Toward a Quantum Theory of Cognition: History, Development and Perspectives

Transcription

1 Toward a Quantum Theory of Cognition: History, Development and Perspectives Tomas Veloz 1,2,3 tveloz@gmail.com 1 Center Leo Apostel for Interdisciplinary Studies, Vrije Universiteit Brussel, Belgium 2 Department of Mathematics, University of British Columbia Okanagan campus, 3333 University Way, Kelowna BC, V1V 1V7, CANADA 3 Instituto de Filosofía y Ciencias de la Complejidad - IFICC. Los Alerces 3024 Ñuñoa, Chile. October 26, 2015 Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

2 Bonjour! These...may be termed conditions of possible experience. When satisfied they indicate that the data may have, when not satisfied they indicate that the data cannot have resulted from an actual observation. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

3 Bonjour! These...may be termed conditions of possible experience. When satisfied they indicate that the data may have, when not satisfied they indicate that the data cannot have resulted from an actual observation. (G. Boole, 1862.) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

4 Bonjour! These...may be termed conditions of possible experience. When satisfied they indicate that the data may have, when not satisfied they indicate that the data cannot have resulted from an actual observation. (G. Boole, 1862.) Topic: Foundations of (Cognitive) Science. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

5 Bonjour! These...may be termed conditions of possible experience. When satisfied they indicate that the data may have, when not satisfied they indicate that the data cannot have resulted from an actual observation. (G. Boole, 1862.) Topic: Foundations of (Cognitive) Science. Related to: PhD(c) Intrerdisciplinary Studies (Quantum Cognition), UBC, Canada. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

6 Bonjour! These...may be termed conditions of possible experience. When satisfied they indicate that the data may have, when not satisfied they indicate that the data cannot have resulted from an actual observation. (G. Boole, 1862.) Topic: Foundations of (Cognitive) Science. Related to: PhD(c) Intrerdisciplinary Studies (Quantum Cognition), UBC, Canada. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

7 Summary The Classical Notion of Experience Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

8 Summary The Classical Notion of Experience The Classical Notion of Experience in Cognition Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

9 Summary The Classical Notion of Experience The Classical Notion of Experience in Cognition Quantum Cognition Decision-Making Paradoxes Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

10 The Classical Notion of Experience The Notion of Possible Experience George Boole introduced the notion of possible experience as part of his investigation of the Laws of Thought [2] Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

11 The Classical Notion of Experience The Notion of Possible Experience George Boole introduced the notion of possible experience as part of his investigation of the Laws of Thought [2] A possible experience constrain the relative frequencies (probabilities) of an observational procedure Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

12 The Classical Notion of Experience The Notion of Possible Experience George Boole introduced the notion of possible experience as part of his investigation of the Laws of Thought [2] A possible experience constrain the relative frequencies (probabilities) of an observational procedure For example, consider an urn with 100 balls and two experiments E A = The ball is red and E B = The ball is wooden, and the outcomes A, B = yes for these experiments. The following constrains must be satisfied µ(ab) µ(a), µ(ab) µ(b), (1) µ(a) + µ(b) µ(ab) 1 (2) where µ( ) represents the probability of an outcome, and AB is the outcome (yes,yes) of the joint experiment E A E B. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

13 The Classical Notion of Experience The Notion of Possible Experience George Boole introduced the notion of possible experience as part of his investigation of the Laws of Thought [2] A possible experience constrain the relative frequencies (probabilities) of an observational procedure For example, consider an urn with 100 balls and two experiments E A = The ball is red and E B = The ball is wooden, and the outcomes A, B = yes for these experiments. The following constrains must be satisfied µ(ab) µ(a), µ(ab) µ(b), (1) µ(a) + µ(b) µ(ab) 1 (2) where µ( ) represents the probability of an outcome, and AB is the outcome (yes,yes) of the joint experiment E A E B. Ex. µ(a) = 0.7, µ(b) = 0.42, and µ(ab) = 0.1 is not a possible experience. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

14 The Classical Notion of Experience The Mathematics of the Conditions of Possible Experience The conditions of possible experience imply the existence of a classical probabilstic representation, and are always satisfied if experiments are performed on a single sample Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

15 The Classical Notion of Experience The Mathematics of the Conditions of Possible Experience The conditions of possible experience imply the existence of a classical probabilstic representation, and are always satisfied if experiments are performed on a single sample They can be deduced for logical system of n experiments, each one having k i outcomes, i = 1,..., n, by means of: Correlation polytopes [3], probabilistic logic (Nilson, 1986), linear programming (Hansen, et. al., 1992), catergory theory (Abramsky, 2013). Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

