LINC12H3 Fall 2024 - Week 2
Semantics: The study of Meaning
Sable Peters
September 12
University of Toronto Scarborough
Welcome Linguists!
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Literal and Non-literal Meaning
Implicatures
Implicatures are a type of inference that can be reached by
reasoning about the literal meaning of a sentence, and other
information from the context and the common ground – common
knowledge that exists between speakers.
(1) (context: after a dinner party)
a. How was the meal?
b. The music was good.
If a speaker replies something like in (b) to the question in (a), we can
likely infer that they wish to communicate something like ‘the meal
was not good’.
Wewill discuss in later weeks how speakers go about this reasoning
process, and how to describe the process in a formal way.
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Implicatures
Implicatures are a type of inference that can be reached by
reasoning about the literal meaning of a sentence, and other
information from the context and the common ground – common
knowledge that exists between speakers.
(1) (context: after a dinner party)
a. How was the meal?
b. The music was good.
If a speaker replies something like in (b) to the question in (a), we can
likely infer that they wish to communicate something like ‘the meal
was not good’.
Wewill discuss in later weeks how speakers go about this reasoning
process, and how to describe the process in a formal way.
2
Implicatures
Unlike entailments, implicatures can be negated/ denied without creating
contradiction.
Context: at a restaurant, asking the waitor
(2) a. Is the fish fresh?
b. The fish is very fresh. # ... but it is not fresh.
(3) a. Is the fish fresh?
b. It’s local!
It’s fresh too too.
▷ ‘The fish is very fresh’ entails ‘the the fish is fresh’; it is contradictory to
negate this because it is an entailment.
▷ ‘The fish is local’ in this context might implicate the that the fish is not
fresh; we can deny this implicature without contradiction however.
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Implicatures
Unlike entailments, implicatures can be negated/ denied without creating
contradiction. Context: at a restaurant, asking the waitor
(2) a. Is the fish fresh?
b. The fish is very fresh. # ... but it is not fresh.
(3) a. Is the fish fresh?
b. It’s local!
It’s fresh too.
▷ ‘The fish is very fresh’ entails ‘the the fish is fresh’; it is contradictory to
negate this because it is an entailment.
▷ ‘The fish is local’ in this context might implicate the that the fish is not
fresh; we can deny this implicature without contradiction however.
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Presuppositions
Sometimes when a sentence is uttered, speakers presuppose
certain things to be true.
Presuppositions are pieces of information that are taken for
granted or presented as taken for granted when a sentence is
uttered. These are things that are assumed to be true prior
uttering the sentence in question.
(4) a. Bolor stopped smoking.
b. Bolor used to smoke.
A speaker who says (a) presupposes that (b) is true.
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Presuppositions
Sometimes when a sentence is uttered, speakers presuppose
certain things to be true.
Presuppositions are pieces of information that are taken for
granted or presented as taken for granted when a sentence is
uttered. These are things that are assumed to be true prior
uttering the sentence in question.
(4) a. Bolor stopped smoking.
b. Bolor used to smoke.
A speaker who says (a) presupposes that (b) is true.
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Presuppositions
More examples of sentences that have salient Presuppositions:
(5) a. Have some more cake!
b. Have you read the brochure again?
c. Nobody knows that I hid the car keys.
d. I do not regret reading that book.
The italicized words are Presupposition Triggers
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Presuppositions & Entailments
Most of the time, Presuppositions are also entailments of a
sentence, but there are some crucial differences
The two behave differently in non-veridical environments
(“non-truthful”)
▷ Entailments are “trapped” in non-veridical environments
▷ Presuppositions project out of non-veridical environments
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Presuppositions & Entailments
Negation is a good environment to test for Entailments vs.
Presuppositions:
(6) a. Heidi is allergic to nuts.
b. Heidi is not allergic to walnuts.
c. Heidi is allergic to walnuts.
c ⇒ a BUT b ⇏ a
If you negate a sentence, you lose all of its entailments
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Presuppositions & Entailments
Negation is a good environment to test for Presuppositions vs.
