CAS LX 502CAS LX 502SemanticsSemantics

2b. A formalism for 2b. A formalism for meaningmeaning

2.5, 3.2, 3.62.5, 3.2, 3.6

Truth and meaningTruth and meaning The basis of formal semantics: knowing The basis of formal semantics: knowing the meaning of a sentence is knowing the meaning of a sentence is knowing under what conditions it is true.under what conditions it is true. Formal semantics, a.k.a. truth conditional Formal semantics, a.k.a. truth conditional semantics, a.k.a. model-theoretic semantics, a.k.a. model-theoretic semantics, related to Montague Grammar.semantics, related to Montague Grammar.

We wish to describe meaning (truth We wish to describe meaning (truth conditions) precisely and in such a conditions) precisely and in such a way as to predict our intuitions about way as to predict our intuitions about meanings—we will do this by using a meanings—we will do this by using a logical language as a logical language as a metalanguagemetalanguage..

Infinite use, finite Infinite use, finite meansmeans

A fundamental property of language is A fundamental property of language is its recursive nature—we can create its recursive nature—we can create unboundedly many new sentences, and unboundedly many new sentences, and understand what they mean.understand what they mean.

““Infinite use of finite means,” one Infinite use of finite means,” one of the main reasons to suppose that of the main reasons to suppose that our knowledge of language is our knowledge of language is systematicsystematic, that language is not a , that language is not a collection of habits and analogy, but collection of habits and analogy, but must be described by a grammar.must be described by a grammar.

Infinite use, finite Infinite use, finite meansmeans

In the domain of syntax, the In the domain of syntax, the task is primarily to task is primarily to describe/explain why some describe/explain why some arrangements of words count as arrangements of words count as sentences of English, others sentences of English, others don’t, and more broadly, how don’t, and more broadly, how this system relates to those this system relates to those underlying other languages, and underlying other languages, and how this system can arise.how this system can arise.

SyntaxSyntax The generally accepted view of syntax The generally accepted view of syntax breaks sentences down into hierarchical breaks sentences down into hierarchical parts. There are nouns, there are verbs, parts. There are nouns, there are verbs, there are units made of verbs and nouns. there are units made of verbs and nouns. New sentences can be created by mixing and New sentences can be created by mixing and matching these components together.matching these components together.

[[SS Pat [ Pat [AuxPAuxP will [ will [VPVP eat [ eat [NP NP the sandwich]]]]the sandwich]]]]

[[SS The students [ The students [AuxP AuxP have [have [VP VP risen [risen [PP PP in in protest]]]]protest]]]]

SemanticsSemantics

We’re not here to study syntax, we’re here to We’re not here to study syntax, we’re here to study semantics, but we’re going to delve a bit study semantics, but we’re going to delve a bit into both.into both.

The syntactic system that defines what are “good The syntactic system that defines what are “good sentences of English” provides hierarchical sentences of English” provides hierarchical structures, but we know not only what sequences structures, but we know not only what sequences of words might be classified as “English” but we of words might be classified as “English” but we know what those sequences of words mean.know what those sequences of words mean.

Just as there must be a grammar that defines Just as there must be a grammar that defines what sequences of words are English, there must what sequences of words are English, there must also be a grammar that tells us how the meanings also be a grammar that tells us how the meanings of the parts contribute to the meaning of the of the parts contribute to the meaning of the whole.whole.

F1F1

To that end, we are going to create To that end, we are going to create a “mini-grammar of English”, a a “mini-grammar of English”, a fragmentfragment. This grammar will provide . This grammar will provide both the syntactic structure of a both the syntactic structure of a small number of English sentences small number of English sentences and the rules by which we can and the rules by which we can understand their meaning. By doing understand their meaning. By doing this, we can start to understand this, we can start to understand what is involved in the grammar of what is involved in the grammar of semantics more generally.semantics more generally.

F1F1

Rewrite rules (the syntax):Rewrite rules (the syntax):

S S N VP N VP N N Pavarotti, Loren, Pavarotti, Loren, BondBond

S S S conj S conj SS

Vi Vi is boring, is is boring, is hungry, is cutehungry, is cute

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 it is not the case thatthat

Using the syntax of F1Using the syntax of F1 We start with We start with SS (we are (we are building a sentence).building a sentence).

