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May 7
Semantics I
Directional Entailing Properties
Consider the following entailment patterns:
(1) a. Every student smokes.
b. #Every Italian student smokes.


Every Italian student smokes.
Every student smokes.
(2) a. #Every student is Italian.
b. Every student is Italian and blond.


Every student is Italian and blond.
Every student is Italian.
With sentences involving every, we get a valid inference whenever we substitute a more specific
common noun for a less specific one, but not vice versa. On the other hand, we get a valid
inference whenever we substitute a less specific VP for a more specific one, but not vice versa.
Now consider some and no:
(3) a. #Some student smokes.
b. Some Italian student smokes.


Some Italian student smokes.
Some student smokes.
(4) a. #Some student is Italian.
b. Some student is Italian and blond.


Some student is Italian and blond.
Some student is Italian.
(5) a. No student smokes.
b. # No Italian student smokes.


No Italian student smokes.
No student smokes.
(6) a. No student is Italian.
b. #No student is Italian and blond.


No student is Italian and blond.
No student is Italian.
With some, we must always infer from a more specific to a less specific phrase. With no, it’s the
opposite.
What’s the semantic property behind these inference patterns?
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(7) a. Right upward entailment
D(A)(B  C)  D(A)(B)
Some student is Italian and blond.  Some student is Italian.
b. Right downward entailment
D(A)(B)  D(A)(B  C)
No students are Italian.  No students are Italian and blond.
c. Left upward entailment
D(B  C)(A)  D(B)(A)
Some Italian student smokes.  Some student smokes.
d.
Left downward entailment
D(B)(A)  D(B  C)(A)
No students smoke.  No Italian students smoke.
In sum,
Some is upward entailing in its both arguments.
No is downward entailing in its both arguments.
Every is…
Few is…
So why do these entailment patterns matter?
Distribution of negative polarity items (any, give a damn, ever, etc.)
(8)
a. *John saw any bird.
b. John did not see any bird.
(9)
a. *Some student gives a damn about Pavarotti.
b. *Every student gives a damn about Pavarotti.
c. No student gives a damn about Pavarotti.
(10)
a. *Some student have ever read a book about Pavarotti.
b. *Every student has ever read a book about Pavarotti.
c. No student has ever read a book about Pavarotti.
(11)
a. *Some student who has ever read a book about Pavarotti would want to meet
him.
b. Every student who has ever read a book about Pavarotti would want to meet him.
c. No student who has ever read a book about Pavarotti would want to meet him.
When are negative polarity items licensed?
Negatives, yes, but not only that (consider every in (11b)).
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If we consider the directional entailing properties discussed above, a negative polarity item is
licensed in a downward entailing environment. (Ladusaw 1979)
What happens with not every?
(10)
(11)
b’. Not every student has ever read a book about Pavarotti.
b’. Not every student who has ever read a book about Pavarotti would want to meet him.
Inclusive or
Crain and Pietroski (2002) suggest downward entailing environments trigger the inclusive
reading of or.
(12)
a. I'll talk to linguists or philosophers.
b. I don't talk to linguists or philosophers.
c. I never talk to linguists or philosophers.
d. No man admires linguists or philosophers.
e. Some man admires linguists or philosophers.
f. Every linguist or philosopher admires Chomsky.
g. Every man admires linguists or philosophers.
Further questions
Negative polarity items can also appear in questions.
(13)
a. Do you give a damn about your health?
b. Do you have any idea?
c. Have you ever been to Paris?
Ladusaw (1979) argues that it is not so much the question itself that licenses the NPI but rather
the expected answer. For yes/no question, NPI can appear when the expected answer is negative.
This analysis might get the right prediction for rhetorical questions, but leaves it unclear why
other questions, where there is less of an expectation about the answer, license NPIs.
(For more discussion on this topic, please see Van Rooy 2003)
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Conservativity
Recall how we defined every, some, and no:
(14)
a. [[every]](Y) = {X  U: Y  X}
b. [[some]](Y) = {X  U: Y  X  }
c. [[no]] (Y) = {X  U: Y  X = }
Given the above, we can evaluate the truth of a sentence (e.g. Every/Some/No student smokes) by
checking to see if all/some/no members of Y (e.g. [[student]]) are in the set corresponding to X
(e.g. [[smoke]]). So, we can always work with the members of the common noun set Y in
checking whether the given relation holds.
However, this regularity is not found in all quantifier-like relations. Consider all-but, and
everyone-except:
(15)
a. All but students smoke.
b. Everyone except students smokes.
To evaluate whether (15a&b) are true, we must look at non-students.
This contrast is stated in the form of conservativity (Barwise and Cooper 1981):
(16)
A determiner is conservative iff
for every X and every Y, X  D(Y) iff X  Y  D(Y).
i.e. D(A)(B)  D(A)(A  B)
Conservative determiners:
(17) a. every:
Every man smokes iff every man is a man who smokes.
b. some:
Some man smokes iff some man is a man who smokes.
c. no:
No man smokes iff no man is a man who smokes.
In fact, all determiners are conservative; if something is not conservative, it’s not a determiner.
Take only:
(18) a. Only students smoke
b. smoke'  D(student)  smoke'  student'
(The set of smokers is a subset of the set of students.)
c. Only students smoke iff only students are students who smoke.
It is not the case that nonconservative relations are conceptually inaccessible or unuseful. All-but
and everyone-except do not share the conservativity property because their common noun does
not specify the range of quantification. So they are not determiners, but we certainly have ways
to express them.
