Quantum Probability, Paraconsistent Logic and Decision Making in Cognitive Domain Sisir Roy National Institute of Advanced Studies,IISC Campus, Bangalore 560012 India e-mail:
[email protected]
Abstract The concept of Quantum probability is used to understand the failure of classical total probability rule in cognitive domain. This is an abstract framework and needs to be contextualized in cognitive domain especially in the context of neuronal architecture. Again as the neurobiological data does not permit to define orthogonal eigenvectors in the functional space of neurons, the use of Hilbert space in quantum probability poses a challenge from realistic point of view. However, quantum analogue of total probability rule within the framework of Paraconsistent Bayesian probability theory is considered as a viable alternative from more realistic perspective. The complementarity principle and the functioning of brain is discussed within the framework of Paraconsistent logic
Keywords: Quantum Probability,Paraconsistent Logic, Bayesian Probability, Complementarity, Bayes’ rule, Decision Making.
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1
Introduction
Bayesian approach to inference has drawn large attention to the community working on broad spectrum of cognitive science. During last decade, many authors addressed the issues like animal learning1 , motor control2 ,visual perception3 , language processing4 , semantic memory5 and acquisition etc, using Bayes’ rule. Many such research programmes have been continuing in order to study these aspects since last decade or so. However, the great challenge is to know about the mysteries, present in the human mind which goes beyond the data, experienced or, in other words, how mind internalizes the external world out of the noisy data, collected through the sensory organs. This is the most challenging aspects of cognition not only from the point of view of computation but also the version of age-old problem of induction in Western thoughts. The same problem lies at the core of the debates which deals with building machines having human-like intelligence robotics. Recent empirical findings6 raise new debate whether classical Bayesian approach is sufficient to build the model for decision making in cognitive science. Several authors7 introduced the concept of Quantum probability as better mathematical tool in building the model for cognitive science. It seems it can explain consistently the observed data so far taken from various observations. Their presumtion is that quantum mechanical formalism is not needed to understand the functioning of the brain but the abstract concept of quantum probability is useful to understand the cognitive function. The reason is that the framework of quntum probability is considered to be quite abstract framework which is devoid of material content like the concept of elementary particle say photon, electron etc. Mittelstaed et al8 developed ”quantum ontology” which is a very comprehensive framework and purely an abstract framework. It can be applied to any branch of knowledge like biology, social science etc.However, one needs contextualization so that the concept of probability can be applied in a particular area (under present consideration) say cognitive domain or more specifically at the level of neuronal architecture. In this type of abstract framework it is necessary to introduce the Hilbert space structure so as to use the quantum probability. Pellinisz and Llinas9 and subsequently Roy and Llinas10 developed a geometric framework attached to functional states of neurons called Dynamic Geometry or Probabilistic Functional Space. However, it is not clear at which level of neuronal hierarchy the structure of Hilbert space is valid. Ofcourse one can build up a geometric space like Hilbert space in the cognitive domain as an abstract space but it immediately faces the same problem like contextualization in the context of neuronal architecture. It is worth mentioning that recently Salazar et al 11 showed the analogue of quantum total probability rule within the Bayesian probability theory12 which appears to be very promising approach in understanding brain function and shown to be extension of logic within Bayesian approach The logic associated to the above mentioned abstract framework of quantum probability is shown to be related to quantum logic which gives rise to contra-
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dictions within the framework of Aristotelian frame work of logic. However, the developments of more generalized logic popularly known as paraconsistent logic can handle the contradictions as mentioned in quantum logic in a logical way. Infact, Salazar et al11 in their above approach considered the framework of Paraconsistent Bayesain Probability theory where analogue of quantum total probability rule can be constructed. We emphasize that instead of considering quantum probability and Hilbert space structure we can start with a Paraconsitent Bayesian probability theory to explain the empirical data in the cognitive level in a biologically more realistic manner. In section II, we discuss Quantum probability and Bayesian probability theory for convenience. The Disjunction effect and Complementarity will be dealt in the framework of Paraconsistent logic and functioning of the brain in section III. Some useful discussions are made in section IV.
