2 # Doing things with monads
4 ## Extended application: Groenendijk, Stokhof and Veltman's *Coreference and Modality*
8 GS = Dynamic Predicate Logic L&P 1991: dynamic binding, donkey anaphora
9 Dynamic Montague Grammar 1990: generalized quantifiers, discourse referents
11 V = epistemic modality
13 What are pegs? A paper by Landman called `Pegs and Alecs'.
15 It might be raining. It's not raining.
16 #It's not raining. It might be raining.
18 ## Two-part assignment functions
22 type refsys = var -> peg
23 type ent = Alice | Bob | Carl
24 type assignment = peg -> ent
27 type world = pred -> ent -> bool
29 type poss = world * pegcount * refsys * assignment
30 type infostate = [poss]
32 So information states track both facts about the world (e.g., which
33 objects count as a man), and facts about the discourse (e.g., how many
36 ref(i, t) = t if t is an individual, and
37 g(r(t)) if t is a variable, where i = (w,n,r,g)
39 s[P(t)] = {i in s | w(P)(ref(i,t))}
41 s[t1 = t2] = {i in s | ref(i,t1) = ref(i,t2)}
45 s[∃xφ] = Union {{(w, n+1, r[x->n], g[n->a]) | (w,n,r,g) in s}[φ] | a in ent}
47 s[neg φ] = {i | {i}[φ] = {}}
49 1. {(w,n,r,g)}[∃x.person(x)]
50 2. {(w,n,r,g)}[∃x.man(x)]
51 3. {(w,n,r,g)}[∃x∃y.person(x) and person(y)]
52 4. {(w,n,r,g)}[∃x∃y.x=y]
56 s[◊φ] = {i in s | s[φ] ≠ {}}
58 1. Alice isn't hungry. #Alice might be hungry.
60 {hungry, full}[Alice isn't hungry][Alice might be hungry]
61 = {full}[Alice might be hungry]
64 2. Alice might be hungry. Alice isn't hungry.
66 = {hungry, full}[Alice might be hungry][Alice isn't hungry]
67 = {hungry, full}[Alice isn't hungry]
70 GSV: a single speaker couldn't possibly be in a position to utter (2).
72 3. Based on public evidence, Alice might be hungry. But in fact she's not hungry.
74 4. Alice might be hungry. Alice *is* hungry.
75 5. Alice is hungry. (So of course) Alice might be hungry.
77 consider: update with the prejacent and its negation must both be non-empty.
81 6. A man^x entered. He_x sat.
82 7. He_x sat. A man^x entered.
84 8. {(w,1,r[x->0],g[0->b])}
86 This infostate contains a refsys and an assignment that maps the
87 variable x to Bob. Here are the facts in world w:
97 9. Someone^x entered. He_x sat.
99 {(w,1,r[x->0],g[0->b])}[∃x.enter(x)][sit(x)]
101 -- the existential adds a new peg and assigns it to each
104 = ( {(w,2,r[x->0][x->1],g[0->b][1->a])}[enter(x)]
105 ++ {(w,2,r[x->0][x->1],g[0->b][1->b])}[enter(x)]
106 ++ {(w,2,r[x->0][x->1],g[0->b][1->c])}[enter(x)])[sit(x)]
108 -- "enter(x)" filters out the possibility in which x refers
109 -- to Alice, since Alice didn't enter
112 ++ {(w,2,r[x->0][x->1],g[0->b][1->b])}
113 ++ {(w,2,r[x->0][x->1],g[0->b][1->c])})[sit(x)]
115 -- "sit(x)" filters out the possibility in which x refers
116 -- to Carl, since Carl didn't sit
118 = {(w,2,r[x->0][x->1],g[0->b][1->b])}
120 existential in effect binds the pronoun in the following clause
122 10. He_x sat. Someone^x entered.
124 {(w,1,r[x->0],g[0->b])}[sit(x)][∃x.enter(x)]
126 -- evaluating `sit(x)` rules out nothing, since (coincidentally)
127 -- x refers to Bob, and Bob is a sitter
129 = {(w,1,r[x->0],g[0->b])}[∃x.enter(x)]
131 -- Just as before, the existential adds a new peg and assigns
134 = {(w,2,r[x->0][x->1],g[0->b][1->a])}[enter(x)]
135 ++ {(w,2,r[x->0][x->1],g[0->b][1->b])}[enter(x)]
136 ++ {(w,2,r[x->0][x->1],g[0->b][1->c])}[enter(x)]
138 -- enter(x) eliminates all those possibilities in which x did
141 = {} ++ {(w,2,r[x->0][x->1],g[0->b][1->b])}
142 ++ {(w,2,r[x->0][x->1],g[0->b][1->c])}
144 = {(w,2,r[x->0][x->1],g[0->b][1->b]),
145 (w,2,r[x->0][x->1],g[0->b][1->c])}
147 11. A man^x entered. He_x sat. He_x spoke.
