CAS LX 502CAS LX 502SemanticsSemantics

3a. A formalism for meaning 3a. A formalism for meaning (cont’d)(cont’d)

3.2, 3.63.2, 3.6

RecapRecap ““F1” = Rules for F1” = Rules for generatinggenerating and and

interpretinginterpreting a small fragment of English. a small fragment of English. Syntax: Phrase structure rulesSyntax: Phrase structure rules

Reviewed on the next slideReviewed on the next slide Idea: All and only sentences generated by the PS rules Idea: All and only sentences generated by the PS rules

are part of the language (F1, approximating English).are part of the language (F1, approximating English). Interpretation: [ ]Interpretation: [ ]MM

Goals:Goals: Assign an interpretation to every node in the structureAssign an interpretation to every node in the structure Arrive at the interpretation compositionallyArrive at the interpretation compositionally

Interpretation is assigned with respect to a model Interpretation is assigned with respect to a model (effectively, the facts about the world: The players [U] (effectively, the facts about the world: The players [U] and their properties [F]).and their properties [F]).

F1: The syntaxF1: The syntax

Phrase Structure rules (the syntax):Phrase Structure rules (the syntax): To be revised…To be revised…

S S N VP N VP N N Pavarotti, Loren, Bond Pavarotti, Loren, Bond

S S S conj S S conj S Vi Vi is boring, is hungry, is is boring, is hungry, is cutecute

S S neg S neg S Vt Vt likes likes

VP VP Vt N Vt N Conj Conj and, or and, or

VP VP Vi Vi Neg Neg it is not the case that it is not the case that

Using the syntax of F1Using the syntax of F1 Starting with Starting with SS, we can , we can

“rewrite it” using the rules “rewrite it” using the rules of the syntax until we get of the syntax until we get to a structure such as this to a structure such as this one.one. S S N VP N VP N N Pavarotti Pavarotti VP VP Vi Vi Vi Vi is boring is boring S S N VP N VP

What is the interpretation What is the interpretation of S? Put another way, of S? Put another way, what is [S]what is [S]MM??

S

N VP

Vi

is boring

Pavarotti

The interpretation of SThe interpretation of S We developed a semantic We developed a semantic

rule that tells us what the rule that tells us what the interpretation of [interpretation of [SS N VP] is: N VP] is: [[SS N VP] N VP]MM = true iff [N] = true iff [N]MM

[VP][VP]MM

Great, are we done? Well, Great, are we done? Well, we would be, if we knew we would be, if we knew what [N]what [N]MM and [VP] and [VP]MM were. were.

What’s [N]What’s [N]MM?? Since meaning is Since meaning is

compositional and N does compositional and N does not branch, [N]not branch, [N]MM is the is the same as [Pavarotti]same as [Pavarotti]MM..

So, what’s [Pavarotti]So, what’s [Pavarotti]MM??

S

N VP

Vi

is boring

Pavarotti

The interpretation of SThe interpretation of S So far:So far:

[[SS N VP] N VP]MM = true iff [N] = true iff [N]MM [VP] [VP]MM

[N][N]MM = [Pavarotti] = [Pavarotti]MM

What’s [Pavarotti]What’s [Pavarotti]MM?? We have a semantic rule that We have a semantic rule that

tells us that:tells us that: [Pavarotti][Pavarotti]MM = F(Pavarotti) = F(Pavarotti)

That is, the interpretation of a That is, the interpretation of a name is the individual from the name is the individual from the model M that the “pointing” (or model M that the “pointing” (or “naming”) function F designates.“naming”) function F designates.

F(Pavarotti) in this model is the F(Pavarotti) in this model is the individual PAVAROTTI.individual PAVAROTTI.

So [Pavarotti]So [Pavarotti]MM = PAVAROTTI. = PAVAROTTI. So [N]So [N]MM = PAVAROTTI. = PAVAROTTI.

