Conjunction and Negation of Natural Concepts: A Quantum-theoretic Modeling
aa r X i v : . [ c s . A I] J un Conjunction and Negation of Natural Concepts: A Quantum-theoreticModeling
Sandro SozzoSchool of ManagementUniversity of LeicesterUniversity Road LE1 7RH Leicester, UKE-Mail: [email protected]
June 28, 2018
Abstract
We perform two experiments with the aim to investigate the effects of negation on the combinationof natural concepts. In the first experiment, we test the membership weights of a list of exemplars withrespect to two concepts, e.g.,
Fruits and
Vegetables , and their conjunction
Fruits And Vegetables . In thesecond experiment, we test the membership weights of the same list of exemplars with respect to thesame two concepts, but negating the second, e.g.,
Fruits and
Not Vegetables , and again their conjunction
Fruits And Not Vegetables . The collected data confirm existing results on conceptual combination,namely, they show dramatic deviations from the predictions of classical (fuzzy set) logic and probabilitytheory. More precisely, they exhibit conceptual vagueness, gradeness of membership, overextension anddouble overextension of membership weights with respect to the given conjunctions. Then, we show thatthe quantum probability model in Fock space recently elaborated to model Hampton’s data on conceptconjunction (Hampton, 1988a) and disjunction (Hampton, 1988b) faithfully accords with the collecteddata. Our quantum-theoretic modeling enables to describe these non-classical effects in terms of genuinequantum effects, namely ‘contextuality’, ‘superposition’, ‘interference’ and ‘emergence’. The obtainedresults confirm and strenghten the analysis in Aerts (2009a) and Sozzo (2014) on the identification ofquantum aspects in experiments on conceptual vagueness. Our results can be inserted within the generalresearch on the identification of quantum structures in cognitive and decision processes.
In the last years there has been a renewed interest in the formulation of a unified psychological theory forrepresenting and structuring concepts. Indeed, traditional approaches to concept theory, mainly, ‘prototypetheory’ (Rosch, 1973; Rosch, 1977; Rosch, 1983), ‘exemplar theory’ (Nosofsky, 1988; Nosofsky, 1992) and‘theory theory’ (Murphy & Medin, 1985; Rumelhart & Norman, 1988) are still facing a crucial difficulty,namely, ‘the problem of how modeling the combination of two or more natural concepts starting fromthe modeling of the component ones’. This ‘combination problem’ has been revealed by several cognitiveexperiments in the last thirty years. More precisely:(i) The ‘Guppy effect’ in concept conjunction, also known as the ‘Pet-Fish problem’ (Osherson & Smith,1981). If one measures the typicality of specific exemplars with respect to the concepts
Pet and
Fish andtheir conjunction
Pet-Fish , then one experimentally finds that an exemplar such as
Guppy is a very typicalexample of
Pet-Fish , while it is neither a very typical example of
Pet nor of
Fish .(ii) The deviation from classical (fuzzy) set-theoretic membership weights of exemplars with respectto pairs of concepts and their conjunction or disjunction (Hampton, 1988a,b). If one measures the mem-1ership weight of an exemplar with respect to a pair of concepts and their conjunction (disjunction), thenone experimentally finds that the membership weight of the exemplar with respect to the conjunction(disjunction) is greater (less) than the membership weight of the exemplar with respect to at least one ofthe component concepts.(iii) The so-called ‘borderline contradictions’ (Alxatib & Pelletier, 2011; Bonini, Osherson, Viale &Williamson, 1999). Roughly speaking, a borderline contradiction is a sentence of the form P ( x ) ∧ ¬ P ( x ),for a vague predicate P and a borderline case x , e.g., the sentence “Mark is rich and Mark is not rich”.If one accepts that concepts are ‘graded’, or ‘fuzzy’, notions (Osherson & Smith, 1982; Zadeh, 1965,1982), as empirical evidence seem to confirm, then one cannot represent the membership weights and typi-calities expressing such gradeness in a classical (fuzzy) set-theoretic model, where conceptual conjunctionsare represented logical conjunctions and conceptual disjunctions are represented by logical disjunctions.These difficulties affect both ‘extensional’ membership-based (Rips, 1995; Zadeh, 1982) and ‘intensional’attribute-based (Hampton, 1988b, 1997; Minsky, 1975). This combination problem is considered so seriousthat many authors maintain that not much progress is possible in the field if no light is shed on this problem(Fodor, 1994; Hampton, 1997; Kamp & Partee, 1995; Komatsu, 1992; Rips, 1995). However no mechanismand/or procedure has as yet been identified that gives rise to a satisfactory description or explanation ofthe effects appearing when concepts combine.Very similar effects and deviations from the predictions of traditional approaches have meanwhile beenexperienced in other domains of cognitive science, specifically, in behavioural economics (Ellsberg, 1961;Machina, 2009) and decision theory (Tversky & Kahneman, 1983; Tversky & Shafir, 1992). These andother difficulties have led various scholars to look for alternative approaches which could provide a moresatisfactory picture of ‘what occurs in human thought in a cognitive or decision process’. Among thepossible alternatives, a major candidate is what has been called ‘quantum cognition’ and it rests theapplication of the mathematical formalism of quantum theory in cognitive and social domains (see. e.g.,Aerts, 2009a,b; Aerts, Broekaert, Gabora & Sozzo, 2013; Aerts & Czachor, 2004; Aerts & Gabora, 2005a,b;Aerts, Gabora & Sozzo, 2013; Aerts & Sozzo, 2011, 2013; Aerts, Sozzo & Tapia, 2014; Busemeyer & Bruza,2012; Busemeyer, Pothos, Franco & Trueblood, 2011; Haven & Khrennikov, 2013; Khrennikov, 2010; Pothos& Busemeyer, 2009, 2013; van Rijsbergen, 2004; Wang, Busemeyer, Atmanspacher & Pothos, 2013).In this paper, we mainly deal with the quantum-theoretic approach to cognitive science elaborated inBrussels. This approach was motivated by a two decade research on the foundations of quantum theory(Aerts, 1999), the origins of quantum probability (Aerts, 1986; Pitowsky, 1989) and the identificationof typically quantum aspects in the macroscopic world (Aerts & Aerts, 1995; Aerts, Aerts, Broekaert& Gabora, 2000). A SCoP formalism was worked out within the Brussels approach which relies on theinterpretation of a concept as an ‘entity in a specific state changing under the influence of a context’ ratherthan as a ‘container of instantiations’ (Aerts & Gabora, 2005a,b), and allowed the authors to provide aquantum representation of the guppy effect (Aerts & Gabora, 2005a,b). Successively, the mathematicalformalism of quantum theory was employed to model the overextension and underextension of membershipweights measured by Hampton (1988a,b). More specifically, the overextension for conjunctions of conceptsmeasured by Hampton (1988a) was described as an effect of quantum interference, superposition andemergence (Aerts, 2009a; Aerts, Gabora & Sozzo, 2013), which also play a primary role in the descriptionof both overextension and underextension for disjunctions of concepts (Hampton, 1988b). Furthermore, aspecific conceptual combination experimentally revealed the presence of another genuine quantum effect,namely, entanglement (Aerts, 2009a,b; Aerts, Broekaert, Gabora & Sozzo, 2012; Aerts, Gabora & Sozzo,2012; Aerts & Sozzo, 2011). Finally, this quantum-theoretic framework was successfully applied to describeborderline vagueness (Sozzo, 2014).More specifically, in the present paper we generalize Aerts (2009a)’s analysis of Hampton’s overextensionfor the conjunction of two concepts, extending it to conjunctions and negations. Negative concepts have2een typically considered as ‘singular concepts’, since they do not have a prototype. Indeed, it is, forexample, easy to determine the membership of a concept such as
Not Fruit , but it does not seem that sucha determination involves similarity with some prototype of
Not Fruit . This is why one is naturally ledto derive the negation of a concept from (fuzzy set) logical operations on the positively defined concept.There has been very little research on how human beings interpret and combine negated concepts. Inthis respect, Hampton (1997) performed a set of experiments in which he considered both conjunctionsof the form
Tools Which Are Also Weapons and conjunctions of the form
Tools Which Are Not Weapons .As expected, his seminal work confirmed overextension in both conjunctions, also showing a violation ofBoolean classical logical rules for the negation. These results were the starting point for our research inthis paper, whose content can be summarized as follows.In Section 2 we describe the two experiments we performed. In the first experiment, we tested themembership weights of four different sets of exemplars with respect to four pairs (
A, B ) of concepts andtheir conjunction ‘ A and B ’. In the second experiment, we tested the membership weights of the same foursets of exemplars with respect to the same four pairs ( A, B ) of concepts, but negating the second concept,hence actually considering A , ‘not B ’ and the conjunction ‘ A and not B ’. We observe that, already atthis level, several exemplars exhibited overextension with respect to both ‘ A and B ’ and ‘ A and not B ’,hence we get a first clue that a deviation from classical (fuzzy set) logic and probability theory is at playin our experiments. A complete analysis of the ‘non-classicality’ underlying the collected data is presentedin Section 3 where we prove two theorems on the representability of a given set of experimental datain a classical Kolmogorovian probability space, thus extending the analysis in (Aerts, 2009a) to negatedconcepts. By applying these theorems, we show that a large part of our data cannot be modeled in aclassical Kolmogorovian space. Moreover, we notice that the deviations from classicality are of two types:(i) overextension of membership weights with respect to both conjunctions ‘ A and B ’ and ‘ A and not B ’,(ii) deviation of the negation ‘not B ’ of the concept B from the classical logical negation. This non-classical behaviour led us to inquire into the possibility of representing our data in a quantum-mechanicalframework. After a brief overview of the rules of a quantum-theoretic modeling in Section 4, we developthis modeling for the combinations ‘ A and B ’ and ‘ A and not B ’ in Section 5, thus extending the analysisin (Aerts, 2009a). Finally, we draw our conclusions in Section 6, where we:(i) prove that a large number of the collected data can be represented in our quantum-theoretic modelingin Fock space;(ii) describe the observed deviations from classicality as a consequence of genuine quantum effects, suchas, ‘contextuality’, ‘interference’, ‘superposition’ and ‘emergence’;(iii) provide a further support to the explanatory hypothesis we have recently put forward for theeffectiveness of a quantum approach in cognitive and decision processes. According to this hypothesis,human thought is the superposition of a ‘quantum emergent thought’ and a ‘quantum logical thought’,and that the quantum modeling approach applied in Fock space enables this approach to general humanthought, consisting of a superposition of these two modes, to be modeled.We observe, to conclude this section, that the results obtained in the present paper confirm those in(Aerts, 2009a) on conceptual conjunction/disjunction and in (Sozzo, 2014) on borderline vagueness. Hencethey can be considered as a further theoretical support towards the identification of quantum structuresin cognition. Hampton identified in his experiments systematic deviations from classical set (fuzzy set) conjunctionsand disjunctions (Hampton, 1988a,b). More explicitly, if the membership weight of an exemplar x withrespect to the conjunction ‘ A and B ’ of two concepts A and B is higher than the membership weight of3 with respect to one concept (both concepts), we say that the membership weight of x is ‘overextended’(‘double overextended’) with respect to the conjunction (by abuse of language, one usually says that x isoverextended with respect to the conjunction). If the membership weight of an exemplar x with respect tothe disjunction ‘ A or B ’ of two concepts A and B is less than the membership weight of x with respect toone concept, we say that the membership weight of x is ‘overextended’ with respect to the disjunction (byabuse of language, one usually says that x is overextended with respect to the disjunction). These were thenon-classical effects detected by Hampton in the combination of two concepts. Similar effects were identifiedby the same author in his experiments on conjunction and negation of two concepts (Hampton, 1997). Theanalysis by Aerts (2009a) evidenced other deviations from classicality in Hampton’s experiments. In thissection we show that very similar devations from classicality can be observed in our experiment on humansubjects. But we first need to describe the experiment.In our experiment, we considered four pairs of natural concepts, namely ( Home Furnishing , Furniture ),(
Spices , Herbs ), (
Pets , Farmyard Animals ) and (
Fruits , Vegetables ). For each pair, we considered 24exemplars and measured their membership with respect to these pairs of concepts and suitable conjunctionsof these pairs. The membership was estimated by using a ‘7-point scale’. The tested subjects were asked tochoose a number from the set { +3 , +2 , +1 , , − , − , − } , where the positive numbers +1, +2 and +3 meantthat they considered ‘the exemplar to be a member of the concept’ – +3 indicated a strong membership, +1a relatively weak membership. The negative numbers − − − − − Home Furnishing , Furniture ), we asked 80 subjects to estimate the membershipof the first set of 24 exemplars with respect to the concepts
Home Furnishing , Furniture and the negation
Not Furniture . Then, we asked 40 subjects to estimate the membership of the same set of 24 exemplarswith respect to the conjunctions
Home Furnishing And Furniture and
Home Furnishing And Not Furniture .Subsequently, we calculated the corresponding membership weights. The results are reported in Tables 1aand 1b .For the conceptual pair (
Spices , Herbs ), we asked 80 subjects to estimate the membership of the secondset of 24 exemplars with respect to the concepts
Spices , Herbs and the negation
Not Herbs . Then, we asked40 subjects to estimate the membership of the same set of 24 exemplars with respect to the conjunctions
Spices And Herbs and
Spices And Not Herbs . Subsequently, we calculated the corresponding membershipweights. The results are reported in Tables 2a and 2b.For the conceptual pair (
Pets , Farmyard Animals ), we asked 80 subjects to estimate the membershipof the third set of 24 exemplars with respect to the concepts
Pets , Farmyard Animals and the negation
Not Farmyard Animals . Then, we asked 40 subjects to estimate the membership of the same set of 24exemplars with respect to the conjunctions
Pets And Farmyard Animals and
Pets And Not FarmyardAnimals . Subsequently, we calculated the corresponding membership weights. The results are reported inTables 3a and 3b.For the conceptual pair (
Fruits , Vegetables ), we asked 80 subjects to estimate the membership of thefourth set of 24 exemplars with respect to the concepts
Fruits , Vegetables and the negation
Not Vegetables .Then, we asked 40 subjects to estimate the membership of the same set of 24 exemplars with respect tothe conjunctions
Fruits And Vegetables and
Fruits And Not Vegetables . Subsequently, we calculated thecorresponding membership weights. The results are reported in Tables 4a and 4b.Pure inspection of Tables 1-4 reveals that several exemplars present overextension with respect to bothconjunctions ‘ A and B ’ and ‘ A and not B ’. For example, the membership weight of Chili Pepper withrespect to
Spices is 0.975, with respect to
Herbs is 0.53125, while its membership weight with respect tothe conjunction
Spices And Herbs is 0.8 (Table 2.a), thus giving rise to overextension. Also, if we considerthe membership weights of
Goldfish with respect to
Pets and
Farmward Animals , we get 0.925 and 0.16875,4espectively, while its membership weight with respect to
Pets And Farmyard Animals is 0.425 (Table 3.a).Even stronger deviations in the combination
Fruits And Vegetables . For example, the exemplar
Broccoli scores 0.09375 with respect to
Fruits , 1 with respect to
Vegetables , and 0.5875 with respect to
Fruits AndVegetables . A similar pattern is observed for
Parsley , which scores 0.01875 with respect to
Fruits , 0.78125with respect to
Vegetables and 0.45 with respect to
Fruits And Vegetables (Tables 4.a).Overextension is also present when one concept is negated, that is, in the combination ‘ A and not B ’.Indeed, the membership weights of Shelves with respect to
Home Furnishing , Not Furniture and
HomeFurnishing And Not Furniture is 0,85, 0,125 and 0.3875, respectively (Table 1.b). Then,
Pepper scores0.99375 with respect to
Spices , 0.58125 with respect to
Not Herbs , and 0.9 with respect to
Spices and NotHerbs (Table 2.b). Finally,
Doberman Guard Dog scores 0.88125 and 0.26875 with respect to
Pets and
Farmyard Animals , respectively, while it scores 0.55 with respect to
Pets And Farmyard Animals (Table3b).Double overextension is also present in various cases and for both conjunctions ‘ A and B ’ and ‘ A and not B ’.For example, the membership weight of Olive with respect to
Fruits And Vegetables is 0.65, which is greaterthan both 0.53125 and 0.63125, i.e. the membership weights of
Olive with respect to
Fruits and
Vegetables ,respectively (Table 4.a). Furthermore,
Prize Bull scores 0.13125 with respect to
Pets and 0.2625 withrespect to
Not Farmyard Animals , but its membership weight with respect to
Pets And Not FarmayardAnimals is 0.275 (Table 3b).Our preliminary analysis above already shows that manifest deviations from classicality occur in theexperiment we performed. When we say ‘deviations from classicality’ we actually mean that the collecteddata behave in such a way that they cannot generally be modeled by using the usual connectives ofclassical fuzzy set logic for conceptual conjunctions, neither the rules of classical probability for theirmembership weights. In order to systematically identify such deviations from classicality we need howevera characterization of the representability of these data in a classical probability space. This is the contentof the next section.
We derive in this section necessary and sufficient conditions for the classicality of experimental data comingfrom the membership weights of two concepts A and B with respect to the conceptual negation ‘not B ’ andthe conjunctions ‘ A and B’ and ‘ A and not B’. More explicitly, we first derive the constraints that shouldbe satisfied by the membership weights µ x ( A ), µ x ( B ) and µ x ( A and B ) of the exemplar x with respect tothe concepts A , B and ‘ A and B’, respectively, in order to represent these data in a classical probabilitymodel satisfying the axioms of Kolmogorov. Then, we derive the constraints that should be satisfied by themembership weights µ x ( A ), µ x (not B ) and µ x ( A and not B ) of the exemplar x with respect to the concepts A , B ,‘not B ’ and ‘ A and not B’, respectively, in order to represent these data in a classical Kolmogorovianprobability model. We follow here mathematical procedures that are similar to those employed in Aerts(2009a) for the classicality of conceptual conjunctions and disjunctions. Let us start by clearly definingwhat we mean by the notion of ‘classical’, or ‘Kolmogorovian’, probability model.Let us start by the definition of a σ -algebra over a set. Definition 1. A σ -algebra over a set Ω is a non-empty collection σ (Ω) of subsets of Ω that is closedunder complementation and countable unions of its members. It is a Boolean algebra, completed to includecountably infinite operations. Definition 2. A measure P is a function defined on a σ -algebra σ (Ω) over a set Ω and taking values inthe extended interval [0 , ∞ ] such that the following three conditions are satisfied:(i) the empty set has measure zero;(ii) if E , E , E , . . . is a countable sequence of pairwise disjoint sets in σ (Ω) , the measure of theunion of all the E i is equal to the sum of the measures of each E i (countable additivity , or σ -additivity) ;(iii) the triple (Ω , σ (Ω) , P ) satisfying (i) and (ii) is then called a measure space, and the members of σ (Ω) are called measurable sets.A Kolmogorovian probability measure is a measure with total measure one. A Kolmogorovian probabilityspace (Ω , σ (Ω) , P ) is a measure space (Ω , σ (Ω) , P ) such that P is a Kolmogorovian probability. The threeconditions expressed in a mathematical way are: P ( ∅ ) = 0 P ( ∞ [ i =1 E i ) = ∞ X i =1 P ( E i ) P (Ω) = 1 (1)Let us now come to the possibility to represent a set of experimental data on two concepts and theirconjunction in a classical Kolmogorovian probability model. Definition 3.
We say that the membership weights µ x ( A ) , µ x ( B ) and µ x ( A and B ) of the exemplar x withrespect to the pair of concepts A and B and their conjunction ‘ A and B ’, respectively, can be represented ina classical Kolmogorovian probability model if there exists a Kolmogorovian probability space (Ω , σ (Ω) , P ) and events E A , E B ∈ σ (Ω) of the events algebra σ (Ω) such that P ( E A ) = µ x ( A ) P ( E B ) = µ x ( B ) and P ( E A ∩ E B ) = µ x ( A and B ) (2)We can prove a useful theorem on the representability of the membership weights with respect to twoconcepts and their conjunction. Theorem 1.
