1. Interpreting Language (with Logic)
Semantics II
Interpreting Language (with Logic)

2. Primary Objectives
Continue our study of how meanings in natural language are
Represented
Constructed
Two examples:
More on adjectives, sets, etc.
How quantification with every, some, etc. is represented

3. Background
Recall that the guiding principle for studying objects larger than single terminals is that meanings are built out of their constituent parts:
Principle of Compositionality: The meaning of a whole is a function of the meaning of the parts and of the way they are syntactically combined. (associated with Frege; cf. Partee reading)
Note crucially that there are two components to this:
What the parts mean
How the parts are combined
We will review these two components with reference to some of the adjective examples we studied last time

4. The Meaning of the Parts
Recall that for some adjectives, we made use of the idea that the interpretation of ADJ N involved intersection:
RULE (informal): When an adjective A modifies a noun N ([A N]), the interpretation of this object is the set defined by the intersection of A’s meaning with N’s meaning
This rule accounts for the interpretation of e.g. red book, as the intersection of two sets.

5. Other types of adjectives
We saw one type of adjective that is not (necessarily) intersective before:
Larry is a skillful artist.
Larry is a chess player
Therefore: Larry is a skillful chess player. (Doesn’t work)
So one thing that we have to know is what kind of adjective we are dealing with
In addition, we’ll need to know what syntactic structure we have
Remember, these are the two components of compositionality

6. Recall… From last time as well: Larry is a poisonous snake
Larry is a chess player.
Therefore: Larry is a poisonous chess player
The phrase poisonous chess player is ambiguous…it can also mean that he’s not poisonous per se, but as a chess player, he is.

7. For example…(again)
So, with poisonous chess player, it seems that some adjectives can be interpreted in either fashion. Recall also:
Larry is a beautiful dancer.
Meaning1: He dances beautifully
Meaning2: He is beautiful, and he is a dancer (he might dance poorly)
Question: Do these differences involve different structures, or just a lexically ambiguous set of adjectives??
I.e., different compositional analyses can be given, which differ in terms of how much they build into the lexical item, or into the structure

8. Still another type Consider a further type of adjective:
John is a former chess player.
Adjectives like former (including alleged, counterfeit, etc.) are:
Not intersective: || former chess player|| is not the intersection of ||former|| and ||chess player||
Not like skillful type adjectives either:
||skillful chess player|| is a subset of ||chess player||
But ||former chess player|| is not a subset of ||chess player||
Aside: John is tall/skillful/*former
All of these things are syntactically Adjectives; but how they combine to create larger meanings is determined in part by how they differ from one another
How to represent such differences goes beyond what we’ll do; at this point, we will examine a second factor, syntactic structure

9. Structure
One simple case illustrating structural differences involves adjectives combining with nouns either (1) in phrases, vs. (2) in compounds.
Example:
Phrase: black board. Meaning: here it is intersective (a thing that is both black and a board)
Compound: bláckboard. Meaning: thing that we write on with chalk. Not intersective! A blackboard could be e.g. green.
So: how things are put together is crucial.
In a sense, this recapitulates what we saw in our study of word structure and syntax (remember ‘unlockable’ and ‘fix the car with a wrench’); the ambiguities correspond to different structures…

10. Other examples of structure
Consider some further examples:
John hammered the metal.
John hammered the metal flat.
In the second example, the adjective flat defines the state that the metal moves towards by being hammered.
Now, how about:
John hammered the metal naked.
In the second example here, we understand the adjective as defining the state that John was in when he undertook the hammering of the metal
However, the structural position of modifiers like the naked adjective is in principle compatible with both subject and object:
John met Bill naked. (=John or Bill)

11. However….
When things like naked appear in the VP, they can be interpreted with either the subject or the object, if it makes sense
Interestingly: Further examples show that the flat type adjectives and the naked type are in different syntactic positions:
John hammered the metal flat naked.
*John hammered the metal naked flat.
The second example is deviant because it seems that the first of the two adjectives must go with the object; and in this case, that doesn’t make sense

12. Interim summary
When it comes to building meanings, two primary factors must be taken into account:
What individual elements (e.g. specific classes of adjectives in the examples above) mean
What syntactic structures these elements appear in
As we have been noting throughout, there are clear correlations between structure and meaning. What we have added in this discussion is the further idea that what the individual words mean can also have an effect on how meanings are derived.

