CAS LX 5029b. Formal semanticsPronouns and quantifiers

Bond is hungry[N]M = F(Bond)[VP]M = [Vi]M = x [ x F(is hungry)][S]M = [VP]M ( [N]M ) = F(Bond) x [ x F(is hungry)] = F(Bond) F(is hungry)USNVPViis hungryBondis hungryBond Loren PavarottiF

Bond is hungry[S]M1 = F1(Bond) F1(is hungry) = Bond {Bond, Loren}

In the specific situation M1.U1SNVPViis hungryBondis hungryBond Loren PavarottiF1

He is hungryWe dont have he in our lexicon yet, but if we did, how should we interpret it?

This sentence could mean different things (have different truth conditions), in the same situation, depending on who were pointing at.USNVPViis hungryHeis hungryBond Loren PavarottiF

He1 is hungryWhen writing a sentence like he is hungry, the standard practice is to indicate the pointing relation by using a subscript on he:He1 is hungry.The idea here is that this is interpreted in conjunction with a pointing function that tells us who 1 points to.USNVPViis hungryHe1is hungryBond Loren PavarottiF1 2 3 g

He1 is hungryWhere different people are being pointed to, we use different subscripts:He1 likes her2, but he3 hasnt noticed.

The pointing function goes by the more official name assignment function, and is generally referred to as g.USNVPViis hungryHe1is hungryBond Loren PavarottiF1 2 3 g1

He1 is hungryThe assignment function g fits into the system much like the valuation function F does. F maps lexical items into the universe of individuals, g maps subscripts into the universe of individuals.

[an]M,g = g(n)USNVPViis hungryHe1is hungryBond Loren PavarottiF1 2 3 g1

He1 is hungry[an]M,g = g(n)

[S]M1,g1 = g1(he1) F1(is hungry) = Bond {Bond, Loren}

U1SNVPViis hungryHe1is hungryBond Loren PavarottiF11 2 3 g1

QuantifiersQuantifiers (everyone, someone, noone) allow us to state generalizations.Someone is boring.Everyone is hungry.When we say everyone is hungry, were saying that for each individual x, x is hungry.We can think of this as follows: Run through the universe of individuals, pointing at each one in turn, and evaluate s/he is hungry. If it is true for every pointing, then everyone is hungry is true.

Bond likes everyoneSo, what were after is something like this:xU [Bond likes x in M]

That is, we have to convert everyone into a pronoun and interpret the S, with a pronoun in it, with every pointing that we can do.

To do this, we will introduce a rule called Quantifier Raising for sentences with quantifiers in them that will accomplish just that.

Syntactic base rules

S NP VPVP Vt NPS S ConjPVP ViConjP Conj SNP Det NCS Neg SNP NP

Det the, a, everyNP 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 hungryNeg it is not the case thatNC book, fish, man, woman

Bond likes every fishNot much new here, except that weve added some words we can play with, including some common nouns, and some determiners (Det) to use to build quantifiers.SNPVPVtlikesNPBondNPDeteveryNCfish

Bond likes every fishBack to the problem of quantifiers.Consider the meaning of Bond likes every fish. It should be something like:

For every x in U that is a fish, Bond likes x. Or: xU [x is a fish in M Bond likes x in M]

Bond likes every fishxU [x is a fish in M Bond likes x in M]

Notice that our sentence is basically here, but with x instead of every fish. The meaning of every fish is kind of factored out of the sentence and used to set the value of x.

In order to get this interpretation, were going to introduce a transformation. A new kind of rule.

TransformationsThe syntactic base rules that we have allow us to construct trees.A transformation takes a tree and alters it, resulting in a new tree.

The particular transformation we are going to adopt here (Quantifier Raising) takes an NP like every fish and attaches it to the top of the tree, leaving an abstract pronoun behind. Then we will write our semantic rules to interpret that structure.

Quantifier RaisingSNPVPVtlikesNPBondNPDeteveryNCfishSNPVPVtlikesNPBondNPDeteveryNCfisht1S1S

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

That is:S NP S ti iSSNP

Interpreting quantifiersNow comes the tricky part: How do we assign a semantic interpretation to the structure? (It is easiernay, possiblenow that we have the QR rule, but lets see why).

Remember, what were after is: xU [x is a fish in M Bond likes x in M]

Interpreting quantifiersLets start with the lower S. We know how to interpret that, it is essentially just Bond likes it1.

< Bond, g(1) > F(likes)SNPVPVtlikesNPBondNPDeteveryNCfisht1S1S

Interpreting quantifiersThe purpose of the 1 node is to make a predicate out of this sentence.

