Hi. I’m Chris Potts. The goal of this screencast is to review our basic theory of the compositional

semantics of adjectives. First, let’s review the typology that we developed

in class, based on the reading by Barbara Partee. We say that an adjective ADJ is intersective

if and only if, when we combine it with a noun N, the resulting meaning is just the intersection of the meaning of ADJ and

the meaning of N. For instance, if we treat the ADJ ‘Swedish’

as denoting the set of all Swedes, that is, if we assume it can have an intersective

analysis, and if we treat the N ‘student’ as denoting

the set of all students, then their semantic combination is just the

intersection — the set of all things that are both Swedish

and students. We say that an adjective ADJ is subsective

if and only if combining ADJ with any noun N returns a set

that is a subset of the original noun N. The adjective ‘skillful’ is a clear example. A skillful spy is certainly a spy. This alone doesn’t say *which* set we end

up with. That has to be built into each adjective meaning. Rather, it just says that we’re constrained

to be somewhere inside the original N. It’s important to note that, by these two definitions, intersective adjectives are just special kinds

of subsective adjectives. This is true because of the set-theoretic

principle that says, for any sets A and B, A intersected with B is a subset of A, and A intersected with B is a subset of B. So all intersective adjectives are subsective. Here’s the beginnings of a typology for these

adjective types, with intersective as a special case of the larger class of subsective adjectives. An adjective ADJ is nonsubsective if and only

if we can find at least one N for which the meaning

of ‘ADJ N’ is not contained in the meaning of N. A clear example of a nonsubsective adjective

is ‘alleged’. An alleged spy is not necessarily a spy, but it might be. This is likely to be the largest class of

adjectives, because this definition is so permissive. First of all, all it takes is one single N for which ‘ADJ N’ isn’t a subset of N semantically. There might be many N’s for which ‘ADJ N’

is inside N. We just find one that isn’t, and the ADJ is classified as nonsubsective. In addition, the core requirement itself is

really weak. It just says that we can find at least one

entity that is in ADJ N but not in N. There might be many that are. In other words, we might fail the subsective

requirement by the smallest margin, and that’s enough for nonsubsective. Here’s how we expand the typology of adjectives

with the nonsubsective ones. Finally, the privative adjectives are a special

case of nonsubsective ones. So we find them inside the space of nonsubsective

adjectives. Here, the condition is that ‘ADJ N’ is outside

the set N. Adjectives like ‘former’ seem to be privative. But probably the clearest case is the prefix

‘non-‘ as in ‘non-student’. It’s not an adjective syntactically, but it’s the best example for getting at the

privative intuition: the students and the non-students seem clearly

to be disjoint. By the typology, every privative adjective

is nonsubsective. Privative is the special case where there is *no* overlap between the two

sets. So privativity is a special, and especially strong, kind of nonsubsectivity. Our next step is to begin to develop an account of how these adjectives work compositionally

— that is, how they combine with nouns to create

new meanings. We have two rules for this. The first handles only intersective adjectives. It basically just repeats the definition of

intersectivity, by saying that the meaning of the modified

noun is the intersective of the meaning of N with

the meaning of ADJ. Things are less transparent for all the other

cases. For them, we have a separate composition rule

that says the meaning of ‘ADJ N’ is the *functional*

meaning of the adjective applied to the noun. That’s what this notation here means. It presumes that ADJ is not a set, but rather

a function, and it applies that function to the N to return

a new set of entities. Which set? That’s determined, not by the rule, but rather by the functional meaning of the

adjective itself as given in the lexicon. To make this a bit more concrete, let’s look at some schematic functional meanings

for non-intersective adjectives. Here’s one. On the left, we have sets of entities. These are the inputs to the function. The arrow shows what happens when each input

is received. It is mapped to a new set: the meaning of

the modified N. So the adjective’s meaning isn’t any particular

set, the way it would be for an intersective adjective. Rather, it’s this big abstract machine of

sorts. The meaning given here satisfies the definition

for being subsective: for each input on the left, the corresponding output set is a subset of

that input. Hence, we’d place this in the set of subsective

adjectives. Here’s a nonsubsective adjective. The modified set is sometimes a subset of

the input, as in the bottom line, but it often isn’t, hence this weak designation. Finally, here’s a privative adjective: the output is always disjoint from the input. So, those are the mechanics. Intersective, that’s easy. All the others seem complex. It’s worth then addressing the question of

why we can’t treat every adjective as intersective. Why not just say all adjectives denote sets? I think the clearest argument is one where we try to assume they can all

be treated as sets and show that it leads to a contradiction. Now assume that Bart Simpson is a skillful

skateboarder, and assume also that he is a student, though not a skillful student. Now, if ‘skillful’ denotes a set, then it’s clear that this amounts to saying that Bart is in the intersection of the skillful

things and the skateboarders. Thus, of course, he has to be in the set ‘skillful’. It thus follows from our assumption that he’s

a student that he’s also a skillful student. This contradicts our previous assumption. Where did we go wrong? It was in assuming that we could define a

set of skillful things independently of the N being modified. Being skillful at one thing doesn’t make you

skillful at all things. But the intersective analysis would force

us to just such an absurd position. The functional meaning for ‘skillful’ avoids

this, because it lets us say, for each input N, what the resulting output set is. Let’s close by seeing how all these pieces

fit together. Here’s a basic syntactic structure for the phrase ‘alleged female Swedish spy’. This has three adjective stacked up. Interpreting it is no problem given the assumptions

we’ve made. If we start with the meaning of ‘spy’ as this

set here, and we modify it with the intersective adjective

‘Swedish’, then the meaning of that first node is just

the set of things that are both Swedish and spies. Next, we can treat ‘female’ as intersective

as well. Thus, we can use the intersective rule again: the meaning of ‘female’ is intersected with

the meaning we just calculated, and the result is the set of entities that

are female and Swedish and spies. Finally, ‘alleged’ isn’t intersective, but rather nonsubsective. It’s looking for sets as inputs. Luckily, it’s got one to its right. So it takes that set as input, and the result is another set, the output. We’ll stop here, but we could continue indefinitely stacking

adjectives. That’s the beauty of semantic composition. From basic assumptions, we can often interpret an endless number of

structures.