CAS LX 502 Semantics 5 b Pronouns assignments

CAS LX 502 Semantics 5 b. Pronouns, assignments, and quantifiers 5. 7(. 1), 6. 1

Names and pronouns n n n 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 n 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. n She is hungry. n n In order to know the conditions under which he is boring is true, we need know who is being “pointed at. ”

Pronoun resolution n Pronouns are a form of a more general class of nouns known as anaphora. n n Resolving pronouns is not a trivial matter, although we will not concern ourselves really with how it happens. n n The primary characteristic of anaphora is that they refer, but their referent is not fixed intrinsically. They get their reference from somewhere else. 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” n n 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 n n 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. n n Pronouns can also “shift” in reference too: n n John lost his keys. Every boy lost his keys. There are two meanings here, a constant one, and a “shifting” one.

Quantifiers n Quantifiers (everyone, someone, noone) allow us to state generalizations. n n 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. n 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. n 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. n

Interpreting quantifiers n Before we get to the technical details, we can think about what a quantifier like only John seems to do. n n This is true when: n n n Only John is hungry 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> n n n 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. n 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> n 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. n Two people are hungry. n Most people are hungry. n n This can’t be how it works.

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

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

Every person is hungry n n 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)] n n 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 n n 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)] n 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 n For something like n n n 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 n 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. n n n

Interpreting quantifiers Only John lost his keys n 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. n n n n 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. n n n 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 n n n n 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 n n n 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. n n 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 S For now, N going back to is hungry Bond as a Vi, for simplicity n n n VP Vi is hungry … Bond Loren Pavarotti … F [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 U

Bond is hungry S N VP Bond Vi is hungry n … is hungry … Bond Loren Pavarotti … F 1 [S]M 1 = BOND is hungry in M 1 = true in the specific situation M 1. U 1

He is hungry S N VP He Vi is hungry n n … is hungry … Bond Loren Pavarotti … F 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. U

He 1 is hungry S N VP He 1 Vi is hungry n n n … is hungry … Bond Loren Pavarotti … F g 1 2 3 … When writing a sentence like he is hungry, the standard practice is to indicate the “pointing” relation by using a subscript on he: He 1 is hungry. The idea here is that this is interpreted in conjunction with a “pointing function” that tells us who “ 1” points to. U

He 1 is hungry S N VP He 1 Vi is hungry n n n F … is hungry … Bond Loren Pavarotti … g 1 1 2 3 … Where different people are being pointed to, we use different subscripts: He 1 likes her 2, but he 3 hasn’t noticed. The “pointing function” goes by the more official name assignment function, and is generally referred to as g. U

He 1 is hungry S N VP He 1 Vi is hungry n n F … is hungry … Bond Loren Pavarotti … g 1 1 2 3 … The 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) U

He 1 is hungry S N VP He 1 Vi is hungry n F 1 … is hungry … Bond Loren Pavarotti … g 1 1 2 3 … [ a n ] M, g = g ( n ) U 1 n [S]M 1, g 1 = g 1(1) is hungry in M 1 = BOND is hungry in M 1 = true in the specific model M 1

Bond likes everyone n n So, what we’re after is something like this: x U [Bond likes x in M] n 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. n 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 n n 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 Bond VP Vt likes DP Det every N fish

Bond likes every fish n Back to the problem of quantifiers. Consider the meaning of Bond likes every fish. It should be something like: n For every x in U that is a fish, Bond likes x. n Or: x U [x is a fish in M Bond likes x in M]

Bond likes every fish n x U [x is a fish in M Bond likes x in M] n 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. n In order to get this interpretation, we’re going to introduce a transformation. A new kind of rule.

