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Linear Algebra and Geometric Approaches to Meaning 4b. Semantics of Questions

Linear Algebra and Geometric Approaches to Meaning 4b. Semantics of Questions. ESSLLI Summer School 2011, Ljubljana August 1 – August 7, 2011. Reinhard Blutner Universiteit van Amsterdam. 1. Reinhard Blutner. 1. Semantics of questions and answers Jäger/Hulstijn question semantics

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Linear Algebra and Geometric Approaches to Meaning 4b. Semantics of Questions

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  1. Linear Algebra and Geometric Approaches to Meaning4b. Semantics of Questions ESSLLI Summer School 2011, Ljubljana August 1 – August 7, 2011 Reinhard Blutner Universiteit van Amsterdam 1 Reinhard Blutner

  2. 1 • Semantics of questions and answers • Jäger/Hulstijn question semantics • Ortho-algebraic question semantics • Answerhood Reinhard Blutner

  3. Understanding Questions • General claim (Groenendijk and Stokhof 1997) • to understand a question is to understand what counts as an answer to that question; • an answer to a question is an assertion or a statement • an assertion is identical with its propositional content • Different approaches that fit into this scheme: • the Groenendijk and Stokhof (1984, 1997) partition theory which defines the meaning of a question as the set of its complete answers. • Hamblin (1973) who identifies a question with the set of propositional contents of its possible answers • Karttunen (1977) for whom it is the smaller set of its true answers 3 Reinhard Blutner

  4. W Partition semantics • Given a domain W of possible worlds • Propositions are described by sets of possible worlds • The semantic value of questions are partitions of W (i.e. a set of pairwise disjoint propositions which cover W. 4 Reinhard Blutner

  5. Two types of Question Theories • Basic distinction between yes/no-questions and wh-questions • Proposition set approach (Hamblin 1973, Karttunen 1977, Groenendijk and Stokhof 1984, 1997). • the answers to wh-questions are identified with the senses of complete sentences • the answers to yes/no questions generates a bipartition of W • Structured meaning approach (Tichy 1978, Krifka 2001, …) • the answers to wh-questions are identified with the senses of noun phrases rather than of sentences. • the answers to yes/no questions generates a bipartition of W decorated with the answers yes or no • Accordingly, the meanings of questions are constructed as functions that yield a proposition when applied to the semantic value of the answer 5 Reinhard Blutner

  6. W W yes no Is the door open? • Proposition set approach: The semantic value of questions are partitions of W • Structured proposition approach:The semantic value of questions are decorated partitions of W 6 Reinhard Blutner

  7. Update semantics • Context dependence of utterances; information states  • I am here • A man comes in. He whistles. • All boys are tall • The meaning of sentences describes their context change potential • Sentences do not only provide data, but also raise issues. In the classical theory these two tasks are strictly divided over two syntactic categories: • declarative sentences provide data • interrogative sentences raise issues. 7 Reinhard Blutner

  8. Limitations of the classical approach • Conditional questions (if Tom is in Berlin where is Mary?) • Unconditionals (Zaefferer 1991) (whether you like it or not, your talk was simply boring) • A proper treatment of hybrid expressions such as disjunctions which act as questions and assertions • Two possible reactions • Modify partition semantics (Jäger 1996, Hulstijn 1997) • Give up partition semantics (see the inquisitive turn) 8 Reinhard Blutner

  9. The present programme • Ortho-algebraic question semantics: Uniform treatment; both observables (questions) and propositions (projections) are analyzed by Hermitian operators • Advantages • Easy to handle variant of the partition semantics • It accounts for conditional questions • It generalizes to attitude questions • In the classical case it correspondents to a structured meaning approach. 9 Reinhard Blutner

  10. 2 • Semantics of questions and answers • Jäger/Hulstijn question semantics • Ortho-algebraic question semantics • Answerhood 10 Reinhard Blutner

  11. Consider the language of propositional logic L, extended with a question operator “?” and a (non-standard) conditional operator “”. QL can be defined as the smallest set containing L and satisfying the following two clauses: a. if   QL then ?  QLb. if ,   QL then ()  QL, ()  QL, and ()  QL The query language QL 11 Reinhard Blutner

  12. Information states • Information states  partition a subset of W • A declarative sentence can be seen as partitioning the set of all worlds that make the proposition true into a partition consisting just of one element: the set of worlds that make the proposition true. • A conditional question partitions the set of all worlds where the antecedent of the conditional is true. • The empty information state 0 partitions the domain W in the empty proposition W. This correspondents to the equivalence relation W2 where all states of W are considered equivalent. 12 Reinhard Blutner