The membership weights µ x ( A ) , µ x ( B ) and µ x ( A and B ) of the exemplar x with respectto concepts A and B and their conjunction ‘ A and B ’, respectively, can be represented in a classicalKolmogorovian probability model if and only if they satisfy the following inequalities: ≤ µ x ( A and B ) ≤ µ x ( A ) ≤ ≤ µ x ( A and B ) ≤ µ x ( B ) ≤ µ x ( A ) + µ x ( B ) − µ x ( A and B ) ≤ Proof. If µ x ( A ) , µ x ( B ) and µ x ( A and B ) can be represented in a classical probability model, then thereexists a Kolmogorovian probability space (Ω , σ (Ω) , P ) and events E A , E B ∈ σ (Ω) such that P ( E A ) = µ x ( A ), P ( E B ) = µ x ( B ) and P ( E A ∩ E B ) = µ x ( A and B ). From the general properties of a Kolmogorovianprobability space it follows that we have 0 ≤ P ( E A ∩ E B ) ≤ P ( E A ) ≤ ≤ P ( E A ∩ E B ) ≤ P ( E B ) ≤
1, which proves that inequalities (3) and (4) are satisfied. From the same general properties of aKolmogorovian probability space it also follows that we have P ( E A ∪ E B ) = P ( E A ) + P ( E B ) − P ( E A ∩ E B ),and since P ( E A ∪ E B ) ≤ P ( E A ) + P ( E B ) − P ( E A ∩ E B ) ≤
1. This proves that inequality65) is satisfied. We have now proved that for the classical conjunction data µ x ( A ) , µ x ( B ) and µ x ( A and B )the three inequalities are satisfied.Now suppose that we have an exemplar x whose membership weights µ x ( A ) , µ x ( B ) , µ x ( A and B ) withrespect to the concepts A and B and their conjunction ‘ A and B ’ are such that inequalities (3), (4) and(5) are satisfied. We prove that, as a consequence, µ x ( A ) , µ x ( B ) and µ x ( A and B ) can be represented in aKolmogorovian probability model. To this end we explicitly construct a Kolmogorovian probability spacethat models these data. Consider the set Ω = { , , , } and σ (Ω) = P (Ω), the set of all subsets of Ω. Wedefine P ( { } ) = µ x ( A and B ) (6) P ( { } ) = µ x ( A ) − µ x ( A and B ) (7) P ( { } ) = µ x ( B ) − µ x ( A and B ) (8) P ( { } ) = 1 − µ x ( A ) − µ x ( B ) + µ x ( A and B ) (9)and further for an arbitrary subset S ⊆ { , , , } we define P ( S ) = X a ∈ S P ( { a } ) (10)Let us prove that P : σ (Ω) → [0 ,
1] is a probability measure. To this end we need to prove that P ( S ) ∈ [0 , S ⊆ Ω, and that the ‘sum formula’ for a probability measure is satisfied to complywith (1). The sum formula for a probability measure is satisfied because of definition (10). What remainsto be proved is that P ( S ) ∈ [0 ,
1] for an arbitrary subset S ⊆ Ω. P ( { } ) , P ( { } ) , P ( { } ) and P ( { } ) arecontained in [0 ,
1] as a direct consequence of inequalities (3), (4) and (5). Further, we have P ( { , } ) = µ x ( A ) , P ( { , } ) = µ x ( B ) , P ( { , } ) = 1 − µ x ( A ) , P ( { , } ) = 1 − µ x ( B ) , P ( { , , } ) = 1 − µ x ( A and B )and P ( { , , } ) = µ x ( A ) + µ x ( B ) − µ x ( A and B ), and all these are contained in [0 ,
1] as a consequenceof inequalities (3), (4) and (5). Consider P ( { , } ) = µ x ( A ) + µ x ( B ) − µ x ( A and B ). From inequality(5) it follows that µ x ( A ) + µ x ( B ) − µ x ( A and B ) ≤ µ x ( A ) + µ x ( B ) − µ x ( A and B ) ≤
1. Further, wehave, following inequalities (3) and (4), µ x ( A and B ) ≤ µ x ( A ) and µ x ( A and B ) ≤ µ x ( B ) and hence2 µ x ( A and B ) ≤ µ x ( A ) + µ x ( B ). From this it follows that 0 ≤ µ x ( A ) + µ x ( B ) − µ x ( A and B ). Hence wehave proved that P ( { , } ) = µ x ( A ) + µ x ( B ) − µ x ( A and B ) ∈ [0 , P ( { , } ) = 1 − µ x ( A ) − µ x ( B ) + 2 µ x ( A and B ) = 1 − P ( { , } ) and hence P ( { , } ) ∈ [0 , P ( { , , } ) = 1 − µ x ( B ) + µ x ( A and B ) = 1 − P ( { } ) ∈ [0 ,
1] and P ( { , , } ) = 1 − µ x ( A ) + µ x ( A and B ) = 1 − P ( { } ) ∈ [0 , P (Ω) = P ( { } ) + P ( { } ) + P ( { } ) + P ( { } ) = 1. We haveverified all subsets S ⊆ Ω, and hence proved that P is a probability measure. Since P ( { } ) = µ x ( A and B ), P ( { , } ) = µ x ( A ) and P ( { , } ) = µ x ( B ), we have modeled the data µ x ( A ), µ x ( B ) and µ x ( A and B ) bymeans of a Kolmogorovian probability space, and hence they are classical conjunction data.Inequalities (3) and (4) hold if and only if the quantity ∆ AB ( x ) = µ x ( A and B ) − min( µ x ( A ) , µ x ( B )) ≤ AB ( x ) is called the ‘conjunction minimum rule deviation’, since it expresses compatibilitywith the ‘minimum rule for the conjunction’ in fuzzy set theory. A situation where ∆ AB ( x ) > k AB ( x ) = 1 − µ x ( A ) − µ x ( B ) + µ x ( A and B ) is calledthe ‘Kolmogorovian conjunction factor’. Its violation is due to a non-classicality that is different from theone entailing the violation ∆ AB ( x ) (Aerts, 2009a). Finally, let us introduce the quantity Doub AB ( x ) =max( µ x ( A ) , µ x ( B )) − µ x ( A and B ). A situation where Doub AB ( x ) > AB ( x ), k AB ( x ) and Doub AB ( x ) for our experimentare reported in Tables 1a, 2a, 3a and 4a.Let us then come to the representability of a set of experimental data on a concept and its negation ina classical Kolmogorovian probability model. 7 efinition 4. We say that the membership weights µ x ( B ) and µ x (not B ) of the exemplar x with respectto the concept B and its negation ‘ not B ’, respectively, can be represented in a classical Kolmogorovianprobability model if there exists a Kolmogorovian probability space (Ω , σ (Ω) , P ) and an event E B ∈ σ (Ω) ofthe events algebra σ (Ω) such that P ( E B ) = µ x ( B ) P (Ω \ E B ) = µ x (not B ) (11)Analogously to the conjunction case, we can prove a useful theorem on the representability of the mem-bership weights with respect to a positive concept A , a negated concept ‘not B ’ and their conjunction‘ A and not B ’. Theorem 2.
The membership weights µ x ( A ) , µ x ( B ) , µ x (not B ) and µ x ( A and not B ) of the exemplar x with respect to the pair of concepts A , B , the negation ‘ not B ’ and the conjunction ‘ A and not B ’,respectively, can be represented in a classical Kolmogorovian probability model if and only if they satisfythe following inequalities: ≤ µ x ( A and not B ) ≤ µ x ( A ) ≤ ≤ µ x ( A and not B ) ≤ µ x (not B ) ≤ µ x ( A ) + µ x (not B ) − µ x ( A and not B ) ≤ − µ x ( B ) − µ x (not B ) = 0 (15) Proof. If µ x ( A ) , µ x (not B ) and µ x ( A and not B ) can be represented in a classical probability model,then there exists a Kolmogorovian probability space (Ω , σ (Ω) , P ) and events E A , E B ∈ σ (Ω) such that P ( E A ) = µ x ( A ), P (Ω \ E B ) = µ x (not B ) and P ( E A ∩ (Ω \ E B )) = µ x ( A and not B ). From the generalproperties of a Kolmogorovian probability space it follows that we have 0 ≤ P ( E A ∩ (Ω \ E B )) ≤ P ( E A ) ≤ ≤ P ( E A ∩ (Ω \ E B )) ≤ P (Ω \ E B ) ≤
1, which proves that inequalities (12) and (13) are satisfied.From the same general properties of a Kolmogorovian probability space it also follows that we have P ( E A ∪ (Ω \ E B )) = P ( E A ) + P (Ω \ E B ) − P ( E A ∩ (Ω \ E B )), and since P ( E A ∪ (Ω \ E B )) ≤ P ( E A ) + P (Ω \ E B ) − P ( E A ∩ (Ω \ E B )) ≤
1. This proves that inequality (14) is satisfied. Finally, wehave P ( E B ) + P (Ω \ E B ) = P (Ω) = 1, in a Kolmogorovian probability space, which proves that inequality(15) is satisfied. We have now proved that for the classical conjunction data µ x ( A ) , µ x ( B ), µ x (not B ) and µ x ( A and not B ) the three inequalities are satisfied.Now suppose that we have an exemplar x whose membership weights µ x ( A ) , µ x (not B ) , µ x ( A and not B )with respect to the concepts A and ‘not B ’ and their conjunction ‘ A and not B ’ are such that inequal-ities (12), (13), (14) and (15) are satisfied. We prove that, as a consequence, µ x ( A ) , µ x (not B ) and µ x ( A and not B ) can be represented in a Kolmogorovian probability model. To this end we explicitlyconstruct a Kolmogorovian probability space that models these data. Consider the set Ω = { , , , } and σ (Ω) = P (Ω), the set of all subsets of Ω. We define P ( { } ) = µ x ( A and not B ) (16) P ( { } ) = µ x ( A ) − µ x ( A and not B ) (17) P ( { } ) = µ x (not B ) − µ x ( A and not B ) (18) P ( { } ) = 1 − µ x ( A ) − µ x (not B ) + µ x ( A and not B ) (19)and further for an arbitrary subset S ⊆ { , , , } we define P ( S ) = X a ∈ S P ( { a } ) (20)8et us prove that P : σ (Ω) → [0 ,
1] is a probability measure. To this end we need to prove that P ( S ) ∈ [0 ,
1] for an arbitrary subset S ⊆ Ω, and that the ‘sum formula’ for a probability measure is satisfied tocomply with (1). The sum formula for a probability measure is satisfied because of definition (20). Whatremains to be proved is that P ( S ) ∈ [0 ,
1] for an arbitrary subset S ⊆ Ω. P ( { } ) , P ( { } ) , P ( { } ) and P ( { } ) are contained in [0 ,
1] as a direct consequence of inequalities (12), (13) and (14). Further, we have P ( { , } ) = µ x ( A ) , P ( { , } ) = µ x (not B ) , P ( { , } ) = 1 − µ x ( A ) , P ( { , } ) = 1 − µ x (not B ) = µ x ( B ),because of Equation (15), P ( { , , } ) = 1 − µ x ( A and not B ) and P ( { , , } ) = µ x ( A ) + µ x (not B ) − µ x ( A and not B ), and all these are contained in [0 ,
1] as a consequence of inequalities (12), (13) and(14). Consider P ( { , } ) = µ x ( A ) + µ x ( B ) − µ x ( A and not B ). From inequality (14) it follows that µ x ( A ) + µ x (not B ) − µ x ( A and not B ) ≤ µ x ( A ) + µ x (not B ) − µ x ( A and not B ) ≤
1. Further, we have,following inequalities (12) and (13), µ x ( A and not B ) ≤ µ x ( A ) and µ x ( A and not B ) ≤ µ x (not B ) and hence2 µ x ( A and not B ) ≤ µ x ( A )+ µ x (not B ). From this it follows that 0 ≤ µ x ( A )+ µ x (not B ) − µ x ( A and not B ).Hence we have proved that P ( { , } ) = µ x ( A ) + µ x (not B ) − µ x ( A and not B ) ∈ [0 , P ( { , } ) = 1 − µ x ( A ) − µ x (not B ) + 2 µ x ( A and not B ) = 1 − P ( { , } ) and hence P ( { , } ) ∈ [0 , P ( { , , } ) = 1 − µ x (not B ) + µ x ( A and not B ) = 1 − P ( { } ) ∈ [0 ,
1] and P ( { , , } ) =1 − µ x ( A ) + µ x ( A and not B ) = 1 − P ( { } ) ∈ [0 , P (Ω) = P ( { } ) + P ( { } ) + P ( { } ) + P ( { } ) = 1. We have verified all subsets S ⊆ Ω, and hence proved that P is a probability measure. Since P ( { } ) = µ x ( A and not B ), P ( { , } ) = µ x ( A ), P ( { , } ) = µ x (not B ), P ( { , } ) = µ x ( B ) and P (Ω \{ , } ) = µ x (not B ), we have modeled the data µ x ( A ), µ x ( B ) and µ x ( A and B )by means of a Kolmogorovian probability space, and hence they are classical conjunction data.Inequalities (12) and (13) hold if and only if the conjunction minimum rule deviation ∆ AB ′ ( x ) = µ x ( A and not B ) − min( µ x ( A ) , µ x (not B )) ≤