S

Using the syntax of F1Using the syntax of F1 We start with We start with SS (we are (we are building a sentence).building a sentence).

Several different rules Several different rules can apply. We can either can apply. We can either rewrite rewrite SS as as N VPN VP, or as , or as S conj SS conj S, or as , or as neg Sneg S. . Let’s pick Let’s pick N VPN VP..

S

N VP

Using the syntax of F1Using the syntax of F1 We start with We start with SS (we are (we are building a sentence).building a sentence).

Several different rules Several different rules can apply. We can either can apply. We can either rewrite rewrite SS as as N VPN VP, or as , or as S conj SS conj S, or as , or as neg Sneg S. . Let’s pick Let’s pick N VPN VP..

Now, Now, N N can be rewritten can be rewritten as as PavarottiPavarotti, , LorenLoren, or , or BondBond..

S

N VP

Bond

Using the syntax of F1Using the syntax of F1 We start with We start with SS (we are (we are building a sentence).building a sentence).

Several different rules Several different rules can apply. We can can apply. We can either rewrite either rewrite SS as as N N VPVP, or as , or as S conj SS conj S, or , or as as neg Sneg S. Let’s pick . Let’s pick N N VPVP..

Now, Now, N N can be rewritten can be rewritten as as PavarottiPavarotti, , LorenLoren, or , or BondBond..

And And VP VP can be rewritten can be rewritten either as either as Vt N Vt N or or ViVi..

S

N VP

ViBond

Using the syntax of F1Using the syntax of F1 We start with We start with SS (we are (we are building a sentence).building a sentence).

Several different rules Several different rules can apply. We can either can apply. We can either rewrite rewrite SS as as N VPN VP, or as , or as S conj SS conj S, or as , or as neg Sneg S. . Let’s pick Let’s pick N VPN VP..

Now, Now, N N can be rewritten can be rewritten as as PavarottiPavarotti, , LorenLoren, or , or BondBond..

And And VP VP can be rewritten can be rewritten either as either as Vt N Vt N or or ViVi..

And And Vi Vi can be rewritten can be rewritten as as is boringis boring, , is hungryis hungry, , or or is cuteis cute..

S

N VP

Vi

is hungry

Bond

Using the syntax of F1Using the syntax of F1

With this little grammar, we can already With this little grammar, we can already create an unbounded number of sentences.create an unbounded number of sentences. It is not the case that Bond is boring or It is not the case that Bond is boring or Loren is hungry.Loren is hungry.

Using the syntax of F1Using the syntax of F1

It is not the case that Bond is boring or Loren is hungry.It is not the case that Bond is boring or Loren is hungry.

S

Neg S

N VP

Bond Vi

is boring

It is notthe case that

S

N VP

Loren Vi

is hungry

SConj

or

Using the syntax of F1Using the syntax of F1

It is not the case that Bond is boring or Loren is hungry.It is not the case that Bond is boring or Loren is hungry.

S

Neg

S

N VP

Bond Vi

is boring

It is notthe case that

S N VP

Loren Vi

is hungry

SConj

or

A word of warningA word of warning Use the rules and only the rules.Use the rules and only the rules.

You may or may not have had experience with You may or may not have had experience with syntax before. And it may or may not have syntax before. And it may or may not have involved trees like the ones we’ve just seen.involved trees like the ones we’ve just seen.

Probably it involved more complicated trees Probably it involved more complicated trees (and for good reasons, which are explored in (and for good reasons, which are explored in the syntax class). But here, it’s fine to the syntax class). But here, it’s fine to just approximate the syntax by using the PS just approximate the syntax by using the PS rules just given.rules just given.

When drawing trees, don’t try to remember When drawing trees, don’t try to remember what you learned about them in LX250 or what you learned about them in LX250 or LX522. Just rewrite the way the rules allow LX522. Just rewrite the way the rules allow you to. No IP, no CP. Just what the rules you to. No IP, no CP. Just what the rules allow.allow.

