CAS LX 50210b. Binding

Syntactic base rules (F2)S NP VP VP Vt NP

S S ConjP VP Vi

ConjP Conj S NP Det NC

S Neg S NP NP

Det the, a, every NP Pavarotti, Loren, Bond, Nemo, Dory, Blinky, Semantics, The Last Juror, hen, shen, itn, himn, hern, himselfn, herselfn, itselfn.

Conj and, orVt likes, hatesVi is boring, is hungry

Neg it is not the case that NC book, fish, man, woman

[Pavarotti]M,g = F(Pavarotti) (any NP)

[is boring]M,g = x [ x F(is boring) ] (any NC or Vi)

[likes]M,g = y [ x [ <x,y> F(likes) ] ] (any Vt)

[and]M,g = y [ x [ x y ] ] (analogous for or)

[it is not the case that]M,g = x [ x ][i]M,g = g(i)

[i]M,g = S [x [ [S]M,g[i/x] ] ][every]M,g = P [ Q [ xU [P(x) Q(x)] ] ][a]M,g = P [ Q [ xU [P(x) Q(x)] ] ]

Pass-Up[ ]M,g = []M,g

Functional application[ ]M,g = []M,g ( []M,g ) or []M,g ( []M,g )

whichever is defined

Quantifier Raising[S X NP Y ] [S NP [S i [S X ti Y ] ] ]

Binding

• Among our NPs, we have a couple of special types:• –self anaphora: himselfn, herselfn, itselfn

• pronouns: himn, hern, itn, hen, shen.

• Unlike Bond, Loren, Pavarotti, these NPs don’t denote a specific individual in U, regardless of the model; they “pick up their reference” from elsewhere.

Binding

• We think of the denotation of these elements as dependent—their reference is provided by something else.

• In the case of pronouns, it is provided by “pointing” (courtesy of the assignment function g).

• With –self anaphora, the reference of …self is bound to the reference of another (generally preceding) NP.

• Bond1 likes himself1.

He1 is hungry

1

Bond1 likes himself1

• It is common, when discussing reference of pronouns and –self anaphora to write sentences with a subscript on each NP under consideration.

• What Bond1 likes himself1 really means is:• The NPs Bond and himself both point to the same

individual (hence the coindexation).• Bond intrinsically names a particular individual.• Himself has no intrinsic reference, it’s all in the

connections (via the index) to other NPs.• Therefore, himself also points to the individual Bond

points to.

Bond1 likes himself1

• This is a useful notation for describing the intended meanings of sentences, but this is not something F2 can give us.• NP Bond• NP himselfn

• The intended meaning of NP himselfn is that NP can be rewritten as himselfn for some index n. Doesn’t matter which, pick one. In fact, pick 1.

• Notice the lack of any NP Bondn rule.

Every fish likes itself

• In fact, –self anaphora (and pronouns too) can pick up their reference from quantifiers like every fish too. We might write this informally as:

• Every fish1 likes itself1.• …although this is just as un-directly-generable by F2

as Bond1 likes himself1 was.

• An example of a bound pronoun, slightly outside of F2, would be every fish1 thinks that Loren likes him1.

Pavarotti likes himself

• Our syntactic rules allow us to build and evaluate this sentence, but with a twist. Let’s see.

S

NP VP

S NP VP

Pavarotti likes himself

• Our syntactic rules allow us to build and evaluate this sentence, but with a twist. Let’s see.

S

NP VP

Vt NP

S NP VPVP Vt NP

Pavarotti likes himself

• Our syntactic rules allow us to build and evaluate this sentence, but with a twist. Let’s see.

S

NP VP

VtNP NP

NP

S NP VPVP Vt NPNP NP

Pavarotti likes himself

• Our syntactic rules allow us to build and evaluate this sentence, but with a twist. Let’s see.

S

NP VP

Vt

likes

NP

Pavarotti

NP

NP

S NP VP Vt likesVP Vt NP NP Pavarotti

NP NP

Pavarotti likes himself

• NP himselfn

• The subscript n means: pick an index, any index.

• Let’s pick 3.

S

NP VP

Vt

likes

NP

Pavarotti

NP

NP

himself3

S NP VP Vt likesVP Vt NP NP Pavarotti

NP NP NP himselfn

Pavarotti likes himself

• What should this mean?

S

NP VP

Vt

likes

NP

Pavarotti

NP

NP

himself3

Pavarotti likes himself

• What should this mean?

