1a) What is Ross's puzzle? Exactly which modal operators does it arise for?
Ross's puzzle is the inference from
You ought to post the letter
to
You ought to post the letter or burn it.
What I am interested in is chiefly epistemic operators, so am abstracting away from some of the peculiarities of deontic operators. My example will be the inference from
Joe might be in the kitchen
to
Joe might be in the kitchen or the attic.
In particular, the latter seems to entail "Joe might be in the kitchen AND Joe might be in the attic." It is this felt entailment which we are trying to explain. Thus it arises for operators which are given truth-conditions in terms of existential quantification over some set of possible worlds. (I wish to leave to others the question of whether we really ought to analyze deontic "ought" and "may" this way.)
The question of the scope of the "or" relative to the modal is under debate here. One thing to note, though, is that the felt entailment does arise for both narrow and wide scope SS's: both
Joe might be in the kitchen or the the attic.
and
Joe might be in the kitchen or he might be in the attic.
...appear to generate the felt entailment.
2) What are the prospects for a simple Gricean, or pragmatic, explanation of the puzzle, according to which the inference is not strictly valid but nevertheless "reasonable in context"?
The prospects are not bad, but they appear to rely on one particular choice of LF. This is the wide-scope LF. We employ the Gricean-inspired principle that a disjunction is appropriate in context iff both disjuncts are epistemically possible for the speaker. We then extrapolate, in the epistemic case, to the possibility of both disjuncts with regard to truthmaking domain of quantification at the context. We will need some account of truth for unembedded epistemic modal clauses at a context for this, and we will need to employ some kind of S5 axiom.
In a nutshell, I think this often works. But it has two weaknesses. The first is that it relies on an undefended syntactic assumption (the wide-scoping of the "or".) The only defense offered for this is that it (pragmatically) validates the free choice inference...but a semantic account might be able to get the inference semantically when the scope is narrow. The second weakness is that it does not generalize well to the deontic cases. So it is both (i) not fully general and (ii) not even a solution to Ross's puzzle in its original form (since the modality involved there was deontic.) The kind of conditions on deontic ideality which would need to hold in order for the inference to be generated pragmatically do not seem reasonable to me: that is, it should not in general follow that when something is possibly permissible, it is permissible.
However, it is worth noting that a major point in favor of the pragmatic analysis is the apparent cancellability of the "free choice inference":
You may have coffee or tea--I don't remember which.
Joe might be in the kitchen or the attic--I'm not telling you which.
3a) What are the prospects for a semantic explanation of the puzzle, according to which the patterns are valid?
I think the prospects are pretty good. But I appear to be in the minority here. The basic idea of the semantic proposal I'm in favor of comes from Mandy Simons. It employs a Hamblin semantics on which an or-coordination denotes a set whose members are its disjuncts, i.e.
[[Larry, Moe or Curly]] = [[Larry or Moe or Curly]] = {Larry, Moe, Curly}
The semantic entry for "or" on such a theory would be:
[[or]] = \lambda x \subseteq D_{\tau} . \lambda y \subseteq D_{\tau} . x \union y
This is a "flexible type" entry: "or" can join nodes of any semantic type, outputting a node in the same semantic type. However, the truth-function requires that the inputs to the semantic entry be sets. The easiest way to accomplish this is to adopt a Hamblin type-shift, according to which nodes denote singletons of their (old) extensional denotations. So, for example, [[Joe]] = {Joe}. We might call our "old" extensions "atoms" or "ur-elements" of the new interpretation function. (Thus one way of looking at this semantics is as "mereological," with atoms, and "or" is the "fusion" operation. )
The question now arises: how does composition proceed? We should use
Hamblin Functional Application: Let \alpha be a branching node with \beta and \gamma its daughters. WOLOG, assume \beta \subseteq D_{\tau} and \gamma \subseteq D{\tau, \pi}. Then [[\alpha]] = {a: Eb \in [[\beta]], g \in [[\gamma]] s.t. a = g(b)}.
While this looks a bit complicated, it's actually quite easy to see how it relates to regular FA.
