CAS LX 503
Semantics 2
Fall 2015
Pronouns, Assignments, and [[ ]] vs. [[ ]] a
Unlike proper names, and, the, etc., pronouns don’t possess constant denotations
across their uses. Rather, (a particular use of) a pronoun will denote whichever
individual we choose to assign to it. One particularly useful way of assigning a
denotation to a pronoun is to point at an individual while uttering the pronoun:
(1)
Rufus bit him.
• utter (1) while pointing at Don: [[ him ]] = Don
• utter (1) while pointing at Frank: [[ him ]] = Frank
At least initially, we will take an “assignment” to be simply an individual, i.e., a
member of De. To specify the denotation of some node α relative to a particular
assignment a, i.e., a particular individual, we write:
[[ α ]]a ≈ ‘the denotation of α relative to the assignment a’
vs.
[[ α ]] ≈ ‘the (assignment-independent, constant) denotation of α’
Since the denotation of a pronoun relative to a particular assignment, i.e., a
particular individual, is simply the assigned individual, we can say that:
[[ him ]]Don = Don
;
[[ him ]]Frank = Frank
Pronouns Rule: if α is a pronoun, then for any assignment a ∈ De , [[ α ]]a = a .
An important consequence of the assignment dependence exhibited by pronouns
is that the denotations of larger expressions containing pronouns also become
assignment dependent:
[[ bit him ]]Don = λx ∈ De . x bit Don
[[ Rufus bit him ]]Don = 1 if R. bit Don,
0 otherwise
[[ bit him ]]Frank = λx ∈ De . x bit Frank
[[ Rufus bit him ]]Frank = 1 if R. bit Frank,
0 otherwise
Let’s think about the step-by-step computation of the truth conditions for Rufus
bit him relative to the assignment Frank:
[[ Rufus bit him ]]Frank = [[ bit him ]]Frank([[ Rufus ]]Frank)
= [[ bit ]]Frank([[ him ]]Frank)([[ Rufus ]]Frank)
(by FA)
(by FA)
CAS LX 503
Semantics 2
Fall 2015
We know that [[ him ]]Frank = Frank. But what are [[ bit ]]Frank and [[ Rufus ]]Frank ?
Assigning a denotation to him shouldn’t affect the denotations for bit and
Rufus—these should continue to denote whatever is listed in the lexicon.
[[ bit ]]Frank should be no different from [[ bit ]]
[[ Rufus ]]Frank
[[ Rufus ]]
(denotation relative to
(constant denotations
assignment for him)
defined in the lexicon)
Assignment-Independent Substitution Rule: for any node α, if [[ α ]] is defined, then
for all assignments a, [[ α ]] = [[ α ]]a .
Picking up with our computation:
[[ Rufus bit him ]]Frank = [[ bit him ]]Frank([[ Rufus ]]Frank)
(by FA)
Frank
Frank
Frank
= [[ bit ]]
([[ him ]]
)([[ Rufus ]]
)
(by FA)
Frank
Frank
= [[ bit ]]
(Frank)([[ Rufus ]]
)
(by Pro. Rule)
= [[ bit ]](Frank)([[ Rufus ]])
(by Subs. Rule)
= [λy ∈ De. [λx ∈ De. x bit y]](Frank)(Rufus) (by TN)
= [λx ∈ De. x bit Frank](Rufus)
= 1 if Rufus bit Frank,
0 otherwise
More generally: we’ve now seen that certain expression only possess
assignment-dependent denotations, for instance, him, bit him, and Rufus bit
him. On the other hand, our Assignment-Independent Substitution Rule guarantees
that any expression that has a constant, assignment-independent denotation will
also (trivially) have an assignment-dependent denotation.
This observation has consequences for how we should approach the task of
calculating the denotation for an arbitrary expression. With the exception of
terminal nodes like bit and Rufus, whose constant denotations are listed in the
lexicon, we cannot assume in advance that an expression will have an
assignment-independent denotation. We can, however, assume that the
expression will have an assignment-dependent denotation, and we might
subsequently realize, via sufficient application of our substitution rule, that the
expression also has an assignment-independent denotation.
