Can quantum probability provide a new direction for cognitive modeling
Summary (11 min read)
1.1. Why move toward quantum probability theory?
- In this article the authors evaluate the potential of quantum probability (QP) theory for modeling cognitive processes.
- Rather, the authors are interested in QP theory as a mathematical framework for cognitive modeling.
- Superposition, entanglement, incompatibility, and interference are all related aspects of QP theory, which endow it with a unique character.
- He served as chief editor of Journal of Mathematical Psychology from 2005 through 2010 and he is currently an associate editor of Psychological Review.
- Instead, QP defines conjunction between incompatible questions in a sequential way, such as “A and then B.”Crucially, the outcome of question A can affect the consideration of question B, so that interference and order effects can arise.
1.2. Why move away from existing formalisms?
- By now, the authors hope they have convinced readers that QP theory has certain unique properties, whose potential for cognitive modeling appears, at the very least, intriguing.
- Many of these findings relate to order/context effects, violations of the law of total probability (which is fundamental to Bayesian modeling), and failures of compositionality.
- On the one hand, there was the strong intuition from classical models (e.g., Newtonian physics, classical electromagnetism).
- This is exactly what makes CP (and QP) models appealing to many theorists and why, as noted, in seeking to understand the unique features of QP theory, it is most natural to compare it with CP theory.
- Note that the authors do not develop an argument that CP theory is unsuitable for cognitive modeling; it clearly is, in many cases.
2.1. The outcome space
- First, a sample space is defined, in which specific outcomes about a question are subsets of this sample space.
- Also, more general emotions, such as happiness, would be represented by subspaces of higher dimensionality.
- To determine the probability of the answer happy, the authors need to project the state represented by |Ψ〉 onto the subspace for “happy” spanned by the vector |happy〉.
- An important feature of QP theory is the distinction between superposition and basis states.
- Therefore, a decision, which causes a person to resolve the indefinite state regarding a question into a definite state, is not a simple read-out from a pre-existing definite state; instead, it is constructed from the current context and question (Aerts & Aerts 1995).
2.2. Compatibility
- Suppose that the authors are interested in two questions, whether the person is happy or not, and also whether the person is employed or not.
- Psychologically, incompatibility between questions means that a cognitive agent cannot formulate a single thought for combinations of the corresponding outcomes.
- Conversely, certainty about employment aligns the state vector with the subspace for employed, which makes the person somewhat uncertain about her happiness (perhaps her job is sometimes stressful).
- Order and context dependence of probability assessments (and, relatedly, the failure of commutativity in conjunction) are some of the most distinctive and powerful features of QP theory.
- Such intuitions can be readily realized in a QP framework through tensor product representations.
2.3. Time evolution
- So far, the authors have seen static QPmodels, whereby they assess the probability for various outcomes for a state at a single point in time.
- It is important to recall that the state vector is a superposition of components along different basis vectors.
- In CP theory, the time-evolved state directly gives the probabilities for the possible outcomes.
- Otherwise, QP theory can produce violations of the law of total probability.
- Suppose that the hypothetical person knows she will find out whether she will be employed or not, before having the inner reflection about happiness (perhaps she plans to think about her happiness after a professional review).
3. The empirical case for QP theory in psychology
- The authors explore whether the main characteristics of QP theory (order/context effects, interference, superposition, entanglement) provide us with any advantage in understanding psychological processes.
- Many of these situations concern Kahneman and Tversky’s hugely influential research program on heuristics and biases (Kahneman et al.
- The authors strategy is to first discuss how the empirical finding in question is inconsistent with CP theory axioms.
- This is not to say that some model broadly based on classical principles cannot be formulated.
- Such illustrations will be simplifications of the correspondingquantummodels.
3.1. Conjunction fallacy
- In a famous demonstration, Tversky and Kahneman (1983) presented participants with a story about a hypothetical person, Linda, who sounded very much like a feminist.
- The important comparison concerned the statements “Linda is a bank teller” (extremely unlikely given Linda’s description) and “Linda is a bank teller and a feminist.”.
- The state vector could not be placed in between the bank teller and feminist subspaces, as this would mean that it is has a high projection to both the bank teller and the feminist outcomes (only the latter is true).
- Psychologically, the QP model explains the conjunction fallacy in terms of the context dependence of probability assessment.
- Also, the QP model is compatible with the representativeness and availability heuristics.
