Linear Algebra and Geometric Approches to Meaning nhard Blutner 1 a. General Introduction ESSLLI Summer School 2011, Ljubljana August 1 – August 7, 2011 Reinhard Blutner Universiteit van Amsterdam 1
nhard Blutner 1 1. Geometric models of meaning 2. Phenomena and puzzles 3. Quantum Probabilities 4. Historical notes 2
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. Reinhard Blutner 3
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 Reinhard Blutner 4
Voronoi Tessellation If the closeness to a prototyp determines class boundaries, then we get a partition of the conceptual space into convex subspaces Reinhard Blutner 5
Concept combination á la Peter Gärdenfors What is the color of a red nose (red flag, red tomato)? skin colors What is the computational mechanism of combination? Reinhard Blutner 6
Natural concepts do not form Boolean algebras |7 • 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 = (A B)+(A B ) Reinhard Blutner 7
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. – The addition of two vectors is an operation which describes the superposition 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 Reinhard Blutner s u 8 A
Superposition Superposing colors Superposing pictures Superposing meanings Reinhard Blutner 9
Superposing colors x Reinhard Blutner 10
x Reinhard Blutner 11
x Reinhard Blutner 12
x Reinhard Blutner 13
Superposition of faces x Reinhard Blutner 14
Structure of the task Left: superposition of female human faces Right: superposition of male human faces Middle: superposition of male and female faces Reinhard Blutner 15
Experiment by Conte et al. (2007) Test II Are the two circles lines of the same size? Reinhard Blutner 16
Comparison Symbolic models Geometric models Formal semantics (Montague 1978); Partee, Kamp Mental spaces (Gärdenfors 2000); Lakoff, Fouconnier Concepts as constructions Natural concepts as convex subspaces set-theoretic Qualitative aspects of meaning, feature and tree structs. Quantitative aspects of meaning, similarity structs. Boolean algebra Orthomodular lattice Compositional architecture Compositionality = big problem Reinhard Blutner 17
2 1. Geometric models of meaning 2. Phenomena and puzzles 3. Quantum Probabilities 4. Historical notes Reinhard Blutner 18
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): - Reinhard Blutner both robins and penguins are clearly birds, but 19
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. Reinhard Blutner 20
Fuzzy set theory as a compositional theory of graded membership 1. m. A (x) membership function for instances x of category A: 0 m. A (x) 1 2. m A (x) = 1–m. A (x) 3. m. A B (x) = min(m. A (x), m. B (x)) 4. m. A B (x) = max(m. A (x), m. B (x)) Reinhard Blutner 21
Why fuzzy sets do not work Chose instance x such that x m. CIRCLE(x) = m. SQUARE(x) 1. Compositionality: m. CIRCLE(x)=m. SQUARE(x) m. ROUND CIRCLE(x)=m. ROUND SQUARE(x) Compositionality is violated 2. Monotonicity: m. ROUND SQUARE (x) m. SQUARE (x) Monotonicity is violated (conjunction puzzle) Reinhard Blutner 22
Why fuzzy sets are not too bad x 3. Boundary Contradiction: m. 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? Reinhard Blutner 23
Borderline contradictions 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 Only 40. 8% judge (*) to be false in that case. borderline case (2) Borderline contradictions are generally found to be quite acceptable Reinhard Blutner 24
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] Reinhard Blutner 25
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) P(A| C) P(A) = 0. 54 = 0. 57 = 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! Reinhard Blutner 26
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/27 Reinhard Blutner
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%) Reinhard Blutner 29
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 – Reinhard Blutner in the geometric framework? 30
3 1. Geometric models of meaning 2. Phenomena and puzzles 3. Quantum Probabilities 4. Historical notes Reinhard Blutner 31
Birkhoff and von Neumann 1936 Hence we conclude that the propositional calculus of quantum mechanics has the same structure as an abstract projective geometry. The logic of quantum mechanics. Annals of Mathematics 37(4), 1936 Reinhard Blutner 32
Geometric model of probabilities • In the vector model, onto the different possibilities are The projections of s pairwise disjoint axes are called represented by pairwise orthogonal subspaces. probability amplitudes. • In the simplest case 1 -dimensional subspaces are re. The square of the amplitudes are the relevant probabilities. • presented up to 1 axes of a Cartesian (Pa s)2 + (Pb s)2 = 1. They sum by the – the length of s : coordinate system • A state is described by a vector s of unit length s b Pb ( s) a Pa ( s) Reinhard Blutner 33
Projections & probabilities s § Projector to subspace A: PA § Projected vector: PA s = u unique vector u such that s = u + u‘ , u U, u‘ U ┴. § u' The length of the projection is written |PA s| = |u |. A 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) Reinhard Blutner 34
Order-dependence of projections a s b |Pa Pb s | | Pa Pb s | Reinhard Blutner Pa Pb s Pb Pa s 35
Asymmetric conjunction • The sequence of projections (Pa ; Pb) corresponds to a Hermitian operator Pa Pb Pa. (Pa ; Pb) =def Pa Pb Pa (Gerd Niestegge’s asymmetric conjunction) • The expected probability for the sequence (Pa ; Pb) is (Pa ; Pb) = |Pb Pa s |2 Reinhard Blutner 36
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) Reinhard Blutner 37
4 1. Geometric models of meaning 2. Phenomena and puzzles 3. Quantum Probabilities 4. Historical notes Reinhard Blutner 38
Quantum Mechanics & Quantum Cognition Heisenberg Aerts 1994 Reinhard Blutner Einstein Conte 1989 Bohr Khrennikov 1998 Pauli Atmanspacher 1994
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 inconsistencies. Empirical problems with certain Reinhard Blutner 40
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 Reinhard Blutner 41
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 42 Reinhard Blutner
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). Reinhard Blutner 43
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 Reinhard Blutner 44
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 Reinhard Blutner 45
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