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TruthCondition 真值條件
 To know meaning of a declarative sentence is to know conditions in which it is true.
Entailment 蘊涵
 p entails q iff every situation in which p is true is a situation in which q is true.
 p => q
 Note: Even if p entails q, there may be situations in which q is true and p is false.
Contradiction 矛盾
 When 2 sentences cannot be true in same situation, they contradict each other.
 p entails q iff p & ¬q are contradictory.
#
It indicates following sentence is semantically unacceptable.
Literal & NonLiteral Meaning （非）字面意思
 2 forms of nonliteral meaning of a sentence are:
 Implicature
 Metaphor
 Not same as truthconditions, which can have both literal and nonliteral meaning.
Implicature 話裡有話
 Nonliteral meaning.
 A type of inference.
 Conclusion reached by reasoning about literal meaning of a sentence.
 Can be denied without creating a contradiction.
Metaphor 隱喻
 Literal meaning of a phrase is used as a symbol for something else.
Presupposition 預設
 Sometimes, when speakers utter certain sentences, they make presuppositions.
 A piece of info that is presented as taken for granted when a speaker utters a sentence.
 It behaves differently in NONVERIDICAL env.
 When putting a clause in a nonveridical env, its entailments are trapped in there. They are not entailments of whole sentence.
 Projected over scope of negation.
 Presuppositions of a sentence are also entailments. But it may NOT.
Negation
 Nonveridical
 If negating a sentence, it loses its entailments, but not PRESUPPOSITION, which is projected over scope of negation.
 e.g.  Did he stop smoking?  He used to smoke.
Speech Act
 Assertion
 Present sentence as being a true and relevant piece of info.
 Give orders
 Ask questions
 Make promises
Set: Basic ({})
 A collection of objects
 {…}
 Using curly braces
 List notation for sets
 List notation is called an extensional definition
 Heterogeneous
 ORDER does not matter.
 {a, b} = {b, a}
 REPETITION does not matter.
 {a, b, b} = {a ,b}
Set: Membership (∈, ∉)
 A relation between a set and something that is a member of this set.
 Symbol: ∈ (“is a member of”)
 e.g. 3 ∈ {1, 2, 3}
 Symbol: ∉ (“is not a member of”)
 e.g. 4 ∉ {1, 2, 3}
Set: Predicate (:)
 {x: x is a natural number}
 Set of all natural numbers
 Set of every x such that x is a natural number
 {x: 3 is a natural number}
 Set of everything
 {x: pi is a natural number}
 Set has no members.
 {x is a natural number}
 Only member of set is formula “x is a natural number”.
Set: Identity (=)
 2 sets are identical iff they have same members
 e.g.
 {1, 2} = {2, 1}
 {1, 2} = {1, 2}
Set: Inclusion (⊆, ⊈, ⊂)
 Let S_1 & S_2 be 2 sets
 S_1 is included in S_2 iff EVERY member of S_1 is a member of S_2.
 If S_1 is included in S_2, then S_1 is a subset of S_2.
 Note: if A = B, then A ⊆ B.
 Note: A = B iff (A ⊆ B and B ⊆ A)
 If A is included in B but A is not identical to B, then A is PROPERLY INCLUDED (⊂) in B.
 If A ⊆ B, then
 A is a subset of B.
 B is a superset of A.
 If A ⊂ B, then
 A is a PROPER subset of B.
 B is a PROPER superset of A.
Set: Union (⋃)
 Union of 2 sets A and B is set that contains every element that is either in A OR B, or both.
 A⋃B = {x: x∈A or x∈B}
 e.g.
 A = {1,2,3}
 B = {3,4,5}
 then A⋃B = {1,2,3,4,5}
Set: Intersection (⋂)
 Intersection fo 2 sets A and B is set that contains every element that is both in A AND B.
 A⋂B = {x: x∈A and x∈B}
 e.g.
 A = {1,2,3}
 B = {3,4,5}
 A⋂B = {3}
Set: Difference ()
 Diff btw a set A and B is set that contains ANY element in A that is NOT in B.
 A  B = {X: x∈A and x∉B}
 e.g.
 A = {1,2,3}
 B = {3,4,5}
 A  B = {1,2}
Set: Empty (∅)
 Empty set is set that hes no member
 IMPORTANT: empty is NOT “nothing”, it is a set.
