introduction to semantics - heim & kratzer 1998, …introduction to semantics heim & kratzer...
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Introduction to semantics
Heim & Kratzer 1998, chapters 1-2
Clemens Mayr
EGG school, Lagodekhi
July 25, 2016
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Overview
1. Tasks of a semantic theory
2. Sets
3. Functions
4. How syntax determines sentence interpretation
5. The three components of compositional interpretation
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Two questions about the meaning of sentences
(1) Tony swims.
1 How should we characterize the meaning of (1)?
2 How does this meaning come about?
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Intuitive answer to the two questions
The meaning of (1) is straightforward: Tony swims
(1) Tony swims.
Tony = the person named Tony
swims = carry out the action of swimming
Tony swims = the person named Tony carries out the action of
swimming
Do we need to say more?
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Intuitive answer to the two questions
The meaning of (1) is straightforward: Tony swims
(1) Tony swims.
Tony = the person named Tony
swims = carry out the action of swimming
Tony swims = the person named Tony carries out the action of
swimming
Do we need to say more?
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Intuitive answer to the two questions
The meaning of (1) is straightforward: Tony swims
(1) Tony swims.
Tony = the person named Tony
swims = carry out the action of swimming
Tony swims = the person named Tony carries out the action of
swimming
Do we need to say more?
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Making question 1 clearer
When we ask,
“How should we characterize the meaning of a sentence S?”
what we as linguists really mean is
“What kind of semantic competence allows a native speaker to
understand S?”
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Truth and falsity
Situation: Your friend Tony is in the swimming pool. He moves his arms
and feet in such a way as to avoid drowning.
XTony swims.
Situation: Your friend Tony is in the swimming pool. He is standing in
the shallow end of the pool drinking a beer.
Tony swims.
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Semantic competence 1
The semantic competence of a native speaker among other things
includes:
To know when a sentence S is true and when S is false.
We want to understand what enables a speaker to know this.
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Semantic competence 2
A speaker knows that when (1) is true, (2) is true as well, but not
necessarily vice versa.
(1) Tony swims.
(2) Someone swims.
A speaker knows which sentences entail other sentences.
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Literal meaning
Literal meaning of S = truth-conditions of S (Alfred Tarski)
Truth-conditions of S in a particular situation s yield the truth-value of S
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Truth-conditions
Tony swims. = true in situation s iff Tony swims
This is not a trivial statement. Languages differ in the expressions they
use, but the truth-conditions remain the same.
Tony plavaet = true in situation s iff Tony swims
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Truth-conditions
Tony swims. = true in situation s iff Tony swims
This is not a trivial statement. Languages differ in the expressions they
use, but the truth-conditions remain the same.
Tony plavaet = true in situation s iff Tony swims
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Truth-conditions and entailment
S entails S ′ iff whenever S is true S ′ is true
(1) entails (⇒) (2)
(1) Tony swims.
(2) Someone swims.
test: contradiction if (2) is false and (1) true.
The truth-conditions for (1) and (2) must guarantee this entailment
relation.
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Making question 2 clearer
When we ask
“How does this meaning—i.e., the truth-conditions—come about?”
what we as linguists really mean is
“What processes guarantee the truth-conditions that we observe?”
How come Tony swims is true iff Tony swims, not, say, iff Tony barks?
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Frege’s principle of compositionality
The truth-conditions of a sentence S are a function of the meaning of
S ’s parts.
Semantic composition consists in function application.
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A semantic theory gives us . . .
The truth-conditions of every possible sentence in a language.
A compositional process determining these truth-conditions.
In order to achieve this, one needs to know at least
1 the meanings of lexical items
2 the compositional rules assigning meanings to complex expressions
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A semantic theory does not give us . . .
Non-literal meaning components such as implicatures.
(3) suggests that not everyone swims.
(3) Some of the boys swim.
This inference is cancelable by continuing with In fact, all do.
Given the cancelability, the inference cannot be part of the
truth-conditions.
Its derivation is the job of a pragmatic theory.
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Overview
1. Tasks of a semantic theory
2. Sets
3. Functions
4. How syntax determines sentence interpretation
5. The three components of compositional interpretation
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Sets
A set is an unordered collection of objects.
These objects are the members/elements of the set.