The predicate will be, in effect, things Bond likes. Goal: 1 [< Bond, g(1) > F(likes)]SNPVPVtlikesNPBondNPDeteveryNCfisht1S1S

Interpreting quantifiersThe interpretation of [1]M,g, then, will be a function that takes a sentence (type ) and returns a predicate (type ).

[1]M,g is type .

[1]M,g = S [x [ [S]M,g[1/x] ] ]SNPVPVtlikesNPBondNPDeteveryNCfisht1S1S

[ ]M,g[1/x]To understand what is going to happen here, we need to introduce one more concept, the modified assignment function g[1/x].Remember that the assignment function maps subscripts to individuals, so that we can interpret pronouns like he2.So, g1 (an assignment function for a particular pointing situation) might map 1 to Pavarotti, 2 to Nemo, 3 to Loren, and so forth.

[ ]M,g[1/x]g1 maps 1 to Pavarotti, 2 to Nemo, 3 to Loren,

A modified assignment function g[i/x] is an assignment function that is just like the original assignment function except that instead of whatever g mapped i to, g[i/x] maps i to x instead. That is:g1(1) = Pavarotti, g1(2) = Nemog1[2/Bond](1) = Pavarotti, g1[2/Bond](2) = Bond

[ ]M,g[1/x]g1(1) = Pavarotti, g1(2) = Nemog1[2/Bond](1) = Pavarotti, g1[2/Bond](2) = Bond

The reason that this is useful is that to interpret every fish, we want to go through all of the fish, and check whether Bond likes it is true when we point to each fish.

It is a pronoun, whose interpretation is dependent on who we are pointing to, so we need to be able to change who we point to (accomplished by modifying the assignment function).

Interpreting quantifiers[1]M,g = S [x [ [S]M,g[1/x] ] ]

[S]M,g = [S]M,g S [x [ [S]M,g[1/x] ] ] x [ [S]M,g[1/x] ] = x [F(likes) ] = x [F(likes)]SNPVPVtlikesNPBondNPDeteveryNCfisht1S1S

Interpreting quantifiersGreat, now we have [S]M,g as a predicate that means things Bond likes.

Now, lets turn to every fish.

Fish is a predicate, true of fish; that is: x [xF(fish)]SNPVPVtlikesNPBondNPDeteveryNCfisht1S1S

Interpreting quantifiersWhat were looking for is a way to verify that every individual that is a fish is also an individual that Bond likes.

That is: xU [x is a fish x is a thing Bond likes]

[S]M,g is the predicate things Bond likes. [NC]M,g is the predicate fish.SNPVPVtlikesNPBondNPDeteveryNCfisht1S1S

Interpreting quantifiersInformally, every takes two predicates, and yields true if everything that satisfies the first predicate also satisfies the second.

[every]M,g = P [ Q [xU [P(x) Q(x)] ] ]SNPVPVtlikesNPBondNPDeteveryNCfisht1S1S

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

[NC]M,g = [fish]M,g = y [yF(fish)]

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

SNPVPVtlikesNPBondNPDeteveryNCfisht1S1S

Interpreting quantifiers[NP]M,g = Q [xU [xF(fish) Q(x)] ]

[S]M,g = y [F(likes)]

[S]M,g = y [F(likes)] Q [xU [xF(fish) Q(x)] ] = xU [xF(fish) x y [F(likes)]] = xU [xF(fish) F(likes) ] SNPVPVtlikesNPBondNPDeteveryNCfisht1S1S

PhewAnd, weve done it. Weve derived the truth conditions for Bond likes every fish:

xU [xF(fish) F(likes) ]

For every individual x in U, if x is a fish, then Bond likes x.

New semantic rulesThe new semantic rules we needed (lexical entries, were still using the same Functional Application and Pass-up rules) were:

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

Loren hates a bookWeve worked out an interpretation for a single quantificational determiner, every, but we can in a parallel way give a meaning to a (as in a book).

The meaning we want for Loren hates a book is that there is some individual x such that x is a book and Loren hates x:

xU [xF(book) F(hates) ]

Loren hates a bookWithout running through all of the steps again, what we want here is for [a]M,g to take two predicates (here, book, and things Loren hates), and be true if there is some individual that satisfies both:

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

Recapstep oneWhen faced with a sentence like A fish likes Nemo, the first thing to do is use the syntactic base rules to construct a tree.S NP VPNP Det NCDet aNC fishVP Vt NPVt likesNP NPNP NemoSNPVPVtlikesNPNemoNPDetaNCfish

Recapstep twoThen, for the quantificational NPs (those with every or a as a determiner), use the QR transformation to attach the quantifier to the top of the tree.