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Quantum Probability and Bayesian Probability Theory
Several attempts13 have been made to understand Quantum probability within Bayesian framework. In Bayesian approach the probability quantifies the degree of belief even for a single trial without having prior connection to frequency interpretation. It was Pierre-Simon Laplace who popularized this approach what is now called Bayesian probability. The interpretation of probability within Bayesian approach can be thought of as an extension of logic that enables reasoning with hypotheses i.e. propositions whose truth or falsity is uncertain, where frequentist inference is typically treated without being assigned a probability. It is generally believed that Bayesian probability theory as an extended logic reproduces many aspects of human mental activities and is thought of as a subjective statement quantifying the knowledge. Thus this theory started to appear as objective during late 1800s and early 1900s. Statistical methods are used to derive things which can be measured in the laboratory, for example, heat. Here, the quantity like heat arises because of probabilistic considerations where probabilities are objective as well. Then came quantum theory which is considered as a complete theory by Bohr and other people of Copenhagen school. According to Copenhagen interpretation all the features of quantum theory should be the objective features of nature. In this paradigm, those probabilities corresponding to quantum states should also be the objective features of nature. On the otherhand, Einstein considered quantum mechanics as incomplete theory where the probabilities are considered as statements of incomplete knowledge. Suppose we consider the probability as a description of uncertainty and ignorance. Then we can take a spectrum of positions. According to statistician I.J. Good14 there exists 46,656 varieties. In this connection Jaynes15 made a remarkable statement We will admit that probabilities are in our heads, my probabilities are in my head, your probabilities are in your head but if I base my probabilities on the same infor3
mation that you base yours on, our two probability assignments should be the same. Conditioned on the information, they should be objective. In the spectrum of 46,656 varieties, this stance is called objective Bayesianism. On the otherside de Finetti considered it a different manner. Fuchs16 elaborated it as: there is no reason whatsoever for my probabilities and yours to match, because mine are based on my experience and yours are based on your experience. The best we can do, in that case, if we think of probabilities as gambling attitudes, is try to make all of our personal gambling attitudes internally consistent. I should do that with mine, and you with yours, but thats the best we can do. Thats what de Finetti meant when he said probability does not exist. He meant, lets take the extreme stance. Instead of saying probabilities are mostly in my head but there are some extra rules that still anchor them to the world, he got rid of the anchor. Recent advances in understanding the quantum reality lead to the proposal of quantum ontology. The wave function as an abstract vector enters into an algorithm. It predicts accurately the results of measurement. However, Griffiths et al 17 tried to construct a consistent quantum ontology based on two main ideas: • The logic, constructed here, is shown to be compatible with the Hilbertspace structure of quantum mechanics. • The quantum evolution is considered to be inherently stochastic process, not just only when measurement takes place. The main advantage of this framework is that it has no measurement problem. The consistent quantum ontology essentially resembles the ontology of classical mechanics represented by deterministic Hamiltonian. It is well known that the complete description of a quantum system is given by a wave function which is defined as a vector in Hilbert space (an abstract vector space). This statement becomes difficult in understanding the concerned problem from realistic point of view. In classical mechanics, in Maxwell theory of electromagnetism, as well as in General Theory of Relativity, the waves act on the particles and guide their motion. In this kind of very case, the term Ontology is used for something which exists independent of human observation or even independent of the existence of human race but whose evolution is governed by laws of physics. Now, let us start with the question: What is really going on in the nature when we are not giving any attention to it or not at all interested to think it is necessary to know or need to have prior knowledge about it? Only when we require definite answer to this kind of inquiry i.e., a reality outside of the thinking or perceiving the very subject, the objective reality arises. Whenever we are asked how and in what sense this so called anything, may exist when we are not looking at it or independent of our interest, only then we are making an ontological inquiry. This means we are getting the act of reality of being as such, without considering the attributes of this or the 4
