Estructura organizacional

The idea that a DP’s denotation is the natural set to be represented by a file card is attractive for those who assume that a sentence is about the sentence topic’s denotation. This is an obvious assumption to make with respect to individual-denoting DPs, since occurrences of sentences such as ‘[Yasma]T is walking’ or ‘[The king]T is walking’ are intuitively about, re-

spectively, Yasma and the contextually salient king; and these are the denotations assigned to these DPs in traditional semantic approaches. I will later argue that the aboutness crite- rion renders the denotational approach deeply implausible.

Candidate 2: Smallest Live-On Set

Another option is to claim that topical DPs are assigned file cards representing their smallest live-on sets. SupposeQD is a typeh1iquantifier, possibly of the formQ0D(A)whereQ0Dis a

typeh1,1iquantifier andAis a set to which it has been applied. WhereBis any set, and the set of alllive-on setsforQD is written asLQD:

Live-on Set Definition:B ∈LQD iff, for allC ⊆D,QD(C)iffQD(B∩C).

25

What this property means is that, to know for any subsetCof a domainDwhether or not it stands in the relevant quantifier relation, you need only look at the part ofCwhich is also a part ofB. The quantifier therefore depends on the setBin an important way, since one need only checkB to establish which other sets are in the quantifier relation. In the following, I will also describe natural language expressions that denote typeh1i quantifiers as having live-on sets, by which I will mean the live-on sets of their extensions.

Next, whereBis any set, and the set ofsmallest live-on setsforQDis written asSLQD:

Smallest Live-on Set Definition:B∈SLQD iffB ∈LQD&¬∃C :C∈LQD &C (B. It should be clear from the definition thatSLQD will always have either a single member or no members; forLQD can contain no disjoint sets, hence it follows that there cannot be more than one set inLQD that lacks a proper subset inLQD.26 Two important properties to note for smallest live-on sets are that, firstly,∅is the smallest live-on set forQD if and only if

QDis trivial (with triviality being a notion I briefly discussed in §(1.2.2), and shall discuss in

more detail in §(4.3.1)). Secondly, where ‘Det’ is a natural language determiner, the smallest live-on set for(_JDetKc)D(A)isA.

27

I will later argue that the smallest live-on set approach is promising with respect to the aboutness criterion, but encounters some difficulties with respect to the anaphora criterion.

Candidate 3: Unique Minimal Witness Set

The next option is to claim that topical DPs are assigned file cards representing their unique minimal witness sets. Take someQD such that the smallest live-on setSLQD has a member

C; then whereBis any set, and the set of allwitness setsforQD is written asWQD:

Witness Set Definition:B ∈WQD iffB ⊆C&B ∈QD.

28

In words,Bis a witness set for a quantifier if and only if it is a subset of the smallest live-on set of the quantifier and furthermore stands in the quantifier relation. For example, consider the witness sets for the following three quantifiers:

W_(Qevery)D(

JNKc)={B:B ⊆JNKc∧JNKc⊆B}

W(Qno)D(JNKc)={B:B ⊆JNKc∧JNKc∩B =∅}

W(Qat−least−f our)D(JNKc) ={B :B ⊆JNKc∧ |JNKc∩B| ≥4}

We therefore see that the only witness set for the quantifier denoted by ‘Every N’ will be

JNKc, since this is the only set that is both a subset of the set of Ns and is furthermore in the

relation defined by the quantifier. The only witness set for the quantifier denoted by ‘No N’ will be∅, since no other subset ofJNKchas an empty intersection withJNKc, as required by

theQnorelation. Finally, the set of witness sets for the quantifier denoted by ‘At least four

Ns’ will consist of all subsets of the set of Ns with a cardinality of four or greater, thus there will often be many such witness sets.

Given that there are cases where there are multiple witness sets with a range of cardinal- ities, it seems that it would be unhelpful to pursue an approach whereby file cards represent

arbitrarywitness sets. This inference derives from a quick assessment of the proposal with respect to the aboutness criterion: it would be highly counter-intuitive for a sentence such as

26_{It is provable that, ifL}

QD containsBandB

0_{, then}

LQD containsB∩B

0_{(see Peters and Westerst˚ahl (2006),}

pp.89-90).SLQDwill be non-empty wheneverDis finite, but is possibly empty whenDis infinite. 27_{Peters and Westerst˚ahl (2006), pp.89-105. More specifically,SL}

QD(A) ={A}whenever the (global) quan-

tifierQhas the properties of isomorphism closure, extension and ‘finite action’; given that every quantifier denoted by a natural language determiner has these properties, it is not necessary to define or discuss them further.

