An inquisitive perspective on meaning
An overview of the inquisitive research program
Point of departure
Floris Roelofsen
• A primary function of language is to exchange information
• Language is used both to provide and to request information
• Sentences have both informative and inquisitive potential
• Semantic theories have focused on informative content,
inquisitive content has received far less attention
Basic aim of the inquisitive research programme
• Develop a semantic framework where meanings capture
both informative and inquisitive content
Workshop on inquisitive logic and dependence logic
Amsterdam, June 17, 2013
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An inquisitive perspective on logic and pragmatics
Today
Inquisitive logic
The basic inquisitive system InqB
• Logic is traditionally concerned with entailment between
informative statements, ruling the validity of argumentation
• General motivation and definition of the semantics (this talk)
• We pursue a generalized notion of entailment, which applies
to both informative and inquisitive sentences, and logical
notions of relatedness, ruling the coherence of conversation
• Logical properties and axiomatization (Ciardelli)
Extensions
• Attentive inquisitive semantics (Westera)
Inquisitive pragmatics
• Suppositional inquisitive semantics (Groenendijk)
• Gricean pragmatics specifies rules for providing information
• See appendix for an overview of other extensions
• We aim to develop a pragmatics of exchanging information,
taking both informative and inquisitive content into account
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The system InqB
The traditional picture
• The system InqB can be seen as the equivalent of
classical logic in the inquisitive logical landscape
• Meaning = informative content
• Providing information = eliminating possible worlds
• The notion of meaning in InqB generalizes the classical notion
in a straightforward and conservative way, capturing both
informative and inquisitive content
• The treatment of the logical constants in InqB, when viewed
algebraically, is exactly the same as in classic logic
• Just like classical logic, InqB forms a common starting point
for many more comprehensive and refined inquisitive logics
• Captures only one type of language use: providing information
• Does not reflect the cooperative nature of communication
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The inquisitive picture
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Propositions in classical logic
• Propositions in classical logic are sets of possible worlds
• Propositions as proposals
• A speaker may propose one or more alternative updates
• Proposals that offer several alternative updates are inquisitive
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p
q
• Intuitively, a proposition carves out a region in logical space
• In asserting a sentence, a speaker provides the information
that the actual world is located in this region
• In this way propositions capture informative content
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Entailment
Algebraic operations
• Propositions are ordered in terms of informative strength
• Every ordered set has a certain algebraic structure,
and comes with certain basic algebraic operations
• One proposition is more informative than another just in case
it locates the actual world within a smaller region
• The set of classical propositions, ordered by entailment,
forms a complete Heyting algebra
A |= B ⇐⇒ A ⊆ B
• This means that there are three basic operations:
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p∧q
p
p∨q
1. Meet (= greatest lower bound w.r.t. entailment)
2. Join (= least upper bound w.r.t. entailment)
3. Relative pseudo-complementation
p ∨ ¬p
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Connectives in classical logic
Propositions in InqB
In classical logic, these basic algebraic operations are taken to be
expressed by conjunction, disjunction, and implication:
• [ϕ ∧ ψ] = [ϕ] ∩ [ψ]
meet
• [ϕ → ψ] = [ϕ] ⇒ [ψ]
relative pseudo-complement
• [ϕ ∨ ψ] = [ϕ] ∪ [ψ]
Worlds and information
• Assume, as before, a universe of possible worlds W
• Information state / piece of information: set of possible worlds
join
Propositions
• A proposition is a non-empty, downward closed set of states
And negation expresses pseudo-complementation relative to ⊥:
• [¬ϕ]
= [ϕ] ⇒ ⊥
• Rooted in seminal work on questions
pseudo-complement relative to ⊥
(Ham’73, Kar’77, GS’84)
• But with a crucial twist: downward closure
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The effect of an utterance
Example
Suppose that ϕ expresses the following proposition:
Common ground
• Body of shared information established in the conversation
• Modeled as an information state
(Stalnaker ’78)
w1
w2
w3
w4
The effect of an utterance
Then, in uttering ϕ, a speaker:
In uttering a sentence ϕ, a speaker:
• Steers the common ground towards a state that is
contained in {w1 , w2 } or in {w1 , w3 }
1. Steers the common ground towards a state in [ϕ]
2. Provides the information that the actual world lies in
S
[ϕ]
• Provides the information that the actual world is
S
located in [ϕ] = {w1 , w2 , w3 }
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Settling propositions and downward closure
Alternatives
• Among all the states in a proposition P, the ones that are
easiest to reach are the ones that contain least information