0. A situation where ∆ AB ′ ( x ) > k AB ′ ( x ) = 1 − µ x ( A ) − µ x (not B ) + µ x ( A and not B ) ≤ l BB ′ ( x ) = 1 − µ x ( B ) − µ x (not B ) = 0 complete the classicality of the conjunction ‘ A and not B ’. Then,let us introduce the quantity Doub AB ′ ( x ) = max( µ x ( A ) , µ x (not B )) − µ x ( A and not B ). A situation whereDoub AB ′ ( x ) > l BB ′ ( x ) = 1 − µ x ( B ) − µ x (not B ) inEquation (15) is a new parameter that must be introduced to represent the negation of a concept in termsof a classical set-theoretic complementation, namely,. A situation where l BB ′ ( x ) = 0 produces a type ofdeviation from classicality and it is due to conceptual negation. The values of the parameters ∆ AB ( x ), k AB ( x ), Doub AB ( x ) and l BB ′ ( x ) for our experiment are reported in Tables 1b, 2b, 3b and 4b.Let us now come back to our experiments. Theorems 1 and 2 are manifestly violated by severalexemplars with respect to both conjunctions ‘ A and B ’ and ‘ A and not B ’. It is however interesting toobserve that the conditions k AB ( x ) > k AB ′ ( x ) > A and B ’ and ‘ A and not B ’ and to a violation of l BB ′ = 0 in the negation not B . For example, theitem Prize Bull has ∆ AB ( x ) = 0 . > Pets And Farmyard Animals , hence it isstrongly overextended with respect to
Pets And Farmyard Animals , and it is even double overextendedwith Doub AB ′ ( x ) = − . < Pets and Not Farmyard Animals . The already mentioned
Broccoli and
Parsley are such that their ∆ AB ( x )s are equal to 0 . . Chili Pepper has ∆ AB ′ ( x ) = 0 . Broccoli has ∆ AB ′ ( x ) = 0 . Fruits And Not Vegetables , hence they are both highly overextended.It is finally interesting to observe that evident deviations from classicality are also due to conceptualnegation. Let us consider some cases. The exemplar
Rug has l BB ′ ( x ) = − . Furniture and its negation
Not Furniture , while
Wall Mirror has l BB ′ ( x ) = − . Sugar and
Chives with l BB ′ ( x ) = − . l BB ′ ( x ) = − . Herbs and
Not Herbs , and
Collie Dog with l BB ′ ( x ) = − . Farmyard Animals and
Not Farmyard Animals .9he results obtained in this section point to a systematic deviation of our experimental data fromthe rules of classical (fuzzy set) logic and probability theory. It is then worth to investigate whetherthe ‘non-classicalities’ identified here are of a quantum-type, and hence they can be described within themathematical formalism of quantum theory. To this end we need to preliminary summarize the essentialsof the quantum mathematics that is needed to employ this quantum formalism for modeling purposes.
We illustrate in this section how the mathematical formalism of quantum theory can be applied to modelsituations outside the microscopic quantum world, more specifically, in the representation of concepts andtheir combinations. We avoid in our presentation superfuous technicalities, but aim to be synthetic andrigorous at the same time.When the quantum mechanical formalism is applied for modeling purposes, each considered entity –in our case a concept – is associated with a complex Hilbert space H , that is, a vector space over thefield C of complex numbers, equipped with an inner product h·|·i that maps two vectors h A | and | B i ontoa complex number h A | B i . We denote vectors by using the bra-ket notation introduced by Paul AdrienDirac, one of the pioneers of quantum theory (Dirac, 1958). Vectors can be ‘kets’, denoted by | A i , | B i , or‘bras’, denoted by h A | , h B | . The inner product between the ket vectors | A i and | B i , or the bra-vectors h A | and h B | , is realized by juxtaposing the bra vector h A | and the ket vector | B i , and h A | B i is also called a‘bra-ket’, and it satisfies the following properties:(i) h A | A i ≥ h A | B i = h B | A i ∗ , where h B | A i ∗ is the complex conjugate of h A | B i ;(iii) h A | ( z | B i + t | C i ) = z h A | B i + t h A | C i , for z, t ∈ C , where the sum vector z | B i + t | C i is called a‘superposition’ of vectors | B i and | C i in the quantum jargon.From (ii) and (iii) follows that inner product h·|·i is linear in the ket and anti-linear in the bra, i.e.( z h A | + t h B | ) | C i = z ∗ h A | C i + t ∗ h B | C i .We recall that the ‘absolute value’ of a complex number is defined as the square root of the productof this complex number times its complex conjugate, that is, | z | = √ z ∗ z . Moreover, a complex number z can either be decomposed into its cartesian form z = x + iy , or into its goniometric form z = | z | e iθ = | z | (cos θ + i sin θ ). As a consequence, we have |h A | B i| = p h A | B ih B | A i . We define the ‘length’ of a ket(bra) vector | A i ( h A | ) as ||| A i|| = ||h A ||| = p h A | A i . A vector of unitary length is called a ‘unit vector’.We say that the ket vectors | A i and | B i are ‘orthogonal’ and write | A i ⊥ | B i if h A | B i = 0.We have now introduced the necessary mathematics to state the first modeling rule of quantum theory,as follows. First quantum modeling rule:
A state A of an entity – in our case a concept – modeled by quantum theoryis represented by a ket vector | A i with length 1, that is h A | A i = 1.An orthogonal projection M is a linear operator on the Hilbert space, that is, a mapping M : H →H , | A i 7→ M | A i which is Hermitian and idempotent. The latter means that, for every | A i , | B i ∈ H and z, t ∈ C , we have:(i) M ( z | A i + t | B i ) = zM | A i + tM | B i (linearity);(ii) h A | M | B i = h B | M | A i (hermiticity);(iii) M · M = M (idempotency).The identity operator maps each vector onto itself and is a trivial orthogonal projection. We saythat two orthogonal projections M k and M l are orthogonal operators if each vector contained in M k ( H ) isorthogonal to each vector contained in M l ( H ), and we write M k ⊥ M l , in this case. The orthogonality ofthe projection operators M k and M l can also be expressed by M k M l = 0, where 0 is the null operator. A10et of orthogonal projection operators { M k | k = 1 , . . . , n } is called a ‘spectral family’ if all projectors aremutually orthogonal, that is, M k ⊥ M l for k = l , and their sum is the identity, that is, P nk =1 M k = .The above definitions give us the necessary mathematics to state the second modeling rule of quantumtheory, as follows. Second quantum modeling rule:
A measurable quantity Q of an entity – in our case a concept – modeledby quantum theory, and having a set of possible real values { q , . . . , q n } is represented by a spectral family { M k | k = 1 , . . . , n } in the following way. If the entity – in our case a concept – is in a state represented bythe vector | A i , then the probability of obtaining the value q k in a measurement of the measurable quantity Q is h A | M k | A i = || M k | A i|| . This formula is called the ‘Born rule’ in the quantum jargon. Moreover, if thevalue q k is actually obtained in the measurement, then the initial state is changed into a state representedby the vector | A k i = M k | A i|| M k | A i|| (21)This change of state is called ‘collapse’ in the quantum jargon.The tensor product H A ⊗ H B of two Hilbert spaces H A and H B is the Hilbert space generated by the set {| A i i ⊗ | B j i} , where | A i i and | B j i are vectors of H A and H B , respectively, which means that a generalvector of this tensor product is of the form P ij | A i i ⊗ | B j i . This gives us the necessary mathematics tointroduce the third modeling rule. Third quantum modeling rule:
A state C of a compound entity – in our case a combined concept – isrepresented by a unit vector | C i of the tensor product H A ⊗ H B of the two Hilbert spaces H A and H B containing the vectors that represent the states of the component entities – concepts.The above means that we have | C i = P ij c ij | A i i ⊗ | B j i , where | A i i and | B j i are unit vectors of H A and H B , respectively, and P i,j | c ij | = 1. We say that the state C represented by | C i is a product state if it isof the form | A i ⊗ | B i for some | A i ∈ H A and | B i ∈ H B . Otherwise, C is called an ‘entangled state’.The Fock space is a specific type of Hilbert space, originally introduced in quantum field theory. Formost states of a quantum field the number of identical quantum entities is not conserved but is a variablequantity. The Fock space copes with this situation in allowing its vectors to be superpositions of vectorspertaining to different sectors for fixed numbers of identical quantum entities. More explicitly, the k -thsector of a Fock space describes a fixed number of k identical quantum entities, and it is of the form H ⊗ . . . ⊗ H of the tensor product of k identical Hilbert spaces H . The Fock space F itself is the directsum of all these sectors, hence F = ⊕ jk =1 ⊗ kl =1 H (22)For our modeling we have only used Fock space for the ‘two’ and ‘one quantum entity’ case, hence F = H ⊕ ( H ⊗ H ). This is due to considering only combinations of two concepts. The sector H is calledthe ‘sector 1’, while the sector H ⊗ H is called the ‘sector 2’. A unit vector | F i ∈ F is then written as | F i = ne iγ | C i + me iδ ( | A i ⊗ | B i ), where | A i , | B i and | C i are unit vectors of H , and such that n + m = 1.For combinations of j concepts, the general form of Fock space expressed in Equation (22) will have to beused.This quantum-theoretic modeling can be generalized by allowing states to be represented by the socalled ‘density operators’ and measurements to be represented by the so called ‘positive operator valuedmeasures’. However, our representation above is sufficient for attaining the results in this paper and wewill use it in the following sections. 11 A quantum model for the combination of two concepts