NB, though, that we will need Regular FA to account for the semantic operation engendered by the "or" itself. (This is by far the most elegant way to account for it, although it expands our repertoire of composition rules.) Otherwise we will get what looks like a type-mismatch when we try to compose the function denoted by "or" with the first of its disjuncts.
3b) What is the intuitive idea behind this semantics?
The intuitive idea behind the semantics, for Hamblin, was that a question denoted a set of possible answers. (For him, the semantic value of a question word like "who" is D_e.) This puts a partition on logical space. The result is that when a question is posed in a context, it partitions the common ground for the audience. The audience's job is then to reduce the common ground by cutting along one of the dotted lines.
For Simons, the thought is similar. When an "or" is asserted in context, its function is to "divide up" (put a partition or cover on) some set. This presumably marks a "dotted line" along which our investigation is to proceed, with the eventual goal of eliminating or choosing one of the disjuncts. However, she can then give a definition for modal operators according to which the modal operators can do something else with the disjuncts: for example, they can universally quantify over them. Here's her entry for the epistemic modal operator "might_e":
[[might]] = \lambda {p1..pn}, p1-pn \in D_{st} . ACC_e \subseteq Union(p1-pn) [Coverage] &
\forall pi, ACC_e \intersect pi \neq \emptyset. [Genuineness]
There are two separate issues here--which, I think, is important for the next step I want to make. FIRST, we want to keep the disjuncts of an or-coordination "separate." By some means, then, we shall give a semantics whereby [[p1 v p2]] \neq [[p]], where [[p]] is the possible-worlds-semantics union of [[p1]] and [[p2]]. (Intuitive examples: [[right hand or left hand]] \neq [[hand]]; [[platypus or echidna]] \neq [[monotreme]], [[nuclear war or nonnuclear war]] \neq [[war]].) There is a large and varied semantics literature reasons to and ways of doing this. [c.f. Zimmermann, Rooth, the association with focus literature, and the closure under entailment problems we looked at in seminar--these problems having to do with giving a semantics for "believes" statements etc.]
4) Can you compare this to other semantic solutions in the literature?
I don't know much about other semantic solutions in the literature (but I'll look at Luis Alonso-Ovalle.) Zimmermann conceives of his solution as a semantic ones, but I'd prefer to classify it as a pragmatic one, for a number of reasons. First, it is a wide-scoping view, and second, it is very far from general. So roughly it seems to have all the disadvantages of the wide-scope view while with none of the advantages (since there's no account of cancellability.) Frankly, Zimmermann's view is quite odd.
What I would like to do is try to sketch out an alternative semantic account, similar in spirit to Simons's, but one that takes up Partee and Rooth (1982)'s suggestion that the semantics of "or" should be assimilated to the semantics of indefinite noun phrases [the suggestion is 27 years old, but to my knowledge nobody has made good on it]. This would be a way of making good on a certain burden imposed on Simons by her account, which is to explain how unembedded disjunctions get interpreted. (To begin drawing the connection, note that (i) there is a deep logical connection between disjunction and existential quantification, and that (ii) indefinites give rise to free choice effects.)
Let me first sketch the problem for Simons, give her response, and then try to sketch the parallel with indefinite noun phrases.
The problem for Simons is that e.g. [[John sang or Jane danced]] = {John sang, Jane danced}. How to assign truth-conditions to this? She writes:
"Recall that the truth conditions for the modal/or sentences require the existence of a set which has two properties: it is related in a specified way to some other semantic object; and it is supercovered by the denotation of the embedded or coordination. Let's suppose that sentences containing or coordinations always have truth conditions of this form. We can achieve the intuitively correct results for [[John sang or Jane danced]] by [identifying the set to be supercovered with a factive common ground]. (18-19)"
What this means is that either (i) the common ground gets into the semantics, as in dynamic semantics; or (ii) the association of a disjunctive sentence with truth conditions occurs in the pragmatics, rather than the semantics. I'm not sure how best to gloss this, or whether this naturally suggestion should really be neutral between these two glosses.
Now I need to make the jump to indefinites. According to Heim, indefinites (just like disjuncts) have no quantificational force of their own. How, then, to account for scope?