S

N VP

Vi

is boring

Pavarotti

[Pavarotti]M =F(Pavarotti) =PAVAROTTI

The The interpretation interpretation

of Sof S So, given that, we have:So, given that, we have: [[SS N VP] N VP]MM = true iff PAVAROTTI = true iff PAVAROTTI

[VP] [VP]MM

Now, what is [VP]Now, what is [VP]MM?? Since meaning is compositional Since meaning is compositional

and VP does not branch, [VP]and VP does not branch, [VP]MM is is the same as [Vi]the same as [Vi]MM..

So, what is [Vi]So, what is [Vi]MM?? Since meaning is compositional Since meaning is compositional

and VP does not branch, [Vi]and VP does not branch, [Vi]MM is is the same as [is boring]the same as [is boring]MM..

We have a semantic rule that We have a semantic rule that tells us that [is boring]tells us that [is boring]MM is the set is the set of individuals from the model M of individuals from the model M that the function F designates.that the function F designates.

So [is boring]So [is boring]MM = F(is boring). = F(is boring).

S

N VP

Vi

is boring

Pavarotti

[N]M = PAVAROTTI

The The interpretation interpretation

of Sof S So far:So far: [[SS N VP] N VP]MM = true iff = true iff

PAVAROTTIPAVAROTTI [VP] [VP]MM

[VP][VP]MM = [Vi] = [Vi]MM

[Vi][Vi]MM = [is boring] = [is boring]MM

[is boring][is boring]MM = F(is boring) = F(is boring) Now, what is F(is boring)?Now, what is F(is boring)? It will depend on the model—who It will depend on the model—who

are the boring individuals in this are the boring individuals in this particular model? F(is boring) will particular model? F(is boring) will be a set of individuals that are be a set of individuals that are boring in this model.boring in this model.

On one particular model, perhaps On one particular model, perhaps F(is boring)= {PAVAROTTI, LOREN}F(is boring)= {PAVAROTTI, LOREN}

In general:In general: F(is boring) = {F(is boring) = {xx: : xx is boring in M} is boring in M}

S

N VP

Vi

is boring

Pavarotti

[N]M = PAVAROTTI

[is boring]M =F(is boring) ={x: x is boring in M}

The The interpretation interpretation

of Sof S Now, we’re basically done.Now, we’re basically done. F(is boring) = {x: x is boring in M}F(is boring) = {x: x is boring in M} [is boring][is boring]MM = F(is boring) = F(is boring) [is boring][is boring]MM = {x: x is boring in M} = {x: x is boring in M} [Vi][Vi]MM = [is boring] = [is boring]MM

[Vi][Vi]MM = {x: x is boring in M} = {x: x is boring in M} [VP][VP]MM = [Vi] = [Vi]MM

[VP][VP]MM = {x: x is boring in M} = {x: x is boring in M} [[SS N VP] N VP]MM = true iff = true iff

PAVAROTTIPAVAROTTI [VP] [VP]MM

[[SS N VP] N VP]MM = true iff = true iffPAVAROTTIPAVAROTTI {x: x is {x: x is

boring in M}boring in M} As desired. Picking the particular As desired. Picking the particular

model where {x: x is boring in M} model where {x: x is boring in M} = {PAVAROTTI, LOREN}, [S]= {PAVAROTTI, LOREN}, [S]MM = = true.true.

S

N VP

Vi

is boring

Pavarotti

[N]M = PAVAROTTI

[is boring]M =F(is boring) ={x: x is boring in M}

Semantic rules of F1Semantic rules of F1

Summarizing the rules we used so far:Summarizing the rules we used so far: [[SS N VP] N VP]MM = true iff [N] = true iff [N]M M [VP] [VP]MM [Pavarotti][Pavarotti]MM = F(Pavarotti) = F(Pavarotti) [is boring][is boring]MM = F(is boring) = F(is boring) F(Pavarotti) = the individual in M named F(Pavarotti) = the individual in M named

by F as “Pavarotti”by F as “Pavarotti” F(is boring) = the set of individuals in M F(is boring) = the set of individuals in M

that are boring = {x: x is boring in M}that are boring = {x: x is boring in M}

Saving ink and expressing a Saving ink and expressing a generalizationgeneralization