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Weak Determiners and Strong Determiners
Some facts:
(19)
a. There is a student.
b. There are three students.
c. There are many students.
d. There are no student.
(20)
a. #There is every student.
b. #There are most students.
c. #There are both students.
d. #There are all students.
e. #There is the student.
Determiners that can occur in existential sentences are called weak determiners.
Determiners that can’t occur in existential sentences are called strong determiners.
Milsark (1977)
Existential sentences assert existence or non-existence denoted by the NP. This assertion clashes
with the use of strong determiners which presuppose the existence of the entity they denote.
Barwise and Cooper (1981)
Terminology first:
(21) a. A determiner D is positive strong for every model M and every A ⊆ U, if D(A) is
defined, then D(A)(A) = 1.
b. A determiner D is negative strong for every model M if every A ⊆ U, if D(A) is
defined, then D(A)(A) = 0.
c. A determiner D is weak if it is neither positive strong nor negative strong.
Examples for (21a):
(22) a. Every student is a student.
b. Three students are students.
Tautology?
Examples for (21b):
(23) a. Neither student is a student.
b. No student is a student.
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Barwise and Cooper’s argument goes like this:
(24)
a. “There is Det CNP” is interpreted as “Det CNP exist(s)”; i.e. D(A)(E)
b. Because of conservativity, D(A)(E) is equivalent to D(A)(A ∩ E).
c. Since A ∩ E = A, this is equivalent to D(A)(A).
d. For positive strong determiners, the formula D(A)(A) is a tautology; it’s never
informative.
e. For negative strong determiners D(A)(A) is a contradiction.
f. Only for weak determiners is it a contingent sentence that can give us information.
g. Thus, only weak determiners are acceptable in existential sentences.
However,
(25) a. *There is every student.
b. Every student exists.
For more discussion on this topic, please see Keenan (2003).
Each and Every (Fiengo 2007)
The English expressions each and every are both represented as .
The two quantifiers are the same in quantificational force, but they differ in manner of
interpretation:
(26) a. Every imposes the manner of Totalizing.
b. Each imposes the manner of Individualizing.
(Totalizing and Individualizing are the terms used by Curme (1935).)
Fiengo suggests these manners should be reflected in logical notation.
(27) a. Tx[P(x)  Q(x)] (“Every P Qs”)
b. Ix[P(x)  Q(x)] (“Each P Qs”)
How the truth value of (27a) is determined:
(28) a. Form the Totality of P in U.
b. Determine whether the Totality of P in U is Q.
c. If the Totality of P in U is Q, then Tx[P(x)  Q(x)] is true, otherwise false.
How the truth value of (27b) is determined:
(29) a. Select an item in U, determine whether it is P, and then, if it is P, determine
whether it is Q.
b. Repeat this procedure until U is exhausted.
c. If all the Ps in U are Q, then Ix[P(x)  Q(x)] is true, otherwise false.
The truth conditions of (27a) and (27b) are the same.
On the other hand, they are distinct in manners by which the truth conditions are derived.
Fiengo claims that since force and manner make separate semantic contributions, they must be
distinguished in logical representation.
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When do we choose to use each over every?
A variety of circumstances affect this choice, but if the number of items is very small, the
Individualizing each instead of the Totalizing every.
(30)
Every student John has ever had has done well.
If we subsequently learn that John has had only two or three students, we may feel misled.
Compare the following. What’s our preference?
(31)
a. You have every prospect of success.
b. You have each prospect of success.
(32)
a. I have every reason to believe you.
b. I have each reason to believe you.
The totality of prospects and the totality of reasons have not been arrived at by exhausting the
prospects or reasons, and determining whether they meet one criterion or another. Rather, they
have been arrived at by paring away their complements.
Consider a standard group portrait of the nine members of the U.S. Supreme Court:
(33)
a. This is a picture of every member of the Supreme Court.
b. This is a picture of each member of the Supreme Court.
How is the truth value of (33b) determined?
For each person, it is to be determined whether he or she is a member of the Supreme Court, and
then, if that person is a member, whether the picture is a picture of him or her.
Compare the following:
(34)
a. Each student is smart.
b. Every student is smart.
c. Each and every student is smart.
Each and every N contains two quantifiers that have distinct modes of presentation and that
determine the same truth conditions. There is no redundancy because there is difference in mode
of presentation.
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By the way, how would you characterize all in contrast with each and every?
Does it impose a specific manner, such as Totalizing or Individualizing?
Also consider the following:
(35)
a. Was each man saved?
b. Yes, each man was saved, but in fact there was only one man aboard.
(36)
a. Was every man saved?
b. Yes, every man was saved, but in fact there was no man aboard.
(37)
a. Were all the man saved?
b. All the men were saved, but in fact there was only one man aboard.
References:
Barwise, J. and R. Cooper. 1981. Generalized Quantifiers and Natural Language. Linguistics
and Philosophy 4: 159-219.
Crain, S and P. Pietroski. 2002. Why language acquisition is a snap. The Linguistic Review
19:163-183
Fiengo, R. 2007. Asking Questions: Using Meaningful Structures to Imply Ignorance. Oxford:
Oxford University Press.
Keenan, E. L. 2003. The definiteness effect: semantic or pragmatic? Natural Language
Semantics 11:187-216
Ladusaw, W. 1979. Negative Polarity as Inherent Scope. University of Texas, Austin: Ph.D.
dissertation.
van Rooy, R. 2003. Negative polarity items in questions: Strength as relevance. Journal of
Semantics 20: 239-273.