linear algebra and geometric approaches to meaning 4b. semantics of questions

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Linear Algebra and Geometric Approaches to Meaning

4b. Semantics of Questions

Reinhard Blutner

Universiteit van Amsterdam

ESSLLI Summer School 2011, Ljubljana

August 1 – August 7, 2011

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1. Semantics of questions and answers

2. Jäger/Hulstijn question semantics

3. Ortho-algebraic question semantics

4. Answerhood

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

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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.

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W

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

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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

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W

W

yes no

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.

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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)Reinhard Blutner

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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.

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2

1. Semantics of questions and answers

2. Jäger/Hulstijn question semantics

3. Ortho-algebraic question semantics

4. Answerhood

The query language QL

• 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

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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.

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Declaratives and questions

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W

W

W

Declaratives

Questions

Conditional Questions

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.

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Jäger/Hulstijn’s question semantics

a. «p» = {(u,v) W2: u(p) and v(p)}

b. «» = {(u,v) W2: (u,u)«» and (v,v)«»}

c. «» = «»«»

d. «?» = {(u,v) : (u,u) «» iff (v,v) «»}

e. «» = {(u,v) «?»:if (u,v) «» then (u,v)

«»«»}

Definitions: () ()

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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}Reinhard Blutner

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Picture of meaning for p

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• Note that a subset of W is partitioned only.

• The partition consists of a single proposition: (p) = {10, 11}

Picture of meaning for ?p

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• The domain W is partitioned into two proposition: (p) = {10, 11} and (p) = {00, 01}

Picture of meaning p ?q

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• The domain W is partitioned into three proposition:(p) = {00, 01}, (pq) = {11}, (pq) = {11}.

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

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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.

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1. Semantics of questions and answers

2. Jäger/Hulstijn question semantics

3. Ortho-algebraic question semantics

4. Answerhood

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)

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Qubits• Assume Hilbert space with 2 dimensions

• Orthonormal base {true, false} corresponding to two independent possible worlds

• Pure qubit state: u = a true + b 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

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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 011 instead of 011 and011 instead of 011.

• Example operators in case of 3 qubits (23 dimensional Hilbert space):

- 000, 001, 010, … . - These operators are pairwise commuting. - They generate a Boolean algebra!

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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.

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The query language QL*

• 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 ”

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Ortho-algebraic semantics

a. «p» = (p) [ assigns projection operators to atoms]

b. «» = i i ai where «» = i i ai (the

spectral decomposition of «»)

c. «» = «» «» (if «» and «» commute)

d. «!»= (ker «»)

e. «?» = y «» + n «»

Definitions: = () = () [Sasaki implication]

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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 ⊦ and v ⊦ - u ⊦ iff (u, u ) ⊦

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p in ortho-algebraic semantics

1

0

«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

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The span of operators (equivalence relation) is depicted

?p in ortho-algebraic semantics

y

n

«p » = (10+11)

«?p »

= y (10+11) + n

(01+00)

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p?q with Sasaki implication

y n

1

«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!

Sasaki: = ()

?p ?q in ortho-algebraic semantics

«?p ?q »

= [y (10+11) + n

(01+00)] [y (11+01) + n

(10+00)]

= yy 11 + yn 10 +

ny 01 + nn 00

nnny

yy yn

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4

1. Semantics of questions and answers

2. Jäger/Hulstijn question semantics

3. Ortho-algebraic question semantics

4. Answerhood

Congruent Answers

Definition (Application function): @(a, i) = ai, where ai is the corresponding projection operator in the spectral decomposition i

i ai of 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 p is a proper answer to ?p, since @(y 1+n 0, n) = 0

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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

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Congruent Answers 2

Definition: is a congruent full answer to a question iff @(«», t) + @(«», 1) = «» for some element t of the spectrum of «».

Examples:• pq 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» =

«pq».

• pq is a proper answer to p?q, analogously.

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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)

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