entity itself. Following the same argument, when questioned about the reality of wave functions, characterizing every possible kind of physical entity, we are really asking the question of ontology.In quantum paradigm, it is very hard to introduce such concept called quantum ontology. Mittelstaedt8 discussed about the existence of possible quantum ontology where no material content of the microscopic domain is used, rather, the language and logic used for the description of microscopic entities.This way, both classical ontology (i.e. the ontology of classical mechanics) and quantum ontology are investigated within this abstract framework. Since this framework is devoid of any material content it can also be applied to any branch of knowledge and hence, can be applied not only to physics but also to the field like Biology, Social science etc. But,here the most challenging issue remains in the question i.e., how to apply it to a particular branch of knowledge. In modern physics, fundamental constants like Planck constant (h), speed of light (c) and gravitational constant (G) play very important role and one should take care of these constants within the above abstract framework of quantum ontology. Mittlestaed discussed this problem within the framework of operational quantum theory, based on the concept of unsharp observables. They showed how quantum ontological framework can be made context dependent so as to include the characteristics of microscopic domain. In principle, it can be applied to the domain of cognitive science. However, ultimate aim, here, is to study the workability of the above framework, connected to context dependence, also in cognitive domain. Mittlesteadt concludes that quantum mechanics is more intuitive than classical mechanics and that classical mechanics can not be taken as the macroscopic limit of quantum mechanics using the concept of unshrp observables. It is now well established that Quantum-mechanical event descriptions are basically context-dependent descriptions. The role of quantum (non-distributive) logic is in the partial ordering of contexts rather than in the ordering of quantum-mechanical events. Not only that the kind of quantum logic displayed by quantum mechanics can easily be inferred from the general notion of contextuality, used in ordinary language. P.A.Heelan18 established that the formalizable core of Bohr’s notion of complementarity is the type of context dependence which can be applied, in principle, to the domain of cognitive science. However, the issue is to make the above framework context dependent specially for cognitive domain. Recently, Tanaka et al19 applied Non-Kolmogorovian approach in order to study the context-dependent systems for which the violations of classical probability law were observed. According to their views, the irreducible contextuality of biosystem is the main source of quantum representation of information about the behavior and dynamics of these systems. They showed that the contextuality and adaptivity of biosystems leads to violation of laws of classical probability theory (based on Kolmogorov axiomatic, 1933). According to their observations these features of quantum probability having non-Kolomogorovian character can be used for the analysis of some phenomena which are connected
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to statistical properties but quite out of the arena of quantum physics. Actually, they did point out the violations of total probability law that can be noticed in experimental findings in biology and also noticed these characteristics also, while studying several contextual phenomena in cognitive domain. However,so far as the author’s knowledge goes, the issue of context dependence in cognitive domain remains one of the most challenging problem as yet to be solved. Quantum probability in quantum ontological framework is linked to density matrices in Hilbert space. Let us now discuss the quantum analogue of total probability rule within the framework of Paraconsistent Bayesian Probability theory11 . The main idea lies in the fact that since according to Bayesian approach the probability theory is an extension of logic, one can get the quantum analogue by considering different logic known as Paraconsistent logic. This logic is designed in such a way that it can handle the theory in which inconsistencies or contradictions might arise without leading to trivialization or logical explosion. It is required to assign probability to the occurrence of contradictions in Paraconsisten Bayesian probability theory. These probabilities enter in the total probability rule depending on their values. Let us describe briefly the total probability rule in this approach in comparison to rule of total probability in quantum theory. Salazar et al11 derived total quantum probability rule as 2
d −1 Y X Y X 1X X Y d+1 X Q( |I)Q( | )− Q( | ) Q( |ρ) = d d k=0
k
k
k
k
This is equivalent to Born’s rule which gives the rule ofPtotal probability allowing us to predict the transition probabilityQto a state P from Q the initial state of the system given by the probabilities Q( k |I) and Q( | k ). This is calculated using the representation of quantum states as 2
2
d −1 −1 Y X Y Y 1 dX d+1 X Q( |I) − Q( |ρ) = d d k=0
k
k
k=0
k
Then they derived the total quantum probability rule using Paraconsistent logic as
P (B|
2 dX −1
k=0
ek , I) d + 1 A d
2 dX −1
k=0
2
d −1 1 X e ek , I) P (Ak |I)P (B|Ak , I) − P (B|A d k=0
Q It is to be noted that in Quantum mechanics the set Q( k |I) provides all the information needed to associate to a quantum state ρ. In Paraconsistent Bayesian Probability theory the set P (Ak |I) represents our state of knowledge about the truth values of the set Ak of complete, possibly contradictory and mutually exclusive propositions. Birkoff and von Neumann20 noticed the relationship between the emergence of contradictions and the nonorthogonality of
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quantum states. For this they changed the rules of classical logic by modifying the distribution identities of conjunction and disjunction in order to resolve the above mentioned contradictions. On the otherhnd, in the framework of Paraconsistent logic the logical contradictions are not eliminated but incorporated in the formalism itself. Next question arises about the applicability of Paraconsistent Bayesian Probability theory in the domain of cognition at the cellular level. This will be discussed in subsequent section.
