LINC12H3 Fall 2024 - Week 2 Semantics: The study of Meaning Sable Peters September 12 University of Toronto Scarborough Welcome Linguists! 0 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. 2 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. 3 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. 3 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. 4 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. 4 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 5 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 6 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 7 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’ 8 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?’ 9 Some Examples 9 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 10 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 10 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 10 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 11 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 11 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 11 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 11 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 12 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 12 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 12 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 12 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 13 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 13 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 13 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. 14 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. 14 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. 14 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! 15 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! 15 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! 15 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 16 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. 17 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. 18 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 19 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. 20 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. 21 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. 21 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 22 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 22 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.’ 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.’ 23 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.’ 24 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.’ 24 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 25 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. 26 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. 27 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. 28 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. 29 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. 30 51作业君版权所有