Course: Modeltheoretic Semantics and the Syntax-Semantics Interface
Eddy Ruys and Yoad Winter
Topics: types, domains, characteristic functions, function application, the category to type matchingIn the first lecture we have talked rather sloppily about models and various sets, functions and truth-values they contain. In this lecture we will go somewhat deeper into the structure of the model: what objects are there in the model, how can we describe them, and does this give us more insight into the relations between syntax and semantics?
As we have said, natural language expressions are taken to denote objects in the model. However, different expressions denote different kinds of things. For instance, we let Tina denote a member of the set of entities E, whereas tall denotes a subset of E and a sentence like Tina is tall denotes a truth value. We would like to have a systematic way to describe these distinctions. For this reason we divide the model into different domains, where each domain has a type: a label describing its internal structure. The three expressions mentioned above denote members of domains with different types.
The simplest types and the simplest domains are the primitive ones.
There are two primitive types, each with its corresponding domain.
One primitive type is called e (for entities) and its corresponding
domain is the non-empty set of entities E, which we from now on denote by
(the e Domain). The other primitive type is called t (for
truth-values) and its corresponding domain is the set
. The proper name Tina takes its denotation
from the domain. In short, we say that proper names are of type e.
A sentence like Tina is tall
denotes a truth value in 
.
All the non-primitive types
are recursively defined on top of these two primitive
types. To see how, let us start with an example. We have said that
predicates like tall denote subsets of .
For instance, consider a model where the set of entities
is 
. Let us
indicate that only Tina and John are tall in this model. We can do that
by letting tall
refer to the set 
, a subset
of . Alternatively,
we can say the same thing in a slightly different way: let
tall, instead of denoting the set T, denote a corresponding
function 
from entities to truth-values, i.e. from
to . For each member of the set of entities, assigns
it true if it is in T and false if it is not in T.
Thus, we have:
is called the characteristic function of the set 
.
For every element x of , the truth value 
indicates
whether x is in T or not.
In general, we may view the denotation of a predicate
like tall either as a subset of the set of entities or as
the characteristic function of such a set.
We can move freely back and forth between these
two perspectives, as they encode precisely the same information
concerning which entities are tall. For the sake of the discussion in this
lecture let us adopt the latter, functional perspective on predicates.
The advantages of this step will become clear as we go along.
We obviously want predicates like tall
to be able to denote any subset of , or, as we now say,
any function from to . Thus, we want the domain from which
tall takes its denotation to be the set of all functions from
to . In set theory this set of functions is denoted by 
.
This domain will be of type (et), for which
we will often use the notation et
as an abbreviation, erasing parentheses whenever this does not cause
confusion.
Here are the eight members in 
for the tripleton domain we are dealing with:
is described by a function in ,
which must correspond to
one of the rows in this array. The predicate tall can
denote any of these eight functions when the set of entities is
the one specified. For instance, the function above
is function 
in this array.
To summarize, a proper name like Tina denotes an element of
, a sentence denotes an element of and a
predicate like tall
denotes a function from the first domain to the latter, i.e.\
an element of . Of course,
when we consider more expressions, we are likely to need more types than
these ones. How do we define them?
What we obviously want is a general strategy for defining types.
The tall example has illustrated the way to do that: we defined
a new type, et, and its corresponding domain on the basis of the given types
e and t and their domains.
Generally, if we are given
the types 
and 
and
their corresponding domains 
and 
, we can define
the new type 
of a domain 
that consists of
all the functions
from to .
In this way we can go on and use the new type and domain we get
to create more new types and domains.
For instance, we can use et and e to create the type (et)e and its
domain 
of all functions from to .
There are infinitely many types we can define using this inductive
procedure.
This is precisely what we do now,
using the following two definitions.
These two simple definitions summarize all we have discussed above.
A model
is the infinite collection
of all typed domains. Note that the primitive domain is given, so
in order to determine a model, all we need to do
is to determine the other primitive domain: . After we have done this, all
the compound domains are inductively fixed by definitions 1 and 2.
Working with functions in this way as the building blocks of compound domains gives us an easy way to describe what is going on compositionally in analyzing a sentence. Consider a maximally simple sentence like Tina smiles. The sentence denotes a truth value simply because the subject denotes an entity of type e and the predicate denotes a function of type et: from e entities to truth-values. Once this function applies to the subject denotation we get a truth value. This operation of function application is crucial: it is the main operation that ``glues'' denotations together into new denotations. Let us exemplify the notation we use in more detail for the simple sentence Tina smiles:
of , or in
short: 
.
.
which is the result of applying the function 
to .
This value is denoted 
.
Note how easy it is to know from two types if function application is allowed between objects in the corresponding domains. When function application is allowed also the outcome type is obvious. For instance:
(the function type) and (the argument type)
a new denotation of type : the result of applying the function to
the argument.
The postulation of function application
also facilitates the typing (=the determination of the type)
of different expressions. For instance, consider the auxiliary verb is in
the sentence Tina is tall.
We have decided to let Tina get the type e and tall get the
type et. The constituent is tall has to denote a function
that applies to and derives a truth-value. Hence, it
must be, like tall, of type et.