Entailments:
(7) a. Heidi stopped buying lottery tickets.
b. Heidi did not stop buying lottery tickets.
c. Heidi bought lottery tickets.
Both a and b still presuppose c!
Presuppositions are ‘projected over the scope of negation’
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Other Non-veridical Environments
▷ The antecedent of conditional clauses:
‘If Heidi stopped buying lottery tickets, she must be saving
money.’
▷ Some modal expressions:
‘Heidi must stop buying lottery tickets.’
▷ Questions:
‘Did Heidi stop buying lottery tickets?’
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Some Examples
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Example 1
(8) a. Gigi is a cellist
b. Gigi is a musician.
Does a entail b?
Negation & Coordination Test: a & not-b
Gigi is a cellist, # and Gigi is not a musician
Evaluation:
a & ¬b results in a contradiction
Conclusion:
b is an entailment of a
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Example 1
(8) a. Gigi is a cellist
b. Gigi is a musician.
Does a entail b?
Negation & Coordination Test: a & not-b
Gigi is a cellist, # and Gigi is not a musician
Evaluation:
a & ¬b results in a contradiction
Conclusion:
b is an entailment of a
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Example 1
(8) a. Gigi is a cellist
b. Gigi is a musician.
Does a entail b?
Negation & Coordination Test: a & not-b
Gigi is a cellist, # and Gigi is not a musician
Evaluation:
a & ¬b results in a contradiction
Conclusion:
b is an entailment of a
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Example 1
(9) a. Gigi is a cellist
b. Gigi is a musician.
Does b entail a?
Negation & Coordination Test: b & not-a
Gigi is a musician, ✓and/but Gigi is not a cellist
Evaluation:
b & ¬a does NOT result in a contradiction.
Conclusion:
a is not an entailment of b
Further Conclusion:
The entailment is unidirectional; the two propositions are not
paraphrases
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Example 1
(9) a. Gigi is a cellist
b. Gigi is a musician.
Does b entail a?
Negation & Coordination Test: b & not-a
Gigi is a musician, ✓and/but Gigi is not a cellist
Evaluation:
b & ¬a does NOT result in a contradiction.
Conclusion:
a is not an entailment of b
Further Conclusion:
The entailment is unidirectional; the two propositions are not
paraphrases
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Example 1
(9) a. Gigi is a cellist
b. Gigi is a musician.
Does b entail a?
Negation & Coordination Test: b & not-a
Gigi is a musician, ✓and/but Gigi is not a cellist
Evaluation:
b & ¬a does NOT result in a contradiction.
Conclusion:
a is not an entailment of b
Further Conclusion:
The entailment is unidirectional; the two propositions are not
paraphrases
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Example 1
(9) a. Gigi is a cellist
b. Gigi is a musician.
Does b entail a?
Negation & Coordination Test: b & not-a
Gigi is a musician, ✓and/but Gigi is not a cellist
Evaluation:
b & ¬a does NOT result in a contradiction.
Conclusion:
a is not an entailment of b
Further Conclusion:
The entailment is unidirectional; the two propositions are not
paraphrases
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Example 1
(10) a. Gigi is a cellist
b. Gigi is a musician.
Does a presuppose b?
Non-veridical environment 1 – Negation:
Gigi is not a cellist.
Non-veridical environment 2 – Antecedent of ‘if’ clause:
If Gigi is a cellist, he’s certainly a good one.
Evaluation:
This requires nothing about whether Gigi is or is not a musician; the
entailment is lost.
Conclusion:
b is not a presupposition of a
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Example 1
(10) a. Gigi is a cellist
b. Gigi is a musician.
Does a presuppose b?
Non-veridical environment 1 – Negation:
Gigi is not a cellist.
Non-veridical environment 2 – Antecedent of ‘if’ clause:
If Gigi is a cellist, he’s certainly a good one.
Evaluation:
This requires nothing about whether Gigi is or is not a musician; the
entailment is lost.