 ∅ = {}
Meaning of Expression (⟦⟧)
 Meaning of common nouns
 ⟦dog⟧ = {x: x is a dog}
 ⟦dog⟧ = set of all dogs
 Meaning of modifiers
 ⟦cute⟧ = {x: x is cute}
 ⟦who smokes⟧ = {x: x smokes}
 e.g.
 ⟦cute dog⟧
 = ⟦cute⟧ ⋂ ⟦dog⟧
 = {x: x is cute} ⋂ {x: x is a dog}
 = {x: x is cute and x is a dog}
 ⟦linguist who smokes⟧
 = ⟦linguist⟧ ⋂ ⟦who smokes⟧
 = {x: x is a linguist} ⋂ {x: x smokes}
 = {x: x is a linguist and x smokes}
 ⟦cute dog⟧
 IMPORTANT: It does not work for ANTIINTERSECTIVE (PRIVATIVE) adjectives.
AntiIntersective Adjective
 aka Privative Adjective
 If A is an antiintersective adjective and N is a noun
 then ⟦A N⟧ ⋂ ⟦N⟧ = ∅
 e.g.
 ⟦fake diamond⟧ ⋂ ⟦diamond⟧ = ∅
 ⟦counterfeit Rembrandt⟧ ⋂ ⟦Rembrandt⟧ = ∅
 Adjs like “fake” and “counterfeit” are antiintersective.
Subset & Superset Inferences
 e.g.
 If Mary bought chocolate, it is in the freezer.
 If Mary bought milk chocolate, it is in the freezer.
 1 entails 2, but not opposite (To subset). Antecedent.
 e.g.
 If Mary went shopping, she bought chocolate.
 If Mary went shopping, she bought milk chocolate.
 2 entails 1, but not opposite (To superset). Consequent.
Downward & Upward Entailments
DE (Downward Entailing)
 Antecedent of a conditional is a DE env.
 An env X_Y is DE iff:
 If A ⊆ B, then XBY entails XAY. (To subset)
UE (Upward Entailing)
 Consequent of a conditional is an UE env.
 An env X_Y is UE iff:
 If A ⊆ B, then XAY entails XBY. (To superset)
Word Analysis: “Any”
 Grammatical in:
 Negative clause
 There isn’t anyone to talk to.
 Antecedents of conditionals
 If we talk to anyone, they will discover our secret.
 Complements of “doubt”
 I doubt that there is anyone to talk to.
 Negative clause
 Ungrammatical in:
 Unembedded positive clauses
 *There is anyone to talk to.
 Consequents of conditional
 * If we go home, there is anyone to talk to.
 Complements of “believe”
 I believe that there is anyone to talk to.
 Unembedded positive clauses
DEness & UEness
 Distribution of “any” if sensitive to DEness of its env.
 DEness is a logical property of envs.
 Grammar is sensitive to logical properties of sentences.
 This supports our use of truthconditions and logical concepts like entailments in our study of natural lnaguage semantics.
 “Exactly 1” is neither UE or DE.
Ambiguity
 Lexical
 Police begin campaign to run down jaywalker.
 run down
 knock down
 chase
 run down
 Safety experts say school bus passengers should be belted.
 belted
 secured with a belt
 beaten with a belt
 belted
 Bank Drivein Window Blocked by Board.
 board
 committee
 piece of wood
 board
 Bar trying to help alcoholic laywers.
 bar
 tribunal
 drinking establishment
 bar
 Police begin campaign to run down jaywalker.
 Structural
 Teacher strikes idle kids.
 (Teacher strikes)//S idle//V kids//O.
 Teacher//S strikes//V (idle//ADJ kids//N)O.
 Squad helps dog bite victim.
 (Squad helps dog) bite victim.
 Squad helps (dogbite victim).
 Bill hit the man with the hammer.
 Bill (hit the man) with the hammer.
 Bill hit (the man with the hammer).
 Teacher strikes idle kids.
 Syntactic (Beyond modification)
 John told the girl that Bill liked the story.
 John told (the girl that Bill liked) the story.
 John told the girl that (Bill liked the story).
 Every student read a book.
 For every student, there is a book that this student read.
 There is a book that all of the students read.
 John told the girl that Bill liked the story.
