Notes on Assignment 4

1 Type shifting

Let's assume that the basic meaning of an adjective (like grey, wet, straight, etc) is the predicative, type [e,t] one. This means that an arbitrary adjective blicky_P has a denotation like the following:

• [[blicky_P]] = [^x in D_e.x is blicky]

Now we want to derive an attributive/modificational, type [et, et] meaning from the basic [e,t] one for an arbitrary adjective. The following rule does the trick:

• [et] --> [et,et]
  If A_P is an adjective such that [[A_P]] is type [e,t], then there is an A_A such that [[A_A]] = [^f in D_[et].[^y in D_e.[[A_P]](y) = 1 and f(y) = 1]].

According to this rule, we are licensed to assume an adjective blicky_A based on blicky_P with the following meaning:

• [[blicky_A]]
  = [^f in D_[et].[^y in D_e.[[blicky_P]](y) = 1 and f(y) = 1]]
  = [^f in D_[et].[^y in D_e.[^x in D_e.x is blicky](y) = 1 and f(y) = 1]] (by lex entry of [[blicky_P]])
  = [^f in D_[et].[^y in D_e.y is blicky and f(y) = 1]] (by lambda simplification)

This is exactly what we want: [[blicky_A]] denotes a function that will combine with a noun meaning [[wug]] and return a function from individuals to truth values that is true of an object y iff y is blicky and y is a wug. This is illustrated by the following derivation.

• [[ blicky_A wug ]]
  = [[blicky_A]]([[wug]]) (by Function Application)
  = [^f in D_[et].[^y in D_e.y is blicky and f(y) = 1]]([[wug]]) (by meaning of blicky_A, as derived above)
  = [^f in D_[et].[^y in D_e.y is blicky and f(y) = 1]]([^z in D_e.z is a wug]) (by lexical entry of wug)
  = [^y in D_e.y is blicky and [^z in D_e.z is a wug](y) = 1] (by lambda simplification)
  = [^y in D_e.y is blicky and y is a wug] (by lambda simplification)

Now assume that the attributive adjective blicky_A is the basic one. We need a rule that takes us from the type [et,et] meaning [^f in D_[et].[^y in D_e.y is blicky and f(y) = 1]] to a type [et] meaning that's equivalent to the one we started out with for blicky_P above. This is a bit more tricky, because instead of adding an argument as above, we need to delete one: the type [et] argument of blicky_A that corresponds to the modified noun. The simplest way to do this is just to saturate the [et] argument with a vacuous function, such as [^x in D_e.1] --- the function that takes an individual and returns back 1:

• [et,et] --> [et]
  If A_A is an adjective such that [[A_A]] is type [et,et], then there is an A_P such that [[A_P]] = [^y in D_e.[[A_A]]([^z in D_e.1])(y) = 1].

The following derivation shows that saturating the [et] argument of blicky_A in this way returns a type [et] function that is equivalent to the meaning of blicky_P that we started out with above --- the modified noun argument is effectively nullified.

• [[blicky_P]]
  = [^y in D_e.[[blicky_A]]([^x in D_e.1])(y) = 1]
  = [^y in D_e.[^f in D_[et].[^x in D_e.x is blicky and f(x) = 1]]([^z in D_e.1])(y) = 1] (lambda simplification)
  = [^y in D_e.[^x in D_e.x is blicky and [^z in D_e.1](x) = 1](y) = 1] (lambda simplification)
  = [^y in D_e.y is blicky and [^z in D_e.1](y) = 1] (lambda simplification)
  = [^y in D_e.y is blicky and 1 = 1] (lambda simplification)

The rules for the prepositions are given below. The strategies for the two cases are identical; the only difference is that the extra type e argument (the complement of the preposition) needs to be taken into account.

• [e,et] --> [e,[et,et]]
  If P_P is a preposition such that [[P_P]] is type [e, et], then there is a P_A such that [[P_A]] = [^x in D_e.[^f in D_[et].[^y in D_e.[[P_P]](x)(y) = 1 and f(y) = 1]]].

• [e,[et,et]] --> [e,et]
  If P_A is a preposition such that [[P_A]] is type [e,[et,et]], then there is a P_P such that [[P_P]] = [^x in D_e.[^y in D_e.[[P_A]](x)([^z in D_e.1])(y) = 1].

Now let's look at and, assuming that the sentential-connective, type [t,tt] meaning below is basic:

• [[and_S]] = [^q in D_t.[^p in D_t.p = 1 and q = 1]]

From this basic type, we want to derive meanings for NP-connnecting and_NP, VP-connecting and_VP, and transitive V-connecting and_TransV. Let's do the second two first, because they're the most straightforward. The following rules do the trick, essentially by taking the conjoined verbs/VPs and supplying them with type e arguments from the rest of the sentence, and using the resulting expressions to supply the type t arguments of and_S.

• [t,tt] --> [et, [et,et]]
  [[and_VP]] = [^g in D_et.[^f in D_et.[^x in D_e.[[and_S]](f(x))(g(x)) = 1]]]
• [t,tt] --> [[e,et], [[e,et],et]]]
  [[and_TransV]] = [^g in D_e,et.[^f in D_e,et.[^y in D_e.[^x in D_e. [[and_S]](f(y)(x))(g(y)(x)) = 1]]]

This is all pretty straightforward. To deal with NP conjunction in an example like Kim and Lee smoke, however, we have to do something we haven't seen before: we have to assign a meaning to and_NP in which a conjoined NP in subject position takes the VP as its argument, rather than the other way around. The type-shifting rule is defined below; if you look at it carefully, you'll see that it assigns the semantic type [et, t] to the conjoined NP.