In this section, we put forward the quantum-theoretic framework that has been employed to model Hamp-ton’s (Hampton, 1988a,b) and Alxatib & Pelletier’s (Alxatib & Pelletier, 2011), applying it to our ex-periment reported in Section 2. We show that this framework, once specified for the given conceptualcombinations, i.e. ‘ A and B ’ and ‘ A and not B ’, enables a complete and successful modeling of thoseexperimental data collected in Section 2.Let us start from the disjunction ‘ A or B ’ of two concepts A and B . When the membership of theexemplar (item) x with respect to A is measured, we represent A by the unit vector | A d ( x ) i of a Hilbertspace H , and describe the decision measurement of a subject estimating whether x is a member of A bymeans of a dichotomic observable represented by the orthogonal projection operator M . The probability µ x ( A ) that x is chosen as a member of A , i.e. its membership weight, is given by the scalar product µ x ( A ) = h A d ( x ) | M | A d ( x ) i . Let A and B be two concepts, represented by the unit vectors | A d ( x ) i and | B d ( x ) i , respectively. To represent the concept ‘ A or B ’ we take the archetypical situation of the quantumdouble slit experiment, where | A i and | B i represent the states of a quantum particle in which only oneslit is open, √ ( | A i + | B i ) represents the state of the quantum particle in which both slits are open, and µ x ( A or B ) is the probability that the quantum particle is detected in a given region of a screen behind theslits. Thus, the concept ‘ A or B ’ is represented by the unit vector √ ( | A d ( x ) i + | B d ( x ) i ), and | A d ( x ) i and | B d ( x ) i are chosen to be orthogonal, that is, h A d ( x ) | B d ( x ) i = 0. The membership weights µ x ( A ) , µ x ( B )and µ x ( A or B ) of an exemplar x for the concepts A , B and ‘ A or B ’ are given by µ x ( A ) = h A d ( x ) | M | A d ( x ) i (23) µ x ( B ) = h B d ( x ) | M | B d ( x ) i (24) µ x ( A or B ) = 12 ( µ x ( A ) + µ x ( B )) + ℜh A d ( x ) | M | B d ( x ) i (25)repsectively, where ℜh A d ( x ) | M | B d ( x ) i is the real part of the complex number h A d ( x ) | M | B d ( x ) i . Thecomplex term ℜh A d ( x ) | M | B d ( x ) i is called ‘interference term’ in the quantum jargon, since it produces adeviation from the average ( µ x ( A ) + µ x ( B )) which would have been observed in the quantum double slitexperiment in absence of interference. We can see that, already at this stage, two genuine quantum effects,namely, superposition and interference, occur in the mechanism of combination of the concepts A and B .The quantum-theoretic modeling presented above correctly describes a large part of data in Hampton(1988b), but it cannot cope with quite some cases – in fact most of all the cases that behave more classicallythan the ones that are easily modeled by quantum interference. The reason is that, if one wants to reproduceHampton’s data within a quantum mathematics model which fully exploits the analogy with the quantumdouble slit experiment, one has to include the situation in which two identical quantum particles areconsidered, both particles passes through the slits, and the probability that at least one particle is detectedin the spot x is calculated. This probability is given by µ x ( A ) + µ x ( B ) − µ x ( A ) µ x ( B ) (Aerts, 2009a).Quantum field theory in Fock space allows one to complete the model, as follows.In quantum field theory, a quantum entity is described by a field which consists of superpositions ofdifferent configurations of many quantum particles (see Section 4). Thus, the quantum entity is associatedwith a Fock space F which is the direct sum ⊕ of different Hilbert spaces, each Hilbert space describinga defined number of quantum particles. In the simplest case, F = H ⊕ ( H ⊗ H ), where H is the Hilbertspace of a single quantum particle (sector 1 of F ) and H ⊗ H is the (tensor product) Hilbert space of twoidentical quantum particles (sector 2 of F ).Let us come back to our modeling for concept combinations. The normalized superposition √ ( | A d ( x ) i + | B d ( x ) i ) represents the state of the new emergent concept ‘ A or B ’ in sector 1 of the Fock space F . In sector2 of F , instead, the state of the concept ‘ A or B ’ is represented by the unit (product) vector | A d ( x ) i ⊗ B d ( x ) i . To describe the decision measurement in this sector, we first suppose that the subject considerstwo identical copies of the exemplar x , pondering on the membership of the first copy of x with respect to A ‘and’ the membership of the second copy of x with respect to B . The probability of getting ‘yes’ in bothcases is, by using quantum mechanical rules, ( h A d ( x ) |h B d ( x ) | ) | M ⊗ M | ( | A d ( x ) i ⊗ | B d ( x ) i ). The probabilityof getting at least a positive answer is instead 1 − ( h A d ( x ) |h B d ( x ) | ) | ( − M ) ⊗ ( − M ) | ( | A d ( x ) i ⊗ | B d ( x ) i ).Hence, the membership weight of the exemplar x with respect to the concept ‘ A or B ’ coincides in sector 2with the latter probability and can be written as 1 − ( h A d ( x ) |h B d ( x ) | ) | ( − M ) ⊗ ( − M ) | ( | A d ( x ) i⊗| B d ( x ) i ) = µ x ( A ) + µ x ( B ) − µ x ( A ) µ x ( B ) = ( h A d ( x ) |h B d ( x ) | ) | M ⊗ + ⊗ M − M ⊗ M | ( | A d ( x ) i ⊗ | B d ( x ) i ).Coming to the Fock space F = H ⊕ ( H ⊗ H ), the global initial state of the concepts is represented bythe unit vector | A or B ( x ) i = m d ( x ) e iλ d ( x ) | A d ( x ) i ⊗ | B d ( x ) i + n d ( x ) e iν d ( x ) √ | A d ( x ) i + | B d ( x ) i ) (26)where the real numbers m d ( x ) , n d ( x ) are such that 0 ≤ m d ( x ) , n d ( x ) and m d ( x ) + n d ( x ) = 1. The decisionmeasurement on the membership of the exemplar x with respect to the concept ‘ A or B ’ is represented bythe orthogonal projection operator M ⊕ ( M ⊗ + ⊗ M − M ⊗ M ), hence the membership weight of x with respect to ‘ A or B ’ is given by µ x ( A or B ) = h A or B ( x ) | M ⊕ ( M ⊗ + ⊗ M − M ⊗ M ) | A or B ( x ) i = m d ( x ) ( µ x ( A ) + µ x ( B ) − µ x ( A ) µ x ( B )) + n x ( x ) ( µ x ( A ) + µ x ( B )2 + ℜh A d ( x ) | M | B d ( x ) i )(27)The simplest Fock space that allows the modeling of ‘ A or B ’ is C ⊕ ( C ⊗ C ). Let us denote by | , , i , | , , i , | , , i the canonical basis of C . Then, let us set a x ( A ) = 1 − µ x ( A ) and b x ( B ) = 1 − µ x ( B )if µ x ( A ) + µ x ( B ) ≤ a x ( A ) = µ x ( A ) and b x ( B ) = µ x ( B ) if µ x ( A ) + µ x ( B ) >
1. In Aerts (2009a) andAerts, Gabora & Sozzo (2012) it has been proved that, independently of the value of µ x ( A ) + µ x ( B ), theinterference term ℜh A d ( x ) | M | B d ( x ) i is given by ℜh A d ( x ) | M | B d ( x ) i = p − a x ( A ) p − b x ( B ) cos θ d ( x ) (28)where θ d ( x ) is the ‘interference angle’. The unit vectors | A d ( x ) i and | B d ( x ) i are instead represented in thecanonical basis of C by | A d ( x ) i = (cid:16)p a x ( A ) , , p − a x ( A ) (cid:17) (29) | B d ( x ) i = e iθ d ( x ) (cid:16)s (1 − a x ( A ))(1 − b x ( B )) a x ( A ) , s a x ( A ) + b x ( B ) − a x ( A ) , − p − b x ( B ) (cid:17) if a x ( A ) = 0(30) | B d ( x ) i = e iθ d ( x ) (0 , ,
0) if a x ( A ) = 0 (31)and the interference angle satisfies the condition θ d ( x ) = arccos (cid:16) n d ( x ) (cid:16) µ x ( A or B ) − m d ( x ) (1 − µ x ( A ) − µ x ( B ) − µ x ( A ) µ x ( B )) (cid:17) − µ x ( A ) − µ x ( B ) p − a x ( A ) p − b x ( B ) (cid:17) (32)if a x ( A ) , b x ( B ) = 1. The angle θ d ( x ) is instead arbitrary if a x ( A ) = 1 or b x ( B ) = 1 (Aerts, 2009a; Aerts,Gabora & Sozzo, 2013).Let us now come to the representation for the conjunction ‘ A and B ’. Here, the decision measurementfor the membership weight of the exemplar x with respect to the concept ‘ A and B ’ is represented in the13ock space F = H ⊕ ( H ⊗ H ) by the orthogonal projection operator M ⊕ ( M ⊗ M ), while the membershipweight of x with respect to ‘ A and B ’ is given by µ x ( A and B ) = h A and B ( x ) | M ⊕ ( M ⊗ M ) | A and B ( x ) i = m c ( x ) µ x ( A ) µ x ( B ) + n c ( x ) ( µ x ( A ) + µ x ( B )2 + ℜh A c ( x ) | M | B c ( x ) i ) (33)where m c ( x ) , n c ( x ) are such that 0 ≤ m c ( x ) , n c ( x ) and m c ( x ) + n c ( x ) = 1. The unit vector | A and B ( x ) i is given by | A and B ( x ) i = m c ( x ) e iλ c ( x ) | A c ( x ) i ⊗ | B c ( x ) i + n c ( x ) e iν c ( x ) √ | A c ( x ) i + | B c ( x ) i ) (34)Also in this case, it has been proved that the interference term ℜh A c ( x ) | M | B c ( x ) i is given by ℜh A c ( x ) | M | B c ( x ) i = p − a x ( A ) p − a x ( B ) cos θ c ( x ) (35)in the Fock space C ⊕ ( C ⊗ C ), θ c ( x ) being the interference angle. The concepts A c ( x ) and B c ( x ) arerespectively represented in the canonical basis | , , i , | , , i , | , , i of C by the unit vectors | A c ( x ) i = (cid:16)p a x ( A ) , , p − a x ( A ) (cid:17) (36) | B c ( x ) i = e iθ c ( x ) (cid:16)s (1 − a x ( A ))(1 − b x ( B )) a x ( A ) , s a x ( A ) + b x ( B ) − a x ( A ) , − p − b x ( B ) (cid:17) if a x ( A ) = 0(37) | B c ( x ) i = e iθ c ( x ) (0 , ,
0) if a x ( A ) = 0 (38)The interference angle satisfies the condition θ c ( x ) = arccos (cid:16) n c ( x ) (cid:16) µ x ( A and B ) − m c ( x ) µ x ( A ) µ x ( B ) (cid:17) − µ x ( A ) − µ x ( B ) p − a x ( A ) p − b x ( B ) (cid:17) (39)if a x ( A ) , b x ( B ) = 1. The angle θ c ( x ) is instead arbitrary if a x ( A ) = 1 or b x ( B ) = 1 (Aerts, 2009a; Aerts,Gabora & Sozzo, 2013).Let us finally particularize Equations (33), (34) and (35) to the conjunctions ‘ A and B ’ and ‘ A and not B ’in Section 2. We have | A and B ( x ) i = m AB ( x ) e iλ AB ( x ) | A ( x ) i ⊗ | B ( x ) i + n AB ( x ) e iν AB ( x ) √ | A ( x ) i + | B ( x ) i ) (40) | A and not B ( x ) i = m AB ′ ( x ) e iλ AB ′ ( x ) | A ( x ) i ⊗ | not B ( x ) i + n AB ′ ( x ) e iν AB ′ ( x ) √ | A ( x ) i + | not B ( x ) i )(41)and µ x ( A and B ) = m AB ( x ) µ x ( A ) µ x ( B ) + n AB ( x ) ( µ x ( A ) + µ x ( B )2 + p − a x ( A ) p − b x ( B ) cos θ AB ( x )) (42) µ x ( A and not B ) = m AB ′ ( x ) µ x ( A ) µ x (not B ) + n AB ′ ( x ) ( µ x ( A ) + µ x (not B )2 + p − a x ( A ) p − b x (not B ) cos θ AB ′ ( x )) (43) The membership weight µ x ( A or B ) could have been calculated from the membership weight µ x ( A and B ) by observingthat the probability that a subject decides for the membership of the exemplar x with respect to the concept ‘ A or B ’ is 1minus the probability of decision against membership of x with respect to the concept ‘ A and B ’.
