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Albert GattLIN3021 Formal SemanticsLecture 9In this lectureNoun phrases as generalised quantifiers: some further concepts

What weve said so farIn a sentence of the form [[Det NP]subj VP] :The determiner itself is a function from sets (N) to sets (V) to truth valuesThe NP subject (once the N combines with Det) is a function from sets to truth valuesThis 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]](et) t[[walk]](et)(the set of walkers)What weve said so farSo 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.The model-theoretic interpretationSuppose 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 walkBut not of skip or fishSo 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.

walkskipfishTomYYNDickYNNharryyyYIn graphicswalkskipfishTomYYNDickYNNharryyyYTomHarryDickwalkfishskipEvery man:Contains walk but not skip or fishPart 1Universal restrictions on Natural Language QuantifiersGQs and semantic universalsOne 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?Observation 1: conservativityWhen 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 Observation 1: conservativityObserve that:Every man walks is true iff every man is a man who walksFew 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]]

Observation 1: conservativityTomHarryDickwalkfishskipEvery man:Contains walk but not skip or fishWe 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)

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 smokeAllnon students smoke =/= allnon students are students who smoke

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?)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 walksOnly athletes run VS athletes only run*the every man VS the only man

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

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]]Observation 3: QuantityWeve 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 U2there 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 M2Observation 3: QuantityWe can specify this by assuming that there is a mapping from each individual N in U1 to each individual N in U2.Tom

Dick

HarryU1Jake

Roger

KeesU2If 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 U2Observation 3: QuantityIf 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 ChomskyPossessives 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.Part 2Generalised quantifiers and negative polarity itemsNegative polarity itemsSome 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.Negative polarity itemsBut 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 etcNegative polarity itemsNPIs 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.

Negative polarity itemsTheres 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.An aside about entailmentsRecall 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 manEntailment from a property (human) to a sub-property (man)Downward entailment (general to specific)

Notice that negation reverses the entailment with hyponymsHow quantifiers come into the pictureA 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.Every N VEvery human being walksFirst 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.

Some N VSome human being walksFirst 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.

Three N VThree human beings walkFirst 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.

No N VNo human beings walkFirst 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.

SummaryFirst PropertySecond PropertyDEUEDEUEEveryYNNYNoYNYNSomeNYNYThreeNYNYNPIs are licensed in DE contextsFirst PropertySecond PropertyDEUEDEUEEveryYNNYNoYNYNSomeNYNYThreeNYNYEvery 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.Important things to noteGeneralised 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 NPIsWhat 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?