Edited by: Gabriel Radvansky, University of Notre Dame, USA
Reviewed by: Ken McRae, University of Western Ontario, Canada; Maria Luisa Dalla Chiara, University of Florence, Italy
*Correspondence: Sandro Sozzo, School of Management, Institute for Quantum Social and Cognitive Science, University of Leicester, University Road, Leicester LE1 7RH, UK
This article was submitted to Cognition, a section of the journal Frontiers in Psychology
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We analyze in this paper the data collected in a set of experiments investigating how people combine natural concepts. We study the mutual influence of conceptual conjunction and negation by measuring the membership weights of a list of exemplars with respect to two concepts, e.g.,
Substantial evidence of presence of quantum structures in processes connected with human behavior and cognition has been put forward in the last decade. More specifically, such quantum structures were identified in situations of decision making and in the structure of language (see e.g., Aerts,
The “combination problem,” that is, the question of how the representation of the combination of two or more natural concepts can be connected to the representation of the component concepts, has been studied experimentally and within classical concept theories in great detail in the last 30 years. The main experimental challenges to traditional modeling approaches to concepts combinations are sketched in the following
The “Guppy effect” in concept conjunction, also known as the “Pet-Fish problem” (Osherson and Smith,
The deviation from classical (fuzzy) set-theoretic membership weights of exemplars with respect to pairs of concepts and their conjunction or disjunction (Hampton,
The existence of “borderline contradictions” in sentences expressing vague properties (Bonini et al.,
What one typically finds in the above situations is a failure of set-theoretic approaches (classical set, fuzzy set, Kolmogorovian probability) to supply satisfactory theoretic models for the experimentally observed patterns. Indeed, all traditional approaches to concept theory [mainly, “prototype theory” (Rosch,
Important results in concept research and modeling have been obtained in the last decade within the approach of quantum cognition in which our research group has substantially contributed. We cannot report in detail the results attained in our approach, for obvious reasons of space limits. We limit ourselves to summarize the fundamentals and attach relevant bibliographic sources in the following.
The structural aspects of the approach rest on the results of older research on the foundations of quantum theory (Aerts,
Continuing in this direction the mathematical formalism of quantum theory was employed to model the overextension and underextension of membership weights measured in Hampton (
This quantum-theoretic framework was successfully applied to describe more complex situations, such as borderline vagueness (Sozzo,
There has been very little research on how people interpret and combine negated concepts. In a seminal study, Hampton (
Let us proceed by steps, summarizing the major findings in this paper, as follows.
In Section 2 we illustrate design and procedure of the four cognitive experiments we performed. In the first experiment, we tested the membership weights of four sets of exemplars with respect to four pairs (
We investigate the representability of the collected data, reported in Appendix A3, in a “single classical Kolmogorovian probability space” (Kolmogorov,
A second major and equally unexpected finding was that the numerical size of the “deviation of classicality pattern” can exactly be predicted in our quantum-theoretic model in two-sector Fock space. And, more, it can be explained by assuming that human reasoning is the superposition of two simultaneous processes, a “logical reasoning” and a “conceptual,” or “emergent,” “reasoning.” Logical reasoning combines cognitive entities (concepts, combinations of concepts, propositions, etc.) by applying the rules of logic, though generally in a probabilistic way. Emergent reasoning instead enables formation of combined cognitive entities as newly emerging entities (new concepts, new propositions, etc.), carrying new meaning, linked to the meaning of the component cognitive entities, but with a connection not defined by the algebra of logic. Emergent reasoning can be modeled in first sector of Fock space and, at variance with widespread beliefs, is dominant in our approach. Logical reasoning can be modeled in second sector of Fock space, hence one expects that classical logical rules hold in this sector, like we explicitly prove here for conceptual conjunctions and negations (see also Aerts et al.,
Our quantum-theoretic model in two-sector Fock space for conceptual negations and conjunctions is elaborated in Section 3. It naturally extends the model in Aerts (
We see in Section 4 that a large amount of data can be faithfully represented in our two-sector Fock space, and construct an explicit representation for some relevant cases that are classically problematical. A complete representation of the data is provided in the Supplementary Material attached to this paper. As we can see the findings presented in this paper provide strong and independent confirmations to our quantum-theoretic framework, and we devote Section 5 to comment on our results and extensively discuss novelties and corroboration of our approach. Technical appendices A4 and A5 complete the paper.
James Hampton identified in his cognitive tests systematic deviations from classical (fuzzy) set predictions for membership weights of exemplars with respect to conjunctions and disjunctions of two concepts, and named these deviations “overextensions” and “underextensions” (Hampton,
Similar effects were identified by Hampton in his experiments on conjunction and negation of two concepts (Hampton,
In the present paper we aim to generalize the results in Sozzo (
The participants to our experimental study—40 persons, chosen among our colleagues and friends—were asked to fill in a questionnaire in which they had to estimate the membership of four different sets of exemplars with respect to four different pairs (
We considered four pairs of natural concepts, namely (
Conceptual membership was estimated by using a “7-point scale.” The participants were asked to choose a number from the set +3, +2, +1, 0, −1, −2, −3, where the positive numbers +1, +2, and +3 meant that they considered “the exemplar to be a member of the concept”—+3 indicated a strong membership, +1 a relatively weak membership. The negative numbers −1, −2, and −3 meant that the participant considered “the exemplar to not be a member of the concept”—−3 indicated a strong non-membership, −1 a relatively weak non-membership.
Although we explicitly measured the “amount of membership” on a 7-point scale, for the scopes of this paper, we only need the data of a sub-experiment, namely the one tested for “membership” or “non-membership”—our plan is to use the “amount of membership data” for a following study leading to a graphical representation of the data, as we did with Hampton's data for the disjunction in earlier work (Aerts et al.,
This experimental study was carried out in accordance with the recommendations of the “University of Leicester Code of Practice and Research Code of Conduct, Research Ethics Committee of the School of Management” with written informed consent from all subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. For each pair (
For the conceptual pair (
For the conceptual pair (
For the conceptual pair (
For the conceptual pair (
A first inspection of tables Tables
The classicality requirements in Theorems 1 and 2 are not symmetric with respect to the exchange of
The conditions above can be further simplified by observing that the membership weights we collected in our experiments are large number limits of relative frequencies, thus all measured quantities are already contained in the interval [0, 1]. Therefore, we have
Now, when Equation (25) is satisfied, we have that from Equations (21) and (22) follows that
This entails that Equations (18) and (19) are satisfied, when Equations (21) and (22) are. Hence, we can amazingly enough formulate Theorem 3 a new, with only five conditions to be satisfied—four conditions expressing the marginal law.
Equations (26–30) express classicality conditions in their most symmetric form. A more traditional way to quantify deviations from classical conjunction in real data is resorting to the following parameters.
In fact, the quantities Δ
Let us now come back to our experiments. Theorems 1–3 are manisfestly violated in several cases, and we report in Appendix A3 the relevant conditions that should hold in a classical setting. Since the conditions
The exemplar
Overextension is present when one concept is negated. More explicitly:
(i) in the conjunction “
(ii) in the conjunction “
When two concepts are negated—“
Double overextension is also present in various cases. For example, the membership weight of
Significant deviations from classicality are also due to conceptual negation, in the form of violation of the marginal law of classical probability theory. By again referring to Tables
We performed a statistical analysis of the data, estimating the probability that the experimentally identified deviations from classicality would be due to chance. We specifically considered the classicality conditions Equations (26–30) with the aim to prove that the deviations
In addition, our data analysis reveals a new, fundamental and a priori unexpected deviation from classicality. The numerical values of it cannot be explained by means of traditional classical probabilistic approaches, since we should have it is “highly stable,” in the sense that the functions is is “systematic,” in the sense that the values of it is “regular,” in the sense that the functions
Observations (i–iv) were for us a clue that
These results could already be considered as crucial for claiming that the violation of classicality occurs at a deep structural conceptual level, but this is not the end of the story. We will see in Section 5 that the stability of this violation can exactly be explained in a quantum-theoretic framework in two-sector Fock space elaborated by ourselves. Hence, we devote Sections 3 and 4 to expose this modeling framework (the essentials of the formalism we apply are reviewed in Appendix A2, and we refer to it for symbols and notation).
