1. Reinhard Blutner 1 Linear Algebra and Geometric Approches to Meaning 5a. Concept Combination Reinhard Blutner Universiteit van Amsterdam ESSLLI Summer School 2011, Ljubljana August 1 – August 7, 2011

2. Reinhard Blutner 2 1 1.Outlook 2.Conditioned probabilities 3.Pitkowsky’s Correlation Polytopes 4.Conjunction and disjunction of natural concepts 5.Borderline contradictions 6.Combining prototypes

3. Vagueness A concept is vague if it does not have precise, sharp boundaries and does not describe a well- defined set. Vagueness is the inevitable result of a knowledge system that stores the centers rather than the boundaries of conceptual categories Vagueness is different from typicality (centrality): -both robins and penguins are clearly birds, but -robins are more typical than penguins as birds Reinhard Blutner 3

4. 4 Why a quantum approach? The geometric approach provides a new theory of vagueness in the spirit of Lipman. “ It is not that people have a precise view of the world but communicate it vaguely; instead, they have a vague view of the world. I know of no model which formalizes this. I think this is the real challenge posed by the question of my title [Why is language vague?]" [Barton L. Lipman, 2001] It is able to solve some hard problems such as the disjunction and the conjunction puzzle It is able to answer the question why boundary contradictions are quite acceptable (x is tall and not tall) Extensional holism coexists with intensional compositionality

5. Vagueness & quantum probability m x (A) = |Ax| 2 degree of membership -Instance x represents a vector state which is projected by the operator A -The squared length of Ax is the probability that x is a member of A Reinhard Blutner x Ax A = 1 5

6. Typicality & quantum probability c x (A) = |a  x| 2 typicality -The vector a represents the prototype of A -the squared length of the projection of x onto the vector a is the probability that x collapses onto a (or a collapses onto x – symmetry) c x (A)  m x (A) Reinhard Blutner x A = 1 a 6

7. Reinhard Blutner 7 2 1.Outlook 2.Conditioned probabilities 3.Pitkowsky’s Correlation Polytopes 4.Conjunction and disjunction of natural concepts 5.Borderline contradictions 6.Combining prototypes

8. Conditioned Probabilities P(A|C) = P(CA)/P(A)  (A|C) =  (CAC)/  (A) (Gerd Niestegge) If the operators commute, Niestegge’s definition reduces to classical probabilities: CAC = CCA = CA Niestegge’s formalism is an adequate way for representing the close connection between interference effects and question order effects (non-commutativity) Introduce ‘sequence’ (C; A) = def CAC Reinhard Blutner 8

9. Interference Effects Classical: P(A) = P(A|C) P(C) + P(A|  C) P(  C) Quantum:  (A) =  (A|C)  (C) +  (A|C  )  (C  ) +  (C, A) where  (C, A) =  (CAC  + C  AC) [Interference Term] Proof Since C+C  = 1, C  C = CC  = 0, we get A = CAC + C  AC  + CAC  + C  AC Reinhard Blutner 9

10. Calculating the interference term In the simplest case (when the propositions C and A correspond to projections of pure states) the interference term is easy to calculate:  (C, A) =  (CAC  + C  AC) = 2  ½ (C; A)  ½ (C  ; A) cos  The interference term introduces one free parameter: The phase shift . Reinhard Blutner 10

11. Solving the Tversky/Shafir puzzle Tversky and Shafir (1992) show that significantly more students report they would purchase a nonrefundable Hawaiian vacation if they were to know that they have passed or failed an important exam than report they would purchase if they were not to know the outcome of the exam  (A|C) = 0.54  (A|C  )= 0.57  (A)= 0.32  (C, A) = [  (A|C)  (C) +  (A|C  )  (C  )]   (A) = 0.23  cos  = -0.43;  = 2.01  231  Reinhard Blutner 11

12. Conjunction Puzzle for probabilities Linda is 31 years old, single, outspoken and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. -Linda is active in the feminist movement. (A)(6,1) -Linda is a bank teller. (C)(3,8) -Linda is a bank teller and is active in the feminist movement. (C&A) (5,1) Reinhard Blutner 12

