Semantics II

1
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 As
  meaning with Ns meaning
• This rule accounts for the interpretation of e.g.
  red book, as the intersection of two sets.

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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.
  (Doesnt work)
• So one thing that we have to know is what kind of
  adjective we are dealing with
• In addition, well need to know what syntactic
  structure we have

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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 ambiguousit
  can also mean that hes not poisonous per se, but
  as a chess player, he is.

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For example

• So, with poisonous chess player, it seems that
  some adjectives can be interpreted in either
  fashion. Here its more transparent
• 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??

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Further considerations

• Could we have contexts like the following?
• A There are lots of beautiful dancers here.
• B Yes, but Mary is the only beautiful beautiful
  dancer.
• If so, which adjective is the one with the
  dances beautifully interpretation, and which
  carries the is a beautiful person meaning?
• Consider further
• John is the only ugly beautiful dancer.
• John is the only beautiful ugly dancer.
• Question (for thought) Does this mean that the
  difference is reducible to structure?

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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 well do at this point, we will examine a
  second factor, syntactic structure

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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

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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.
• 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)

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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 doesnt make
  sense

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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.

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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.

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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 this ambiguity,
  we will look at some simple logic

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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. Well introduce a restricted world below.
  
• Quantifiers dont 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.

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A restricted domain

• Lets 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)

Bert couldnt make it.
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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

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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, ifthen 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
  quantifiers

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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

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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

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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, well look at
• Everyone saw someone
• Which has the same ambiguity
• In our logic, the two readings have unambiguous
  statements

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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?