N is the set of natural numbers
2 ∈ N (2 is a member/element of N)
1/2 6∈ N (1/2 is not a member/element of N)
Set without members is the empty set, ∅.
The members are enclosed by curly brackets: e.g. {a, b, c}
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Why sets?
Semantic composition and therefore truth-conditions are characterizable
by sets and set-membership, which are in turn characterized by functions
and function application, respectively.
swim = set of swimmers
Tony = Tony
Tony swims is true iff Tony is a member of the set of swimmers
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Set equivalence
Sets A and B are equivalent iff A and B have exactly the same elements.
{a, b, c} = {a, b, c}
Since sets are unordered
{a, b, c} = {b, c , a} = {b, a, c} = {c , a, b} = {c , b, a} . . .
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Subset
A is a subset of B iff all members of A are members of B.
{a, b, c} ⊆ {a, b, c , d}
Each set is a subset of itself.
{a, b, c} ⊆ {a, b, c}
A is a proper subset of B iff A is a subset of B but not equivalent to B
{a, b, c} ⊂ {a, b, c , d}
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Intersection
The intersection A and B, A ∩ B, is the C with exactly those elements
which are shared by A and B.
{a, b, c} ∩ {a, c , d} = {a, c}
da cb
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Overlap and disjointness
A and B overlap iff A ∩ B 6= ∅.
{a, b, c , d} 6= {d , e, f }
A and B are disjoint iff A ∩ B = ∅.
{a, b, c} 6= {d , e, f }
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Union
The union of A and B, A ∪ B, is the C with all the members of A and B
and nothing else.
{a, b, c} ∪ {d , e, f } = {a, b, c , d , e, f }
a b c d e f
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Complement
The complement of A in B, B −A, is the C with all elements of B which
are not in A and nothing else.
{a, b, c , d} − {a, b} = {c , d}
{a, b, c} − {a, b, d} = {c}
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Defining sets
A := . . .
‘Let A be that set, . . . ’
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Defining sets via listing
A := {a, b, c}
‘Let A be that set, whose elements are a, b, c , and nothing else.’
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Abstraction
Sets can be defined via abstraction so that not all members have to be
listed, which is helpful when there is an infinite number of members or
the members are not known.
A := {x : x is a president}
x is a variable
All elements in A must satisfy requirement to the right of :
‘Let A be that set such that for all x ∈ A it holds that x is a president’
‘Let A be the set of presidents’
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Set membership
To determine whether object a is a member of A, the variable is set to a.
If the result is true, a ∈ A
If the result is false, a 6∈ A.
A := {x : x is a president}
Obama is a president is true
Obama ∈ A
Sarkozy is a president is false
Sarkozy 6∈ A
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Notation
x , y , z are variables ranging over objects
a, b, c are names of objects
X ,Y ,Z are variables ranging over sets
A,B,C are names of sets
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Overview
1. Tasks of a semantic theory
2. Sets
3. Functions
4. How syntax determines sentence interpretation
5. The three components of compositional interpretation
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Why functions?
Semantic composition and therefore truth-conditions are characterizable
by sets and set-membership, which are in turn characterized by functions
and function application, respectively. Functions are types of sets.
The meaning of loves is applied to the meaning of John. The result, the
meaning of VP is applied to the meaning of Mary.
S
Mary VP
loves John
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Ordered pairs
Pairs of objects are ordered
a in 〈a, b〉: left component of the pair 〈a, b〉
b in 〈a, b〉: right component of the pair 〈a, b〉
〈a, b〉 6= 〈b, a〉 (except for when a = b)
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Relations
A (binary) relation is a set of ordered pairs.
{〈a, b〉, 〈c , d〉}
Relations are defined by two conditions on two variables to the right of :
R := {〈x , y〉 : x is an author and y is a book of x}
〈Thomas Mann, Buddenbrooks〉 ∈ R〈Leo Tolstoy, War & Peace〉 ∈ R
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Functions
Functions are relations satisfying the uniqueness condition: each left
component of an ordered pair in it has exactly one right component.
A relation f is a function iff for every x , such that there are y and z such
that 〈x , y〉 ∈ f and 〈x , z〉 ∈ f , y = z .
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Domain and range I
Each function has a domain and a range.