CompositionalityCompositionality A fundamental assumption about how it A fundamental assumption about how it is that we can know what novel is that we can know what novel sentences mean is that meaning is sentences mean is that meaning is compositionalcompositional..

The meaning of the whole is derived The meaning of the whole is derived from the meaning of the parts and how from the meaning of the parts and how the parts are arranged.the parts are arranged.

The syntax gives us the parts and how The syntax gives us the parts and how they are arranged, now we must they are arranged, now we must approach the question of how the approach the question of how the meaning is assigned to the parts and meaning is assigned to the parts and from there to the whole.from there to the whole.

The meaning of namesThe meaning of names

We talked about a meaning for names We talked about a meaning for names (like (like BondBond, say) as being something , say) as being something like “pointing to an individual that like “pointing to an individual that exists in the world.”exists in the world.”

We need a way to formalize this kind of We need a way to formalize this kind of intuitive idea: a model.intuitive idea: a model.

A A modelmodel contains two relevant things: A contains two relevant things: A set of the individuals in the universe, set of the individuals in the universe, and a “pointing function” that and a “pointing function” that associates names with those associates names with those individuals.individuals.

Models and pointingModels and pointing We’ll call the set of We’ll call the set of individuals in the universe individuals in the universe “U” (for “Universe”), and “U” (for “Universe”), and the pointing function “F” the pointing function “F” (for “function”—or maybe (for “function”—or maybe “finger”). Both of those “finger”). Both of those together constitute a model, together constitute a model, which we will often call “M” which we will often call “M” (for “model”).(for “model”). MM = < = <UU, , FF>.>.

So, to evaluate the meaning So, to evaluate the meaning of a name, we see which of a name, we see which individual in the world the individual in the world the name points to.name points to. FF((PavarottiPavarotti), then, is ), then, is “the individual named (in “the individual named (in this model) by this model) by PavarottiPavarotti”.”.

U

…is hungry…BondLorenPavarotti…

F

Evaluating the meaning Evaluating the meaning of bits of treeof bits of tree

Our goal in F1 is to create simple Our goal in F1 is to create simple sentence structures with the sentence structures with the syntax, and a assign a meaning syntax, and a assign a meaning (compositionally) to the whole (compositionally) to the whole sentence that matches our sentence that matches our intuitions.intuitions.

So, we need to evaluate the meaning So, we need to evaluate the meaning of individual nodes in the tree as of individual nodes in the tree as well.well.

Evaluating meaning: [ ]Evaluating meaning: [ ]MM

Translating from a node in a Translating from a node in a syntactic structure to a semantic syntactic structure to a semantic meaning is accomplished by what we meaning is accomplished by what we call an call an evaluation functionevaluation function. Given a . Given a syntactic node, its result is the syntactic node, its result is the semantic interpretation of that semantic interpretation of that node.node. The interpretation depends on the The interpretation depends on the model, so we also need to specify with model, so we also need to specify with respect to what model a node is being respect to what model a node is being evaluated.evaluated.

Simplest caseSimplest case

The simplest case would be The simplest case would be evaluating the meaning of the a evaluating the meaning of the a node like node like PavarottiPavarotti at the bottom at the bottom of the tree.of the tree. Evaluating the node: [Evaluating the node: [PavarottiPavarotti]]MM

The meaning of names:The meaning of names:[Pavarotti][Pavarotti]MM = = FF(Pavarotti) = (Pavarotti) = PavarottiPavarotti

The ultimate interpretation assigned The ultimate interpretation assigned to this node is the to this node is the individualindividual PavarottiPavarotti..

Predicates/propertiesPredicates/properties So we have a meaning So we have a meaning assigned for one node assigned for one node in the tree.in the tree.

How about the “verb” How about the “verb” is hungryis hungry??

What is [is hungry]What is [is hungry]MM?? A way we can think of A way we can think of properties is as properties is as something that something that divides the universe divides the universe of individuals into of individuals into two groups, those two groups, those that have the that have the property and those property and those that do not.that do not.