16 The Classical Notion of Experience The Mathematics of the Conditions of Possible Experience The conditions of possible experience imply the existence of a classical probabilstic representation, and are always satisfied if experiments are performed on a single sample They can be deduced for logical system of n experiments, each one having k i outcomes, i = 1,..., n, by means of: Correlation polytopes [3], probabilistic logic (Nilson, 1986), linear programming (Hansen, et. al., 1992), catergory theory (Abramsky, 2013). Mathematical interests: Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

17 The Classical Notion of Experience The Mathematics of the Conditions of Possible Experience The conditions of possible experience imply the existence of a classical probabilstic representation, and are always satisfied if experiments are performed on a single sample They can be deduced for logical system of n experiments, each one having k i outcomes, i = 1,..., n, by means of: Correlation polytopes [3], probabilistic logic (Nilson, 1986), linear programming (Hansen, et. al., 1992), catergory theory (Abramsky, 2013). Mathematical interests: 1. Representation 2. Satisfiability 3. Bounds 4. Optimization Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

18 The Classical Notion of Experience in Cognition Conditions of Possible Experience in Cognition Since the conditions of possible experience bridge logic, probability, and experimentation, they also entail the foundations of most cognitive theories Remarkable cases: Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

19 The Classical Notion of Experience in Cognition Conditions of Possible Experience in Cognition Since the conditions of possible experience bridge logic, probability, and experimentation, they also entail the foundations of most cognitive theories Remarkable cases: 1. Expert Systems (Traditional AI) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

20 The Classical Notion of Experience in Cognition Conditions of Possible Experience in Cognition Since the conditions of possible experience bridge logic, probability, and experimentation, they also entail the foundations of most cognitive theories Remarkable cases: 1. Expert Systems (Traditional AI) 2. Fuzzy set theory and Possibility Theory (Categorization) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

21 The Classical Notion of Experience in Cognition Conditions of Possible Experience in Cognition Since the conditions of possible experience bridge logic, probability, and experimentation, they also entail the foundations of most cognitive theories Remarkable cases: 1. Expert Systems (Traditional AI) 2. Fuzzy set theory and Possibility Theory (Categorization) 3. Decision theories (Economics) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

22 The Classical Notion of Experience in Cognition Conditions of Possible Experience in Cognition Since the conditions of possible experience bridge logic, probability, and experimentation, they also entail the foundations of most cognitive theories Remarkable cases: 1. Expert Systems (Traditional AI) 2. Fuzzy set theory and Possibility Theory (Categorization) 3. Decision theories (Economics) 4. Bayesian Models (Modern AI) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

23 The Classical Notion of Experience in Cognition Conditions of Possible Experience in Cognition Since the conditions of possible experience bridge logic, probability, and experimentation, they also entail the foundations of most cognitive theories Remarkable cases: 1. Expert Systems (Traditional AI) 2. Fuzzy set theory and Possibility Theory (Categorization) 3. Decision theories (Economics) 4. Bayesian Models (Modern AI) 5. Database Consistency (Computer Science) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

24 The Classical Notion of Experience in Cognition Conditions of Possible Experience in Cognition Since the conditions of possible experience bridge logic, probability, and experimentation, they also entail the foundations of most cognitive theories Remarkable cases: 1. Expert Systems (Traditional AI) 2. Fuzzy set theory and Possibility Theory (Categorization) 3. Decision theories (Economics) 4. Bayesian Models (Modern AI) 5. Database Consistency (Computer Science) 6. ETC. However, there is a body of experimental evidence that challenges the validity of the conditions of possible experience in cognition Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

25 The Classical Notion of Experience in Cognition Example: Overextension of Conjunction Given two concepts A and B, and a set of exemplars Σ, we can test the membership µ k of each exemplar x k Σ w.r.t A, B and their conjunction A and B. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

26 The Classical Notion of Experience in Cognition Example: Overextension of Conjunction Given two concepts A and B, and a set of exemplars Σ, we can test the membership µ k of each exemplar x k Σ w.r.t A, B and their conjunction A and B. From a (Fuzzy) logical perspective, we expect that for all x k Σ the following holds: µ k (A and B) µ k (A), and µ k (A and B) µ k (B) (3) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

