Albert Gatt LIN3021 Formal Semantics Lecture 9. In this lecture Noun phrases as generalised...
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- Slide 1
- Albert Gatt LIN3021 Formal Semantics Lecture 9
- Slide 2
- In this lecture Noun phrases as generalised quantifiers: some
further concepts
- Slide 3
- What weve said so far In a sentence of the form [[Det NP] subj
VP] : The determiner itself is a function from sets (N) to sets (V)
to truth values The NP subject (once the N combines with Det) is a
function from sets to truth values This function takes as argument
a set-denoting expression (namely, the denotation of the VP) It
returns a truth value (the entire construction is a proposition)
[[every man]] (e t) t [[walk]] (e t) (the set of walkers)
- Slide 4
- What weve said so far So the GQ (the NP) in a sentence of the
form [[Det NP] subj VP] can be thought of as a set of sets (a
property of properties). Were thinking of every man as saying: the
property it combines with (walks) describes every man in our
model.
- Slide 5
- The model-theoretic interpretation Suppose there are three men:
Tom, Dick and Harry. Lets think of every man as describing the set
of all the properties that belong to each and every one of them In
this model: Every man does describe the (extension of) the property
walk But not of skip or fish So every man walks is true just in
case the property (set) walk is a member of the set of things
described by every man. Alternatively, its true just in case every
man is indeed a man who walks. walkskipfish TomYYN DickYNN
harryyyY
- Slide 6
- In graphics walkskipfish TomYYN DickYNN harryyyY Tom Harry Dick
walk fish skip Every man: Contains walk but not skip or fish
- Slide 7
- Part 1 Universal restrictions on Natural Language
Quantifiers
- Slide 8
- GQs and semantic universals One of the questions that
semanticists have asked is: What are the properties of Natural
Language quantifiers, and what restrictions can there be on the way
languages express quantified NPs? Are there generalisations that we
can make which apply cross- linguistically?
- Slide 9
- Observation 1: conservativity When we interpret a GQ of the
form [Det NP] in a model, we are normally not interested in the
whole of U. To verify every man runs, were only interested in the
men, not in the women. (Whether the women run or not is
irrelevant.) One of the properties that GQs apparently have is
conservativity
- Slide 10
- Observation 1: conservativity Observe that: Every man walks is
true iff every man is a man who walks Few athletes take drugs is
true iff few athletes are athletes who take drugs So in [[DET N]
V], it is always [[N]] that sets the scene, and were interested in
that part of [[V]] which intersects with [[N]] What the examples
suggest is that [[Det N] subj V] is being interpreted as a relation
between [[N]] and [[V]], and the property [[N]] is being carried
over into the interpretation of [[V]]
- Slide 11
- Observation 1: conservativity Tom Harry Dick walk fish skip
Every man: Contains walk but not skip or fish We could make this
generalisation (assuming a model, as usual): If N, V are subsets of
U then: DET(N)(V) is true iff DET(N)(N V)
- Slide 12
- What does conservativity tell us? Barwise and Cooper (1981)
hypothesise the following universal: Every NL Quantifier is
conservative. Lets imagine a quantifier allnon, where allnon N
means everything which is not an N. Therefore: allnon(N)(V) is true
iff (U-N) V (Note: this is a bit like every, but every N V would
require that N V) This quantifier violates conservativity. Compare:
All students smoke = all students are students who smoke Allnon
students smoke =/= allnon students are students who smoke
- Slide 13
- What does conservativity tell us? The example of allnon
suggests that: Its not difficult to think of determiners that
violate conservativity. Therefore, the conservativity universal
makes a very strong empirical claim, one that has surprising
consequences. (Why should quantifiers like allnon be excluded from
Natural Languages?)
- Slide 14
- What does conservativity tell us? One consequence of
Conservativity (if its true) is that some things frequently
regarded as determiners might not be determiners after all. Only
athletes run. We cant verify this only by taking into account the
intersection of athletes and running things. We need to check that
it is only athletes that run, so we need to look at non- running
and non-athlete things as well. So only violates Conservativity.
