Reinhard Blutner1

Linear Algebra and Geometric Approches to Meaning

1a. General Introduction

Reinhard Blutner

Universiteit van Amsterdam

ESSLLI Summer School 2011, Ljubljana

August 1 – August 7, 2011

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1. Geometric models of meaning

2. Phenomena and puzzles

3. Quantum Probabilities

4. Historical notes

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Geometric models in Cognitive Psychology

• Basic claim: An understanding of problem solving, categorization, memory retrieval, inductive reasoning, and other cognitive processes requires that we understand how humans assess similarity.

• W. S. Torgerson (1965): Multidimensional scaling of similarity. Psychometrika 30: 379–393.

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Geometric models in Cognitice Psychology II

• A. Tversky (1977): Features of similarity. Psychological Review 84: 327–352.

• P. Gärdenfors: The Geometry of Thought (2000)Concepts as convex spaces

• D. Widdows: Geometry and Meaning (2004)Distributional semantics

Voronoi Tessellation

If the closeness to a prototyp determines class boundaries, then we get a partition of the conceptual space into convex subspaces

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Concept combination á la Peter Gärdenfors

What is the computational mechanism of combination?

What is the color of a red nose

(red flag, red tomato)?

skin colors

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| 7Natural concepts do not form

Boolean algebras

• P. Gärdenfors: The Geometry of Thought (2000) Concepts as convex spaces

• The intersection of convex sets is convex again, but union and complement are not.

• Hans Primas: Convex sets form orthomodular lattices.

• Orthomodularity: If A B then A = (AB)+(AB

)

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Possible worlds and vectors• Possible worlds: Isolated entities which are used

for modeling propositions (sets of possible worlds)

• Vector states: abstract objects which form vector spaces. s

uA

– The addition of two vectors is an operation which describes the super-position of possibly conflicting states

– The scalar product is an operation which describes the similarity of two states

– Projections are operators that map vector spaces onto certain subspaces

Superposition

Superposing colors

Superposing pictures

Superposing meanings

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Superposing colors

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Superposition of faces

x

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Structure of the task

Left: superposition of female human facesRight: superposition of male human facesMiddle: superposition of male and female

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Experiment by Conte et al. (2007)

Test II

Are the two circles of the same size?

Test I

Are the two lines of the same size?

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Comparison

Symbolic models Geometric models

Formal semantics (Montague 1978); Partee, Kamp

Mental spaces (Gärdenfors 2000); Lakoff, Fouconnier

Concepts as set-theoretic constructions

Natural concepts as convex subspaces

Qualitative aspects of mean-ing, feature and tree structs.

Quantitative aspects of meaning, similarity structs.

Boolean algebra Orthomodular lattice

Compositional architecture

Compositionality= big problem

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1. Geometric models of meaning

2. Phenomena and puzzles

3. Quantum Probabilities

4. Historical notes

Puzzle 1: 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 is language vague?

“ 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]

The geometric approach provides a new theory of vagueness in the spirit of Lipman. It is able to solve some hard problems such as the disjunction and the conjunction puzzle.

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Fuzzy set theory as a compositional theory of graded

membership

1. mA (x) membership function for

instances x of category A: 0 mA (x)

1

2. mA (x) = 1–mA (x)

3. mAB (x) = min(mA (x), mB (x))

4. mAB (x) = max(mA (x), mB (x))

Why fuzzy sets do not work

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Chose instance x such that x

mCIRCLE(x) = mSQUARE(x)

1. Compositionality: mCIRCLE(x)=mSQUARE(x) mROUND CIRCLE(x)=mROUND SQUARE(x)

Compositionality is violated

2. Monotonicity:

mROUND SQUARE (x) mSQUARE (x)

Monotonicity is violated (conjunction puzzle)

Why fuzzy sets are not too bad

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x

3. Boundary Contradiction: mSQUARE SQUARE (x) > 0

Empirical counter-evidence by Bonini (1999), Alxatib & Pelletier (2011), Ripley (2011), Sauerland (2011).