27 The Classical Notion of Experience in Cognition Example: Overextension of Conjunction Given two concepts A and B, and a set of exemplars Σ, we can test the membership µ k of each exemplar x k Σ w.r.t A, B and their conjunction A and B. From a (Fuzzy) logical perspective, we expect that for all x k Σ the following holds: µ k (A and B) µ k (A), and µ k (A and B) µ k (B) (3) However, psychological findings show strong overextensions (inversions of (3)) in experimental data (Osherson & Smith, 1981; Hampton, 1988). Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

28 The Classical Notion of Experience in Cognition Example of Data Consider the concepts A = Furniture, B = Household Appliances, and their conjunction AB= Furniture and Household Appliances Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

29 The Classical Notion of Experience in Cognition Example of Data Consider the concepts A = Furniture, B = Household Appliances, and their conjunction AB= Furniture and Household Appliances Participants estimate membership µ( ) for 16 objects (e.g., ashtray, vacuum cleaner, coffee table,etc) [6] Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

30 The Classical Notion of Experience in Cognition Example of Data Consider the concepts A = Furniture, B = Household Appliances, and their conjunction AB= Furniture and Household Appliances Participants estimate membership µ( ) for 16 objects (e.g., ashtray, vacuum cleaner, coffee table,etc) [6] We would expect µ(ab) µ(x ), X = A, B. However, all objects are overextended (red points are doubly overextended) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

31 The Classical Notion of Experience in Cognition Other Examples of Non-classical Cognitive Data Borderline Contradiction: John is tall and not tall...makes sense! Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

32 The Classical Notion of Experience in Cognition Other Examples of Non-classical Cognitive Data Borderline Contradiction: John is tall and not tall...makes sense! µ(x and x) > 0 Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

33 The Classical Notion of Experience in Cognition Other Examples of Non-classical Cognitive Data Borderline Contradiction: John is tall and not tall...makes sense! µ(x and x) > 0 Conjunction Fallacy Linda is more likely to be a bank-teller and a feminist, than a bank-teller only Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

34 The Classical Notion of Experience in Cognition Other Examples of Non-classical Cognitive Data Borderline Contradiction: John is tall and not tall...makes sense! µ(x and x) > 0 Conjunction Fallacy Linda is more likely to be a bank-teller and a feminist, than a bank-teller only P(BT, F ) > P(BT ) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

35 The Classical Notion of Experience in Cognition Other Examples of Non-classical Cognitive Data Borderline Contradiction: John is tall and not tall...makes sense! µ(x and x) > 0 Conjunction Fallacy Linda is more likely to be a bank-teller and a feminist, than a bank-teller only P(BT, F ) > P(BT ) Order Effects answer to first question modify answer to second question Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

36 The Classical Notion of Experience in Cognition Other Examples of Non-classical Cognitive Data Borderline Contradiction: John is tall and not tall...makes sense! µ(x and x) > 0 Conjunction Fallacy Linda is more likely to be a bank-teller and a feminist, than a bank-teller only P(BT, F ) > P(BT ) Order Effects answer to first question modify answer to second question P(A, B) P(B, A) Ellsberg and Machina Paradox Uncertainty about irrelevant variables matter Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

37 The Classical Notion of Experience in Cognition Other Examples of Non-classical Cognitive Data Borderline Contradiction: John is tall and not tall...makes sense! µ(x and x) > 0 Conjunction Fallacy Linda is more likely to be a bank-teller and a feminist, than a bank-teller only P(BT, F ) > P(BT ) Order Effects answer to first question modify answer to second question P(A, B) P(B, A) Ellsberg and Machina Paradox Uncertainty about irrelevant variables matter P(A) P(A, B) + P(A, B) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

38 The Classical Notion of Experience in Cognition ETC! Tomas Veloz (CLEA) Quantum Cognition October 26, / 25 Other Examples of Non-classical Cognitive Data Borderline Contradiction: John is tall and not tall...makes sense! µ(x and x) > 0 Conjunction Fallacy Linda is more likely to be a bank-teller and a feminist, than a bank-teller only P(BT, F ) > P(BT ) Order Effects answer to first question modify answer to second question P(A, B) P(B, A) Ellsberg and Machina Paradox Uncertainty about irrelevant variables matter P(A) P(A, B) + P(A, B)

39 The Classical Notion of Experience in Cognition Violations of Possible Experience and Quantum Probability Quantum phenomena violate the conditions of possible experience because it is contextual, non-compositional, and vague Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