Some of these rules are very specific. Some of these rules are very specific. Rather than add a new rule for each Rather than add a new rule for each individual and predicate…individual and predicate… [Bond][Bond]MM = F(Bond) = F(Bond) [Loren][Loren]MM = F(Loren) = F(Loren) [is hungry][is hungry]MM = F(is hungry) = F(is hungry) [is cute][is cute]MM = F(is cute) = F(is cute)

……we can abstract out the pattern here and we can abstract out the pattern here and write a more general rule:write a more general rule: [X][X]MM = F(X) where X is a terminal node (has no = F(X) where X is a terminal node (has no

children, does not appear on the LHS of a PS children, does not appear on the LHS of a PS rule in the syntax)rule in the syntax)

The role of FThe role of F

This perhaps also clarifies the role of F.This perhaps also clarifies the role of F. F is essentially the thing that translates F is essentially the thing that translates

the the object languageobject language (English, say) into (English, say) into the the metalanguagemetalanguage in terms of the in terms of the modelmodel..

F is responsible for assigning the F is responsible for assigning the interpretations to the terminal nodes.interpretations to the terminal nodes.

The semantic rules are responsible for The semantic rules are responsible for assigning the interpretations to the assigning the interpretations to the combinations.combinations.

Continuing with the Continuing with the semantic rulessemantic rules

We can also generate trees with Neg that we need to We can also generate trees with Neg that we need to assign an interpretation to as well.assign an interpretation to as well. Notice that we have written one of the S nodes as SNotice that we have written one of the S nodes as S. This is . This is

like painting one blue and one red—we just want to be able to like painting one blue and one red—we just want to be able to refer to each one separately. As far as the rules are refer to each one separately. As far as the rules are concerned, it is just a normal S.concerned, it is just a normal S.

We know what [SWe know what [S]]MM is, we just is, we justjust worked that out.just worked that out.

We know what we wantWe know what we want[S][S]MM to be—false when to be—false when[S[S]]MM is true, and true when is true, and true when[S[S]]MM is false. is false.

S

Neg

N VP

Pavarotti Vi

is boring

It is notthe case that

S

Neg SNeg S Goal: [Goal: [SS Neg SNeg S]]MM = = false false if [if [SS]]MM = = truetrue, , true true if if

[[SS]]MM = = falsefalse..

What interpretation must we assign to [What interpretation must we assign to [NegNeg]]MM to to arrive at this result?arrive at this result?

Let’s try to make this look like Let’s try to make this look like is hungry is hungry in a in a certain sense. A property of truth values, in this certain sense. A property of truth values, in this case the property of being false.case the property of being false.

[Neg][Neg]MM = {false} = {false}

Neg SNeg S Goal: [Goal: [SS Neg SNeg S]]MM = = false false if [if [SS]]MM = = truetrue, , true true if if

[[SS]]MM = = falsefalse..

[Neg][Neg]MM = {false} = {false} So [Neg]So [Neg]MM is a set of truth values (like [is hungry] is a set of truth values (like [is hungry]MM is a is a

set of individuals).set of individuals).

Now we can define an interpretation rule very Now we can define an interpretation rule very much like our previous [much like our previous [SS N VP] N VP]MM rule. rule.

[[SS Neg S Neg S]]MM = true iff [S = true iff [S]]MM [Neg] [Neg]MM

It is not the case thatIt is not the case thatPavarotti is boringPavarotti is boring

[S][S]MM = [ = [SS Neg S Neg S]]MM

[[SS Neg SNeg S]]MM = true iff [S = true iff [S]]MM [Neg] [Neg]MM

[Neg][Neg]MM = {false} = {false} [S[S]]MM = true iff= true iff

PAVAROTTIPAVAROTTI {x: x is boring in M} {x: x is boring in M} [S][S]MM = true iff = true iff

[PAVAROTTI[PAVAROTTI {x: x is boring in M}]{x: x is boring in M}]

[S][S]MM = true iff = true iffPAVAROTTIPAVAROTTI {x: x is boring in M}{x: x is boring in M}