Principle of Modification
 If an expression X (constituent or word) modifies an expression Y, then X must ccommand Y.
CCommand (Constituent Command)
 Node A ccommands node B if every node dominating A also dominates B, and neither A nor B dominates the other.
Test of Ambiguity
 Type
 Not ambiguous but context dependent
 I am hungry.
 Diff contexts, diff speakers.
 Not ambiguous but (lexical) underspecification
 I am going to visit my aunt.
 My aunt: My father’s sister, or my mother’s sister.
 Not ambiguous but vague
 John is bald.
 How much hair loss counts as baldness?
 Not ambiguous but context dependent
 Test
 TruthConditions
 If a sentence can both be true and false in same situation, it is AMBIGUOUS.
 John hit the man with the hammer.
 Situation: hammer was used as an instrument.
 Sentence is true in 1 reading, false in another.
 Thus AMBIGUOUS.
 I am hungry
 No matter who speaker is, I refers to speaker.
 Thus NOT AMBIGUOUS.
 John hit the man with the hammer.
 If a sentence can both be true and false in same situation, it is AMBIGUOUS.
 Ellipsis (?)
 An elided expression must hae same meaning as its antecedent.
 If an elided expression and its antecedent can have diff meanings, diff in meaning is not a case of ambiguity.
 TruthConditions
SDCH (Scope/Domain Correspondence Hypothesis)
 Semantic scope of a linguistic operator in a sentence corresponds to its ccommand domain in some syntactic representation of sentence.
Scope Ambiguity
 John didn’t leave because he was sick.
 Meaning
 John didn’t leave, because he was sick.
 John didn’t (leave because he was sick).
 Expression
 Negation
 Subordinating conjunction “because”
 Meaning
 We need single men or women.
 Meaning
 We need (single men) or women.
 We need single (men or women).
 Meaning
DetQ (Determiner Quantifier)
 Determiner quantifiers are expressions like
 Type
 every
 ⟦Every cat is cute⟧
 T iff ⟦cat⟧ ⊆ ⟦cute⟧
 some
 ⟦Some cats are cute⟧
 T iff ⟦cat⟧ ⋂ ⟦cute⟧ ≠ ∅
 most
 all
 every
 e.g.
 Every child read some book.
 Meaning
 For every child, there is some book that this child read. (Surface Scope)
 There is some book such that every child read this book. (Inverse Scope)
 Situations

 is true in abc

 is true in bc only

 Meaning
 Every child read some book.
 Type
LF (Logical Form)
 In order to keep a strict correspondence between ccommand and scope, we assume that there is a level of syntactic representation where quantifiers can be moved according to their scope.
 This level of syntactic representation is called Logical Form.
Logical Form Scope Principle
 At level of LF, an element α has scope over β iff β is in domain of α.
 “Domain” means ccommand domain.
Entailments e.g.
 DE
 Every DOG is a pet.
 At most 5 STUDENTS watched the video.
 At most 5 students WATCHED THE VIDEO.
 Every cat who ate BANANAS got sick.
 No CAT got sick.
 No cat GOT SICK.
 UE
 Every dog is A PET.
 Some cat who ate BANANAS got sick.
 Neither
 Exactly one DOG is a pet.
 The CAT ate the canary.
Ambiguity e.g.
 Lexical
 Syntactical
 Both
Principle of Compositionality 複合性原理
 Meaning of a composite expression is a function of meaning of its immediate constituents and way these constituents are put together.
Mental Representation 心智表徵
Reference
 The meaning of the expression X as a relation btw this expression and the real object called X.
 The name X refers to (denotes) the object called X.
Referent (Denotation)
 The object that is called X is the referent (or denotation) of the name X.
Definite Description
 Formed by combining a definite determiner with an NP.
 Possessive phrases are another type of definite descriptions.
 It describes its denotation, while proper names do not.
CN (Common Noun)
 Singular common nouns refer to SETS.
 e.g.