13. Conjunction effect  (A) = 0.38 (Linda is a bank teller)  (C) = 0.61(Linda is a feminist)  (C ; A) = 0.51 (Linda is a feminist bank teller) Quantum:  (C ; A)   (A) =  (  (C  ; A) +  (C, A)) Given example:  (C; A)   (A) = +0.13 (sign.)  cos  =  0.7,  = 2.35  270  Reinhard Blutner 13

14. Approaching vagueness Graded membership: To what degree is x a member of A ? answer:  x (A) = | Ax | 2 -Instances x represent vector states which are projected by the operator A -The squared length of Ax is the probability that x is a member of A Conjunctions are represented by sequences: (C ; A) = def CAC Disjunctions are represented by the orthomodular dual of sequences: (C  ; A  )  x Ax A = 1 Reinhard Blutner 14

15. Reinhard Blutner 15 3 1.Outlook 2.Conditioned probabilities 3.Pitkowsky’s Correlation Polytopes 4.Conjunction and disjunction of natural concepts 5.Borderline contradictions 6.Combining prototypes

16. Kolmogorov Probabilities Monotonicity X  Y  P(X) ≤ P(Y) Additivity P(X)+P(Y) = P(X  Y)+ P(X  Y) X X  Y Y X  Y Y X Reinhard Blutner 16

17. Pitkowsky diamond Conjunction P(A  B) ≤ min(P(A),P(B)) P(A)+P(B)  P(A  B) ≤ 1 Disjunction P(A  B) ≥ max(P(A),P(B)) P(A)+P(B)  P(A  B) ≤ 1 Reinhard Blutner 17

18. Reinhard Blutner 18 4 1.Outlook 2.Conditioned probabilities 3.Pitkowsky’s Correlation Polytopes 4.Conjunction and disjunction of natural concepts 5.Borderline contradictions 6.Combining prototypes

19. Hampton 1988: judgement of membership AB Furniture Food Weapon Building Machine Bird Household appliances Plant Tool Dwelling Vehicle Pet A and B overextension AB Home furnishing Hobbies Spices Instruments Pets Sportswear Fruits Household appliances Furniture Games Herbs Tools Farmyard animals Sports equipment Vegetables Kitchen utensils A or B underextension, no additivity Reinhard Blutner 19

20. Conjunction ‘building and dwelling’ Classical : cave, house, synagogue, phone box. Non-classical : tent, library, apartment block, jeep, trailer. Example ‘overextension’  library (building) =.95  library (dwelling) =.17  library (b_  d_) =.31 cf. Aerts 2009 Reinhard Blutner 20

21. Disjunction ‘fruits or vegetables’ Classical: green pepper, chili pepper, peanut, tomato, pumpkin. Non-classical: olive, rice, root ginger, mushroom, broccoli. Example ‘additivity’  olive (fruit) =.5  olive (vegetable) =.1  olive (f_  v_) =.8 cf. Aerts 2009 Reinhard Blutner 21

22. Reinhard Blutner 22 5 1.Outlook 2.Conditioned probabilities 3.Pitkowsky’s Correlation Polytopes 4.Conjunction and disjunction of natural concepts 5.Borderline contradictions 6.Combining prototypes

23. Alxatib & Pelletier 2011 Pictures with 5 persons of different size are presented. (Order of persons randomized) Subjects have to judge forms with four sentences as True/False/Can’t Tell. (Order of questions randomized) Reinhard Blutner 23 #3 is tall True ❏ False ❏ Can’t Tell ❏ #3 is not tall True ❏ False ❏ Can’t Tell ❏ #3 is tall and not tall True ❏ False ❏ Can’t Tell ❏ #3 is neither tall nor not tall True ❏ False ❏ Can’t Tell ❏

24. % judged true Data Reinhard Blutner 24 x is tall x is neither tall nor not tall x is not tall x is tall and not tall

25. Tensor product as conjunction Reinhard Blutner 25 x is tall and not tall theoretical prediction (1 parameter fitted) % judged true A and B : A  B  x  x (A  B) =  x (A)   x (B)