Conclusion:
b is not a presupposition of a
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Example 1
(10) a. Gigi is a cellist
b. Gigi is a musician.
Does a presuppose b?
Non-veridical environment 1 – Negation:
Gigi is not a cellist.
Non-veridical environment 2 – Antecedent of ‘if’ clause:
If Gigi is a cellist, he’s certainly a good one.
Evaluation:
This requires nothing about whether Gigi is or is not a musician; the
entailment is lost.
Conclusion:
b is not a presupposition of a
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Example 1
(10) a. Gigi is a cellist
b. Gigi is a musician.
Does a presuppose b?
Non-veridical environment 1 – Negation:
Gigi is not a cellist.
Non-veridical environment 2 – Antecedent of ‘if’ clause:
If Gigi is a cellist, he’s certainly a good one.
Evaluation:
This requires nothing about whether Gigi is or is not a musician; the
entailment is lost.
Conclusion:
b is not a presupposition of a
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Example 2
(11) a. My sister is an astronomer.
b. I have a sister.
Does a entail b?
Negation & Coordination Test: a & not-b
My sister is an astronomer, # but I don’t have a sister
Evaluation:
a & ¬b results in a contradiction
Conclusion:
b is an entailment of a
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Example 2
(11) a. My sister is an astronomer.
b. I have a sister.
Does a entail b?
Negation & Coordination Test: a & not-b
My sister is an astronomer, # but I don’t have a sister
Evaluation:
a & ¬b results in a contradiction
Conclusion:
b is an entailment of a
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Example 2
(11) a. My sister is an astronomer.
b. I have a sister.
Does a entail b?
Negation & Coordination Test: a & not-b
My sister is an astronomer, # but I don’t have a sister
Evaluation:
a & ¬b results in a contradiction
Conclusion:
b is an entailment of a
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Example 2
(12) a. My sister is an astronomer.
b. I have a sister.
Does a presuppose b?
Non-veridical environment 1 – Negation:
My sister is not an astronomer,
# and I don’t have a sister.
Non-veridical environment 2 – Antecedent of ‘if’ clause:
If my sister were an astronomer, I’d ask her what that star is.
??/# but I don’t have a sister.
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Example 2
(12) a. My sister is an astronomer.
b. I have a sister.
Does a presuppose b?
Non-veridical environment 1 – Negation:
My sister is not an astronomer,
# and I don’t have a sister.
Non-veridical environment 2 – Antecedent of ‘if’ clause:
If my sister were an astronomer, I’d ask her what that star is.
??/# but I don’t have a sister.
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Example 2
(12) a. My sister is an astronomer.
b. I have a sister.
Does a presuppose b?
Non-veridical environment 1 – Negation:
My sister is not an astronomer,
# and I don’t have a sister.
Non-veridical environment 2 – Antecedent of ‘if’ clause:
If my sister were an astronomer, I’d ask her what that star is.
??/# but I don’t have a sister.
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Example 2
Non-veridical environment 1 – Negation:
My sister is not an astronomer,
# and I don’t have a sister.
Non-veridical environment 2 – Antecedent of ‘if’ clause:
If my sister were an astronomer, I’d ask her what that star is.
??/# but I don’t have a sister.
Evaluation:
In both environments it is semantically odd to follow up by negating the
proposition that we are testing to be a presupposition; i.e. a still requires b to
be true even in a non-veridical environment
Conclusion:
b is a presupposition of a
 Some tests like the second non-veridical environment test above aredifficult to interpret – it is important to test multiple environments!
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Example 2
Non-veridical environment 1 – Negation:
My sister is not an astronomer,
# and I don’t have a sister.
Non-veridical environment 2 – Antecedent of ‘if’ clause:
If my sister were an astronomer, I’d ask her what that star is.
??/# but I don’t have a sister.
Evaluation:
In both environments it is semantically odd to follow up by negating the
proposition that we are testing to be a presupposition; i.e. a still requires b to
be true even in a non-veridical environment
Conclusion:
b is a presupposition of a
 Some tests like the second non-veridical environment test above aredifficult to interpret – it is important to test multiple environments!