So, we should follow the overall strategy of stating our semantic composition
rules in terms of assignment-dependent denotations like [[ him ]]Frank and
[[ bit ]]Frank, and then substituting assignment-independent denotations like [[ bit ]]
wherever possible.
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CAS LX 503
Semantics 2
Fall 2015
Back to Relative Clauses…
DP
NP
the
CP
woman
whom
S
Arturo
VP
married
t
λx ∈ De . Arturo married x
Question #1: How do we get the trace to contribute a variable to the denotation
of the entire relative clause?
Let us assume that traces, like pronouns, only denote relative to an assignment:
[[ t ]]Bruno = Bruno ;
[[ t ]]Nigel = Nigel
Pronouns & Traces Rule (P&T)
If α is a pronoun or a trace, then for any assignment a , [[ α ]]a = a .
What is [[ t ]]x ?
[[ t ]]x = x (a variable over individuals)
• traces differ from pronouns in that they are not assigned particular
individuals, but rather variables over individuals
• syntactic function of traces: to provide a “placeholder” that represents the
origin site of a moved expression
• semantic function of traces: to provide a “placeholder” that represents a
function’s empty argument slot
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CAS LX 503
Semantics 2
Fall 2015
Step-by-step computation of the denotation for Arturo married t relative to the
assignment x:
[[ Arturo married t ]]x = [[ married t ]]x([[ Arturo ]]x)
(by FA)
x
x
x
= [[ married ]] ([[ t ]] )([[ Arturo ]] )
(by FA)
x
x
= [[ married]] (x)([[ Arturo ]] )
(by Pro. Rule)
= [[ married ]](x)([[ Arturo ]])
(by Subs. Rule)
= [λy ∈ De. [λz ∈ De. z married y]](x)(Arturo) (by TN)
= [λz ∈ De. z married x](Arturo)
= 1 if Arturo married x,
huh?
0 otherwise
DP
NP
the
CP
woman
whom
S
Arturo
VP
married
t
λx ∈ De . Arturo married x
Question #2: How do we get the relative pronoun to convert a sentence (type t)
into a one-place predicate (type <e,t>?
Let us assume that the presence of a relative pronoun triggers the application of
the following semantic composition rule:
Predicate Abstraction (PA)
If α is a branching node whose daughters are a relative pronoun and β,
then for any assignment a, [[ α ]]a = λx ∈ De . [[ β ]]x .
(α)
CP
whom
(rel. pro.)
S
(β)
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CAS LX 503
Semantics 2
Fall 2015
Applying our new composition rule yields the following result:
[[ whom Arturo married t ]]Bruno = λx ∈ De . [[ Arturo married t ]]x
(by PA)
= λx ∈ De . 1 if Arturo married x, (see above)
0 otherwise
= λx ∈ De . Arturo married x
(abbreviation)
Voila!
Stepping back a bit, we can make a couple of observations:
1. The relative pronoun whom does not itself have a denotation—nowhere have
we defined [[ whom ]], or even [[ whom ]]a. Rather, the presence of whom
triggers the application of a particular semantic composition rule, Predicate
Abstraction, which has the effect of creating a one-place predicate. In other
words, whom is “syncategorematic”: it does not possess its own denotation,
but its presence determines how we calculate the denotation of the
immediately higher node.
2. Predicate Abstraction forces us to calculate the denotation of the S
node Arturo married t (which corresponds to β, in the rule) relative to the
assignment x. In other words, the rule “overwrites” whatever previous
assignment may have been made (Bruno, in the above calculation). It
simultaneously prefaces the result of this calculation with λx ∈ De , in order
to create a function from individuals from truth values. This ensures that in
the final λ-expression, the argument variable introduced by λ matches the
variable contributed by the trace.
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