14n the Fock space C ⊕ ( C ⊗ C ).We have thus completed our quantum mathematics representation of the concepts A , B , the negation‘not B ’ and the conjunctions ‘ A and B ’ and ‘ A and not B ’ in Fock space. In the next section we will seehow this representation works for the experimental data in Section 2. Equations (27) and (33) in Section 5 contain the quantum probabilistic expressions allowing the modelingof a major part of Hampton’s data (1988a,b). Moreover, we have showed in Sozzo (2014) that Equation(33) can also model (Alxatib & Pelletier, 2012)’s data on borderline vagueness. We show in this sectionthat almost all the data collected in our experiments on ‘ A and B ’ and ‘ A and not B ’ can be modeled inthe same Fock space framework.Let us start from the conjunction ‘ A and B ’. Tables 5a, 6a, 7a and 8a report, for each exemplar x , thevalues of the interference angle θ AB ( x ) and the weights m AB ( x ) and n AB ( x ) which satisfy Equation (42),together with the representation of the unit vectors | A AB ( x ) i and | B AB ( x ) i in C satisfying Equations(36), (37) and (38). Let us consider the exemplar Olive which was double overextended in Section 2, sinceit scored a membership weight µ x ( A ) = 0 . Fruits , µ x ( B ) = 0 . Vegetables , and µ x ( A and B ) = 0 .
65 with respect to
Fruits And Vegetables . As we can see from Table 8a,
Olive can be modeled in the Fock space C ⊕ ( C ⊗ C ) with an interference angle θ AB ( x ) = 60 . ◦ , a weight m AB ( x ) = 0 . n AB ( x ) = 0 . Fruits and
Vegetables are represented by the unit vectors | A AB ( x ) i = (0 . , , . | B AB ( x ) i = e i . ◦ (0 . , . , − .
61) in C . An exemplar that in Section 2 had a big overextension with respect to Pets And Farmyard Animals was
Goldfish . Goldfish scored µ x ( A ) = 0 .
925 with respect to
Pets , µ x ( B ) = 0 . Farmyard Animals and µ x ( A and B ) = 0 .
425 with respect to
Pets And Farmyard Animals . It can bemodeled in C ⊕ ( C ⊗ C ) with θ AB ( x ) = 99 . ◦ , m AB ( x ) = 0 .
23 and n AB ( x ) = 0 .
77. The concept
Pets is represented by | A AB ( x ) i = (0 . , , . Farmyard Animals is represented by | B AB ( x ) i = e i . ◦ (0 . , . , − .
91) (Table 7a). Another non-classical exemplar was
Parsley with respectto
Fruits And Vegetables . In our Fock space representation, it is possible to model µ x ( A ) = 0 . µ x ( B ) = 0 . µ x ( A and B ) = 0 .
45 of
Parsley with θ AB ( x ) = 45 . ◦ , m AB ( x ) = 0 .
07 and n AB ( x ) =0 .
93. Hence, the decision process of a subject estimating whether
Parsley belongs to
Fruits , Vegetables and
Fruits And Vegetables occurs prevalently in sector 1 of the Fock space C ⊕ ( C ⊗ C ). The concepts Fruits and
Vegetables are represented by | A AB ( x ) i = (0 . , , .
14) and | B AB ( x ) i = e i . ◦ (0 . , . , − . Shelves had a membership weight of µ x ( A ) = 0 .
85 with respect to
Home Furnishing , µ x ( B ) = 0 . Furniture , and µ x ( A and B ) = 0 . Home FurnishingAnd Furniture . Shelves can be represented in C ⊕ ( C ⊗ C ) with θ AB ( x ) = 101 . ◦ , m AB ( x ) = 0 .
42 and n AB ( x ) = 0 .
58. The concept
Home Furnishing is represented by | A AB ( x ) i = (0 . , , .
39) and the conceptis represented by | B AB ( x ) i = e i . ◦ (0 . , . , − .
26) with respect to the exemplar
Shelves (Table 5a).Let us now come to the modeling of the conjunction ‘ A and not B ’. Tables 5b, 6b, 7b and 8b report, foreach exemplar x , the values of the interference angle θ AB ′ ( x ) and the weights m AB ′ ( x ) and n AB ′ ( x ) whichsatisfy Equation (43), together with the representation of the unit vectors | A AB ′ ( x ) i and | not B AB ′ ( x ) i in C satisfying Equations (36), (37) and (38). Let us start from the data that are classically very problematical.The exemplar Prize Bull was double overextended with respect to
Pets And Not Farmyard Animals , sinceit scored µ x ( A ) = 0 . Pets , µ x (not B ) = 0 . Not Farmyard Animals and µ x ( A and not B ) = 0 .
275 wit respect to
Pets And Not Faryard Animals . The exemplar
Prize Bull can be modeled in Fock space with an interference angle θ AB ′ ( x ) = 45 . ◦ and weights m AB ′ ( x ) = 0 . n AB ′ ( x ) = 0 .
82 for sector 1. The concepts
Pets and
Not FarmyardAnimals are represented by | A AB ′ ( x ) i = (0 . , , .
36) and | not B AB ′ ( x ) i = e i . ◦ (0 . , . , − .
51) withrespect to the exemplar
Prize Bull (Table 7b). The exemplar
Shelves scored a high overextension withrespect to
Home Furnishing And Not Furniture , since it gave µ x ( A ) = 0 . µ x (not B ) = 0 .
125 and µ x ( A and not B ) = 0 . θ AB ′ ( x ) = 87 . ◦ and weights m AB ′ ( x ) = 0 .
29 and n AB ′ ( x ) = 0 .
71. The concepts
Home Furnishing and
Not Furniture arerepresented by | A AB ′ ( x ) i = (0 . , , .
92) and | not B AB ′ ( x ) i = e i . ◦ (0 . , . , − .
35) with respect tothe exemplar
Shelves (Table 5b). A similar pattern can be observed for the exemplar
Doberman GuardDog which scored µ x ( A ) = 0 . µ x (not B ) = 0 . µ x ( A and not B ) = 0 .
55. Our quantummodel works for this exemplar with θ AB ′ ( x ) = 74 . ◦ and weights m AB ′ ( x ) = 0 .
25 and n AB ′ ( x ) =0 .
75. The concepts
Pets and
Not Farmyard Animals are represented by | A AB ′ ( x ) i = (0 . , , .
34) and | not B AB ′ ( x ) i = e i . ◦ (0 . , . , − .
86) with respect to the exemplar
Doberman Guard Dog (Table 7b).Also in this case, the ‘classical data’ can be modeled as well. For example, the exemplar
Yam scored µ x ( A ) =0 .
375 with respect to
Fruits , µ x (not B ) = 0 . Not Vegetables and µ x ( A and not B ) =0 . Fruits And Not Vegetables . Yam has an interference angle θ AB ′ ( x ) = 94 . ◦ andweights m AB ′ ( x ) = 0 .
64 and n AB ′ ( x ) = 0 .
36. This means that the decision process of a subject estimatingwhether
Yam belongs to
Fruits , Vegetables and
Fruits And Vegetables occurs prevalently in sector 2 of theFock space C ⊕ ( C ⊗ C ). The concept Fruits is represented by | A AB ′ ( x ) i = (0 . , , . NotVegetables is represented by | not B AB ′ ( x ) i = e i . ◦ (0 . , . , − .
66) in the Hilbert space C (Table 8b).Our analysis above, together with Tables 5-8, allow one to conclude that our quantum-mechanical modelin Fock space satisfactorily represents the majority of experimental data collected in our experiments onconcepts and their combinations, which are classically problematical, as we have observed in Section 3.Moreover, our quantum-theoretic framework describes the deviations of these data from classical (fuzzyset) logic and probability theory in terms of genuinely quantum effects. Indeed, a quantum probabilisticmodel is needed for the whole set of data, which entails the presence of ‘contextuality’ (Aerts, 1986). Also,both ‘superposition’ and ‘interference’ are manifestly present between concepts, both in the conjunction‘ A and B ’ and in the conjunction ‘ A and not B ’. And quantum field-theoretic notions, i.e. sector, Fockspace, tensor product (Section 4) are required to model our data. For what instead concerns ‘emergence’,‘emergent dynamics’ – which also strongly occurs – deserves a more detailed analysis and it is connectedwith our explanatory hypothesis we have recently provided to cope with such deviations from classicality incognitive and decision processes (Aerts, 2009a; Aerts, Gabora & Sozzo, 2013). We have indeed proposed amechanism that explains the effectiveness of a quantum-theoretic modeling. It is exactly the quantum effectof emergence which comes into play. More precisely, whenever a given subject is asked to estimate whethera given exemplar x belongs to the vague concepts A , B , ‘ A and B ’ (‘ A and not B ’), two mechanismsact simultaneously and in superposition in the subject’s thought. A ‘quantum logical thought’, which is aprobabilistic version of the classical logical reasoning, where the subject considers two copies of the exemplar x and estimates whether the first copy belongs to A and the second copy of x belongs to B (‘not B ’). Butalso a ‘quantum conceptual thought’ acts, where the subject estimates whether the exemplar x belongs tothe newly emergent concept ‘ A and B ’ (‘ A and not B ’). The place whether these superposed processescan be suitably structured is the Fock space. Sector 1 of Fock space hosts the latter process, while sector2 hosts the former, while the weights m AB ( x ) and n AB ( x ) ( m AB ′ ( x ) and n AB ′ ( x )) measure the amount of‘participation’ of sectors 2 and 1, respectively. But, what happens in human thought during a cognitivetest is a quantum superposition of both processes. As a consequence of this explanatory hypothesis, aneffect, a deviation, or a contradiction, are not failures of classical logical reasoning but, rather, they are amanifestation of the presence of a superposed thought, quantum logical and quantum emergent thought.It is important to remark, to conclude, that we did not inquire into the relationships between therepresentation of a concept A and that of ‘not A ’in the present article. We indeed observe that quantum16ogical rules should hold in Sector 2 of Fock space – this was implicitly assumed in the modeling of both theconjunction ‘ A and B ‘ and the disjunction ‘ A or B ’ in Section 5. Logical coherence would then lead us toassume that the representation of ‘not A ’ should be constructed from the representation of A by requiringthat quantum logical rules – the rules of quantum logical negation, in this case – are valid in Sector 2 ofFock space. We believe that this should be the case, but we also think that a complete analysis of thissituation is only possible if data on A , B , ‘not B ‘, ‘ A and B ’, ‘ A and not B ’, but also ‘not A ‘, ‘not A and B ’and ‘not A and not B ’ are simultaneously collected. We are presently working on the elaboration of thesedata and we plan to deal with this interesting aspect in a forthcoming paper (Aerts, Sozzo & Veloz, 2014). Acknowledgments.
