Preclass: Recaps and Discussion
Exercises from Notes 5:
Exercise: Can you think of a way of preserving the relationships
implicit in (a) but achieving the interpretation implicit in (b)?
a. DP b. DP
/\ /\
/ \ / \
/ \ / \
DP REL / \
/\ /\ / NP
/ \ : : Det /\
/ \ : : | / \
Det NP : : | / \
| | : : | NP REL
| | : : | | : :
| | : : | | : :
every fish that Bill catches every fish that Bill catches
Compare Bach and Cooper 1978. A method that Robin
Cooper used to deal with adjoined relative clauses in Hittite was adapted for
English: a hidden variable over relative clause meanings is introduced
optionally, then bound at the level of NP's (DP's). The general technique has
come to be called "Cooper store/storage" and has been widely used as one
technique that in a sense circumvents strict compositionality.
Optional Exercise: discuss seek, compare with treatments in other
frameworks.
The Montague treatment involves two aspects. First, meaning postulates are
used to induce extensional interpretations for verbs (other than verbs like
seek) which require the existence of their objects. The intensional
meaning of sentences with verbs like seek is generated by derivations
with the DP in situ. This compares with most transformational
derivations, where quantifier phrases require Quantifier Raising to be
interpreted (or in Generative Semantics treatments Quantifier Lowering).
Meaning postulates will be discussed later in class.
Optional Exercise: show what happens with and without tenses:
Example: Every president runs.
Example: Every president will run.
Discussion: General point: quantifying in versus generation of quantifier
phrases in situ can interact with tenses and negation. Since the first
example is a simple present tense there will be no difference in truth
conditions, but with the second, the temporal location of the president will
vary. Note that this does not depend on there being a tense associated with
the nominal. Sketch derivations in class.
Exercise: lambda-abstractions (or some equivalent) have played a crucial role
in explicating the two readings of sentences like these:
- Sally washed her car and so did Nancy.
- Harry loves his house more than Ned does.
The two readings of sentences like these are provided by binding or not
binding a pronoun/variable in the verb-phrase: roughly:
λx[wash x's car(x)] vs. λx[wash y's car(x)].
- Quantification So Far
So far we've just followed the standard story about quantification in first
order logic, which is what Montague put into his PTQ. That is, Montague
showed how to build a fragment of a natural language which reproduced the
usual machinery of existential and universal quantification over a domain of
individuals and included Bertrand Russell's analysis of definite descriptions.
The way in which it was worked out, using the interpretation of DP's as
generalized quantifiers had a lasting effect on semantic theory and led to a
whole cartload of interesting results and questions. One such was the
important paper by Jon Barwise and Robin Cooper (1981) on "Generalized
quantifiers and natural languages." In it they stated a universal to the
effect that every natural language has a syntactic category of NP's (now =
DP's perhaps) which are interpreted as generalized quantifiers.
- Troubles in Firstorderland
Review the semantics of quantification given earlier in these notes. Note
especially the role of assignments, as in the rule for universal
quantification.
- Most etc
Generalized quantifiers can be used to interpret natural
language generalizations that go beyond the power of first-order languages.
It's easy to see this with DP's using the determiner most:
- Most swans are white.
There is no way of capturing the meaning of this sentence in first-order logic
with the universal and existential quantifiers. There is a strict proof of
this result (Barwise and Cooper 1981). Practically, we can do a "proof by
exhaustion": try it! (Recall meaning of "first-order.")
- ∀x[swan'(x) ? white'(x)]
- ∃x[swan'(x) ? white'(x)]
What could ? possibly be? What is the problem? One trouble with first-order
quantification is that you have to quantify over the whole domain of
individuals. But to understand most you need to restrict the domain to
whatever the set is that is being referred to with the generalized quantifier.
So what is needed is some way of doing restricted quantification. And
the system of generalized quantifiers allows just that.
This still isn't good enough, because most has to jump up to second
order interpretations. This sentence seems to be making a claim about the
cardinality of the set of swans that are white in relation to the cardinality
of the set of swans. So it would be true if there are six million swans, say,
and four million of them are found in the Nothern Hemisphere. (There's
another interpretation which takes swans to refer to species or kinds
of swans. We'll take this point up in our next session.) Letting |M| mean
the cardinality of the set M, we want something like this as the final
intepretation of our sentence:
- |Swan ∩ White| > |Swan|/2
(taking most to mean `more than half')
Exercise: using lambda notation give a representation of the denotation of
most as a determiner.
- David Lewis's Cases
Think about these sentences:
- Last evening, many commuters were delayed because of signal failures.
- Cats always tolerate the suppliers of their food and shelter.
- Frequently, a commuter reads a newspaper.
- Usually, a quadratic equation has two solutions.
