Semantics I - Summary
1. AMBIGUITY
1.1 Meanings
1. Have to be combinable and the combinations have to be meanings as well, both on the sentential and
sub-sentential level
2. Compositionality = meaning of a sentence is determined by the words + syntax (how it is composed)
3. Ambiguity = when a sentence has at least two different readings and there are situations in which one
reading is true and the other is false
a. Lexical ambiguity
b. Structural ambiguity
Every cat does not sleep.
- Readings:
1. Not every cat is sleeping.
2. No cats are sleeping.
- Kind of ambiguity: structural
- Paraphrases:
1. It’s not the case that every cat is sleeping.
2. Every cat is such that they are not sleeping.
1.2 Grammar = semantic rules + syntactic rules -> generates a set of sentences paired with meanings
Semantic rules take as input
a. meanings of two or more expressions
b. syntactic structure in which they are combine
and give as output the meaning of the combinations.
2. TRUTH-CONDITIONS AND TRUTH-VALUES
There is a difference between knowing the truth-conditions of a sentence and knowing whether or not a
sentence is true. By combining them, we can determine the truth-values of a sentence (true or false). This
is only for sentences, not for non-sentential expressions. When we combine these with facts of the world,
we get extensions = entities that expressions stand for.
- Individual = ‘Anna’
- Set of entities (all cats) = ‘cat’
- Relation = ‘above’, ‘brother’
The truth-value of a sentence is determined by extensions of the words in the sentence, and the syntax of
the sentence.
word meanings + facts about the world = extensions
extensions + syntax = truth-value
3. ENTAILMENT
3.1 Entailment
a entails b, when the situation that makes a true also makes b true.
a. All cars are green.
b. All sportscars are green.
a ⊨ b, because when all cars are green, it also means that all sportscars are green.
3.2 Synonymy
a and b are synonymous, when they entail each other.
a. John gave Mary a cake.
b. John gave a cake to Mary.
, 3.3 Not entailment
a does not entail b when there is a situation in which a is true and b is false.
a. John owns a blue sweater.
b. John does not own a sweater.
If John owns a blue sweater, then it is false that he does not own a sweater.
4. SETS
4.1 Characteristics of a set
A set is a collection of distinct members, so repeating elements are only counted once.
∈ = is a member of
∉ = is not a member of
A set can be described as list notation: {a, b, c}
Or as predicate notation: {x : x is a cat} ‘the set of all x, such that x is a cat’ the set of all cats
A set can be an element of another set: {2, 5, 3 {3, 4}}
An empty set is a set without members = {} = ø.
This is not the same as {ø}, as this would be {{}}: a set that has a set as a member, which doesn’t have any
members.
4.2 Combining sets
1. ∪ = union = the set of the members of two sets combined.
a. ({a, b, c} ∪ {1, 2}) = {a, b, c, 1, 2}
b. ({x : x is smart} ∪ {y : y is a linguist}) = {z : z is smart or a linguist}
c. A = {ø, {cat}} and B = {{ø}}; (A ∪ B) = {ø, {ø}, {cat}}
2. ∩ = intersection = the set of the members that the two sets have in common
a. ({a, b, c,} ∩ {b, c, d}) = {b, c}
b. ({x : x rides a bike} ∩ {y : y lives in Leiden}) = {z : z rides a bike and lives in Leiden}
c. A = {ø, {cat}} ({{}, {cat}}) and B {{ø}} ({{{}}} = a set that has a set as a member. This member is
a set, which doesn’t have any members.) they have nothing in common. A ∩ B = {} = ø
The cardinality of a set is the number of members in the set. We write it between | |.
a. |{x}| = 1
b. |{1, a, {cat}, 2}| = 4
c. |ø| = 0 (a set without members)
d. |{ø}| = 1
4.3 Relations among sets
1. A ⊆ B means that all elements of set A are also elements of set B.
A is a subset of B and B is a superset of A
a. {a, b, c} ⊆ {a, b, c, d, e}
b. {x : x lives in Leiden} ⊆ {x : x lives in the Netherlands}
c. ø ⊆ {a, b, c}
d. {y : y likes cats and dogs} ⊆ {y : y likes cats}
2. A = B means that set A and B are identical. They are still subsets of each other, they just have the same
elements.
