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. 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 - PowerPoint PPT PresentationTRANSCRIPT
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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
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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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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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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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