In Aerts (
To model conceptual negations we also need a new theoretical step which was not necessary in our previous formulations, namely, the introduction of “entangled states” in second sector of Fock space to formalize situations where the membership weights are not independent. This introduction, together with the application of quantum logical rules in second sector of Fock space, are compatible with previous formulations, but they make our generalization in this paper highly non-obvious. We will extensively discuss the novelties of the present modeling in the next sections. Let us first proceed with our mathematical construction.
Let us denote by μ(
The decision measurement testing whether a specific exemplar
The conceptual negations
Let us first analyze the situation where we look for a modeling solution in the Hilbert space
Let us analyze in detail the aspects of this situation with the aim of resulting in a view on the possible solutions. Geometric considerations induce to observe that, if we look for a solution in the complex Hilbert space ℂ8, we will find the most general type of solution. Indeed, since we consider four orthonormal vectors |
By using standard rules for quantum probabilities we have that the membership weights for the conjunctions corresponding to the measured data should satisfy the following equations:
In Aerts (
Is it possible to introduce some “type of entanglement” in second sector ℂ8 ⊗ ℂ8? This question is interesting, since it is reasonable to believe that the outcomes of experiments for
Suppose that the concept “
Also the experimental data collected on
Theorem 4 implies that the tensor product Hilbert space model (second sector of Fock space) has exactly the same generality as the most general classical conditions for conjunction and negation. More specifically, given any data that satisfy the five classicality conditions of Theorem 3′, we can construct an entangled state such that in second sector “exactly” these classicality conditions are satisfied. Moreover, it clarifies that entangled states in our general Fock space modeling of data on conceptual conjunction and negation play a fundamental unexpected role in the combination of human concepts. The fact that classical logical rules are satisfied, in a probabilistic form, in second sector of Fock space provides an important confirmation to our two-sector quantum framework, as we will see in Section 5.
In Section 3.1 we have considered the situation of first sector of Fock space, representing the starting concepts
Following Aerts (
Let us denote, following our analysis in Section 3.2, such a general entangled state in ℂ8 by means of
The state vector representing the concept “
What is the procedure corresponding to emergent and quantum logical parts of human thought when we also take into account negations, i.e., when we consider the conjunctions “
In second sector of Fock space, we however have a specific situation to solve. Namely, exactly as we did for the conjunction, we need to identify what is the quantum logical structure related with negation, independent of its provoking the emergence of a new concept, i.e., the negation of the original concept. In second sector of Fock space we indeed only express the quantum logical reasoning in human thought and not the emergent reasoning. For the conjunction “
Our theoretic proposal is that:
(1) the first expression describes what happens in a human mind when emergent thought is dominant with respect to a concept (2) the second expression describes what happens in a human mind when quantum logical thought is dominant with respect to a concept
Expressions (1) and (2) are two structurally speaking subtle deeply different possibilities of reasoning related to a concept and its negation.
Our third theoretic proposal is that:
(3) human thought, when confronted with this situation, follows a dynamics described by a quantum superposition of the two modes (1) and (2).
We will see in the following that the mathematical structure of Fock space enables modeling this in an impecable way.
Indeed, expression (1) will be modeled in first sector of our Fock space, and it is mathematically realised by making
The above conceptual analysis makes it possible for us to write the complete Fock space formulas for the other combinations. More specifically, if we represent the concept “
Equations (86), (88), (90), and (92) contain the probabilistic expressions for simultaneously representing experimental data on conjunctions and negations of two concepts in a quantum-theoretic framework. These equations express the membership weights of the conjunctions “ the angles ϕ the pairs of convex coefficients ( the normalized coefficients
As we can see our two-sector Fock space framework is able to cope with conceptual negation in a very natural way. In fact, the latter negation is modeled by using the general assumption that emergent aspects of a concept are represented in first sector of Fock space, while logical aspects of a concept are represented in second sector. This will be made explicit in Section 5. It is however important to stress that, for a given experiment
The conclusion we draw from the analysis above is that finding solutions for a given set of experimental data in our quantum-theoretic modeling it is highly non-obvious, which makes the results in the next section even more significant.
Most of these data in Tables
Let us start with exemplars that are double overextended.
The interference angles ϕ
The interference angles ϕ
The interference angles
Let us now come to the exemplars that present complete overextension, that is, exemplars that are overextended in all experiments.
The interference angles ϕ
The interference angles ϕ
Let us then illustrate some relevant exemplars that either cannot be modeled in a pure Hilbert space framework, or cannot be represented by product states in second sector of Fock space.
However, a complete representation satisfying Equations (86), (88), (90), and (92) can only be worked out in the Fock space ℂ8 ⊕ (ℂ8 ⊗ ℂ8). This occurs for interference angles
However, a complete representation satisfying Equations (86), (88), (90), and (92) can only be worked out in the Fock space ℂ8 ⊕ (ℂ8 ⊗ ℂ8). This occurs for interference angles
Let us finally describe the quantum-theoretic representation of an exemplar that does not present overextension in any conjunction, but still does not admit a representation in a classical Kolmogorovian probability framework.
The interference angles ϕ
The theoretic analysis on the representatibility of the data in Tables
Our experimental data on conjunctions and negations of natural concepts confirm that classical probability does not generally work when people combine concepts, as we have seen in the previous sections. And, more, we have proved here that the deviations from classicality cannot be reduced to overextension and underextension, but they also include a very strong and fundamental pattern of violation. On the other side, our quantum-theoretic framework in Fock space has received remarkable corroboration. Thus, we think it worth to summarize and stress the novelties that have emerged in this article with respect to our approach.
We have recently put forward an explanatory hypothesis with respect to the deviations from classical logical reasoning that have been observed in human cognition (Aerts et al.,
Now, if one reflects on how we represented conceptual negation in Section 3, one realizes at once that its modeling directly and naturally follows from the general hypothesis stated above. Indeed, suppose that a person is asked to estimate whether a given exemplar
The second confirmation of our quantum-conceptual framework comes from the significantly stable deviations from classicality in Equations (26–30). We have seen in Section 2.4 that these deviations occur at a different, deeper, level than overextension and underextension. We think we have identified a general mechanism determining how concepts are formed in the human mind. And this would already be convincing even without mentioning a Fock space modeling. But, this very stable pattern can exactly be explained in our two-sector Fock space framework by assuming that emergence plays a primary role in the human reasoning process, but also aspects of logic are systematically present. Indeed, suppose that, for every exemplar
The third strong confirmation of this two-layered structure of human thought and its representation in two-sector Fock space comes from the peculiarities of conceptual negation. Indeed, being pushed to cope with conceptual conjunction and negation simultaneously, we have found a new insight which we had not noticed before, namely, the emergent non-classical properties of the conjunction
The fourth corroboration derives from the fact that our Fock space indicates how and why introducing entanglement. In our previous attempts to model conceptual combinations, we had not recognized that by representing the combined concept by a tensor product vector |
The discussion above shows, in our opinion, that the merits of our two-sector Fock space framework go beyond faithful representation of one or more sets of experimental data. It captures some fundamental aspects of the mechanisms through which concepts are formed, combine and interact in human cognition.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The Supplementary Material for this article can be found online at:
We introduce in this section the elementary measure-theoretic notions that are needed to express the classicality of experimental data coming from the membership weights of two concepts
Let us start by the definition of a σ-algebra over a set.
Measure structures are the most general classical structures devised by mathematicians and physicists to structure weights. A Kolmogorovian probability measure is such a measure applied to statistical data. It is called “Kolmogorovian,” because Andrey Kolmogorov was the first to axiomatize probability theory in this way (Kolmogorov,
the empty set has measure zero; if the triple (Ω, σ(Ω),
A
We illustrate in this section how the mathematical formalism of quantum theory can be applied to model situations outside the microscopic quantum world, more specifically, in the representation of concepts and their combinations. As in Appendix A1, we will limit technicalities to the essential.