The domain of a function f is the set of arguments for which f is defined.
The range of a function f is the set of values f can take when applied to
an argument.
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Domain and range II
If f is a function, then
the domain of f = {x : there is a y such that 〈x , y〉 ∈ f },the range of f = {x : there is a y such that 〈y , x〉 ∈ f }.
If A is the domain and B the range of f , then f is from A onto B.
If C a superset of the range of f , then f is from A (in)to C .
f : A→ B
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Argument and value
Because of uniqueness, it holds true that for every f applied to x (or f of
x), f (x) := the unique y such that 〈x , y〉 ∈ f
f (x) = ‘the value of f for the argument x ’
We say f maps x to y
f (x) = y ≡ 〈x , y〉 ∈ f
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Defining functions I
We can list the elements
F := {〈a, b〉, 〈b, c〉, 〈c , a〉}
=
F :=
a→ bb → cc → a
=
Let F be that function f with domain {a, b, c} such that
f (a) = b, f (b) = c , f (c) = a.
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Defining functions II
Functions with large or infinite domain are defined with conditions.
F := f : A→ BFor every x ∈ A, f (x) = . . .
A is domain
B is value
F is a function from A to B
F is the function f from A to B such that every argument x in A is
mapped by f to the value . . .
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Defining functions III
Let F+1 be that function f such that
f : N→ N, and for every x ∈ N, f (x) = x + 1
=
F+1 := f : N→ NFor every x ∈ N, f (x) = x + 1
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Overview
1. Tasks of a semantic theory
2. Sets
3. Functions
4. How syntax determines sentence interpretation
5. The three components of compositional interpretation
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Assumptions about the syntax
Sentences are represented as phrase-structure trees
(4) Mary smokes.
S
NP
N
Mary
VP
V
smokes
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Denotations of sentences and proper names
S
NP
N
Mary
VP
V
smokes
Denotation of S: truth-value – 1 (true) / 0 (false)
Denotation of proper name: individual – Mary
What are the denotations of the remaining nodes?
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Denotations of sentences and proper names
S
NP
N
Mary
VP
V
smokes
Denotation of S: truth-value – 1 (true) / 0 (false)
Denotation of proper name: individual – Mary
What are the denotations of the remaining nodes?
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Denotations of sentences and proper names
S
NP
N
Mary
VP
V
smokes
Denotation of S: truth-value – 1 (true) / 0 (false)
Denotation of proper name: individual – Mary
What are the denotations of the remaining nodes?
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Non-branching nodes in NP
S
NP
N
Mary
VP
V
smokes
N inherits denotation from Mary: Mary
NP inherits denotation from N: Mary
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Non-branching nodes in NP
S
NP
N
Mary
VP
V
smokes
N inherits denotation from Mary: Mary
NP inherits denotation from N: Mary
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Generalizing to all non-branching nodes
A non-branching node inherits the denotation of its immediate daughter.
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Branching nodes
S
NP
N
Mary
VP
V
smokes
S denotes a truth-value
NP denotes an individual
What does VP denote?
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Semantic composition: Frege’s conjecture
Semantic composition is function(al) application.
VP must denote a function.
Because of the non-branching node generalization, all of VP’s daughters
denote that same function.
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Nodes in VP
S
NP
N
Mary
VP
V
smokes
S denotes a truth-value, NP an individual.
VP denotes a function from individuals to truth-values.
V and smokes do so, too.
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Overview
1. Tasks of a semantic theory
2. Sets
3. Functions
4. How syntax determines sentence interpretation
5. The three components of compositional interpretation
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Component I: inventory of denotations
Elements of D: set of actual individuals
Elements of {0, 1}: set of truth-values
Functions from D to {0, 1}
. . .
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Component II: lexical entries
Proper names
[[Mary]] = Mary
[[Bill]] = Bill
. . .
Intransitive verbs
[[dances]] = f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x dances
[[smokes]] = f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes
[[ ]] is the interpretation function
[[ ]] assigns denotation/extension to nodes
[[α]] is the denotation/extension of α in situation s
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Component II: lexical entries
Proper names
[[Mary]] = Mary
[[Bill]] = Bill
. . .