3.2. Failures of commutativity in decision making
- The authors next consider failures of commutativity in decision making, whereby asking the same two questions in different orders can lead to changes in response (Feldman & Lynch 1988; Schuman & Presser 1981; Tourangeau et al.
- QP theory can accommodate order effects inGallup polls, in a way analogous to how the conjunction fallacy is explained.
- The two sets of basis vectors are not entirely orthogonal; the authors assume that if a person considers Clinton honest, then that person is a little more likely to consider.
- It can be seen that the direct projection is less, compared to the projection via the |Gore yes〉 vector.
- Trueblood and Busemeyer (2011) proposed a QP model for two such situations, a jury decision-making task (McKenzie et al. 2002) and a medical inference one (Bergus et al. 1998).
3.3. Violations of the sure thing principle
- The model Trueblood and Busemeyer (2011) developed is an example of a dynamic QP model, whereby the inference process requires evolution of the state vector.
- When participants were told that the opponent was going to cooperate, they decided to defect; and when they were told that the opponent was defecting, they decided to defect as well.
- Tversky and Shafir (1992) described such violations of the sure thing principle as failures of consequential reasoning.
- The same unitary operator also embodied the idea of wishful thinking, rotating the state vector so that the amplitudes for the “cooperate–cooperate” and “defect–defect” combinations for participant and opponent actions increased.
- Note that this quantum model is more complex than the ones considered previously.
3.4. Asymmetry in similarity
- The authors have considered how the QP explanation for the conjunction fallacy can be seen as a formalization of the representativeness heuristic (Tversky & Kahneman 1983).
- In some cases, the similarity of A to B would not be the same as the similarity of B to A. Tversky’s (1977) findings profoundly challenged the predominant approach to similarity, whereby objects are represented as points in a multidimensional space, and similarity is modeled as a function of distance.
- Pothos and Busemeyer (2011) proposed that different concepts in their experience correspond to subspaces of different dimensionality, so that concepts for which there is more extensive knowledge were naturally associated with subspaces of greater dimensionality.
- This is set so that it is neutral with respect to the A and B subspaces (i.e., prior to the similarity comparison, a participant would not be thinking more about A than about B, or vice versa).
- Tversky’s proposal was that symmetry is violated, because the authors have more extensive knowledge about China than about Korea, and, therefore, China has more distinctive features relative to Korea.
4.1 Can the psychological relevance of CP theory be
- It is always possible to augment a model with additional parameters or mechanisms to accommodate problematic results.
- Moreover, deviations from CP predictions in judgment could be explained by introducing assumptions of how participants interpret the likelihood of statements in a particular hypothesis, over and above what is directly stated (e.g., Sher & McKenzie 2008).
- Also, the introduction of post-hoc parameters will lead to models that are descriptive and limited in insight.
- Therefore, when obtaining psychological evidence for a formal framework, the authors do not just support the particular principles under scrutiny.
- There is a clear sense that if one wishes to pursue a formal, probabilistic approach for the Tversky, Kahneman type of findings, then CP theory is not the right choice, even if it is not actually possible to disprove the applicability of CP theory to such findings.
4.2. Heuristics vs. formal probabilistic modeling
- The critique of CP theory by Tversky, Kahneman and collaborators can be interpreted in a more general way, as a statement that the attempt to model cognition with any axiomatic set of principles is misguided.
- Many of these proposals sought to relate generic memory or similarity processes to performance in decision making (e.g., the availability and representativeness heuristics; Tversky & Kahneman 1983).
- Other researchers have developed heuristics as individual computational rules.
- Likewise, failures of consequential reasoning in prisoner’s dilemma (Tversky & Shafir 1992) can be formalized with quantum interference effects.
- The contrast between heuristic and formal probabilistic approaches to cognition is a crucial one for psychology.
4.3. Is QP theory more complex than CP theory?
- The authors have discussed the features of QP theory, which distinguish it from CP theory.
- Dynamic QP models must obey the law of double stochasticity, while CP Markov models can violate this law.
- More generally, a fundamental constraint of QP theory concerns Gleason’s theorem, namely that probabilities have to be associated with subspaces via the equation Prob(A|c) = ‖PA|cl‖2. Finding that Gleason’s theorem is psychologically implausible would rule out quantum models.
