Inquisitive Semantics and Logic
Ivano Ciardelli
Milano, 30 settembre 2011
www.illc.uva.nl/inquisitive-semantics
What is formal semantics?
What is meaning?
In order to say what a meaning is, we may first ask what a
meaning does, and then find something that does that.
David Lewis, General Semantics
Information states
• An information state is a set of models for the language
(possible worlds).
• We think of an information state s as the set of configurations
that the subject considers possible for the actual world.
• s is more informed than t in case s ⊆ t.
• We will think of sentences as acting not on the participants’
private information states, but on the common ground of the
conversation.
The traditional picture
What meaning does is to provide information.
• Meaning = informative content
• Providing information = eliminating possible worlds
• Update: s [ϕ] = s ∩ |ϕ| = {w ∈ s | w |= ϕ}
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Ignorance state W
The traditional picture
What meaning does is to provide information.
• Meaning = informative content
• Providing information = eliminating possible worlds
• Update: s [ϕ] = s ∩ |ϕ| = {w ∈ s | w |= ϕ}
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W [p ]
The traditional picture
What meaning does is to provide information.
• Meaning = informative content
• Providing information = eliminating possible worlds
• Update: s [ϕ] = s ∩ |ϕ| = {w ∈ s | w |= ϕ}
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W [p ][¬q]
Correspondence with classical logic
• ϕ |= ψ in case ϕ is at least as informative as ψ
• ⇐⇒ for any s, s [ϕ] ⊆ s [ψ]
• ⇐⇒ for any s, s ∩ |ϕ| ⊆ s ∩ |ψ|
• ⇐⇒ |ϕ| ⊆ |ψ|
• ⇐⇒ for any w, w |= ϕ ⇒ w |= ψ
Limitations of the classical picture
1. Even in a purely informative dialogue, utterance serve other
purposes than to provide information: in particular, crucially,
they can request information.
2. Does not provide the means to analyze the coherence of a
dialogue.
3. Updates are imposed on the common ground: does not reflect
the cooperative nature of communication.
Mission statement
Inquisitive semantics
• Meaning is traditionally identified with informative content
• Our main aim is to develop a notion of meaning that
captures both informative and inquisitive content
Inquisitive logic
• Logic is traditionally concerned with entailment, which rules
the validity of argumentation
• We aim to develop logical notions of relatedness, which
govern the coherence of conversation
Inquisitive pragmatics
• Gricean pragmatics specifies rules for providing information
• We aim to develop a pragmatics of exchanging information,
taking both informative and inquisitive content into account
Mission statement
Inquisitive semantics
• Meaning is traditionally identified with informative content
• Our main aim is to develop a notion of meaning that
captures both informative and inquisitive content
Inquisitive logic
• Logic is traditionally concerned with entailment, which rules
the validity of argumentation
• We aim to develop logical notions of relatedness, which
govern the coherence of conversation
Inquisitive pragmatics
• Gricean pragmatics specifies rules for providing information
• We aim to develop a pragmatics of exchanging information,
taking both informative and inquisitive content into account
Mission statement
Inquisitive semantics
• Meaning is traditionally identified with informative content
• Our main aim is to develop a notion of meaning that
captures both informative and inquisitive content
Inquisitive logic
• Logic is traditionally concerned with entailment, which rules
the validity of argumentation
• We aim to develop logical notions of relatedness, which
govern the coherence of conversation
Inquisitive pragmatics
• Gricean pragmatics specifies rules for providing information
• We aim to develop a pragmatics of exchanging information,
taking both informative and inquisitive content into account
Today
• Motivations
X
• Propositions as proposals
• Propositional inquisitive semantics
Other themes
• Inquisitive logic
• Inquisitive pragmatics
• Relatedness, compliance
• Attentive semantics
Worlds, possibilities, and propositions
• Start with a universe of possible worlds
• Possibility: set of possible worlds
• Proposition: set of possibilities
Illustration
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worlds
possibility
proposition
How to think of propositions?
• Traditionally, a proposition is simply a set of possible worlds
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• We think of such a proposition A as providing the information
that the actual world corresponds to one of the worlds in A
How to think of propositions?
• Now, a proposition is a set of possibilities
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• How should we think of such propositions?
• What is the information that they provide?
• Could we think of them as representing something else
besides informative content? If so, what exactly?
