CAS LX 502 Semantics

CAS LX 502Semantics 5b. Pronouns, assignments,and quantifiers 5.7(.1), 6.1

Names and pronouns • We’re modeling names as referring to an individual in the universe of individuals. We use names to refer to individuals. • Pavarotti is boring is true whenever the individual that the name Pavarotti refers to is in the set of individuals for which the property is boring holds. • Turning our attention to another class of words, the pronouns, they seem to be much like names in that they pick out an individual in the world.

Names and pronouns • There is an additional dimension of complexity to pronouns—what individual they pick out depends on who you are pointing at as the time. • He is boring. • She is hungry. • In order to know the conditions under which he is boring is true, we need know who is being “pointed at.”

Pronoun resolution • Pronouns are a form of a more general class of nouns known as anaphora. • The primary characteristic of anaphora is that they refer, but their referent is not fixed intrinsically. They get their reference from somewhere else. • Resolving pronouns is not a trivial matter, although we will not concern ourselves really with how it happens. • Mary told Sue that she won first prize, and then she congratulated her. • Rather, we will assume it does happen, and take as given a list of “pointings” that go with the pronouns.

Caution about the term “anaphora” • Pronouns are a form of anaphora. • Other things are also anaphora. • The most salient other type are the -self-type words (herself, himself). • Very often (particularly in syntax), one uses the term “anaphor” to refer only to the self-anaphora. Pronouns are not called “anaphors” in those situations, but rather just “pronouns.” We will follow that practice here.

Shifty reference • One of the interesting things about pronouns is that there is a kind of ambiguity with them. • A pronoun can refer to anyone you could (under the right circumstances) point at. This is always a possible meaning for pronoun. • John lost his keys. • Pronouns can also “shift” in reference too: • Every boy lost his keys. • There are two meanings here, a constant one, and a “shifting” one.

Quantifiers • Quantifiers (everyone, someone, noone) allow us to state generalizations. • Someone is boring. • Everyone is hungry. • When we say everyone is hungry, we’re saying that for each individual x, x is hungry. • We can think of this as follows: Run through the universe of individuals (people at least), 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. • If it is true for at least one pointing, then someone is hungry is true. If it is true for none of the pointings, then noone is hungry is true.

What quantifiers tell us about pronouns • In a sentence like He is hungry, the referent of the pronoun is just determined by the context. • In a sentence like Every boy lost his keys, the referent of pointing doesn’t exactly come from the context. There is something grammatical going on that “points” to each individual in turn.

Interpreting quantifiers • Before we get to the technical details, we can think about what a quantifier like only John seems to do. • Only John is hungry • This is true when: • The property is hungry holds of John. • The property is hungry does not hold of anyone who is not John. • So, it goes through the people, “points” at each one, and checks to see if the property holds. And then sees if the results match certain conditions.

Quantifiers are not type <e> • Consider everyone is hungry. • The property is hungry is true of individuals—so it is initially tempting to suppose that this is true if some individual, referred to by everyone, is hungry. • This could be more or less like the people are hungry. • We discussed how this would work already. If we suppose people is a plural, a collection of collections, we can take the to pick the biggest one, and then we attribute is hungry (or is hungry) to that collection. Interpreted distributively, this gets the meaning right.

Quantifiers are not type <e> • But try as you might, finding such an individual (or group/collection) for other quantifiers won’t lead to a satisfying conclusion. • Nobody is hungry. • Two people are hungry. • Most people are hungry. • This can’t be how it works.

Quantifiers relate properties • If we think about what every person is hungry does, it seems to be the following: • Consider the individuals for which person is true. • Consider the individuals for which is hungry is true. • If you find an individual in the first set, that individual will be in the second set.

Some logical notation • We didn’t cover logic very systematically, but it is now going to start to be relevant, so let’s work through it a bit as a reminder. • The relation of if…then is indicated with the  symbol. • For any two propositions p and q, the proposition p q is true under the following conditions: • p is false and q is anything. • p is true and q is also true. • Another way you can read  is as “implies.”

Every person is hungry • So, to say “if an individual has the person property, it also has the is hungry property” we can write: • person(x)  hungry(x) • For this to be true, being a person must imply being hungry.

Quantifiers in logic • There are two quantifiers in logic that we will make use of in describing at least the parallel quantifiers in natural language. •  : universal quantifier (“for all”) •  : existential quantifier (“there exists a”) • In action: • x[person(x)] • This is true if every individual you pick, and call “x”, results in “person(x)” being true. • Everything is a person. • x[person(x)] • This is true if there is some individual you could pick, call it “x”, that will result in “person(x)” being true. • Something is a person.