- Even if at a broad level CP and QP theories are subject to analogous constraints, a critic may argue that it is still possible that QP models are more flexible (perhaps because of their form).
- The models could still differ with respect to their complexity.
5. The rational mind
- Beginning with Aristotle and up until recently, scholars have believed that humans are rational because they are capable of reasoning on the basis of logic.
- Considerable evidence accumulated that naïve observers do not typically reason with classical logic (Wason 1960); therefore, classical logic could not be maintained as a theory of thinking.
- Finally, optimality is a key aspect of Anderson’s (1990) rational analysis and concerns the accuracy of probabilistic inference.
- Classical theory would assume that this story generates a sample space for all possible characteristic combinations for Linda, including unfamiliar ones such as feminist bank teller.
- Note that the perspective dependence of probabilistic assessment in QP theory may seem to go against an intuition that “objective” probabilities are somehow more valid or correct.
6.1 Theoretical challenges
- The results of Tversky, Kahneman, and colleagues (e.g., Tversky & Kahneman 1974) preclude a complete explanation of cognitive processes with CP theory.
- The authors have suggested that QP theory is the appropriate framework to employ for cases in which CP theory fails.
- In exploring such a proposal, the first step should be to identify the precise boundary conditions between the applicability and failure of CP principles in cognitive modeling.
- The results of Tversky et al. (1974) reveal situations in which this reliance breaks down.
- There is a further, potentially relevant literature on quantum information theory (Nielsen & Chuang 2000), which concerns the processing advantages of probabilistic inference based on QP theory.
6.2. Empirical challenges
- So far, the quantum program has involved employing quantum computational principles to explain certain, prominent empirical findings.
- Rather, the authors discussed results that have presented ongoing challenges and have resisted explanation based on classical principles.
- Trueblood and Busemeyer (2011) developed a model to accommodate order effects in the assessment of evidence in McKenzie et al.’s (2002) task.
- The model successfully described data from both the original conditions and a series of relevant extensions.
- Overall, understanding the quantum formalism to the extent that surprising, novel predictions for cognition can be generated is no simple task (in physics, this was a process that took several decades).
6.3. Implications for brain neurophysiology
- An unresolved issue is how QP computations are implemented in the brain.
- The authors have avoided a detailed discussion of this research area because, although exciting, is still in its infancy.
- The most controversial (Atmanspacher 2004; Litt et al. 2006) perspective is that the brain directly supports quantum computations.
- For quantum computation to occur, a system must be isolated from the environment, as environmental interactions cause quantum superposition states to rapidly decohere into classical states.
- Overall, in cognitive science it has been standard to initially focus on identifying the mathematical principles underlying cognition, and later address the issue of how the brain can support the postulated computations.
6.4. The future of QP theory in psychology
- There is little doubt that extensive further work is essential before all aspects of QP theory can acquire psychological meaning.
- But this does not imply that current QP models are not satisfactory.
- The purpose of this article is to argue that researchers attracted to probabilistic cognitive models need not be restricted to classical theory.
- Rather, quantum theory provides many theoretical and practical advantages, and its applicability to psychological explanation should be further considered.
Projectors (or projection operators)
- For a onedimensional subspace, corresponding, for example, to the |happy〉 ray, the projector is a simple outer product, Phappy = |happy〉 〈happy|.
- Given the above subspace for “happy,” the probability that a person is happy is given by ‖Phappy|cl‖2 = ‖happyl khappy|cl‖2.
- 〈happy|Ψ〉 is the standard dot product and |happy〉 is a unit length vector.
- The single lines on the right hand side denote the modulus of a complex number.
Composite systems
- Two subspaces can be combined into a composite space in two ways: one way is by forming a tensor product space (as in Figure 1b) and the other way is by forming a space from a direct sum.
- First consider the formation of a tensor product space.
- Suppose |happy〉, |∼happy〉 are two basis vectors that span the subspace H, representing the possibility of happiness, and suppose |employed〉, |∼employed〉 are two basis vectors that span the subspace E, representing the possibility of employment.
- Then, the tensor product space equals the span of the four basis vectors formed by the tensor products {|happyl⊗ |employedl, |happyl⊗ | employedl, | happyl⊗ |employedl, | happyl⊗ | employedl}.
- Next consider the formation of a space by direct sum.
Time dependence
- The quantum state vector changes over time according to Schrödinger’s equation, d dt |c(t)l = −i ·H · |c(t)l where H is a Hermitian linear operator.