40 The Classical Notion of Experience in Cognition Violations of Possible Experience and Quantum Probability Quantum phenomena violate the conditions of possible experience because it is contextual, non-compositional, and vague The notion of possible experience strongly depend on the properties of the observation process...how is the cognitive observation process? Contextual, Non-compositional, Vague?...Quantum? Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

41 Quantum Cognition Quantum Cognition is an area that dedicates to identify quantum structure and develop quantum-like modeling in domains such as Cognitive Science, Economics, Artificial Intelligence, etc. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

42 Quantum Cognition Quantum Cognition is an area that dedicates to identify quantum structure and develop quantum-like modeling in domains such as Cognitive Science, Economics, Artificial Intelligence, etc. Q-structure? Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

43 Quantum Cognition Quantum Cognition is an area that dedicates to identify quantum structure and develop quantum-like modeling in domains such as Cognitive Science, Economics, Artificial Intelligence, etc. Incompatible Measurements, Q-structure? Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

44 Quantum Cognition Quantum Cognition is an area that dedicates to identify quantum structure and develop quantum-like modeling in domains such as Cognitive Science, Economics, Artificial Intelligence, etc. Q-structure? Incompatible Measurements, State Superposition/Interference Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

45 Quantum Cognition Quantum Cognition is an area that dedicates to identify quantum structure and develop quantum-like modeling in domains such as Cognitive Science, Economics, Artificial Intelligence, etc. Q-structure? Incompatible Measurements, State Superposition/Interference Entanglment Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

46 Quantum Cognition Quantum Cognition is an area that dedicates to identify quantum structure and develop quantum-like modeling in domains such as Cognitive Science, Economics, Artificial Intelligence, etc. Q-structure? Incompatible Measurements, State Superposition/Interference Entanglment Etc. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

47 Quantum Cognition Quantum Cognition is an area that dedicates to identify quantum structure and develop quantum-like modeling in domains such as Cognitive Science, Economics, Artificial Intelligence, etc. Q-structure? Incompatible Measurements, State Superposition/Interference Entanglment Etc. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

48 Quantum Cognition Quantum Cognition is an area that dedicates to identify quantum structure and develop quantum-like modeling in domains such as Cognitive Science, Economics, Artificial Intelligence, etc. Q-structure? Q-Models Incompatible Measurements, State Superposition/Interference Entanglment Etc. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

49 Quantum Cognition Quantum Cognition is an area that dedicates to identify quantum structure and develop quantum-like modeling in domains such as Cognitive Science, Economics, Artificial Intelligence, etc. Q-structure? Q-Models Incompatible Measurements, State Superposition/Interference Entanglment Etc. Categorization (Aerts, Sozzo, Veloz 2015) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

50 Quantum Cognition Quantum Cognition is an area that dedicates to identify quantum structure and develop quantum-like modeling in domains such as Cognitive Science, Economics, Artificial Intelligence, etc. Q-structure? Q-Models Incompatible Measurements, State Superposition/Interference Entanglment Etc. Categorization (Aerts, Sozzo, Veloz 2015) Decision Making (Sozzo 2011, Busemeyer 2013) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

51 Quantum Cognition Quantum Cognition is an area that dedicates to identify quantum structure and develop quantum-like modeling in domains such as Cognitive Science, Economics, Artificial Intelligence, etc. Q-structure? Q-Models Incompatible Measurements, State Superposition/Interference Entanglment Etc. Categorization (Aerts, Sozzo, Veloz 2015) Decision Making (Sozzo 2011, Busemeyer 2013) Information Retrieval (Van Rijsbergen 2004) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

52 Quantum Cognition Quantum Cognition is an area that dedicates to identify quantum structure and develop quantum-like modeling in domains such as Cognitive Science, Economics, Artificial Intelligence, etc. Q-structure? Q-Models Incompatible Measurements, State Superposition/Interference Entanglment Etc. Categorization (Aerts, Sozzo, Veloz 2015) Decision Making (Sozzo 2011, Busemeyer 2013) Information Retrieval (Van Rijsbergen 2004) etc. (QI Proceedings, ). Quantum Cognition does not follow or take a position w.r.t. Quantum brain hypothesis Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

53 Summary of Quantum Cognition: Key steps Quantum Cognition models assume that Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