S

Neg

N VP

Pavarotti Vi

is boring

It is notthe case that

S

Transitive verbsTransitive verbs

The syntax of F1 also generates trees with The syntax of F1 also generates trees with transitive verbs, like transitive verbs, like likeslikes.. S S N VP N VP VP VP Vt N Vt N Vt Vt likes likes

We want to be able to evaluate [We want to be able to evaluate [SS N VPN VP]]MM the same way whether the same way whether VPVP is built from a is built from a transitive verb or an intransitive verb. That transitive verb or an intransitive verb. That is, we want [is, we want [VPVP]]MM to be a predicate, a set to be a predicate, a set of individuals in either case.of individuals in either case.

Transitive verbsTransitive verbs Essentially, we want [Essentially, we want [likes Bondlikes Bond]]MM to be a to be a

set of those individuals that like set of those individuals that like Bond Bond in in MM..

However, we need a definition for [However, we need a definition for [likeslikes]]MM (we already have one for [(we already have one for [BondBond]]MM). It ). It should be something that creates a set of should be something that creates a set of individuals that depends on the individual individuals that depends on the individual next to it in the structure.next to it in the structure.

[[VPVP likes Bond] likes Bond]MM = {x: x likes Bond in M} = {x: x likes Bond in M}

Transitive verbsTransitive verbs

A transitive verb A transitive verb relatesrelates two individuals. They two individuals. They stand in an (asymmetrical) relationship.stand in an (asymmetrical) relationship.

Suppose that this is expressed in the model as a Suppose that this is expressed in the model as a set of pairsset of pairs that are involved in the relationship. that are involved in the relationship. For example, if For example, if PP likes likes LL, , LL likes likes BB and that’s all the and that’s all the

liking in this situation, then F(liking in this situation, then F(likeslikes) = { <) = { <PP,,LL>, <>, <LL,,BB> }> }

We could express this as follows, to use a We could express this as follows, to use a (metalanguage) shorthand:(metalanguage) shorthand: [[likeslikes]]MM = { < = { <xx,,yy> : > : xx likes likes yy in in MM } }

Transitive verbsTransitive verbs And then, we define a rule that will interpret And then, we define a rule that will interpret

the VP in a sentence with a transitive verb:the VP in a sentence with a transitive verb: [[VP VP Vt NVt N]]MM = { = {xx : < : < xx, [, [NN]]MM > > [ [VtVt]]MM } }

If [If [NN]]MM = = BondBond, [, [VPVP Vt NVt N]]MM is the set containing is the set containing those individuals who like those individuals who like Bond Bond in M.in M. For example For example Loren likes BondLoren likes Bond: If in a particular : If in a particular

model M1, [model M1, [likeslikes]]M1M1 = {<P,L>, <L,B>}, then = {<P,L>, <L,B>}, then[[VPVP Vt N] Vt N]M1M1 = {L}, and [S] = {L}, and [S]M1M1 = true. = true.

In general, [S]In general, [S]MM = true iff = true iffF(F(LorenLoren) ) { {xx: <: <xx, F(, F(BondBond)> )> F( F(likeslikes)})}= true iff <F(= true iff <F(LorenLoren), F(), F(BondBond)> )> F( F(likeslikes).).

Sentence coordinationSentence coordination We also need a way to interpret We also need a way to interpret oror and and andand.. Two options: New rule for ternary branching and Two options: New rule for ternary branching and

symmetric relations. Or recast as binary branching.symmetric relations. Or recast as binary branching.

S

Neg

S

N VP

Pavarotti Vi

is boring

It is notthe case that

S N VP

Loren Vi

is hungry

SConj

or

Thoughts on coordinationThoughts on coordination Like transitive verbs, Like transitive verbs, or or and and and and express a kind of express a kind of

relation (between truth values, rather than between relation (between truth values, rather than between individuals).individuals).

The relation expressed by The relation expressed by oror and and andand is symmetrical, is symmetrical, order does not seem to affect the relation.order does not seem to affect the relation.

But some transitive verbs are like this too (e.g. But some transitive verbs are like this too (e.g. resembleresemble).).

