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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)
-
In type-theoretical terms: every man denotes an (et)t function.
Application of this function to the VP denotation:
- (3)
-
``every member of the set is a member of the set ''
- (4)
- Some man ran.
- (5)
-
``there is an entity that is a member of both and ''
- (6)
- No man ran.
- (7)
-
- (8)
- exactly five men:
- (9)
- most men:
Terminology:
We call (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)
-
``the set of runners is in the set of subsets of E that contain
''
- (12)
-
- a.
- Some man ran quickly
Some man ran
- b.
- Some man ran quickly
Some man ran
- (13)
-
- a.
- No man ran quickly
No man ran
- b.
- No man ran quickly
No man ran
- (14)
-
- a.
- Exactly five men ran quickly
Exactly five men ran
- b.
- Exactly five men ran quickly
Exactly five men ran
Some man is called an upward monotone (mon) noun phrase.
No man is called a downward monotone (mon) noun phrase.
Exactly five men is called a non-monotone (neither mon
nor mon) noun phrase.
Determiners like the ones considered above denote functions from sets to
GQs:
- (15)
-
Thus, determiners are functions of type (et)((et)t).
Alternatively, we can view them as relations between subsets of
E:
- (16)
- iff
- (17)
- iff
- (18)
- iff
Terminology:
We call (a relation between subsets of E)
a determiner over E.
- (19)
-
- a.
- Some blue car arrived
Some car arrived
- b.
- Some blue car arrived
Some car arrived
- (20)
-
- a.
- Every blue car arrived
Every car arrived
- b.
- Every blue car arrived
Every car arrived
- (21)
-
- a.
- Exactly five blue cars arrived
Exactly five cars arrived
- b.
- Exactly five blue cars arrived
Exactly five cars arrived
Fact: A determiner D is mon (mon) iff for every
: D(A) is a mon (mon)
quantifier.
Some is called an upward left-monotone (mon) determiner
(it is also mon).
Every is called a downward left-monotone (mon) determiner (but it is mon).
Exactly five men is not left-monotone (nor is it right-monotone).
- (22)
- Every man ran Every man is a man who ran
- (23)
- Some man ran Some man is a man who ran
- (24)
- Exactly five men ran Exactly five men are men who ran
... and so on for all determiners!
Universal: All natural language determiners (simple and complex)
are conservative.
Exercises
- 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.
- For each NP in 1 say whether it is mon, mon or
non-monotone.
- 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.
- For each determiner in 3 specify its left and right monotonicity.
- 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 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.
- Show that the following artificial determiner is not
conservative:
NC(A)(B) iff
Hint: give A and B such that NC(A)(B) and 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