# Assignment 6

### Exercise 6, p. 105

Each of the sentences in this exercise have quite a few different logical representations, but many of them are truth-conditionally equivalent. Below I have only given representations for the different interpretations of the English sentences (rather than all the possible translations), so you should be aware that your translations may look different from mine. What is important is that you come up with the same number of truth-conditionally distinct representations, and that your logical representations accurately capture the (possible) meanings of the sentences. (I.e., make sure that your connectives are used correctly, and that all variables bound by a quantifier appear inside the quantifier's scope.)

i. A professor gave every student a book.

This sentence has two interpretations in which the universal quantifier has intermediate scope with respect to the two existential quantifiers:

ExAyEz[Student(y) --> [(Professor(x) & Book(z)) & Gave(x,y,z)]]

SCOPE RELATIONS: APROF - EVERY - ABOOK

The relative scopes of the quantifiers in this reading most closely mirrors the surface syntactic structure of the English sentence. Under this reading, a single professor gave every student a book, though the books might have been different for each student.

EzAyEx[Student(y) --> [(Professor(x) & Book(z)) & Gave(x,y,z)]]

SCOPE RELATIONS: ABOOK - EVERY - APROF

This logical representation corresponds to a reading in which every student got the same book, but it may have been given by different professors.

ExEzAy[Student(y) --> [(Professor(x) & Book(z)) & Gave(x,y,z)]]

SCOPE RELATIONS: ABOOK - APROF - EVERY

On this interpretation, the universal quantifier has narrow scope with respect to both existentials: a single professor gave every student the same book.

AyExEz[Student(y) --> [(Professor(x) & Book(z)) & Gave(x,y,z)]]

SCOPE RELATIONS: EVERY - ABOOK - APROF

Finally, (i) has a reading in which the universal quantifier takes wide scope with respect to both existentials. This logical form represents the interpretation of (i) in which different professors give every student different books.

Note that the two other logically possible orderings of the quantifiers (EzExAy and AyEzEx) don't affect the truth conditions because they differ only in the relative order of the two existential quantifiers.

#### Entailments

Since the Laws of Quantifier Independence (see deSwart p. 87) tell us that formulas of the form ExAy(PHI) entail those of the form AyEx(PHI), but not the other way around, and that reordering quantifiers of the same type doesn't change truth conditions, we can conclude that reading 3 entails all of the other readings, and readings 1 and 2 entail reading 4, but reading 4 doesn't entail any of the others.

ii. No professor gave a student two books.

A big problem with this sentence is, of course, that we don't have a quantifier that corresponds to the English word two. We can, however, get this meaning using two occurrences of an an existential quantifier. The basic idea can be informally described as follows. If John gave Bill two books, then:

ExEy[(John gave Bill x & x is a book) & (John gave Bill y & y is a book) & (x is not equal to y)]

This pseudo-logical representation (and the more precise one that we could build from it) wreak havoc upon the English sentence we started out with, but it accurately characterizes the sentence's truth conditions. In the discussion that follows (and the discussion of example (iii)), I will assume this sort of approach to the truth conditions for two.

~EwExEyEz[(Professor(w) & Student(x)) & (Book(y) & Book (z)) & (Gave(w,x,y) & Gave(w,x,z)) & ~(y = z)]

SCOPE RELATIONS: NO - A - TWO

This is the reading that mirrors the English syntax: there is no professor who is such that he gave a student two books. It might be the case that there are professors who gave a student one book, but no one gave away two. It's worth pointing out that this formula also prohibits professors from giving away more than two books, which seems right: without contrastive stress on two, if (ii) is true under this reading, it means that no professor gave away more than two books. (If a professor gave a studnent 3 books, then he also gave the student 2 books.) The addition of contrastive stress on two allows this sort of interpretation (No professor gave a student TWO books; every professor gave a student THREE books), but this is an instance of what Horn (see Horn's book A Natural History of Negation) would call ``metalinguistic negation'', which is a slightly different phenomenon.

EyEz~EwEx[(Professor(w) & Student(x)) & (Book(y) & Book (z)) & (Gave(w,x,y) & Gave(w,x,z)) & ~(y = z)]

SCOPE RELATIONS: TWO - NO - A

This logical form represents a reading in which two books (speaking as though two books were actually part of the logical representation!) has wide scope with respect to negation and a student has narrow scope with respect to negation: two books are such that no professor gave them to any student. There might be other books that were given out, but there is some pair (say, Syntactic Structures and Tropic of Cancer) that didn't get distributed. This logical representation allows for the possibilty that every student did in fact receive two books, it's just that none of them received the same two books.

Ex~Ew EyEz[(Professor(w) & Student(x)) & (Book(y) & Book (z)) & (Gave(w,x,y) & Gave(w,x,z)) & ~(y = z)]

SCOPE RELATIONS: A - NO - TWO

This reading is the mirror image of reading 2: there is some student who is such that no professor gave him two books. So this logical representation allows for the possibility that most (possibly all but one!) students did in fact receive two books, but there's one student out there who didn't.

EyEzEx~Ew[(Professor(w) & Student(x)) & (Book(y) & Book (z)) & (Gave(w,x,y) & Gave(w,x,z)) & ~(y = z)]

SCOPE RELATIONS: TWO - A - NO

This reading is very similar to reading 2, differing only in that a student has wide scope with respect to negation: two books are such that some (particular) student wasn't given them by any professor.

