Linear Algebra and Geometric Approches to Meaning 5a. Concept Combination

1. Linear Algebra and Geometric Approches to Meaning5a. Concept Combination ESSLLI Summer School 2011, Ljubljana August 1 – August 7, 2011 Reinhard Blutner Universiteit van Amsterdam 1 Reinhard Blutner

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

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 3 Reinhard Blutner

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 Reinhard Blutner

5. Vagueness & quantum probability x Ax A = 1 5 mx(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

6. Typicality & quantum probability a x A = 1 6 cx(A) = |a  x|2typicality 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) cx(A)  mx(A) Reinhard Blutner

7. 2 • Outlook • Conditioned probabilities • Pitkowsky’s Correlation Polytopes • Conjunction and disjunction of natural concepts • Borderline contradictions • Combining prototypes 7 Reinhard Blutner

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) =defCAC 8 Reinhard Blutner

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 9 Reinhard Blutner

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 . 10 Reinhard Blutner

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 11 Reinhard Blutner

12. Conjunction Puzzle for probabilities 12 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

13. Conjunction effect 13 (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

14. Approaching vagueness x A = 1 Ax 14 Graded membership: To what degree is x a member of A ? answer: x(A) = | Ax |2 • Instances x represent vector stateswhich 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) =defCAC Disjunctions are represented by the orthomodular dual of sequences: (C; A) Reinhard Blutner

15. 3 • Outlook • Conditioned probabilities • Pitkowsky’s Correlation Polytopes • Conjunction and disjunction of natural concepts • Borderline contradictions • Combining prototypes 15 Reinhard Blutner

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

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 17 Reinhard Blutner

18. 4 • Outlook • Conditioned probabilities • Pitkowsky’s Correlation Polytopes • Conjunction and disjunction of natural concepts • Borderline contradictions • Combining prototypes 18 Reinhard Blutner

19. Hampton 1988: judgement of membership A or B underextension, no additivity A and B overextension 19 Reinhard Blutner

20. Conjunction ‘building and dwelling’ 20 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

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 21 Reinhard Blutner

22. 5 • Outlook • Conditioned probabilities • Pitkowsky’s Correlation Polytopes • Conjunction and disjunction of natural concepts • Borderline contradictions • Combining prototypes 22 Reinhard Blutner

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) #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 ❏ 23 Reinhard Blutner

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

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

26. Extension of the formalism 26 Reinhard Blutner 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. Two arguments 27 Reinhard Blutner 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. 6 • Outlook • Conditioned probabilities • Pitkowsky’s Correlation Polytopes • Conjunction and disjunction of natural concepts • Borderline contradictions • Combining prototypes 28 Reinhard Blutner

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 29 Reinhard Blutner

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 • cx(A&B) > cx(B) • In case of "incompatible conjunctions" such as pet fish or striped apple the conjunction effect is greater than in "compatible conjunctions“ (red apple). • cx(A‘ & B) – cx(B‘ ) > cx(A & B) – cx(B) (if A invites B but A' does not invite B') 30 Reinhard Blutner

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. 31 Reinhard Blutner

32. Prototypes as superposed instances • . • |ai|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 32 Reinhard Blutner

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

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

35. Conjunction effect apple striped striped apple 35 Reinhard Blutner

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

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

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

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

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

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

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

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

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

45. Conclusions 45 Reinhard Blutner 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. General Conclusions 46 Reinhard Blutner 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