Every person is hungry • To complete the example from before, to say every person is hungry in the fancy-looking logic notation, we want to say that “for every individual x there is, being a person implies being hungry.” • x[person(x)  hungry(x)] • The “syntax” of the logical  and  is quite like the “syntax” of . • To be perhaps a bit more precise, we might write this:xU [person(x)  hungry(x)] (U being the universe of individuals). If not written, it’s implicit.

A person is hungry • Likewise, to say some person is hungry, we want to say that “there exists some individual x such that it is both a person and hungry.” • x[person(x)  hungry(x)] • You can say a person in a way that is synonymous to some person. This is what we’ll usually talk about in the context of quantifiers. Notice that this version of a is not semantically “empty.”

Interpreting quantifiers • For something like • Only John lost his keys • We might continue to think of this as talking about a property that John has and nobody else does. What is that property? • Well, it seems to be something like “having lost one’s own keys.” • True of x where x lost x’s keys: • x [ x lost x’s keys in M]

Interpreting quantifiers • Only John lost his keys • x [ x lost x’s keys in M] • It appears that his is interpreted somehow as having the same value as the subject. • The reference of his depends entirely on the reference of the subject. • We check this by trying different values for the subject. • This kind of dependent reference is known as being bound. The reference of the pronoun is “tied” to the reference of the subject. It is a bound pronoun.

Interpreting quantifiers • Only John lost his keys • x [ x lost x’s keys in M] • How do we get this to come out? • The basic idea goes like this: When you have a quantifier, you “shift” it out of the sentence in order to form the property. • Take only John out and put it at the beginning. • Add x and brackets around the sentence. • Put x where the quantifier was. • For any pronouns interpreted as bound, replace them with x. • Every girl finished her sandwich. • Every girl x [ x finished x’s sandwich in M] • Combining these is the next task. This recipe above is what we will come to call “quantifier raising.”

Interpreting quantifiers • No girl finished her sandwich. • No girl x [ x finished x’s sandwich in M] • We have here a quantifier (no girl) and a property (to have finished one’s own sandwich). • We need to check to see if everyone (among the girls) has this property. • Is no girl the argument of the property (like it would be in John finished a sandwich)? • Well, no, that won’t work. (Cf. earlier, not type <e>). • It must be that the property is taken as an argument to the quantifier, actually.

Interpreting quantifiers • And it makes sense for the quantifier to take a property. • No girl takes a property P. • It is true if, when you have checked all of the individuals (among the girls), none of them had the property P. • No girl (x [ x finished x’s sandwich in M]) • P [ for no girl, x: P(x)] • P [ for every girl, x:P(x)] • Now, let’s make these ideas more precise…

Bond is hungry … is hungry … BondLorenPavarotti … S • [N]M = F(Bond) = BOND • [VP]M = [Vi]M = x [ x is hungry in M] • [S]M = [VP]M ( [N]M ) =x [x is hungry in M] (BOND) =BOND is hungry in M For now,going backto is hungryas a Vi, forsimplicity VP N Bond Vi is hungry F U

Bond is hungry … is hungry … BondLorenPavarotti … S • [S]M1 = BOND is hungry in M1 =true in the specific situation M1. VP N Bond Vi is hungry F1 U1

He is hungry … is hungry … BondLorenPavarotti … S • We don’t 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 we’re pointing at. VP N He Vi is hungry F U

… is hungry … BondLorenPavarotti … He1 is hungry F S • When 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. VP N He1 Vi 123… g is hungry U

… is hungry … BondLorenPavarotti … He1 is hungry F S • Where different people are being pointed to, we use different subscripts: • He1 likes her2, but he3 hasn’t noticed. • The “pointing function” goes by the more official name assignment function, and is generally referred to as g. VP N He1 Vi 123… g1 is hungry U

… is hungry … BondLorenPavarotti … He1 is hungry F S • The assignment function g fits into the system much like the valuation function F does. F maps lexical itemsinto the universe of individuals, g maps subscriptsinto the universe of individuals. • [an]M,g = g(n) VP N He1 Vi 123… g1 is hungry U

… is hungry … BondLorenPavarotti … He1 is hungry F1 S • [an]M,g = g(n) • [S]M1,g1 = g1(1) is hungry in M1 =BOND is hungry in M1 =true in the specific model M1 VP N He1 Vi 123… g1 is hungry U1

Bond likes everyone • So, what we’re after is something like this: • xU [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.

