Contextualizing
Greg Kobele
Semantics II
Winter 2015
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Introduction
Pronouns, at least in their deictic usage, seem to require a context of use for
their interpretation. Once we have specified a context, a pronoun will denote an
individual. We will model this by taking a pronoun to denote a function from
contexts to individuals. A sentence containing a pronoun used deicticly, such as
He smiled, seems not to be true or false, unless we specify a particular context
in which it was uttered. We will model this by taking a sentence to denote, not
a truth value, but rather a function from contexts to truth values.
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Contexts
A context is for us at the moment whatever we need to resolve the interpretation
of pronouns, and of sentences containing them. In general, we assume that we
have a set C of contexts (whatever they turn out to be), and that contexts have
type c. Our discussion below does not depend on any particular choice of what
contexts should be, but, for the sake of concreteness, we can take contexts to be
lists of individuals, which we can think of intuitively as a description of which
things there are which can be referred to, ranked according to how salient they
are.
A pronoun, like he, will denote a function from contexts to individuals; a
function of type ce. We will not be assuming anything in particular about which
functions these are, or how they work, other than that they operate on some
model of the discourse context. Much work in natural language processing has
been done on algorithms for pronoun resolution; we are here providing a basic
framework for incorporating this and other work into compositional semantic
interpretation.
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Context dependent types
Our decision to treat sentences as being of type ct, and pronouns of type ce,
forces us to systematically alter the types we have previously assigned to expressions. For example, an intransitive verb, such as sings, previously of type
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et, must now combine with its argument to build a sentence of type ct. It must
also be able to combine with a pronoun of type ce.
3.1
The wrong way
While we could combine an expression of type ect with one of type ce using
function composition (the B combinator), this would result in something (of
type cct) which is waiting for two contexts. A transitive verb, now for the
moment of type eect, could be composed with two pronoun denotations to result
in something of type ccct; this is something which is waiting for three contexts,
one for the subject pronoun, one for the object pronoun, and one representing
the context in which the sentence was uttered. This is intuitively on the wrong
track, as what we would seem to want is that the pronouns are interpreted in
the context of utterance, not in potentially arbitrary other contexts.
3.2
The right way
Instead, we will assume that predicates combine with pronoun denotations via
function application. Thus, an intransitive verb denotation will be of type (ce)ct,
and a transitive verb denotation will be of type (ce)(ce)ct. This forces us to
change the types of generalized quantifiers from (et)t to ((ce)ct)ct, so that a
DP like every boy can combine with a property via function application. This
in turn will force us to change the types of prepositions (as they combine with
DPs), and, more generally, anything that takes part in the complex web of
interactions postulated by our current semantic analyses.
The most straightforward way of achieving this is to do it systematically.
Indeed, looking at the change in a generalized quantifier type, from (et)t to
((ce)ct)ct, we see that the type t is replaced everywhere by the type ct, and the
type e by ce.
This is a type homomorphism, which can be represented as follows (writing
αc for the type which results from α):
Case
ac
(αβ)c
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Output
ca
(αc β c )
Building context-sensitive meanings
Now that we have systematically changed the types of semantic expressions,
we need to decide how to change the meanings we had previously assigned to
expressions so as to make them have the new type. Beginning with a concrete
example, we would like the sentence He smiled to be interpreted in a context c
as true just in case whoever he refers to in c smiled; in other words, just in case
[[smile]] ([[he]] (c)) = 1. Similarly, He praised her should be true in a context c
just in case whoever he picks out in c praised whoever her picks out in c; in
other words, just in case [[praise]] ([[her]] (c))([[he]] (c)) = 1. The most natural
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denotation to give to praised is thus λx, y, c.praise(x c)(y c); similarly, smiled
should denote λx, c.smile(x c). More generally, given any expression k of type
en t, its context-sensitive denotation should be λx1 , . . . , xn , c.k(x1 c) . . . (xn c).
It is important to observe that we are not changing the interpretation function of the model, which assigns to each word a meaning. Rather, we are showing
how to repackage the standard meaning of a word so as to ‘fit’ in a structure
which is expecting a context-sensitive meaning.