• Supervaluation & probability theory both fail

• Is there a solution to all three puzzles of vagueness based on a uniform theory?

Borderline contradictions

• Only 40.8% judge (*) to be false in that case.

• Borderline contradictions are generally found to be quite acceptable

Alxatib & Pelletier 2011 asked subjects to judge sentences such as

(*) x is tall and not tall

• 44.7% of their subjects judge (*) to be true for the borderline case (2)

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Puzzle 2: Probability judgements violate the Kolmogorov axioms

• A Boolean algebra over W is a set ℱ of subsets of W [events, possibilities] that contains W and is closed under union and complementation (intersection)

• Normalized additive measure function

– P(A B) = P(A) + P(B) for disjoint sets A and B

– P(W) = 1

• Consequences:

– P(A B) P(A), P(A B) P(A) [monotonicity]

– P(A) + P(B) P(A B) ≤ 1 [additivity I]

– P(A) + P(B) P(A B) ≤ 1 [additivity II]

Disjunction 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

• P(A|C) = 0.54P(A|C) = 0 .57P(A) = 0 .32

• P(A) = P(A|C) P(C) + P(A|C) P(C)since CA (C)A = A (distributivity)

The ‘sure thing principle’ is violated empirically!

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Puzzle 3: Complementarity• Two Boolean descriptions are said to be

complementary if they cannot be embedded into a single Boolean description.

• Unicity: In classical probability theory, a single sample space is proposed which provides a complete and exhaustive description of all events that can happen in an experiment. – If unicity is valid, then complementarity does not exist– If complementarity exists then unicity cannot be valid

• Examples: physical time/mental time; physical/ mental objectivity (mind/body); Jung’s rational/ irrational functions.

Cartographic map projections

All cartographic maps are valid only in the small, i.e. locally.

Adapted from Primas (2007)28

Puzzle 4: Question order effects for attitude

questions

Is Clinton honest? (50%) Is Gore honest? (68%)

Is Gore honest? (60%) Is Clinton honest? (57%)

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Puzzle 5: Asymmetric similarities

• Korea is similar to China vs. China is similar to Korea

• Chicago's linebackers are like tigers vs. *Tigers are like Chicago's linebackers

• From a classical perspective this is puzzling:

– sim (X, Y) = f (distance (X, Y))

– sim (Y, X) = f (distance (Y, X))

But distance is a symmetric function

• How to express the basic cognitive operations – asymmetric similarity, asymmetric conjunction –

in the geometric framework?Reinhard Blutner30

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1. Geometric models of meaning

2. Phenomena and puzzles

3. Quantum Probabilities

4. Historical notes

Birkhoff and von Neumann1936

Hence we conclude that the propositional calculus of quantum mechanics has the same structure as an abstract projective geometry.

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The logic of quantum mechanics. Annals of Mathematics 37(4), 1936

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Geometric model of probabilities

• In the vector model, pairwise disjoint possibilities are represented by pairwise orthogonal subspaces.

• In the simplest case 1-dimensional subspaces are re-presented by the axes of a Cartesian coordinate system

• A state is described by a vector s of unit length• The projections of s onto

the different axes are called probability amplitudes.

• The square of the amplitudes are the relevant probabilities.

• They sum up to 1 – the length of s : (Pa s)2 + (Pb s)2 = 1.

b

Pa(s)

s

a

Pb(s)

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Projections & probabilitiess

u

u'

A

Projector to subspace A: PA

Projected vector: PA s = u

unique vector u such thats = u + u‘ , uU, u‘U ┴.

The length of the projection is written |PA

s| = |u |. The probability that s is about A (that s collapes onto A) is the square of the length of the corresponding projection: |PA s| 2 (Born rule)

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Order-dependence of projections

a

bs

Pb Pa s

Pa Pb s|Pa Pb s | |Pa Pb s |

Asymmetric conjunction

• The sequence of projections (Pa ; Pb) corresponds to a Hermitian operator Pa a Pb Pa .