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 〈 〈 〈
From (ii) and (iii) follows that inner product 〈·|·〉 is linear in the ket and anti-linear in the bra, i.e., (
We recall that the “absolute value” of a complex number is defined as the square root of the product of this complex number times its complex conjugate, that is,
We have now introduced the necessary mathematics to state the first modeling rule of quantum theory, as follows.
An orthogonal projection 〈
The identity operator 𝟙 maps each vector onto itself and is a trivial orthogonal projection. We say that two orthogonal projections
The above definitions give us the necessary mathematics to state the second modeling rule of quantum theory, as follows.
The tensor product
The above means that we have
The Fock space is a specific type of Hilbert space, originally introduced in quantum field theory. For most states of a quantum field the number of identical quantum entities is not conserved but is a variable quantity. The Fock space copes with this situation in allowing its vectors to be superpositions of vectors pertaining to different sectors for fixed numbers of identical quantum entities. More explicitly, the
Tables
Now suppose that
Let us now prove a result which is useful for our purposes. Following (5) and (6) we have that μ(
The equality
Hence, we have proved the Theorem 2. □
A further simplification is possible. Indeed, (5), (6) and (7) are equivalent with (14), (15), (16), (17) and (A11). We have proved above that (5), (6) and (7) imply (A11). Let us prove the inverse. Hence suppose that (14), (15), (16), (17) and (A11) are satisfied, and let us proof (7). We have
We have thus proved Theorem 3, stating a new and more symmetric set of classicality conditions. □
Let us prove the other implication. Hence suppose that we have available data satisfying the classicality conditions (26)–(30), let us prove that we can find a state |
In this section we make explicit the conditions that should be satisfied by experimental data in order to be represented in Fock space. In our analysis, we distinguish between the first sector representation in ℂ8 and the complete two-sector representation in ℂ8 ⊕ (ℂ8 ⊗ ℂ8).
Let us start from the ℂ8 representation. We first analyze whether or not the solution of (52)–(65) is compatible with (70)–(73). To this end note that the right hand side of (70)–(73) correspond to the average of the probabilities of the former concepts, plus the so called “interference term,” which depends on (i) how the vectors representing the former concepts in the combination, when restricted to the subspace determined by
Let us now come to the complete representation in ℂ8 ⊕ (ℂ8 ⊗ ℂ8), and let us consider the data collected in the experiments
The analysis we made in the foregoing sections makes it possible for us to propose a general modeling procedure. For what concerns solutions that can be found on first sector alone, we determined the intervals of solutions as explained in (A20), (A21), (A22) and (A23). We can now easily determine the general intervals of solutions, including the extra solutions made possible by second sector. Therefore, we need to consider the following quantities
We can analyze the other combinations in an equivalent way. Let us start with the combination “
For the combination “
Finally, for the combination “not
Mantelpiece | 0.9 | 0.61 | 0.12 | 0.5 | 0.71 | 0.75 | 0.21 | 0.21 | 0.1 | 0.25 | 0.09 | 0.09 | −0.89 | −0.56 | −0.31 | −0.31 | −0.46 |
Window Seat | 0.5 | 0.48 | 0.47 | 0.55 | 0.45 | 0.49 | 0.39 | 0.41 | −0.03 | −0.01 | −0.08 | −0.06 | −0.74 | −0.44 | −0.36 | −0.33 | −0.35 |
Painting | 0.8 | 0.49 | 0.35 | 0.64 | 0.64 | 0.6 | 0.33 | 0.38 | 0.15 | −0.04 | −0.03 | 0.03 | −0.94 | −0.44 | −0.48 | −0.35 | −0.33 |
Light Fixture | 0.88 | 0.6 | 0.16 | 0.51 | 0.73 | 0.63 | 0.33 | 0.16 | 0.13 | 0.11 | 0.16 | 0 | −0.84 | −0.48 | −0.45 | −0.33 | −0.28 |
Kitchen Counter | 0.67 | 0.49 | 0.31 | 0.62 | 0.55 | 0.54 | 0.38 | 0.33 | 0.06 | −0.08 | 0.06 | 0.01 | −0.79 | −0.42 | −0.44 | −0.39 | −0.24 |
Bath Tub | 0.73 | 0.51 | 0.28 | 0.46 | 0.59 | 0.59 | 0.36 | 0.29 | 0.08 | 0.13 | 0.08 | 0.01 | −0.83 | −0.45 | −0.44 | −0.37 | −0.41 |
Deck Chair | 0.73 | 0.9 | 0.27 | 0.2 | 0.74 | 0.41 | 0.54 | 0.18 | 0.01 | 0.21 | 0.27 | −0.03 | −0.86 | −0.42 | −0.38 | −0.44 | −0.39 |
Shelves | 0.85 | 0.93 | 0.24 | 0.13 | 0.84 | 0.39 | 0.53 | 0.08 | −0.01 | 0.26 | 0.29 | −0.05 | −0.83 | −0.38 | −0.43 | −0.36 | −0.34 |
Rug | 0.89 | 0.58 | 0.18 | 0.61 | 0.7 | 0.68 | 0.41 | 0.21 | 0.13 | 0.07 | 0.24 | 0.04 | −1 | −0.48 | −0.54 | −0.45 | −0.28 |
Bed | 0.76 | 0.93 | 0.26 | 0.11 | 0.79 | 0.36 | 0.61 | 0.14 | 0.03 | 0.26 | 0.35 | 0.03 | −0.9 | −0.39 | −0.48 | −0.49 | −0.39 |