S

N VP

Vi

is hungry

Bond

[Bond]M =F(Bond) =Bond [is hungry]M

Predicates/propertiesPredicates/properties One simple and intuitive One simple and intuitive way to implement this is way to implement this is to say that []to say that []MM of a of a property is a set property is a set containing those containing those individuals that have the individuals that have the property.property.

Like we did for names of Like we did for names of individuals, we can individuals, we can suppose that the name of suppose that the name of a property “points” to a property “points” to the set of individuals the set of individuals that has the property.that has the property.

That is, this can be part That is, this can be part of the job that of the job that FF does. does.

S

N VP

Vi

is hungry

Bond

[Bond]M =F(Bond) =Bond [is hungry]M

Predicates/propertiesPredicates/properties Suppose Bond and Suppose Bond and Pavarotti are the Pavarotti are the hungry ones in the hungry ones in the universe of universe of individuals in this individuals in this model.model.

FF(is hungry) =(is hungry) ={Bond, Pavarotti}{Bond, Pavarotti}

Great, 2 down, 4 to Great, 2 down, 4 to go.go.

S

N VP

Vi

is hungry

Bond

[Bond]M =F(Bond) =Bond [is hungry]M =

F(is hungry) ={Bond, Pavarotti}

Bond is Bond is hungryhungry

[N][N]MM = = FF((BondBond)) [VP][VP]MM = [Vi] = [Vi]MM = = FF(is hungry)(is hungry){ { x: x x: x is hungry in M}is hungry in M}

[S][S]MM = true iff [N] = true iff [N]M M [VP] [VP]MM = =truetrue iff iff FF((BondBond) ) { { x: x x: x is is hungry in M}hungry in M}

U

S

N VP

Vi

is hungry

Bond

…is hungry…BondLorenPavarotti…

F

Bond is Bond is hungryhungry

[S][S]MM11 = = FF11((Bond)Bond) FF11((is hungryis hungry) =) =BondBond { {BondBond, , LorenLoren}}

In the specific situation In the specific situation MM11..

U1

S

N VP

Vi

is hungry

Bond

…is hungry…BondLorenPavarotti…

F1

[ ][ ]MM

We now need to assign We now need to assign interpretations to the interpretations to the rest of the nodes of the rest of the nodes of the tree.tree.

There are no new There are no new meaningful elements, so meaningful elements, so the meanings will all be the meanings will all be formed on the basis of formed on the basis of BondBond or or is hungryis hungry or or both.both.

Meaning is Meaning is compositional.compositional.

So, what’s [N]So, what’s [N]MM??

S

N VP

Vi

is hungry

Bond

[Bond]M =F(Bond) =Bond [is hungry]M =

F(is hungry) ={Bond, Pavarotti}

[ ][ ]MM

Based on the principle Based on the principle of compositionality, we of compositionality, we can assume/deduce the can assume/deduce the that nodes above share that nodes above share the same denotation as the same denotation as the nodes below, in the nodes below, in cases where there is no cases where there is no combination happening.combination happening.

[N][N]MM = = BondBond

S

N VP

Vi

is hungry

Bond

[Bond]M =F(Bond) =Bond [is hungry]M =

F(is hungry) ={Bond, Pavarotti}

[ ][ ]MM

Now, to determine the Now, to determine the meaning of the S as a meaning of the S as a whole…whole…

What do we What do we wantwant?? Well, this should be Well, this should be true only when Bond true only when Bond is hungry.is hungry.

And that’s true if And that’s true if Bond is in the Bond is in the FF(is (is hungry) set.hungry) set.

That is, [S]That is, [S]MM = = truetrue just in case [N]just in case [N]MM is is in the set [VP]in the set [VP]MM..

S

N VP

Vi

is hungry

Bond

[Bond]M =F(Bond) =Bond [is hungry]M =

F(is hungry) ={Bond, Pavarotti}

[ ][ ]MM

We can define a We can define a semantic rule for semantic rule for interpretation that interpretation that says just that:says just that:

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

[N][N]MM [VP][VP]MM,,otherwise otherwise falsefalse..

