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Course: Modeltheoretic Semantics and the Syntax-Semantics Interface
Eddy Ruys and Yoad Winter

Lecture 4:
Generalized Quantifiers (1)

Topics: NP denotations as sets of sets, NP monotonicity, determiner denotations as relations between sets, determiner monotonicity
Reading: Keenan, E. L. 1996: The Semantics of Determiners, in S. Lappin (ed.), The Handbook of Contemporary Semantic Theory, Blackwell.
(1)
Every man ran.

Let every man denote a set of sets: the set of subsets of E that include the set of men:
(2)
tex2html_wrap_inline208
In type-theoretical terms: every man denotes an (et)t function.
Application of this function to the VP denotation:
(3)
tex2html_wrap_inline212
``every member of the set tex2html_wrap_inline214 is a member of the set tex2html_wrap_inline216''
(4)
Some man ran.
(5)
tex2html_wrap_inline218
``there is an entity that is a member of both tex2html_wrap_inline214 and tex2html_wrap_inline216''
(6)
No man ran.
(7)
tex2html_wrap_inline224
(8)
exactly five men: tex2html_wrap_inline226
(9)
most men: tex2html_wrap_inline228
Terminology: We call tex2html_wrap_inline230 (a set of subsets of E) a generalized quantifier (GQ) over E.
Also proper names like Tina denote GQs from now on:
(10)
Tina ran.
(11)
tex2html_wrap_inline236
``the set of runners is in the set of subsets of E that contain tex2html_wrap_inline240''
tex2html_wrap_inline242
(12)
a.
Some man ran quickly tex2html_wrap_inline244 Some man ran
b.
Some man ran quickly tex2html_wrap_inline246 Some man ran

(13)
a.
No man ran quickly tex2html_wrap_inline248 No man ran
b.
No man ran quickly tex2html_wrap_inline250 No man ran

(14)
a.
Exactly five men ran quickly tex2html_wrap_inline248 Exactly five men ran
b.
Exactly five men ran quickly tex2html_wrap_inline246 Exactly five men ran

Some man is called an upward monotone (montex2html_wrap_inline256) noun phrase.
No man is called a downward monotone (montex2html_wrap_inline258) noun phrase.
Exactly five men is called a non-monotone (neither montex2html_wrap_inline256 nor montex2html_wrap_inline258) noun phrase.
definition122
Determiners like the ones considered above denote functions from sets to GQs:
(15)
tex2html_wrap_inline282
Thus, determiners are functions of type (et)((et)t).
Alternatively, we can view them as relations between subsets of E:
(16)
tex2html_wrap_inline288 iff tex2html_wrap_inline290
(17)
tex2html_wrap_inline292 iff tex2html_wrap_inline294
(18)
tex2html_wrap_inline296 iff tex2html_wrap_inline298
Terminology: We call tex2html_wrap_inline300 (a relation between subsets of E) a determiner over E.
(19)
a.
Some blue car arrived tex2html_wrap_inline244 Some car arrived
b.
Some blue car arrived tex2html_wrap_inline246 Some car arrived

(20)
a.
Every blue car arrived tex2html_wrap_inline248 Every car arrived
b.
Every blue car arrived tex2html_wrap_inline250 Every car arrived

(21)
a.
Exactly five blue cars arrived tex2html_wrap_inline248 Exactly five cars arrived
b.
Exactly five blue cars arrived tex2html_wrap_inline246 Exactly five cars arrived


definition145
Fact: A determiner D is montex2html_wrap_inline256 (montex2html_wrap_inline258) iff for every tex2html_wrap_inline366: D(A) is a montex2html_wrap_inline256 (montex2html_wrap_inline258) quantifier.
Some is called an upward left-monotone (tex2html_wrap_inline256mon) determiner (it is also montex2html_wrap_inline256).
Every is called a downward left-monotone (tex2html_wrap_inline258mon) determiner (but it is montex2html_wrap_inline256).
Exactly five men is not left-monotone (nor is it right-monotone).
(22)
Every man ran tex2html_wrap_inline382 Every man is a man who ran
(23)
Some man ran tex2html_wrap_inline382 Some man is a man who ran
(24)
Exactly five men ran tex2html_wrap_inline382 Exactly five men are men who ran

... and so on for all determiners!
definition158
Universal: All natural language determiners (simple and complex) are conservative.

Exercises

  1. Give denotations for the following NPs: more than four women, at most five children, between four and ten cows, all but five students, less than half the men, not every goat, Tina and Mary.
  2. For each NP in 1 say whether it is montex2html_wrap_inline256, montex2html_wrap_inline258 or non-monotone.
  3. Give a relational denotation for the determiners in 1: more than four, at most five, between four and ten, all but five, less than half the, not every.
  4. For each determiner in 3 specify its left and right monotonicity.
  5. Arrange the determiners mentioned in the lecture notes and in exercise 3 in a table according to the nine possibilities for monotonicity in both arguments (9 = 3 possibilities for L-monotonicity tex2html_wrap_inline398 3 for R-monotonicity). Try to fill in the ``holes'' in the table (=logical possibilities that no determiner realizes) with natural language determiners, however syntactically artificial these may be. You may consult Keenan's paper for examples.
  6. Show that the following artificial determiner is not conservative:
    NC(A)(B) iff tex2html_wrap_inline402
    Hint: give A and B such that NC(A)(B) and tex2html_wrap_inline410 do not have the same truth-value.
    Can you think of a syntactic element with the meaning of NC in a sentence of the form students ran? After you find such an element: give syntactic arguments for whether it is of the same category of the word all or not. Conflicting arguments are welcome - you don't have to decide on this matter.



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Yoad Winter
Fri Oct 10 13:36:30 MET DST 1997