But maybe only isnt really a determiner at all! Note that this
semantic argument is supported by syntactic distribution: Every man
walks. VS *man every walks Only athletes run VS athletes only run
*the every man VS the only man
- Slide 15
- Observation 2: Extension (constancy) The truth of the
proposition every woman sneezes in a model M = is independent of
the size of U. If we extend our model to U, by adding non-woman and
non-sneezing things to U, the truth is unaffected. So we can ignore
U-[[N]] and U-[[V]] in our interpretation. So: If [[N]], [[V]] are
subsets of U and U is a subset of U then: DET(N)(V) is true in U
iff DET(N)(V) is true in U
- Slide 16
- Observation 2: Extension (constancy) If [[N]], [[V]] are
subsets of U and U is a subset of U then: DET(N)(V) is true in U
iff DET(N)(V) is true in U But not all determiners pass this test
(its not a universal property). Some determiners are very sensitive
to the size of the domain: Many women sneeze = many things which
are women sneeze (and some women dont) = the number of women who
sneeze is greater than the number of other (non- women) things that
sneeze. On this interpretation, we do need to consider U-[[N]] and
U-[[V]]
- Slide 17
- Observation 3: Quantity Weve said that extending the domain U
to include things not in [[N]] and [[V]] should have no effect on
[[DET N] V] With some determiners, we do care about how many Ns and
Vs there are: Several men walk does require us to consider the
number of things in [[man]]. Note: it doesnt matter who the men
are! Suppose: we have two models M1 and M2 with domains U1 and U2
there are equal numbers of men in U1 and U2 (i.e. For every man in
U1, there is a corresponding man in U2) The models are otherwise
identical (including the interpretation of predicates) then the
truth of several men walk should stay the same in the two models,
if we have a mapping from each man in M1 to a man in M2
- Slide 18
- Observation 3: Quantity We can specify this by assuming that
there is a mapping from each individual N in U1 to each individual
N in U2. Tom Dick Harry U1 Jake Roger Kees U2 If its possible to
map each N in U1 to each N in U2 and vice versa (we dont care about
identity of the men), then we can determine that we have equally
sized sets. We specify a bijection F: U1 to U2, s.t. For each x in
U1, there is a unique F(x) in U2
- Slide 19
- Observation 3: Quantity If F is a bijection from U1 to U2,
then: DET(N)(V) is true in M1 iff DET(F(A))(F(B)) is true in M2
Some so-called determiners in English dont satisfy Quantity. Every
students book is by Chomsky Possessives like every students are
often viewed as determiners (syntactically, they seem to behave the
same) The problem here is that interpreting every students book
needs to take into account the individual, specific relation
between a student and their book.
- Slide 20
- Part 2 Generalised quantifiers and negative polarity items
- Slide 21
- Negative polarity items Some expressions seem to be biased
towards negative shades of meaning: Nobody has ever been there. No
person in this room has any money. *The people have ever been
there. *Jake has any money.
- Slide 22
- Negative polarity items But in some non-negative contexts, an
NPI seems ok: *The people have ever been there. Have people ever
been there? If people have ever been there, Ill be very surprised.
*Jake has any money. Has Jake got any money? If Jake has any money
he should pay for his own drinks. The relevant contexts include
questions, if-clauses etc
- Slide 23
- Negative polarity items NPIs are also ok with certain
generalised quantifiers, but not others. *The people have ever been
there. Every person whos ever been here was stunned. *Some person
whos ever been here was stunned. *Jake has any money. Every man who
has any money should buy a round. *Some man who has any money
should buy a round.
- Slide 24
- Negative polarity items Theres a difference between an NPI
within the GQ and an NPI in the VP: NPI inside quantifier: Every
man who has ever met Mary loved her. No man who has ever met Mary
loved her. *Some man who has ever met Mary loved her. *Three men
who have ever met Mary loved her. NPI inside predicate *Every man
has ever met Mary. No man has ever met Mary. *Some man has ever met
Mary. *Three men have ever met Mary loved her.
- Slide 25
- An aside about entailments Recall the definition of hyponymy: I
am a man I am a human being. Entailment from a property (man) to a
super-property (human) Upward entailment (specific to general) I am
not a human being I am not a man Entailment from a property (human)
to a sub-property (man) Downward entailment (general to specific)
Notice that negation reverses the entailment with hyponyms
- Slide 26
- How quantifiers come into the picture A quantifier is doubly
unsaturated: it combines with two properties: (DET N) V Were
interested in how quantifiers behave with respect to upward and
downward entailments.
- Slide 27
- Every N V Every human being walks First property (inside the
GQ) DE ok Every man walks. UE blocked: Every animal walks. Second
property (VP): DE blocked: Every human being walks fast. (not
downward entailing) UE ok Every human being moves.
- Slide 28
- Some N V Some human being walks First property (inside the GQ)
DE blocked: Some man walks. UE OK: Some animal walks. Second
property (VP): DE blocked: Some human being walks fast. UE ok: Some
human being moves.
- Slide 29
- Three N V Three human beings walk First property (inside the
GQ) DE blocked: Three men walk. UE OK: Three animals walk. Second
property (VP): DE blocked: Three human beings walk fast. UE ok:
Three human beings move.
- Slide 30
- No N V No human beings walk First property (inside the GQ) DE
Ok: No men walk. UE blocked: No animals walk. Second property (VP):
DE OK: No human beings walk fast. UE blocked: No human beings
move.
- Slide 31
- Summary First PropertySecond Property DEUEDEUE EveryYNNY NoYNYN
SomeNYNY ThreeNYNY
- Slide 32
- NPIs are licensed in DE contexts First PropertySecond Property
DEUEDEUE EveryYNNY NoYNYN SomeNYNY ThreeNYNY Every man who has ever
seen Mary loved her. *Every man has ever seen Mary. No man who has
ever seen Mary loved her. No man has ever seen Mary. *Some man who
has ever seen Mary loved her. *Some man has ever seen Mary. *Three
men who have ever seen Mary loved her. *Three men have ever seen
Mary.
- Slide 33
- Important things to note Generalised quantifiers have upward or
downward entailments. This depends on the quantifier itself. The
entailments apply in the NP and/or the VP (the fact that the
quantifier has entailments in both suggests were correct in
adopting the generalised quantifier analysis, where the quantifier
is a relation between N and V) The entailments of quantifiers seem
to play a causal role in where they are allowed to appear in
combination with NPIs
- Slide 34
- What about other languages? Can you think of the DE/UE
entailments in languages such as Italian and English? What are the
counterparts of NPIs in these languages? Are they licensed in the
same way? Could we claim that this is a semantic universal?