of Explanation
Since initial contexts appear in the induction as a basis case, the relevant definition from (48) is repeated here.
(71) Initial contextsA minimal context is K =hW2,∅,∅, T,{Pi, Pj}i,
an initial context of ignorance and indifference. And now the induction.
92The definition is only partially successful in this regard, since one way forKto license
φ! is ifK |=φ! orK |=¬φ!, i.e., ifφ! is not informative inK (including the case where
φ! is a tautology ofP L) or ifφ! is not consistent withK (including the case whereφ! is a contradiction ofP L). This complicates the characterization of answerhood in terms of licensing, given in§4.6.
(72) a. All initial contexts are admissible.
b. IfK =hC, τ, κ, T, Pi is an admissible context, (i) hC0, τ;φ?, κ, T, Piwhere
•φ? is pertinent and
•C0 ={hw, vi ∈C:Jφ?Kw,K =Jφ?Kv,K} is an admissible context; and
(ii) hC0, τ;φ!, κ;λ, T, Pi where
•φ! is pertinent,
•C0 ={hw, vi ∈C:w, v∈Jφ!KK}, and
•λ= ψ1?;. . . is the largest sequence of interrogatives
disjoint with κsuch that – λ≺τ and
–for eachψi?∈λ, there is someχ! such thatJχ!KK ∈ Jψi?KK and hw, wi ∈C
0 ⇒w∈
Jχ!KK. is an admissible context
The significance of this long-awaited definition is that an interrogative φ? has identical effects onK regardless of whether or not it is explanatory, and likewise forφ!. Hopefully it is obvious that (72-b) gives the effect of updating K with φ? in (72-b-i) and of updating K with φ! in (72-b-ii). Henceforth the effect of updatingK with a formula φis writtenK[φ]. Significantly, all updates retain the so-called classical update property.
(73) For K = hC, τ, κ, T, Pi, K[φ] = hC0, τ;φ, κ;λ, T, Pi, for possibly emptyλ, withC0 ⊆C.93
Moreover, since Ψ = {ψ1, . . . : RhJψαKK,hJφKK,Jχ?KKii} is assumed to be of some finite sizen, there is guaranteed to be a sentence ofEP Lthat gives the complete explanation ofφrelative to χ? inK.
(74) Fact: For any admissible context K and explanatory interrog- ative of the form ?χ?φ such that K[?χ?φ] is an admissible con-
text, there is some ξ! of the form
m−1 V i=1 (φ χ?ψi) ∧ n V i=m ¬(φ χ?ψi)
such that K[?χ?φ][ξ!] = hC, τ, κ, T, Pi with ?χ?φ ∈ κ. In fact,
Jξ!KK ∈J?χ?φKK.
Such a conjunction of explanatory indicatives and their negations is a com- plete, exhaustive explanation ofφrelative toχ? inK; thus it is a complete, exhaustive answer to ?χ?φ. (Hence, it is essential to the possibility of stating
93
an exhaustive answer to a why interrogative inEQL that Ψ be finite.) Updates with non-explanatory formulae affectC as in Groenendijk (1999). Figure 2 illustrates how updating a context K with explanatory formu- lae affects C. (Note that the structured common ground for the context K[φ1]. . .[φn] is written C[φ1]. . .[φn]; thatψ1, ψ2, and ψ1∧ψ2 express the
only propositionsK-relevant to ?χ?φ; and thatξ:= (φ
χ?ψ1)∧ ¬(φχ?ψ2).) C P a P b C[?χ?φ] C0 P a φψ1 P a φψ2 P a φ(ψ1∧ψ2) P a ¬(ψ1∨ψ2) P b φψ1 P b φψ2 P b φ(ψ1∧ψ2) P b ¬(ψ1∨ψ2) C0[ξ] C00 P a φψ1 P b φψ1 C[?χ?φ][ξ]
Figure 2: UpdatingK with explanatory formulae
In Figure 2, the initial K = hC, τ, κ, T, Pi is such that ?χ?φ is pertinent.
This reveals many facts aboutK:
(75) a. By the definition of pertinence, ?χ?φis licensed inK, so
(i) K |=φand
representative of each proposition relevant to φ relative toχ?, K 2¬( n W i=1 ψi).
b. Also by definition of pertinence, ?χ?φis not entailed by K, i.e.,
K 2?χ?φ. That is, there is some hw, vi ∈C withJ?χ?φKw,K 6= J?χ?φKv,K.
(i) By definition of J?χ?φKK, then, there is no θ! such that Jθ!KK ∈J?χ?φKK and hw, wi ∈C⇒w∈Jθ!KK. So ?χ?φis not inκ.
(ii) Neither, by (72), is ?χ?φinτ.
c. Again by definition of pertinence, ?χ?φ is consistent with K,
hence countenanced byT. This means that all non-logical sym- bols inχ? andφ appear inT.
(75) lists the conclusions that can be drawn about K from the fact that ?χ?φis pertinent inK: the pertinence inK of a why interrogative not only
requires that it arises in K in van Fraassen (1980)’s sense, but also reveals something about τ, κ, and T. (P, for its part, contributes the worldviews of the players, which settles theK-intensions of explanatory formulae.) Of course, Figure 2 also depicts a situation whereξ= (φ
χ?ψ1)
∧ ¬(φ
χ?ψ2) is
pertinent inK0:=K[?χ?φ]. The fact thatξis licensed inK0actually follows
from ξ being consistent with K0, ξ not being entailed by K0, and the fact that ?χ?φwas not entailed by K, given theK-intensions ofξ and ?χ?φ. The
relationship between licensing and the other facts just mentioned underlies the definition of answerhood in terms of licensing, finally to be made explicit in§4.6.
But first, the logical notions of §4.4 will be further illuminated.
(76) Fact: If φ! is countenanced by T but not consistent with K =
hC, τ, κ, T, Pi, then {hw, vi ∈C :w, v∈Jφ!KK}=∅.
Attempting to update a context with an inconsistent (though countenanced by T) φ! would eliminate all worlds, because it would be inconsistent with the information already provided by previous utterances inτ. Attempting to update a context with a formula entailed by it, on the other hand, would be redundant or superfluous with respect toC.
(77) a. Fact: If K = hC, τ, κ, T, Pi |= φ!, then {hw, vi ∈ C : w, v ∈
Jφ!KK}=C.
Jφ?Kv,K}=C.
While the (77-b) just states one direction of the definition of entailment for interrogatives, (77-a) shows that effectively the same thing holds for entailed indicatives. Importantly, (76) and (77) amount to the fact that, if the per- tinence requirements were removed from (72), the game could go wrong by eliminating all worlds from the structured common ground or by allowing moves that have no effect on the structured common ground. Thus for a context to be accessible according to (72), it must be played according to the rules of the game. In Groenendijk (1999), by way of contrast, contexts are defined independently of the logical notions that govern the game, and players obey the rules of the game only when all of their moves are perti- nent.94
It is of note that, from (72), the desired facts about informativeness and inquisitiveness hold.
(78) Fact: Only indicatives can be informative, and only interrogatives can be inquisitive.
Before discussing licensing in §4.6, note that the appearance of τ and κ within K allows for the following fact about indifferent contexts:
(79) Fact: A contextK =hC, τ, κ, T, Piis indifferent if all interrogatives φ?∈τ are also∈κ.