A〉, then theprobability of obtaining the value qk in a measurement of themeasurable quantity Q is 〈A|Mk|A〉 = ||Mk|A〉||2. This formulais called the “Born rule” in the quantum jargon. Moreover,if the value qk is actually obtained in the measurement, thenthe initial state is changed into a state represented by thevector|Ak〉 =Mk|A〉||Mk|A〉||(A1)This change of state is called “collapse” in the quantum jargon.The tensor productHA⊗HB of twoHilbert spacesHA andHBis the Hilbert space generated by the set {|Ai〉 ⊗ |Bj〉}, where |Ai〉and |Bj〉 are vectors ofHA andHB, respectively, whichmeans thata general vector of this tensor product is of the form∑ij |Ai〉 ⊗|Bj〉. This gives us the necessary mathematics to introduce thethird modeling rule.A.3. Third Quantum Modeling RuleA state C of a compound entity—in our case a combinedconcept—is represented by a unit vector |C〉 of the tensor productHA ⊗ HB of the two Hilbert spaces HA and HB containingthe vectors that represent the states of the component entities—concepts.The above means that we have |C〉 = ∑ij cij|Ai〉 ⊗ |Bj〉,where |Ai〉 and |Bj〉 are unit vectors of HA and HB, respectively,and∑i,j |cij|2 = 1. We say that the state C represented by|C〉 is a product state if it is of the form |A〉 ⊗ |B〉 for some|A〉 ∈ HA and |B〉 ∈ HB. Otherwise, C is called an “entangledstate.”The Fock space is a specific type of Hilbert space, originallyintroduced in quantum field theory. For most states of a quantumfield the number of identical quantum entities is not conservedbut is a variable quantity. The Fock space copes with thissituation in allowing its vectors to be superpositions of vectorspertaining to different sectors for fixed numbers of identicalquantum entities. MoreA explicitly, the k-th sector of a Aockspace describes a fixed number of k identical quantum entities,Frontiers in Psychology | www.frontiersin.org 20 March 2016 | Volume 7 | Article 418Aerts et al. Generalizing Prototype Theoryand it is of the form H ⊗ . . . ⊗ H of the tensor product of kidentical Hilbert spaces H. The Aock space A itself is the directsum of all these sectors, henceA = ⊕jk= 1 ⊗kl= 1 H (A2)Aor our modeling we have only used Aock space for the “two”and “one quantum entity” case, hence A = H ⊕ (H ⊗ H). Thisis due to considering only combinations of two concepts. Thesector H is called the “first sector,” while the sector H ⊗ H iscalled the “second sector.” A unit vector |F〉 ∈ F is then writtenas |F〉 = neiγ |C〉 + meiδ(|A〉 ⊗ |B〉), where |A〉, |B〉 and |C〉 areunit vectors ofH, and such that n2 +m2 = 1. For combinationsof j concepts, the general form of Fock space in Equation (A2)should be used.The quantum modeling above can be generalized by allowingstates to be represented by the so called “density operators”and measurements to be represented by the so called “positiveoperator valued measures.” However, for the sake of brevity wewill not dwell on this extension here.Frontiers in Psychology | www.frontiersin.org 21 March 2016 | Volume 7 | Article 418

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