54 Summary of Quantum Cognition: Key steps Quantum Cognition models assume that 1. Entites exist in a state rather than being a container of instantiations 2. Entities are observed in one particular state, usually observation-dependent Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

55 Summary of Quantum Cognition: Key steps Quantum Cognition models assume that 1. Entites exist in a state rather than being a container of instantiations 2. Entities are observed in one particular state, usually observation-dependent These assumptions lead to the introduction of collapse, observer effect, and non-symmetry (non-classical probability) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

56 Summary of Quantum Cognition: Key steps Quantum Cognition models assume that 1. Entites exist in a state rather than being a container of instantiations 2. Entities are observed in one particular state, usually observation-dependent These assumptions lead to the introduction of collapse, observer effect, and non-symmetry (non-classical probability) Next, assuming that 3 Entities can be combined, and states of combined entities can be different to product states of the former entities states Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

57 Summary of Quantum Cognition: Key steps Quantum Cognition models assume that 1. Entites exist in a state rather than being a container of instantiations 2. Entities are observed in one particular state, usually observation-dependent These assumptions lead to the introduction of collapse, observer effect, and non-symmetry (non-classical probability) Next, assuming that 3 Entities can be combined, and states of combined entities can be different to product states of the former entities states lead to interference and entanglement effects, and to the modelling of entities with quantum states (bra-ket and wave function) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

58 Summary of Quantum Cognition: Key steps Quantum Cognition models assume that 1. Entites exist in a state rather than being a container of instantiations 2. Entities are observed in one particular state, usually observation-dependent These assumptions lead to the introduction of collapse, observer effect, and non-symmetry (non-classical probability) Next, assuming that 3 Entities can be combined, and states of combined entities can be different to product states of the former entities states lead to interference and entanglement effects, and to the modelling of entities with quantum states (bra-ket and wave function) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

59 Example: Two modes of reasoning about conjunction Consider the experimental situation of a participant estimating the membership weight of banana with respect to the concepts Fruit, and Vegetable, and their conjunction Fruit and Vegetable. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

60 Example: Two modes of reasoning about conjunction Consider the experimental situation of a participant estimating the membership weight of banana with respect to the concepts Fruit, and Vegetable, and their conjunction Fruit and Vegetable. Two kinds of reasoning can be identified: Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

61 Example: Two modes of reasoning about conjunction Consider the experimental situation of a participant estimating the membership weight of banana with respect to the concepts Fruit, and Vegetable, and their conjunction Fruit and Vegetable. Two kinds of reasoning can be identified: 1. banana being an exemplar of the concept Fruit and Vegetable, Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

62 Example: Two modes of reasoning about conjunction Consider the experimental situation of a participant estimating the membership weight of banana with respect to the concepts Fruit, and Vegetable, and their conjunction Fruit and Vegetable. Two kinds of reasoning can be identified: 1. banana being an exemplar of the concept Fruit and Vegetable, 2. banana being an exemplar of the concept Fruit, and the concept Vegetable, separately. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

63 Example: Two modes of reasoning about conjunction Consider the experimental situation of a participant estimating the membership weight of banana with respect to the concepts Fruit, and Vegetable, and their conjunction Fruit and Vegetable. Two kinds of reasoning can be identified: 1. banana being an exemplar of the concept Fruit and Vegetable, 2. banana being an exemplar of the concept Fruit, and the concept Vegetable, separately. In the first case, a single instance of banana is taken into consideration, and the membership is estimated with respect to the meaning of a single concept Fruit and Vegetable. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

64 Example: Two modes of reasoning about conjunction Consider the experimental situation of a participant estimating the membership weight of banana with respect to the concepts Fruit, and Vegetable, and their conjunction Fruit and Vegetable. Two kinds of reasoning can be identified: 1. banana being an exemplar of the concept Fruit and Vegetable, 2. banana being an exemplar of the concept Fruit, and the concept Vegetable, separately. In the first case, a single instance of banana is taken into consideration, and the membership is estimated with respect to the meaning of a single concept Fruit and Vegetable. In the second case, two instances of banana are taken into consideration, one for with respect to the meaning of Fruit, and the other with respect to Vegetable. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

65 Example: Two modes of reasoning about conjunction Consider the experimental situation of a participant estimating the membership weight of banana with respect to the concepts Fruit, and Vegetable, and their conjunction Fruit and Vegetable. Two kinds of reasoning can be identified: 1. banana being an exemplar of the concept Fruit and Vegetable, 2. banana being an exemplar of the concept Fruit, and the concept Vegetable, separately. In the first case, a single instance of banana is taken into consideration, and the membership is estimated with respect to the meaning of a single concept Fruit and Vegetable. In the second case, two instances of banana are taken into consideration, one for with respect to the meaning of Fruit, and the other with respect to Vegetable. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