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Paraconsistent Logic and Decision making in Cognitive Domain
Macnamara21 discussed logic and cognition in a comprehensive manner and opined that two subjects i.e. logic and cognition are either remotely connected or not connected at all. Logic which deals with the specifying correct inference belongs to the domain of philosophy where as the other one i.e. cognition is dealt in the domain of psychology as it is usually viewed as the mental activity. But the debate lies in the relation between the logic and psychology. One group maintains that foundations of logic are based on psychological basis. Jacob Fries Friedrich22 and John Stuart Mill23 shared this view. On the contrary Edmond Husserl24 in his book Logical investigations argued that • Logic does not derive its basic principles from psychology. • Logic does not describe psychological states and events. • Susan Hacck25 took the third position, i.e.,Logic has nothing to do with the mental processes. Macnamara21 proposed that logic and psychology constrains each other like the mathematics and physics constraints each other. For example, calculus was discovered as a mathematical language for analysis of continuum which has been used to understand Newtonian law of gravitation. Similarly, logic is considered as a mathematical language to understand the cognition processes. It is wellknown fact that the works on decision making consider the logic to be an integral part of understanding in the decision making process. George Boole26 , a English mathematician developed a system of logical thought known as Boolean logic. AND, OR and NOT are common Boolean operations used in this logical system. Truth tables and Boolean algebra are used to describe logical expressions where the principle of formal logic is the principle of identity. Boolean logic is extensively used in computer science. True or false are two values which can be taken by a variable within Boolean framework. The applicability of the Boolean algebra is effective, especially; to that particular formal machinery where it as allowed to logically combine this kind of truth values. Thus the mechanism of Boolean operations offer a framework by which it is possible to handle the questionnaire and also the data in
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such a way so that this can be handled by common binary kid (Yes or No). In various decision making processes, for example, clinical decisions, even measurements, probabilities, etc., it is needed to have the data, very often, to be dichotomized (binarized). Many common examples can be found in many instruments, say, digital computer, large number of electronic switches like on and off which are connected in a Boolean manner. However, the framework of Boolean operations and also their logical combinations underlie logical checking of rule-based decision support systems for, say, inconsistencies, incompleteness and redundancy. Recently, Aerts et al27 , Busemeyers and others28 performed a series of experiments, taking human as subjects. They observed that, in plenty of cases, there exists a clear violation of traditional probability theory. Intensive literature survey also indicates, the classical probability theory fails to explain human cognition modeling, beyond a certain limit. The results of the above mentioned experiments can be classified into following six groups as a cognition spectrum of human mind: • Disjunction effect • Categorization decision interaction • Perception of ambiguous figures • Conjunction and Disjunction fallacies • Overextension of Category membership • Memory recognition over-distribution effect. Fallacies over-distribution effect. . From the above experiments, based on gambling ,the law of additivity of probabilities of occurring of two mutually exclusive events (particle aspect or wave aspect) having found to be violated i.e. total probability PAB 6= PA + PB for two mutually exclusive events A and B. This is similar to the existence of interference effects observed in Double slit experiment in Quantum Theory to detect particle and or wave. This formula is known as Formula of Total Probability (FTP). So many authors emphasize to use the concept of Quantum Probability to explain the data in the above experiments. Again, they mentioned that the formalism of quantum theory as such should not be considered to describe the functioning of the brain function where as an abstract framework of quantum probability can be used to describe the new situation. Ofcourse, the Hilbert space geometry has to be assumed to be valid either in describing the neuronal architecture of Central Nervous System(CNS) or in an abstract space
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for cognition. In both the situations we need to understand the functioning of brain and cognitive activities in a more realistic manner. It is generally believed that the cerebellum’s function is to help the brain to coordinate movement29 . However, the recent neurophysiological evidence challenges this dogma30 , instead, indicates that apart from being just a specialized control box, the cerebellum participates in many activities of the brain including cognition. It becomes, then, necessary and crucially important to investigate the correspondence between the response and percept domains. Roy and Kafatos31 emphasized that in this case also, there exists a principle similar to the wellknown complementarity principle, present in quantum mechanics that operates for response and percept domains. Here, it is assumed that geometry of cerebellum’s is nothing but Hilbert space geometry. It is worth mentioning that Pribram32 constructed a geometry of neurodynamics so that the neural wave functions can be described by a Hilbert space similar to the wave-function in Hilbert space in ordinary quantum mechanics. He also emphasized two important aspects which should be considered in brain processes, i. e, both a nonlocality similar to the quantum nonlocality and the Hilbert space description for the geometric description. In this Hilbert space approach, the cortex is assumed to possess a system of eigenvectors forming a complete orthogonal system in the processing sense. However, much controversy arose during the last decade regarding the space-time representation in the brain, particularly for the cerebellum. Pellionisz and Llinas9 discussed the problems of space-time representation for the cerebellum in a series of publications. In fact, Braitenberg33 made a pioneering attempt towards our understanding of the space-time representation in the brain. Pellionisz and Llinas critically analyzed the situation and found difficulty in using relativistic coordinates. Pellionisz and Llinas9 and subsequently Roy and Llinas10 developed the dynamic geometry associated to Central Nervous System(CNS) to understand the functioning of the brain. A central question concerning present day neuroscience is that of understanding the rules for the embedding of universals into intrinsic functional space. One needs to understand the fundamental structure of this internal space or functional geometry before going into the details of its applications. They can be stated as follows: • For any geometry one needs to define the smoothness property of the manifold.Thus in functional geometry the existence of derivatives associated with the sensory covariant vector and motor contravariant vector is the prerequisite of the functional manifold, and so a definition of differentiability in the functional space becomes necessary. • Non-orthogonal coordinate axes have been considered to be associated with covariant and contravariant vectors based on physiological observations34,35 . So one must consider the non-orthogonal frame of references in this type of functional space. • The analysis of the contravariant character of the forces exerted by the 9
