GAME SHOW: There are 100 boxes and five of them contain money. The rest are empty. The show’s host knows which boxes contain money. Contestants are given one chance to choose a box. If a contestant chooses a box with money, they win the money. The host provides hints with the common understanding that she will not reveal all the information she has.
Since the host is not expected to share all relevant information, the Maxim of Quantity is deactivated. But, if the host utters (1), there is an exhaustification implicature that there is not money in both boxes:
(1) There is money in box 20 or box 25.
I will reply to Fox (2014). Contra Fox, I deny that the pragmatic approach must always appeal to the Maxim of Quantity in order to generate scalar implicatures. First, I will show that Grice’s Cooperative Principle is active in GAME SHOW.
The Cooperative Principle demands that you “[m]ake your conversational contribution such as is required…by the accepted purpose or direction of the talk exchange in which you are engaged” (Grice, 1989, p.26, my emphasis). In GAME SHOW, the accepted purpose of the conversation is to advance the game show fairly. But then the deactivation of the Maxim of Quantity does not entail the deactivation of the Cooperative Principle. Sometimes, sharing all relevant information would undermine the accepted purpose of a conversation!
Griceans can explain why (1) has an exhaustification implicature. They need only accept the Maxim of Secrecy:
Maxim of Secrecy: Do not reveal information such that were you to share it, the goal of the conversation would be undermined.
Since the host should not reveal information that would indicate a winning strategy, the Maxim of Secrecy is active in GAME SHOW. If both box 20 and box 25 have money in them, then a winning strategy for the contestants would be to choose box 20 or box 25. One way to indicate this winning strategy (i.e. make it salient) is for the host to utter (1). For by uttering p ∨ q, one makes both p and q salient. By uttering (1), the host would cause the contestants to think that they should choose either box 20 or 25. But making a winning strategy salient to contestants would undermine the goal of the conversation. This is evidenced by the infelicitousness of (2) when uttered by the host:
(2)#There is money in box 20.
So, if there is money in both box 20 and box 25, the host should not utter (1). Therefore, Griceans can explain why (1) has an exhaustification implicature.
To further defend the pragmatic approach, I will derive the “possibility” implicatures associated with (1):
(3) There might be money in box 20.
(4) There might be money in box 25.
Let □ regiment ‘In all worlds in which all beliefs are true’ and ■ regiment ‘In all worlds in which all commitments are true’. If the host utters (1), we can derive the implicatures (3) and (4):
1. ■(p ∨ q) Host’s utterance, Maxim of Quality Goal: ■◊p∧■◊q
2. ¬■p 1, Maxim of Secrecy Axiom 5*: ♦φ→■◊φ
3. ¬■q 1, Maxim of Secrecy Assumption: All instances of
4. ♦¬p 2, Modal Logic Axiom 5* are true.
5. ♦¬q 3, Modal Logic
6. ♦p 1, 5, Modal Logic
7. ♦q 1, 4, Modal Logic
8. ■◊p 6, Axiom 5*, Propositional Logic
9. ■◊q 7, Axiom 5*, Propositional Logic
10. ■◊p∧■◊q 8, 9, Propositional Logic