66 Modes of Thought Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

67 Quantum Modeling of the two Modes of Thought Each concept exist in a different state ( A, B H), and a membership operator M is introduced for each exemplar. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

68 Quantum Modeling of the two Modes of Thought Each concept exist in a different state ( A, B H), and a membership operator M is introduced for each exemplar. µ(a) = A M A, µ(a) = B M B Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

69 Quantum Modeling of the two Modes of Thought Each concept exist in a different state ( A, B H), and a membership operator M is introduced for each exemplar. µ(a) = A M A, µ(a) = B M B First Mode (Non-classical): The combination is a new emergent state AB 1 = 1 2 ( A + B ) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

70 Quantum Modeling of the two Modes of Thought Each concept exist in a different state ( A, B H), and a membership operator M is introduced for each exemplar. µ(a) = A M A, µ(a) = B M B First Mode (Non-classical): The combination is a new emergent state AB 1 = 1 2 ( A + B ) µ(ab 1 ) = AB 1 M AB 1 = 1 (µ(a) + µ(b)) + R( A M B ). 2 Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

71 Quantum Modeling of the two Modes of Thought Each concept exist in a different state ( A, B H), and a membership operator M is introduced for each exemplar. µ(a) = A M A, µ(a) = B M B First Mode (Non-classical): The combination is a new emergent state AB 1 = 1 2 ( A + B ) µ(ab 1 ) = AB 1 M AB 1 = 1 (µ(a) + µ(b)) + R( A M B ). 2 Second Mode (Classical): The combination is a compositional state AB 2 = A B H H (can be an entangled state) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

72 Quantum Modeling of the two Modes of Thought Each concept exist in a different state ( A, B H), and a membership operator M is introduced for each exemplar. µ(a) = A M A, µ(a) = B M B First Mode (Non-classical): The combination is a new emergent state AB 1 = 1 2 ( A + B ) µ(ab 1 ) = AB 1 M AB 1 = 1 (µ(a) + µ(b)) + R( A M B ). 2 Second Mode (Classical): The combination is a compositional state AB 2 = A B H H (can be an entangled state) µ(ab) = AB M M AB = µ(a)µ(b). Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

73 Quantum Modeling of the two Modes of Thought Each concept exist in a different state ( A, B H), and a membership operator M is introduced for each exemplar. µ(a) = A M A, µ(a) = B M B First Mode (Non-classical): The combination is a new emergent state AB 1 = 1 2 ( A + B ) µ(ab 1 ) = AB 1 M AB 1 = 1 (µ(a) + µ(b)) + R( A M B ). 2 Second Mode (Classical): The combination is a compositional state AB 2 = A B H H (can be an entangled state) µ(ab) = AB M M AB = µ(a)µ(b). Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

74 Fock Space and Superposition of Modes of Thought Definition The Fock space is the Hilbert space made from the direct sum of tensor products of single-particle Hilbert spaces: F = n=0 H n Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

75 Fock Space and Superposition of Modes of Thought Definition The Fock space is the Hilbert space made from the direct sum of tensor products of single-particle Hilbert spaces: F = n=0 H n In particular, we can model the two interpretations underlying the combination of concepts in H (H H) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

76 Fock Space and Superposition of Modes of Thought Definition The Fock space is the Hilbert space made from the direct sum of tensor products of single-particle Hilbert spaces: F = n=0 H n In particular, we can model the two interpretations underlying the combination of concepts in H (H H) The conjuntion concept is described by the superposition of modes of thought AB = neiφ 2 ( AB n 2 e iθ AB 2 Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

77 A Simple Model Illustrating the General Scheme The membership operator is M F = M (M M). Then µ(ab) = AB M F AB = n 2 ( µ(a) + µ(b) 2 ) + R( A M B ) + (1 n 2 )µ(a)µ(b). (4) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

78 A Simple Model Illustrating the General Scheme The membership operator is M F = M (M M). Then µ(ab) = AB M F AB = n 2 ( µ(a) + µ(b) 2 ) + R( A M B ) + (1 n 2 )µ(a)µ(b). (4) This model has been successfully applied to represent data on conjunctions and disjunctions of concepts (Aerts, 2009) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