muscles that such a geometry must accommodate includes an overcomplete set of possible dynamic configurations associated with a non-orthogonal frame of reference leading to similar motor execution. Accordingly, the CNS must calculate an inverse solution via the mathematical indeterminacy inherent in the CNS in a manner similar to that implemented in Moore-Penrose based generalized inverse solutions 36 . • The motion of an object in the external world does not engender simultaneity of space and time in its counterpart functional space because the conduction speeds through various axons are different for a given stimulus. Let us consider the possible geometrical structures for the CNS. To define any metric tensor one needs to address a well-defined distance function that satisfies all the axioms of metric tensors. From a global point of view, the anatomy of the brain does not present a smooth and linear representation of the external world. So a distance function must be constructed which can be defined as functionally isotropic. In fact, the multidimensional spaces of the CNS are, for the most part, not definable in wellknown geometries such as Euclidean or Riemannian spaces. Indeed, if we consider, for instance the olfactory space with more than 10,000 different categories, we find that this space is defined by the chemical and biochemical properties of the odorant substance. This, in turn, can only be defined by the requirements of the organism. Accordingly, unlike Euclidean or Riemannian spaces, this olfactory space cannot be defined independently of the measurement instruments. It is well known in the tensor approach, that the metric tensor is gij = ei .ej where ei and ej are the unit vectors along the non-orthogonal axes. Here, if the angle between these vectors be uncertain, the metric will be uncertain. The weakly chaotic nature of neuronal oscillations 37 may give rise to the uncertainty in the angle which will be the stochastic nature of the functional space since the metric will be stochastic. Now the question is whether we can construct a distance function associated with this kind of stochastic space for the above type of functional space. Starting with the local chaotic behavior, which causes indeterminacy of the angles and hence the stochastic nature of the metric, the neuronal architecture at the global level (for which there exists synchronous oscillations) will support a smooth metric structure at this level. It is clear from this analysis that it is not possible to think of Hilbert space type of geometry with orthogonal basis vectors. It is not yet clear at which level of hierarchy pf neuronal architecture one think of orthogonality. In such situation, the quantum analog of probability sum rule in Praconsistent Bayesian Probability may be considered as a viable framework.
3.1
Bayesian Approach and Brain Function
Bayesian approach to inference has drawn large attention to the community working on broad spectrum of cognitive science. Bayesian rules, quite contrary to other variants of probability theory, help cognitive scientists to define the rules 10
for rationality. This rule updates the belief of the agents from new data in the light of information as well as the prior knowledge. In Bayesian models, probability computations are applied for explaining learning and reasoning instead of hypothesis space of possible concepts, word meanings, or causal laws. The structure of the learners hypothesis space reflects their domain specific prior knowledge, while the nature of the probability computations depends on the domain-general statistical principles. Bayesian models of cognition thus combine both of the two approaches that have historically been kept separate for their philosophical differences, providing a way to combine structured representations and domain specific knowledge with domain-general statistical learning. Battaliga et al 38 made a comprehensive overview of Bayesian modeling and Bayesian networks. According to their findings, the use of sensory information is found to be satisfactorily efficient, in making judgments and for the guidance of successful actions in the world. Added to this, it has been argued that the brain must represent and make use of the information gathered about uncertainty, both for perception and action in its computations. Applications of this technique have been successfully made in building computational theories for perception as well as for sensorimotor control. Not only that, sufficient evidences has been provided in case of psychophysics which has been putting examples and proof that Bayes optimal, the only answer to human perceptual computations. With these specific characteristics, the Bayesian coding hypothesis states that the brain represents sensory information probabilistically i.e., in the form of probability distributions. This type of approach, of course, puts special emphasis on the viability and degree of success connected to these schemes, with the aim, for making their model successful in dealing with the populations of neurons. Neurophysiological data on this hypothesis, however, is almost nonexistent. But, for neuroscientists, this situation poses a major challenge to test these ideas experimentally, i.e., how to determine through which possible process neurons code information about sensory uncertainty will be applicable with success. They mainly focused on three types of information processing, especially are: on inference, parameter learning and structure learning. These three types of operations are discussed in the context of Bayesian networks and human cognition. These types of Bayesian networks become more and more popular in the field of artificial intelligence and human cognition since the factorization of a joint distribution is expressed by graph theory where a network contains nodes, edges and probability distributions. It is to be noted that the models, developed, following Bayesian framework, do not follow the usual algorithmic or process level. On the otherhand, this characterizes more the usual traditional cognitive modeling but in the spirit of Marr’s computational theory.
