CAS LX 50211b. Summarizing the

fragment analysis, relativeclauses

(F3) S NP VP VP Vt NPS S ConjP VP ViConjP Conj S NP Det NC

S Neg S NP NP

VP Vbe NPpred NC Adj NC

NPpred Detdummy NC

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, hatesAdj boring, hungry

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

Detdummy a Vbe is

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

[boring]M,g = x [ x F(boring) ] (any NC or Adj 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 ][every]M,g = P [ Q [ xU [P(x) Q(x)] ] ][a]M,g = P [ Q [ xU [P(x) Q(x)] ] ][i]M,g = g(i) [is]M,g = —

[i]M,g = S [x [ [S]M,g[i/x] ] ] [DETdummy a]M,g = —

Pass-UpIf a node has only one daughter with a semantic value, , then []M,g = []M,g

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

or []M,g ( []M,g )

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

Predicate modification[ ]M,g = z [ []M,g(z) []M,g(z) ]where and are predicates

Once more from the top

• Our semantic system (the fragment) is designed to take a sentence and provide a characterization of the situations in which the sentence would be true.

• Bond is hungry• We determine whether this is true by comparing models

of the world, where a model is primarily a specification of the individuals in the world and their properties.

• This is something that people can do, and by concentrating on a small fragment of English, we are attempting to understand something about the mechanisms by which we compute this.

Models

• The model is based primarily on individuals (like Bond, Pavarotti, Loren, Nemo). An individual is a basic type of thing, we refer to it as <e> (generally writing types between pointed brackets).

• The model of the situation provides not only a list of the individuals in the situation, but also their properties in the situation. Some of them are hungry, some of them are boring. The model specifies who the boring individuals are and who the hungry individuals are.

Models

• We represent a property like hungry in the model by naming it (hungry) and listing the individuals for which the property holds.

• We use the notation F(hungry) to indicate the collection of individuals for which the named property holds. F is the denotation function in the situation.

• So in a particular situation, where F7 is the denotation function for that situation, we might have:

• F7(hungry) = {Bond, Pavarotti}.

Structures

• Sentences have hierarchical structures, parts that contain other parts. The sentence as a whole (S) contains a noun phrase (the subject) and a verb phrase, and in some cases the verb phrase may contain a verb and another noun phrase (the object).

• The syntactic base rules tell us what kinds of structures are valid sentences in this fragment (first slide; Bond likes Nemo can be assigned a structure and is valid, Bond and likes every cannot and is not).

Interpretation

• The semantic rules tell us, given a syntactic structure (tree), how we can determine the conditions under which the sentence is true.

• The most fundamental assumption is that meaning is compositional: that the meaning of the whole is derived from the meanings of its component parts (and how they are arranged). So, to determine the meaning (semantic value) of S, we need first to determine the meanings of its component parts (NP and VP).

• At the bottom of the tree (the leaves) are the lexical items, that have no component parts, but a basic meaning of their own.

Terminal nodes

• The “leaves” or terminal nodes can be divided into several categories.

• Some terminal nodes are simply proper names that denote specific individuals. So, Bond denotes the individual F(Bond), “the individual we refer to by the name Bond.” Individuals, again, we say have a basic type of <e>.

Terminal nodes

• Other terminal nodes denote properties (predicates) that can either hold or not hold of individuals. For example, hungry.

• Our semantic rules treat these as functions. You tell hungry who you’re talking about, it will tell you whether the individual is hungry or not. If the individual is hungry, the function returns “true”, if the individual is not hungry, the function returns “false”.

• The simplest kind of sentence would be something like Bond is hungry. This sentence would be true in any situation where F(Bond) is in the set F(hungry).

Sentences

• A sentence in this system is either true or false, depending on the situation. True and false are taken to be a different kind of thing than Bond is. We say that true and false are truth values, and we assign the basic type of <t> to truth values.

• Since the meaning of the sentence is derived from the meanings of its (meaningful) component parts, whether Bond is hungry is true or false depends on the meanings of Bond and hungry.

Sentences

• The meaning of Bond is F(Bond) (“the individual we refer to as Bond in the situation”), an individual, basic type <e>.

• The intuition is that the whole thing will be true depending on whether F(hungry) contains the individual F(Bond). So, we say that the meaning of hungry is such that, if you give it an individual that is in F(hungry), the result is true.

Predicates• The translation of hungry, then is that it will take an individual and

yield true or false:• [hungry]M,g = x [ x F(hungry) ]

• The x signifies that this is a function. It is “waiting for something to call x”. When it gets something, the result is whatever is in the brackets. We provide this function with an individual by writing the individual in parentheses after the function: x [ x F(hungry) ] ( F(Bond) )

• This means we’re giving the function what it is waiting for; it should call F(Bond) x. The result boils down to:• F(Bond) F(hungry)

• This could be true, it could be false; it depends on the situation (more specifically, it depends on F).

Alphabetic variants• One thing about functions like this is that it doesn’t matter

what we call the thing it is waiting for. We could call it x: x [ x F(hungry) ]

• We could call it y: y [ y F(hungry) ]

• We could call it Edward: Edward [ Edward F(hungry) ]

• We could call it : [ F(hungry) ]

• These are all the same function.