Transformations n n n 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 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 Raising S S DP Bond DP VP Vt likes DP Det every N fish S Det N every fish 1 S DP Bond VP Vt likes t 1

Quantifier Raising n Quantifier Raising [S X DP Y ] [S DP [S i [S X ti Y ] ] ] n That is: S … DP … S DP S i S … ti …

Interpreting quantifiers n 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). n Remember, what we’re after is: x U [x is a fish in M Bond likes x in M]

Interpreting quantifiers n Let’s start with the lower S. We know how to interpret that, it is essentially just Bond likes it 1. n S DP S Det N every fish True if BOND likes g(1) in M 1 S DP Bond VP Vt likes t 1

Interpreting quantifiers n n n The purpose of the 1 node is to make a property out of this sentence. The property will be, in effect, things Bond likes. Goal: 1 [BOND likes g(1) in M] S DP S Det N every fish 1 S DP Bond VP Vt likes t 1

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

[ ]M, g[1/x] n n n 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 he 2. So, g 1 (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] n g 1 maps 1 to Pavarotti, 2 to Nemo, 3 to Loren, … n 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: g 1(1) = Pavarotti, g 1(2) = Nemo g 1[2/Bond](1) = Pavarotti, g 1[2/Bond](2) = Bond n n

[ ]M, g[1/x] n n g 1(1) = Pavarotti, g 1(2) = Nemo g 1[2/Bond](1) = Pavarotti, g 1[2/Bond](2) = Bond n 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. n 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 n [1]M, g = S [ x [ [S]M, g[1/x] ] ] S DP Det n every [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 N fish 1 S DP Bond VP Vt likes t 1

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

Interpreting quantifiers n n n 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 Det N every fish 1 S DP Bond VP Vt likes t 1

Interpreting quantifiers n n n Informally, every takes two predicates, and yields true if everything that satisfies the first predicate also satisfies the second. <<e, t>, t>> S DP S Det N every fish [every]M, g = P [ Q [ x [P(x) Q(x)] ] ] 1 S DP Bond VP Vt likes t 1

Interpreting quantifiers n n [every]M, g = P [ Q [ x [P(x) Q(x)] ] ] S [N]M, g = [fish]M, g = y [y is a fish in M] DP Det n [DP]M, g = every [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 N fish 1 S DP Bond VP Vt likes t 1

Interpreting quantifiers n n n [DP]M, g = Q [ x [x is a fish Q(x)] ] S [S ]M, g = y [BOND likes y] DP S Det N every fish [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 likes x] 1 S DP Bond VP Vt likes t 1

Phew n n n 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.

New semantic rules n The new semantic rules we needed (lexical entries, we’re still using the same Functional Application and Pass-up rules) were: [every]M, g = P [ Q [ x [P(x) Q(x)] ] ] n [i]M, g = S [ x [ [S]M, g[i/x] ] ] n

Loren hates a book n We’ve 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). n 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: n x [x is a book LOREN hates x ]

Loren hates a book n n Without 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 [ x [P(x) Q(x)] ] ]

Comments about QR n Quantifier Raising [S X DP Y ] [S DP [S i [S X ti Y ] ] ] n As it is stated, QR applies to any DP, 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. n n A transitive verb needs something of type <e>. A quantifier needs something of type <e, t> (and is itself of type <<e, t>, t>>). Neither one can take the other as an argument.

Comments about QR n Quantifier Raising [S X DP Y ] [S DP [S i [S X ti Y ] ] ] n As it is stated, QR applies to any DP, 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 hadn’t done QR. So, you don’t really need QR to interpret the structure. n

Comments about QR n Quantifier Raising [S X DP Y ] [S DP [S i [S X ti Y ] ] ] n As it is stated, QR applies to any DP, whether it is a quantifier or not. So, when do you use QR? In fact, it turns out that when you have a quantificational DP as a subject (as in Every fish hates The Last Juror), you don’t actually need QR in order to interpret the structure either. n n 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 QR n Quantifier Raising [S X DP Y ] [S DP [S i [S X ti Y ] ] ] n As it is stated, QR applies to any DP, whether it is a quantifier or not. So, when do you use QR? Moreover, if you apply QR to a DP, you still have an NP—you could apply QR again to that same NP if you wanted. Again, like with names, this won’t affect the ultimate interpretation, it will just increase the amount of effort necessary to work out the truth conditions. n

Comments about QR n Quantifier Raising [S X DP Y ] [S DP [S i [S X ti Y ] ] ] n As it is stated, QR applies to any DP, whether it is a quantifier or not. So, when do you use QR? n 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 book n Some 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 don’t require QR in order to be interpretable. n 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. n 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).