And we might want to consider And we might want to consider if if a kind of a kind of coordinator—but for coordinator—but for ifif, order , order does does matter.matter.

Let’s consider symmetry an accidental property, due Let’s consider symmetry an accidental property, due to the definition of the word in question (according to the definition of the word in question (according to F), and not a property inherent in a new type of to F), and not a property inherent in a new type of semantic combination.semantic combination.

Breaking the structural Breaking the structural symmetrysymmetry

In order to reduce symmetrical In order to reduce symmetrical andand and and oror to a binary-branching (and to a binary-branching (and therefore necessarily asymmetrical) therefore necessarily asymmetrical) structure, we modify the syntax structure, we modify the syntax slightly:slightly:

S S S ConjP S ConjP ConjP ConjP Conj S Conj S

Revised structure for or:Revised structure for or: Thus:Thus:

S

Neg

S

N VP

Pavarotti Vi

is boring

It is notthe case that

S

N VP

Loren Vi

is hungry

SConj

or

ConjP

OrOr For For oror we need to consider pairs of we need to consider pairs of

sentences. We want sentences. We want SS11 or S or S22 to be to be false false when when SS11 is is false false and and SS22 is is false false , and , and true true under any under any other circumstance.other circumstance.

Goal:Goal: [[SS SS11 [[ConjPConjP or S or S2 2 ]]]]MM = true iff [S = true iff [S11]]MM [S [S22]]MM..

The combination occurs in two stages, first The combination occurs in two stages, first with Swith S22, to yield a property then applied to S, to yield a property then applied to S11..

OrOr On the model of transitive verbs, suppose that On the model of transitive verbs, suppose that

F(F(oror) is a set of relations between true values:) is a set of relations between true values: F(or) = {<true, true>, <true, false>, <false, true>}F(or) = {<true, true>, <true, false>, <false, true>}

And a rule of combination just like that for [And a rule of combination just like that for [VPVP Vt Vt N]:N]: [[ConjP ConjP Conj SConj S]]MM = { = {xx : < : < xx, [, [SS]]MM > > [ [ConjConj]]MM } }

Does it work?Does it work? What’s F(and)?What’s F(and)? What would be involved in adding What would be involved in adding ifif??

Semantic rules of F1Semantic rules of F1

Summarizing the rules we used so far:Summarizing the rules we used so far: [[SS N VP] N VP]MM = true iff [N] = true iff [N]M M [VP] [VP]MM

[[SS SS11 Conj S Conj S22]]MM = true iff {[S = true iff {[S11]]MM, [S, [S22]]MM}} [Conj] [Conj]MM

[[SS Neg SNeg S]]MM = true iff [S = true iff [S]]MM [Neg] [Neg]MM

[X][X]MM = F(X) where X is a terminal node = F(X) where X is a terminal node F(It is not the case that) = {false}F(It is not the case that) = {false} F(or) = {{true, true}, {false, true}}F(or) = {{true, true}, {false, true}} F(and) = {{true, true}}F(and) = {{true, true}}

Note the change for and, or, not (ultimately assigned Note the change for and, or, not (ultimately assigned by F)by F)

Full summary of F1Full summary of F1S S N VP N VP [[SS N VPN VP]]MM = true = true iff [iff [NN]]MM [ [VPVP]]MM

S S Neg S Neg S [[SS Neg S´Neg S´]]MM = true iff [ = true iff [S´S´]]MM [ [NegNeg]]MM

S S S ConjP S ConjP [[SS S ConjPS ConjP]]MM = true iff [ = true iff [SS]]MM [ [ConjPConjP]]MM

ConjP ConjP Conj S Conj S [[ConjPConjP Conj SConj S]]MM = = {{xx : < : < xx, [, [SS]]MM > > [[ConjConj]]MM } }

VP VP Vt N Vt N [[VP VP Vt NVt N]]MM = { = {xx : < : < xx, [, [NN]]MM > > [ [VtVt]]MM } }

