## Diamond implicature

If I say "", you would often take me to have asserted both "" and "". A quick internet search didn't come up with any useful literature on this, so I'd be grateful for pointers.

Examples:

• You may have beer or wine.
[Implies: you may have beer and you may have wine.]
• For all I know, she might be in the pub or in the library.
[Implies: for all I know, she might be in the pub, and for all I know, she might be in the library.]
• It is physically possible that a dropped cup goes sideways or even upwards.
[Implies: it is physically possible that a dropped cup goes sideways and that it goes upwards.]
• If I were to drop this cup, it might go sideways or even upwards.
[Implies: if I were to drop the cup, it might go sideways, and if I were to drop the cup, it might go upwards.]

The same happens with existential quantification instead of disjunction:

• You may have one of the drinks on the top shelf.
• She might be in one of the pubs in Civic.

Arguably, it also occurs when the antecedent in some other way leaves relevant details open:

• You're allowed to install software on your office computer.
[Implies: you're allowed to install any (reasonable, legal) software on the computer, not just, say, Microsoft Word.]
• She might be married and have children.
[Implies that there are several people p and numbers n and functions f from n to sexes such that I cannot rule out that she married p and has n children with sex f(1)..f(n).]

I'm currently interested in this phenomenon because I've been thinking about counterfactuals a little, and it seems that here it shows up not only in the consequent of might-counterfactuals, but also in the antecedent of counterfactuals quite generally:

• If I had arrived 5 or 10 minutes later, I would still have caught the train.
[Implies: if I had arrived 5 minutes later, I would still have caught the train, and if I had arrived 10 minutes later, I would still have caught the train.]
• If she had drunk from one of the bottles on the top shelf, she would now be dead.
[Implies: for any bottle on the top shelf, if she had drunk from it, she would be dead.]
• If the dart had landed on a red field or anywhere on the left-hand side of the board, I would have won.
[Implies: I would have one in any of these cases.]
• If I had been run over by a car yesterday, I would be either dead now or severely injured.
[Implies: on any (reasonable) way I could have been run over by a car, I would now be dead or severely injured. To see the implication, drop one of the disjuncts in the consequent and notice how it sounds false.]

(Why does the antecedent behave like a diamond context in this respect? Maybe because it is a negated compound in a box context?)

I've found an old article by Barry Loewer, "Counterfactuals with disjunctive antecedent" (JoP 1976, pp.531-537), where he attempts a Gricean explanation of cases with disjunctive antecedents. It goes as follows: suppose I assert a counterfactual "if A or B, then C". Case 1: It isn't known which of A and B is closer to actuality. Then I normally shouldn't have made the assertion unless I believed it was true in either case. So you take me to believe both. Case 2: it is known which of A and B is closer, say A. Then saying "if A then C" would have been saying something shorter and stronger, so I should rather have said that; as I didn't, you re-interpret what I said to be the conjunction of the two counterfactuals. But this doesn't sound very convincing. Why would you re-interpret me in exactly this way? And actually, strengthening the antecedent doesn't strengthen a conditional; "if A then C" is usually not stronger than "if A or B then C". So at least the most straight-forward Gricean explanation doesn't seem to work here.

# on 09 January 2007, 12:03

This is actually a topic that has received quite a bit of attention recently in formal semantics. Check out for example Luis Alonso-Ovalle's dissertation (at http://alonso-ovalle.net/index.php?page_id=28), Bart Geurts' JoS paper (at http://www.ru.nl/ncs/bart/papers/disjunction.pdf), and Rob van Rooij's JoS paper (http://dx.doi.org/10.1093/jos/ffl004).

# on 10 January 2007, 12:42

Thanks Kai, that seems to be the kind of thing I've been looking for!

# on 13 January 2007, 19:00

Hi Wo,

It was a long time ago but I think the idea was that supposing Lewis' account then we can see for Gricean like reasons that if someone asserts (Disj) (AvB)>C then it is reasonable to infer that she believes A>C and B>C even though Disj doesn't entail either of these. The reason is that (generally) if she doesn't know which of A worlds, B worlds are more similar to the actual world then she will know (Disj) only if she knows the conjunction (the conjunction does imply Disj). If she does know say that A worlds are more similar to the actual world than any B world then she ought to say A>C. It is shorter and typically more relevant...you are right that A>C is not stronger than (AvB)>C ... (it is not weaker either...these are Lewis counterfactuals not material conditionals) .. the discussion is against the background that if (Disj) does logically imply (as opposed to conversationally imply) each of the disjuncts then > is not variably strict...Lewis' account would collapse... it has been argued by some that > isnt variably strict but i dont find these aruments convincing.

Barry

# on 15 January 2007, 09:00

Thanks Barry,

I agree that this kind of Gricean explanation is worth pursuing. My worry about the second case (where it is known that A-worlds are closer than B worlds) is mainly that I don't see by what reasoning we should re-interpret the speaker to assert the conjunction of conditionals merely because what she literally said could have been expressed a little shorter.

I also find it quite hard to cancel the implicature in many cases, e.g. in the dart case above. And I would like to have an explanation that works for the other diamond contexts, especially for "it is physically possible that a dropped cup goes sideways or even upwards". (The epistemic and deontic cases are easier.) And I would like to have an explanation why the phenomenon never occurs with box contexts, as far as I can see (including might-counterfactuals on the non-expistemic reading):

"You must drink beer or wine" doesn't entail you must drink wine.

"I'm sure she has left a message or called him" doesn't entail I'm sure she has left a message.

"If I had arrived 5 or 10 minutes later, I might have met Fred" doesn't entail that I might have met him if I had arrived 5 minutes later.

# trackback from on 27 January 2007, 11:01

I've thought a little more about this thing I called 'diamond implicature', and I've come up with the following explanation. I don't know if it's original, and unfortunately, ...