Settling a proposition
• Visually, these states are easy to identify:
they are the maximal elements of P
• We say that a piece of information α settles a proposition P
iff mutual acceptance of α leads the cg to a state in P
• We call these states the alternatives in P, and from now on,
when visualizing propositions, we will only depict alternatives
Downward closure
• We assume that if a proposition P is settled by α, then it is
also settled by any stronger piece of information β ⊂ α
Proposition:
• Therefore, propositions are required to be downward closed
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w1
w2
w3
w4
Alternatives:
w1
w2
w3
w4
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Informativeness
Inquisitiveness
• In uttering ϕ, a speaker provides the information that the
S
actual world is contained in [ϕ]
• We call
• In uttering ϕ, a speaker steers the cg towards a state in [ϕ]
• Sometimes, all that is needed to reach such a state
is mutual acceptance of info(ϕ)
S
[ϕ] the informative content of ϕ, info(ϕ)
• We say that ϕ is informative iff info(ϕ) , W
⇒ This is the case if info(ϕ) ∈ [ϕ]
w1
w2
w1
w2
w1
w2
w1
w2
w3
w4
w3
w4
w3
w4
w3
w4
+informative
−informative
+informative
• Otherwise, additional information needs to be provided
⇒ In this case, i.e., if info(ϕ) < [ϕ], we say that ϕ is inquisitive
Inquisitiveness and alternatives
If W is finite (which is the case in all our examples):
−informative
• ϕ is inquisitive ⇐⇒ [ϕ] contains at least two alternatives
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Informativeness and inquisitiveness
Entailment
Two natural conditions
In order for ϕ to entail ψ:
Summary
• ϕ is informative ⇔ info(ϕ) , W
• ϕ is inquisitive ⇔ info(ϕ) < [ϕ] ⇔fin at least two alternatives
1. ϕ must be at least as informative as ψ: info(ϕ) ⊆ info(ψ)
2. ϕ must be at least as inquisitive as ψ: [ϕ] ⊆ [ψ]
w1
w2
w1
w2
w1
w2
w1
w2
w3
w4
w3
w4
w3
w4
w3
w4
+informative
−inquisitive
+informative
+inquisitive
−informative
+inquisitive
(every piece of information that settles [ϕ] also settles [ψ])
Simplification
• The second condition implies the first
−informative
−inquisitive
• So: ϕ |= ψ ⇐⇒ [ϕ] ⊆ [ψ]
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Algebraic operations
Connectives
As before, conjunction, disjunction, implication & negation
can be taken to express these basic algebraic operations:
• Just as in the classical setting, the set of all propositions,
ordered by entailment, forms a complete Heyting algebra
• [ϕ ∧ ψ] = [ϕ] ∩ [ψ]
meet
• [ϕ → ψ] = [ϕ] ⇒ [ψ]
relative pseudo-complement
join
• [ϕ ∨ ψ] = [ϕ] ∪ [ψ]
• This means that we have the same basic operations:
1. Meet
• [¬ϕ]
2. Join
pseudo-complement relative to ⊥
= [ϕ] ⇒ ⊥
⇒ We enriched the notion of meaning, but we preserved
3. Relative pseudo-complementation
the essence of the classical treatment of the connectives
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Atomic sentences
Negation
Definition
Definition
S
• [¬ϕ] B [ϕ] ⇒ ⊥ = ℘ [ϕ]
• [p ] B ℘{w ∈ W | w (p ) = 1}
• First take the union of all the states in [ϕ];
then take the complement;
then the powerset
• the set of all states that make p true everywhere
Examples
Example, negation of an atomic sentence:
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[p ]
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[q]
[p ]
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[¬p ]
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Negation
Negation
Definition
S
Definition
• [¬ϕ] B [ϕ] ⇒ ⊥ = ℘ [ϕ]
S
• [¬ϕ] B [ϕ] ⇒ ⊥ = ℘ [ϕ]
• First take the union of all the states in [ϕ];
then take the complement;
then the powerset
• First take the union of all the states in [ϕ];
then take the complement;
then the powerset
Example, negation of an inquisitive sentence:
Atoms, negation, and inquisitiveness
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• [p ] and [¬ϕ] always contain a single alternative
• Thus, atoms and negations are never inquisitive
• We refer to non-inquisitive sentences as assertions
[¬ϕ]
[ϕ]
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Disjunction
Disjunction
Definition
Definition
• [ϕ ∨ ψ] B [ϕ] ∪ [ψ]
• ?ϕ
• [ϕ ∨ ψ] B [ϕ] ∪ [ψ]
B ϕ ∨ ¬ϕ
• ?ϕ
Examples
B ϕ ∨ ¬ϕ
Disjunction, inquisitiveness, and questions
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• Disjunctions are typically inquisitive
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• ∨ is the only source of inquisitiveness in our language
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p∨q
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• ?ϕ is typically inquisitive, and never informative
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• We refer to non-informative sentences as questions
?p
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Double negation
Conjunction
(1)
• Double negation always preserves the informative content of
a sentence, but removes inquisitiveness
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¬(p ∨ q)
p∨q
Carla speaks Spanish and she speaks French.
w1
w3
(2)
¬¬(p ∨ q)
w3
• Clearly, !ϕ is always an assertion
w4
∩
w2
w1
w2
w3
w4
=
w3
w4
Does Carla speak Spanish, and does she speak French?
w1
• Therefore, we abbreviate ¬¬ϕ as !ϕ
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w1
∩
w2
w1
w2
w3
w4
=
w3
w4
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Implication
(3)
Dependence
If John goes to the party, Mary will go as well.