Intransitive verbs
[[dances]] = f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x dances
[[smokes]] = f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes
[[ ]] is the interpretation function
[[ ]] assigns denotation/extension to nodes
[[α]] is the denotation/extension of α in situation s
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Component II: lexical entries
Proper names
[[Mary]] = Mary
[[Bill]] = Bill
. . .
Intransitive verbs
[[dances]] = f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x dances
[[smokes]] = f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes
[[ ]] is the interpretation function
[[ ]] assigns denotation/extension to nodes
[[α]] is the denotation/extension of α in situation s
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Component 3: semantic rules for non-terminal nodes
S1 If α has the form S
β γ
, then [[α]] = [[γ]]([[β]]).
S2 If α has the form NP
β
, then [[α]] = [[β]].
S3 If α has the form VP
β
, then [[α]] = [[β]].
S4 If α has the form N
β
, then [[α]] = [[β]].
S5 If α has the form V
β
, then [[α]] = [[β]].
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Component 3: semantic rules for non-terminal nodes
S1 If α has the form S
β γ
, then [[α]] = [[γ]]([[β]]).
S2 If α has the form NP
β
, then [[α]] = [[β]].
S3 If α has the form VP
β
, then [[α]] = [[β]].
S4 If α has the form N
β
, then [[α]] = [[β]].
S5 If α has the form V
β
, then [[α]] = [[β]].
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Component 3: semantic rules for non-terminal nodes
S1 If α has the form S
β γ
, then [[α]] = [[γ]]([[β]]).
S2 If α has the form NP
β
, then [[α]] = [[β]].
S3 If α has the form VP
β
, then [[α]] = [[β]].
S4 If α has the form N
β
, then [[α]] = [[β]].
S5 If α has the form V
β
, then [[α]] = [[β]].
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Proof of truth-conditions I
S
NP
N
Mary
VP
V
smokes
Assume [[[S . . . ]]] = 1 iff Mary smokes
S1 If α has the form S
β γ
, then [[α]] = [[γ]]([[β]]).
[[[S . . . ]]] = [[[VP . . . ]]]([[[NP . . . ]]])
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Proof of truth-conditions I
S
NP
N
Mary
VP
V
smokes
Assume [[[S . . . ]]] = 1 iff Mary smokes
S1 If α has the form S
β γ
, then [[α]] = [[γ]]([[β]]).
[[[S . . . ]]] = [[[VP . . . ]]]([[[NP . . . ]]])
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Proof of truth-conditions I
S
NP
N
Mary
VP
V
smokes
Assume [[[S . . . ]]] = 1 iff Mary smokes
S1 If α has the form S
β γ
, then [[α]] = [[γ]]([[β]]).
[[[S . . . ]]] = [[[VP . . . ]]]([[[NP . . . ]]])
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Proof of truth-conditions II
S
NP
N
Mary
VP
V
smokes
S2 If α has the form NP
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[VP . . . ]]]([[[N . . . ]]])
S4 If α has the form N
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[VP . . . ]]]([[Mary]])
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Proof of truth-conditions II
S
NP
N
Mary
VP
V
smokes
S2 If α has the form NP
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[VP . . . ]]]([[[N . . . ]]])
S4 If α has the form N
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[VP . . . ]]]([[Mary]])
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Proof of truth-conditions II
S
NP
N
Mary
VP
V
smokes
S2 If α has the form NP
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[VP . . . ]]]([[[N . . . ]]])
S4 If α has the form N
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[VP . . . ]]]([[Mary]])
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Proof of truth-conditions II
S
NP
N
Mary
VP
V
smokes
S2 If α has the form NP
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[VP . . . ]]]([[[N . . . ]]])
S4 If α has the form N
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[VP . . . ]]]([[Mary]])
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Proof of truth-conditions II
S
NP
N
Mary
VP
V
smokes
S2 If α has the form NP
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[VP . . . ]]]([[[N . . . ]]])
S4 If α has the form N
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[VP . . . ]]]([[Mary]])
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Proof of truth-conditions III
S
NP
N
Mary
VP
V
smokes
S3 If α has the form VP
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[V . . . ]]]([[Mary]])
S5 If α has the form V
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[smokes]]([[Mary]])
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Proof of truth-conditions III
S
NP
N
Mary
VP
V
smokes
S3 If α has the form VP
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[V . . . ]]]([[Mary]])
S5 If α has the form V
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[smokes]]([[Mary]])
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Proof of truth-conditions III
S
NP
N
Mary
VP
V
smokes
S3 If α has the form VP
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[V . . . ]]]([[Mary]])
S5 If α has the form V
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[smokes]]([[Mary]])
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Proof of truth-conditions III
S
NP
N
Mary
VP
V
smokes
S3 If α has the form VP
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[V . . . ]]]([[Mary]])
S5 If α has the form V
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[smokes]]([[Mary]])
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Proof of truth-conditions III
S
NP
N
Mary
VP
V
smokes
S3 If α has the form VP
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[[V . . . ]]]([[Mary]])
S5 If α has the form V
β
, then [[α]] = [[β]].