Informative and inquisitive content
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• We think of a proposition A as representing a proposal to
update the common ground in one or more ways
• Each possibility in A embodies one of the proposed updates
• A provides the information that the actual world is contained
in at least one of the possibilities in A
• At the same time, A requests a response that establishes at
least one of the proposed updates
Informative and inquisitive content
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• We think of a proposition A as representing a proposal to
update the common ground in one or more ways
• Each possibility in A embodies one of the proposed updates
• A provides the information that the actual world is contained
in at least one of the possibilities in A
• At the same time, A requests a response that establishes at
least one of the proposed updates
⇒ a single semantic object embodies both informative and inquisitive content
Informative content
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• A proposition A provides the information that the actual world
is contained in at least one of the possibilities in A
• So, the informative content of A , info(A ), is determined
by the union of all the possibilities in A :
info(A ) =
[
A
• We say that A is informative in case its informative content is
non-trivial, i.e. if info(A ) , W .
Inquisitive proposals
• A proposition A requests a response that establishes at least
one of the updates that A proposes.
• Sometimes it suffices to accept the information provided by A .
• If additional information is required, we call A inquisitive.
• Formally, A is inquisitive in case info(A ) < A .
• ⇐⇒ A has a greatest possibility.
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non-inquisitive
inquisitive
inquisitive
Questions, assertions, and hybrids
• An assertion is a non-inquisitive proposition.
• A question is a non-informative proposition.
• An hybrid is a proposition that is informative and inquisitive.
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Assertion
Question
Hybrid
Entailment
Definition (Entailment)
A |= B iff any α ∈ A is included in some β ∈ B.
Problem
Problem: entailment is not antisymmetric.
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≡
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A restriction on propositions
In order to obtain a unique representation of meanings, we put the
following restriction on propositions.
Definition
A proposition is a downward closed non-empty set of possibilities.
Denote the set of propositions by Ω.
Now entailment is antisymmetric and boils down to:
Definition (Entailment)
A |= B ⇐⇒ A ⊆ B
Today
• Motivations
X
• Propositions as proposals
X
• Propositional inquisitive semantics
Other themes
• Inquisitive logic
• Inquisitive pragmatics
• Relatedness, compliance
• Attentive semantics
A Propositional Language
Basic Ingredients
• Set of proposition letters P
• Connectives ⊥, ∧, ∨, →
Defined connectives
• Negation:
• Non-inquisitive closure:
¬ϕ B ϕ → ⊥
!ϕ B ¬¬ϕ
• Non-informative closure: ?ϕ B ϕ ∨ ¬ϕ
Problem
How to give give meaning to these connectives?
Algebraic foundations of classical logic
Classical propositions
• Sets of possible worlds.
• Entailment ordering: A ⊆ B
In the poset (℘(W ), ⊆) we have:
1. Top and bottom element: W , ∅;
2. Least upper bound (join) of A and B: A ∪ B;
3. Greatest lower bound (meet) of A and B: A ∩ B;
4. Complement of A : C (A ) = {a ∈ A | a < A };
5. Implication (rel. presudo-complement) A ⇒ B := C (A ) ∪ B.
Algebraic foundations of classical logic
These algebraic operations on ℘(W ) provide meaning to the
connectives of classical logic.
Classical truth
1. |p | = {w | w (p ) = 1}
2. |⊥| = ∅;
3. |ϕ ∧ ψ| = |ϕ| ∩ |ψ|;
4. |ϕ ∨ ψ| = |ϕ| ∪ |ψ|;
5. |¬ϕ| = C(|ϕ|);
6. |ϕ → ψ| = |ϕ| ⇒ |ψ|.
Inquisitive algebra
Let’s look at what structure the algebra of inquisitive meaning has.
In the poset (Ω, ⊆) we have:
• Top and bottom elements: ℘(W ), {∅}.
• Join of A and B, given by A ∪ B.
• Meet of A and B, given by A ∩ B.
Inquisitive algebra
• Except for ℘(W ), no element has a complement: if
A ∪ B = ℘(W ) then W must be in either A or B, and by
downward closure A = ℘(W ) or B = ℘(W ).
• However, the implication A ⇒ B always exists, and is given by
A ⇒ B = {γ | for all χ ⊆ γ, if χ ∈ A then χ ∈ B )}
Propositional inquisitive semantics
The algebraic operations on (Ω, ⊆) provide meaning to the
connectives.