To get this, the denotation of is will be a function of type (et)(et):
when it applies to
an et object like 
it will give us an et object. In fact, is will
denote the identity function on predicates: the function that for
each et predicate assigns the same predicate itself. Thus, the denotation of
is tall is the same as the denotation of tall.
These typing considerations are summarized in figure 1. Note that now it is natural to use a more realistic structure than the one that was tentatively used in the first lecture for expository purposes.
Figure 1: compositionality and typing
What we have done above is a kind of puzzle solving. We were given parts of
the puzzle: the types for the denotations of the words Tina and
tall and the type
for the whole sentence. We had the structure of the sentence
and sought types for the word is and the category PRED that would
match the other types by allowing function application to work compositionally.
Because we knew that the denotation of PRED would have to apply to an e
denotation to produce a t denotation, we concluded
that this category
should be of type et. To generate this type, the denotation
has to apply to the et type denotation of tall and give
another et function. Thus,
it has to be of type (et)(et).
We should stress the relations between typing and sentence
structure. For instance,
the (et)(et) typing of is is motivated by the structure
in figure 1, but not necessarily by different structures that
might have been postulated. Suppose tentatively that
we gave the sentence the implausible structure of figure 2,
in which Tina and is are assumed to form a constituent.
In this structure
the typing of is could not have remained (et)(et): functions
of this type apply to et type objects and not to e type entities.
A possibility would be to change the typing of is
to ee: a function from
entities to entities. This would allow the denotation to apply to
and produce an e type denotation for the category FUNNY, to
which the predicate could apply and produce a truth value.
This simple example shows that in assigning a type to a particular expression, structural considerations play a major role: the syntactic configurations in which an expression appears often affect the semantic type we choose to assign to it.
So far, types have just put some order in things that were already done in the previous lecture. But the idea to let expressions denote typed objects is very helpful when we want to define denotations of new expressions. For instance, what does the transitive verb see denote? Consider a sentence like Tina saw Mary. Compositionally, the constituent saw Mary is going to have the type et: like intransitive predicates (e.g. tall) its denotation applies to entities to produce a truth value. Since we have decided to let Mary be of type e there is no question as for what type the verb see has to get: the type of functions from e objects to et objects. In short: e(et). Similarly, a ditransitive predicate like show, as in Tina showed Mary John denotes objects of type e(e(et)): functions from entities to functions from entities to functions from entities to truth values. It is surely convenient that we have so many types at our disposal!
Note that in the discussion above we have made two implicit assumptions:
) of one of two sisters, the other sister must
either be a function over this type (i.e. 
) or
an argument of this function (i.e. 
). Furthermore, the mother
must be either of type 
(in the first case) or
of type 
(in the latter case). These two possibilities are illustrated
in figure 3.
Figure 3: possibilities with function application
Such a strict organization of the grammar is challenging, and therefore theoretically desired: it leaves us with only a small amount of possibilities to develop syntactic theories with suitable typings of lexical items. At the present stage of semantic theory, however, it is debatable whether such a strict configuration can succeed as a linguistic program. These restrictions may be too strict for natural language, and attempts are being made to relax them in principled ways. Nevertheless; since these two constraints are simple and methodologically convenient, we will adopt them throughout the first part of this course as an illustration of the power of modeltheoretic techniques.
A note on characteristic functions: We have shown above that
we can freely switch between
a set denotation of a predicate like tall and the corresponding
characteristic function of type et. The situation is in fact more general.
Consider the transitive predicate see. Intuitively, such a
predicate can be viewed as a two-place relation between entities:
a subset of the cartesian product 
.
The relation will specify which
entities see which entities. For instance,
in case 
, we can describe
a situation like the one given in (1) by letting see denote the
relation S in (2).
,
and vice versa. For instance,
the e(et) function corresponding to S is:
assigns each entity x in a function in that
characterizes the set of entities that saw x. For instance, the VP saw
Mary denotes the characteristic function of the set of entities that saw
Mary, namely: 
.
The reasoning behind this choice is that the denotation of see
compositionally applies to the denotation of the
object in the sentence. The resulting denotation of the VP
applies to the subject denotation. Hence, the first argument of the
denotation of the verb should be the object denotation. Generally, the
characteristic function of a relation S is defined by:
iff 
Note that 
is itself a characteristic function. In our case,
the one characterizing the set of
entities that saw x. The successive application 
is
sometimes written in short 
.
This switch in the intuitive order of the arguments is confusing upon first
encounter. However, it is necessary if we want to be able to speak of the
denotation of constituents like saw Mary: it is much more natural then to
think of Mary as the first argument of the verb's denotation and
not as in the intuitive relation S.
After some exercising you will surely get used to this and realize that
the method is completely innocuous. Moreover, the procedure we have used
for moving back and forth
between the intuitively convenient sets/relations and their
corresponding characteristic functions is easily extended to many other
useful types.
Exercises
. Write down all the members of 
, 
and 
. Notice the difference between the last two domains.
. Why don't you need to know
in order to do that?
.
describing a denotation of the predicate run.
describing a denotation of the verb
kiss.
describing a denotation of the verb
show. Read 
as ``x showed y z''.
(iii)?