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Example 2
Non-veridical environment 1 – Negation:
My sister is not an astronomer,
# and I don’t have a sister.
Non-veridical environment 2 – Antecedent of ‘if’ clause:
If my sister were an astronomer, I’d ask her what that star is.
??/# but I don’t have a sister.
Evaluation:
In both environments it is semantically odd to follow up by negating the
proposition that we are testing to be a presupposition; i.e. a still requires b to
be true even in a non-veridical environment
Conclusion:
b is a presupposition of a
 Some tests like the second non-veridical environment test above aredifficult to interpret – it is important to test multiple environments!
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Logic and Arguments
Logic and Arguments
In Semantics, we use the term Argument in a specific way. An
Argument is a sequence of premises with a conclusion
Logic is the study of Valid Arguments: Whenever the
premises are true, the conclusion is true
This means: the conclusion is entailed by the premises
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Valid Arguments
We can categorise certain structures of arguments that we
recognise as Valid.
Modus ponens – affirming the antecedent
1. If P, then Q.
2. P
3. Therefore, Q
Example:
If it’s snowing, it’s cold.
It’s snowing.
∴ It is cold.
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Valid Arguments
We can categorise certain structures of arguments that we
recognise as Valid.
Modus tollens – Denying the consequent
1. If P, then Q.
2. ¬Q
3. Therefore, ¬P
Example:
If there’s fire, there will be smoke.
There is no smoke.
∴ There is no fire.
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Invalid Arguments
Some structures of arguments are always invalid
Affirming the consequent

1. If P, then Q.
2. Q
3. Therefore, P
Example:
If it’s snowing, it’s cold.
It’s cold.
× It’s snowing
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Invalid Arguments
Some structures of arguments are always invalid
Denying the Antecedent

1. If P, then Q.
2. ¬P
3. Therefore, ¬Q
Example:
If it’s snowing, it’s cold.
It’s not snowing.
× It’s not cold.
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A (Logical) Metalanguage
Propositional Logic
Propositional Logic can be thought of as a simple language that uses
logical operators to write formulas describing propositions. We can then
evaluate e.g. whether arguments in these formulas are valid, etc.
PL only has two types of expressions:
▷ symbols that represent atomic propositions
▷ logical operators on these propositions
Formulas that have no structure are called atomic : they have no proper
parts that independantly have Truth Values (TV)
(13) Bold is tall.
This sentence has no structure; it as a whole is either true or false. We can
represent it as an atomic proposition with a letter, e.g. p, q, r, s, p′, q′ etc.
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Propositional Logic
Propositional Logic can be thought of as a simple language that uses
logical operators to write formulas describing propositions. We can then
evaluate e.g. whether arguments in these formulas are valid, etc.
PL only has two types of expressions:
▷ symbols that represent atomic propositions
▷ logical operators on these propositions
Formulas that have no structure are called atomic : they have no proper
parts that independantly have Truth Values (TV)
(13) Bold is tall.
This sentence has no structure; it as a whole is either true or false. We can
represent it as an atomic proposition with a letter, e.g. p, q, r, s, p′, q′ etc.
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Logical Operators
The simplest operator is negation:
Negation: “not”, ¬,
p = ‘It is snowing.’
¬p = ‘It is not snowing.’
Negation is a unary or one-place operator
There are five more operators we will use, which are all binary
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Logical Operators
The simplest operator is negation:
Negation: “not”, ¬,
p = ‘It is snowing.’
¬p = ‘It is not snowing.’
Negation is a unary or one-place operator
There are five more operators we will use, which are all binary
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Logical Operators
Conjunction: “and”, ∧, &
p = ‘It is snowing.’
q = ‘It is windy.’ p ∧ q = It is snowing and it is windy.’
Disjunction: “or”, ∨
p = ‘It is hot.’
q = ‘It is raining.’ p ∨ q = It is hot or it is raining.’
