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An interval based semantics for negative degree questions MÁRTA ABRUSÁN (Institut Jean Nicod, ENS) & BENJAMIN SPECTOR (Harvard Society of Fellows) JSM :: Toulouse :: April 4, 2008

Introduction Goal: To give a new account for negative islands with degree questions: (1) Negative Islands a. Who didn’t John invite to the party? b. *How tall isn’t John? ..and their obviation (Fox and Hackl 2005): (2) How much radiation are we not allowed to expose our workers to? (3) * How much radiation are we not required to expose our workers to?

The core ideas to be developed:
INTERVALS: degree predicates denote relations between individuals and intervals: (Schwarzschild and Wilkinson (2002), Schwarzschild (2004), Heim (2006))
If degree questions range over intervals, the presupposition that a question should have a MAXIMALLY INFORMATIVE TRUE ANSWER (Dayal 1996) can never be met, and (1)b and (3) result in a presupposition failure.

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The plan 1

A brief overview of the analysis

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Comparison with Fox and Hack’s (2005) and Rullmann’s (1995) account

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Our analysis in more detail

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A problem? Apparent cases of overgeneration

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The solution to our problem

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Conclusion

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Appendix 1: cases of undergeneration: the Π-operator

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Appendix 2: fast vs. slow

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Overview of the account Dayal (1996) has proposed that a question presupposes that it has a most informative true answer. (4) Maximal Informativity Hypothesis (Dayal 1996) A question presupposes that it has a maximally informative true answer Intervals:
Schwarzschild and Wilkinson (2002), Schwarzschild (2004), Heim (2006): degree predicates denote relations between individuals and intervals (sets of degrees): (5) A set of degrees D is an interval iff: For all d, d’, d’’: if d∈D & d’’∈D & d≤d’≤d’’, then d’∈D (6)
tall=λI. λxe. x’s height ∈I (7)
How tall is John?w = λp.∃I [I∈DI ∧ p=λw’. John’s height ∈I in w’] ‘For what interval I, John’s height is in I?’ We predict that examples of negative islands such as (9) result in a presupposition failure, given the above two assumptions: (8) Who didn’t John invite to the party? (9) *How tall isn’t John 3

Positive and Negative degree questions Positive degree questions (10)
How tall is John?w

= λp. ∃ I [I∈Dintervals & p=λw’. John’s height ∈ I in w’] ‘For what interval I, John’s height is in I?’

(11) -------------------------------[---(---J’s height ---)---]--------------------------------Most informative answer : take I = {John’s height} Negative degree questions (12)
*How tall isn’t John?w = λp. ∃ I [I∈Dintervals & p=λw’. John’s height ∉ I in w’] ‘For what interval I, John’s height is not in I?’ (13) ---------[----------------------] I2-- J’s height ----------{-------------------------}I3--Intuitively:
We are looking for the maximal interval among the intervals in which John’s height is not contained. Given the negative context, if an interval K covers interval I, the truth of John’s height ∉ K will entail the truth of John’s height ∉I.
The problem is that there is no such maximal interval. 4

Negative degree questions, more precisely (14)
*How tall isn’t John?w = λp. ∃ I [I∈Dintervals & p=λw’. John’s height ∉ I in w’] ‘For what interval I, John’s height is not in I?’ (15) ---[----------------------] I2------dj ---------{--------------}I3--More precisely: (i) Let John’s height be any non-zero degree d. (ii) The set of all intervals that do not include John’s height (=N) contains exactly two exclusive sets of intervals:
all the intervals fully below d, contained in [0, d[ (=A) and
all the intervals fully above d, contained in ]d, +∞) (=B). (iii) It is easy to see that for any interval I included in A, the (true) proposition that John’s height is not in I, does not entail that John’s height is not in B, and vice versa. (iv) Hence, there is no interval I in N such that the true proposition that John’s height is not in I entails all the true propositions of the same form in N. Dayal’s (1996) condition cannot be met, and we predict a presupposition failure. 5

Modal obviation Fox and Hackl (2005) observe that existential modals below negation improve negative islands: (16) a. b.

How fast are we not allowed to drive? For what I, it is not allowed that our speed be in I?

This fact is straightforwardly predicted by the present account:
While wrt. (9) it was a fact about the world that John’s height is a single degree, the degrees of speed with which we are allowed to drive might correspond to an interval, e.g. ]0, 120km/h], if the law states that our speed must be at most 120 km/h.
Now any I that is entirely above 120km/h is such that it is not allowed that our speed be in I.
In this case, there is a strongest true proposition among the H/K alternatives to (16): the true proposition that the speed with which we are not allowed to drive is ∈ ]120, +∞).
Dayal’s condition is met Same for universal modals above negation: How fast are we required not to drive? 6

Modal obviation (contd.) Existential modals above negation however do not obviate negative islands: (17) a. # How fast are we allowed not to drive? b. For what I, it is allowed that our speed be not in I? Suppose we have some obligations as to what our speed should be: Call S the set of all the speeds such that our speed is required to be one of them: E.g: (i)

0------ allowed not--------[d--------[--allowed not--]------------>

S: required
For any subset of S which does not equal S, it is allowed that our speed is not in it.
Also, for any subset of the complement set of S, it is allowed that our speed is not in it.
The interval which covers all the intervals such that our speed is allowed not to be in it is therefore [0,+∞[. However this interval cannot be the maximally informative interval, because it will also cover the interval(s) for which it is required that our speed be in it, therefore Dayal’s presupposition is not met. If we have no obligations for our speed, the maximal interval for which it is allowed that our speed is not in it is [0,+∞[. Dayal’s presupposition is met, however, the answer means that there is a possible world in which our speed is not in [0,+∞[, which is impossible. 7

Comparison with Fox & Hackl (2005) Replacing Maximality with Maximal Informativity (cf. Beck and Rullmann 1989) ( this is similar to our account)
Dayal’s assumption A question presupposes that it has a unique maximally informative answer, i.e. a true answer that entails all the other true answers.
A question asks for the maximally informative answer But given a monotonic semantics for degree predicates negative islands are not predicted: (18)

a.* How tall isn’t Jack? b. For what d, Jack isn’t d-tall


The maximally informative answer should ask for the smallest degree d such that Jack is not dtall. There should be no problem if there is such a degree. F&H’s solution: Dense scales
The smallest degree d such that Jack is not d-tall does not exist.