VP VP Vi Vi [ [X] ][ [X] ]MM = [X] = [X]MM for any X for any X

[[XX]]MM = F( = F(XX) where X is a terminal node) where X is a terminal node

N N Pavarotti, … Pavarotti, … F(F(PavarottiPavarotti) = PAVAROTTI) = PAVAROTTI

Vi Vi is boring, … is boring, … F(F(is boringis boring) = {) = {xx: : xx is boring in is boring in MM}}

Vt Vt likes likes F(F(likeslikes) = { <) = { <xx,,yy> : > : xx likes likes yy in in MM } }

Conj Conj and, or and, or F(and) = {<true, true>}, F(or) =F(and) = {<true, true>}, F(or) ={<true, true>, <true,false>, {<true, true>, <true,false>, <false,true>}<false,true>}

Neg Neg it is not… it is not… F(iintct) = {false}F(iintct) = {false}

What we haveWhat we have We have created a little fragment describing a We have created a little fragment describing a

(very small) subset of English, generating (very small) subset of English, generating structural descriptions of syntactically valid structural descriptions of syntactically valid sentences and providing the means to determine sentences and providing the means to determine the truth conditions of these sentences.the truth conditions of these sentences.

We did this by formulating a set of syntactic We did this by formulating a set of syntactic rewrite rules, each accompanied by a semantic rewrite rules, each accompanied by a semantic rule of interpretation, such that every syntactic rule of interpretation, such that every syntactic step can be interpreted compositionally.step can be interpreted compositionally.

One step more generalOne step more general Looking over the rules that we have, there are basically just two Looking over the rules that we have, there are basically just two

kinds:kinds: [[SS N VP] N VP]MM = true iff [N] = true iff [N]MM [VP] [VP]MM

[[SS S ConjP] S ConjP]MM = true iff [S] = true iff [S]MM [ConjP] [ConjP]MM

[[SS Neg S Neg S]]MM = true iff [S = true iff [S]]MM [Neg] [Neg]MM

[[VPVP Vt N] Vt N]MM = {x: <x,[N] = {x: <x,[N]MM>> [Vt] [Vt]M M }} [[ConjPConjP Conj S] Conj S]MM = {x: <x,[S] = {x: <x,[S]MM>> [Conj] [Conj]M M }}

More generally:More generally: [A B][A B]MM = true iff [A] = true iff [A]MM [B] [B]MM

(where [B](where [B]MM is a set of [A] is a set of [A]MM-type things)-type things) [A B][A B]MM = {x: <x,[A] = {x: <x,[A]MM>}>} [B] [B]MM

(where [B](where [B]MM is a set of pairs, the second member being an [A] is a set of pairs, the second member being an [A]MM-type -type thing)thing)

[ [A] ][ [A] ]MM = [A] = [A]MM

This will cover our other rules… and make it easier to extend This will cover our other rules… and make it easier to extend our syntax as well.our syntax as well.

One step further…?One step further…? If we have these rules:If we have these rules:

[A B][A B]MM = true iff [A] = true iff [A]MM [B] [B]MM (where [B](where [B]MM is a set of [A] is a set of [A]MM-type things)-type things)

[A B][A B]MM = {x: <x,[A] = {x: <x,[A]MM>> [B] [B]MM } } (where [B](where [B]MM is a set of pairs, the second member being an [A] is a set of pairs, the second member being an [A]MM-type thing)-type thing)

[ [A] ][ [A] ]MM = [A] = [A]MM

It feels as if we still have a kind of specific rule: the first looks kind It feels as if we still have a kind of specific rule: the first looks kind of like a “special case” of the second. But how can we reduce of like a “special case” of the second. But how can we reduce them to one rule?them to one rule?