• [t, tt] --> [e,[e,[et,t]]]
  [[and_NP]] = [^y in D_e.[^x in D_e.[^f in D_et.[[and_S]](f(x))(f(y)) = 1]]]

The reason we have to do this is because we're starting out with a meaning for and that works on two expressions of type t. The rule above gets things right by basically turning the conjoined NP into an expression that takes the VP as its argument and then applies the VP to each of the two conjoined NPs, finally supplying the resulting propositions as the arguments of the original and_S. If instead we had made the conjoined NP type e (which seems quite natural; in this case, and_NP would be type [e,ee]), there would be no way to give it a meaning based on the original type [t,tt] and_S. The best we could do would be to supply the conjoined NP as the argument of the VP twice, but that would end up giving us a meaning along the lines of Kim and Lee smoke and Kim and Lee smoke, which is obviously not what we want. Instead, we end up with something that is equivalent to Kim smokes and Lee smokes, which --- for this example, at least --- is exactly what we want.

An interesting question, though, is whether we can find evidence that we do sometimes want to treat conjoined NPs as type e --- as special kinds of individuals. The ambiguity of an example like the following, which has the two paraphrases listed below it, suggests that we do:

• Kim and Lee lifted a piano.
  Kim lifted a piano and Lee lifted a piano. (two piano lifting events)
  Kim and Lee lifted a piano together. (one piano lifting event)

Our semantics for and_NP gets the first reading, but it doesn't get the second one.

Finally, our analysis will NOT extend to examples like Kim saw Lee and Pat, in which the conjoined NP is the direct object of a verb. Saying that a conjoined NP is type [et,t] works OK for subjects, because something of type [et,t] can combine with a type [et] VP by Function Application, returning a truth value for the sentence. This is totally normal; it's just that the order of application is the opposite of what we've grown used to. However, we have no way at present to put together a type [et,t] NP and a type [e,et] transitive verb. So we're going to have to think of something else for conjoined NPs in object --- or any non-subject --- position.

2 Some (in)valid arguments

The easiest way to make the argument in (4) on the handout is to insert negation:

• Socrates is not smarter than every modern philosopher.
  Frege is a modern philosopher.
  /Socrates is not smarter than Frege.

It is clear that the first two premises can be true and the conclusion false. As long as there is one modern philosopher that Socrates is not smarter than, then the first premise is true. But that particular philosopher need not be Frege, so (assuming he is a modern philosopher), we can get true premises and a false conclusion.

It is not merely the addition of negation that renders the argument invalid, however. Consider the following example, which also includes negation, but which is clearly valid.

• Every senator thinks that he is not required to pay parking tickets.
  Kennedy is a senator.
  /Kennedy thinks that he is not required to pay parking tickets.

The crucial difference between the two examples is the relation between every NP and negation: in the first argument, negation `takes scope' over every, producing a not ... every interpretation; in the latter example, things are the other way around, giving a every ... not meaning. The semantic scope relations in these examples mirror the syntactic form: negation c-commands every NP in the first (invalid) argument; every NP c-commands negation in the second (valid) argument. The meaning we get is not alwasy reflected in the surface form, however, as shown by the following example:

• Every senator didn't vote for the bill.
  Kennedy is a senator.
  /Kennedy didn't vote for the bill.

The first premise of this example is ambiguous: it can either mean that no senator voted for the bill (the every ... not interpretation), or it can mean that not every senator voted for the bill (the not ... every interpretation). The argument is valid on the first reading and invalid on the second one, just as in the examples above. But this example also indicates that the mapping from form to meaning is more abstract than we have been assuming: we get both meanings, even though every NP c-commands negation in the surface form.

3 Antecedent-contained deletion

The following examples illustrate 'good' and 'bad' cases of antecedent-contained deletion, respectively.

• Jones [_VP1 read every book that Smith did [_VP2 -- ]].
• *Jones [_VP1proved that Smith did [_VP2 -- ]].

The bad example follows from our assumption that VP-ellipsis is licensed by a (syntactic or semantic) identity condition: there is no possible way for VP₂ to be identical to VP₁ when the latter contains the former, since that would require it to contain an occurrence of itself, resulting in infinite regress. However, according to this reasoning, the good example should also be bad, because the same structural configuration obtains: VP₂ is contained in the direct object of VP₁; since VPs contain their direct objects, it follows that VP₂ is contained in VP₁. So why is ellipsis ok here?

This is a question that we will answer in some detail in the coming weeks; for now, let me just point out two differences between these examples, which were observed by many people in the class. First, the that-clauses that contain the elided VPs in the two examples are syntactically and semantically distinct. In the bad example, the elided VP is contained in a clausal complement of the verb prove; in the good example, the elided VP is contained in a relative clause modifier of the noun book. Second, the good example involves quantification, and when we paraphrase its meaning, we can see that there is a certain independence between the two VPs:

• For every x such that x is a book and Smith read x, Jones read x.

This paraphrase shows that there is a link between the two VPs --- the one in the relative clause is being used to restrict the denotation of the noun phrase that provides the argument to the matrix VP --- but there is no actual antecedent containment in the paraphrase. This fact will be central to our analysis.

That said, it is clear that the analysis is not going to be simple. As many people pointed out, if we 'undo' ellipsis in good cases of ACD, we get a VP that is clearly NOT identical to its antecedent. This is illustrated by the following structure, which spells out the details of the relative clause, by inserting a relative pronoun (which is phonologically null in this example) and the trace in object position that it binds:

• Jones [_VP1 read every book [_CP wh₁ that Smith did [_VP2 read t₁]].

So our task is a difficult one: saying how VP₂ --- read t --- counts as identical to VP₁ --- read every book that Smith read t. As we will see, the answer to this question has a great deal of significance for our assumptions about the relation between syntax and semantics.