• So, we need himself3 to denote Pavarotti.

• Our interpretation rules say it refers to g(3)—whoever “3” points to.

S

NP VP

Vt

likes

NP

Pavarotti

NP

NP

himself3

[i]M,g = g(i)[himself3]M,g = g(3)

Pavarotti likes himself

• In less formal circumstances, this interpretation is written like:

Pavarotti3 likes himself3.

Where we give the proper name Pavarotti a matching index.

• In F2, we can’t give Pavarotti a subscript. First, no rule provides for that. Second, the interpretation of anything3 is g(3). We lose the connection to Pavarotti. Right?

S

NP VP

Vt

likes

NP

Pavarotti

NP

NP

himself3

[i]M,g = g(i)[himself3]M,g = g(3)

Pavarotti likes himself

• So that’s the puzzle: How can Pavarotti still refer to this guy and simultaneously bind the anaphor himself3 by matching its index?

• It turns out, there’s a straightforward way to do this, and we can do it with the F2 we already have.

S

NP VP

Vt

likes

NP

Pavarotti

NP

NP

himself3

[i]M,g = g(i)[himself3]M,g = g(3)

Pavarotti likes himself

• The answer? QR.

• All that QR says is: findan NP inside an S, pickan index, attach the indexand the NP to the top, andreplace the original NPwith a t that shares the index.Pavarotti is an NP. So, hey, why not?

• Quantifier Raising[S X NP Y ] [S NP [S i [S X ti Y ] ] ]

S

t3 VP

Vt

likes

NP

NP

himself3

S

S

3

NP

NP

Pavarotti

Pavarotti likes himself

• Using a little foresight, we’ll pick 3 for our index when we do QR as well.

• So, what does that leave us with?

• [i]M,g = g(i)• So, [himself3]M,g and [t3]M,g

both = g(3).• Using the rules as usual, the

lower [S]M,g is true if g(3) likes g(3).

S

t3 VP

Vt

likes

NP

NP

himself3

[S]M,g = <g(3), g(3)> F(likes)<t>

S

S

3

NP

NP

Pavarotti

Pavarotti likes himself

• We have a node that is true if g(3) likes g(3).

• We want to combine it with the index node 3.

• We want the result to be the property of liking oneself.

• We can then attribute to Pavarotti the property of liking oneself, and the sentence will be true if Pavarotti likes himself.

S

t3 VP

Vt

likes

NP

NP

himself3

[S]M,g = <g(3), g(3)> F(likes)<t>

S

S

3

NP

NP

Pavarotti

Pavarotti likes himself

• [i]M,g = S [x [ [S]M,g[i/x] ] ]• [3]M,g = S [x [ [S]M,g[3/x] ] ]

• We know that[S]M,g = <g(3), g(3)> F(likes)

• What if we evaluate that not with the assignment function g but with g[3/x] instead? That is, with an assignment function where 3 points to x?• g[3/x] (1) = g(1);• g[3/x] (2) = g(2);• g[3/x] (3) = x;• g[3/x] (4) = g(4); …

S

t3 VP

Vt

likes

NP

NP

himself3

[S]M,g = <g(3), g(3)> F(likes)<t>

S

S

3

NP

NP

Pavarotti

We know this for sure;that’s what g[3/x] means.

Pavarotti likes himself

• [i]M,g = S [x [ [S]M,g[i/x] ] ]• [3]M,g = S [x [ [S]M,g[3/x] ] ]

• We know that[S]M,g = <g(3), g(3)> F(likes)

• [S]M,g[3/x] =

<g[3/x] (3), g[3/x] (3)> F(likes) =

<x, x> F(likes)

S

t3 VP

Vt

likes

NP

NP

himself3

[S]M,g = <g(3), g(3)> F(likes)<t>

S

S

3

NP

NP

Pavarotti

Pavarotti likes himself

• [i]M,g = S [x [ [S]M,g[i/x] ] ]• [3]M,g = S [x [ [S]M,g[3/x] ] ]

• [S]M,g[3/x] = <x, x> F(likes)

• So, now, we can figure out what [S]M,g is:[3]M,g is a function, it takes an S as its argument.