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Fox & Hackl (2005) (Contd.) Dense scales Let d be Jack’s height:
Suppose that d’ = d + ε is the smallest degree such that Jack is not d’-tall.
Take d’’ = d +ε/2. Clearly, Jack is not d’’-tall; but d’’ is smaller than d’, which is contradictory. Intuitively: Let Jack’s height be 180 cm. The set of all true propositions of the form Jack is not dtall is the following: (19) {…,Jack is not 180,000001 cm –tall, ..., Jack is not 180,05 cm-tall,…., Jack is not 181 cm-tall,….}

It will be apparent that there is no minimal degree d such that Jack is not d-tall. This is simply because for any d > 180cm, there is a d’ such that d > d’ > 180cm. Dayal’s condition is not met. Discrete scales F & H must extend this account even to cases where the domain of degrees is not “intuitively” dense, such as cardinality measures (a certain sort of degrees), as in: (20) *How many children doesn’t Jack have? 9

Fox and Hackl (2005) on Discrete scales, modularity and blindness to contextual information (21) *How many children doesn’t Jack have? Account: suppose Jack has exactly 3 children. Then he does not have 4 children, but he also does not have 3.5 children, or 3.00001 children… A natural objection: even granting that it makes sense to say that Jack has 3.5. children, yet the exact number of children someone has is always an integer. So Jack does not have 3+ε children is known to be equivalent to Jack does not have 4 children, and there is in fact a true answer that entails all the other ones. Modularity F & H have to assume a very strong modularity assumption: presumably, the knowledge that the number of children someone has is an integer is a form of lexical/encyclopedic knowledge. According to them however, this knowledge is not purely logical, given some reasonable notion of logicality (one that is blind to lexical semantics/encyclopedic knowledge). F & H’s central claim is that Dayal’s condition is computed only on the basis of the purely logical meaning of the question, i.e. is blind to contextual, encyclopedic or lexical information.  while there might be arguments for this view, we would like to explore an alternative 10

Fox and Hackl (2005): Accounting for the modal obviation facts Existential modal under negation (22) How fast are we not allowed to drive?
Before, it was a fact about the world that predicates such as λd. we are d-fast denote a closed interval, hence its complement denoted an open interval given the assumption that scales are dense. As a result, Dayal’s condition could not be met.
Here, the predicates of the form λd. POSSIBLE (P(d)) can denote open intervals, and therefore their complements, i.e. predicates of the form λd.¬POSSIBLE(P(d)) can denote closed intervals. Likewise for λd.NEC(¬P(d)). Illustration Suppose that the law states that our speed should be lower than 65 km/h, and says nothing more. It follows that the set of worlds compatible with the law is {w: our speed is lower than 65 km/h}. So for any speed v below 65 km/h (however close to 65 km/h), there is a permissible world in which our speed is v. Hence for any speed lower than 65km/h, we are allowed to drive at that speed. On the other hand, we are not allowed to drive at 65km/h. Hence 65 km/h is the lowest speed v such that we are not allowed to drive at speed v.  Dayal’s condition can be met. 11

Returning to our account: Discrete scales Nothing special needs to be said about discrete scales. (23) A. b.

How many children doesn’t Jack have? For what interval I of integers, the number of children Jack has is not in I?


Suppose Jack has exactly 4 children:
Then the set of intervals that correspond to a true answer are all the intervals strictly below 4 and every interval strictly above 4.
But no interval in this set corresponds to a maximally informative answer (as before)

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Argument for the interval reading That the interval reading indeed exists is shown by the following example: (24) How fast are we required to drive? Suppose that one the highway we should drive between 60km/h and 120 km/h. Then the complete answer is predicted to state exactly this. Scenario Jack and Peter are devising the perfect Republic. They argue about speed limits on highways. Jack believes that people should be required to drive at a speed between 60km/h and 120km/h. Peter believes that they should be required to drive at a speed between 60km/h and 140 km/h. Therefore… (25) Jack and Peter do not agree on how fast people should be required to drive on highways The disagreement is about the maximal speed, not the minimal speed, as the standard account would have it

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The overgeneration problem Problem # 1 (26) How fast are we required to drive? • Case a: there is a minimal speed s: hence the maximally informative answer is the interval [s, +∞). This corresponds to our intuitions • Case b: there is a maximal speed s. There is a maximally informative answer, namely the one based on [0, s]. But in fact (26) is unnatural if it is known that there is a maximal speed and no minimal speed. This is an overgeneration problem, in the sense that our account so far predicts a question like (26) to be felicitous in too many contexts. • (Note that the corresponding question is perfectly fine in a language like French, where instead of how fast/how slow one finds At what speed) Problem #2 (27) How fast are we not allowed to drive on this highway? Normally, one understands from (27) that one is not allowed to drive too fast. But suppose that in fact the only obligation we have is to drive at a speed higher than 60 km/h, and there is no maximal speed. Then the most informative interval I such that we are not allowed to drive at a speed in I is [0, 60], and so Dayal’s condition is met. Again we fail to predict that a question like (27) imposes a further restriction, namely implicates that there is a maximal speed. 14

Solving the overgeneration problem Assumption : Let HowI{S,