- This is the QP theory equivalent of the Kolmogorov forward equation for Markov models in CP theory.
- The two (obviously related) operators H and U(t) contain all the information about the dynamical aspects of a system.
- Thus, the effect of U(t) on a state vector is to rotate it in a way that captures some dynamical aspect of the situation of interest.
An example of how interference can arise in QP theory
- Consider a situation whereby a person tries to assess whether she is happy or not, depending upon whether she is employed or not.
- Note that so far the situation is identical to what the authors would have had if they were applying a CP theory Markov model.
- This nonlinearity in QP theory can lead to interference terms that produce violations of the law of total probability.
- Therefore, regardless of the outcome regarding employment, the evolved state will be a state that is not a superposition one.
ACKNOWLEDGMENTS
- Research relevant to this work has been supported by the MIUR grant “Problem solving and decision making: Logical, psychological and neuroscientific aspects within criminal justice” (PRIN, n.2010RP5RNM_006) and by Grant CR 409/1-1 from the Deutsche Forshungsgemeinshaft (DFG) as part of the prority program New Frameworks of Rationality (SPP 1516).
- The debate, the evidence, and the future doi:10.
Institute for Frontier Areas of Psychology, D-79098 Freiburg, Germany; Collegium Helveticum, CH-8092 Zurich, Switzerland.
- It was an old idea by Niels Bohr, one of the founding architects of quantum physics, that central features of quantum theory, such as complementarity, are also of pivotal significance beyond the domain of physics.
- The proper framework for a logic of incompatible propositions is a partial Boolean lattice (Primas 2007), where locally Boolean sublattices are pasted together in a globally non-Boolean way – just like an algebra of generally non-commuting operations may contain a subset of commuting operations.
- The authors use the notion of “quantum probability” for psychological and cognitive models and their predictions (cf. Gudder 1988; Redei & Summers 2007).
- Whereas Kolmogorov probabilities refer to events for a single condition, quantum probabilities refer to the entire set of incompatible conditions, necessary for a comprehensive description of the experiment.
13083-886 SP, Brazil.
- Pothos & Busemeyer’s (P&B’s) query about whether quantum probability can provide a foundation for the cognitive modeling embodies so many underlying implications that the subject is far from exhausted, also known as Abstract.
- Quantum superposition is commonly considered to be a mapping of two bit states into one.
- In their target article, Pothos & Busemeyer (P&B) elegantly argue that there may be quantum principles – notably superposition and entanglement – at play in the context of human cognitive behavior.
- From the point of view of the process of subsuming information, the material meaningfully incorporated within an individual’s cognitive structure is never lost, but a process called “forgetting” takes place in a much more spontaneous manner, because it is a continuation of the very process of associative subsumption by which one learns.
- In support of this idea, Todd (1999) also advocated that the unit of information embedded in Brookes’ theory is a concept derived from Ausubel’s learning theory.
Pittsburgh, PA 15213.
- Quantum probability (QP) theory provides an alternative account of empirical phenomena in decision making that classical probability (CP) theory cannot explain, also known as Abstract.
- Here, the authors argue that cognitive architectures, a modeling approach with a long history in the cognitive sciences, may also address the outlined challenges.
- Whereas a formal probability theory such as QP and CP represent the latter as a logical conjunction, they are represented as independent instances in memory in a computational theory such as ACT-R.
- When eschewing a formal probabilistic framework in favor of a computational account, apparent impossibilities simply dissolve in light of the cognitive processes used to actually produce the decisions.
NOTE
- ACT-R code, publications, and models are available at http://act-r. psy.cmu.edu.
- Attributes and associations to the options in the Linda problem.
- 286 BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3 Quantum probability and comparative cognition doi:10.
United Kingdom.
- The authors concentrate on two aspects of the article by Pothos & Busemeyer (P&B): the relationship between classical and quantum probability and quantum probability as a basis for rational decisions.
- From a mathematical point of view, CP is embedded as a special case in the more general non-commutative (also referred to as “quantum”) probability theory.
- It is worth noting that the decisions of nonhuman animals violate the principles of rational decision making (Houston et al. 2007b).
- With these examples, the authors are trying to illustrate that the macroscopic world of decisions is more complex than traditional models of decision theory assume.
Montreal, QC H3A 1B1, Canada.