[3]M,g ( [S]M,g ) = x [ [S]M,g[3/x] ] =

x [<x, x> F(likes) ]

S

t3 VP

Vt

likes

NP

NP

himself3

[S]M,g = x [ <x, x> F(likes) ]<e,t>

S

S

3

NP

NP

Pavarotti

Pavarotti likes himself

• A point of pedantic clarification:

• Strictly speaking, [S]M,g is theargument, so in…[3]M,g = S [x [ [S]M,g[3/x] ] ]…S is really going to be [S]M,g.

• So, what we’re evaluating is really[ [S]M,g ]M,g[3/x]. We take [S]M,g tomean the value of [S] under the current model and assignment function. If we are evaluating the whole sentence under M,g, that’s the current model and assignment function. But by evaluating [S]M,g under M,g[3/x], we’ve changed the “current assignment function” inside the brackets to be g[3/x]. So,this is effectively the same as [ [S]M,g [3/x]]M,g[3/x]. You can safely ignore this technicality, and just do things as they were done on the previous slide. This is here only for completeness.

S

t3 VP

Vt

likes

NP

NP

himself3

[S]M,g = x [ <x, x> F(likes) ]<e,t>

S

S

3

NP

NP

Pavarotti

Pavarotti likes himself

• [S]M,g = x [<x, x> F(likes) ]

• Perfect, now we’re set.

• Combining the <e,t> functionrepresented by [S]M,g andthe <e> individual representedby the top NP, we have:

• [S]M,g = [S]M,g ( [NP]M,g ) =x [<x, x> F(likes) ] ( F(Pavarotti) ) =<F(Pavarotti), F(Pavarotti) > F(likes)

S

t3 VP

Vt

likes

NP

NP

himself3

[S]M,g = x [ <x, x> F(likes) ]<e,t>

S

S

3

NP

NP

Pavarotti

Every manlikes himself

• What’s more, we can easily now interpret Every man likes himself. Same structure, only now with every man instead of Pavarotti. We perform QR just as we did, we interpret every node up to the S just as we did—so [S]M,g is the property of liking oneself.

• [every]M,g takes first the predicate man, then the predicate likes oneself, and the sentence is true if being a man implies liking oneself.

S

t3 VP

Vt

likes

NP

NP

himself3

[S]M,g = x [ <x, x> F(likes) ]<e,t>

S

S

3

NP

NC

man

Det

every

Every manlikes himself

• [every]M,g = P [ Q [ xU [P(x) Q(x)] ] ]

• [NC]M,g = x [ x F(man) ]

• [NP]M,g = [Det]M,g ( [NC]M,g )

• In terms of the definition ofevery, we will replace P with [NC]M,g.

• We also need P(x), which will be [NC]M,g (x), orx [ x F(man) ] (x),so (replacing xes with xes),P(x) is just x F(man).

• [NP]M,g = Q [ xU [x F(man) Q(x)] ]

S

t3 VP

Vt

likes

NP

NP

himself3

[S]M,g = x [ <x, x> F(likes) ]<e,t>

S

S

3

NP

NC

man

Det

every

Every manlikes himself

• [NP]M,g = Q [ xU [x F(man) Q(x)] ] <<e,t>,t>

• Now, the last step, combiningthe NP and the S.

• [S]M,g is the predicatelikes oneself, type <e,t>.

• [NP]M,g needs a predicate to call Q. So, we replace Q with [S]M,g. We also need Q(x), which is x [ <x, x> F(likes) ] (x), or (replacing xes with xes), <x, x> F(likes) .

• [S]M,g = [NP]M,g ( [S]M,g ) =xU [x F(man) <x, x> F(likes)]

S

t3 VP

Vt

likes

NP

NP

himself3

[S]M,g = x [ <x, x> F(likes) ]<e,t>

S

S

3

NP

NC

man

Det

every

Himselfn, herselfn, itselfn

• Additional restrictions on –self anaphors:• Himself must denote a man, herself must denotes a

woman, itself must denote something that is neither a man nor a woman.• *Pavarotti3 likes herself3.

• There is also a restriction that the thing that has a matching index must be “nearby” and “higher in the tree”:• *Pavarotti3 is hungry and Loren likes himself3.• *Himself3 likes Pavarotti3.

Binding theory• The “nearby” condition, approximately:

• A –self anaphor must have an index that matches the index of something within the smallest S that contains it.

• LX522: Binding theory, Principle A.• The “higher in the tree” condition, approximately:

• The node with an index matching a –self anaphor must be the sister of a node that contains the –self anaphor.

