Relations between determiner and determiner-phrase quantifiers

D. Terence Langendoen

April 23, 2009

As I showed in my March
2009 *everyone* are necessity modal operators, and existential DPQs like *someone* are possibility modal operators.[1] As
such they exhibit the pattern of truth-preserving implication in 1-4; the abbreviations DAX (for 'distribution of and with
box'), etc. are as in my presentation.

*DAX: everyone sang and danced <==> everyone sang and everyone danced**NOX:**everyone sang or everyone danced ==> everyone sang or danced**everyone sang or danced =/=> everyone sang or everyone danced**NAD:**someone sang and danced ==> someone sang and someone danced**someone sang and someone danced =/=> someone sang and danced**DOD: someone sang or someone danced <==> someone sang or danced*

On the other hand, the universal
determiner quantifier (DQ) *every* and
existential DQ *some*, as defined on p.
84 of Keenan & Faltz, *Boolean
Semantics for Natural Language*, are not modal operators. The denotational
equivalence in 5 shows that *every*
satisfies DAX, but the denotational pattern of implication in 6 shows that it does not satisfy NOX; instead it
satisfies the dual of NOX. Similarly, the equivalence in 8 shows that *some*
satisfies DOD, but the pattern of implication in 7 shows that it does not satisfy NAD; instead it
satisfies the dual of NAD.

- DAX: every singer and dancer <==> every singer and every dancer
- dual of NOX:
- every singer or every dancer =/=> every singer or dancer
- every singer or dancer ==> every singer or every dancer
- dual of NAD:
- some singers and dancers =/=> some singers and some dancers
- some singers and some dancers ==> some singers and dancers
- DOD: some singers or some dancers <==> some singers or dancers

In my presentation, I
also showed that the cardinal exceptive DPQs such as *all but four people* (understood as 'all but at most four people')
and plural cardinal DPQs *four people*
(understood as 'at least four people') are quasimodal quantifiers (respectively
quasinecessity and quasipossibility operators) having the pattern of
truth-preserving implication in 9-12.

- NAD:
- all but four people sang and danced ==> all but four people sang and all but four people danced
- all but four people sang and all but four people danced =/=> all but four people sang and danced
- NOX:
- all but four people sang or all but four people danced ==> all but four people sang or danced
- all but four people sang or danced =/=> all but four people sang or all but four people danced
- NAD:
- four people sang and danced ==> four people sang and four people danced
- four people sang and four people danced =/=> four people sang and danced
- NOX:
- four people sang or four people danced ==> four people sang or danced
- four people sang or danced =/=> four people sang or four people danced

On the other hand, the
cardinal exceptive DQs such as *all but
four* (understood as 'all but at most four') and plural cardinal DQs such as
*four* (understood as 'at least four') have
the pattern of denotational entailment in 13-16. Not only is this pattern different from that of the
non-modal universal DQ *every *and
existential DQ *some* in 5-8 (13 and 16 differ from 5 and 8, respectively) but it is also different from that of
their quasimodal DPQ counterparts such as *all
but four people* and *four people*
in 9-12 (14 and 15 differ from 10 and 11 respectively).

- NAD:
- all but four men and women ==> all but four men and all but four women
- all but four men and all but four women =/=> all but four men and women
- dual of NOX:
- all but four men or all but four women =/=> all but four men or women
- all but four men or women ==> all but four men or all but four women
- dual of NAD:
- four men and women =/=> four men and four women
- four men and four women ==> four men and women
- NOX:
- four men or four women ==> four men or women
- four men or women =/=> four men or four women

Yet another pattern of
implication is exhibited by fractional DQs like *half* (*of*) 'at least half
of' *two-thirds* (*of*), etc., all of which are self-dual operators (i.e. where *Qx* <==> *~Q~x*). Their DPQ counterparts such as *half of the people* are quasimodals that satisfy NAD and NOX, as
shown in 17-18; cf. 9-10 and 11-12.

- NAD:
- half of the people sang and danced ==> half of the people sang and half of the people danced
- half of the people sang and half of the people danced =/=> half of the people sang and danced
- NOX:
- half of the people sang or half of the people danced ==> half of the people sang or danced
- half of the people sang or danced =/=> half of the people sang or half of the people danced

However the fractional DQs, unlike the other DQs that I have described, satisfy the duals of NAD and NOX, as shown in 19-20.

- dual of NAD:
- half of the men and half of the women ==> half of the men and women
- half of the men and women =/=> half of the men and half of the women
- dual of NOX:
- half of the men or women ==> half of the men or half of the women
- half of the men or half of the women =/=> half of the men or women

Finally, if a DQ is
itself a modal operator, then the corresponding DPQ is also. For example, the
generic DQ, represented here by *G* (it
is phonologically null when it co-occurs with a plural NP), is a necessity
modal, and so exhibits the pattern of inference in 21-22 (cf. 1-2).

- DAX: G men and women <==> G men and G women
- NOX:
- G men or women ==> G men or G women
- G men or G women =/=> G men or women

The generic DPQ pattern in 23-24 is the same.

- DAX: G people sang and danced <==> G people sang and G people danced
- NOX:
- G people sang or danced ==> G people sang or G people danced
- G people sang or G people danced =/=> G people sang or danced

The dual ~*G*~ of the DQ *G* and ~*G people~ *of the
DPQ *G people *show the same pattern of
implication as the possibility modal in 3-4.

Comparison of all the DQ and DPQ patterns described here leads to the conclusions in 25-29.

- If a DQ satisfies DAX and either NOX or the dual of NOX, then the corresponding DPQs are necessity modals that satisfy DAX and NOX.
- If a DQ satisfies DOD and either NAD or the dual of NAD, then the corresponding DPQs are possibility modals that satisfy NAD and DOD.
- If a DQ satisfies NAD and the dual of NOX, then the corresponding DPQs are quasinecessity quasimodals that satisfy NAD and NOX.
- If a DQ satisfies NOX and the dual of NAD, then the corresponding DPQs are quasipossibility quasimodals that satisfy NAD and NOX.
- If a DQ satisfies the dual of NAD and the dual of NOX, then the corresponding DPQs are self-dual quasimodals that satisfy NAD and NOX.

[1] I thank Massimo Piattelli-Palmarini for asking me at the colloquium about the relation between my account of quantifiers and Ed Keenan's. At the time, I didn't have an answer, and so decided to try to find out what it is.