Lewis's idea (Lewis 1975) about sentences like these was to say that they could be
interpreted with an appropriate number of variables and an unselective
binder that could bind an indefinite number of variables. Each sequence
of unbound variables constitutes an instance or case. For example,
sentence (5) just given has a core involving commuters and newspapers, so we
could represent the whole sentence as meaning something like this:
- frequently(x,y)[commuter'(x) & newspaper'(y) & read'(y)(x)]
Excursus: notice another problem with this sentence: we would not consider it
verified by a world in which every commuter buys a newspaper on his/her way
home and sits down with a cup of tea to read it after getting home. We
construct an appropriate understood context such as "in the tube" or whatever.
The refinement needed is like finding appropriate contextualizations for
sentences like these:
- Cats alway land on their feet.
- Londoners generally tailgate.
- Birds lay eggs.
- Everyone enjoyed the party.
Consideration of Lewis's ideas and a look at quantification across a variety
of languages led to a distinction between two major patterns:
D-quantification (D for determiner) and A-quantification (A for
Adverbial, but also possibly other types, which fortuitously are all also
associated with the letter "A": Auxiliary, Affix). (See Bach et al. 1995:
Introduction, and several papers in that volume.)
Exercise: Extend one or more of the examples to show that there is no
reasonable way to say just how many variables you need for Lewis's analysis.
- Donkeys and their problems
The problem sentences exhibited here (unfortunately) all have to do with
donkeys and their sad relations with their masters. The problem was discussed
originally by Geach (1962), in modern times at least (Geach was steeped in the
history of logic,and often took off from medieval discussions, for example):
- Every man who owns a donkey beats it.
- If a man owns a donkey he always beats it.
There are two troubles with these sentences:
a. How do you account for the binding of the pronoun it?
b. What is the interpretation of the indefinite DP a donkey?
In general, it is impossible to interpret sentences with quantified
term-phrases inside relative clauses as taking wide scope outside their local
domain. And even if we could if we interpret sentences like (8) and (9) as
having wide scope existential quantification for a donkey the result
doesn't accord with our intuitions about what the sentences mean. Geach
[check me on this as I don't have the original handy EB] thought that we need
to interpret the indefinite noun phrase as having universal import. But that
doesn't seem right either. Trying to cope with such sentences led to a
whole-sale overhaul of ideas about modeling indefinites. The two main
original proponents of these new ideas were Irene Heim and Hans Kamp, both
following David Lewis's lead.
We should note that there has been a lot of discussion about what these
sentences exhibiting "donkey-anaphora" actually mean: Is there a
uniqueness presupposion? If some donkey owners have several donkeys do they
have to beat all of them? Can the meaning be captured by thinking of it
as a `pronoun of laziness' (Geach) that stands for a definite description:
`the donkey that he owns'? Etc.
- Quantification and Binding: Revisions and Additions
One of the most active areas in formal semantics of the last decade has been
the investigation of questions of quantification, interpretation of pronouns,
and the like. We start with a brief look at two innovations which have spun
off a lot of activity.
- Discourse Representation Theory and File Change Semantics
We are not going to go into much technical detail in the following
exposition of the theories of Hans Kamp and Irene Heim, developed
independently at about the same time (Kamp 1981, Heim 1982). It would take
much more time than we have available here to do justice to their theories.
Instead I will try to give an intuitive picture and then we will go on to look
at Chierchia's theory of Dynamic Interpretation which (imho) takes the
insights of both to a different level.
Lauri Karttunen's 1976 paper on "Discourse Referents" should be given major
credit here, as it showed that a major part of language understanding relied on
the existence of a set of entities that are salient or available for reference
in a discourse situation. This basic insight informs all of the approaches
outlined in these notes.
Both start from the same kind of problem: apparent mismatches between
structure and scope in the interpretation of indefinites, as in the Donkey
sentences exhibited above.
Both Heim and Kamp depart from the interpretation of indefinite DP's as
embodying existential quantification directly (as Generalized Quantifiers).
Both depart from a straightforward compositional treatment of English.
Besides some of the problem sentences exhibited above, both provide a solution
to the problem presented by sequences of sentences like these:
- A policeman came into the bank. He looked around.
- ??Every policeman came into the bank. He looked around.
- Every candidate walked to the front of the room. He or she
shook the dean's hand.
(Actually, I'm not sure they deal with the last example, but they could.)
Chierchia (1995: 11) summarizes the assumptions of what he calls "classical
DRT" (comprising both Kamp's and Heim's formulations) as follows:
- Indefinites have no quantificational force on their own. They are treated
like free variables.
- The quantificational force is determined by the first available binder
(determiner or adverb of quantification).
- A binder Q sets up a tripartite structure of the form Q[A][B],
where A is the restriction of the binder and B its
(nuclear) scope.