  13. W W W Declaratives and questions Declaratives Questions Conditional Questions 13 Reinhard Blutner

  14. Basic semantic notions Information states are modeled by equivalence relations over the logical space, W 2. Information change potential «» of sentences of QL are functions that map inf. states onto inf. states Span: (u, v ) ⊦  iff (u, v )  «» [equivalent states] Truth: u⊦iff (u, u )  «» [for assertions ] Entailment 1:  |=  iff «» «» = «» for all information states  Entailment 2:  |=  iff 0«» «» = 0«» , where 0 = W2 is the empty information state. 14 Reinhard Blutner

  15. «p» = {(u,v)  W2: u(p) and v(p)} «» = {(u,v)  W2: (u,u)«» and (v,v)«»} «» = «»«» «?» = {(u,v) : (u,u) «» iff (v,v) «»} «» = {(u,v) «?»: if (u,v) «» then (u,v) «»«»} Definitions: () () Jäger/Hulstijn’s question semantics 15 Reinhard Blutner

  16. Examples • Consider fragment with two atomic formulas: p, q • Identifying possible worlds with functions assigning the truth values 1 (true) and 0 (false) to the atoms, we get four possible worlds abbreviated by 10, 11, 01, 00 • (p) = {10, 11}, (q) = {01, 11} • Initial information state: 0 = W2. This information state describes a partition of W consisting of a single proposition: the whole set of possible worlds W. • 0«p» = {(u,v)  W2: u {10, 11} and v {10, 11} 16 Reinhard Blutner

  17. Picture of meaning for p • Note that a subset of W is partitioned only. • The partition consists of a single proposition: (p) = {10, 11} 17 Reinhard Blutner

  18. Picture of meaning for ?p • The domain W is partitioned into two proposition: (p) = {10, 11} and (p) = {00, 01} 18 Reinhard Blutner

  19. Picture of meaning p ?q • The domain W is partitioned into three proposition:(p) = {00, 01}, (pq) = {11}, (pq) = {11}. 19 Reinhard Blutner

  20. Velissaratou’s example A: If Mary reads this book will she recommend it to Peter? B: Mary does not read this book. • The Jäger/Hulstijn approach predicts that the answer given by (B) should count as a (complete) answer, having the same status as the two other possible answer, namely “yes, he will” and “no, he will not” • Isaacs and Rawlins (2005): Responses like (B) do not resolve the issue raised by the question. Instead, they indicate a species of presupposition failure • p?p comes out as semantically equivalent with ?p 20 Reinhard Blutner

  21. Preliminary conclusions • The Jäger/Hulstijn approach has conceptual and empirical problems • The conceptual flaws are mainly related to the need of two different definitions of conditionals, one relating to the usual material implication, the other to the interrogative conditional. • The empirical problems are due to the uniformity of the classical partition semantics which gives all elements (blocks) of a partition the same status. 21 Reinhard Blutner

  22. 3 • Semantics of questions and answers • Jäger/Hulstijn question semantics • Ortho-algebraic question semantics • Answerhood 22 Reinhard Blutner

  23. Observables in physics • A substantial part of Quantum Theory relates to a theory of questions (or ‘observables’ in the physicist’s jargon) • Typical observables: • what is the polarization of the photon? • Is the photon polarized in -direction? • If photon 1 is -polarized, what is the polarization of photon 2? • The spectral theorem provides a decorated partition theory of questions/observables (structured propositions) 23 Reinhard Blutner

  24. Qubits • Assume Hilbert space with 2 dimensions • Orthonormal base {true, false} corresponding to two independent possible worlds • Pure qubit state: u =  true +  false • Pure states u are uniquely related to certain projection operators Pu simply written as u • Example operators: false,true, I (identity),  (zero) • I = true + false. All the operators true, false, I and are commuting with each other. Note: Instead of true/false we sometimes write 1/0 24 Reinhard Blutner

  25. Tensor product • In quantum theory complex systems are built by using the tensor product . • This operation applies both to vectors of the Hilbert space u v and to linear operators a  b. • Write 011instead of 011 and011instead of 011. • Example operators in case of 3 qubits (23 dimensional Hilbert space): • 000, 001, 010, … . • These operators are pairwise commuting. • They generate a Boolean algebra! 25 Reinhard Blutner