Quantifier Raising [S X NP Y ] [S NP [S i [S X ti Y ] ] ]SNPVPVtlikesNPNemoNPDetaNCfishSSt11

Recapstep threeAnd then work your way up the tree, evaluating first the lexical entries at the bottom of the tree:

[Nemo]M,g = Nemo

[fish]M,g = x[xF(fish)][t1]M,g = g(1)[likes]M,g = y[x[F(likes)]][a]M,g = P [ Q [ xU [P(x) Q(x)] ] ][1]M,g = S [x [ [S]M,g[1/x] ] ]SNPVPVtlikesNPNemoNPDetaNCfishSSt11

Recapstep threeWork out parent nodes by applying functions to arguments

[VP]M,g = [Vt]M,g ( [NP]M,g ) = Nemo y[x[F(likes)]] = x[F(likes)][S]M,g = [VP]M,g ( [t1]M,g ) = g(1) x[F(likes)] = F(likes)SNPVPVtlikesNPNemoNPDetaNCfishSSt11

Recapstep threeWork out parent nodes by applying functions to arguments

[NP]M,g = [Det]M,g ( [NC]M,g ) = z[zF(fish)] P [ Q [ xU [P(x) Q(x)] ] ] = Q [ xU [x z[zF(fish)] Q(x)] ] ] = Q [ xU [xF(fish) Q(x)] ] ][S]M,g = [1]M,g ( [S]M,g ) = F(likes) S [x [ [S]M,g[1/x] ] ] = x [F(likes) ] = x [F(likes) ]SNPVPVtlikesNPNemoNPDetaNCfishSSt11

Recapstep threeWork out parent nodes by applying functions to arguments

Notice that you can change the names of variables freelyto ensure that they dont conflict.

[S]M,g = [NP]M,g ( [S]M,g ) = z [F(likes) ] Q [ xU [xF(fish) Q(x)] ] = xU [xF(fish) x z [F(likes) ] ] = xU [xF(fish) F(likes) ]SNPVPVtlikesNPNemoNPDetaNCfishSSt11

Comments about QRQuantifier Raising [S X NP Y ] [S NP [S i [S X ti Y ] ] ]

As it is stated, QR applies to any NP, whether it is a quantifier or not. So, when do you use QR?There are certain situations (e.g., Nemo hates every book) where the structure cannot be interpreted without QR. When the quantifier is the object of a transitive verb, for example.A transitive verb needs something of type . A quantifier needs something of type (and is itself of type ). Neither one can take the other as an argument.

Comments about QRQuantifier Raising [S X NP Y ] [S NP [S i [S X ti Y ] ] ]

As it is stated, QR applies to any NP, whether it is a quantifier or not. So, when do you use QR?On the other hand, if you apply QR to a name like Loren, the semantic interpretation becomes more complicated to work out, but the end result (the truth conditions) are the same as if you hadnt done QR. So, you dont really need QR to interpret the structure.

Comments about QRQuantifier Raising [S X NP Y ] [S NP [S i [S X ti Y ] ] ]

As it is stated, QR applies to any NP, whether it is a quantifier or not. So, when do you use QR?In fact, it turns out that when you have a quantificational NP as a subject (as in Every fish hates The Last Juror), you dont actually need QR in order to interpret the structure either.The VP hates The Last Juror is a predicate. Every applies to fish and then can apply to the VP, resulting in the same truth conditions you would have if you applied QR.

Comments about QRQuantifier Raising [S X NP Y ] [S NP [S i [S X ti Y ] ] ]

As it is stated, QR applies to any NP, whether it is a quantifier or not. So, when do you use QR?Moreover, if you apply QR to an NP, you still have an NPyou could apply QR again to that same NP if you wanted. Again, like with names, this wont affect the ultimate interpretation, it will just increase the amount of effort necessary to work out the truth conditions.

Comments about QRQuantifier Raising [S X NP Y ] [S NP [S i [S X ti Y ] ] ]

As it is stated, QR applies to any NP, whether it is a quantifier or not. So, when do you use QR?

So, the answer is: Use QR as necessary, where it will result in a different interpretation from not using QR (or when not using QR prevents the structure from being interpreted at all).

Every fish likes a bookSome sentences have more than one quantifier. We know that, because a book is the object of a transitive verb, we need to apply QR to a book. We could refrain from applying QR to every fish (the subject) because subjects dont require QR in order to be interpretable.

The result will be, paraphrasing: There is a book x such that for every y, if y is a fish, then y likes x. This is certainly something the sentence can mean.

However, the sentence can also mean: For every y, if y is a fish, then there is a book x such that y likes x. To get this meaning, you must also apply QR to the subject (after applying QR to the object).

[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 [ 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 [b a]M,g = [a]M,gFunctional application [g a b ]M,g = [b]M,g ( [a]M,g ) or [a]M,g ( [b]M,g ) whichever is defined

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