Veltman style modalities for a propositional language (1) It might be sunny. But it’s not sunny (easy update) (2) # It’s not sunny. But it might (for all I know) be sunny. • for atomic formulas: λ ik ( p ∧ k ( i + p )), where i + p is interpreted as i ∩ � p � . • for ¬ d : λ P λ ik ( ¬ P ∧ k ( i − Pik )). • for ∧ d : λ P λ Q λ ik Pi ( λ i ′ Qi ′ k ) • for � d : λ P λ ik ki , with the presupposition that i 1 + P � 0 Dynamic Diamonds presuppose that their test succeeds. The at issue content of might intu- itively is to bring a particular epiistemic possi- bility to mind. 1
The first order case We have used two notions of left context— Veltman modalities require left contexts to be sets of worlds or something similar, while modal subordination requires left contexts to contain lists of accessible referents. Put these together, adding another parameter w for a set of worlds that a continuation needs. 2
A few details Our first combination rule for modalized sen- tences: � S 1 . S 2 � = λ i 1 i 2 wk 1 k 2 f . � S 1 � i 1 i 2 w (3) ( λ i ′ 1 i ′ 2 . � S 2 � i ′ 1 , i 2 k 1 k 2 Π 1 ) k 2 w f This gives us: (4) λ � ( ∃ x ( W ( x ) ∧ � ( ( E ( sel ( x :: i 1 ∪ i 2 ) , u ) ∧ ⊤ Second combination rule for non modalized second sentences a ff ects w (5) � S 1 . S 2 � = λ i 1 i 2 k 1 k 2 w f . � S 1 � i 1 i 2 k 1 ( λ i ′ 1 i ′ 2 . � S 2 � i ′ 1 , i ′ 2 k 1 k 2 Π 2 ) w + S 2 f 3
Independent but anaphorically linked modal- ities (6) Sam wants to marry an Italian. He hopes she will be rich. (7) Hob thinks a witch has blighted his mare, and Nob thinks she has killed his cow. First attempt: ∃ x s → e B h (witch( ∨ x ) ∧∃ ! u ( h ′ s mare( u ) ∧ blighted( ∨ x , u ))) B (8) ∧ killed( x , u ))) The value of x within Nob’s belief worlds may be disjoint from the value of x in Hob’s belief worlds and this we don’t want. 4
Dependent individual concepts in TY2: (9) x is a dependent concept relative to a ’s and b ’s beliefs i ff ∀ w ′ ∈ B a , w ∃ w ′′ ∈ B b , w x ( w ′ ) = x ( w ′′ ) and ∀ w ′′ ∈ B b , w ∃ w ′ ∈ B a , w x ( w ′′ ) = x ( w ′ ) (10) λ w ∃ x dic ( h , n ) ∀ w ′ ∈ B h , w (witch( x ( w ′ ) , w ′ ) ∧ ∃ ! u ( h ′ s mare( u , w ′ ) ∧ blighted( x ( w ′ ) , u ) , w ′ )) ∧ ∀ w ′′ ∈ B n , w ( ∃ ! v ( h ′ s cow( v , w ′′ ) ∧ killed( x ( w ′′ ) , u , w ′′ ))) (11) ∃ x dic ( h , n ) B h (((witch( ∨ x ) ∧∃ ! u ( h ′ s mare( u ) ∧ blighted( ∨ x , u ))) ∧ B n ( ∃ ! v ( h ′ s cow( v , ′ ) ∧ killed( ∨ x , u ))) 5
3 minute sketch of SDRT Three tasks: • segmenting a text into E(lementary) D(iscourse) U(nit)s • computing attachment points of EDUs in a discourse structure • computing one or more discourse relations between an EDU and its attachment point(s) A discourse structure or SDRS results from these computations and may contain complex con- stituents containing multiple EDUs. An SDRS is a discourse logical form & can be represented in various ways, even as a λ term in a De Groote continuation style semantics (see appendix for some thoughts) 6
An Example (12) π 1 . John had a great evening last night. π 2 . He had a great meal. π 3 . He ate salmon. π 4 . He devoured lots of cheese. π 5 . He then won a dancing competition. (12 ′ ) � A , F , Last � , where: A = { π 0 , π 1 , π 2 , π 3 , π 4 , π 5 , π 6 , π 7 } F ( π 0 ) = Elaboration ( π 1 , π 6 ) F ( π 6 ) = Narration ( π 2 , π 5 ) ∧ Elaboration ( π 2 , π 7 ) F ( π 7 ) = Narration ( π 3 , π 4 ) Last = π 5 7
a DRS- like representation: π 0 π 1 , π 6 π 1 : K π 1 π 2 , π 5 , π 7 π 2 : K π 2 , π 5 : K π 5 Narration ( π 2 , π 5 ) π 3 , π 4 π 0 : π 6 : π 7 : π 3 : K π 3 , π 4 : K π 4 , Narration ( π 3 , π 4 ) Elaboration ( π 2 , π 7 ) Elaboration ( π 1 , π 6 ) 8
SDRT continuation style (13) ( π 1 ) A man walked in. ( π 2 ) He coughed. Here we compute a discourse relation between π 1 and π 2 . I’ll assume it’s Narration. The De Groote rule works nicely here: • λ io π 1 : ∃ x ( Mx ∧ Wx ∧ o ( i + x )) i [ λ i ′ π 2 : C sel( i ′ ) ∧ o ( i ′ )] −→ β • λ io π 1 : ∃ x ( Mx ∧ Wx ∧ π 2 : C sel( i + x ) ∧ Narration( π 1 , π 2 ) ∧ o ( i + x )) 9
Using Asher-Pogodolla... Let’s now consider a di ff erent second sentence: (14) ( π 1 ) A man walked in. ( π 3 ) He was fat. Let’s suppose that π 3 attaches to π 1 with E-Elab. Intuitively, we would like to leave open both for the possibility of continuing the elaboration or description of the man or by talking about something that is linked to the first constituent. (15) A man walked in. He sported a hat. (16) A man walked in. He sported a hat. He wanted to buy a new suit. (17) A man walked in. He sported a hat. Then a woman walked in. She was wearing a fur coat. These possibilities indicate that we will want to use a di ff erent discourse rule for the second case. Roughly, we want two continuations that 10
we can use—one for a high attachment and one for a low attachment: (18) λ i 1 o 1 io π 1 : ∃ x ( Mx ∧ Wxo 1 ( i 1 + x ) ∧ π 2 : ∃ vHv ∧ S v sel( i + x + v ) ∧ E-Elab( π 1 , π 2 ) ∧ o ( i + x + v )) 11
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