 CAT refers to set of cats
 ⟦X is cute⟧ = 1 iff ⟦X⟧ \in {x: x is a cute INDIVIDUAL}
Word Analysis: “Every”
 ⟦Every boy is happy⟧ = 1 iff ⟦boy⟧ ⊆ ⟦is happy⟧
 ⟦Every NP VP⟧ = 1 iff ⟦NP⟧ ⊆ ⟦VP⟧
Word Analysis: “Some”
 ⟦Some NP VP⟧ = 1 iff ⟦NP⟧ ⋂ ⟦VP⟧ ≠ ∅
Word Analysis: “A(n)”
 ⟦Some NP VP⟧ = 1 iff ⟦NP⟧ ⋂ ⟦VP⟧ ≠ ∅
Relational Noun
 A class of nouns that do not refer to sets of individuals
 e.g.
 mother
 friend
 side
 owner
 birthplace
Functional Noun
 Relations that satisfy the following condition are functions:
 A 2 place relation R is a function iff \forall x,u,v: if both <x,u> \in R and <x,v> \in R, then u=v.
Transitivity
 0: Intransitive
 1: Transitive
 2: Ditransitive
Intransitive Verb
 Like CNs, they denote sets of individuals
 ⟦smokes⟧ = the set of individuals who smoke
 ⟦is sleeping⟧ = the set of individuals who are sleeping
 ⟦was sleeping⟧ = the set of individuals who were sleeping
 ⟦slept⟧ = the set of individuals who slept
 Predication as set membership
 ⟦A is sleeping⟧ = 1 iff ⟦A⟧ \in ⟦is sleeping⟧
 ⟦A is sleeping⟧ = 1 iff ⟦A⟧ \in {x: x is sleeping}
Transitive Verb
 Similar to that of relational nouns.
 A likes B.
 ⟦likes⟧ = {<A, B>, <A, C>, …}
 Order: <SUBJECT, OBJECT>
 ⟦A likes B⟧ = 1 iff <A, B> \in ⟦likes⟧
 Not symmetric
 A likes B but B might not like A.
Ditransitive Verb
 Have triples in its extensions.
 ⟦A gave B to C⟧ 1 iff <A, B, C> \in ⟦gave⟧
 ⟦gave⟧ = {<A, B, C>, <D, E, F>, …}
Meaning of Expression
 Extension
 Intension
Extension
 Denotation of these (Proper Name, CN, etc.) expressions are called their extensions.
 Extension of an expression is its denotation.
 e.g.
 Extension of sentence “A likes B” is the truth value ‘true’.
 Extension of sentence “A does not like B” is the truth value ‘false’.
Review for Midterm
 Truth Condition
 Entailment
 p entails q iff sentence "p and not q" IS contradictory.
 Situation:
 Test
1. p && q
2. p && !q
 If 1 && !2 => p entails q
 If 1 && 2 => p does not entail q
 If !1 => p and q are contradictory
 Implicature
 Cancellation Test (.... Indeed, ...)
 I saw two elks. I didn't see three elks. => I saw two elks. Indeed, I saw three of them. (No contradiction: Implicature)
 I saw two elks. I saw more than one elk. => \*I saw two elks. Indeed, I didn't see more than one elk. (Contradiction: Entailment)
 Presupposition
 Projection Test
 NonVerdical Environment
 Antecedent of Conditionals
 Negation
 Complement of "Doubt"
 Question
 Inclusion
 ⊆: Inclusion
 ⊂: Proper Inclusion
 Union
 A∪B = {x: x∈A or x∈B}
 Intersection
 A∩B = {x: x∈A and x∈B}
 Difference
 AB = {x: x∈A and x∉B}
 Ambiguity
 Test
 TruthConditions
 Ellipsis
 Constituent Command
 Extension (Set of ...)
 Intension (Concept of ...)
 Set Theory
 \{\{1,2,3,6},{4,5,8\}\}  {4,5,8} = \{\{1,2,3,6},{4,5,8\}\}
 \{\{1,2,3,6},{4,5,8\}\}  \{\{4,5,8\}\} = \{\{1,2,3,6\}\}
 {1,2,3,4,5,6}  {4,5,8} = {1,2,3,6}
 \{\{\{2,6\}\}\}
 \{\{8},\{\{2,6\}\}\}  {2,6,{8\}\} = \{\{\{2,6\}\}\}
 \{\{8},\{\{2,6\}\}\}  \{\{2,6},{8\}\} = \{\{\{2,6\}\}\}
 \{\{8},\{\{2,6\}\}\}  \{\{\{2,6\}\},{8\}\} = empty