Might and Must in Questions Jakob Piribauer and Joannes B. Campell - - PowerPoint PPT Presentation

Might and Must in Questions

Jakob Piribauer and Joannes B. Campell May 3, 2017

Institute for Logic, Language and Computation - University of Amsterdam 1

Outline

Background Inquisitive Semantics Desiderata Modelling Proposal General Idea Basic Definitions The Semantics Results

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Background

Formal Semantics

• Rooted in Linguistic, Logic and Philosophy.
• Using formal methods to investigate and model the meaning
• f natural language.
• Example: Dynamic Predicate Logic: A logic that is

compositional and can deal with anaphora. There is a man. He wears a hat. (∃x)(Mx ∧ Hx) Not compositional. (∃x)(Mx) ∧ Hx Reference of x?

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Inquisitive Semantics

• Model the informative and inquisitive content of sentences.
• Support conditional semantics.
• Information states: set of possible worlds.
• Issues: non-empty, downward closed set of information states.
• Sentences are modelled as issues and contain all the states

that resolve the issues raised by the sentence.

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Inquisitive Semantics

p, q p, ¯ q ¯ p, q ¯ p, ¯ q

(a)

p, q p, ¯ q ¯ p, q ¯ p, ¯ q

(b)

p, q p, ¯ q ¯ p, q ¯ p, ¯ q

(c)

(a) The declarative sentence that q. (b) The closed question whether p or q. (c) The question whether q.

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Goal

Incorporate the epistemic modalities might and must. Reading:

• ‘might’ introduces a relevant possibility.

Example: “Have you thought about taking an umbrella? It might rain in the evening.”

• ‘must’ indicates what is presumably the case.

Example: “The keys must be in the car, I have looked everywhere else.”

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Desideratum 1: must and might in Questions

The use of must and might in questions seems to allow answers along the following lines.

• 1. Might Dr. Jekyll and Mr. Hyde be the same person?

1.1 Yes, they might be the same person. 1.2 No, they must be different people.

• 2. Must Dr. Jekyll and Mr. Hyde be different people?

2.1 Yes, they must be different people. 2.2 No, they might be the same person.

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Desideratum 2: Epistemic Contradictions

might-sentences and must-sentences can lead to contradictions. In particular the following four sentences seem contradictory:

• 1. It is not raining and it might be raining.
• 2. It is not raining and it must be raining.
• 3. It must not be raining and it might be raining.
• 4. It must not be raining and it must be raining.

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Desideratum 3: Free Choice

It has frequently been observed that might sentences allow for free choice (e.g. see Kamp, 1973). An example of this effect is that the following three sentences are equivalent.

• 1. It might rain or snow.
• 2. It might rain or it might snow.
• 3. It might rain and it might snow.

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Live Possibilities

• Introduced by Willer, 2013.
• To consider something as a live possibility is “[. . . ] a

disposition to take the possibility of p’s being true into serious consideration whenever it is of practical or theoretical pertinence.” (Willer, 2013)

• Roelofsen 2016 integrates live possibilities into inquisitive

semantics in order to capture the attentive content of a sentence conveyed by might.

• The information states of this semantics have two

components: information and live possibilities.

• There is no treatment of ‘must’ and the negation of ♦p is ¬p.

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Getting a ‘must’ from the ‘might’?

• On the surface, ‘must’ and ‘might’ should work as duals in the

sense that ¬♦p ≡ ¬p; (1) ¬p ≡ ♦¬p. (2)

• Attempts to define the semantics of ‘must’ using this

framework lead to undesirable interplay with negation.

• In particular, we should have failure of contraposition, as e.g.:

p ♦p, but (3) ¬p ¬p. (4)

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Live Necessities

• Introduced by Willer, 2016.
• Live necessities “[impose] an upper bound on the region of

logical space to be taken seriously in discourse and reasoning [. . . ].” (Willer, 2016)

• Live necessities reflect our reading of ‘must’.

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Proposal

Information States

Our information states model three aspects of an agent’s epistemic state:

• Information. Modelled as a set of possible worlds.
• Live necessity. Modelled in the same way as information.
• Live possibilities. Modelled as an upward closed collection of

sets of worlds.

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Information States

Information state: An information state s is a triple i, n, l with i, n ∈ P(W ) and l ∈ Up(P(W )) that satisfies the following conditions:

• 1. n ⊆ i.
• 2. n ∈ l.
• 3. If n = ∅ then ∀j ∈ l : j ∩ n = ∅.

Information states are ordered as follows: For states s and t: s ≤ t iff s1 ⊆ t1, s2 ⊆ t2 and s3 ⊇ t3. So an extension s of t is a state with a stronger information, a stronger live necessity and more live possibilities.

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Information States

p, q ¯ p, q p, ¯ q ¯ p, ¯ q

Information, live necessity, and live possibilities of a state.

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Issues and Support

Issues: An issue is a non-empty set of pairwise ≤-incomparable information states. The elements of an issue are called alternatives. An issue with more than one alternative is called inquisitive. Support: An information state s supports an issue I, in symbols s I, iff there is a t ∈ I with s ≤ t. Entailment: An issue I entails a issue J , in symbols I J , if s I implies s J for all states s.

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Syntax

We consider formulas built up from the atomic sentences using the binary connectives ∧ and

• as well as the unary connectives ¬, ♦,

and !. Further we use the abbreviation ?ϕ for ϕ

• ¬ϕ.

ϕ ::= p | ¬ϕ | ♦ϕ | ϕ | !ϕ | ϕ ∧ ϕ | ϕ

• ϕ.