‘At least four kings are walking’, uttered relative to a context where the extension of ‘kings’ includes fifty members, to be about (say) twenty seven kings; yet if the DP’s file card were to represent an arbitrary witness set for the quantifier, then a set with a cardinality of 27 may well turn out to be represented. Rather, a prima facie promising approach would be to pick thesmallestwitness set to be represented by the file card, to prevent occurrences of sentences from being about witness sets with gratuitous members.

In order to develop this third approach, we define the set of minimal witness sets as follows. Take someQD such that the smallest live-on set SLQD has a member, meaning

WQD also has members; then whereB is any set, and the set of allminimal witness sets for

QDis written asM WQD:

Minimal Witness Set Definition: B ∈ M WQD iff B ∈ WQD &¬∃C : C ∈ WQD &

C(B.29

In words,Bis a minimal witness set for a quantifier if it is among the smallest witness sets. Clearly, there may be many such minimal witness sets, if there are multiple disjoint witness sets each lacking proper subsets that are witnesses. For example, the minimal witness sets for ‘At least four kings’ will consist of every set of contextually salient kings with a cardi- nality of four. However, whenever the cardinality ofM WQD is 1, we may say thatQDhas a

unique minimal witness set.

In order to clarify these ideas, note the following list of DPs’ minimal witness sets:

M W(Qevery)D(JNKc)={JNKc} M W_(Qsome)D( JNKc)={B⊆JNKc:|B|= 1} M W_{(Qf our)D(} JNKc)={B ⊆JNKc:|B|= 4} M W(Qat−least−f our)D(JNKc)={B ⊆JNKc:|B|= 4} M W(Qno)D(JNKc) ={∅} M W_{(Qat−most−f our)D(} JNKc)={∅}

The third approach to constructing file cards would involve representing the unique mini- mal witness set of the quantifier denoted by a DP. However, a problem associated with this approach is immediately obvious: not all quantifiers have unique minimal witness sets. In- deed, in the above list, only ‘Every N’, ‘No N’ and ‘At most four N’ denote quantifiers with unique minimal witness sets. I will later argue that this presents difficulties with respect to the aboutness criterion.

Candidate 4: Arbitrary Minimal Witness Set

In light of the fact that some quantifiers lackuniqueminimal witness sets, the final approach consists of representing anarbitrary minimal witness set on the relevant file card. Indeed, this is the approach that Ebert (2009) (p.236) settles on, claiming that ‘a good representative

would be an element of the quantifier which does not contain any ‘disturbing’ elements. A

minimal set... is a set that meets this requirement’.

In cases where the minimal witness set is empty, this approach commits itself to the view that a file card associated with DPs headed by these determiners would represent the empty set. It is worth briefly delineating the class of quantifiers that have an empty minimal wit- ness set, which is the class of right monotone decreasing quantifiers. Monotonicityproper- ties clarify the entailment relations between quantificational claims, and a quantifier’s being

right monotone decreasing means that when an ordered pair of sets stand in the quantifier relation, any subset of the second member of that pair will also stand in that relation:

Right Monotone Decreasing Definition for Typeh1,1iQuantifiers:WhereQis a type

h1,1iquantifier andDis a domain,QD isright monotone decreasingiff: ifB0 ⊆B ⊆D,

thenQD(A)(B)impliesQD(A)(B0). Qis right monotone decreasing if eachQD is.

I shall often describe a determiner as ‘right monotone decreasing’ if the quantifier it denotes has the property in question, and I will describe a DP formed from a right monotone de- creasing determiner as simply ‘monotone decreasing’ (since the quantifier the DP denotes has only one argument for which monotonicity properties may vary). Right monotone de- creasing determiners include: ‘not all’, ‘no’, ‘at mostn’ and ‘few’. The reason that the unique minimal witness set of all right monotone decreasing quantifiers will be empty is that the empty set will be a subset of the smallest live-on set of any such quantifier (due to its being a subset of every set), and will also be in the relevant quantifier relation (due to the monotone decreasing property), meeting the two conditions for being a minimal witness set. In the following section, I will argue that this observation presents some surmountable difficulties for the current approach with respect to the aboutness criterion. However, I will claim that the anaphora criterion provides strong evidence in favour of it.