A ⊆ B and B ⊆ A
a. {a, b, c} ⊆ {a, b, c}
3. A ⊂ B means that all elements of set A are elements of B, but not all elements of set B are elements of A.
A is a proper subset of B: A ⊆ B, but not B ⊆ A
(see examples from 1)
1. AMBIGUITY
1.1 Meanings
1. Have to be combinable and the combinations have to be meanings as well, both on the sentential and
sub-sentential level
2. Compositionality = meaning of a sentence is determined by the words + syntax (how it is composed)
3. Ambiguity = when a sentence has at least two different readings and there are situations in which one
reading is true and the other is false
a. Lexical ambiguity
b. Structural ambiguity
Every cat does not sleep.
- Readings:
1. Not every cat is sleeping.
2. No cats are sleeping.
- Kind of ambiguity: structural
- Paraphrases:
1. It’s not the case that every cat is sleeping.
2. Every cat is such that they are not sleeping.
1.2 Grammar = semantic rules + syntactic rules -> generates a set of sentences paired with meanings
Semantic rules take as input
a. meanings of two or more expressions
b. syntactic structure in which they are combine
and give as output the meaning of the combinations.
2. TRUTH-CONDITIONS AND TRUTH-VALUES
There is a difference between knowing the truth-conditions of a sentence and knowing whether or not a
sentence is true. By combining them, we can determine the truth-values of a sentence (true or false). This
is only for sentences, not for non-sentential expressions. When we combine these with facts of the world,
we get extensions = entities that expressions stand for.
- Individual = ‘Anna’
- Set of entities (all cats) = ‘cat’
- Relation = ‘above’, ‘brother’
The truth-value of a sentence is determined by extensions of the words in the sentence, and the syntax of
the sentence.
word meanings + facts about the world = extensions
extensions + syntax = truth-value
3. ENTAILMENT
3.1 Entailment
a entails b, when the situation that makes a true also makes b true.
a. All cars are green.
b. All sportscars are green.
a ⊨ b, because when all cars are green, it also means that all sportscars are green.
3.2 Synonymy
a and b are synonymous, when they entail each other.
a. John gave Mary a cake.
b. John gave a cake to Mary.
, 3.3 Not entailment
a does not entail b when there is a situation in which a is true and b is false.
a. John owns a blue sweater.
b. John does not own a sweater.
If John owns a blue sweater, then it is false that he does not own a sweater.
4. SETS
4.1 Characteristics of a set
A set is a collection of distinct members, so repeating elements are only counted once.
∈ = is a member of
∉ = is not a member of
A set can be described as list notation: {a, b, c}
Or as predicate notation: {x : x is a cat} ‘the set of all x, such that x is a cat’ the set of all cats
A set can be an element of another set: {2, 5, 3 {3, 4}}
An empty set is a set without members = {} = ø.
This is not the same as {ø}, as this would be {{}}: a set that has a set as a member, which doesn’t have any
members.
4.2 Combining sets
1. ∪ = union = the set of the members of two sets combined.
a. ({a, b, c} ∪ {1, 2}) = {a, b, c, 1, 2}
b. ({x : x is smart} ∪ {y : y is a linguist}) = {z : z is smart or a linguist}
c. A = {ø, {cat}} and B = {{ø}}; (A ∪ B) = {ø, {ø}, {cat}}
2. ∩ = intersection = the set of the members that the two sets have in common
a. ({a, b, c,} ∩ {b, c, d}) = {b, c}
b. ({x : x rides a bike} ∩ {y : y lives in Leiden}) = {z : z rides a bike and lives in Leiden}
c. A = {ø, {cat}} ({{}, {cat}}) and B {{ø}} ({{{}}} = a set that has a set as a member. This member is
a set, which doesn’t have any members.) they have nothing in common. A ∩ B = {} = ø
The cardinality of a set is the number of members in the set. We write it between | |.
a. |{x}| = 1
b. |{1, a, {cat}, 2}| = 4
c. |ø| = 0 (a set without members)
d. |{ø}| = 1
4.3 Relations among sets
1. A ⊆ B means that all elements of set A are also elements of set B.
A is a subset of B and B is a superset of A
a. {a, b, c} ⊆ {a, b, c, d, e}
b. {x : x lives in Leiden} ⊆ {x : x lives in the Netherlands}
c. ø ⊆ {a, b, c}
d. {y : y likes cats and dogs} ⊆ {y : y likes cats}
2. A = B means that set A and B are identical. They are still subsets of each other, they just have the same
elements.
A ⊆ B and B ⊆ A
a. {a, b, c} ⊆ {a, b, c}
3. A ⊂ B means that all elements of set A are elements of B, but not all elements of set B are elements of A.
A is a proper subset of B: A ⊆ B, but not B ⊆ A
(see examples from 1)