We repackage the meaning of a determiner, every, as follows. As the contextdependent type of every is ((et)(et)t)c = (((ce)ct)((ce)ct)ct), it must take two
context-dependent property denotations (of type (ce)ct), and a context as arguments. In order to obtain a λ-term of this type from our original constant
every of type (et)(et)t, we must determine how to obtain a term of type t from
one of type (ce)ct, one of type c, and one of type e. This can be done in the
following manner, where P : (ce)ct, d : c, and x : e: P (λb.x) d. Then we can
give the context-sensitive meaning of every in terms of its original meaning as
follows:
λP, Q, d.every(λx.(P (λb.x)d))(λy.(Q(λb.x)d))
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Automatically deriving context-sensitive meanings
There is a general principle at work to obtain the context-sensitive meanings of
our lexical items from their usual ones. To give it, we borrow from Lebedeva
[1]. Given a constant k : α, we will construct a term Conα (λd.k) : αc . More
generally, we define an operation Conα : (cα)αc for each α, with the help of
a family of operations Nocα : αc cα. Intuitively, Conα turns a non-contextsensitive function of type cα into a context-sensitive one of type αc , and Nocα
takes a context-sensitive function of type αc together with a context d, and
returns a non-context-sensitive function of type α.
Cone (a) = a
Cont (φ) = φ
Conαβ (f ) = λA.Conβ (λd.f d (Nocα A d))
Noce (A) = A
Noct (Φ) = Φ
Nocαβ (F ) = λd, a.Nocβ (F (Conα (λb.a))) d
We can examine the behaviour of Con(et)(et)t on every:
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Con(et)(et)t (λg.every)
= λA.Con(et)t (λd.(λg.every) d (Nocet A d))
= λA.Con(et)t (λd.every (Nocet A d))
= λA.λB.Cont (λg.(λd.every (Nocet A d)) g (Nocet B g))
= λA, B.Cont (λg.every (Nocet A g) (Nocet B g))
= λA, B.λg.every (Nocet A g) (Nocet B g)
= λA, B, g.every (Nocet A g) (Nocet B g)
= λA, B, g.every ((λd, a.Noct (A (Cone λb.a) d)) g) (Nocet B g)
= λA, B, g.every (λa.Noct (A (Cone λb.a) g)) (Nocet B g)
= λA, B, g.every (λa.Noct (A (λb.a) g)) (Nocet B g)
= λA, B, g.every (λa.Noct (A (λb.a) g)) ((λd, a.Noct (B (Cone λb.a) d)) g)
= λA, B, g.every (λa.(A (λb.a) g)) ((λd, a.Noct (B (Cone λb.a) d)) g)
= λA, B, g.every (λa.(A (λb.a) g)) ((λa.Noct (B (Cone λb.a) g))
= λA, B, g.every (λa.(A (λb.a) g)) (λa.Noct (B (λb.a) g))
= λA, B, g.every (λa.(A (λb.a) g)) (λa.(B (λb.a) g))
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Pronouns and assignment functions
What we are calling contexts are standardly called assignment functions, and are
commonly written with letters like g and h. Assignment functions are infinite
sequences of individuals; we write gi to denote the ith element in the sequence
g. The standard treatment of pronouns involves writing something like the
following, where i is a number, and g an assignment:
g
[[hei ]] = g(i)
Of note is the fact that the denotation function ([[·]]) is superscripted with an
assignment function. Indeed, it is standard to describe function application in
the following terms:
g
g
g
[[[α β]]] = [[α]] ([[β]] )
This is tantamount to writing instead:
[[α]] ⊕ [[β]] = S [[α]] [[β]] = λg. [[α]] (g)([[β]] (g))
In other words, the superscripted item on the denotation function is ‘hiding’
the fact that the expression being evaluated is a function which is taking the
superscripted item as an argument.
With this in mind, we can see that [[hei ]] is a function from contexts/assignments
to individuals. (Namely, the one which maps a context to its ith element.) From
the perspective of pronouns as denoting pronoun resolution algorithms, this is
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implementing a family of very poor pronoun resolution algorithms (each just
picks out a single point in a context). No one thinks that this is a good algorithm for resolving pronouns. Instead, the idea is that the important work of
pronoun resolution should be done in the syntax.
From our perspective, we can see this standard approach as just one way of
implementing our more general approach to context-sensitivity.
References
[1] Ekaterina Lebedeva. Expression de la dynamique du discours à l’aide de
continuations. PhD thesis, Université de Lorraine, 2012.
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