The author is greatly indebted with Prof. Diederik Aerts for reading the manuscript and providing anumber of valuable remarks and suggestions.
References
Aerts, D. (1986). A possible explanation for the probabilities of quantum mechanics.
Journal of Mathe-matical Physics 27 , 202–210.Aerts, D. (1999). Foundations of quantum physics: A general realistic and operational approach.
Inter-national Journal of Theoretical Physics 38 , 289–358.Aerts, D. (2009a). Quantum structure in cognition.
Journal of Mathematical Psychology 53 , 314–348.Aerts, D. (2009b). Quantum particles as conceptual entities: A possible explanatory framework forquantum theory.
Foundations of Science 14 , 361–411.Aerts, D., & Aerts, S. (1995). Applications of quantum statistics in psychological studies of decisionprocesses.
Foundations of Science 1 , 85-97.Aerts, D., Aerts, S., Broekaert, J., & Gabora, L. (2000). The violation of Bell inequalities in themacroworld.
Foundations of Physics 30 . 1387–1414.Aerts, D., Broekaert, J., Gabora, L., & Sozzo, S. (2013). Quantum structure and human thought.Behavioral and Brain Sciences 36, 274–276.Aerts, D., & Czachor, M. (2004). Quantum aspects of semantic analysis and symbolic artificial intelligence.
Journal of Physics A: Mathematical and Theoretical 37 , L123–L132.Aerts, D., & Gabora, L. (2005a). A theory of concepts and their combinations I: The structure of thesets of contexts and properties.
Kybernetes 34 , 167–191.Aerts, D., & Gabora, L. (2005b). A theory of concepts and their combinations II: A Hilbert spacerepresentation.
Kybernetes 34 , 192–221.Aerts, D., Gabora, L., & S. Sozzo, S. (2013). Concepts and their dynamics: A quantum–theoretic modelingof human thought.
Topics in Cognitive Science 5 , 737–772.Aerts, D., & Sozzo, S. (2011). Quantum structure in cognition. Why and how concepts are entangled.
Quantum Interaction. Lecture Notes in Computer Science 7052 , 116–127.17erts, D., & Sozzo, S. (2013). Quantum entanglement in conceptual combinations.
International Journalof Theoretical Physics . DOI 10.1007/s10773-013-1946-z (in print).Aerts, D., Sozzo, S., & Tapia, J. (2014). Identifying quantum structures in the Ellsberg paradox.
Inter-national Journal of Theoretical Physics . DOI DOI 10.1007/s10773-014-2086-9 (in print).Aerts, D., Sozzo, S., & Veloz, T. (2014). Negation of natural concepts and the foundations of humanreasoning (in preparation).Alxatib, S., & Pelletier, J. (2011). On the psychology of truth gaps. In Nouwen, R., van Rooij, R., Sauer-land, U., & Schmitz, H.-C. (Eds.),
Vagueness in Communication (pp. 13–36). Berlin, Heidelberg:Springer-Verlag.Bonini, N., Osherson, D., Viale, R., & Williamson, T. (1999). On the psychology of vague predicates.Mind and Language, 14, 377–393.Busemeyer, J. R., & Bruza, P. D. (2012).
Quantum Models of Cognition and Decision . Cambridge:Cambridge University Press.Busemeyer, J. R., Pothos, E. M., Franco, R., & Trueblood, J. S. (2011). A quantum theoretical explanationfor probability judgment errors.
Psychological Review 118 , 193–218.Dirac, P. A. M. (1958).
Quantum mechanics , 4th ed. London: Oxford University Press.Ellsberg, D.(1961). Risk, ambiguity, and the Savage axioms.
Quarterly Journal of Economics 75 . 643-669.Fodor, J. (1994) Concepts: A potboiler.
Cognition 50 , 95–113.Hampton, J. A. (1988a). Overextension of conjunctive concepts: Evidence for a unitary model for concepttypicality and class inclusion.
Journal of Experimental Psychology: Learning, Memory, and Cognition14 , 12–32.Hampton, J. A. (1988b). Disjunction of natural concepts.
Memory & Cognition 16 , 579–591.Hampton, J. A. (1997). Conceptual combination: Conjunction and negation of natural concepts.
Memory& Cognition 25 , 888–909.Haven, E., & Khrennikov, A. Y. (2013).
Quantum Social Science . Cambridge: Cambridge UniversityPress.Kamp, H., & Partee, B. (1995). Prototype theory and compositionality.
Cognition 57 , 129–191.Komatsu, L. K. (1992). Recent views on conceptual structure.
Psychological Bulletin 112 , 500–526.Khrennikov, A. Y. (2010).
Ubiquitous Quantum Structure . Berlin: Springer.Kolmogorov, A. N. (1933).
Grundbegriffe der Wahrscheinlichkeitrechnung , Ergebnisse Der Mathematik;translated as
Foundations of Probability . New York: Chelsea Publishing Company, 1950.Machina, M. J. (2009). Risk, ambiguity, and the darkdependence axioms.
American Economical Review99 . 385-392.Minsky, M. (1975). A framework for representing knowledge. In P. H. Winston (Ed.)
The Psychology ofComputer Vision (211–277). New York: McGraw-Hill.18urphy, G. L., & Medin, D. L. (1985). The role of theories in conceptual coherence.
Psychological Review92 , 289-16.Nosofsky, R. (1988). Exemplar-based accounts of relations between classification, recognition, and typi-cality.
Journal of Experimental Psychology: Learning, Memory, and Cognition 14 , 700708.Nosofsky, R. (1992). Exemplars, prototypes, and similarity rules. In Healy, A., Kosslyn, S., & Shiffrin, R.(Eds.),
From learning theory to connectionist theory: Essays in honor of William K. Estes . HillsdaleNJ: Erlbaum.Osherson, D., & Smith, E. (1981). On the adequacy of prototype theory as a theory of concepts.
Cognition9 , 35–58.Osherson, D. N., Smith, E. (1982). Gradedness and Conceptual Combination.
Cognition 12 , 299–318.Osherson, D. N., & Smith, E. (1997). On typicality and vagueness.
Cognition 64 , 189–206.Pitowsky, I. (1989).
Quantum Probability, Quantum Logic . Lecture Notes in Physics vol. . Berlin:Springer.Pothos, E. M., & Busemeyer, J. R. (2009). A quantum probability explanation for violations of ‘rational’decision theory.
Proceedings of the Royal Society B 276 , 2171–2178.Pothos, E. M., & Busemeyer, J. R. (2013). Can quantum probability provide a new direction for cognitivemodeling?
Behavioral and Brain Sciences 36 . 255–274.Rips, L. J. (1995). The current status of research on concept combination.
Mind and Language 10 ,72–104.Rosch, E. (1973). Natural categories,
Cognitive Psychology 4 , 328–350.Rosch, E. (1978). Principles of categorization. In Rosch, E. & Lloyd, B. (Eds.),
Cognition and catego-rization . Hillsdale, NJ: Lawrence Erlbaum, pp. 133–179.Rosch, E. (1983). Prototype classification and logical classification: The two systems. In Scholnick, E. K.(Ed.),
New trends in conceptual representation: Challenges to Piaget theory? . New Jersey: LawrenceErlbaum, pp. 133–159.Rumelhart, D. E., & Norman, D. A. (1988). Representation in memory. In Atkinson, R. C., Hernsein, R.J., Lindzey, G., & Duncan, R. L. (Eds.),
Stevenshandbook of experimental psychology . New Jersey:John Wiley & Sons.Sozzo, S. (2014). A quantum probability explanation in Fock space for borderline contradictions.
Journalof Mathematical Psychology 58 , 1–12.Tversky, A. & Kahneman, D. (1983). Extension versus intuitive reasoning: The conjunction fallacy inprobability judgment.
Psychological Review 90 , 293–315.Tversky, A., & Shafir, E. (1992). The disjunction effect in choice under uncertainty.
Psychological Science3 , 305–309.Van Rijsbergen, K. (2004).
The Geometry of Information Retrieval , Cambridge: Cambridge UniversityPress. 19ang, Z., Busemeyer, J. R., Atmanspacher, H., & Pothos, E. (2013). The potential of quantum probabilityfor modeling cognitive processes.
Topics in Cognitive Science 5
Information & Control 8 , 338–353.Zadeh, L. (1982). A note on prototype theory and fuzzy sets.