Wall-Hangings | 0.87 | 0.46 | 0.21 | 0.68 | 0.55 | 0.71 | 0.35 | 0.24 | 0.09 | 0.03 | 0.14 | 0.03 | −0.85 | −0.39 | −0.44 | −0.38 | −0.27 |
Space Rack | 0.38 | 0.43 | 0.63 | 0.62 | 0.41 | 0.49 | 0.43 | 0.58 | 0.04 | 0.11 | 0 | −0.04 | −0.9 | −0.53 | −0.41 | −0.37 | −0.44 |
Ashtray | 0.74 | 0.4 | 0.32 | 0.64 | 0.49 | 0.6 | 0.36 | 0.39 | 0.09 | −0.04 | 0.04 | 0.07 | −0.84 | −0.34 | −0.45 | −0.43 | −0.35 |
Bar | 0.72 | 0.63 | 0.37 | 0.51 | 0.61 | 0.61 | 0.4 | 0.4 | −0.01 | 0.11 | 0.03 | 0.03 | −1.03 | −0.51 | −0.39 | −0.43 | −0.51 |
Lamp | 0.94 | 0.64 | 0.15 | 0.49 | 0.75 | 0.7 | 0.4 | 0.2 | 0.11 | 0.21 | 0.25 | 0.05 | −1.05 | −0.51 | −0.51 | −0.45 | −0.41 |
Wall Mirror | 0.91 | 0.76 | 0.13 | 0.45 | 0.83 | 0.66 | 0.44 | 0.14 | 0.07 | 0.21 | 0.31 | 0.01 | −1.06 | −0.58 | −0.51 | −0.45 | −0.35 |
Door Bell | 0.75 | 0.33 | 0.32 | 0.79 | 0.5 | 0.64 | 0.34 | 0.51 | 0.17 | −0.11 | 0.02 | 0.19 | −0.99 | −0.39 | −0.51 | −0.53 | −0.36 |
Hammock | 0.62 | 0.66 | 0.41 | 0.41 | 0.6 | 0.5 | 0.56 | 0.31 | −0.02 | 0.09 | 0.16 | −0.09 | −0.98 | −0.48 | −0.5 | −0.47 | −0.41 |
Desk | 0.78 | 0.95 | 0.31 | 0.09 | 0.78 | 0.33 | 0.75 | 0.15 | −0.01 | 0.24 | 0.44 | 0.06 | −1 | −0.32 | −0.58 | −0.59 | −0.39 |
Refrigerator | 0.74 | 0.73 | 0.26 | 0.41 | 0.66 | 0.55 | 0.46 | 0.25 | −0.06 | 0.14 | 0.21 | −0.01 | −0.93 | −0.47 | −0.4 | −0.46 | −0.39 |
Park Bench | 0.53 | 0.66 | 0.59 | 0.46 | 0.55 | 0.29 | 0.56 | 0.39 | 0.02 | −0.17 | −0.03 | −0.07 | −0.79 | −0.31 | −0.45 | −0.36 | −0.22 |
Waste Paper Basket | 0.69 | 0.54 | 0.36 | 0.63 | 0.59 | 0.41 | 0.46 | 0.49 | 0.04 | −0.22 | 0.1 | 0.13 | −0.95 | −0.31 | −0.51 | −0.59 | −0.27 |
Sculpture | 0.83 | 0.46 | 0.34 | 0.66 | 0.58 | 0.73 | 0.46 | 0.36 | 0.11 | 0.07 | 0.13 | 0.03 | −1.13 | −0.48 | −0.58 | −0.49 | −0.43 |
Sink Unit | 0.71 | 0.57 | 0.34 | 0.58 | 0.6 | 0.56 | 0.38 | 0.38 | 0.03 | −0.01 | 0.04 | 0.04 | −0.91 | −0.46 | −0.41 | −0.41 | −0.36 |
Molasses | 0.36 | 0.13 | 0.67 | 0.84 | 0.24 | 0.54 | 0.25 | 0.73 | 0.11 | 0.18 | 0.12 | 0.06 | −0.75 | −0.41 | −0.36 | −0.31 | −0.43 |
Salt | 0.67 | 0.04 | 0.36 | 0.92 | 0.24 | 0.69 | 0.09 | 0.6 | 0.19 | 0.02 | 0.04 | 0.24 | −0.61 | −0.26 | −0.28 | −0.33 | −0.37 |
Peppermint | 0.67 | 0.93 | 0.38 | 0.1 | 0.7 | 0.38 | 0.55 | 0.15 | 0.03 | 0.28 | 0.18 | 0.05 | −0.78 | −0.41 | −0.33 | −0.33 | −0.43 |
Curry | 0.96 | 0.28 | 0.04 | 0.78 | 0.54 | 0.88 | 0.16 | 0.21 | 0.26 | 0.1 | 0.13 | 0.18 | −0.79 | −0.45 | −0.42 | −0.34 | −0.31 |
Oregano | 0.81 | 0.86 | 0.21 | 0.13 | 0.79 | 0.4 | 0.5 | 0.08 | −0.03 | 0.28 | 0.29 | −0.05 | −0.76 | −0.38 | −0.43 | −0.36 | −0.35 |
MSG | 0.44 | 0.12 | 0.59 | 0.85 | 0.23 | 0.58 | 0.24 | 0.73 | 0.11 | 0.13 | 0.12 | 0.13 | −0.76 | −0.36 | −0.34 | −0.37 | −0.45 |
Chili Pepper | 0.98 | 0.53 | 0.05 | 0.56 | 0.8 | 0.9 | 0.28 | 0.13 | 0.27 | 0.34 | 0.23 | 0.08 | −1.1 | −0.73 | −0.54 | −0.35 | −0.46 |
Mustard | 0.65 | 0.28 | 0.39 | 0.71 | 0.49 | 0.65 | 0.23 | 0.46 | 0.21 | 0 | −0.05 | 0.08 | −0.83 | −0.49 | −0.44 | −0.3 | −0.41 |
Mint | 0.64 | 0.96 | 0.43 | 0.09 | 0.79 | 0.31 | 0.64 | 0.11 | 0.14 | 0.23 | 0.21 | 0.03 | −0.85 | −0.46 | −0.47 | −0.32 | −0.34 |
Cinnamon | 1 | 0.49 | 0.02 | 0.51 | 0.69 | 0.79 | 0.21 | 0.15 | 0.19 | 0.28 | 0.19 | 0.13 | −0.84 | −0.48 | −0.41 | −0.34 | −0.43 |
Parsley | 0.54 | 0.9 | 0.54 | 0.09 | 0.68 | 0.26 | 0.73 | 0.18 | 0.14 | 0.18 | 0.19 | 0.09 | −0.84 | −0.4 | −0.5 | −0.36 | −0.35 |
Saccarin | 0.34 | 0.14 | 0.68 | 0.88 | 0.24 | 0.54 | 0.24 | 0.8 | 0.1 | 0.19 | 0.1 | 0.12 | −0.81 | −0.43 | −0.34 | −0.36 | −0.46 |
Poppy Seeds | 0.82 | 0.47 | 0.29 | 0.54 | 0.59 | 0.66 | 0.31 | 0.28 | 0.12 | 0.13 | 0.02 | −0.02 | −0.84 | −0.43 | −0.43 | −0.29 | −0.4 |
Pepper | 0.99 | 0.47 | 0.1 | 0.58 | 0.7 | 0.9 | 0.18 | 0.14 | 0.23 | 0.32 | 0.08 | 0.04 | −0.91 | −0.61 | −0.41 | −0.21 | −0.46 |
Turmeric | 0.88 | 0.53 | 0.11 | 0.43 | 0.74 | 0.69 | 0.28 | 0.21 | 0.21 | 0.26 | 0.16 | 0.1 | −0.91 | −0.54 | −0.49 | −0.38 | −0.47 |
Sugar | 0.45 | 0.34 | 0.59 | 0.77 | 0.35 | 0.56 | 0.25 | 0.65 | 0.01 | 0.11 | −0.09 | 0.06 | −0.81 | −0.46 | −0.26 | −0.31 | −0.44 |
Vinegar | 0.3 | 0.11 | 0.76 | 0.88 | 0.15 | 0.41 | 0.26 | 0.83 | 0.04 | 0.11 | 0.16 | 0.07 | −0.65 | −0.26 | −0.31 | −0.33 | −0.36 |
Sesame Seeds | 0.8 | 0.49 | 0.3 | 0.59 | 0.59 | 0.7 | 0.34 | 0.29 | 0.1 | 0.11 | 0.04 | −0.01 | −0.91 | −0.49 | −0.44 | −0.33 | −0.4 |
Lemon Juice | 0.28 | 0.2 | 0.74 | 0.81 | 0.15 | 0.43 | 0.39 | 0.81 | −0.05 | 0.15 | 0.19 | 0.07 | −0.78 | −0.3 | −0.34 | −0.46 | −0.43 |
Chocolate | 0.27 | 0.21 | 0.78 | 0.8 | 0.2 | 0.46 | 0.38 | 0.78 | −0.01 | 0.19 | 0.16 | −0.01 | −0.81 | −0.39 | −0.36 | −0.37 | −0.44 |
Horseradish | 0.61 | 0.67 | 0.48 | 0.28 | 0.61 | 0.4 | 0.53 | 0.33 | 0 | 0.12 | 0.04 | 0.04 | −0.86 | −0.4 | −0.47 | −0.37 | −0.44 |
