Notes on Assignment 2

And

1. There are various ways of handling this example; here I will lay out one of the simplest (if not the most general), since the bigger questions raised by Part 2 are independent of particular assumptions in Part 1 (mostly). The first step is to assume a lexical entry for and whereby it denotes the identity function on sets.

• For any set X, [[and]](X) = X

And is therefore a 1-place set-theoretic operation, and can combine with its complement in (1b) using 1-Place Application.

We now adopt a version of the 'Modification' composition rule discussed in class:

• Modification
  If A is a constituent with daughters B, C such that [[B]] and [[C]] are sets, then [[A]] = [[B]] intersect [[C]].

Putting all this together lets us interpret (1b) as follows:

• [[ [Kim [ smoke [ and [ drink ] ] ] ] ]] = 1 iff
  [[Kim]] is in [[ [ smoke [ and [ drink ] ] ] ]] (Predication)
  Kim is in [[ [ smoke [ and [ drink ] ] ] ]] (lexical entry of 'Kim')
  Kim is in [[smoke]] intersect [[ [ and [ drink ] ] ]] (Modification)
  Kim is in {x | x smoke} intersect [[ [ and [ drink ] ] ]] (lex: 'smoke')
  Kim is in {x | x smoke} intersect [[and]]([[drink]]) (1-place application
  Kim is in {x | x smoke} intersect [[drink]] (lex: 'and')
  Kim is in {x | x smoke} intersect {y | y drink} (lex: 'drink')

2. These assumptions cannot handle the structure in (2b). The central problem is that we have three kinds of 'type mismatch', which are listed below along with potential fixes:

1. In (2b), 'and' combines with the sentence 'Kim drink'. The semantic type of a sentence is a truth value, but the denotation of 'and' provided is defined only for arguments that are sets. We could fix this by redefining 'and' so that it denotes the identity function for any input: [[and]](x) = x for any x.
2. Even if we gave 'and' a meaning that would allow it to combine with something that denotes a truth value, we wouldn't be able to interpret the constituent [and [ Kim drink ] ] in (2b) because our 1-place application rule is defined only for set denoting expressions and expressions that denote set-theoretic operations. We could fix this by generalizing 1-place application along the following lines: if A is a constituent consisting of daughters B and C, and [[B]] is an operation on expressions of type [[C]], then [[A]] = [[B]]([[C]]).
3. Even after thesse first two fixes, we still don't have a way of interpreting the whole thing. The result of our modifications will be that [and [ Kim drink ] ] denotes a truth value, so the whole coordination consists of two truth value denoting expressions. We don't have a composition rule for handling this, though we could in principle add one that gets the results we want: If A is a constituent consisting of B, C such that [[B]], [[C]] denote truth values, then [[A]] = 1 iff [[B]] = 1 and [[C]] = 1.

It should be fairly easy to see that (2b) and (1b) are going to be true in the same situations: (2b) is going to be true just in case [[ Kim smoke ]] and [[ Kim drink ]] are true, according to our new rule; but this will be the case only if Kim is in the set of smokers and Kim is in the set of drinkers, given Predication. This is equivalent to saying that Kim is in the interesection of the set of smokers and drinkers, which is what the truth conditions of (1b) require.

Now, this solution works for this particular case, but how general is it? In some sense, it seems plausible: treating 'and' as denoting the identity function has the advantage of predicting that it should in principle be able to combine with constituents of any type, which does seem to be the case. Furthermore, our new rule in (3) will handle constituents consisting of two sentences that are not conjoined by 'and' in the same way as sentences that are. Some people have argued that discourses should be thought of in this way, e.g. examples like the following:

• Kim smokes. Kim drinks.

Whether it's right to think of these as instances of 'and-less' conjunction or not, one thing is clear: the whole discourse is 'true' (this actually makes sense only if we do treat the discourse as and-less conjunction, but beaer with me) just in case the two indpendent clauses are true. So we could in principle extend the analysis to such cases.

On the other hand, we are going to need a bunch of different composition rules to handle different types of conjunctions (conjoined VPs, conjoined Ss, conjoined NPs, ...). These rules are going to be very similar in character, so we would seem to end up missing a generalization that the particular set of rules we end up with is consistently associated with the word 'and' --- which itself (on this analysis) has no meaning! This doesn't seem like the best result, and instead suggests an alternative way of thinking about the problem: rethink the analysis of 'and'. And here the goal would be to come up with some sort of general meaning for the word that 1) allows it to combine with constituents of different semantic types, and 2) 'does the right thing' (produces the right semantic output) based on the types of constituents in combines with. We'll consider various options in the coming weeks.

That said, there is one other possibility that may be worth considering: perhaps our assumptions about the syntactic representations associated with (1a) and (2a) are wrong. If we could make the case that these two sentences actually have the same syntactic representations, then we could have one denotation for 'and' and one (relevant) composition rule. Is this a plausible option?