- Pothos & Busemeyer (P&B) present the Dirac formalism of quantum probability (DQP) as a potential direction for cognitive modeling.
- P&B do not show how the framework could be used to build predictive theories: all the examples listed are post hoc descriptive models.
- Local and global hidden variable theories cannot be distinguished, and a classical explanation cannot be ruled out.
- DQP is restricted to closed systems, but the mind is an open system, and, therefore, NQP yields a better modeling framework.
Kingdom.
- Quantum probability models for choice in large worlds may be motivated pragmatically – there is no third theory – or metaphysically: statistical processing in the brain adapts to the true scale-relative structure of the universe.
- The premise that the CP-based theory of decision –which is better characterized as a theory of incentive response – has been refuted by experimental evidence is questionable.
- This would invite the question as to whether there is some general feature of the world that explains why both fundamental physical structure and fundamental cognitive structure follow QP rather than CP.
- Many philosophers presume a kind of atomism that is inconsistent with quantum physics (Ladyman & Ross 2007).
Los Angeles, CA 90032.
- Pothos & Busemeyer (P&B) argue that classical probability (CP) fails to describe human decision processes accurately and should be supplanted by quantum probability, also known as Abstract.
- To use Baron’s (2004) terminology, CP may be useful as a prescriptive theory of behavior, but not as a descriptive theory.
- The focus is on how the answers interplay: happiness might be more probable if a person is employed (because he or she has money) or less probable (because she hates her job).
- The authors are too focused on mathematics at the expense of usefulness.”.
3G1, Canada.
- Abstract:Quantum probability (QP) theory can be seen as a type of vector symbolic architecture (VSA): mental states are vectors storing structured information and manipulated using algebraic operations.
- This allows existing biologically realistic neural models to be adapted to provide a mechanistic explanation of the cognitive phenomena described in the target article by Pothos & Busemeyer (P&B).
- If HAPPY is a particular 500-dimensional vector, and EMPLOYED is a different 500-dimensional vector, then HAPPY⊛EMPLOYED gives a new 500-dimensional vector (a tensor product would give a 250,000-dimensional vector).
- Resolving this ambiguity will be a key test of QP.
R1. Beyond classical probability (CP) theory: The potential of quantum theory in psychology
- As the authors mentioned in their main article, quantum probability theory simply refers to the theory for assigning probabilities to events developed in quantum mechanics, without any of the physics (cf. Aerts, Broekaert, Gabora, & Sozzo [Aerts et al.]).
- Moreover, a sense of probabilistic determinism can arise in quantum theory in a way analogous to that of classical theory: in quantum theory, if it is likely that thinking that Gore is honest makes Clinton likely to be honest too, then the subspaces for the corresponding outcomes are near to each other.
- It will not always be the case that a person thinking that Gore is honest will also think that Clinton is honest, but, on average, this will be the case.
- Contrary to what Lee & Vanpaemel suggest, the objective of quantum cognitive models is exactly to provide insight into those aspects of cognitive process for which classical explanation breaks down.
- Gonzalez & Lebiere rightly point out that cognitive architectures, such as Adaptive Character of Thought – Rational (ACT-R), go some way toward addressing their criticisms of approaches based on individual heuristics.
R2. Misconceptions on limitations
- Even given the broad description of the theory in the target article, the authors were impressed that some commentators were able to develop their own variations of quantum models.
- It is true that the authors motivate properties of quantum theory, notably incompatibility, partly by appeal to the unrealistic demands from the principle of unicity.
- Processing one question plausibly 312 BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3 interferes with knowledge about another; the available empirical results strongly indicate this to be the case, at least in some cases (e.g., the conjunction fallacy; Busemeyer et al. 2011).
- Equally, the specific characteristics of quantum evolution in quantum models do not always map well onto cognitive processes, and in some cases classical models appear more successful (Busemeyer et al. 2006).
- Note first that Probability (Clinton honest) ∗ Probability (Gore honest |Clinton honest) is the probability of deciding that Clinton is honest and Gore is honest.
R3. Empirical and theoretical extensions
- Aerts et al. point out that it is not just QP that is relevant.
- Rather, there many aspects of quantum theory that are potentially relevant to the modeling of cognition.
- Their demonstrations could be extended in a way such that the question of interest can have an answer along a continuum, rather than a binary yes–no.
