Relations between determiner and determiner-phrase quantifiers

D. Terence Langendoen

April 23, 2009

 

As I showed in my March 2009 University of Arizona Linguistics Colloquium presentation http://dingo.sbs.arizona.edu/~langendoen/Quantification&Modality-plain.pdf, universal determiner phrase quantifiers (DPQs) like 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.

  1. DAX: everyone sang and danced <==> everyone sang and everyone danced
  2. NOX:
    1. everyone sang or everyone danced ==> everyone sang or danced
    2. everyone sang or danced =/=> everyone sang or everyone danced
  3. NAD:
    1. someone sang and danced ==> someone sang and someone danced
    2. someone sang and someone danced =/=> someone sang and danced
  4. 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.

  1. DAX: every singer and dancer <==> every singer and every dancer
  2. dual of NOX:
    1. every singer or every dancer =/=> every singer or dancer
    2. every singer or dancer ==> every singer or every dancer
  3. dual of NAD:
    1. some singers and dancers =/=> some singers and some dancers
    2. some singers and some dancers ==> some singers and dancers
  4. 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.

  1. NAD:
    1. all but four people sang and danced ==> all but four people sang and all but four people danced
    2. all but four people sang and all but four people danced =/=> all but four people sang and danced
  2. NOX:
    1. all but four people sang or all but four people danced ==> all but four people sang or danced
    2. all but four people sang or danced =/=> all but four people sang or all but four people danced
  3. NAD:
    1. four people sang and danced ==> four people sang and four people danced
    2. four people sang and four people danced =/=> four people sang and danced
  4. NOX:
    1. four people sang or four people danced ==> four people sang or danced
    2. 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).

  1. NAD:
    1. all but four men and women ==> all but four men and all but four women
    2. all but four men and all but four women =/=> all but four men and women
  2. dual of NOX:
    1. all but four men or all but four women =/=> all but four men or women
    2. all but four men or women ==> all but four men or all but four women
  3. dual of NAD:
    1. four men and women =/=> four men and four women
    2. four men and four women ==> four men and women
  4. NOX:
    1. four men or four women ==> four men or women
    2. 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.

  1. NAD:
    1. half of the people sang and danced ==> half of the people sang and half of the people danced
    2. half of the people sang and half of the people danced =/=> half of the people sang and danced
  2. NOX:
    1. half of the people sang or half of the people danced ==> half of the people sang or danced
    2. 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.

  1. dual of NAD:
    1. half of the men and half of the women ==> half of the men and women
    2. half of the men and women =/=> half of the men and half of the women
  2. dual of NOX:
    1. half of the men or women ==> half of the men or half of the women
    2. 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).

  1. DAX: G men and women <==> G men and G women
  2. NOX:
    1. G men or women ==> G men or G women
    2. G men or G women =/=> G men or women

 

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

  1.  DAX: G people sang and danced <==> G people sang and G people danced
  2. NOX:
    1. G people sang or danced ==> G people sang or G people danced
    2. 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.

  1. 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.
  2. 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.
  3. If a DQ satisfies NAD and the dual of NOX, then the corresponding DPQs are quasinecessity quasimodals that satisfy NAD and NOX.
  4. If a DQ satisfies NOX and the dual of NAD, then the corresponding DPQs are quasipossibility quasimodals that satisfy NAD and NOX.
  5. 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.