class: center, middle ### PLIN3004/PLING218 # Advanced Semantic Theory ### Week 4 ### 27 October, 2016 --- # Agenda 1. Semantic Types 2. Transitive Verbs 3. Connectives --- class: center, middle # Semantic Types --- # Semantic Types There are two kinds of primitive semantic types: - Type **e**: the semantic type of individuals/entities - Type **t**: the semantic type of truth-values, i.e. 0 and 1 Everything else in our semantics is a function. The semantic type of a function always looks like **⟨σ,τ⟩** where, σ is the type of input and τ is the type of output Examples: - Type **⟨e,t⟩**: the semantic type of functions from individuals to truth-values - Type **⟨e,⟨e,t⟩⟩**: the semantic type of functions from individuals to functions from individuals to truth-values --- # Recursive Definition of semantic types Formally, we define semantic types to be the following: - e is a semantic type. - t is a semantic type. - Whenever σ and τ are semantic types, ⟨σ,τ⟩ is a semantic type. - Nothing else is a semantic type. There are infinitely many semantic types (why?): - e, t, ⟨e,t⟩, ⟨t,t⟩, ⟨e,⟨e,t⟩⟩, ⟨⟨e,⟨e,t⟩⟩,t⟩, ⟨⟨e,t⟩,t⟩, etc. are types. - ⟨e⟩, ⟨t⟩, ⟨et⟩, ⟨e,e,t⟩, ⟨t,⟨e,t⟩,⟨e,t⟩⟩ etc. are not types (why not?) (We sometimes write 'et' for '⟨e,t⟩') --- # Domains of Semantic Types Each semantic type τ has a **domain** D
τ
. - D
e
= the set of individuals/entities in the model - D
t
= {0,1} = the set of truth-values - For any other semantic type ⟨σ,τ⟩,
D
⟨σ,τ⟩
= the set of functions from D
σ
to D
τ
= {f| f:D
σ
→D
τ
}
Examples:
D
⟨e,t⟩
= the set of functions from D
e
to D
t
= {f| f:D
e
→D
t
}
D
⟨e,⟨e,t⟩⟩
= the set of functions from D
e
to D
⟨e,t⟩
= {f| f:D
e
→D
⟨e,t⟩
} = {f| f:D
e
→{g| g:D
e
→D
t
}}
--- # Intransitive verbs
⟦
smokes
⟧
M
= λx∈D. 1 iff x smokes in M
= λx∈D
e
. 1 iff x smokes in M
- ⟦
smokes
⟧
M
takes an individual x (an object of type e) and returns a truth-value (an object of type t) - So it is a function of type ⟨e,t⟩ - Or equivalently, ⟦
smokes
⟧
M
∈D
⟨e,t⟩
--- # Exercise 1: Semantic Types What are the semantic types of the following things? Also for 3-6, rewrite the functions in the lambda notation. 1. Truth-value 1 2. John 3. the function that takes an individual x in M and returns 1 iff x has a cat in M 4. the function that takes any married individual x in M and returns x's spouse in M 5. the function that takes a truth-value u in M and returns John if u=1 and returns Mary if u=0 6. the function that takes an individual x in M and returns a function that takes an individual y in M and returns 1 iff x=y --- class: center, middle # Adding Transitive Verbs --- # Review |
**Lexicon**
||| |:--|:--|:--| | ⟦
smokes
⟧
M
|=|λx∈D
e
. 1 iff x smokes in M| | ⟦
left
⟧
M
|=|λx∈D
e
. 1 iff x left in M| | ⟦
John
⟧
M
|=|some individual in M| | ⟦
Mary
⟧
M
|=|some individual in M| |
**Compositional Rules**
| |:--| |
Branching Node Rule
:| | ⟦
[
A
B C ]
⟧
M
= ⟦
B
⟧
M
(⟦
C
⟧
M
) or ⟦
[
A
B C ]
⟧
M
= ⟦
C
⟧
M
(⟦
B
⟧
M
)| |
Non-Branching Node Rule
: ⟦
[
A
B ]
⟧
M
= ⟦
B
⟧
M
| Today, we add **transitive verbs** (verbs that take an object and a subject), e.g.