148 12. He_x sat. A man^x entered. He_x spoke.
150 consider: new peg requires object not in the domain of the assignment fn
152 13. If a woman entered, she sat.
154 ## Interactions of binding with modality
156 (∃x.enter(x)) and (sit(x)) ≡ ∃x (enter(x) and sit(x))
158 but (∃x.closet(x)) and (◊guilty(x)) ≡/≡ ∃x (closet(x) and ◊guilty(x))
161 There are three sons, Bob, Carl, and Dave.
162 One of them broke a vase.
163 Bob is known to be innocent.
164 Someone is hiding in the closet.
168 --------------- ---------------
177 GSV observe that (∃x.closet(x)) and (◊guilty(x)) is true if there is
178 at least one possibility in which a person in the closet is guilty.
179 In this scenario, world w is the verifying world. It remains possible
180 that there are closet hiders who are not guilty in any world. Carl
181 fits this bill: he's in the closet in world w', but he is not guilty
184 14. Someone^x is in the closet. He_x might be guilty.
186 {(w,0,r,g), (w',0,r,g}[∃x.closet(x)][◊guilty(x)]
188 -- existential introduces new peg
190 = ( {(w,1,r[x->0],g[0->b])}[closet(x)]
191 ++ {(w,1,r[x->0],g[0->c])}[closet(x)]
192 ++ {(w,1,r[x->0],g[0->d])}[closet(x)]
193 ++ {(w',1,r[x->0],g[0->b])}[closet(x)]
194 ++ {(w',1,r[x->0],g[0->c])}[closet(x)]
195 ++ {(w',1,r[x->0],g[0->d])}[closet(x)])[◊guilty(x)]
197 -- only possibilities in which x is in the closet survive
199 = {(w,1,r[x->0],g[0->d]),
200 (w',1,r[x->0],g[0->c])}[◊guilty(x)]
202 -- Is there any possibility in which x is guilty?
203 -- yes: for x = Dave, in world w Dave broke the vase
205 = {(w,1,r[x->0],g[0->d]),
206 (w',1,r[x->0],g[0->c])}
208 Now we consider the second half:
210 14. Someone^x is in the closet who_x might be guilty.
212 {(w,0,r,g), (w',0,r,g)}[∃x(closet(x) & ◊guilty(x))]
214 -- existential introduces new peg
216 = {(w,1,r[x->0],g[0->b])}[closet(x)][◊guilty(x)]
217 ++ {(w,1,r[x->0],g[0->c])}[closet(x)][◊guilty(x)]
218 ++ {(w,1,r[x->0],g[0->d])}[closet(x)][◊guilty(x)]
219 ++ {(w',1,r[x->0],g[0->b])}[closet(x)][◊guilty(x)]
220 ++ {(w',1,r[x->0],g[0->c])}[closet(x)][◊guilty(x)]
221 ++ {(w',1,r[x->0],g[0->d])}[closet(x)][◊guilty(x)]
223 -- filter out possibilities in which x is not in the closet
224 -- and filter out possibilities in which x is not guilty
225 -- the only person who was guilty in the closet was Dave in
228 = {(w,1,r[x->0],g[0->d])}
230 The result is different, and more informative.
232 ## Binding, modality, and identity
234 The fragment correctly predicts the following contrast:
236 15. Someone^x entered. He_x might be Bob. He_x might not be Bob.
237 (∃x.enter(x)) & ◊x=b & ◊not(x=b)
238 -- This discourse requires a possibility in which Bob entered
239 -- and another possibility in which someone who is not Bob entered
241 16. Someone^x entered who might be Bob and who might not be Bob.
242 ∃x (enter(x) & ◊x=b & ◊not(x=b))
243 -- This is a contradition: there is no single person who might be Bob
244 -- and who simultaneously might be someone else
248 Let α be a term which differs from x. Then α is an identifier if the
249 following formula is supported by every information state:
253 The idea is that α is an identifier just in case there is only one
254 object that it can refer to. Here is what GSV say:
256 A term is an identifier per se if no mattter what the information
257 state is, it cannot fail to decie what the denotation of the term is.
259 ## Why have a two-part assignment function?
261 In the current system, variables are associated with values in two
264 Variables Pegs Entities
265 --------- r ---- g --------
270 Here, r is a refsys mapping variables to pegs, and g is an assignment function mapping pegs to entities