Interpreting sisters

• The basic principle behind our interpretation rules (once we get past the lexical items), is that when two nodes are sisters, the meaning of the unit containing the two is the result of providing one sister as the argument to the other. This is encoded as the Functional Application rule.

Types again• A function like y [ y F(hungry) ]

is one that is waiting for anindividual, and the thing inside thebrackets will be either true or falsedepending on the situation.

• We say that the function has type <e,t>: it is waiting for an <e> and when it gets it, the combination will be of type <t>.

• Because meaning is assigned using Function Application, we can also read these complex types as <sister-type, mother-type>. That is, even if we didn’t know already that hungry was <e,t>, we could deduce that it is given that the sister is type <e> and the mother is type <t>.

S <t>

Bond<e>

hungry<e,t>

Relations• Another category of lexical item wehave are relations. These are things likelike, and, or every which relate twomeanings (truthfully or not, dependingon the situation). Because our structuresonly ever have binary branches,relations need to proceed in two stages.

• And is waiting for a truth value(provided by S1), and providessomething that is waiting for a (second)truth value (provided by S2), the resultof which will be true iff both S1 and S2

are true.• [and]M,g = y [ x [ x y ] ]• [ConjP]M,g = y [ x [ x y ] ] ([S1]M,g) = x [ x [S1]M,g ]• [S3]M,g = x [ x [S1]M,g ] ([S2]M,g) = [S2]M,g [S1]M,g

S3 <t>

S2

<t>ConjP <t,t>

Conjand<t,<t,t>>

S1

<t>

Relations• Likes works the same way; it is waitingfor an individual (provided by Loren),and provides something that is waitingfor a (second) individual (provided byBond), the result of which will be trueiff Bond likes Loren in the situation;that is, <F(Bond), F(Loren)> F(likes).

• [likes]M,g = y [ x [ <x,y> F(likes) ] ]• [VP]M,g = y [ x [ <x,y> F(likes) ] ] ( F(Loren) )

= x [ <x, F(Loren)> F(likes) ]• [S]M,g = x [ <x, F(Loren)> F(likes) ] ( F(Bond) )

= <F(Bond), F(Loren)> F(likes)

S <t>

NP2

Bond<e>

VP <e,t>

Vtlikes<e,<e,t>>

NP1

Loren<e>

Relations• every is one step more complicated, but itsbasic structure is just likes and and and; it iswaiting for a predicate (provided by fish),and provides something that is waiting fora (second) predicate (provided by hungry),the result of which will be true iff for allindividuals, being a fish implies being hungry;that is, xU [xF(fish) xF(hungry)]

• (Note: et is shorthand for <e,t>)

• [every]M,g = P [ Q [ xU [P(x) Q(x)] ] ]• [NP]M,g = P [ Q [ xU [P(x) Q(x)] ] ] ( y[y F(fish)] )

= Q [ xU [ y[y F(fish)](x) Q(x)] ]= Q [ xU [ x F(fish) Q(x)] ]

• [S]M,g = Q [ xU [ x F(fish) Q(x)] ] ( y[y F(hungry)] )= xU [ x F(fish) y[y F(hungry)](x) ] = xU [ x F(fish) x F(hungry)]

S <t>

VPhungry<e,t>

<et,t> NP

Detevery<et,<et,t>>

NC

fish<e,t>

Object quantifiers

• When a quantificational NP like every book is in the object position, there is a type mismatch and without anything further, the structure can not be interpreted.

• [NP]M,g is waiting for <e,t>, but [Vt]M,g is not <e,t>.

• [Vt]M,g is waiting for <e>, but [NP]M,g is not <e>.

VP ???

Vtlikes<e,et>

NP <et,t>

Detevery<et,<et,t>>

NC

book<e,t>

Quantifier Raising

• Quantifier Raising (QR) is a solution to this problem.

• Among other things, it replaces the NP every book with tn for some index n (here, 6) the semantic value of which is an individual.

• [n]M,g = g(n)• [t6]M,g = g(6)

VP ???

Vtlikes<e,et>

NP <et,t>

Detevery<et,<et,t>>

NC

book<e,t>

VP <e,t>

Vtlikes<e,et>

t6 <e>

Quantifier Raising

• Quantifier Raising replaces the original position of every book with tn for some index n (here, 6), and puts every book up at the top of the structure.

• Notice that every book can’t simply be the sister to S. S is type <t>, and we would still have a type mismatch.

• We need the sister of every book to be type <e,t>. This is what the index (6) node provides. 6 converts a sentence<t> into a predicate <e,t>.

S2 ???

S1

…t6…<t>

<et,t> NP

Detevery<et,<et,t>>

NC

fish<e,t>

S2 <t>

S1

…t6…<t>

<et,t> NP

Detevery

<et,<et,t>>

NC

fish<e,t>

S <e,t>

6<t,et>

Quantifier Raising

• If the sentence is Bond likesevery fish, then QR resultsin S1 being Bond likes t6.