love
,
saw
, etc. --- # Transitive Verbs in Syntax First we need to enrich the syntax so that the syntax can generate sentences containing transitive verbs. We do this by adding new rules. -
S
→
DP VP
-
DP
→
John
|
Mary
-
VP
→
smokes
|
left
|
loves DP
|
saw DP
It's easy to add more transitive verbs, of course. Now, we can generate sentences like
. --- # The Semantic Type of Transitive Verbs
We can infer what the semantic types of transitive verbs should be: 1. The sentence denotes a truth-value, and
[
DP
John]
denotes an individual. 2. Given Frege's Conjecture,
[
VP
loves [
DP
Mary]]
should denote a function of type ⟨e,t⟩. 3. Then
loves
must denote a function that takes the denotation of
[
DP
Mary]
and produces the denotation of
[
VP
loves [
DP
Mary]]
. ⟦
loves
⟧
M
is a function of type ⟨e,⟨e,t⟩⟩ that takes an individual (e.g. Mary) and returns a function of type ⟨e,t⟩, which in turn takes another individual (i.e. John) and returns a truth-value. --- # Denotations of Transitive Verbs Given that ⟦
loves
⟧
M
is of type ⟨e,⟨e,t⟩⟩, it will look like:
[λx∈D
e
. [λy∈D
e
. ??????]]
- The first argument (x) should be the object and the second argument (y) should be the subject of
loves
, because the syntax says the object combines with the verb first. -- - So it should be: > ⟦
loves
⟧
M
=λ
x
∈D
e
. [λ
y
∈D
e
. 1 iff
y
loves
x
in M] E.g. ⟦
loves
⟧
M
(
Mary
)(
John
) = 1 iff
John
loves
Mary
in M --- # The Role of Syntax ||| |:--|:--| | ⟦
loves
⟧
M
| = [λ
x
∈D
e
. [λ
y
∈D
e
. 1 iff
y
loves
x
in M]]| | | ≠ [λ
y
∈D
e
. [λ
x
∈D
e
. 1 iff
y
loves
x
in M]]| The latter is wrong because it makes the object the person who loves the other poerson, and the subject the person who is loved. This is because we have the syntactic structure on the left. If the structure on the right were the correct structure, we could use the second function above.
--- # Other Transitive Verbs Other transitive verbs can be analyzed in the same way - ⟦
saw
⟧
M
= [λx∈D
e
.[λy∈D
e
. 1 iff y saw x in M]] - ⟦
killed
⟧
M
= [λx∈D
e
.[λy∈D
e
. 1 iff y killed x in M]] - ⟦
met
⟧
M
= [λx∈D
e
.[λy∈D
e
. 1 iff y met x in M]] --- # Summary |
**Lexicon**
|||| |:--|:--|:--|:--| | ⟦
John
⟧
M
|=some individual in M| | ⟦
Mary
⟧
M
|=some individual in M| | ⟦
saw
⟧
M
|= λx∈D
e
.[λy∈D
e
. y saw x in M]| | ⟦
left
⟧
M
|=λx∈D
e
. 1 iff x left in M| | ⟦
loves
⟧
M
|= λx∈D
e
.[λy∈D
e
. y loves x in M]| | ⟦
smokes
⟧
M
|=λx∈D
e
. 1 iff x smokes in M| |
**Compositional Rules**
| |:--| |
Branching Node Rule
: ⟦
[
A
B C ]
⟧
M
= ⟦
B
⟧
M
(⟦
C
⟧
M
) or ⟦
[
A
B C ]
⟧
M
= ⟦
C
⟧
M
(⟦
B
⟧
M
)| |
Non-Branching Node Rule
: ⟦
[
A
B ]
⟧
M
= ⟦
B
⟧
M
| --- # Exercise 2: Computation Compute the meaning of the following sentence in model M (either bottom-up or top-down). Let's assume ⟦
Mary
⟧
M
= m, ⟦
John
⟧
M
= j and m didn't see j in M.
