# Speak with the vulgar.

Think with me.

## Questions

I’ve been thinking a bit about the semantics of questions. I know hardly any of the literature on this, but I’ve worked out a little view that seems to have some nice features. If you know more I’d be interested to hear what you think.

The semantic value (SV) of a question has two jobs to do. First, it should fit nicely into the rest of our semantics: it should help us get the right truth conditions for sentences with embedded questions. Second, it should fit nicely into the rest of our pragmatics: it should help us explain what a speaker does when she asks a question. Ideally both of these should require minimal revision to the rest of what we were doing in those projects.

As I see it, the standard account (the SV of a question is a set of propositions that partition logical space) does a mediocre job on both counts. You can get things to work, but the account doesn’t really make it easy, and you end up having to build a lot of new machinery in other places, like attitude verbs. I think I might be able to do better.

### Semantics

• Alice knows who won.

This evidently means something close to the following:

• For some person x, Alice knows that x won.

So here’s the first interesting observation: it looks like the embedded question introduces a quantifier, and it looks like the quantifier takes scope over the attitude verb.

Chris Barker and Chung-Chieh Shan give a nice way of doing this sort of thing compositionally, with a handy notation. Semantic values are represented by “towers”, where the higher levels of the tower take scope over the lower levels. It looks like this:

• $\left( \text{Alice knows []} \right) \left( \frac{\exists x . []}{x \text{ won}} \right)$

(The empty brackets are a convenient shorthand: $\cdots [] \cdots$ abbreviates $\lambda X . \cdots X \cdots$, where $X$ has the appropriate type.) We compose the towers of the same height by combining (by function application) the expressions at each level of the tower. There are also two type-shifters, “Lift”, which adds a $[]$ to the top of a tower, and “Lower” which applies an upper-story function to the argument beneath it. (There are syntactic constraints on when to do these things, but I’m leaving that out for simplicity. I’m also leaving out Barker and Shan’s slightly more complicated story about binding.)

Here’s how it works in the example: Lift the tower on the left to get

• $\left( \frac{[]}{\text{Alice knows []}}\right) \left( \frac{\exists x . []}{x \text{ won}}\right)$.

Then compose the two to yield $\frac{\exists x . []}{\text{Alice knows } x \text{ won}}$, and Lower this to get $\exists x . \text{Alice knows } x \text{ won}$. (The underlying machinery for this notation uses continuations. This can all be worked out in regular old typed lambda calculus.)

That’s almost the complete account. There’s one extra ingredient: the question clause “who won” also carries the presupposition that someone won.

So here’s the theory. Suppose a question clause has the underlying form $?x. \phi(x)$. Here $?x$ is a slightly idealized version of “what” (it doesn’t assume the answer is inanimate). Then $?x.\phi(x)$ has the SV $\frac{\exists x.[]}{\phi(x)}$, and $?x.\phi(x)$ carries the presupposition $\exists x . \phi(x)$. That’s the account.

We handle “Alice guessed who won” in the same way as “knows”: we get the presupposition $\exists x . x \text{ won}$, and the content $\exists x . \text{Alice guessed } x \text{ won}$. The nice thing to note is that the x of Alice’s guess need not be the same as the x who actually won. (I’m told Groenendijk and Stokhof have some trouble with this.)

Different question words can be understood in terms of the idealized “what”: “who” means “what person”, “when” means “at what time”, and “whether” means “what truth value”. The last case is worth fleshing out: the SV of “whether p” is $\frac{\exists v \in \{T, F\} . []}{p \text{ has truth-value } v}$. Effectively, “whether p” means “p or not p”—except that the disjunction can take scope over operators that govern the question. So consider the sentence “It’s unclear whether p”. The semantic value for “unclear” splits into a top-floor “un-” and a bottom-floor “clear” (this should be well-motivated for other reasons). The result:

• $\frac{\neg []}{\text{clear}[]} \frac{\exists v \in \{T, F\} . []}{p \text{ has truth-value } v} = \frac{\neg \exists v \in \{T, F\}. []}{\text{clear}(p \text{ has truth-value } v)}$. Lowering this gives $\neg \Big( \text{clear}(p \text{ is true}) \lor \text{clear}(p \text{ is false}) \Big)$, or equivalently, $\neg \text{clear}(p) \land \neg \text{clear}(\neg p)$.

I think this is a nice result, and it comes out very cleanly.

### Pragmatics

A reasonable pragmatics of questions falls out of this story, too.

Consider an unembedded question: “Who won?” Rendering this as $?x. x \text{ won}$, it gets the SV $\exists x. x \text{ won}$, and it carries the presupposition $\exists x. x \text{ won}$. So “who won?” turns out to be a peculiar kind of assertion, one that presupposes exactly what it asserts. The question is guaranteed not to add any new information to the common ground. What conversational purpose can it serve? Well, by the maxim of quantity, if I know Bill won, I shouldn’t just say “Someone won”—I should say “Bill won”. So if I say “Someone won”, this lets my interlocutors infer that I don’t know who won. So on my account, when I ask “Who won?”, I am adding no new information to the common ground—in that sense I’m not really asserting anything—but I’m still conveying the information that I don’t know who won. Finally, the question intonation has the effect of a hopeful pleading look: “Help me out here!” I am pointing out a lacuna in my state of information, and then I wait for someone to helpfully fill it.