79 A Simple Model Illustrating the General Scheme The membership operator is M F = M (M M). Then µ(ab) = AB M F AB = n 2 ( µ(a) + µ(b) 2 ) + R( A M B ) + (1 n 2 )µ(a)µ(b). (4) This model has been successfully applied to represent data on conjunctions and disjunctions of concepts (Aerts, 2009) Can we test if these new conditions of possible experience apply in other experiments in cognition? Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

80 Step Forward: Conjunctions and Negations of Concepts An experiment (40 participants) carried by Sandro Sozzo tested the conjunction and negation of concepts (Aerts, Sozzo, Veloz, 2015) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

81 Step Forward: Conjunctions and Negations of Concepts An experiment (40 participants) carried by Sandro Sozzo tested the conjunction and negation of concepts (Aerts, Sozzo, Veloz, 2015) Example: estimate the memberships of x = tomato given by µ x (A), µ x (Ā), µ x (B), µ x ( B), µ x (AB), µ x (A B), µ x (ĀB), µ x (Ā B), where Ȳ = not Y, with A = Fruit, B = Vegetable. 4 pairs of concepts, 24 exemplars each pair. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

82 Step Forward: Conjunctions and Negations of Concepts An experiment (40 participants) carried by Sandro Sozzo tested the conjunction and negation of concepts (Aerts, Sozzo, Veloz, 2015) Example: estimate the memberships of x = tomato given by µ x (A), µ x (Ā), µ x (B), µ x ( B), µ x (AB), µ x (A B), µ x (ĀB), µ x (Ā B), where Ȳ = not Y, with A = Fruit, B = Vegetable. 4 pairs of concepts, 24 exemplars each pair. Conditions of possible experience: I A = µ(a) µ(ab) µ(a B) = 0, (5) IĀ = µ(ā) µ(āb) µ(ā B) = 0, (6) I B = µ(b) µ(ab) µ(āb) = 0, (7) I B = µ( B) µ(a B) µ(ā B) = 0, (8) I ABĀ B = 1 µ(ab) µ(āb) µ(a B) µ(ā B) = 0. (9) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

83 Experiment Results 95% confidence intervals of I X, X = {A, B, Ā, B} calculated for all exemplars. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

84 Experiment Results 95% confidence intervals of I X, X = {A, B, Ā, B} calculated for all exemplars. Extreme values of the intervals show that In red: Negated concepts; In blue: Non-negated concepts Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

85 Experiment Results 95% confidence intervals of I X, X = {A, B, Ā, B} calculated for all exemplars. Extreme values of the intervals show that In red: Negated concepts; In blue: Non-negated concepts Boole s conditions of possible experience violated in a stable manner Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

86 Experiment Results 95% confidence intervals of I X, X = {A, B, Ā, B} calculated for all exemplars. Extreme values of the intervals show that In red: Negated concepts; In blue: Non-negated concepts Boole s conditions of possible experience violated in a stable manner Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

87 Comparing different datasets Comparing the model s performance w.r.t data set on conjunction (Hampton), and on conjunction and negation (Aerts, Sozzo, Veloz) Data compatible with Fock space model choosing n XY 0.8 Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

88 Conclusions and Further Questions Boole s conditions of possible experience are profoundly related to Cognitive Science Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

89 Conclusions and Further Questions Boole s conditions of possible experience are profoundly related to Cognitive Science Strong evidence for the non-satisfaction of these conditions in cognitive phenomena Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

90 Conclusions and Further Questions Boole s conditions of possible experience are profoundly related to Cognitive Science Strong evidence for the non-satisfaction of these conditions in cognitive phenomena Quatum-inspired models propound robust alternative Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

91 Conclusions and Further Questions Boole s conditions of possible experience are profoundly related to Cognitive Science Strong evidence for the non-satisfaction of these conditions in cognitive phenomena Quatum-inspired models propound robust alternative Introduced modes of thought and a quantum-inspired framework for mind and language Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

92 Conclusions and Further Questions Boole s conditions of possible experience are profoundly related to Cognitive Science Strong evidence for the non-satisfaction of these conditions in cognitive phenomena Quatum-inspired models propound robust alternative Introduced modes of thought and a quantum-inspired framework for mind and language Quantum conditions of possible experience in cognition? Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

