Linguist 130a – Semantic composition 3: Semantic grammar
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Linguist 130a – Semantic composition 3: Semantic grammar

August 23, 2019

Hi. I’m Chris Potts. This is the third of three screencasts on
compositional semantic grammars. The first covered some technical preliminaries. The second reviewed the semantic lexicon that serves as the basis for the grammar presented
here. You can think of that lexicon as the raw ingredients, and the rules presented here combine those
ingredients into complex meanings. Our semantic grammar consists of a series
of rules that tell you what to do with various local
syntactic structures. Here’s the most basic rule. It’s called “NB” for “non-branching”. It says that, if you see a non-branching node
in the syntax, you should just pass up the meaning of the
child to its parent. This is kind of a book-keeping rule. You’re just passing up the meaning unmodified. Our second rule is more interesting. It’s called S because it represents the most
basic way of obtaining a sentence meaning from a proper
name (a PN) combining with a verb phrase (a VP) to get the meaning of the whole structure. The semantic part of the rule says, “apply the VP meaning to the PN meaning”. The third rule is “A” for “adjective. It explains how to handle adjectival modifiers
of NPs. The rule is similar to S on the semantic side: we again do function application, but here it’s the left child rather than the
right child that acts as the functor. This is because, in our lexicon, we defined adjectives as functions looking
for noun-type meanings as their arguments. Rule N, for negation. This is a bit of a hack. We just assume that the negation is tacked
on to the VP. It acts as the functor on that VP. Of course, we don’t actually say things like
“Kim not run”. Rather, we say “Kim doesn’t run”. As semanticists, maybe we get a little leeway
in terms of ignoring the fact that “do” appears here in English. Next up is rule TV. It too derives VP meanings. “TV” stands for “transitive verb”. The rule says that transitive verb meanings
apply to the PN objects semantically. The result is a one-place predicate, just like both of the VP nodes in rule N. Finally, we have two rules for handling quantificational
determiners, since they have two arguments. Rule Q1 says that a determiner D takes its
restriction as an argument semantically. And Rule Q2 says that the output of Q1 applies
to the VP to create the overall meaning of the sentence. So we have two ways of making an S meaning. Notice that they differ in which item is the
function and which is the argument. For the rule on the left, the VP is the functor. For the rule on the right, it’s the QP. We also have two ways of creating VPs, and they are potentially inter-connected. For instance, I can use rule TV to create
a VP node, and then rule N can negate it. And of course rule N is also happy to take
simple intransitive VPs like ‘run’ as its lower VP, VP-sub-i. In other words, a transitive verb with its
object argument in is just like an intransitive verb semantically. Let’s now walk through two derivations aimed at showing how the grammar rules work
together. We start with this simple sentence ‘Bart skateboards’. We’ll use the grammar to interpret it compositionally
— every node will have an independent meaning, and that meaning will either come from the
lexicon, in the case of the leaf nodes, or else be derived from it by its children
according to a grammar rule. We start with the meaning of the word ‘skateboards’. Now rule (NB) tells us how to handle this
non-branching structure. The non-branching structure matches the rule’s
syntactic template, and this tells us what to do meaning-wise. In the syntax, we have one more non-branching
structure here, so we again use rule NB. Here, to reveal more of what it’s happening, I unpacked the meaning of ‘skateboard’ into
a lambda expression. So we’re operating under the highly simplified
assumption that the skateboarders are just Bart and Homer. Now we process the subject. We begin with Bart. Rule NB tells us to pass him up unmodified, because we again match NB’s template. Finally, we get to use a new rule: S. The matching is triggered by the S node branching
to PN and VP. Notice that we cannot use rule Q2, which requires a QP as its left argument. On the meaning side, the rule says, “apply the VP meaning to the NP meaning”. So we do that. We can further reduce the term by substituting
Bart in for x, and that takes us to the final value — which happens to be true. Let’s do one more example. We’ll again derive an S meaning, but now we’ll use rule Q2 to do it. This example is more involved than the previous
one. We start with ‘every child skateboards’ in
the syntax. We have to apply rule NB a bunch of times
to process the non-branching nodes. Here I’ve done all those steps. The only noteworthy thing is that I also unpacked
the meaning of ‘every’, since we want to see how it functions as the
primary piece of the composition. We first put together the determiner ‘every’
with its restriction ‘child’. That’s an application of Q1. Intuitively, what happens is that the meaning
of ‘child’ knocks out the “lambda f” in ‘every’, and goes in for f in the body of that expression, which delivers the entire meaning of the subject
‘every child’. Finally, we use rule Q2 to get the sentence
meaning, because we match its template syntactically. The subject meaning is the functor. It applies to the VP meaning [[skateboards]]. which knocks out the “lambda g”, and goes in for the variable g in the body
of the expression, which delivers this final expression. Notice that this is just saying we have a
value of T if the children (more precisely, the characteristic
set of the ‘child’ function) is a subset of the skateboarders. So we have derived the final meaning, which corresponds to a claim about the world, as we would expect given our intuitions about
the sentence we’re analyzing.

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