S

N VP

Vi

is hungry

Bond

[Bond]M =F(Bond) =Bond [is hungry]M =

F(is hungry) ={Bond, Pavarotti}

[ ][ ]MM

Thus, we end up with an Thus, we end up with an interpretation of this interpretation of this sentence that goes like sentence that goes like this:this: [S][S]MM = = true true iffiffFF((BondBond) ) FF((is is hungryhungry), otherwise ), otherwise falsefalse..

Given this particular Given this particular model, that boils down tomodel, that boils down to [S][S]MM = = true true iff iff Bond Bond {{BondBond, , PavarottiPavarotti}, }, otherwise otherwise falsefalse..

(True in this situation)(True in this situation)

S

N VP

Vi

is hungry

Bond

[Bond]M =F(Bond) =Bond [is hungry]M =

F(is hungry) ={Bond, Pavarotti}

A semantic rule for A semantic rule for every structural ruleevery structural rule

Our goal is to design a Our goal is to design a semantics for F1 that can semantics for F1 that can provide an interpretation (truth provide an interpretation (truth conditions) for any structure conditions) for any structure that the syntax can provide.that the syntax can provide.

So, we also need rules for So, we also need rules for structures like structures like S conj SS conj S, , neg Sneg S, , Vt NVt N..

Neg SNeg S For For Neg SNeg S, we want it to be , we want it to be false false whenever whenever SS is is truetrue, and , and truetrue whenever whenever S S is is falsefalse..

[[Neg SNeg S]]MM = = false false if [if [SS]]MM = = truetrue,,true true if [if [SS]]MM = = falsefalse..

However, this is not quite enough—we want However, this is not quite enough—we want to have an interpretation for every node to have an interpretation for every node in the tree. This gives us an in the tree. This gives us an interpretation of interpretation of [[SS Neg S] Neg S], but what is , but what is the interpretation of the interpretation of NegNeg??

NegNeg What What Neg Neg does is takes the truth does is takes the truth value of the value of the SS it is next to and it is next to and reverses it.reverses it.

It is a It is a functionfunction—it takes the truth —it takes the truth value of the value of the SS it is next to as an it is next to as an argumentargument, and , and returns returns a truth value a truth value (the opposite one).(the opposite one).

[[it is not the case thatit is not the case that]]MM = = truetrue falsefalse

false false true true

Neg SNeg S

[[SS [ [NegNeg It is not the case that] [ It is not the case that] [S´S´ Pavarotti is Pavarotti is boring]].boring]].

[[NegNeg]]M M = [= [It is not the case thatIt is not the case that]]MM = = truetrue falsefalse

false false true true

[[S´S´]]MM = = truetrue iff [ iff [NN]]M M [ [VPVP]]MM, otherwise , otherwise falsefalse= = truetrue iff [ iff [PavarottiPavarotti]]MM [ [ViVi]]MM, otherwise , otherwise falsefalse= = truetrue iff [ iff [PavarottiPavarotti]]MM [ [is boringis boring]]MM, , otherwise otherwise falsefalse= = FF((PavarottiPavarotti) ) FF((is boringis boring), otherwise ), otherwise falsefalse

Neg SNeg S

[[SS [ [NegNeg It is not the case that] [ It is not the case that] [S´S´ Pavarotti is boring]].Pavarotti is boring]].

And, so [And, so [SS Neg S´Neg S´]]MM = [ = [NegNeg]]MM ( [ ( [S´S´]]MM ). ).

Resulting in:Resulting in:

[[SS]]MM = = false false if if FF((PavarottiPavarotti) ) FF((is is boringboring),),otherwise otherwise truetrue..

AndAnd For dealing with For dealing with andand and and oror, we also , we also want to define a function. We want want to define a function. We want SS11 and Sand S22 to be to be true true when when SS11 is is true true and and SS22 is is truetrue, and , and false false under any other under any other circumstance.circumstance.

[[SS SS11 Conj S Conj S22]]MM = [ = [ConjConj]]MM ( < [ ( < [SS11]]MM, [, [SS22]]MM > > ))

[[andand]]MM = = < < truetrue, , true true > > truetrue< < truetrue, , falsefalse > > falsefalse< < falsefalse, , truetrue > > falsefalse< < falsefalse, , false false > > falsefalse

OrOr For dealing with For dealing with andand and and oror, we also , we also want to define a function. We want want to define a function. We want SS11 or or SS22 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 other under any other circumstance.circumstance.