26. Reinhard Blutner 26 Extension of the formalism So far, we have two notions for the conjunction –Asymmetric conjunction (A; B) = ABA –Tensor product A  B  x (A;  A) = 0 ;  x  x (A   A) =  x (A)  (1-  x (A)) Aerts (2009) proposes to combine both methods using the Fock space. (allowing states such as (x + x  x)) In the Fock-space, then ‘A and B ’ corresponds to the operator ABA + A  B

27. Reinhard Blutner 27 Two arguments Aerts 2009: The combination is required for fitting the Hampton data of category membership Sauerland 2010: Borderline contradictions are not extensional in the sense of fuzzy logic, i.e.  x (A) =  x (B)   x (A and  A)   x (A and  B)

28. Reinhard Blutner 28 6 1.Outlook 2.Conditioned probabilities 3.Pitkowsky’s Correlation Polytopes 4.Conjunction and disjunction of natural concepts 5.Borderline contradictions 6.Combining prototypes

29. Effect of contrast classes A collie is a dog, but a tall collie is not a tall dog Red nose red flag red beans Striped apple stone lion Reinhard Blutner 29

30. Conjunction Effect of Typicality x=guppy is a poorish example of a fish, and a poorish example of a pet, but it's quite a good example of a pet fish –c x (A&B) > c x (B) In case of "incompatible conjunctions" such as pet fish or striped apple the conjunction effect is greater than in "compatible conjunctions“ (red apple). –c x (A‘ & B) – c x (B‘ ) > c x (A & B) – c x (B) (if A invites B but A' does not invite B') Reinhard Blutner 30

31. Compositional Semantics and Global Effects Fregean Formal Semantics is based on the Principal of Compositionality Global effects: The meaning of one part can influence the meaning of another part. Context as a global (hidden) parameter Frege (1884) took this as an argument against compositionality in Natural Language Quantum Cognition can explain the global contextual effects without giving up compositionality because the different constituents can be entangled. Reinhard Blutner 31

32. Prototypes as superposed instances. |a i | 2 is the probability for selecting if the prototype is not one of the presented instances it is still recognized as such. Modification rule + recalibrating to unit length Reinhard Blutner 32

33. Typicality of conjoined concepts For conjoined concepts A  B perform the following steps: 1.Build the corresponding vectorsand 2.Construct the tensor product 3.Perform the compression operation in order to build an entangled state 4.The typicality of instance is the quantum probability that the entangled state collapses into Reinhard Blutner 33

34. The compression operator Definition Modification  [ ] = The resulting state is entangled, i.e. Reinhard Blutner 34

35. Conjunction effect striped apple striped apple Reinhard Blutner 35

36. Striped apple 2 Form Texture Apple striped Reinhard Blutner 36

37. Concept combination: a geometrical model (Peter Gärdenfors) Reinhard Blutner 37

38. Red Nose General Distribution Red Color Distribution Noses Conjoined Concept Red Nose Reinhard Blutner 38

39. Red and White Beans General Distribution Red Color Distribution Beans Color Distribution Red Beans Reinhard Blutner 39

40. Red and White Beans Color Distribution White Beans Color Distribution Beans General Distribution White Reinhard Blutner 40

41. Tall Boy tallboy tall boy Reinhard Blutner 41

42. Red apple: color of peel red apple red  apple Kullback-Leibler information = 0.25 Reinhard Blutner 42

43. Red apple: color of pulp red apple red  apple Kullback-Leibler information = 0.06 Reinhard Blutner 43

44. Stone lion stone stone  lion lion Kullback-Leibler Information   Reinhard Blutner 44

45. Reinhard Blutner 45 Conclusions Several examples of context-sensitivity can be treated in a straightforward way by using a compositional operation on conceptual states. Since conceptual states contain (frozen) usage information, they combine semantic and pragmatic information. It makes superfluous ‘truth-conditional pragmatics’ as an inferential theory. The present account is non-inferential; it is ‘as direct as perception’ (cf. Millikan, Recanati).

46. Reinhard Blutner 46 General Conclusions Asymmetric conjunction accounts for interference effects –Explaining probability judgments. If quantum probabilities are rational constructs then this kind of rationality conforms to the judgment data –Describing the combination of vague concepts –Problems with borderline contradictions can be overcome by using the Fock space. The combination of prototypes likewise is using the Fock space and particular compression operations Extensional holism coexists with intensional compositionality