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Semantics

The semantics assigns an issue [ [ϕ] ] to a formula ϕ according to the following definition:

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Semantics

The semantics assigns an issue [ [ϕ] ] to a formula ϕ according to the following definition: [ [p] ] :=

• |p|, |p|, |p|↑

.

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Semantics

The semantics assigns an issue [ [ϕ] ] to a formula ϕ according to the following definition: [ [p] ] :=

• |p|, |p|, |p|↑

. [ [♦ϕ] ] :=

• W , W ,

t∈[ [ϕ] ] t3

• .

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Semantics

The semantics assigns an issue [ [ϕ] ] to a formula ϕ according to the following definition: [ [p] ] :=

• |p|, |p|, |p|↑

. [ [♦ϕ] ] :=

• W , W ,

t∈[ [ϕ] ] t3

• .

[ [ϕ] ] :=

• W ,

t∈[ [ϕ] ] t2, t∈[ [ϕ] ] t3

• .

The union in the third component is motivated by the intuition that a sentence like ‘It must be the case that it might rain’ seems to require taking the possibility that it rains seriously.

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Semantics

[ [ϕ∧ψ] ] := {⌊t1 ∩ s1, t2 ∩ s2, t3 ∪ s3⌋ : t ∈ [ [ϕ] ], s ∈ [ [ψ] ]}\non-maximal. Given a triple s ∈ P(W ) × P(W ) × Up(P(W )) we define

s as

the (unique) greatest extension of s which is a state, i.e. the greatest information state t with t ≤ s.

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Semantics

[ [ϕ∧ψ] ] := {⌊t1 ∩ s1, t2 ∩ s2, t3 ∪ s3⌋ : t ∈ [ [ϕ] ], s ∈ [ [ψ] ]}\non-maximal. Given a triple s ∈ P(W ) × P(W ) × Up(P(W )) we define

s as

the (unique) greatest extension of s which is a state, i.e. the greatest information state t with t ≤ s. [ [ϕ

• ψ]

] := [ [ϕ] ] ∪ [ [ψ] ] \ non-maximal.

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Semantics

[ [ϕ∧ψ] ] := {⌊t1 ∩ s1, t2 ∩ s2, t3 ∪ s3⌋ : t ∈ [ [ϕ] ], s ∈ [ [ψ] ]}\non-maximal. Given a triple s ∈ P(W ) × P(W ) × Up(P(W )) we define

s as

the (unique) greatest extension of s which is a state, i.e. the greatest information state t with t ≤ s. [ [ϕ

• ψ]

] := [ [ϕ] ] ∪ [ [ψ] ] \ non-maximal. [ [!ϕ] ] :=

• s∈[

[ϕ] ] s1, s∈[ [ϕ] ] s2, s∈[ [ϕ] ] s3

• .

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Semantics

[ [¬ϕ] ] :=

• s∈[

[ϕ] ]

s1, s1, s1↑ ∪

• W , W , s2↑

: s2 = s1

• ∪
• W , i, i↑

: i ∈ s3 and i ⊇ s2 . Each alternative s of ϕ has to be rejected:

• A state can establish the complement of info(s).
• If the alternative s is such that nec(s) is more specific than

info(s), then establishing the complement of nec(s) as a live possibility rejects s.

• In case that s contains live possibilities that are not at most

as specific as its necessity, then establishing the complement as a live necessity rejects s.

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Results

Results

Negated Modalities: [ [¬♦p] ] = {W , |p|, |p|

↑} = [

[¬p] ], [ [¬p] ] = {W , W , |p|

↑} = [

[♦¬p] ]. must and might in Questions: [ [?p] ] = [ [p] ] ∪ [ [♦¬p] ] and [ [?♦p] ] = [ [♦p] ] ∪ [ [¬p] ]. This corresponds to the intuitions mentioned in the desiderata. Failure of Contraposition: The failure of contraposition is predicted by our semantics, as we have that p p and ¬p ¬p.

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Results

Free Choice: We have narrow scope and wide scope free choice for might-sentences: [ [♦(p

• q)]

] = [ [!(♦p

• ♦q)]

] = [ [♦p ∧ ♦q] ] = {W , W , |p|↑ ∪ |q|↑}. Note that (p

• q) entails ♦(p
• q) and hence entails ♦p ∧ ♦q.

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Results

Epistemic contradictions: [ [p ∧ ♦¬p] ] = {|p|, ∅, P(P(W ))} . This gives us the following with respect to entailments: p ∧ ♦¬p ♦ϕ, for all formulas ϕ, ϕ, for all formulas ϕ, p, ¬p, q, for any q = p.

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References

• Ciardelli, Ivano, Jeroen Groenendijk, and Floris Roelofsen (2009).

“Attention! ‘Might’ in inquisitive semantics”. In: Semantics and Linguistic Theory Vol. 19: pp. 91-108.

• Groenendijk, Jeroen and Floris Roelofsen (2009). “Inquisitive semantics

and pragmatics”. Presented at the Workshop on Language, Communication, and Rational Agency at Stanford, available via www.illc.uva.nl/inquisitive-semantics.

• Kamp, Hans (1973). “Free choice permission”. In: Proceedings of the

Aristotelian Society, Vol. 74: pp. 57-74.

• Roelofsen, Floris (2016). “First sketch of inquisitive live possibility

semantics”. Available via www.illc.uva.nl/inquisitivesemantics/courses/amsterdam-2016.

• Willer, Malte (2013). “Dynamics of Epistemic Modality”. In:

Philosophical Review, Vol. 122, No. 1: pp. 45-92.

• Willer, Malte (2016). “Lessons from Sobel Sequences”. Forthcoming in

Semantics and Pragmatics.

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