(5)
w1
w3
(4)
w1
w2
w4
⇒
w2
w1
w2
w3
w4
?p → ?q
=
w3
w4
Whether John goes depends on whether Mary goes.
w1
w2
w3
w4
If John goes to the party, will Mary go as well?
w1
w3
w1
w2
w4
⇒
w2
w1
w2
w3
w4
(6)
=
w3
w4
Who goes depends on who has a car.
?x .Gx → ?x .Cx
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Questions, assertions, hybrids, and tautologies
? and ! as projection operators
• ϕ is a question iff it is not informative
Questions
• ϕ is an assertion iff it is not inquisitive
Hybrids
• ϕ is a hybrid iff it is both informative and inquisitive
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?p
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p
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p∨q
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ϕ ϕ ≡ ?ϕ ∧ !ϕ
?ϕ
• ϕ is a tautology iff it is neither informative nor inquisitive
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Tautologies
!ϕ
Assertions
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!(p ∨ ¬p )
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Support
Two facts about support
Fact 1: recursive characterization of support
Propositions and informative content relative to a state
1. s |= p
B {t ∈ [ϕ] | t ⊆ s }
S
• infos (ϕ) B [ϕ]s
• [ϕ]s
2. s |= ⊥
• ϕ is informative in s iff infos (ϕ) , s
iff s = ∅
3. s |= ϕ ∧ ψ
iff s |= ϕ and s |= ψ
5. s |= ϕ → ψ
iff ∀t ⊆ s : if t |= ϕ then t |= ψ
4. s |= ϕ ∨ ψ
Informativeness and inquisitiveness relative to a state
iff ∀w ∈ s : w (p ) = 1
iff s |= ϕ or s |= ψ
Fact 2: support, propositions, and entailment
• ϕ is inquisitive in s iff infos (ϕ) < [ϕ]s
1. s |= ϕ iff s ∈ [ϕ]
Support
2. ϕ |= ψ iff every state that supports ϕ also supports ψ
• s |= ϕ iff ϕ is neither informative nor inquisitive in s
⇒ we could also take the recursive characterization of support
as basic, and define propositions and entailment in terms of it
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Summary and looking ahead
Summary and looking ahead
Looking ahead
Summary
• Ivano Ciardelli will discuss the logical properties of InqB in
more detail
• InqB can be seen as the equivalent of classical logic
in the inquisitive logical landscape
• Matthijs Westera will discuss a system where meanings
capture informative and attentive content, and discuss a
pragmatic application of the notion of entailment in this system
• Its notion of meaning generalizes the classical notion,
capturing both informative and inquisitive content
• Jeroen Groenendijk will present an extension of InqB where
proposals are not only characterized in terms of the states
that settle/support it, but also in terms of the states that reject
it, and ones that dismiss one of its suppositions.
• Its treatment of the logical constants, when viewed
algebraically, is exactly the same as in classical logic
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Thank you
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Appendix: overview of extended IS systems
Attentive/unrestricted IS
• Defines propositions as arbitrary sets of states,
not necessarily downward closed.
• Besides informative and inquisitive content, these type of
propositions are taken to capture attentive content as well.
• Ref: Ciardelli ’09, Ciardelli et.al. ’09, Roelofsen ’11a/b,
Westera ’12a/b
Presuppositional IS
• Defines meanings as functions from states to propositions.
Presuppositional meanings are modeled as partial functions,
not yielding a proposition in every state.
www.illc.uva.nl/inquisitive-semantics
• Ref: Ciardelli et.al. ’12 ’13
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Appendix: overview of extensions
Appendix: overview of extensions
Radical IS
Inquisitive Dynamic Epistemic Logic (IDEL)
• Defines propositions in terms of support and reject conditions.
• Ref: Groenendijk & Roelofsen ’10, Lojko ’12, Sano ’12
• Integrates basic notions of InqB with epistemic logic.
• Introduces a dynamic announcement operator that models the
conversational effects of questions and assertions in a
uniform way.
Suppositional IS
• Extension of Radical IS, adding supposition failure conditions.
• Ref: Ciardelli & Roelofsen ’12
• Ref: Groenendijk & Roelofsen ’13
• Closely related: an inquisitive solution of the Gettier problem
for the notion of knowledge as justified true belief
Deontic IS
• Ref: Uegaki ’12
• Extension of Radical IS, adding deontic modalities.
• Ref: Aher ’12 ’13
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Appendix: overview of extensions
IS with witnesses
• Strengthens the support-conditions for existentials: in order to
support ∃xPx a state has to contain a witness for P. This is
done in order to get at a better logical notion of relatedness.
• Ref: Ciardelli et. al. ’12
IS with highlighting / discourse referents
• Assumes sentences highlight certain alternatives. When a
sentence is uttered, these alternatives are made available for
subsequent anaphoric reference in the discourse. Main
applications worked out so far: polarity particles (yes/no),
superlative modifiers (at least/at most).
• Ref: Roelofsen & van Gool ’10, Pruitt & Roelofsen ’11,
Farkas & Roelofsen ’12, Coppock & Brochhagen ’13
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