Vanilla | 0.76 | 0.51 | 0.3 | 0.49 | 0.63 | 0.61 | 0.33 | 0.35 | 0.11 | 0.13 | 0.03 | 0.05 | −0.91 | −0.48 | −0.44 | −0.38 | −0.48 |
Chives | 0.66 | 0.89 | 0.43 | 0.26 | 0.76 | 0.28 | 0.64 | 0.31 | 0.1 | 0.02 | 0.21 | 0.06 | −0.99 | −0.38 | −0.51 | −0.53 | −0.33 |
Root Ginger | 0.84 | 0.56 | 0.23 | 0.44 | 0.69 | 0.59 | 0.41 | 0.23 | 0.13 | 0.14 | 0.18 | −0.01 | −0.91 | −0.43 | −0.54 | −0.41 | −0.37 |
Goldfish | 0.93 | 0.17 | 0.12 | 0.81 | 0.43 | 0.91 | 0.18 | 0.43 | 0.26 | 0.1 | 0.06 | 0.31 | −0.94 | −0.41 | −0.43 | −0.48 | −0.53 |
Robin | 0.28 | 0.36 | 0.71 | 0.64 | 0.31 | 0.35 | 0.46 | 0.46 | 0.04 | 0.08 | 0.1 | −0.18 | −0.59 | −0.39 | −0.41 | −0.22 | −0.18 |
Blue-tit | 0.25 | 0.31 | 0.76 | 0.71 | 0.18 | 0.39 | 0.44 | 0.56 | −0.08 | 0.14 | 0.13 | −0.15 | −0.56 | −0.31 | −0.3 | −0.24 | −0.24 |
Collie Dog | 0.95 | 0.77 | 0.03 | 0.35 | 0.86 | 0.56 | 0.25 | 0.11 | 0.09 | 0.21 | 0.23 | 0.09 | −0.79 | −0.48 | −0.34 | −0.34 | −0.33 |
Camel | 0.16 | 0.26 | 0.89 | 0.75 | 0.2 | 0.31 | 0.51 | 0.68 | 0.04 | 0.16 | 0.26 | −0.08 | −0.7 | −0.36 | −0.46 | −0.3 | −0.24 |
Squirrel | 0.3 | 0.39 | 0.74 | 0.65 | 0.28 | 0.26 | 0.46 | 0.59 | −0.03 | −0.04 | 0.07 | −0.06 | −0.59 | −0.24 | −0.34 | −0.31 | −0.2 |
Guide Dog for Blind | 0.93 | 0.33 | 0.13 | 0.69 | 0.55 | 0.73 | 0.16 | 0.33 | 0.23 | 0.03 | 0.04 | 0.2 | −0.76 | −0.35 | −0.39 | −0.36 | −0.36 |
Spider | 0.31 | 0.39 | 0.73 | 0.63 | 0.31 | 0.31 | 0.44 | 0.51 | 0 | 0 | 0.05 | −0.12 | −0.58 | −0.31 | −0.36 | −0.23 | −0.19 |
Homing Pigeon | 0.41 | 0.71 | 0.61 | 0.34 | 0.56 | 0.25 | 0.59 | 0.34 | 0.16 | −0.09 | −0.03 | 0 | −0.74 | −0.41 | −0.44 | −0.31 | −0.25 |
Monkey | 0.39 | 0.18 | 0.65 | 0.79 | 0.2 | 0.49 | 0.29 | 0.61 | 0.03 | 0.09 | 0.11 | −0.04 | −0.59 | −0.29 | −0.31 | −0.25 | −0.31 |
Circus Horse | 0.3 | 0.48 | 0.74 | 0.6 | 0.34 | 0.35 | 0.53 | 0.48 | 0.04 | 0.05 | 0.04 | −0.13 | −0.69 | −0.39 | −0.38 | −0.26 | −0.23 |
Prize Bull | 0.13 | 0.76 | 0.88 | 0.26 | 0.43 | 0.28 | 0.83 | 0.34 | 0.29 | 0.14 | 0.06 | 0.08 | −0.86 | −0.57 | −0.49 | −0.28 | −0.35 |
Rat | 0.2 | 0.36 | 0.85 | 0.68 | 0.21 | 0.28 | 0.54 | 0.63 | 0.01 | 0.08 | 0.18 | −0.05 | −0.65 | −0.29 | −0.39 | −0.31 | −0.23 |
Badger | 0.16 | 0.28 | 0.88 | 0.73 | 0.14 | 0.26 | 0.44 | 0.66 | −0.03 | 0.1 | 0.16 | −0.07 | −0.5 | −0.24 | −0.3 | −0.23 | −0.19 |
Siamese Cat | 0.99 | 0.5 | 0.05 | 0.53 | 0.74 | 0.75 | 0.18 | 0.24 | 0.24 | 0.23 | 0.13 | 0.19 | −0.9 | −0.5 | −0.41 | −0.36 | −0.46 |
Race Horse | 0.29 | 0.7 | 0.71 | 0.39 | 0.51 | 0.31 | 0.65 | 0.31 | 0.23 | 0.03 | −0.05 | −0.08 | −0.79 | −0.54 | −0.46 | −0.26 | −0.24 |
Fox | 0.13 | 0.3 | 0.86 | 0.68 | 0.18 | 0.29 | 0.46 | 0.59 | 0.04 | 0.16 | 0.16 | −0.09 | −0.51 | −0.33 | −0.34 | −0.19 | −0.19 |
Donkey | 0.29 | 0.9 | 0.78 | 0.15 | 0.56 | 0.18 | 0.81 | 0.23 | 0.28 | 0.03 | 0.04 | 0.08 | −0.78 | −0.45 | −0.48 | −0.26 | −0.25 |
Field Mouse | 0.16 | 0.41 | 0.83 | 0.59 | 0.23 | 0.24 | 0.43 | 0.58 | 0.06 | 0.08 | 0.02 | −0.01 | −0.46 | −0.3 | −0.24 | −0.18 | −0.23 |
Ginger Tom-cat | 0.82 | 0.51 | 0.21 | 0.54 | 0.59 | 0.58 | 0.26 | 0.29 | 0.08 | 0.03 | 0.05 | 0.08 | −0.71 | −0.34 | −0.34 | −0.34 | −0.32 |
Husky in Slead team | 0.64 | 0.51 | 0.37 | 0.53 | 0.56 | 0.51 | 0.44 | 0.29 | 0.06 | −0.01 | 0.07 | −0.08 | −0.8 | −0.43 | −0.49 | −0.36 | −0.28 |
Cart Horse | 0.27 | 0.86 | 0.76 | 0.15 | 0.53 | 0.2 | 0.84 | 0.23 | 0.26 | 0.05 | 0.08 | 0.08 | −0.79 | −0.46 | −0.5 | −0.31 | −0.28 |
Chicken | 0.23 | 0.95 | 0.8 | 0.06 | 0.58 | 0.11 | 0.81 | 0.18 | 0.34 | 0.05 | 0.01 | 0.11 | −0.68 | −0.46 | −0.44 | −0.19 | −0.23 |
Doberman Guard Dog | 0.88 | 0.76 | 0.14 | 0.27 | 0.8 | 0.55 | 0.45 | 0.23 | 0.04 | 0.28 | 0.31 | 0.09 | −1.03 | −0.47 | −0.49 | −0.54 | −0.51 |
Apple | 1 | 0.23 | 0 | 0.82 | 0.6 | 0.89 | 0.13 | 0.18 | 0.38 | 0.07 | 0.13 | 0.18 | −0.79 | −0.49 | −0.5 | −0.3 | −0.24 |
Parsley | 0.02 | 0.78 | 0.99 | 0.25 | 0.45 | 0.1 | 0.84 | 0.44 | 0.43 | 0.08 | 0.06 | 0.19 | −0.83 | −0.53 | −0.51 | −0.29 | −0.29 |
Olive | 0.53 | 0.63 | 0.47 | 0.44 | 0.65 | 0.34 | 0.51 | 0.36 | 0.12 | −0.11 | 0.04 | −0.08 | −0.86 | −0.46 | −0.53 | −0.41 | −0.26 |
Chili Pepper | 0.19 | 0.73 | 0.83 | 0.35 | 0.51 | 0.2 | 0.68 | 0.44 | 0.33 | 0.01 | −0.06 | 0.09 | −0.83 | −0.53 | −0.46 | −0.29 | −0.29 |
Broccoli | 0.09 | 1 | 0.94 | 0.06 | 0.59 | 0.09 | 0.9 | 0.25 | 0.49 | 0.03 | −0.04 | 0.19 | −0.83 | −0.58 | −0.49 | −0.21 | −0.28 |
Root Ginger | 0.14 | 0.71 | 0.81 | 0.33 | 0.46 | 0.14 | 0.71 | 0.43 | 0.33 | 0 | 0 | 0.1 | −0.74 | −0.46 | −0.46 | −0.33 | −0.24 |
Pumpkin | 0.45 | 0.78 | 0.51 | 0.26 | 0.66 | 0.21 | 0.63 | 0.18 | 0.21 | −0.05 | 0.11 | −0.09 | −0.68 | −0.43 | −0.51 | −0.29 | −0.13 |
Raisin | 0.88 | 0.27 | 0.13 | 0.76 | 0.53 | 0.75 | 0.25 | 0.34 | 0.26 | −0.01 | 0.12 | 0.21 | −0.86 | −0.39 | −0.51 | −0.46 | −0.33 |