3.2
Paraconsistent Logic and Complementarity
In 1927 Bohr introduced the concept of complementarity in quantum me-
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chanics. Let us start with Max Jammer’s statement A given theory T admits a complementarity interpretation if the following conditions are satisfied: • T contains (at least) two descriptions D1 and D2 of its substance-matter; • D1 and D2 refer to the same universe of discourse U (in Bohr’s case, microphysics); • neither D1 nor D2, if taken alone, accounts exhaustively for all phenomena of U; • D1 and D2 are mutually exclusive in the sense that their combination into a single description would lead to logical contradictions. It is further emphasized: Complementarity denotes the logical relation, of quite a new type, between concepts which are mutually exclusive, and which therefore cannot be considered at the same time that would lead to logical mistakes but which nevertheless must both be used in order to give a complete description of the situation. However, the advancement of Paraconsistent logic show us how to incorporate the contradictions or inconsistentcies in a logical way instead of discarding them. It immediately gives rise to the issue like how these complementary aspects or contradictions can be understood in cognitive domain or more specifically in the brain. Grossberg39 made a challenging attempt to understand how complementary aspects of the physical world are translated into complementary brain designs for coping with this world. The main idea lies in the fact that brains are organized into parallel processing streams with complementary properties. Now the interactions between parallel steams create behavioral representations. The present author along with his collaborator31 suggested a complementary principle for Response and Percept domain. The recent neurophysiological results suggest that cerebellar output projects via the thalamus into multiple cortical areas, including the premotor and prefrontal cortex, as well as the primary motor cortex. In addition, the projections to different cortical areas appear to originate from distinct regions of the deep cerebellar nuclei. Some neurobiologists40 have challenged this kind of distinct behavior of the cerebellum by studying some patients. However, they have acknowledged that cognitive and motor functions are integral constituents of the mechanism governing animal behavior. They analyzed substantial amounts of data that support the view that, if the cerebellum is important for executing a specific behavior, it also participates in any long or short-term modifications of the characteristics of the behavior. It is clear that there are many ambiguities regarding the various experimental results. More refined experiments and critical analysis of the data are needed so as to clarify our understanding of the cognition and control of movement. The problem arises regarding the integration process so as to coordinate dentate regions that are responsible for motor and cognitive actions. We want to emphasize that the generalized complementarity principle proposed here might play the role of 12
coordinator. It is clear from the neurophysiological data that there exist distinct regions responsible for mutually exclusive behaviors such as motor and cognitive but there may also exist other regions (as pointed out by Bloedel and Bracha’I’”), which are responsible for both kind of behaviors.
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Discussions
The above analysis clearly shows that the Bayesian probability theory based on Paraconsistent logic helps us to explain the observational data in cognitive domain using its quantum analogue of total probability rule rather than using quantum probability based on using Hilbert space structure. In such a stuation, we do not need to consider the applicability of quantum formalism to cognitive domain or brain function or the usual quantum probability concept related to Hilbert structure. The neurophysiological data indicates that the functioning of brain or mind is governed by Paraconsistent logic instead of Boolean logic. This will shed new light on the issue of applicability of quantum paradigm in cognitive science which will be discussed in subsequent publications. Acknowledgement: The author acknoledges the grant from , SERB,Department of Science and Technology, Government of India under the project No.SB/S4/MS:844/2013.
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