AwExEyEz[(Student(x)) & (Book(y) & Book (z)) & ~(x = y) & [Professor(w) --> ~(Gave(w,x,y) & Gave(w,x,z))]]]

This reading is nearly the same as reading 1, with one important exception: it requires the existence of books and students, while reading 1 does not. The important difference is that the existential quantifiers that bind the ``book'' and ``student'' variables are outside the scope of negation, which means that there have to be books and students in order for the sentence to be true. We can construct a series of different interpretations along this line, that correspond in quantifier ordering to readings 2-3 above:

EyEzAwEx[(Student(x)) & (Book(y) & Book (z)) & ~(x = y) & [Professor(w) --> ~(Gave(w,x,y) & Gave(w,x,z))]]] (analogous to reading 2)

ExEyEzAw[(Student(x)) & (Book(y) & Book (z)) & ~(x = y) & [Professor(w) --> ~(Gave(w,x,y) & Gave(w,x,z))]]] (analogous to reading 3)

These representations have truth conditions similar to those of readings 2 and 3 above; they differ in requiring the existence of students and books, respectively. This affects entailment relations, as discussed below.

#### Other representations

This sentence has two other (relevant) logical representations, which correspond to readings 1 and 4 above, but don't change the truth conditions, since they only involve shifting the relative ordering of existential quantifiers (exclusive of negation). They are, respectively: ~EwEyEzEx[(Professor(w) & Student(x)) & (Book(y) & Book (z)) & (Gave(w,x,y) & Gave(w,x,z)) & ~(y = z)]

SCOPE RELATIONS: NO - TWO - A

ExEyEz~Ew[(Professor(w) & Student(x)) & (Book(y) & Book (z)) & (Gave(w,x,y) & Gave(w,x,z)) & ~(y = z)]

SCOPE RELATIONS: A - TWO - NO

#### Entailments

~E is logically equivalent to A~ (by the rule of quantifier negation; see deSwart p. 73-74), and formulas of the form ExAy(PHI) entail those of the form AyEx(PHI), but this does not allow us to make an inference from an ordering like A NO, which corresponds to E A ~, to NO A. The problem is that the latter ordering actually corresponds to A~ E, which is not the same as A E~ (the entailment that we actually can get via the laws of quantifier independence). As a result, none of the readings of (ii) in 1-4 entail the other.

In particular, the first reading isn't entailed by the others, for the following important reason: reading 1 can be true if there aren't any books or students, while readings 2-4 require the existence of one or the other (or both).

Because the logical representations in readings 5-7 do have existential entailments, however, they also participate in the normal sorts of entailment relations we expect for quantifier orderings. Specifically, readings 6-7 entail reading 5; reading 4 entails reading 5; and reading 4 entails readings 6-7.

iii. Some delegate from every city attended two meetings.

Unfortunately, I didn't have time to finish this one before I left. The analysis is going to be very similar to that of (ii), with a few extra wrinkles. In particular, in order to account for readings in which all the delegates are distributed across two meetings (though neither meeting may have independently hosted all the delegates), we can use disjunction of the main predicate instead of conjunction, as above, to deal with the interpretation of two. For example, the reading of (iii) in which there are two meetings such that every city's delegate attended one of them is represented by the following logical representation:

EyEzAxEw[City(x) --> [(Delegate(w) & From(w,x)) & (Meeting(y) & Meeting(z)) & (Attend(w,y) v Attend(w,z))]]

iv. No student knows every linguist who wrote a book on semantics.

This sentence is quite similar to the ones Farkas discusses in her paper. What's interesting about it is that even though the indefinite NP a book on semantics is deeply embedded (it's contained in a relative clause), it can participate in scope interactions with the other quantifiers. (Recall that a universal quantifier in the same position can't scope out of the clause in which it appears---the scope of universal quantifiers is, for the most part, clause-bounded.)

For the following discussion, let S(x) = `x is a student', K(x,y) = `x knows y', L(x) = `x is a linguist', W(x,y) = `x wrote y', and B(x) = `x is a book on semantics'.

~ExAyEz[S(x) & [(L(y) & W(y,z) & B(z)) --> K(x,y)]]

SCOPE RELATIONS: NO - EVERY - A

No student is that for every linguist who wrote any kind of book on semantics, the student knows the linguist. This reading allows for cases in which there are students who know some linguists who wrote books on semantics; it disallows cases in which there are students who know all the linguists who wrote books on semantics.

~ExEzAy[S(x) & [(L(y) & W(y,z) & B(z)) --> K(x,y)]]

SCOPE RELATIONS: NO - A - EVERY

No student is such that for some book on semantics, he knows all of its (linguist) authors. This reading holds books constant for groups of linguists.

Ez~ExAy[S(x) & [(L(y) & W(y,z) & B(z)) --> K(x,y)]]

SCOPE RELATIONS: A - NO - EVERY

A book on semantics is such that no student knows every linguist who wrote it. This reading holds books constant across students and linguists; it's the so-called ``specific'' reading of the indefinite NP.

In principle, this sentence should also have readings in which the universal quantifier takes scope over negation (in readings 1-3, the universal is below negation). From my perspective, at least, these readings are either impossible or extremely hard to get. Here is what they would look like:

AyEz~Ex[S(x) & [(L(y) & W(y,z) & B(z)) --> K(x,y)]]

SCOPE RELATIONS: EVERY - A - NO

According to this logical representation, every linguist who wrote a book on semantics (not necessarily the same book) is known by no student. In other words, all the linguist-book-writers are unknown by students. I find this reading of (iv) to be virtually impossible, even though nothing we have said so far rules it out.

EzAy~Ex[S(x) & [(L(y) & W(y,z) & B(z)) --> K(x,y)]]

SCOPE RELATIONS: A - EVERY - NO

The reading corresponding to this representation differs from the previous one only in that a book on semantics has a specific interpretation: some book on semantics is such that every linguist who wrote it is known by no students. Again, I don't think that the EVERY - NO scope relation is a possible interpretation of (iv).

Do people agree or disagree with my intuitions on this? Let me know, and I'll report the judgments in class.

The answers to questions 7 and 8 will be posted later this week.