- By contrast, if the emphasis is on ensembles (e.g., how the behavior of a whole group of people changes), then perhaps mixed states are more appropriate (Franceschetti & Gire).
- The authors knowledge space would be populated with several possible subspaces, corresponding to questions that relate to their world knowledge.
R4. Empirical challenges
- Whether researchers accept the quantum framework as a viable alternative to CP theory is partly an empirical issue.
- If the authors were to adopt a representation analogous to the one for the Linda problem, they would need an initial state that exists within a subspace for pet fish, which corresponds to the concept of guppy.
- This is one examination for one particular quantum model, but this examination does provide the only available evidence and this evidence does its small bit toward undermining a claim that quantum models are in general more flexible than matched classical ones.
- Information with regard to each question can be evaluated without the appeal to the other” to conclude that these questions cannot be incompatible.
- Exactly howprecise the specification of subspaces and the state vector is will depend on exactly how precise the authors require the predictions to be.
R5. Neural basis
- It is possible to utilize quantum theory to build models of neural activity.
- The issue of quantum neural processing, as described previously, is distinct to that of the neural implementation of quantum cognitive models.
- Noori & Spanagel argue that brain neurobiology is deterministic and, moreover, that it deterministically specifies behavior.
- But in this comment, he advances his ideas by providing a detailed discussion for how his orchestrated objective reduction (Orch OR) model can be extended to incorporate subspaces and projections for complex thoughts, as would be required in quantum cognitive models.
- It is currently unclear whether such dimensionality increases present a problem in relation to neural implementation.
R6. Rationality
- The question of whether an account of human rationality (or not) should emerge from quantum cognitive models partly relates to their intended explanatory level.
- Aerts, D. (1986) A possible explanation for the probabilities of quantum mechanics.
- Proceedings of the 32nd Annual Conference of the Cognitive Science Society, ed. S. Ohlsson & R. Cattrambone, pp. 2188–93, also known as In.
- Trueblood, J. S. & Busemeyer, J. R. (2011) A comparison of the belief-adjustment model and the quantum inference model as explanations of order effects in human inference.
- Tversky, A. & Kahneman, D. (1974) Judgment under uncertainty: Heuristics and biases.
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Frequently Asked Questions (9)
Q2. What are the future works mentioned in the paper "Can quantum probability provide a new direction for cognitive modeling?" ?
There is little doubt that extensive further work is essential before all aspects of QP theory can acquire psychological meaning. In fact, the authors argue that the quantum approach to cognition embodies all the characteristics of good cognitive theory: it is based on a coherent set of formal principles, the formal principles are related to specific assumptions about psychological process ( e. g., the existence of order/context effects in judgment ), and it leads to quantitative computational models that can parsimoniously account for both old and new empirical data. Rather, quantum theory provides many theoretical and practical advantages, and its applicability to psychological explanation should be further considered.
Q3. What is the only system that works in physics?
For the real, noisy, confusing, ever-changing, chaotic world, QP is the only system that works in physics and, the authors strongly suspect, in psychology as well.
Q4. What is the reason why probability is interpreted as similarity?
Because in QP theory probability is computed from the overlap between a vector and a subspace, it is naturally interpreted as similarity (Sloman 1993).
Q5. What is the example of a holism in memory research?
Another example from memory research is Bruza et. al.’s (2009) application of quantum entanglement (which implies a kind of holism inconsistent with classical notions of causality) to explain associativememoryfindings,which cannot beaccommodated within the popular theory of spreading activation.
Q6. What is the influential demonstration of similarity?
In one of the most influential demonstrations in the similarity literature, Tversky (1977) showed that similarity judgments violate all metric axioms.
Q7. What is the probability of defecting when the opponent is known to cooperate?
the probability of defecting when the opponent is known to cooperate is based on the projection Pparticipant to D |Ψopponent known C〉. But, in the unknown case, the relevant state vector is the superposition 1 2√ |copponent known Dl+ 1 2√ |copponent known Cl.
Q8. What is the effect of deciding that gore is honest?
In other words, deciding that Gore is honest increases the probability that Clinton is judged to be honest as well (and, conversely,deciding that Clinton is honest first, reduces the probability that Gore is judged as honest).
Q9. Who was known to believe that aspects of quantum theory could provide insight about cognitive process?
one of the founding fathers of quantum theory, was known to believe that aspects of quantum theory could provide insight about cognitive process (Wang et al., in press).