Acorn | 0.59 | 0.4 | 0.49 | 0.64 | 0.46 | 0.49 | 0.38 | 0.51 | 0.06 | −0.1 | −0.03 | 0.02 | −0.84 | −0.36 | −0.44 | −0.39 | −0.36 |
Mustard | 0.07 | 0.39 | 0.87 | 0.6 | 0.29 | 0.23 | 0.55 | 0.75 | 0.22 | 0.16 | 0.16 | 0.15 | −0.81 | −0.44 | −0.45 | −0.43 | −0.38 |
Rice | 0.12 | 0.46 | 0.9 | 0.52 | 0.21 | 0.23 | 0.59 | 0.59 | 0.09 | 0.11 | 0.13 | 0.07 | −0.61 | −0.32 | −0.34 | −0.28 | −0.29 |
Tomato | 0.34 | 0.89 | 0.64 | 0.19 | 0.7 | 0.2 | 0.74 | 0.23 | 0.36 | 0.01 | 0.1 | 0.04 | −0.86 | −0.56 | −0.55 | −0.33 | −0.24 |
Coconut | 0.93 | 0.32 | 0.17 | 0.7 | 0.56 | 0.69 | 0.2 | 0.34 | 0.24 | −0.01 | 0.03 | 0.17 | −0.79 | −0.33 | −0.44 | −0.37 | −0.33 |
Mushroom | 0.12 | 0.66 | 0.9 | 0.38 | 0.33 | 0.13 | 0.66 | 0.5 | 0.21 | 0.01 | 0 | 0.12 | −0.61 | −0.33 | −0.33 | −0.26 | −0.24 |
Wheat | 0.17 | 0.51 | 0.8 | 0.52 | 0.34 | 0.21 | 0.61 | 0.56 | 0.17 | 0.04 | 0.11 | 0.04 | −0.73 | −0.38 | −0.44 | −0.38 | −0.26 |
Green Pepper | 0.23 | 0.61 | 0.81 | 0.41 | 0.49 | 0.24 | 0.61 | 0.43 | 0.26 | 0.01 | 0 | 0.02 | −0.76 | −0.5 | −0.49 | −0.23 | −0.26 |
Watercress | 0.14 | 0.76 | 0.89 | 0.25 | 0.49 | 0.1 | 0.79 | 0.35 | 0.35 | −0.04 | 0.03 | 0.1 | −0.73 | −0.45 | −0.51 | −0.24 | −0.2 |
Peanut | 0.62 | 0.29 | 0.48 | 0.75 | 0.48 | 0.55 | 0.25 | 0.53 | 0.18 | −0.07 | −0.04 | 0.05 | −0.8 | −0.41 | −0.43 | −0.3 | −0.33 |
Black Pepper | 0.21 | 0.41 | 0.81 | 0.61 | 0.38 | 0.21 | 0.5 | 0.63 | 0.17 | 0.01 | 0.09 | 0.01 | −0.71 | −0.38 | −0.46 | −0.31 | −0.23 |
Garlic | 0.13 | 0.79 | 0.88 | 0.24 | 0.53 | 0.1 | 0.75 | 0.45 | 0.4 | −0.03 | −0.04 | 0.21 | −0.83 | −0.5 | −0.49 | −0.33 | −0.31 |
Yam | 0.38 | 0.66 | 0.71 | 0.43 | 0.59 | 0.24 | 0.65 | 0.44 | 0.21 | −0.14 | −0.01 | 0.01 | −0.91 | −0.45 | −0.58 | −0.38 | −0.24 |
Elderberry | 0.51 | 0.39 | 0.54 | 0.61 | 0.45 | 0.41 | 0.46 | 0.48 | 0.06 | −0.09 | 0.07 | −0.07 | −0.8 | −0.36 | −0.52 | −0.39 | −0.28 |
Almond | 0.76 | 0.29 | 0.28 | 0.72 | 0.48 | 0.61 | 0.24 | 0.48 | 0.18 | −0.11 | −0.04 | 0.19 | −0.8 | −0.33 | −0.42 | −0.43 | −0.37 |
Lentils | 0.11 | 0.66 | 0.89 | 0.38 | 0.38 | 0.11 | 0.7 | 0.53 | 0.26 | 0 | 0.04 | 0.15 | −0.71 | −0.38 | −0.41 | −0.33 | −0.26 |
Mantelpiece | 5.61E-09 | Molasses | 9.73E-07 | Goldfish | 2.52E-05 | Apple | 1.78E-08 |
Window Seat | 1.08E-05 | Salt | 7.01E-04 | Robin | 1.78E-06 | Parsley | 5.14E-07 |
Painting | 4.19E-07 | Peppermint | 5.57E-06 | Blue-tit | 3.56E-06 | Olive | 1.98E-05 |
Light Fixture | 4.20E-06 | Curry | 6.51E-05 | Collie Dog | 7.90E-06 | Chili Pepper | 1.85E-07 |
Kitchen Counter | 6.10E-05 | Oregano | 1.78E-06 | Camel | 5.92E-05 | Broccoli | 4.35E-09 |
Bath Tub | 4.17E-06 | MSG | 5.93E-05 | Squirrel | 5.23E-04 | Root Ginger | 1.44E-06 |
Deck Chair | 5.35E-06 | Chili Pepper | 4.00E-12 | Guide Dog for Blind | 6.72E-04 | Pumpkin | 9.35E-06 |
Shelves | 2.20E-06 | Mustard | 2.26E-05 | Spider | 4.19E-05 | Raisin | 1.13E-06 |
Rug | 1.81E-09 | Mint | 1.56E-05 | Homing Pigeon | 4.87E-05 | Acorn | 1.04E-05 |
Bed | 5.81E-07 | Cinnamon | 8.90E-08 | Monkey | 7.21E-04 | Mustard | 6.05E-07 |
Wall-Hangings | 7.75E-07 | Parsley | 2.05E-05 | Circus Horse | 3.71E-07 | Rice | 7.87E-05 |
Space Rack | 2.02E-08 | Saccarin | 1.64E-06 | Prize Bull | 2.02E-08 | Tomato | 1.77E-07 |
Ashtray | 2.73E-06 | Poppy Seeds | 5.63E-05 | Rat | 1.05E-03 | Coconut | 1.61E-03 |
Bar | 3.07E-08 | Pepper | 2.70E-07 | Badger | 3.79E-04 | Mushroom | 3.40E-05 |
Lamp | 4.80E-08 | Turmeric | 1.62E-08 | Siamese Cat | 4.20E-06 | Wheat | 3.76E-06 |
Wall Mirror | 1.95E-10 | Sugar | 1.52E-07 | Race Horse | 1.47E-07 | Green Pepper | 3.96E-07 |
Door Bell | 5.94E-07 | Vinegar | 5.93E-04 | Fox | 2.26E-05 | Watercress | 2.60E-07 |
Hammock | 2.35E-06 | Sesame Seeds | 5.49E-07 | Donkey | 1.79E-06 | Peanut | 7.62E-05 |
Desk | 2.94E-05 | Lemon Juice | 2.79E-05 | Field Mouse | 1.20E-05 | Black Pepper | 6.54E-06 |
Refrigerator | 2.41E-07 | Chocolate | 3.06E-06 | Ginger Tom-cat | 9.79E-06 | Garlic | 2.38E-07 |
Park Bench | 1.09E-06 | Horseradish | 1.55E-06 | Husky in Slead team | 1.56E-05 | Yam | 3.68E-08 |
Waste Paper Basket | 2.41E-06 | Vanilla | 4.28E-07 | Cart Horse | 2.62E-08 | Elderberry | 8.82E-05 |
Sculpture | 1.43E-06 | Chives | 7.42E-06 | Chicken | 8.62E-08 | Almond | 1.04E-04 |
Sink Unit | 3.97E-07 | Root Ginger | 5.02E-06 | Doberman Guard Dog | 1.29E-04 | Lentils | 9.59E-07 |
Mantelpiece | 1.09E-05 | Molasses | 4.21E-06 | Goldfish | 1.89E-07 | Apple | 9.00E-07 |
Window Seat | 1.89E-05 | Salt | 9.37E-05 | Robin | 8.27E-07 | Parsley | 9.14E-08 |
Painting | 2.20E-07 | Peppermint | 2.50E-04 | Blue-tit | 2.90E-06 | Olive | 6.81E-08 |
Light Fixture | 2.99E-06 | Curry | 8.55E-06 | Collie Dog | 2.00E-06 | Chili Pepper | 1.60E-07 |
