Proceedings of the 13 ESSLLI Student Session - Multiple Choices ...

Though every element of an information state is a salient possibility (except the empty
set, which is present to simplify the definition of an answer to a question), the sets in an
information state do not exhaust its salient possibilities. Rather, the salient possibilities
in an information state are generated by closing it under union and intersection. Salient
possibilities are defined this way because, intuitively, if P1 and P2 are salient possibilities
in a context, then if they are not mutually exclusive it is also possible that they both obtain.
Thus, their intersection should count as a salient possibility as well. Similar reasoning
supports considering the union of salient possibilities to be a salient possibility.
DEFINITION 3. Let Γ be an information state. Then 〈Γ〉, the set of salient possibilities in
Γ, is defined as the the smallest set such that:
(i) If P ∈ Γ, then P ∈ 〈Γ〉 (ii) If P1, P2 ∈ 〈Γ〉, then P1 ∪ P2 ∈ 〈Γ〉
(iii) If P1, P2 ∈ 〈Γ〉, then P1 ∩ P2 ∈ 〈Γ〉.
We need one more concept in order to define the semantics of wh-questions. On our
analysis, wh-questions introduce salient possibilities corresponding to each of their possible
answers into an information state. To represent the possible answers to a wh-question,
we use the relations defined in definition 5 (Definition 4 is a standard account of satisfaction,
which is necessary for articulating definition 5):
DEFINITION 4. Let φ ψ ∈ L1, let i be an index, and let g be a variable assignment function.
(i) If φ = Qt1...tn, then i |= φ [g] iff 〈[t1] i,g ,...,[tn] i,g 〉 ∈ i(Q)
(ii) i |= φ ∧ ψ [g] iff i |= φ [g] and i |= ψ [g] (iii) i |= ¬φ [g] iff i �|= φ [g]
DEFINITION 5. Let φ ∈ L1, and let i and j be indices. We say that i ≡ j (mod φ) if for all
assignments g, i |= φ [g] iff i |= ψ [g]
Given a formula φ, definition 6 defines the conditions under which two indices give
the same answer to the question ?φ. For a sentence φ of L1, i ≡ j (mod φ) will hold as
long as i and j assign φ the same truth value. But for a formula of L1 with free variables,
congruence modulo φ requires that the indices assign the same denotations (or just similar
denotations if the formula contains both free variables and constants) to predicates that
occur in φ. The following examples illustrates how this definition works.
EXAMPLE 1.
Proceedings of the 13 th ESSLLI Student Session
(i) Let ?φ = ?Px (Who came to the party?) i ≡ j (mod φ) if i(P) = j(P), or informally, if the
same people came to the party according to indices i and j.
(ii) Let ?φ = ?Ibx (Who did Bill invite to the party?) i ≡ j (mod φ) if
{d ∈ D| 〈d, i(b)〉 ∈ i(P)} = {d ∈ D| 〈d, j(b)〉 ∈ j(P)}.
(iii) Let ?φ = ?p (Did Alice help Bill?) i ≡ j (mod φ) if i |= p iff j |= p.
In our update semantics, the effect of a formula on an information state will be defined
in terms of the effects it has on certain elements of the information state. Thus, to state
our update semantics for information states we require an update semantics for sets of
indices as well. The update semantics for sets indices is fairly simple, and is roughly the
same as that given in Veltman (1996).
DEFINITION 6. Let φ ∈ L1 ∪ L2 be a sentence, and let P be a set of indices. We define the
update of P with φ, written P[φ], as follows:
(i) P[p] = {i ∈ P | i |= p} (ii) P[φ ∧ ψ] = P[φ][ψ]
(iii) P[¬φ] = {i ∈ P | i �∈ P[φ]} (iv) P[⋄φ] = P if P[φ] �= ∅
(v) P[⋄φ] = ∅ if P[φ] = ∅
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