One option:One option: Redefine F(is boring) as, e.g., {<Bond,true>,<Loren,false>,…}Redefine F(is boring) as, e.g., {<Bond,true>,<Loren,false>,…} Define {true} as true and {false} as false.Define {true} as true and {false} as false. Redefine F(likes) as, e.g., {<Bond,<Loren,true>>, <Loren,<Bond,false>>,Redefine F(likes) as, e.g., {<Bond,<Loren,true>>, <Loren,<Bond,false>>,

…}…} See how it works? But it’s confusing…See how it works? But it’s confusing…

Exploring the option…Exploring the option… The option:The option:

Redefine F(is boring) as, e.g., {<Bond,true>,<Loren,false>,…}Redefine F(is boring) as, e.g., {<Bond,true>,<Loren,false>,…} Define {true} as true and {false} as false.Define {true} as true and {false} as false. Redefine F(likes) as, e.g., {<Bond,<Loren,true>>, <Loren,<Bond,false>>,…}Redefine F(likes) as, e.g., {<Bond,<Loren,true>>, <Loren,<Bond,false>>,…}

What we have to do is, for properties: redefine the set so that there is a What we have to do is, for properties: redefine the set so that there is a pair for each individual, with true or false depending on whether the pair for each individual, with true or false depending on whether the individual has the property.individual has the property.

But, wait. What we just defined is in fact a But, wait. What we just defined is in fact a functionfunction. The first member of . The first member of the pair is the argument, the second is the return value.the pair is the argument, the second is the return value.

Is-boring(x) = true iff x is boring.Is-boring(x) = true iff x is boring. Ah. It would be less confusing if we just wrote it as a function.Ah. It would be less confusing if we just wrote it as a function.

F(is boring) = the function f such that f(x)=true iff X is boring (in M)F(is boring) = the function f such that f(x)=true iff X is boring (in M) Or, using the Or, using the -notation we saw before:-notation we saw before:

F(is boring) = F(is boring) = x[x is boring in M]x[x is boring in M] This is the same thing as a set of pairs, the first of which is an individual, This is the same thing as a set of pairs, the first of which is an individual,

and second of which is true if the individual is boring in M and false and second of which is true if the individual is boring in M and false otherwise. But thinking of it as a function is more graspable. It’s something otherwise. But thinking of it as a function is more graspable. It’s something that needs an individual and provides a truth value. Type <e,t>. See where that needs an individual and provides a truth value. Type <e,t>. See where the notation comes from?the notation comes from?

Exploring the option…Exploring the option… As for transitive verbs:As for transitive verbs:

Redefine F(likes) as, e.g., {<Bond,<Loren,true>>, Redefine F(likes) as, e.g., {<Bond,<Loren,true>>, <Loren,<Bond,false>>,…}<Loren,<Bond,false>>,…}

What we want is a function that applies to the object and What we want is a function that applies to the object and returns a property.returns a property.

A property is a function that applies to an individual and A property is a function that applies to an individual and returns a truth value.returns a truth value. F(likes) = F(likes) = y[y[x[x likes y in M]]x[x likes y in M]] F(is boring) = F(is boring) = x[x is boring in M]x[x is boring in M]

Well, that’s much more compact.Well, that’s much more compact. So, combining So, combining likeslikes with with BondBond yields: yields:

[likes Bond][likes Bond]MM

= = y[y[x[x likes y in M]](Bond)x[x likes y in M]](Bond)= = x[x likes Bond in M]x[x likes Bond in M](the property of liking Bond)(the property of liking Bond)

What this buys usWhat this buys us Defining things in terms of functions allows us to reduce our Defining things in terms of functions allows us to reduce our

semantic rules to just two:semantic rules to just two:

Functional application:Functional application:[[ ]]MM = [ = []]MM ([ ([]]MM ) or [ ) or []]MM ([ ([]]MM),),

whichever is defined.whichever is defined. Pass up:Pass up:

[[ ]]MM = [ = []]MM

This will be the basis of F2, which we will define fully next This will be the basis of F2, which we will define fully next time and then move on to the connection with “theta-roles.”time and then move on to the connection with “theta-roles.” By the time we’re done, there will be one more semantic rule, to By the time we’re done, there will be one more semantic rule, to

interpret “modification” relations like adjectives and adverbs.interpret “modification” relations like adjectives and adverbs. We will also consider an alternative version in terms of “events” We will also consider an alternative version in terms of “events”

and “states” in future classes.and “states” in future classes.