The nice thing about this story is that it involves absolutely no new pragmatic machinery: it just applies regular stuff about presupposition, common ground, and implicature to the independently-motivated semantics.

### Subject matter

I only really dealt with one kind of context for embedded question: contexts like those under “know”, “guess”, and “clear” that also accept that-clauses. But question clauses also occur places where that-clauses can’t. I’m not sure what the best way of approaching this is.

One strategy is to try to uncover propositional attitudes in these cases. For instance, “Alice wonders who won” might turn out to mean something like “Alice wishes to know who won”. The SV of “wonders” would be something like $\frac{\lambda y.y \text{ wishes } []}{y \text{ knows } []}$, which yields the interpretation “Alice wishes that (presupposing someone won) some x is such that Alice knows that x won”. This seems ok, but it’s not clear how far we can push the strategy.

(Quine in “Quantifiers and Propositional Attitudes” pulls similar shenanigans to deal with sentences like “I want a sloop” and “Ctesias is hunting unicorns”: superficially the attitude verbs take NP-complements, but they turn out to disguise propositional attitudes. I remember being annoyed at this when I first read it, and I thought you should just let wanting take a quantifier as its argument. But there’s a certain admirable economy in Quine’s approach, constraining our psychological repertoire to just propositional attitudes. Similar considerations weigh against positing new sorts of attitudes that have a question-y object.)

The case of “wonder” looks like it might be assimilated with what I’ll call subject-matter contexts.

• The argument was about who won.
• They discussed who won.
• I have no opinion as to who won.

I think the standard account of questions was aimed primarily at these kinds of cases. Indeed, David Lewis initially presented partitions of logical space as a theory of subject matter, not a semantic account of questions. And I don’t really want to start from scratch on a new theory of subject matter. Rather, I ought to be able to retrieve subject matters from the SVs of question clauses.

And indeed I can. Consider this conversion function:

• $\frac{\lambda q. []}{q \text{ is the proposition that } []}$.

If we apply this to the SV of a question (and Lower), we get back the set of propositions that comprise the corresponding subject matter. For instance, applying it to the SV of $?x. x \text{ won}$, we get $\lambda p. \exists x . p \text{ is the proposition that } x \text{ won}$.

Accordingly, we can give an account of “about” that applies to my SVs for question clauses. Here’s Lewis’s definition of “p is (entirely) about Q” (where p is a proposition and Q is a set of propositions): for each q in Q, q determines p—i.e., either q entails p or q entails not-p. (This isn’t really quite what we want—for most purposes we want something more like “partly about” rather than “entirely about”. But the point here is just to show how the basic pieces fit together.) Putting this together with the subject-matter conversion, we get this SV for “about”:

• $\frac{\lambda p. \forall q .(\text{if } [] \text{, then } q \text{ determines }p)}{q \text{ is the proposition that } []}$

Applying this function to a question clause (and Lowering) returns the property of being a proposition about the subject matter of the question.

This part is a little more cumbersome than the standard account, but it isn’t all that bad. One way to proceed from here would be to try to reduce the various subject-matter contexts to combinations of propositional attitudes and propositional aboutness. Maybe it’s more complicated than that, but it seems worth a shot.

[Does anyone know how to do display-style formulas in wordpress?]

Written by Jeff

May 25, 2009 at 8:42 am

### 4 Responses

1. One place to look at if you’re interested in logical aspects of question posing is A. Wisniewski’s work. For instance, his “The posing of questions”:

also, D. Harrah’s stuff on the logic of questions is quite interesting:

I’m not sure if that’s exactly what you’re looking for, though.

Rafal

June 20, 2009 at 6:27 pm

2. Thanks for the references, Rafal. I look forward to checking those out. (Also, sorry for the very long lag in replying.)

Jeff

August 27, 2009 at 12:22 am

3. It seems plausible that the analysis of “Alice knows who won” should use a universal quantifier, not an existential (especially if there is a presupposition that someone won). Consider a contest between 6 people (a, b, c, d, e, f), with 2 winners (a and f).

It seems to follow from the claim that Alice knows who won the contest that:
1) Alice knows that a won the contest.
and
2) Alice knows that f won the contest.

I am not sure if changing that existential to a universal would have an impact on the rest of the results.

Lewis Powell

January 28, 2010 at 10:17 pm

4. Hi Lewis,

I’m not sure what the best approach is to this issue (I think it’s the one discussed under the heading of “mention-some” vs. “mention-all” questions), but one thought is that in this case the “who” should be read as plural. Then the semantics predicts that “Alice knows who won” says, presupposing that some people won, there are some people such that Alice knows they won. I think we’ll have to look at pragmatics to see why in some cases we get the extra feeling that the answer should be *complete*.

In support of the plural hypothesis, we can consider languages that explicitly mark plural “who”. And indeed, in Spanish this would be said, “Alice sabe quienes ganaron”, rather than “Alice sabe quién ganó”.

Jeff

January 29, 2010 at 6:05 pm