93 Thank you!...questions? Boole, G. On the Theory of Probabilities. Philosophical Transactions of the Royal Society of London (1962) Boole, G. The Laws of Thought. New York: Dover Edition (1958) Pitowsy, I. George Boole s Conditions of Possible Experience and the Quantum Puzzle. Brit. J. Phil. Sci. (1995) Van Rijsbergen, C. J. (2004). The geometry of information retrieval (Vol. 157). Cambridge: Cambridge University Press. Proceedings of the n-th Quantum Interaction Symposium-Qi-n, n=2007,...,2015. These and the other references can be requested to me, space problems to put them all! Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

94 Mind-Language and Quantum Cognition Hypothesis Thesis Mind Creates Artificial Mind Not part of the thesis proposal Unstructured Meaning That as becomes Structured Meaning Is a trace of Collapses to To model Natural Language Is a trace of is the basis for a theory of Semantic Estimations Concept SCOP Has a probabilistic realization in Allows to perform Hilbert/Fock Space Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

95 Decision-Making Paradoxes Sure Thing Principle - The Ellsberg Paradox Rational maximization of expected utility approach implies that P(A) P(B) P(A or C) P(B or C) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

96 Decision-Making Paradoxes Sure Thing Principle - The Ellsberg Paradox Rational maximization of expected utility approach implies that P(A) P(B) P(A or C) P(B or C) This has been tested experimentally and strong violations have been found Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

97 Decision-Making Paradoxes Sure Thing Principle - The Ellsberg Paradox Rational maximization of expected utility approach implies that P(A) P(B) P(A or C) P(B or C) This has been tested experimentally and strong violations have been found Example (Ellsberg Paradox): Consider an urn with 30 red balls and 60 balls either black or yellow (unknown proportion). Assume the following bets: 1. Bet on Red 2. Bet on Black 3. Bet on Red or Yellow 4. Bet on Black or Yellow Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

98 Decision-Making Paradoxes Sure Thing Principle - The Ellsberg Paradox Rational maximization of expected utility approach implies that P(A) P(B) P(A or C) P(B or C) This has been tested experimentally and strong violations have been found Example (Ellsberg Paradox): Consider an urn with 30 red balls and 60 balls either black or yellow (unknown proportion). Assume the following bets: 1. Bet on Red 2. Bet on Black 3. Bet on Red or Yellow 4. Bet on Black or Yellow Most people prefer 1 over 2, but 4 over 3! which contradicts STP. Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

99 Decision-Making Paradoxes Quantum Model of Ellsberg Paradox Several explanations of the Ellsberg paradox have been given in Economics literature Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

100 Decision-Making Paradoxes Quantum Model of Ellsberg Paradox Several explanations of the Ellsberg paradox have been given in Economics literature A quantum model have been also developed: The urn is in a quantum state Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

101 Decision-Making Paradoxes Quantum Model of Ellsberg Paradox Several explanations of the Ellsberg paradox have been given in Economics literature A quantum model have been also developed: The urn is in a quantum state Ells = 1 ) (e iθr, ρ y e iθy, ρ b e iθ b, ρ 2 y + ρ 2 b = 1 3 One dimensional projectors P x represent bets on one type x = r, y, b of ball Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

102 Decision-Making Paradoxes Quantum Model of Ellsberg Paradox Several explanations of the Ellsberg paradox have been given in Economics literature A quantum model have been also developed: The urn is in a quantum state Ells = 1 ) (e iθr, ρ y e iθy, ρ b e iθ b, ρ 2 y + ρ 2 b = 1 3 One dimensional projectors P x represent bets on one type x = r, y, b of ball Appying the Born rule, probabilities exhibit interference. This interferece models the deviation from rationality Tomas Veloz (CLEA) Quantum Cognition October 26, / 25

103 Decision-Making Paradoxes Quantum Model of Ellsberg Paradox Several explanations of the Ellsberg paradox have been given in Economics literature A quantum model have been also developed: The urn is in a quantum state Ells = 1 ) (e iθr, ρ y e iθy, ρ b e iθ b, ρ 2 y + ρ 2 b = 1 3 One dimensional projectors P x represent bets on one type x = r, y, b of ball Appying the Born rule, probabilities exhibit interference. This interferece models the deviation from rationality It has been recently shown that the Machina Paradox (a more complex situation similar to the Ellsberg Paradox) is incompatible with all economic literature models, but the quantum approach still models it (Aerts et. al. 2013) Tomas Veloz (CLEA) Quantum Cognition October 26, / 25