Material implication: “if... then”,→
p = ‘It is raining’
q = ‘The ground is wet.’ p→ q = If it is raining, (then) ground
is wet.’
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Logical Operators
Conjunction: “and”, ∧, &
p = ‘It is snowing.’
q = ‘It is windy.’ p ∧ q = It is snowing and it is windy.’
Disjunction: “or”, ∨
p = ‘It is hot.’
q = ‘It is raining.’ p ∨ q = It is hot or it is raining.’
Material implication: “if... then”,→
p = ‘It is raining’
q = ‘The ground is wet.’ p→ q = If it is raining, (then) ground
is wet.’
23
Logical Operators
Conjunction: “and”, ∧, &
p = ‘It is snowing.’
q = ‘It is windy.’ p ∧ q = It is snowing and it is windy.’
Disjunction: “or”, ∨
p = ‘It is hot.’
q = ‘It is raining.’ p ∨ q = It is hot or it is raining.’
Material implication: “if... then”,→
p = ‘It is raining’
q = ‘The ground is wet.’ p→ q = If it is raining, (then) ground
is wet.’
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Logical Operators
Biconditional : “iff, if and only if”,↔
p = ‘You’ve passed your courses.’
q = You will graduate.’ p↔ q =If you’ve passed your courses,
and only if you’ve passed your courses, you will graduate.
Exclusive Disjunction: “either... or...”, XOR
p = ‘It is hot’
q = ‘It is cold.’ p XOR q = Either it is hot or it is cold.’
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Logical Operators
Biconditional : “iff, if and only if”,↔
p = ‘You’ve passed your courses.’
q = You will graduate.’ p↔ q =If you’ve passed your courses,
and only if you’ve passed your courses, you will graduate.
Exclusive Disjunction: “either... or...”, XOR
p = ‘It is hot’
q = ‘It is cold.’ p XOR q = Either it is hot or it is cold.’
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Truth tables: negation
Negation ¬ (“not”)
p = ‘The sun is shining’
¬p = ‘The sun is not shining’
To see how an operator changes the truth conditions of a
proposition, we will use truth tables:
p ¬p
T F
F T
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Truth tables: Conjunction
Conjunction ∧ (“and”)
p = ‘The sun is shining’
q = ‘It’s raining’
p ∧ q = ‘The sun is shining and it’s raining’
p q p ∧ q
T T T
T F F
F T F
F F F
Unless both members of the conjunction are true, the whole
expression is false.
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Truth tables: Disjunction
Disjunction ∨ (“or”)
p = ‘The sun is shining’
q = ‘It’s raining’
p ∨ q = ‘The sun is shining or it’s raining’
p q p ∨ q
T T T
T F T
F T T
F F F
As long as one member of the disjunction is true, the whole
expression is true.
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Truth tables: Material implication
Material implication→ (“if ... then”)
p = ‘The sun is shining’
q = ‘I’ll buy you ice-cream’
p→ q = ‘If the sun is shining, (then) I’ll buy you ice-cream’
p q p→ q
T T T
T F F
F T T
F F T
If the sun is not shining (p is F), I am not breaking my promise,
therefore p→ q will be true. The only case when p→ q is false is
when the sun is shining but I don’t buy you ice-cream.
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Truth tables: Biconditional
Biconditional↔ (“iff” = “if and only if”)
p = ‘The sun is shining’
q = ‘I’ll buy you ice-cream.’
p↔ q = ‘I’ll buy you ice-cream if and only if the sun is shining
(otherwise I won’t).’
p q p↔ q
T T T
T F F
F T F
F F T
p↔ q is true iff p and q have the same truth value.
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Truth tables: Exclusive disjunction
Exclusive disjunction (XOR)
p = ‘The sun is shining’
q = ‘It’s raining’
p XOR q = ‘Either the sun is shining or it’s raining’
p q p XOR q
T T F
T F T
F T T
F F F
p XOR q is false when both p and q are T or both are F. To become
true, one and only one of them can be true.