--- class: middle, center # Connectives --- # Adding Connectives
Let us now add
and
and
or
to our grammar. We assume the binary branching structure. To generate such sentences, we enrich the syntax as follows. -
S
→
DP VP
|
S S'
-
S'
→
and S
|
or S
-
DP
→
John
|
Mary
-
VP
→
smokes
|
left
|
loves DP
|
saw DP
--- # Recursion Notice that we can now generate infitnitely many sentences. -
S
can be expanded into
S S'
and then to
S and/or S
. - This process can apply to each of the new
S
's recursively. As we will see, our compositional semantics will be capable of dealing with all sentences so generated. --- # Semantic type of
and
What should ⟦
and
⟧
M
be? Let's start with its semantic type. Its first argument is ⟦
S
⟧
M
, which we know is a truth-value. So the semantic type of ⟦
and
⟧
M
should look like ⟨t,σ⟩. -- Then the result ⟦
and S
⟧
M
also combines with the denotation of
S
and produces the denotation of the sentence. So
S'
needs to denote a function of type ⟨t,t⟩. Then, ⟦
and
⟧
M
should be **⟨t,⟨t,t⟩⟩**. --- # The denotation of
and
Since the type of
and
is ⟨t,⟨t,t⟩⟩,
⟦
and
⟧
M
looks like [λu∈D
t
. [λv∈D
t
. ???]]
We will model the meaning of
and
based on the logical conjunction **∧** from Propositional Logic: p∧q = 1 iff p=1 and q=1 The idea is that the conjunction of two sentences is true iff both of the sentences are true. -- > ⟦
and
⟧
M
= [λu∈D
t
. [λv∈D
t
. 1 iff u=1 and v=1]] --- # The denotation of
or
Or
can be analyzed in a simialr way, based on the logical disjunction **⋁**: p⋁q = 1 iff p=1 or q=1 > ⟦
or
⟧
M
= [λu∈D
t
. [λv∈D
t
. 1 iff u=1 or v=1]] --- # Summary
**Lexicon**
- ⟦
John
⟧
M
= some individual in M - ⟦
Mary
⟧
M
= some individual in M - ⟦
smokes
⟧
M
=λx∈D
e
. 1 iff x smokes in M - ⟦
left
⟧
M
=λx∈D
e
. 1 iff x left in M - ⟦
loves
⟧
M
= λx∈D
e
.[λy∈D
e
. y loves x in M] - ⟦
saw
⟧
M
= λx∈D
e
.[λy∈D
e
. y saw x in M] - ⟦
and
⟧
M
= [λu∈D
t
. [λv∈D
t
. 1 iff u=1 and v=1]] - ⟦
or
⟧
M
= [λu∈D
t
. [λv∈D
t
. 1 iff u=1 or v=1]] --- # Summary (cont.) |
**Compositional Rules**
| |:--| |
Branching Node Rule
: ⟦
[
A
B C ]
⟧
M
= ⟦
B
⟧
M
(⟦
C
⟧
M
) or ⟦
[
A
B C ]
⟧
M
= ⟦
C
⟧
M
(⟦
B
⟧
M
)| |
Non-Branching Node Rule
: ⟦
[
A
B ]
⟧
M
= ⟦
B
⟧
M
| --- # Exercise 3: Computation Compute the denotation of the following sentence in model N. Let's assume that ⟦
John
⟧
N
= a, ⟦
Mary
⟧
N
= b, a didn't leave in N, but b saw a in N.
--- # Next Time - Do Assignment 4 and submit it on Moodle - Read Lecture Notes for Week 5