[[[S . . . ]]] = [[smokes]]([[Mary]])
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Proof of truth-conditions IV
[[Mary]] = Mary lexicon
[[smokes]] = f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes lexicon
[[[S . . . ]]] = [[smokes]]([[Mary]])
[[[S . . . ]]] =
[f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes
](Mary)
[f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes
](Mary) = 1 iff Mary smokes
[[[S . . . ]]] = 1 iff Mary smokes
End of proof
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Proof of truth-conditions IV
[[Mary]] = Mary lexicon
[[smokes]] = f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes lexicon
[[[S . . . ]]] = [[smokes]]([[Mary]])
[[[S . . . ]]] =
[f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes
](Mary)
[f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes
](Mary) = 1 iff Mary smokes
[[[S . . . ]]] = 1 iff Mary smokes
End of proof
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Proof of truth-conditions IV
[[Mary]] = Mary lexicon
[[smokes]] = f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes lexicon
[[[S . . . ]]] = [[smokes]]([[Mary]])
[[[S . . . ]]] =
[f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes
](Mary)
[f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes
](Mary) = 1 iff Mary smokes
[[[S . . . ]]] = 1 iff Mary smokes
End of proof
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Proof of truth-conditions IV
[[Mary]] = Mary lexicon
[[smokes]] = f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes lexicon
[[[S . . . ]]] = [[smokes]]([[Mary]])
[[[S . . . ]]] =
[f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes
](Mary)
[f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes
](Mary) = 1 iff Mary smokes
[[[S . . . ]]] = 1 iff Mary smokes
End of proof
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Truth-conditions of a sentence
Lexicon defines extension of an intransitive verb as a condition
[[smokes]] = f : D → {0, 1}For all x ∈ D, f (x) = 1 iff x smokes
Together with the rules we get truth-conditions as meaning of S .
This seems adequate regarding the intuitions of speakers with respect to
the meaning of [[smokes]].
Normally we do not have full information about who smokes.
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Truth-value of a sentence
Assume D = {John, Mary, Sue} in Situation s
Extension of smokes in s represented by table
[[smokes]] =
John → 0
Mary → 1
Sue → 0
Delivers as denotation of S a truth-value
[[[S . . . ]]] =
John → 0
Mary → 1
Sue → 0
(Mary) = 1
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Truth-value of a sentence
Assume D = {John, Mary, Sue} in Situation s
Extension of smokes in s represented by table
[[smokes]] =
John → 0
Mary → 1
Sue → 0
Delivers as denotation of S a truth-value
[[[S . . . ]]] =
John → 0
Mary → 1
Sue → 0
(Mary) = 1
58 / 58
![Page 83: Introduction to semantics - Heim & Kratzer 1998, …Introduction to semantics Heim & Kratzer 1998, chapters 1-2 Clemens Mayr EGG school, Lagodekhi July 25, 2016 1/58 Overview 1. Tasks](https://reader034.vdocuments.us/reader034/viewer/2022052314/5f0a95b77e708231d42c5a9e/html5/thumbnails/83.jpg)
Truth-value of a sentence
Assume D = {John, Mary, Sue} in Situation s
Extension of smokes in s represented by table
[[smokes]] =
John → 0
Mary → 1
Sue → 0
Delivers as denotation of S a truth-value
[[[S . . . ]]] =
John → 0
Mary → 1
Sue → 0
(Mary) = 1
58 / 58