[[SS SS11 Conj S Conj S22]]MM = [ = [ConjConj]]MM ( < [ ( < [SS11]]MM, [, [SS22]]MM > > ))

[[oror]]MM = = < < truetrue, , true true > > truetrue< < truetrue, , falsefalse > > truetrue< < falsefalse, , truetrue > > truetrue< < falsefalse, , false false > > falsefalse

Transitive verbsTransitive verbs The one piece of the model that we The one piece of the model that we have not addressed yet are transitive have not addressed yet are transitive verbs, like 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 N VPVP]]MM the same way whether the same way whether VPVP is built is built from a transitive verb or an from a transitive verb or an intransitive verb. That is, we want intransitive verb. That is, we want [[VPVP]]MM to be a predicate, a set of to be a predicate, a set of individuals.individuals.

Transitive verbsTransitive verbs Essentially, we want [Essentially, we want [likes Bondlikes Bond]]MM to be a set of those individuals to be a set of those individuals that like 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 should be something ). It should be something that creates a set of individuals that creates a set of individuals that depends on the individual next that depends on the individual next to it in the structure. A function to it in the structure. A function again.again.

Transitive verbsTransitive verbs

Like Like andand, , likeslikes relates two things, although relates two things, although likeslikes relates two individuals, and relates two individuals, and andand relates two sentences.relates two sentences.

So, we build a two-place predicate, in the So, we build a two-place predicate, in the same way:same way:

[[likeslikes]]MM = { < = { <xx,,yy> : > : xx likes likes yy in in MM } }

For example, if For example, if PP likes likes LL, , LL likes likes BB and that’s and that’s all the liking in this situation, then [all the liking in this situation, then [likeslikes]]MM = = { <{ <PP,,LL>, <>, <LL,,BB> }> }

Transitive verbsTransitive verbs

And then, we define a rule that And then, we define a rule that will interpret the VP in a will interpret the VP in a sentence with a transitive verb:sentence with a transitive verb:

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

So if [So if [NN]]MM = = BondBond, then [, then [VPVP Vt NVt N]]MM is the set containing those is the set containing those individuals who like individuals who like Bond Bond in in MM..

S S N VP N VP [[SS N VPN VP]]MM = = true true iff [iff [NN]]MM [ [VPVP]]MM, , otherwise otherwise falsefalse

S S S Conj S S Conj S [[SS SS11 Conj S Conj S22]]MM = [ = [ConjConj]]MM ( < [ ( < [SS11]]MM, [, [SS22]]MM > )> )

S S Neg S Neg S [[SS Neg S´Neg S´]]MM = [ = [NegNeg]]MM ( [ ( [S´S´]]MM ). ).

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

VP VP Vi Vi

N N Pavarotti, Pavarotti, ……

[[PavarottiPavarotti]]MM = F( = F(PavarottiPavarotti))

Vi Vi is is boring, …boring, …

[[is boringis boring]]MM = { = {xx: : xx is boring in is boring in MM}}

Vt Vt likes likes [[likeslikes]]MM = { < = { <xx,,yy> : > : xx likes likes yy in in MM } }

Conj Conj and, … and, … [and][and]MM = {<<true,true>,true>, = {<<true,true>,true>, <true,false>,false>, …}<true,false>,false>, …}

Neg Neg it is it is not the case not the case thatthat

[iintct][iintct]MM = {<true,false>, = {<true,false>, <false,true>}<false,true>}

What we haveWhat we have We have created a little fragment We have created a little fragment describing a (very small) subset of describing a (very small) subset of English, generating structural English, generating structural descriptions of syntactically valid descriptions of syntactically valid sentences and providing the means to sentences and providing the means to determine the truth conditions of these determine the truth conditions of these sentences.sentences.

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

One step more generalOne step more general Looking over the rules that we have, we Looking over the rules that we have, we can actually go a step further in can actually go a step further in generalizing our semantic rules (helpful generalizing our semantic rules (helpful as we expand our fragment’s coverage).as we expand our fragment’s coverage).

