## Diamond Implicature II

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, I don't see how exactly it applies to the antecedent of counterfactuals, which is what I am most interested in.

The explanandum is that in many contexts, appears to imply . For example,

1) you may have beer or wine

implies

2) you may have beer and you may have wine.

As I mentioned in the last post, I don't think this has anything in particular to do with "or", for the same phenomenon occurs whenever the embedded sentence explicitly characterizes a range of possibilities, like "you have beer or wine" and "you have a drink from one of the bottles on the top shelf".

What lies behind these cases is, I believe, that when we predicate a 'range predicate' of several things, we usually implicate that the range is distributed over those things. This is best understood by examples.

3) The rooms are between 12.7 and 26.5 square meters.

In a suitable context, this implies that some room has 12.7 square meters and some other 26.5. Likewise,

4) The best things in life are either illegal, immoral or fattening

implies that some of the best things in life are illegal, some are immoral, and some are fattening. "being between 12.7 and 16.5 square meters" and "being either illegal, immoral or fattening" are range predicates in the sense that they present a range of alternative ways things might be.

The implication is strongest if the utterer knows how the relevant things distribute over the range. If the utterer of (3) doesn't know how big the individual rooms are, but knows that 12.7 is the legal minimum and 26.5 the legal maximum for room sizes, then one can hear (3) as saying just that all the rooms fall somewhere within this range, which is true even if all of them have exactly 18 square meters. By contrast, someone who knows that the rooms are all 18 square meters normally wouldn't say the weaker and more roundabout (3).

Not surprisingly, the implicature is still present if the relevant things are introduced by quantification:

5) I looked at some rooms between 12.7 and 26.5 square meters.
6) Some of the guests gave only 1 or 2 dollars.
7) Sometimes she brought mushrooms or flowers.

These are diamond contexts. However, what's important is actually not the diamond, but the plural. Replacing "sometimes" with "at a few times", "at many times", "often", "usually" or "always" doesn't destroy the implicature; replacing it with "on at least one occasion" does. If there's only one thing, it can hardly cover the whole range of incompatible alternatives.

The diamond cases ("sometimes") are more striking than others (like "usually") because they license or-to-and inferences, from (7) to

8) sometimes she brought mushrooms and sometimes she brought flowers.

By contrast,

9) she usually brought mushrooms or flowers

doesn't imply

10) she usually brought mushrooms and she usually brought flowers.

But in fact, the implicature is still there: Like (7), (9) implies that the range is distributed over the cases (times) in question. This isn't captured by (10); it is captured by (8), and in a suitable context, (9) does indeed imply (8).

The "always" case is slightly different. First, the implicature is weaker here: "it always either rains or doesn't rain" sounds more like a tautology than "it sometimes either rains or doesn't rain". More importantly, unlike "always", "sometimes" (and "at many times" and "often") introduces a plurality of times and makes it available for backreference and operations in the matrix sentence:

11) sometimes, the sun was shining and then (at those times) she always brought mushrooms or flowers.

This doesn't work with "always".

The possibility of backreference also shows that (7) cannot be analysed as (8). For there is no plausible analysis along these lines for (11). (11) doesn't mean the same as

12) sometimes, the sun was shining and then she always brought mushrooms, and sometimes, the sun was shining and then she always brought flowers,

nor does it mean the same as

13) sometimes, the sun was shining and she brought mushrooms, and sometimes, the sun was shining and she brought flowers.

Instead of analysing (7) as (8), it is better to leave (7) as it is and explain the inference to (8) as the 'range' implicature illustrated by (3) -- (6).

The application to "possibly", "may", "might" etc. should be obvious: like "sometimes", these operators introduce a collection of possibilities that an embedded range proposition is taken to distribute over.

As before, the implication is strongest if we assume that the speaker knows for each of the introduced possibilities where it lies in the range. And as before, there are examples where we backrefer or operate on the selected collection which a conjunctive analysis can't handle:

14) she might have gone to the forest, and then she will certainly return with mushrooms or flowers.

And again, the implicature isn't limited to diamond contexts. But in other contexts, it doesn't surface as or-to-and inferences. Rather, it shows up as the implicature from

15) the PhD thesis must be published within 2 years after the final exam

to the negation of

16) the PhD thesis must be published within 1 year after the final exam.

(The PhD regulations of the University of Bielefeld contain both (15) and (16), and it is of some importance to me that this is naturally read as a contradiction.)

So far, so good. As I said, I don't see clearly why the implicature occurs in the antecedent of counterfactuals, and, more generally, in clauses restricting universal quantifiers:

17) All rooms between 12 and 16 square meters are available for rent.

This implies that some room with 12 square meters is available for rent. Perhaps these are really different cases that need a different explanation?

# on 03 February 2007, 09:24

I'm not sure if this is relevant, but you might consider examples like the following:

You may have beer or wine, but I don't know which.

(This kind of example is due, I think, to Hans Kamp.) If the logical form of this sentence is:

[(You may have beer) and (you may have wine)] and [(I don't know that you may have beer) and (I don't know that you may have wine)]

we have a Moorean paradox (a variation highlighted in recent work by Seth Yalcin) because this is equivalent to:

[(You may have beer) and (I don't know that you may have beer)] and [(you may have wine) and (I don't know that you may have wine)].

Unfortunately, things aren't much better if the logical form of the sentence is:

[(You may have beer) or (you may have wine)] and [(I don't know that you may have beer) and (I don't know that you may have wine)]

since we can get Moorean paradoxes out of this as well. So, it seems to me that either the sentence in question is meaningless or this is a case where we have an irreducibly disjunctive permission, i.e. because of the epistemic modality in the second part of the sentence, the disjunctive permission in the first part of the sentence is not equivalent either to a conjunction of permissions or a disjunction of permissions.