Notes on Assignment 5

1 Type shifting

A. Assume that the sentential-connective, type [t,tt] denotation for and given below is basic:

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

Let's deal with the transitive V-connecting and_TransV in (3b) first. For this case, the following rule does the trick, essentially by taking the conjoined verbs 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] --> [[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]]]

To deal with NP conjunction in an example like (3a) 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.

B. Unfortunately, this analysis will NOT extend to examples like Kim kissed 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: we get a type-mismatch between the (complex) direct object (the conjoined NP Lee and Pat) and the verb. So we need another option.

One option would be to just shift and to an even more complex type, one that has the effect of making a conjoined NP in object position a function that is looking for two arguments --- a transitive verb and a subject --- both of which get 'plugged in' to the two propositional arguments associated with the basic denotation of and along with the NPs that are understood as direct objects. The following denotation does the trick; I will leave it as an exercise to you to actually run through a derivation proving that it works.

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

C. Both of the analyses of NP-conjoining and that I presented here agree in one important respect: neither treats the conjoined NP as an expression of type e. Instead, both essentially claim that conjoined NPs become the main functional expresions in a sentence, and end up 'distributing' the other elements of the sentence across the subconsituents of the conjunct. This approach has the positive result that it lets us analyze different uses of and in terms of one basic denotation (plus type-shifting).

However, we should also ask whether there is any evidence that we sometimes need to treat conjoined NPs as type e expressions after all. If we do, then clearly type-shifting is not enough; we would really need (at least) two distinct meanings for and. The ambiguity of an example like the following, which has the two paraphrases listed below it, suggests that we may in fact need to say a bit more than we have so far:

• 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 clearly derives the first reading, since it assigns truth conditions in terms of conjoined propositions, but it's not so clear that it gets the second one. Solid evidence that it doesn't in fact get the second one comes from examples like the one on the assignment, which is perfectly acceptable, but if we try to spell it out in terms of the truth conditions we would assign, we see that we should predict it to be bad:

• Kim and Lee are a good team.
  #Kim is a good team and Lee is a good team.
  Kim and Lee are a good team together.

Clearly, the first paraphrase is not what we want. The problem here is that (be) a good team is a so-called 'collective predicate': it is something that can only apply to pluralities, as show by the following examples:

• The boys are a good team.
• #The boy is a good team.

The contrast between the singular and plural subjects in the previous example, and the acceptability of conjoined subjects, indicates that we need some other meaning for NP₁ and NP₂, one in which it denotes a plurality of entities rather than an expression that derives two conjoined propositions. The conclusion, then, is that even though type-shifting might be a useful way of capturing the distribution of and relative to one kind of meaning (the 'propositional connective' meaning), it won't help us relative to another kind of meaning (the 'plurality of individuals' meaning). So we need to posit an ambiguity after all.

Of course, it's also possible that we started from the wrong 'basic' meaning. Perhaps if we had been thinking in terms of pluralities all along, we would have had better luck with our generalized type-shifting analysis...?

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.