• LX522: The binder must c-command the –self anaphor.• You can verify for yourself that these conditions hold, and

you’ll hear more about them in LX522. These are not semantic conditions—our system can interpret sentences violating these conditions just fine. These are syntactic.

Pronouns vs. –self anaphors• Often, when you want co-reference but can’t use a –self

anaphor, you can use a pronoun instead:• Pavarotti3 is hungry and Loren likes him3.• *Pavarotti3 likes him3.

• F2 does not distinguish between pronouns and –self anaphors, the interpretation is the same: [him3]M,g = [himself3]M,g = g(3).

• So, where a pronoun is appropriate, where a –self anaphor is appropriate, these are governed by separate constraints on what structures are valid as structures—syntactic constraints.

VP anaphora

• In a manner somewhat similar to the way herself picks up the referent of a higher NP, English has a phrase do too that seems to pick up the denotation of a preceding VP.

• Loren likes Bond and Pavarotti does too.• The VP in the first sentence is a predicate, meaning is a

Bond-liker: x [ <x, F(Bond)> F(like) ]• The VP in the second sentence, does too, is interpreted as

if it were just the same as the VP in the first sentence:• Loren is a Bond-liker and Pavarotti is a Bond-liker.

VP anaphora

• Every man likes Loren and Nemo does too.• Being a man implies being a Loren-liker, and

Nemo is a Loren-liker.

• A fish likes every book.• There is a fish z such that for every book x, z likes x• For every book x, there is a fish z such that z likes x

A fish likes every book

• Every book is a quantifier in object position (type <<e,t>,<<e,t>,t>>) and must undergo QR in order to be interpretable.

• A fish can either undergo QR or not. If it does not, we have the one-uncritical-fish interpretation. If it does, we have the to-each-its-own interpretation.

and Loren does too• A fish likes every book.

• [A fish [ 1 [every book [2 [t1 likes t2]]]]• [every book [ 2 [a fish likes t2 ]]]

• Loren likes every book.• [every book [ 2 [Loren likes t2 ]]]

• A fish likes every book and Loren does too.

• Parallelism: Where a VP is anaphoric to another, the two sentences must have a parallel structure.

• Scope economy: Only do QR if it makes a difference in the meaning.

And we’re done!We’ve described English!

• Well, not exactly. Actually, there are plenty of English sentences we don’t have a formal interpretation procedure for yet.

• Here’s one:

• Most fish are boring.

Most fish are boring

• The first steps are simple enough. Most looks like it is a determiner like every, so we can add Det mostto our syntactic base rules.

• And it seems to mean something kind of similar. It should take two predicates and return a truth value; that is, most takes fish then are boring and says something about individuals for which fish is true being individuals for which is boring is true too.

Most fish are boring

• Every says: given predicates P and Q, for each individual x in U, P(x) Q(x).

• Most is less than every, so maybe:• Given predicates P and Q, for over half of the individuals x in U,

P(x) Q(x).• That was easy. But wait.• What if there are 2 fish in U and 47 books? Certainly if x is a

book, fish(x) Q(x) is true for any Q, given that falseanything is true.

• If that’s the case, then every fish is happy can be true, even while most fish are unhappy is true (after all, 47 of 49 xes are such that being a fish implies being unhappy). Hmm.

Most fish are boring

• What we need is to be checking only the fish, not all of the individuals in the universe. We don’t care about non-fish when we are evaluating most. Of the fish, are most such that they are also boring?

Thinking in terms of sets

• A predicate like is boring or fish defines a set, the set of boring individuals, or the set of fish.

• In fact, the F function already gave us that set, and our interpretation rule turned it into a function.

• [is boring]M,g = x [ x F(is boring) ]

Most fish are boring

• When we think interms of sets, whatmost fish are boringis saying is that theboring fish outnumberthe non-boring fish.• The boring fish are

individuals in bothF(fish) and F(is boring).

• The nonboring fish areindividuals in F(fish)but not in F(is boring).

• Most(A,B) = |AB| > |A-B|U

F(fish)F(is boring)

Most fish are boring

• For predicates P and Q,• P = {x: P(x)}• Q = {x: Q(x)}• P Q = {x: P(x) Q(x)}• P - Q = {x: P(x) Q(x)}

• Most(A,B) =|AB| > |A-B|

• So, we can write most as:• [most]M,g =

P [ Q [ | {x: P(x) Q(x) } | > | {x: P(x) Q(x) } | ] ]

U

F(fish)F(is boring)