- A rule of existential closure
assigns existential force to
indefinites that are not otherwise quantified.
- Hans Kamp's little boxes
Kamp develops a system of Discourse Representation Structures, which may be
thought of as models of situations. His interpretation is a two stage
process: a representation is built for each of the subclauses of an English
sentence, with variables and small conditions. The representations are
usually given in the form of boxes enclosing the ingredients for the partial
interpretations. The theory is usually referred to as DRT. For example,
suppose we want to interpret the Geach sentence given here:
If a man owns a donkey he always beats it.
We need two little boxes and a superbox:
_____________________________________________
| ______________ ______________ |
| | x y | | | |
| | | | | |
| | man(x) | => | beat(x,y) | |
| | donkey(y) | always | | |
| | own(x,y) | |______________| |
| |______________| |
|_____________________________________________|
Given the representation of the situations involved and the links of identity,
truth in a model is then given by stating a condition for the =>always
linking. The sentence represented is true iff whenever the antecedent is
embedded into the model then the consequent can be embedded. It is easy to see
the connection between this representation and Lewis type unselective
binders. (For details consult Kamp 1981, Kamp and Reyle 1993.)
If you don't like boxes, you can follow the linear notation introduced in
Chierchia (1995). Instead of boxes use square brackets, write the variables
just in front of their bracket-box and link separate lines by a conjunction
sign. Write in the connectives where they occur. So the DRS pictured above would look like this:
x,y[[man(x) ∧ donkey(y) ∧ own(x,y)] =>always
[beat(x,y)]]
- Irene Heim's filing cabinets
Heim's formulation works from a level of Logical Form and carries out the
operations and interpretations outlined by Chierchia above. The metaphor here
is that of keeping a file with cards for each entity and updating the cards as
a discourse proceeds. Definite DP's are also stripped of their
quantificational force and are just represented by free variables (with
conditions: donkey(x) as in DRT. Definites require finding a "card"
already there and salient in the filebox. Besides the principles outlined
Heim includes a "novelty condition" that accounts for the fact that
indefinites introduce new entities in the discourse, so for indefinites you
have to fill out a new card. Parallel to the largest boxes of Kamp enclosing
a whole discourse, Heim provides a unit of Texts (above sentences) which can
provide a domain for binding across sentences.
- Dynamic Semantics
A common thread throughout the preceding innovations has been the idea that
sentences can be thought of as ways of updating contexts or information
states. The formal setups we've been following up to now have studiously avoided
certain problems that come from expressions that need to refer to the context
at which expressions are evaluated. We've not gone into tenses at all nor have we
considered a whole class of of other expressions that need to find their
values from context: first and second person pronouns, words like here,
now, so called indexicals. For these we need to supplement our
interpretation procedure by a pragmatic component (in one sense of
"pragmatic"). We actually have everything there for this move, but first we
should take our grammar a little bit further into real English.
- Fragment IV: Tenses, Modalities, Negation
PTQ includes rules for tenses and negation, and includes one sentence level
adverb necessarily, to show how to use possible worlds to explicate
necessity, possibility, etc. (from Kripke). The tense rules provide for
future and present perfect, as in the next two examples, and negation as in
the third:
- Harry has seen a fish.
- Mary will seek the unicorn.
- Sam does not admire every horse.
I will not spell out how these sentences are derived in PTQ. They involve
several different "Rules of Tense and Sign." The interpretations of the
tenses go by way of sentence operators W and H in the
Intensional Logic with standard temporal interpretation. (13) counts as true
at a world i and time j just in case Harry saw a fish at a time
j' strictly before j, and the mirror image future interpretation
of sentences like (13) works as expected. Negation follows the standard truth
conditional interpretation. As noted before interpretation is carried out
relative to a world and a time. So there is already a little bit of context
coming in with tenses and modalities, and it is possible to think of the
assignments of values to variables as building in some more context.
There is a whole big literature on tense logic. And there are many interesting
issues to think about. I'll just mention one here: in PTQ times are
independent of worlds. This means that you can identify times across
different possible worlds. Is this right?
Many systems of formal semantics for natural language split the interpretation
process into two stages. A first stage builds up a "meaning" (let's call it)
which is then subject to a pragmatic component which fills in values
for the contextually dependent elements: speaker, hearer, time of evaluation,
location, assigments of values to free variables, and so on. The result is
then a full interpretation which is evaluated in the standard way. Note that
there is something of this in the setups of Heim and Kamp we just skimmed
through.
Going back now to Dynamic Binding and Dynamic Semantics, Chierchia (1995,
following the lead of Groenendijk and Stokhof 1990, 1991), interprets
sentences as functions from contexts to contexts. A sentence is
taken to denote a set of worlds or situations plus a kind of "hook" or
place-holder for further specifications of the set.