(Pa ; Pb) =def Pa a Pb Pa (Gerd Niestegge’s

asymmetric conjunction)

• The expected probability for the sequence (Pa ; Pb) is (Pa ; Pb) = |Pb Pa s |2

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Solving the puzzles

• The asymmetric conjunctions account for interference effects, which partly can explain the puzzles

• Many possible applications:– Vagueness and probability judgements (Puzzles 1

& 2)It does not account for borderline contradictions!

– Complementarity and uncertainty principle (Puzzle 3)

– Order effects for questions (Puzzle 4)– Asymmetric similarity (Puzzle 5)

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1. Geometric models of meaning

2. Phenomena and puzzles

3. Quantum Probabilities

4. Historical notes

Quantum Mechanics & Quantum Cognition

Heisenberg Einstein Bohr Pauli

Aerts1994

Conte1989

Khrennikov1998

Atmanspacher

1994Reinhard Blutner

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Sommerfeld-Bohr atomic model

• Electrons move on discrete orbits• Electrons emit photons when jumping from one

orbit to the next• Problems: many stipulations and conceptual in-

consistencies. Empirical problems with certain spectra.

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Heisenberg‘s matrics mechanics

• Heisenberg 1925: New solution to old puzzle of spectral lines of hydrogen. The electrons do not move on orbits

• “It was about three o' clock at night when the final result of the calculation lay before me. At first I was deeply shaken. I was so excited that I could not think of sleep. So I left the house and awaited the sunrise on the top of a rock.”

• Max Born: Linear algebra. Eigenvalue problem

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The pioneers and applications to the

macroworld• William James was the first who introduced the

idea of complementarity into psychology“It must be admitted, therefore, that in certain persons, at least, the total possible consciousness may be split into parts which coexist but mutually ignore each other, and share the objects of knowledge between them. More remarkable still, they are complementary” (James, the principles of psychology 1890, p. 206)

• Nils Bohr introduced it into physics (Complementarity of momentum and place) and proposed to apply it beyond physics to human knowledge.

• Beim Graben & Atmanspacher gave a systematic treatment of complementarity in the macroworld.

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Interference effects and puzzles of bounded rationality

• Conjunction and disjunction fallacy: Aerts et al.

(2005), Khrennikov (2006), Franco (2007), Conte et al. (2008), Blutner (2008), Busemeyer et al. (2011).

• Prisoner’s dilemma: Pothos and Busemeyer (2009).

• Order effects: Trueblood and Busemeyer (in press).

• Categorization: Aerts and Gabora (2005), Aerts (2009), Busemeyer, Wang, and Lambert-Mogiliansky (2009).

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The recent geometric turn • Quantum Interaction workshop at Stanford in 2007

organized by Bruza, Lawless, van Rijsbergen, and Sofge (as part of the AAAI Spring Symposium)

• Workshops at Oxford (England) in 2008, • Workshop at Vaxjo (Sweden) in 2008, • Workshop at Saarbrücken (Germany) in 2009, • AAAI meeting Stanford in 2010. • Busemeyer (Indiana University) et al. organized a special

issue on Quantum Cognition, which was published in the October (2009), issue of Journal of Mathematical Psychology

• Busemeyer & Bruza (forthcoming): Quantum cognition and decision. Cambridge University Press.

www.quantum-cognition.de

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Conclusions• The inclusion of the concept of probability into

traditional geometric models is the beginning of a real break through: it resolves a series of serious puzzles and long-standing problems

• Concepts of natural language semantics such as similarity and logical operations do not directly correspond to standard operations in the ortho-modular (vector) framework but rather indirectly

• The problem of conceptual combination can be solved in the new framework. Extensional holism coexists with intensional compositionality.