Inquisitive meaning
1. [p ] = ↓{|p |} = {α | w (p ) = 1 for all w ∈ α}
2. [⊥] = {∅};
3. [ϕ ∧ ψ] = [ϕ] ∩ [ψ];
4. [ϕ ∨ ψ] = [ϕ] ∪ [ψ];
5. [ϕ → ψ] = [ϕ] ⇒ [ψ].
Treatment of information
Proposition
In IS, the classical treatment of information is preserved.
info(ϕ) :=
[
[ϕ] = |ϕ|
Thus, Inquisitive Semantics is a conservative refinement of
classical dynamic semantics.
Corollary
ϕ is an assertion iff [ϕ] = ↓{|ϕ|}.
Tautologies and questions
• Recall that a formula is a question in case it is
non-informative, i.e. in case info(ϕ) = W .
• Given that info(ϕ) = |ϕ|, this yields the following:
Corollary
ϕ is a question iff it is a classical tautology.
• In a classical setting, any non-informative sentence is
tautological, i.e., insignificant.
• In inquisitive semantics, classical tautologies come to form a
new class of meaningful sentences, namely questions
• Questions are non-informative, but they may be inquisitive.
The semantics at work: atoms
[p ] = ↓{|p |}
An atom is an assertion providing the information that p.
(1)
John speaks Russian.
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[p ]
The semantics at work: negation
[¬ϕ] = [ϕ → ⊥] = {γ | ∀χ ⊆ γ, if χ ∈ [ϕ] then χ ⊆ ∅} = ↓{|¬ϕ|}
¬ϕ is an assertion providing the information that ϕ does not hold.
(2)
John does not speak Russian or French.
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[ϕ]
[¬ϕ]
The semantics at work: disjunction
[ϕ ∨ ψ] = [ϕ] ∪ [ψ]
Disjunction is a source of inquisitiveness.
(3)
John speaks Russian or he speaks French.
(4)
Does John speak Russian?
Examples:
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p∨q
?p (B p ∨ ¬p )
The semantics at work: conjunction.
Conjunction applies uniformly to questions and assertions
(5)
John speaks Russian and he speaks French.
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(6)
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=
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Does John speak Russian, and does he speak French?
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∩
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The semantics at work: implication
Implication applies uniformly to questions and assertions
(7)
If John goes to the party, Mary will go as well.
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(8)
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=
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If John goes to the party, will Mary go as well?
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⇒
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The semantics at work: implication
Conditional questions with disjunctive antecedents
(9)
If John or Fred goes to the party, will Mary go as well?
There are four possibilities for this sentence,
corresponding to the following responses:
(10)
a.
b.
c.
d.
Yes, if John or Fred goes, Mary will go as well.
No, if John or Fred goes, Mary won’t go.
If J goes, M will go as well, but if F goes, M won’t go.
If F goes, M will go as well, but if J goes, M won’t go.
Non-inquisitive closure
[!ϕ] = [¬¬ϕ] = ↓{|ϕ|}
• !ϕ is an assertion providing the information that ϕ
• The operator ! makes ϕ into an assertion preserving
informative content
• ϕ is an assertion iff ϕ ≡!ϕ
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p∨q
!(p ∨ q)
Non-informative closure
[?ϕ] = [ϕ ∨ ¬ϕ] = [ϕ]∪ ↓{|¬ϕ|}
• ?ϕ is always a question
• ϕ is a question iff ϕ ≡?ϕ
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?p
?(p ∨ q)
Pure components decompositions
Questions
For any ϕ
• ?ϕ is a question
?ϕ
ϕ ≡!ϕ∧?ϕ
• !ϕ is an assertion
• ϕ ≡ ?ϕ ∧ !ϕ
Assertions
!ϕ
Overview
Today
• Motivations
X
• Propositions as proposals
X
• Propositional inquisitive semantics
Other themes
• Inquisitive logic
• Inquisitive pragmatic
• Relatedness, compliance
• Attentive semantics
X
Some references
Inquisitive semantics and pragmatics
Jeroen Groenendijk and Floris Roelofsen (2009)
Stanford workshop on Language, Communication and Rational Agency
Inquisitive semantics and intermediate logics
Ivano Ciardelli (2009)
Master of Logic series of ILLC
Algebraic foundations for inquisitive semantics
Floris Roelofsen (2011)
Logics for Rational Interaction
Inquisitive logic
Ivano Ciardelli and Floris Roelofsen (2011)
Journal of Philosophical Logic 40(1), 55–94.