Kitchen Counter | 5.12E-06 | Oregano | 1.25E-06 | Camel | 1.35E-07 | Broccoli | 1.22E-06 |
Bath Tub | 1.43E-06 | MSG | 1.79E-06 | Squirrel | 5.85E-06 | Root Ginger | 7.24E-06 |
Deck Chair | 1.07E-04 | Chili Pepper | 2.20E-07 | Guide Dog for Blind | 4.55E-06 | Pumpkin | 1.17E-07 |
Shelves | 7.84E-07 | Mustard | 5.57E-06 | Spider | 6.15E-05 | Raisin | 4.75E-08 |
Rug | 4.99E-09 | Mint | 1.80E-06 | Homing Pigeon | 1.18E-06 | Acorn | 9.32E-08 |
Bed | 2.73E-06 | Cinnamon | 5.03E-06 | Monkey | 2.95E-05 | Mustard | 4.61E-08 |
Wall-Hangings | 3.93E-06 | Parsley | 7.52E-06 | Circus Horse | 2.27E-05 | Rice | 9.76E-06 |
Space Rack | 2.16E-08 | Saccarin | 2.63E-06 | Prize Bull | 2.17E-06 | Tomato | 4.63E-07 |
Ashtray | 8.83E-07 | Poppy Seeds | 3.05E-06 | Rat | 4.08E-06 | Coconut | 3.58E-06 |
Bar | 3.07E-05 | Pepper | 4.70E-06 | Badger | 8.59E-05 | Mushroom | 1.39E-04 |
Lamp | 8.94E-07 | Turmeric | 2.06E-07 | Siamese Cat | 1.98E-05 | Wheat | 5.03E-07 |
Wall Mirror | 1.05E-06 | Sugar | 2.45E-06 | Race Horse | 5.03E-06 | Green Pepper | 8.15E-07 |
Door Bell | 6.27E-07 | Vinegar | 1.89E-05 | Fox | 9.66E-05 | Watercress | 1.26E-07 |
Hammock | 4.82E-06 | Sesame Seeds | 6.89E-07 | Donkey | 2.41E-06 | Peanut | 3.58E-06 |
Desk | 5.97E-06 | Lemon Juice | 1.64E-06 | Field Mouse | 1.29E-03 | Black Pepper | 3.97E-07 |
Refrigerator | 3.82E-05 | Chocolate | 9.12E-06 | Ginger Tom-cat | 1.64E-06 | Garlic | 1.15E-05 |
Park Bench | 2.16E-08 | Horseradish | 1.31E-05 | Husky in Slead team | 1.01E-07 | Yam | 1.62E-09 |
Waste Paper Basket | 7.21E-08 | Vanilla | 1.22E-07 | Cart Horse | 2.26E-07 | Elderberry | 1.98E-08 |
Sculpture | 4.46E-08 | Chives | 3.81E-07 | Chicken | 4.98E-05 | Almond | 2.17E-06 |
Sink Unit | 3.64E-06 | Root Ginger | 1.02E-08 | Doberman Guard Dog | 2.40E-06 | Lentils | 7.69E-06 |
Mantelpiece | 2.38E-05 | Molasses | 6.35E-05 | Goldfish | 1.51E-05 | Apple | 9.93E-05 |
Window Seat | 4.88E-04 | Salt | 1.63E-04 | Robin | 2.85E-03 | Parsley | 3.03E-05 |
Painting | 6.06E-05 | Peppermint | 3.01E-03 | Blue-tit | 4.54E-04 | Olive | 1.33E-06 |
Light Fixture | 5.64E-04 | Curry | 7.06E-04 | Collie Dog | 1.39E-05 | Chili Pepper | 3.47E-05 |
Kitchen Counter | 1.55E-05 | Oregano | 1.33E-04 | Camel | 5.17E-04 | Broccoli | 7.60E-03 |
Bath Tub | 9.61E-05 | MSG | 2.91E-05 | Squirrel | 5.10E-05 | Root Ginger | 1.71E-04 |
Deck Chair | 2.96E-04 | Chili Pepper | 8.75E-05 | Guide Dog for Blind | 2.75E-05 | Pumpkin | 2.55E-04 |
Shelves | 6.06E-05 | Mustard | 2.14E-03 | Spider | 1.06E-02 | Raisin | 7.20E-06 |
Rug | 1.09E-05 | Mint | 2.62E-03 | Homing Pigeon | 8.10E-04 | Acorn | 1.68E-05 |
Bed | 3.00E-05 | Cinnamon | 3.65E-04 | Monkey | 5.59E-03 | Mustard | 1.75E-06 |
Wall-Hangings | 1.12E-04 | Parsley | 1.33E-04 | Circus Horse | 5.80E-04 | Rice | 4.90E-03 |
Space Rack | 3.28E-06 | Saccarin | 6.42E-06 | Prize Bull | 3.59E-04 | Tomato | 4.41E-04 |
Ashtray | 1.55E-05 | Poppy Seeds | 2.05E-03 | Rat | 1.29E-03 | Coconut | 1.86E-04 |
Bar | 2.34E-05 | Pepper | 7.60E-03 | Badger | 2.48E-02 | Mushroom | 2.62E-03 |
Lamp | 2.26E-05 | Turmeric | 5.96E-04 | Siamese Cat | 2.71E-04 | Wheat | 2.76E-05 |
Wall Mirror | 5.58E-06 | Sugar | 2.81E-04 | Race Horse | 1.28E-03 | Green Pepper | 2.75E-02 |
Door Bell | 1.64E-05 | Vinegar | 5.31E-05 | Fox | 9.46E-02 | Watercress | 1.41E-03 |
Hammock | 2.04E-04 | Sesame Seeds | 1.17E-03 | Donkey | 6.94E-03 | Peanut | 1.20E-05 |
Desk | 1.36E-05 | Lemon Juice | 1.35E-07 | Field Mouse | 2.22E-02 | Black Pepper | 1.10E-03 |
Refrigerator | 2.55E-05 | Chocolate | 3.03E-05 | Ginger Tom-cat | 2.79E-04 | Garlic | 6.14E-05 |
Park Bench | 2.93E-05 | Horseradish | 2.80E-06 | Husky in Slead team | 3.22E-05 | Yam | 1.13E-06 |
Waste Paper Basket | 7.87E-09 | Vanilla | 2.07E-06 | Cart Horse | 2.98E-04 | Elderberry | 1.58E-05 |
Sculpture | 2.41E-06 | Chives | 1.08E-06 | Chicken | 2.66E-02 | Almond | 6.63E-06 |
Sink Unit | 7.90E-06 | Root Ginger | 7.32E-04 | Doberman Guard Dog | 2.64E-07 | Lentils | 6.54E-05 |
Mantelpiece | 1.09E-06 | Molasses | 3.14E-07 | Goldfish | 2.89E-07 | Apple | 1.46E-05 |
Window Seat | 8.03E-05 | Salt | 1.56E-05 | Robin | 2.23E-02 | Parsley | 5.26E-05 |
Painting | 5.38E-05 | Peppermint | 1.56E-05 | Blue-tit | 1.63E-04 | Olive | 4.77E-04 |
Light Fixture | 1.16E-03 | Curry | 1.83E-04 | Collie Dog | 2.06E-04 | Chili Pepper | 5.16E-05 |
Kitchen Counter | 3.02E-03 | Oregano | 1.11E-03 | Camel | 5.93E-03 | Broccoli | 1.83E-04 |
Bath Tub | 4.21E-06 | MSG | 2.60E-07 | Squirrel | 2.64E-03 | Root Ginger | 2.14E-03 |
Deck Chair | 4.45E-06 | Chili Pepper | 1.71E-07 | Guide Dog for Blind | 5.83E-04 | Pumpkin | 5.09E-03 |
Shelves | 5.06E-04 | Mustard | 4.08E-06 | Spider | 2.57E-02 | Raisin | 2.38E-05 |