Bond likes every fish • Add a few (obvious things), so we can make better sentences. • Not much new here, except that we’ve added some words we can play with, including some common nouns, and some determiners (Det) to use to build quantifiers. S DP VP Bond Vt DP N likes Det every fish

Bond likes every fish • Back 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:xU [x is a fish in M Bond likes x in M]

Bond likes every fish • xU [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, we’re going to introduce a transformation. A new kind of rule.

Transformations • The 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 fishand 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 Raising S S DP S DP VP N 1 S Det Bond Vt DP every fish DP VP N likes Det Bond Vt t1 every fish likes

Quantifier Raising • Quantifier Raising[SX DP Y ]  [S DP [S i [SXtiY ] ] ] • That is: S DP S S i S … DP … … ti …

Interpreting quantifiers • Now comes the tricky part: How do we assign a semantic interpretation to the structure? (It is easier—nay, possible—now that we have the QR rule, but let’s see why). • Remember, what we’re after is:xU [x is a fish in M Bond likes x in M]

Interpreting quantifiers • Let’s start with thelower S. We knowhow to interpretthat, it is essentiallyjust Bond likes it1. • True if BOND likes g(1) in M S DP S N 1 S Det every fish DP VP Bond Vt t1 likes

Interpreting quantifiers • The purpose of the 1node is to make aproperty out of thissentence. • The property willbe, in effect,things Bond likes. • Goal:1 [BOND likes g(1) in M] S DP S N 1 S Det every fish DP VP Bond Vt t1 likes

Interpreting quantifiers • The interpretation of [1]M,g,then, will be a functionthat takes a sentence(type <t>) and returns apredicate (type <e,t>). • [1]M,g is type <t,<e,t>>. • [1]M,g =S [x [ [S]M,g[1/x] ] ] S DP S N 1 S Det every fish DP VP Bond Vt t1 likes

[ ]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) = Nemo • g1[2/Bond](1) = Pavarotti, g1[2/Bond](2) = Bond

[ ]M,g[1/x] • g1(1) = Pavarotti, g1(2) = Nemo • g1[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 [x [ [S]M,g[1/x] ] ]([S]M,g) =x [ [S]M,g[1/x] ] =x [BOND likes g[1/x](1) ] =x [BOND likes x] S DP S N 1 S Det every fish DP VP Bond Vt t1 likes

Interpreting quantifiers • Great, we have [S]M,g as a predicate that means things Bond likes. • Now to every fish. • Fishis a property, true of fish; that is:x [x is a fish in M] S DP S N 1 S Det every fish DP VP Bond Vt t1 likes

Interpreting quantifiers • What we’re looking for is a way to verify that every individual that is a fish is also an individual that Bond likes. • That is:x [x is a fish x is a thing Bond likes] • [S]M,g is the predicate things Bond likes. [N]M,g is the predicate fish. S DP S N 1 S Det every fish DP VP Bond Vt t1 likes

Interpreting quantifiers • Informally, every takes twopredicates, and yields trueif everything that satisfiesthe first predicate alsosatisfies the second. • <<e,t>,<<e,t>,t>> • [every]M,g =P [ Q [x [P(x) Q(x)] ] ] S DP S N 1 S Det every fish DP VP Bond Vt t1 likes

Interpreting quantifiers • [every]M,g =P [ Q [x [P(x) Q(x)] ] ] • [N]M,g = [fish]M,g =y [y is a fish in M] • [DP]M,g =[every]M,g ( [fish]M,g ) =P[Q [x [P(x) Q(x)] ] ] (y[y is a fish] )=Q [x [y [y is a fish](x)Q(x)] ] =Q [x [x is a fish Q(x)] ] S DP S N 1 S Det every fish DP VP Bond Vt t1 likes

Interpreting quantifiers • [DP]M,g =Q [x [x is a fish Q(x)] ] • [S]M,g =y [BOND likes y] • [S]M,g = Q [x [x is a fish Q(x)] ] (y [BOND likes y]) =x [x is a fish  y [BOND likes y](x)] =x [x is a fish  BOND likesx] S DP S N 1 S Det every fish DP VP Bond Vt t1 likes

Phew • And, we’ve done it. We’ve derived the truth conditions for Bond likes every fish: • x [x is a fish in M  BOND likes x ] • For every individual x, if x is a fish, then Bond likes x.

CAS LX 502 Semantics