  26. Boolean algebras as special ortho-algebras • The formalism of Hermitian linear operators constitutes a question theory • The theory allows the combination of assertions and questions such as in conditional questions • The semantics of “decorated partitions” is a straight-forward consequence of the spectral theorem • By considering commuting observables a ‘classical’ partition semantics results, which can directly be compared with standard possible world frameworks. 26 Reinhard Blutner

  27. Consider the language of propositional logic L, extended with a question operator “?” and declarative operator “!” . QL* can be defined as the smallest set containing L and satisfying the following two clauses: a. if   QL* then ?  QL* and !  QL* b. if ,   QL* then ()  QL*, and ()  QL* The “flat fragment ” The query language QL* 27 Reinhard Blutner

  28. Ortho-algebraic semantics • «p» = (p) [ assigns projection operators to atoms] • «» = i i ai where «» = i i  ai (the spectral decomposition of «») • «» = «» «» (if «» and «» commute) • «!»= (ker «») • «?» = y  «» + n  «» Definitions:    = ()   =   () [Sasaki implication] 28 Reinhard Blutner

  29. Basic semantic notions Truth: u⊦ iff «» u = u [for assertions ] Span: (u, v ) ⊦  iff «»u = u and «»v = v for some 0 Entailment:  |=  iff «» «» = «» Facts: • The span of any expression  of QL* forms an equivalence relation • For assertive expressions it holds: • (u, v )⊦ iff u⊦  andv⊦  • u⊦  iff (u, u ) ⊦  29 Reinhard Blutner

  30. p in ortho-algebraic semantics The span of operators (equivalence relation) is depicted «p » = (10+11) 10 and 11 are the eigenvectors with eigenvalue 1 The null-space null is spanned by the eigenvectors 00 and 01 The null-space is suppressed 1 0 30 Reinhard Blutner

  31. y n ?p in ortho-algebraic semantics «p » = (10+11) «?p » = y (10+11) + n (01+00) 31 Reinhard Blutner

  32. y n 1 p?q with Sasaki implication Sasaki:    =   () «p  ?q» = (10+11) + (10+11)(y (01+11)+n (10+00)) = (00+01)+(y 11+n 10) Same partition as in JH-approach. However, the partition is decorated!

  33. yy yn ny nn ?p?qin ortho-algebraic semantics «?p?q» = [y (10+11) + n (01+00)] [y (11+01) + n (10+00)] = yy11 + yn10 + ny 01 + nn00 33 Reinhard Blutner

  34. 4 • Semantics of questions and answers • Jäger/Hulstijn question semantics • Ortho-algebraic question semantics • Answerhood 34 Reinhard Blutner

  35. Congruent Answers Definition (Application function): @(a, i) = ai, where ai is the corresponding projection operator in the spectral decomposition i i  aiof a. Definition:  is a congruent full answer to a question  iff @(«», t) = «» for some element t of the spectrum of «». Examples: • p is a proper answer to ?p, since @(y 1+n 0, y) = 1 • pis a proper answer to ?p, since @(y 1+n 0, n) = 0 35 Reinhard Blutner

  36. Conditional questions and answers A: If Mary reads this book will she recommend it to Peter? B: Yes. If Mary reads this book, she will recommend it to Peter B: *Yes. Mary reads this book, and she will recommend it to Peter • Conditional answers are not predicted by the Jäger/Hulstijn approach • How to handle them in ortho-algebraic semantics? • Proper conception of answerhood 36 Reinhard Blutner

  37. Congruent Answers 2 Definition:  is a congruent full answer to a question  iff @(«», t) + @(«», 1) = «» for some element t of the spectrum of «». Examples: • pq is a proper answer to p?q, • «p?q» = «p» + «p»«?q» = 1«p» + y«p»«q» + n«p»«q». • @(«p?q», y) = «p»«q», @(«p?q», 1) = «p» • @(«p?q», y) + @(«p?q», 1) =«p» + «p»«q» = «pq». • pq is a proper answer to p?q, analogously. 37 Reinhard Blutner

  38. Conclusions • Ortho-algebraic semantics conforms to a decorated partition theory (structured propositions) • It explains why informationally equivalent questions like “is the door open?” and “is the door closed?” have different meanings • It overcomes some conceptual and empirical problems of the Jäger/Hulstijn approach. • It resolves the biggest puzzle of this approach, which counter-intuitively predicts conjunctive answers for conditional questions • In the classical case of commuting operators it is equivalent to the structured meaning approach • Generalization to attitude questions possible (non-commuting operators) 38 Reinhard Blutner

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