linear algebra and geometric approches to meaning 5a. concept combination

Reinhard Blutner1

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

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11. Outlook

2. Conditioned probabilities

3. Pitkowsky’s Correlation Polytopes

4. Conjunction and disjunction of natural concepts

5. Borderline contradictions

6. Combining prototypes

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

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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

Vagueness & quantum probability

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

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x

Ax

A = 1

5

Typicality & quantum probability

cx(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)

cx(A) mx(A)

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x

A = 1

a

6

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21. Outlook






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

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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, CC = CC = 0, we get

A = CAC + CAC + CAC + C AC

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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 + CAC)

= 2 ½ (C; A) ½ (C; A) cos

• The interference term introduces one free parameter: The phase shift .

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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 231Reinhard Blutner11

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)

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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

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Approaching vagueness • 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) =def CAC

• Disjunctions are represented by the orthomodular dual of sequences: (C; A)

x

AxA = 1

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31. Outlook






Kolmogorov Probabilities

Monotonicity

XY P(X) ≤ P(Y)

Additivity

P(X)+P(Y) = P(XY)+P(XY)

X XY Y

X Y

YX

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Pitkowsky diamond

ConjunctionP(AB) ≤ min(P(A),P(B)) P(A)+P(B)P(AB) ≤ 1

DisjunctionP(AB) ≥ max(P(A),P(B))P(A)+P(B)P(AB) ≤ 1

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41. Outlook






Hampton 1988: judgement of membership

A B

Furniture

FoodWeaponBuildingMachineBird

Household appliancesPlantToolDwellingVehiclePet

A and Boverextension

A B

Home furnishingHobbiesSpicesInstrumentsPetsSportswearFruitsHousehold appliances

FurnitureGamesHerbsToolsFarmyard animalsSports equipmentVegetablesKitchen utensils

A or Bunderextension, no additivity

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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 2009Reinhard Blutner

20

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 2009Reinhard Blutner21

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51. Outlook






Alxatib & Pelletier 2011Pictures 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)

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#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 ❏

% judged true

1 2 3 4 5

20

40

60

80

Data

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x is tall

x is neither tall nor not tall

x is not tall

x is tall and not tall

1 2 3 4 5

0

10

20

30

40

Tensor product as conjunction

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x is tall and not tall

theoretical prediction

(1 parameter fitted)

% judged true

A and B : A B xx(A B) = x(A) x(B)

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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 ; xx(A A) = x(A) (1-x(A))

• Aerts (2009) proposes to combine both methods using the Fock space.

(allowing states such as (x + xx))

• In the Fock-space, then ‘A and B ’ corresponds to the operator

ABA + AB

21

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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)

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61. Outlook






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 applestone lion

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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')

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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.

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Prototypes as superposed instances

iii baba )(

i ii naa

• .

• |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 lengthReinhard Blutner

32

in

Typicality of conjoined concepts

For conjoined concepts A B perform the following steps:

1. Build the corresponding vectors and

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

a

b

ba

)( ba

ii nn

in

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The compression operator

• Definition

• Modification

[ ] =

• The resulting state is entangled, i.e.

iii iiij

jiij nnXnnX

)]([

)( baba

yxba

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Conjunction effect

2 4 6 8 10 12 14Instances

0.05

0.1

0.15

0.2

0.25

Typicality

striped apple

striped apple

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Striped apple 2

Form

Texture

Apple

striped

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Concept combination: a geometrical model (Peter

Gärdenfors)

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Red Nose

General Distribution

Red

Color Distribution

NosesConjoined Concept

Red Nose

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Red and White Beans


Red

Color Distribution

Beans

Color Distribution

Red Beans

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Red and White Beans

Color Distribution

White Beans

Color Distribution

Beans


White

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Tall Boy

5 10 15 20In s ta n c e s o rd e re d w ith s ize

0 .05

0 .10

0 .15

T yp ic a lity

tallboytallboy

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5 10 15 20In s ta n c e s o rd e re d w ith c o lo r o f p e e l

0 .05

0 .10

0 .15

0 .20

T yp ic a lity

Red apple: color of peel

red

apple

redapple

Kullback-Leibler information = 0.25

dxx

xxKLI )

)(

)(log()(

0

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5 10 15 20In s ta n c e s o rd e re d w ith c o lo r o f pu lp

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6T yp ic a lity

Red apple: color of pulp

red

apple

redapple

Kullback-Leibler information = 0.06

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Stone lion

5 10 15 20Ins tances

0 .05

0 .10

0 .15

0 .20

0 .25

0 .30

Typ ical i ty

stone

stone lion

lion

Kullback-Leibler Information

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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).

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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

linear algebra and geometric approches to meaning 5a. concept combination

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