• [S1]M,g = < F(Bond), g(6) > F(likes)

• 6 converts this into a predicate (“something Bond likes”) by:• Putting S1 inside x[…]• Rewriting g(6) as x.

• [S]M,g = x [ < F(Bond), x > F(likes) ]• [S2]M,g says: being a fish implies being something Bond likes.

S2 <t>

S1

…t6…<t>

<et,t> NP

Detevery

<et,<et,t>>

NC

fish<e,t>

S <e,t>

6<t,et>

Quantifier Raising

• [S1]M,g = < F(Bond), g(6) > F(likes) • 6 converts this into a predicate

(“something Bond likes”) by:• Putting S1 inside x[…]• Rewriting g(6) as x.

• The way 6 does this is by guaranteeing that the pointing function will point to x for index 6. We use a pointing function that does that (namely g[6/x]) instead of using g. We know that (no matter what g is): g[6/x] (6) = x

• So, if we evaluate S1 with the pointing function g[6/x] instead of g, we’ll effectively replace all g(6)es with xes.

• Policy: From now on, you do not need to write out what [n]M,g is for any index n. Just write [S]M,g according to above (embed S1 in x[…] and replace g(n) with x).

S2 <t>

S1

…t6…<t>

<et,t> NP

Detevery

<et,<et,t>>

NC

fish<e,t>

S <e,t>

6<t,et>

Where to go from here

• That’s a summary of the interpretation system for the English fragment F3.

• There are a number of things that we have not yet incorporated into F3, if we want to have a more complete interpretation system.

• For example, tense. Bond was hungry is true under different conditions from Bond is hungry. Or Bond will have been hungry.• A common approach to this is to treat the time as another

part of the situation; i.e., [S]M,g,t is true iff S is true at time t in the situation M with “pointing function” g.

Where to go from here

• Another thing we might add are verbs like say, think, believe, where the “object” of these verbs is another (embedded) sentence:• Bond believes that Loren likes Nemo.• Bond said that every fish is hungry.

• Notice that Bond believes that Loren likes Nemo can be true even if Loren likes Nemo is false. We need to evaluate Loren likes Nemo with respect to Bond’s “beliefs” and not with respect to the actual world, again something we could add to the “situation”: [S]M,g,t,w is true iff S is true at time t in the situation M in possible world w with “pointing function” g.

Where to go from here

• When we start dealing with “Bond’s beliefs” and possible worlds in general, we can start considering a formalization of modals:• Bond must like Nemo.• Bond might be hungry.

• Bond might be hungry will be true iff in some possible world w (in the set of worlds comprising the modal base), [S]M,g,t,w is true. Bond must like Nemo will be true iff in all possible worlds w (in the modal base), [S]M,g,t,w is true.

Where to go from here

• As hinted at the end of the last handout, we can also add relative clauses to our fragment, allowing for:• The fish that likes Bond is hungry.• A man that every fish likes likes the woman

that likes a book a fish likes.• We’ll take this last step, as a final

augmentation of our fragment.

Relative clauses

• A relative clause acts something like an adjective:• a happy fish• A fish that Bond likes

• The first holds of an x such that x is a fish and x is happy.

• The second holds of an x such that x is a fish and Bond likes x.

Forming relative clauses

• The way we will form relative clauses is to effectively take a sentence:• Bond likes every fish

• And turn it into a noun phrase:• Every fish that Bond likes

• Which can then go anywhere an NP can go.

• To do this, we are going to need another transformation, like Quantifier Raising.

The base structure

• We start by saying that we can expand NP not only to NP or to Det NC, but to Rel S:• NP NP

• NP Det NC

• NP Rel S• Rel that

Relative clauses• The goal is to

get: Every fish that is hungry is happy.

S

NP

VP

Vbe

is

S

Adj

hungry

Rel

that NP

Det

every

NC

fish

VP

Vbe

is

Adj

happy

Relative clauses

• Our existing interpretation rules can interpret this structure, once we add:

• [that]M,g = —

S

NP

VP

Vbe

is

S

Adj

hungry

Rel

that t4

VP

Vbe

is

Adj

happy

NP

Det

every NC

fish

NC

4

N

Relative clauses

• The relative clause formation rule (above) must apply whenever the conditions are met.

S

S

VP

Vbe

is

S

Adj

hungry

Rel

that t4

VP

Vbe

is

Adj

happy

NP

Det

every NC

fish

NC

4

S

[NP that [S X [NP Det NC] Y ] ][NP Det [Nc NC [S i [S that [S X ti Y ] ] ] ] ]

Homework 8• Homework 8: Due Tuesday 4/6.• 1. Work out the truth conditions for Every fish that is

hungry is happy using the structure on the previous slide.• 2. Draw the structure for Every hungry fish is happy and

work out the truth conditions for it. They should end up being the same as those you got in part 1 for Every fish that is hungry is happy.

• Note: You do not need to explicitly write the semantic value of index nodes (like 4 on the preceding tree). See “policy” a few slides back.