There are basically two kinds of rules we There are basically two kinds of rules we have: Those that combine meanings of have: Those that combine meanings of adjacent (sister) nodes in the syntactic adjacent (sister) nodes in the syntactic structure, and those that define intrinsic structure, and those that define intrinsic (non-compositional) meanings.(non-compositional) meanings.

Semantic typeSemantic type The entire semantics that we are The entire semantics that we are creating here depends on two types of creating here depends on two types of things, things, individuals individuals and and truth valuestruth values..

We can label individuals as being of We can label individuals as being of type “type “ee” (traditional, think ” (traditional, think “entity”), and truth values as being “entity”), and truth values as being of type “of type “tt”.”.

In these terms, names like In these terms, names like Bond Bond are are of type of type <e><e>, and sentences like , and sentences like Bond Bond is hungry is hungry are of type are of type <t><t>..

Characteristic Characteristic functionsfunctions

For predicates like For predicates like is hungryis hungry, we have considered , we have considered these to be sets of individuals (e.g., those that these to be sets of individuals (e.g., those that are hungry in the model).are hungry in the model).

We can look at those same individuals in a We can look at those same individuals in a slightly different way, using the slightly different way, using the characteristic characteristic functionfunction of the set.of the set.

A characteristic function is a function that, A characteristic function is a function that, given an argument, will return true iff the given an argument, will return true iff the argument was a member of the set, and false argument was a member of the set, and false otherwise. The same otherwise. The same information contentinformation content as the as the set.set.

Predicates as functionsPredicates as functions

So, without losing information, we can So, without losing information, we can view predicates from the perspective of view predicates from the perspective of their characteristic functions and their characteristic functions and define define is hungryis hungry to instead be a to instead be a function that, given an individual, function that, given an individual, will return will return truetrue if the individual is if the individual is hungry in the model.hungry in the model.

[[is hungryis hungry]]M M == xx true true if if xx is is hungry in hungry in MM

xx falsefalse otherwiseotherwise

Semantic typeSemantic type

Predicates like Predicates like is hungry is hungry can can then be said to have semantic then be said to have semantic type type <e,t><e,t>. That is, a function . That is, a function from individuals to truth values.from individuals to truth values.

Similarly, Similarly, it is not the case it is not the case that that can be taken to be of type can be taken to be of type <t,t><t,t>, a function from truth , a function from truth values to truth values.values to truth values.

Transitive verbsTransitive verbs For transitive verbs, what we want is a For transitive verbs, what we want is a relation between two individuals, relation between two individuals, resulting in a truth value. The way we resulting in a truth value. The way we have it set up now, a verb like have it set up now, a verb like likes likes will will combine with the object to form a simpler combine with the object to form a simpler predicate like predicate like likes Bondlikes Bond, at which point , at which point it acts just like it acts just like is boringis boring..

So, here, we want So, here, we want likes likes to take an to take an argument of type argument of type <e><e> and return a and return a predicate of type predicate of type <e,t><e,t>. So, we define it . So, we define it as a function of type as a function of type <e,<e,t>><e,<e,t>>..

Transitive verbsTransitive verbs

That is, we can define [That is, we can define [likeslikes]]MM as as something like this:something like this:

[[likeslikes]]MM = = xx ff where where ff is a function from is a function from individuals to truth values and individuals to truth values and ff((yy) = ) = true true iff iff yy likes likes xx in in MM, otherwise , otherwise falsefalse..

That is, [That is, [likeslikes]]MM is a function from is a function from individuals to functions (from individuals individuals to functions (from individuals to truth values): semantic type to truth values): semantic type <e,<e,t>><e,<e,t>>..

Why we’re doing thisWhy we’re doing this Once we have defined things in terms of Once we have defined things in terms of semantic type, and in terms of functions semantic type, and in terms of functions and arguments, we can collapse a number of and arguments, we can collapse a number of our semantic interpretation rules into our semantic interpretation rules into more general rules.more general rules.

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

whichever is defined.whichever is defined. Pass up:Pass up:[[ ]]MM = [ = []]MM