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 2
Background
Formal Semantics • Rooted in Linguistic, Logic and Philosophy. • Using formal methods to investigate and model the meaning of 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 ? 3
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. 4
Inquisitive Semantics p , q p , q p , q p , ¯ q p , ¯ q p , ¯ q ¯ ¯ p , ¯ ¯ p , ¯ ¯ ¯ p , ¯ ¯ p , q q p , q q p , q q (a) (b) (c) (a) The declarative sentence that q . (b) The closed question whether p or q . (c) The question whether q . 5
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.” 6
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. 7
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. 8
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. 9
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 . 10
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) 11
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’. 12
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. 13
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 s 1 ⊆ t 1 , s 2 ⊆ t 2 and s 3 ⊇ t 3 . So an extension s of t is a state with a stronger information, a stronger live necessity and more live possibilities. 14
Information States p , q ¯ p , q p , ¯ q p , ¯ ¯ q Information, live necessity, and live possibilities of a state. 15
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 . 16
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. � 17
Semantics The semantics assigns an issue [ [ ϕ ] ] to a formula ϕ according to the following definition: 18
Semantics The semantics assigns an issue [ [ ϕ ] ] to a formula ϕ according to the following definition: �� | p | , | p | , | p | ↑ �� [ [ p ] ] := . 18
Semantics The semantics assigns an issue [ [ ϕ ] ] to a formula ϕ according to the following definition: �� | p | , | p | , | p | ↑ �� [ [ p ] ] := . �� �� [ [ ♦ ϕ ] ] := W , W , � ] t 3 . t ∈ [ [ ϕ ] 18
Semantics The semantics assigns an issue [ [ ϕ ] ] to a formula ϕ according to the following definition: �� | p | , | p | , | p | ↑ �� [ [ p ] ] := . �� �� [ [ ♦ ϕ ] ] := W , W , � ] t 3 . t ∈ [ [ ϕ ] �� �� [ [ � ϕ ] ] := W , � ] t 2 , � ] t 3 . t ∈ [ [ ϕ ] t ∈ [ [ ϕ ] 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. 18
Semantics [ [ ϕ ∧ ψ ] ] := {⌊� t 1 ∩ s 1 , t 2 ∩ s 2 , t 3 ∪ s 3 �⌋ : t ∈ [ [ ϕ ] ] , s ∈ [ [ ψ ] ] }\ non-maximal . � as Given a triple s ∈ P ( W ) × P ( W ) × Up ( P ( W )) we define � s the (unique) greatest extension of s which is a state, i.e. the greatest information state t with t ≤ s . 19
Semantics [ [ ϕ ∧ ψ ] ] := {⌊� t 1 ∩ s 1 , t 2 ∩ s 2 , t 3 ∪ s 3 �⌋ : t ∈ [ [ ϕ ] ] , s ∈ [ [ ψ ] ] }\ non-maximal . � as Given a triple s ∈ P ( W ) × P ( W ) × Up ( P ( W )) we define � s 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 . � 19
Semantics [ [ ϕ ∧ ψ ] ] := {⌊� t 1 ∩ s 1 , t 2 ∩ s 2 , t 3 ∪ s 3 �⌋ : t ∈ [ [ ϕ ] ] , s ∈ [ [ ψ ] ] }\ non-maximal . � as Given a triple s ∈ P ( W ) × P ( W ) × Up ( P ( W )) we define � s 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 1 , � ] s 2 , � ] s 3 . s ∈ [ [ ϕ ] s ∈ [ [ ϕ ] s ∈ [ [ ϕ ] 19
Semantics � �� s 1 , s 1 , s 1 ↑ �� �� W , W , s 2 ↑ � � � [ [ ¬ ϕ ] ] := ∪ : s 2 � = s 1 ∪ s ∈ [ [ ϕ ] ] �� W , i , i ↑ � � � : i ∈ s 3 and i �⊇ s 2 . 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 . 20
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 . 21
Results Free Choice: We have narrow scope and wide scope free choice for might-sentences: ] = {� W , W , | p | ↑ ∪ | q | ↑ �} . [ [ ♦ ( p q )] ] = [ [!( ♦ p ♦ q )] ] = [ [ ♦ p ∧ ♦ q ] � � Note that � ( p q ) entails ♦ ( p q ) and hence entails ♦ p ∧ ♦ q . � � 22
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