Cognition 12 , 291–297.20 =Home Furnishing, B =FurnitureExemplar µ x ( A ) µ x ( B ) µ x ( A and B ) ∆ AB ( x ) k AB ( x ) Doub AB ( x ) Mantelpiece
Window Seat
Painting
Light Fixture
Kitchen Counter
Bath Tub
Deck Chair
Shelves
Rug
Bed
Wall-Hangings
Space Rack
Ashtray
Bar
Lamp
Wall Mirror
Door Bell
Hammock
Desk
Refrigerator
Park Bench
Waste Paper Basket
Sculpture
Sink Unit
Table 1a. Membership weights with respect to the concepts
Home Furnishing , Furniture and their con-junction
Home Furnishing And Furniture . 21 =Home Furnishing, B =FurnitureExemplar µ x ( A ) µ x (not B ) µ x ( A and not B ) ∆ AB ′ ( x ) k AB ′ ( x ) Doub AB ′ ( x ) l BB ′ ( x ) Mantelpiece
Window Seat
Painting
Light Fixture
Kitchen Counter
Bath Tub
Deck Chair
Shelves
Rug
Bed
Wall-Hangings
Space Rack
Ashtray
Bar
Lamp
Wall Mirror
Door Bell
Hammock
Desk
Refrigerator
Park Bench
Waste Paper Basket
Sculpture
Sink Unit
Table 1b. Membership weights with respect to the concepts
Home Furnishing , Not Furniture and theirconjunction
Home Furnishing And Not Furniture . 22 =Spices, B =HerbsExemplar µ x ( A ) µ x ( B ) µ x ( A and B ) ∆ AB ( x ) k AB ( x ) Doub AB ( x ) Molasses
Salt
Peppermint
Curry
Oregano
MSG
Chili Pepper
Mustard
Mint
Cinnamon
Parsley
Saccarin
Poppy Seeds
Pepper
Turmeric
Sugar
Vinegar
Sesame Seeds
Lemon Juice
Chocolate
Horseradish
Vanilla
Chives
Root Ginger
Table 2a. Membership weights with respect to the concepts
Spices , Herbs and their conjunction
SpicesAnd Herbs . 23 =Spices, B =HerbsExemplar µ x ( A ) µ x (not B ) µ x ( A and not B ) ∆ AB ′ ( x ) k AB ′ ( x ) Doub AB ′ ( x ) l BB ′ ( x ) Molasses
Salt
Peppermint
Curry
Oregano
MSG
Chili Pepper
Mustard
Mint
Cinnamon
Parsley
Saccarin
Poppy Seeds
Pepper
Turmeric
Sugar
Vinegar
Sesame Seeds
Lemon Juice
Chocolate
Horseradish
Vanilla
Chives
Root Ginger
Table 2b. Membership weights with respect to the concepts
Spices , Not Herbs and their conjunction
SpicesAnd Not Herbs . 24 =Pets, B =Farmayard AnimalsExemplar µ x ( A ) µ x ( B ) µ x ( A and B ) ∆ AB ( x ) k AB ( x ) Doub AB ( x ) Goldfish
Robin
Blue-tit
Collie Dog
Camel
Squirrel
Guide Dog for Blind
Spider
Homing Pigeon
Monkey
Circus Horse
Prize Bull
Rat
Badger
Siamese Cat
Race Horse
Fox
Donkey
Field Mouse
Ginger Tom-cat
Husky in Slead Team
Cart Horse
Chicken
Doberman Guard Dog
Table 3a. Membership weights with respect to the concepts
Pets , Farmyard Animals and their conjunction
Pets And Farmayard Animals . 25 =Pets, B =Farmyard AnimalsExemplar µ x ( A ) µ x (not B ) µ x ( A and not B ) ∆ AB ′ ( x ) k AB ′ ( x ) Doub AB ′ ( x ) l BB ′ ( x ) Goldfish
Robin
Blue-tit
Collie Dog
Camel
Squirrel
Guide Dog for Blind
Spider
Homing Pigeon
Monkey
Circus Horse
Prize Bull
Rat
Badger
Siamese Cat
Race Horse
Fox
Donkey
Field Mouse
Ginger Tom-cat
Husky in Slead team
Cart Horse
Chicken
Doberman Guard Dog
Table 3b. Membership weights with respect to the concepts
Pets , Not Farmyard Animals and their con-junction
Pets And Not Farmyard Animals . 26 =Fruits, B =VegetablesExemplar µ x ( A ) µ x ( B ) µ x ( A and B ) ∆ AB ( x ) k AB ( x ) Doub AB ( x ) Apple
Parsley
Olive
Chili Pepper
Broccoli
Root Ginger
Pumpkin
Raisin
Acorn
Mustard
Rice
Tomato
Coconut
Mushroom
Wheat
Green Pepper
Watercress
Peanut
Black Pepper
Garlic
Yam
Elderberry
Almond
Lentils
Table 4a. Membership weights with respect to the concepts
Fruits , Vegetables and their conjunction
FruitsAnd Vegetables . 27 =Fruits, B =VegetablesExemplar µ x ( A ) µ x (not B ) µ x ( A and not B ) ∆ AB ′ ( x ) k AB ′ ( x ) Doub AB ′ ( x ) l BB ′ ( x ) Apple
Parsley
Olive
Chili Pepper
Broccoli
Root Ginger
Pumpkin
Raisin
Acorn
Mustard
Rice
Tomato
Coconut
Mushroom
Wheat
Green Pepper
Watercress
Peanut
Black Pepper
Garlic
Yam
Elderberry
Almond
Lentils
Table 4b. Membership weights with respect to the concepts
Fruits , Not Vegetables and their conjunction
Pets And Not Vegetables . 28 =Home Furnishing, B =FurnitureExemplar µ x ( A ) µ x ( B ) µ x ( A and B ) θ AB ( x ) m AB ( x ) n AB ( x ) | A AB ( x ) i e − iθ AB ( x ) | B AB ( x ) i Mantelpiece
Window Seat
Painting
Light Fixture
Kitchen Counter
Bath Tub
Deck Chair
Shelves
Rug
Bed
Wall-Hangings
Space Rack
Ashtray
Bar
Lamp
Wall Mirror
Door Bell
Hammock
Desk
Refrigerator
Park Bench
Waste Paper Basket
Sculpture
Sink Unit
Table 5a. Representation of A , B and ‘ A and B ’ in the case of the concepts Home Furnishing and
Furniture .Note that the angles are expressed in degrees. A =Home Furnishing, B =FurnitureExemplar µ x ( A ) µ x (not B ) µ x ( A and not B ) θ AB ′ ( x ) m AB ′ ( x ) n AB ′ ( x ) | A AB ′ ( x ) i e − iθ AB ′ ( x ) | not B AB ′ ( x ) i Mantelpiece
Window Seat
Painting
Light Fixture
Kitchen Counter
Bath Tub
Deck Chair
Shelves
Rug
Bed
Wall-Hangings
Space Rack
Ashtray
Bar
Lamp
Wall Mirror
Door Bell
Hammock
Desk
Refrigerator
Park Bench
Waste Paper Basket
Sculpture
Sink Unit
Table 5b. Representation of A , ‘not B ’ and ‘ A and not B ’ in the case of the concepts Home Furnishing and
Furniture . Note that the angles are expressed in degrees.29 =Spices, B =HerbsExemplar µ x ( A ) µ x ( B ) µ x ( A and B ) θ AB ( x ) m AB ( x ) n AB ( x ) | A AB ( x ) i e − iθ AB ( x ) | B AB ( x ) i Molasses
Salt
Peppermint
Curry
Oregano
MSG
Chili Pepper
Mustard
Mint
Cinnamon
Parsley
Saccarin
Poppy Seeds
Pepper
Turmeric
Sugar
Vinegar
Sesame Seeds
Lemon Juice
Chocolate
Horseradish
Vanilla
Chives
Root Ginger
Table 6a. Representation of A , B and ‘ A and B ’ in the case of the concepts Spices and
Herbs . Note thatthe angles are expressed in degrees. A =Spices, B =HerbsExemplar µ x ( A ) µ x (not B ) µ x ( A and not B ) θ AB ′ ( x ) m AB ′ ( x ) n AB ′ ( x ) | A AB ′ ( x ) i e − iθ AB ′ ( x ) | not B AB ′ ( x ) i Molasses
Salt
Peppermint
Curry
Oregano
MSG
Chili Pepper
Mustard
Mint
Cinnamon
Parsley
Saccarin
Poppy Seeds
Pepper
Turmeric
Sugar
Vinegar
Sesame Seeds
Lemon Juice
Chocolate
Horseradish
Vanilla
Chives
Root Ginger
Table 6b. Representation of A , ‘not B ’ and ‘ A and not B ’ in the case of the concepts Spices and
Herbs .Note that the angles are expressed in degrees. 30 =Pets, B =Farmyard AnimalsExemplar µ x ( A ) µ x ( B ) µ x ( A and B ) θ AB ( x ) m AB ( x ) n AB ( x ) | A AB ( x ) i e − iθ AB ( x ) | B AB ( x ) i Goldfish
Robin
Blue-tit
Collie Dog
Camel
Squirrel
Guide Dog for Blind
Spider
Homing Pigeon
Monkey
Circus Horse
Prize Bull
Rat
Badger
Siamese Cat
Race Horse
Fox
Donkey
Field Mouse
Ginger Tom-cat
Husky in Slead team
Cart Horse
Chicken
Doberman Guard Dog
Table 7a. Representation of A , B and ‘ A and B ’ in the case of the concepts Pets and
Farmyard Animals .Note that the angles are expressed in degrees. A =Pets, B =Farmyard AnimalsExemplar µ x ( A ) µ x ( B ) µ x ( A and B ) θ AB ( x ) m AB ( x ) n AB ( x ) | A AB ( x ) i e − iθ AB ( x ) | B AB ( x ) i Goldfish
Robin
Blue-tit
Collie Dog
Camel
Squirrel
Guide Dog for Blind
Spider
Homing Pigeon
Monkey
Circus Horse
Prize Bull
Rat
Badger
Siamese Cat
Race Horse
Fox
Donkey
Field Mouse
Ginger Tom-cat
Husky in Slead Team
Cart Horse
Chicken
Doberman Guard Dog
Table 7b. Representation of A , ‘not B ’ and ‘ A and not B ’ in the case of the concepts Pets and
FarmyardAnimals . Note that the angles are expressed in degrees.31 =Fruits, B =VegetablesExemplar µ x ( A ) µ x ( B ) µ x ( A and B ) θ AB ( x ) m AB ( x ) n AB ( x ) | A AB ( x ) i e − iθ AB ( x ) | B AB ( x ) i Apple
Parsley
Olive
Chili Pepper
Broccoli
Root Ginger
Pumpkin
Raisin
Acorn
Mustard
Rice
Tomato
Coconut
Mushroom
Wheat
Green Pepper
Watercress
Peanut
Black Pepper
Garlic
Yam
Elderberry
Almond
Lentils
Table 8a. Representation of A , B and ‘ A and B ’ in the case of the concepts Fruits and
Vegetables . Notethat the angles are expressed in degrees. A =Fruits, B =VegetablesExemplar µ x ( A ) µ x (not B ) µ x ( A and not B ) θ AB ′ ( x ) m AB ′ ( x ) n AB ′ ( x ) | A AB ′ ( x ) i e − iθ AB ′ ( x ) | not B AB ′ ( x ) i Apple
Parsley
Olive
Chili Pepper
Broccoli
Root Ginger
Pumpkin
Raisin
Acorn
Mustard
Rice
Tomato
Coconut
Mushroom
Wheat
Green Pepper
Watercress
Peanut
Black Pepper
Garlic
Yam
Elderberry
Almond
Lentils
Table 8b. Representation of A , ‘not B ’ and ‘ A and not B ’ in the case of the concepts Fruits and