Logical Notions

As a last step before the admissible contexts can be characterized, this section treats the logical notions that govern the progression of the game, culminating in a logical notion of pertinence. Since the logical notions in- volve semantic as well as syntactic elements, and since the semantics of EQL is contextual, the logical notions are context-relative. This is a nec- essary departure from Groenendijk (1999), where the logical notions were defined relative to a sequence of QL-formulae τ. In fact, the departure is also necessary to take into account the effect of the background theory T, which appears in (66).

The first logical notion is consistency.

(66) a. φ! isconsistent withK =hC, τ, κ, T, Pi iff (i) ∃hw, wi ∈C s.t. w∈_Jφ!_KK, and

(ii) φ! is countenanced byT.

b. φ? isconsistentwithK=hC, τ, κ, T, Pi ifφ? is countenanced by T.

The idea behind (66), which is supposed to represent the Gricean maxim of quality, is that a move in the game should be credible. Therefore it must be both consistent with previous information provided and expressed in the language of the background theory. In the case of interrogatives, which are taken to provide no information, consistency reduces to being expressed in the language of the background theory. Forcing the game to be governed by this notion, however, leads to a limitation of the model. Where why- questions are motivated by surprise, like (34) and (35) in§3.3.2, the answers

are likely not to be consistent with their contexts. The implementation of a sufficiently sophisticated mechanism to deal with revision of the common ground would, therefore, go hand in hand with a revision of this notion of consistency.

The next logical notion is entailment.

(67) a. K =hC, τ, κ, T, Pi |=φ! iff,∀hw, wi ∈C,w∈_Jφ!_KK.

b. K =hC, τ, κ, T, Pi |=φ? iff{hw, vi ∈C:_Jφ?_Kw,K =Jφ?Kv,K}= C.

The idea behind (67), which is supposed to represent the Gricean maxim of quantity, is that moves in the game should be non-redundant and non- superfluous. The notion of entailment gives rise, as in Groenendijk (1999), to notions of informativeness and inquisitiveness.

(68) a. K =hC, τ, κ, T, Pi₂φ! iff∃hw, wi ∈C such thatw /∈_Jφ!_KK.

b. K = hC, τ, κ, T, Pi 2 φ? iff ∃hw, vi ∈ C such that Jφ?Kw,K 6= Jφ?Kv,K.

An indicative φ! such that K 2 φ! will have the effect on C of removing

hw, wi from C if w /∈ _Jφ!_KK. A formula φ is informative iff there is some

K such that φ has that effect on K; in such K, it is informative in K. An interrogative φ? such that K 2 φ? will have the effect on C of remov-

ing hw, vi from C if _Jφ?_Kw,K 6= Jφ?Kv,K. A formula φ is inquisitive iff there is someK such thatφhas that effect on K; in suchK, it is inquisi- tive in K. To verbify these adjectives, an informative (inK) φ!provides information (inK), and an inquisitive (inK)φ?raises an issue (inK). Next is licensing.

(69) a. K = hC, τ, κ, T, Pi licenses φ! iff, ∀hw, vi ∈ C, if w /∈ _Jφ!_KK,

thenv /∈_Jφ!KK.

b. K =hC, τ, κ, T, Pi licenses φ? iff (i) φ? is of the form ?~xψ or (ii) φ? is •of the form ?(φ χ?ψ) or ?χ?φ, •K |=φ, and •where Ψ is as in (65) above, K 2¬( n W i=1 ψi).

The idea behind (69), which is supposed to represent the Gricean maxim of relation, is that moves should be responsive to previous moves. For the witness, this means that his responses should address the interrogator’s questions.92 For the interrogator, it means his questions should be such as to actually arise in the context. For the interrogatives of QL, this is al- ways taken to be the case. But for explanatory interrogatives, van Fraassen (1980)’s account of the arising in a context of a why-question has been incor- porated. The requirement thatK|=φis van Fraassen’s first presupposition, viz., that the topic of a why-question be true. By our implementation of contrast-classes, satisfaction of the first presupposition automatically guar- antees satisfaction of the second, viz., that the topic’s contextual contrasts be false. The requirement that K 2 ¬(

n

W

i=1

ψi) is his third presupposition,

viz., that it not be known that a why-question has no contextually relevant answer.

These three notions are combined in the notion of pertinence. (70) φis pertinent inK iff

a. φis consistent with K (Quality), b. φis not entailed by K (Quantity), and c. φis licensed byK (Relation)

Pertinence appears in the inductive definition of admissible contexts, which can now be given.