13. Quantifiers: More Ambiguities
Thus far we have seen different ways in which an ambiguity may arise:
Structural ambiguity:
Unlockable = [[un lock] able] or [un [lockable]
Fix the car with a hammer = PP modifies VP, or PP modifies NP
Lexical ambiguity (from homophony/polysemy)
The pool made the party a lot better.
= swimming pool
= game of pool
= entertaining rain puddle
Another type of ambiguity is found in (certain) sentences with more than one quantifier (like every, etc.):
Every student read some book.

14. The ambiguity Every student read some book.
Reading1: Every student read some book or other (different books)
Reading2: Every student read the same book
Such ambiguities arise in other cases as well; consider:
A student is certain to solve this problem.
Reading1: Some student or other is going to solve this problem
Reading 2: A particular student is going to solve this problem (e.g. Mary)
(think about it…)
These are often called scope ambiguities; see below
In order to explain the nature of such ambiguities, we will look at some simple logic

15. Introducing logic We already saw some basic logic in the last slides
In effect, we talked about the operation of intersection, which we associated with AND
In order to make our logic powerful enough to talk about the ambiguity introduced above, we’ll have to add something to it

16. Interpreting Quantifiers
Understanding the nature of the problem here requires some assumptions about quantifiers.
In logical analysis, quantifiers are interpreted with respect to some domain; think of this as a world. We’ll introduce a restricted world below.
Quantifiers don’t seem to refer to things in the way that things like cat do. Consider:
No students went to the library.
What would no students refer to??
In order to see how quantifiers are interpreted, it is useful to have a small domain (think of it like a model) to look at what our logical statements mean.
I choose….

17. A restricted domain
Let’s illustrate with respect to a simple domain how the quantifiers work.
We have a domain (in this example, a set of characters from Sesame Street):
Who is missing…?

18. Back to reality: some basic logic
In our logic, we need names for individuals:
ernie  Ernie
bb  Big bird
elmo  Elmo
Etc…
We also need predicates, which are sets of individuals: e.g., red, blue, googly-eyed; these apply to one argument (see below)
These predicates represent sets, like in our adjective examples; in this world:
||blue|| = {grover, cookie monster…}
||googly-eyed} = {cookie monster}
Etc.
We can then write simple statements, and judge whether or not they are true with respect to our model

19. Example statements Some things that we could say (with truth value)
Blue(cm) ‘cookie monster is blue’; true
Red(bb) ‘big bird is red’; false
And so on
We can also have predicates with two places; e.g. Taller(x,y) for ‘x is taller than y’:
Taller(bb,cm) true (big bird is taller than cookie monster)
Within this system, we can also define “and”, “or”, “not”, “if…then”; e.g. blue(cm) AND red(elmo)
How are we going to say things like ‘some things are red’, ‘no thing is an NBA player’, and so on? This is where we need a way of representing quantification

20. Two quantifiers
Quantifiers come with variables, presented here as x, y, etc.
Existential Quantification:
This is written with a ‘backwards E’
It is read as ‘there exists an x such that…’
Example: x BLUE(x)
This means ‘there exists an x such that x is blue’
In our model, this is true; we can find individuals in the denotation of BLUE
The other quantifier we need is one that says ‘every…’

21. Universal Quantification
Universal Quantification: Represents in logical the meaning of ‘every’ or ‘all’
This is written with an upside-down ‘A’
Example: (let the predicate Ses be ‘is a Sesame Street character’)
x Ses(x)
This is read as ‘for all x, x is a Sesame Street character’
This is true in our model, but not in other models, e.g. the real word.
Meanings like ‘no’ involve the quantifiers above and negation

22. Now, Returning to two Quantifiers
Remember that we launched into this investigation of logic in order to understand the two meanings of examples like Every student read some book. Simplifying:
Every student read some book.
To simplify, we’ll look at:
Everyone saw someone
Which has the same ambiguity
In our logic, the two readings have unambiguous statements

23. Representing the readings
Reading1: Everyone saw some person or other
 x y (Saw(x,y))
Read as: ‘For all x, there exists some y such that x saw y’
Reading 2: Everyone saw the same person.
y  x (Saw(x,y))
Read as: ‘There exists a y such that for all x, x saw y’
The question for research in natural language semantics is how a single sentence/structure like that of Everyone saw someone can have or correspond to these distinct logical representations
That is, why is it that the single sentence, with someone as object, can correspond to a meaning in which the existential quantifier is outside of the universal?