Rug | 1.17E-05 | Mint | 1.13E-04 | Homing Pigeon | 9.81E-03 | Acorn | 1.37E-05 |
Bed | 4.44E-05 | Cinnamon | 5.03E-07 | Monkey | 9.47E-03 | Mustard | 3.07E-05 |
Wall-Hangings | 7.21E-04 | Parsley | 6.14E-05 | Circus Horse | 1.87E-03 | Rice | 6.15E-04 |
Space Rack | 1.38E-06 | Saccarin | 6.59E-08 | Prize Bull | 4.22E-04 | Tomato | 5.09E-04 |
Ashtray | 5.03E-06 | Poppy Seeds | 1.87E-04 | Rat | 1.14E-02 | Coconut | 2.81E-04 |
Bar | 1.74E-08 | Pepper | 3.52E-04 | Badger | 1.18E-02 | Mushroom | 4.84E-04 |
Lamp | 2.24E-07 | Turmeric | 1.32E-06 | Siamese Cat | 3.69E-05 | Wheat | 1.45E-04 |
Wall Mirror | 1.52E-04 | Sugar | 2.60E-06 | Race Horse | 7.53E-03 | Green Pepper | 4.63E-03 |
Door Bell | 5.64E-06 | Vinegar | 1.50E-05 | Fox | 1.97E-02 | Watercress | 1.68E-03 |
Hammock | 2.32E-04 | Sesame Seeds | 3.41E-05 | Donkey | 4.90E-03 | Peanut | 1.16E-04 |
Desk | 6.37E-05 | Lemon Juice | 1.16E-06 | Field Mouse | 1.18E-02 | Black Pepper | 3.76E-03 |
Refrigerator | 1.71E-05 | Chocolate | 8.26E-08 | Ginger Tom-cat | 1.20E-03 | Garlic | 4.62E-05 |
Park Bench | 1.04E-04 | Horseradish | 7.46E-08 | Husky in Slead team | 2.62E-03 | Yam | 3.20E-05 |
Waste Paper Basket | 2.88E-06 | Vanilla | 1.36E-06 | Cart Horse | 3.42E-04 | Elderberry | 3.72E-04 |
Sculpture | 1.28E-04 | Chives | 6.15E-05 | Chicken | 1.92E-03 | Almond | 2.73E-06 |
Sink Unit | 1.89E-04 | Root Ginger | 6.54E-05 | Doberman Guard Dog | 1.15E-05 | Lentils | 6.94E-05 |
Mantelpiece | 4.34E-09 | Molasses | 4.33E-07 | Goldfish | 3.98E-09 | Apple | 1.24E-08 |
Window Seat | 5.12E-06 | Salt | 4.04E-05 | Robin | 1.09E-05 | Parsley | 1.03E-08 |
Painting | 1.07E-07 | Peppermint | 7.51E-07 | Blue-tit | 3.54E-07 | Olive | 1.44E-08 |
Light Fixture | 5.01E-07 | Curry | 3.31E-06 | Collie Dog | 1.55E-07 | Chili Pepper | 2.81E-09 |
Kitchen Counter | 4.63E-06 | Oregano | 1.95E-07 | Camel | 1.19E-05 | Broccoli | 6.15E-09 |
Bath Tub | 1.13E-07 | MSG | 5.67E-07 | Squirrel | 1.99E-05 | Root Ginger | 1.79E-06 |
Deck Chair | 3.04E-07 | Chili Pepper | 2.19E-10 | Guide Dog for Blind | 2.43E-05 | Pumpkin | 6.60E-07 |
Shelves | 7.84E-08 | Mustard | 5.67E-07 | Spider | 5.24E-05 | Raisin | 7.37E-09 |
Rug | 8.18E-09 | Mint | 9.21E-08 | Homing Pigeon | 5.23E-06 | Acorn | 8.26E-09 |
Bed | 3.69E-08 | Cinnamon | 2.42E-08 | Monkey | 5.05E-04 | Mustard | 9.61E-08 |
Wall-Hangings | 1.09E-06 | Parsley | 2.61E-07 | Circus Horse | 7.10E-07 | Rice | 3.68E-06 |
Space Rack | 3.77E-08 | Saccarin | 1.20E-07 | Prize Bull | 1.27E-08 | Tomato | 1.82E-09 |
Ashtray | 9.08E-08 | Poppy Seeds | 1.24E-06 | Rat | 5.07E-05 | Coconut | 6.54E-07 |
Bar | 2.27E-09 | Pepper | 8.33E-07 | Badger | 3.01E-04 | Mushroom | 5.77E-06 |
Lamp | 1.87E-08 | Turmeric | 5.34E-08 | Siamese Cat | 1.34E-07 | Wheat | 9.21E-08 |
Wall Mirror | 2.20E-09 | Sugar | 5.03E-07 | Race Horse | 5.77E-08 | Green Pepper | 6.54E-07 |
Door Bell | 1.62E-07 | Vinegar | 3.40E-05 | Fox | 9.66E-04 | Watercress | 6.57E-08 |
Hammock | 7.17E-07 | Sesame Seeds | 1.07E-07 | Donkey | 2.38E-06 | Peanut | 3.51E-07 |
Desk | 7.94E-07 | Lemon Juice | 4.30E-07 | Field Mouse | 8.23E-04 | Black Pepper | 9.33E-07 |
Refrigerator | 5.49E-07 | Chocolate | 5.18E-08 | Ginger Tom-cat | 6.79E-07 | Garlic | 2.51E-07 |
Park Bench | 1.39E-08 | Horseradish | 5.03E-08 | Husky in Slead team | 3.99E-08 | Yam | 1.60E-10 |
Waste Paper Basket | 1.38E-09 | Vanilla | 6.49E-08 | Cart Horse | 8.26E-09 | Elderberry | 1.60E-07 |
Sculpture | 7.78E-09 | Chives | 1.80E-08 | Chicken | 1.53E-06 | Almond | 9.09E-08 |
Sink Unit | 2.66E-07 | Root Ginger | 6.10E-08 | Doberman Guard Dog | 3.86E-08 | Lentils | 3.47E-07 |
1One typically gains insight into how people combine concepts by gathering data on “typicalities” or “membership weights.” To obtain data on “typicalities,” participants are given a concept, and a list of instances or exemplars, and asked to pick which exemplar they consider most typical of the concept. A membership weight is instead obtained by asking people to estimate the membership of specific exemplars with respect to a concept. This estimation can, e.g., be quantified by using 7-point (Likert) scale and then converted into a relative frequency and then into a probability called the “membership weight.” We have worked on both typicality measurements, as in the analysis of the Guppy effect, and membership weights measurements, as in the analysis of Hampton's experiments and in the present paper.
2Remark that, if we set
3We introduce in this model a superposition vector with equal weights on the two vectors. The general case of a weighted superposition can be considered in future investigation, and it is an interesting line of research in itself, as the interpretation of the weights is not trivial.
4We remind that